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\section{INTRODUCTION} Interest in analysing and forecasting time series is thousands of years old. This field of research has witnessed two breakthroughs over the second-half of the twentieth century in the development of the Box-Jenkins methodology (\cite{box2015}) and the derivation of the Kalman filter (\cite{kalman1960}). While the ARIMA model underpinning the Box-Jenkins methodology has since become a standard in graduate curricula worldwide, the Kalman filter has found a great number of applications in addition to time series forecasting, including, but not limited to, object tracking, navigation and computer vision. \subsection{Popular Approaches} The Box-Jenkins methodology for analysing time series consists of three steps. Firstly, trends and seasonalities are removed from the training dataset by differentiating the time series iteratively\footnote{That is the following time series are constructed from the original time series $y_k$: $\Delta^{(1)} y_k := y_k - y_{k-1}$, \dots, $\Delta^{(i+1)} y_k := \Delta^{(i)}y_k - \Delta^{(i)}y_{k-1}$ etc...} until an ergodic-stationary time series is found. Secondly, the ergodic-stationary time series is assumed to be a realisation of an ARMA process whose parameters are estimated, typically by maximizing the likelihood. Finally, the calibrated model is checked by analysing the sample autocorrelation of residuals. Both the ARIMA model and the Kalman filter can be represented as discrete time \textit{linear Gaussian state space models} (see \cite[][chap. 3]{durbin12} or \cite[][chap. 10]{comkoop}). Discrete time state space models aim at inferring the state $x_k \in \mathbb{R}^p$ of a latent dynamic system, that is assumed to evolve according to Markovian dynamics, from some noisy observations $y_k \in \mathbb{R}^q$. The models are characterised by a probability distribution for the initial latent state of the system $p(x_0)$, a Markovian transition dynamics $p(x_{k} \vert x_{k-1})$, and a measurement model $p(y_k \vert x_k)$. It is also assumed that the observations are independent from each other and from other latent states conditional on their associated latent states (i.e. $ \forall i \neq j ~y_i \perp y_j |x_i$ and $y_i \perp x_j |x_i$). The forecasting problem then consists of recursively determining $p(x_{k+n} \vert y_1, \dots, y_k)$ for some $n>0$. In linear Gaussian state space models (LGSSM), the three distributions above are taken to be Gaussian, and the transition and measurement distributions have covariance matrices that do not depend on the state process. Moreover, it is assumed that \begin{align} \text{E}\left( x_k \vert x_{k-1}\right)=F_kx_{k-1} + B_k u_k, E\left(y_k \vert x_k \right) = H_k x_k, \nonumber \end{align} where $u_k$, $B_k$, $F_k$ and $H_k$ are given (or estimated off-line). Under this class of models the forecasting problem can be solved exactly and very efficiently using the Kalman filter equations. LGSSMs can be extended in several ways. Alternatives have been proposed, including the popular \textit{extended Kalman filter}, that relax the linearity assumption in LGSSMs by postulating that \begin{align} \text{E}\left( x_k \vert x_{k-1}\right)=f(x_{k-1}, u_k) \text{, } E\left(y_k \vert x_k \right) = h(x_k),\nonumber \end{align} for some (possibly nonlinear) functions $f$ and $h$, while retaining the Gaussian assumptions on the initial, transition and measurement distributions. Other approaches extend LGSSMs by relaxing the foregoing Gaussian assumptions. For instance, discrete measurement probability distributions such as the Poisson distribution or the negative binomial distribution would allow for count (as opposed to real-valued) observations (\cite{west85}), while leptokurtic probability distributions such as the student-t distribution or the Laplace distribution, used as measurement distribution, might improve the robustness of forecasts to outliers. For a more comprehensive review of Box-Jenkins models and state space methods for time series analysis, we refer the reader to \cite{comkoop}, \cite{durbin12} and \cite{ham94}. More generally, time series forecasting can be regarded as a regression problem, where one is interested in predicting future values and associated confidence bounds from historical observations. Gaussian process regression or GPR \cite[][chap. 2]{rasswill} provides a flexible Bayesian nonparametric framework for that purpose. The GPR approach to time series modelling consists of regarding the time series as arising from the sum of a continuous time Gaussian process $(z_t)_{t \geq 0}$ with mean function $m$ and covariance function $k_{\theta}$, and an independent Gaussian white noise: \begin{align} &(z_t)_{t \geq 0} \sim \mathcal{GP}\left(m(.), k_{\theta}(.,.)\right), ~ (\epsilon_t)_{t \geq 0} \sim \mathcal{GWN}(\sigma^2), \nonumber \\ &\forall t_i, ~ y_{t_i} = z_{t_i} + \epsilon_{t_i}\nonumber. \end{align} As a result, $(y_t)_{t \geq 0}$ is also a Gaussian process (GP), and the noise variance $\sigma^2$ and all other hyper-parameters can easily be inferred from some historical data $\{y_{t_0}, \dots, y_{t_k}\}$ by maximizing the corresponding multivariate Gaussian likelihood. Predictive distributions may then be obtained in closed-form. More recently, online \textit{passive-aggressive} algorithms have been proposed by \cite{pajmlr}, that allow for online linear and basis function regression with strong worst-case loss bounds. \subsection{Limitations of Popular Approaches} The critical assumption underpinning the Box-Jenkins methodology is that any time series, after enough iterative differentiations, will become \textit{ergodic-stationarity}, or more precisely will pass standard ergodic-stationarity statistical tests. The fundamental problem with this approach is that in practice, one will have access to finite samples that might not be sufficiently large or informative for the time series to pass any ergodic-stationarity test. It is in this spirit that \cite[][\S 3.10.1]{durbin12} noted that `in the economic and social fields, real series are never stationary however much differencing is done'. The ergodic-stationarity assumption is not a major limitation per se, but relying on the assumption that sufficient data have been collected to fully characterise the latent process (including testing for ergodic-stationarity) is a major limitation of the Box-Jenkins methodology. In that regards, \cite{kpss92} noted that `most economic time series are not very informative about whether or not there is a unit root'. The state space approach on the other hand does not require that the sample be informative enough to test for stationarity, which is why \cite{comkoop} and \cite{durbin12} argued it should be preferred to the Box-Jenkins methodology. Additionally, the scalability of state space models is very appealing. However, the dynamics of the latent time series, characterised by $H_k$ ($h$ in the nonlinear case) is usually assumed to be known. Although this assumption is fairly mild in many engineering applications where the dynamics are derived from the laws of physics, when analysing time series arising from complex systems where the dynamics are not known, $H_k$ (resp. $h$) fully characterize the covariance function of the latent time series $(x_t)_{t \geq 0}$. Thus, hand-picking $H_k$ or $h$ would be problematic as the covariance function of the latent process should be flexibly learned from the data to appropriately uncover and exploit patterns. Unfortunately, no state space dynamics has been proposed in the literature, to the best of our knowledge, that guarantees that the implied covariance functions of the latent processes $(x_t)_{t \geq 0}$ can approximate arbitrarily well any stationary covariance function. Families of covariance functions that are dense in the class of all continuous bounded covariance functions have recently been proposed by \cite{samo_gsk}, and can readily be used within the GPR framework. Unfortunately, with a cubic time complexity and a squared memory requirement, GPR scales poorly with the number of training samples. Sparse approximations have been proposed that scale linearly with the number of observations (\cite{quincand, sparsespectrum}). However, those methods are not good enough for streaming applications as they require loss of information through windowing to be of practical interest. Passive-aggressive (PA) algorithms (\cite{pajmlr}) have constant time complexity and memory requirement for online linear regression problems. However, their kernelized versions have memory requirement and computational time that depend on the number of `support vectors', and might increase unboundedly with the number of observations (\cite{wang10}). Thus, PA algorithms in their original form are not appropriate for flexibly forecasting time series in a high throughput setting. \cite{wang10} have introduced PA algorithms with an additional memory or CPU `budget' constraint, but the authors restricted themselves to classification problems. Moreover, unlike probabilistic approaches, PA algorithms do not provide uncertainty around predictions. A Bayesian PA-like online learning approach has been proposed by \cite{shi14}. However, like all other kernel-PA algorithms, the kernel is assumed to be given and online learning of its parameters is not addressed by the authors. \textbf{Our contibutions:} In this paper we build on the work of \cite{samo_gsk} to propose a state space model for analysing and forecasting time series that we prove is equivalent to GPR, under a family of kernels for the latent GP that is dense in the family of all stationary kernels, and that allows decoupling the flexibility of the kernel from the degree of smoothness of the latent GP. More importantly, we derive a novel scheme for online learning of hyper-parameters as a solution to a convex optimization problem, which happens to be a \textit{passive-aggressive} algorithm (\cite{pajmlr}) with \textit{stochastic gradient descent} updates (\cite{bottou98}). Overall, our model has constant time complexity and constant memory requirement, both for online learning of the hyper-parameters and for online prediction. The rest of the paper is structured as follows. In section \ref{sct:bk} we provide the intuition behind our approach and the related mathematical background. In section \ref{sct:mdl} we formally introduce our model. We illustrate that our approach outperforms competing approaches with some experiments in section \ref{sct:exp}. Finally, we discuss possible extensions and conclude in section \ref{sct:con}. \section{INTUITION AND BACKGROUND} \label{sct:bk} In this paper we consider modelling real-valued time series. Our working assumption is that each sample $\{y_{t_0}, \dots, y_{t_N} \}$ of a time series arises as noisy observations of a `smooth'\footnote{By smooth we mean that the process is at least mean square continuous, and higher order derivatives might exist.} latent stochastic process $(z_t)_{t \geq 0}$ pertubated by an independent white noise $(\epsilon_t)_{t \geq 0}$: \begin{align} &\forall t, ~y_t = z_t + \epsilon_t, \label{eq:cond1}\\ &(\epsilon_t)_{t \geq 0} \sim \mathcal{WN}(\sigma^2), ~ (\epsilon_t)_{t \geq 0} \perp (z_t)_{t \geq 0}.\label{eq:cond2} \end{align} In financial econometrics applications for instance, the process $(z_t)_{t \geq 0}$ may represent the logarithm of the equilibrium or fair price of an asset, while $(\epsilon_t)_{t \geq 0}$ accounts for short term shocks, microstructure noise and so-called `fat-fingers'. More generally, $(z_t)_{t \geq 0}$ denotes the value of the time series of interest, after taking out hazards such as side effects, measurement noise, recording errors and so forth, accounted for by $(\epsilon_t)_{t \geq 0}$. Moreover, we will assume that the latent process $(z_t)_{t \geq 0}$ is a trend-stationary Gaussian process, i.e. a Gaussian process with a translation-invariant covariance function and an arbitrary mean function: \begin{align} (z_t)_{t \geq 0} \sim \mathcal{GP}(m(.), k(.)).\label{eq:cond3} \end{align} It is our working assumption that the class of trend-stationary Gaussian processes is large enough for our purpose. We refer the reader to \ref{app:hyp_disc} for a discussion on why this is a relatively mild assumption. From a Bayesian perspective, we are assuming \textit{a priori} that the process evolves about a deterministic trend, and that our \textit{prior} uncertainty on how the process might deviate from the trend is invariant by translation. In applications, the trend $m$ will typically be taken to lie in a parametric family of functions. Furthermore, we assume that the white noise process $(\epsilon_t)_{t \geq 0}$ is Gaussian: \begin{align} (\epsilon_t)_{t \geq 0} \sim \mathcal{GWN}(\sigma^2).\label{eq:cond4} \end{align} We will discuss possible extensions to more robust noise models in section \ref{sct:con}. Our setup thus far is the same as GPR with a translation-invariant covariance function. Our next objective is to investigate whether we could exploit the scalability of the state space approach by providing an `equivalent' state space representation. More precisely, by `equivalent' we mean one in which the observable is the noisy time series value $y_t \in \mathbb{R}$, and for every collection of observation times $\{ t_0, \dots, t_N\}$, the joint distribution over $\left(y_{t_0}, \dots, y_{t_k}\right)$ as per the state space representation is the same as that implied by Equations (\ref{eq:cond1}) to (\ref{eq:cond4}). Thus, we need to construct an appropriate Markov process $(x_t)_{t \geq 0}, x_t \in \mathbb{R}^p$ that is related to $(z_t)_{t \geq 0}$ and from which the state space dynamics will be deduced. We will then derive the measurement dynamics from Equation (\ref{eq:cond1}). As $(x_t)_{t \geq 0}$ should be a Markov process, intuitively $x_t$ should contain all information about present and past values of the latent time series $(z_t)_{t \geq 0}$ if we want the state space representation to be equivalent to Equations (\ref{eq:cond1}) to (\ref{eq:cond4}). Thus, it seems natural to consider that $(z_t)_{t \geq 0}$ is a coordinate process of the vector state process $(x_t)_{t \geq 0}$. \underline{Case 1}: $\forall t, ~ x_t = z_t$ If we take the state process to be the latent time series itself, $(z_t)_{t \geq 0}$ needs to be Markovian. However, this would imply that the latent time series will either not be mean square differentiable, or, as the following lemma states, it will almost surely be equal to the trend plus a constant, which might be too restrictive. \begin{lemma}If a real-valued trend-stationary Gaussian process with differentiable mean function is mean square differentiable and Markovian, then it has a constant covariance function (or equivalently it is almost surely equal to its mean function plus a constant). \end{lemma} \begin{proof} See \ref{sct:only_smooth_mgp_}. \end{proof} Intuitively, the memorylessness of the Markov assumption conflicts with the memoryfulness of the differentiability assumption, so that the choice $x_t = z_t$ might only be appropriate when assuming that the smooth latent time series $(z_t)_{t \geq 0}$ is mean square continuous but not differentiable. An example covariance function yielding a continuous trend-stationary Markov Gaussian process is the Mat\'{e}rn-$\frac{1}{2}$ kernel $k_{\text{exp}}$ \cite[see][Appendix \S B.2]{rasswill}. The equivalent state space representation is easily derived from standard Gaussian identities as: \begin{equation*} \begin{cases} z_{t_0} \sim \mathcal{N}\left(0, k_{\text{exp}}(0)\right),\\ z_{t_k} = m(t_k) + \alpha_k z_{t_{k-1}} + \sqrt{k_{\text{exp}}(0)\left(1- \alpha_k^2\right)}\xi_{t_k}\\ y_{t_k} = z_{t_k} + \epsilon_{t_k} \end{cases} \end{equation*} where $\alpha_k = \frac{k_{\text{exp}}(t_k, t_{k-1})}{k_{\text{exp}}(0)}$, and $(\xi_t)_{t \geq 0} \sim \mathcal{GWN}(1)$. \underline{Case 2}: $\forall t, ~ x_t = \left(z_t, z_t^{(1)}, \dots, z_t^{(p)}\right)$ In many applications, we might be interested in postulating that the latent time series $(z_t)_{t \geq 0}$ is smoother, for instance $p$ times mean square differentiable. In this case, we derive our intuition for what might be good candidates to complete the state process into an appropriate Markov vector process from the theory of analytic functions underpinning implementations of scientific functions in most programming languages. Conceptually, the aforementioned theory implies that under mild conditions, the value of an infinitely smooth function and its derivatives $f(t_k), f^{(1)}(t_k), \dots, f^{(i)}(t_k), \dots$ at a given point $t_k$ are sufficient to fully characterise the entire function elsewhere on the domain. Although we only require $(z_t)_{t \geq 0}$ to be $p$ times differentiable, it is our hope that by augmenting the state process with the $p$ consecutive derivatives of $(z_t)_{t \geq 0}$, we could ensure that (i) each state variable $x_t$ contains all information about past values of the latent time series $(z_u)_{u \leq t}$ or equivalently the state process $(x_t)_{t \geq 0}$ is Markovian, (ii) the latent time series $(z_t)_{t \geq 0}$ is as smooth as desired, and (iii) the covariance function of the latent time series is not strongly restricted. When $(z_t)_{t \geq 0}$ is a Gaussian process that is $p$ times continuously differentiable in the mean square sense, we denote as \textit{p-derivative Gaussian process} ($p$DGP) the vector Gaussian process $(x_t)_{t \geq 0}$ where $x_t=\left(z_t, z^{(1)}_t, \dots, z^{(p)}_t\right)$ and $(z^{(i)})_{t \geq 0}$ is the $i$-th order mean square derivative of $(z_t)_{t \geq 0}$. We say that $(z_t)_{t \geq 0}$ is a $p$-Markov Gaussian process ($p$M-GP) when it is $p$ times continuously differentiable and the corresponding $p$DGP is Markovian.\footnote{A $0$M-GP ($p=0$) is a Gaussian Markov process.} Stationary $p$M-GPs have been extensively studied in the seminal paper by \cite{doob44}.\footnote{In Doob's paper a $p$M-GP is denoted a t.h.G.M.$_{(p-1)}$ process.} In particular, the fundamental theorem below \cite[see][Theorem 4.9 ii]{doob44} provides necessary and sufficient conditions on our covariance function $k$ for the augmented state process to be Markovian when the trend is constant. \begin{theorem} A real-valued stationary Gaussian process $(z_t)_{t \geq 0}$ with integrable covariance function $k$ is $p$-Markov if and only if $k$ admits a spectral density of the form \begin{equation} \label{eq:pm_cond} \forall s> 0, ~S(s) := \int k(\tau) e^{-2\pi i s\tau}d\tau = \frac{c}{|A(is)|^2}, \end{equation} where $c>0$ and $A$ is a polynomial of degree $p+1$ with real-valued coefficients and no purely imaginary root. \end{theorem} \begin{corollary}\label{coro:coro} A stationary Gaussian process with covariance function the Mat\'{e}rn kernel \begin{equation*} k_{\text{ma}}(\tau; k_0, l, \nu) = k_0 \frac{2^{1-\nu}}{\Gamma(\nu)} \left(\frac{\sqrt{2\nu} \vert \tau \vert}{l} \right)^\nu K_\nu \left( \frac{\sqrt{2\nu} \vert \tau \vert}{l}\right), \end{equation*} where $k_0, l >0$, $\Gamma$ is the Gamma function, $K_\nu$ is the modified Bessel function of second kind, and with $\nu = p + \frac{1}{2}, ~p \in \mathbb{N}$ is $p$-Markov. \end{corollary} \begin{proof} The spectral density of a Mat\'{e}rn-$(p+\frac{1}{2})$ covariance function is of the form \[\frac{c}{(\alpha + 4\pi^2 s^2)^{p+1}}= \frac{c}{\vert (\sqrt{\alpha} + 2\pi i s)^{p+1}\vert^2}\] with $c, \alpha>0$ \cite[see][]{rasswill}. \end{proof} Before deriving the state space representation of trend-stationary GPs with Mat\'{e}rn-$(p+\frac{1}{2})$ covariance functions, we recall that a $p$DGP is a vector Gaussian process and that in the trend-stationary case \begin{equation*} \forall ~0 \leq i, j \leq p, ~\text{cov}\left(z_u^{(i)}, z_v^{(j)} \right) = (-1)^j k^{(i+j)}(u-v), \end{equation*} where we use the superscript $^{(i)}$ to denote $i$-th order derivative or the orginal function (or process) when $i=0$ \cite[see][\S 2.6]{adlertaylor}. If we further denote \[K_{u,v} := \left\{\text{cov}\left(z_u^{(i)}, z_v^{(j)} \right)\right\}_{0 \leq i,j \leq p}\] the covariance matrix between $x_u$ and $x_v$, as \[K_{u\vert v} := K_{u, u} - K_{u, v} K_{v, v}^{-1}K_{v, u}\] the auto-covariance matrix of $x_u$ conditional on $x_v$, and as $L_{u\vert v}$ any squared matrix satisfying\footnote{The Cholesky factorisation and the SVD provide such a decomposition.} \[K_{u\vert v} = L_{u\vert v}L_{u\vert v}^T,\] then we obtain the following equivalent LGSSM representation (see \ref{sct:pmat_lgssm} for details): \begin{equation*} \begin{cases} x_{t_0} \sim \mathcal{N}(0, K_{t_0, t_0}) \\ x_{t_k} = F_{t_k} x_{t_{k-1}} + L_{t_k \vert t_{k-1}}\xi_{t_k}\\ y_{t_k} = m(t_k) + H^Tx_{t_k} + \epsilon_{t_k} \end{cases} \end{equation*} with $F_{t_k} = K_{t_k, t_{k-1}} K_{t_{k-1}, t_{k-1}}^{-1}, ~H=(1, 0, \dots, 0),$ and where $(\xi_t)_{t \geq 0}$ is now a $(p+1)$-dimensional standard Gaussian white noise. The idea of using derivative information to construct a scalable alternative to GPs for MCMC inference has recently been developed in \cite{samo_sgp}. The idea of representing GPR as a state space model with a Markovian dynamics involving derivatives has been explored by \cite{sarkka13}, where the authors proposed approximating the spectral density of a given covariance function by a function in the form of Equation (\ref{eq:pm_cond}). Unfortunately, deriving such approximations might be tedious in certain cases. More importantly, postulating that the covariance function has spectral density of the form Equation (\ref{eq:pm_cond}) is still not flexible enough as it creates a link between the degree of differentiability of the latent process and the number of modes of the spectral density. To see why, we note that critical points (or local extrema) of $\frac{c}{|A(is)|^2}$ and $|A(is)|^2$ are the same, and that $|A(is)|^2$ being a polynomial of degree $2p+2$ in $s$, it can have at most $2p +1$ critical points. This means that if for instance we would like the latent function to be at most once differentiable ($p=1$), the approach of \cite{sarkka13} would restrict the spectral density to have at most $3$ modes, rather than flexibly learning the number of modes from the data. This also holds when the spectral density is approximated by a rational function in the extension suggested by \cite{sarkka13}, as the degree of the numerator needs to be smaller than the degree of the denominator, and the later is linked to the degree of differentiability of the latent GP. Families of kernels, namely \textit{spectral Mat\'{e}rn kernels} (\cite{samo_gsk}), do exist that are dense in the family of all stationary covariance functions, and allow postulating or learning the degree of differentiability of the corresponding GP. In the next section, we build on the state space representation of $p$-Markov Mat\'{e}rn Gaussian processes to provide an exact state space representation of our time series model (Equations (\ref{eq:cond1}) to (\ref{eq:cond4})) when the latent time series $(z_t)_{t \geq 0}$ is assumed to have a \textit{spectral Mat\'{e}rn-($p+\frac{1}{2}$)} covariance function (\cite{samo_gsk}). Moreover, we derive a novel \textit{passive-aggressive} algorithm for online learning of the hyper-parameters of the state space model, and that has constant memory requirement and constant time complexity. \section{OUR MODEL} \label{sct:mdl} We start by introducing a state space time series model we refer to as the $p$-Markov Gaussian process filter ($p$M-GP filter) that we prove is equivalent to Gaussian process regression under a \textit{spectral Mat\'{e}rn} kernel. \subsection{The $p$-Markov Gaussian Process Filter} \begin{definition}\label{def:pm-gpf}($p$M-GP filter) We denote $p$-Markov Gaussian process filter ($p$M-GP filter) a state space model of the form: \begin{equation*} \begin{cases} \perp \left\{{}^{i}_{c}x_{t_0}, {}^{i}_{s}x_{t_0}\right\}_{i=0}^n, \perp \left\{({}^i_{c}\xi_{t})_{t \geq 0}, ({}^i_{s}\xi_{t})_{t \geq 0}\right\}_{i=0}^n \\ \forall i, ~ {}^i_{c}x_{t_{k-1}} \perp {}^i_{c}\xi_{t_k},~ {}^i_{s}x_{t_{k-1}} \perp {}^i_{s}\xi_{t_k} \\ \forall i, ~ {}^i_{c}x_{t_{k}} \perp \epsilon_{t_k},~ {}^i_{s}x_{t_{k}} \perp \epsilon_{t_k} \\ \forall i, ~{}^i_{c}x_{t_0}, {}^i_{s}x_{t_0} \sim \mathcal{N}(0, {}^iK_{t_0, t_0})\\ \forall i, ~{}^i_{c}x_{t_k} = {}^iF_{t_k} {}^i_{c}x_{t_{k-1}} + {}^iL_{t_k \vert t_{k-1}}{}^i_{c}\xi_{t_k}\\ \forall i, ~{}^i_{s}x_{t_k} = {}^iF_{t_k} {}^i_{s}x_{t_{k-1}} + {}^iL_{t_k \vert t_{k-1}}{}^i_{s}\xi_{t_k}\\ y_{t_k} = H^T\sum_{i=0}^{n} \left(\cos(\omega_i t_k){}^i_{c}x_{t_k} + \sin(\omega_i t_k){}^i_{s}x_{t_k} \right)\\ ~~~~~ + m(t_k) + \epsilon_{t_k} \end{cases} \end{equation*} where $\perp$ denotes mutual independence, $({}^i_{c}\xi_{t})_{t \geq 0}$ and $({}^i_{s}\xi_{t})_{t \geq 0}$ are $(p+1)$-dimensional standard Gaussian white noises, $(\epsilon_t)_{t \geq 0}$ is a scalar Gaussian white noise with variance $\sigma^2$, $^{i}K_{u, v}$ is the cross-covariance matrix of a $p$DGP with Mat\'{e}rn kernel $k_{\text{ma}}(\tau; k_{0i}, l_i, p+\frac{1}{2})$, and where $H, {}^iF_{t_k}, {}^iL_{t_k \vert t_{k-1}}$ are as in the previous section (replacing $K_{u, v}$ by $^{i}K_{u,v}$). \end{definition} The following proposition establishes that the $p$M-GP filter is equivalent to GPR under a spectral Mat\'{e}rn kernel, and a possibly non-constant mean function. \begin{proposition}\label{prop:eqv}Let $(\hat{z}_t)_{t \geq 0}$ be a trend-stationary Gaussian process with mean function $m$ and \textit{spectral Mat\'{e}rn} covariance function \begin{equation} k_{\text{sma}}(\tau) = \sum_{i=0}^{n} k_{\text{ma}}\left(\tau; k_{0i}, l_i, p + \frac{1}{2}\right)\cos(\omega_i \tau). \end{equation} Let $(\hat{\epsilon}_t)_{t \geq 0}$ be a Gaussian white noise with variance $\sigma^2$ that is independent from $(\hat{z}_t)_{t \geq 0}$, and $(\hat{y}_t)_{t \geq 0}$ the process defined as \[\forall t \geq 0, ~\hat{y}_t = \hat{z}_t + \hat{\epsilon}_t.\] Finally, let $(y_t)_{t \geq 0}$ be the observation process of the $p$M-GP filter with the same parameters $m, n, p, \sigma, \{k_{0i}, l_i, \omega_i\}_{i=0}^n$. Then, the processes $(y_t)_{t \geq 0}$ and $(\hat{y}_t)_{t \geq 0}$ have the same law, or equivalently: \[\forall t_0 < \dots < t_N, ~ (y_{t_0}, \dots, y_{t_N}) \sim (\hat{y}_{t_0}, \dots, \hat{y}_{t_N}).\] \end{proposition} \begin{proof} See \ref{sct:pm_gp_fil}. \end{proof} The above proposition has profound implications. Firstly, the $p$M-GP filter is the first equivalent state space representation of the Gaussian process regression model (on a unidimensional input space) under a family of covariance functions that is dense in the family of all stationary covariance functions. This representation allows combining the great flexibility of GPR, with the unmatched scalability of state space models, without resorting to approximations. Secondly, the model allows controlling the degree of differentiability of the latent time series independently from the flexibility of the covariance function of the latent time series. Finally, unlike models such as ARIMA that assume equally spaced observations, as any GPR model, the $p$M-GP filter is inherently asynchronous and naturally copes with missing data. \subsection{Solution to the Forecasting Problem} We note that the $p$M-GP filter can be rewritten as \begin{equation} \label{eq:solu} \begin{cases} \bm{x}_{t_0} \sim \mathcal{N}(0, \bm{K}_{t_0, t_0})\\ \bm{x}_{t_k} = \bm{F}_{t_k}\bm{x}_{t_{k-1}} + \bm{L}_{t_k \vert t_{k-1}} \bm{\xi}_{t_k}\\ y_{t_k} = m(t_k) + \bm{H}_{t_k}^T \bm{x}_{t_k} + \epsilon_{t_k}. \end{cases} \end{equation} Here, $\bm{x}_{t_k}$ (resp. $\bm{\xi}_{t_k}$) is the $2(p+1)(n+1)$ vector obtained by stacking up the the vectors $\{ \dots, {}^i_{c}x_{t_k}, {}^i_{s}x_{t_k}, \dots\}$ (resp. $\{ \dots, {}^i_{c}\xi_{t_k}, {}^i_{s}\xi_{t_k}, \dots\}$). Similarly, $\bm{K}_{t_0, t_0}$ (resp. $\bm{F}_{t_k}, \bm{L}_{t_k \vert t_{k-1}}, \bm{K}_{t_k \vert t_{k-1}}$) is the $2(p+1)(n+1) \times 2(p+1)(n+1)$ block diagonal matrix whose $2i$-th and $(2i + 1)$-th diagonal blocks are both ${}^iK_{t_0, t_0}$ (resp. ${}^iF_{t_k}, {}^iL_{t_k \vert t_{k-1}}, {}^iK_{t_k \vert t_{k-1}}$). Finally, $\bm{H}_{t_k}$ is the $2(p+1)(n+1)$ vector with coordinates $0$ except at indices multiple of $p+1$, and $\forall i \in [0 \dots n], ~\bm{H}_{t_k}[2i(p+1)] = \cos(\omega_i t_k)$ and $\bm{H}_{t_k}[(2i+1)(p+1)] = \sin(\omega_i t_k)$. Thus the $p$M-GP filter is a LGSSM, and the forecasting problem can be solved exactly, iteratively and in closed-form, and with memory requirement and time complexity both constant in the total number of observations. Using standard Kalman filter and Gaussian processes techniques (see \ref{sct:solu_fill}) we obtain: \begin{equation} \label{eq:solu2} \begin{cases} \forall t> t_{k-1}, ~\bm{x}_t \vert y_{t_0 : t_{k-1}} &\sim \mathcal{N}(\bm{m}_t^{-}, \bm{P}_t^{-})\\ \forall t> t_{k-1}, ~y_t \vert y_{t_0 : t_{k-1}} &\sim \mathcal{N}(m(t) + \bm{H}_t^T\bm{m}_t^{-}, v_t^{-})\\ \forall t> t_{k-1}, ~z_t \vert y_{t_0 : t_{k-1}} &\sim \mathcal{N}(m(t) + \bm{H}_t^T\bm{m}_t^{-},v_t^{-}-\sigma^2 )\\ \forall t \geq t_{k-1}, ~\bm{x}_t \vert y_{t_0 : t} &\sim \mathcal{N}(\bm{m}_t, \bm{P}_t)\\ \forall t \geq t_{k-1}, ~z_t \vert y_{t_0 : t} &\sim \mathcal{N}(m(t) + \bm{H}_t^T\bm{m}_t, v_t) \end{cases} \end{equation} with $y_{t_0: t_{k-1}} = \{y_{t_0}, \dots, y_{t_{k-1}}\}$, $z_t = m(t) + \bm{H}_t^T \bm{x}_t$ and \begin{align} &\underline{\text{Prediction step:}}\nonumber\\ &\bm{m}_{t_0}^{-} = 0, ~ \bm{P}_{t_ 0}^{-} = \bm{K}_{t_0, t_0} \label{eq:pred} \\ &\forall k \geq 1, t> t_{k-1} \begin{cases} \bm{m}_t^{-} &= \bm{F}_t \bm{m}_{t_{k-1}}\\ \bm{P}_t^{-} &= \bm{F}_t \bm{P}_{t_{k-1}} \bm{F}_t^T + \bm{K}_{t \vert t_{k-1}} \end{cases}\nonumber \\ &\underline{\text{Update step:}}\nonumber\\ &\forall t \begin{cases} v_t^{-} &= \bm{H}_t^T \bm{P}_{t}^{-} \bm{H}_t + \sigma^2 \\ \bm{e}_t^{-} &= y_t - m(t) - \bm{H}_t^T\bm{m}_t^{-} \\ \bm{G}_t &= \frac{1}{v_t^{-}}\bm{P}_t^{-}\bm{H}_t \\ \bm{m}_t &= \bm{m}_t^{-} + \bm{e}_t^{-}\bm{G}_t \\ \bm{P}_t &= \bm{P}_t^{-} - v_t^{-}\bm{G}_t\bm{G}_t^T\\ v_t &= \bm{H}_t^T\bm{P}_t\bm{H}_t \end{cases}.\label{eq:update} \end{align} \subsection{Online Learning of Hyper-Parameters} Noting that $y_{t_0} \sim \mathcal{N}(m(t_0), v_{t_0}^{-})$, and that \begin{equation*} \log p(y_{t_0:t_N}) = \log p(y_{t_0}) + \sum_{k=1}^T \log p(y_{t_k} \vert y_{t_0:t_{k-1}}), \end{equation*} it follows that maximum likelihood inference of the hyper-parameters of $m$, $\sigma$, and $\{k_{0i}, l_i, \omega_i\}_{i=0}^n$ may be achieved with linear time complexity $\mathcal{O}(T)$ and with constant memory requirement using Equations (\ref{eq:solu2}, \ref{eq:pred}, \ref{eq:update}). However, to be of practical interest in streaming applications, this time complexity requires loss of information through windowing, which might be detrimental to forecasting performance. We herein propose an alternative online approach that has constant time complexity. So far we have assumed that the hyper-parameters are constant over time. However, if we extend the model to iteration-specific hyper-parameters, the solution to the forecasting problem (Equations (\ref{eq:solu2}, \ref{eq:pred}, \ref{eq:update})) will provably remain unchanged. The intuition behind our approach to online learning of the hyper-parameters is borrowed from maximum likelihood inference when the hyper-parameters are constant, and online passive-aggressive algorithms for regression, classification and semi-supervised learning (\cite{pajmlr,wang10,chang10}). Let $\bm{\theta}_{t_k}$ be the vector of hyper-parameters (or log hyper-parameters for positive hyper-parameters) prevailing at time $t_k$. We recall that when hyper-parameters are constant, we may write \begin{align} \log p_{{\bm{\theta}}}\left(y_{t_0:t_T}\right) = \log p_{{\bm{\theta}}}\left(y_{t_0}\right) + \sum_{k=1}^{T} \log p_{{\bm{\theta}}}\left(y_{t_k}\vert y_{t_0:t_{k-1}}\right).\nonumber \end{align} Let us define \begin{equation*} \mathcal{L}^{l}_{t_k}(\bm{\theta}) := \begin{cases} \log p_{\bm{\theta}}\left(y_{t_0}\right) & \text{if } k=0 \\ \log p_{\bm{\theta}}\left(y_{t_k}\vert y_{t_0:t_{k-1}}\right) & \text{if } k \neq 0 \end{cases}. \end{equation*} At time $t_k$, $\mathcal{L}^{l}_{t_k}(\bm{\theta})$ represents the (log) likelihood that $\bm{\theta}$ explains the new observation $y_{t_k}$ given all previous observations (and hyper-parameters) and thus can be regarded as a \textit{local utility} function that we should aim to maximize. However, we should also be mindful of the `utility' of $\bm{\theta}$ in accounting for all previously observed data $(y_{t_0}, \dots, y_{t_{k-1}})$, which can be achieved by ensuring that the new update is not be too far away from $\bm{\theta}_{t_{k-1}}$. Deriving online learning updates as solutions to a constrained optimization problem that provides a trade-off between retaining information from previous iterations and locally reducing a loss function has been advocated for a long time in the online learning literature (\cite{littlestone89, kivinen97,helmbold99,pajmlr,wang10,chang10}). The approach we adopt is largely inspired by the PA-II algorithm of \cite{pajmlr}. We first consider the problem: \begin{align} \label{prob:1} \begin{cases} \underset{\bm{\theta}}{\text{min}} ~|| \bm{\theta} - \bm{\theta}_{t_{k-1}}||^2 + c_k \xi^2\\ \text{ s.t. } \max\left(-\epsilon -\mathcal{L}^{l}_{t_k}(\bm{\theta}), 0\right) \leq \xi \end{cases}, \end{align} where $\epsilon \geq 0$ and $c_k>0$. $c_k$ can be regarded as an aggressiveness parameter: the larger $c_k$ the more weight is given to increasing the local utility compared to retaining information from previous iterations. $\epsilon$ on the other hand is a margin or tolerance parameter that allows for some degree of local sup-optimality, thereby helping to avoid overfitting. Although this problem provides a suitable trade-off, it is tedious to solve analytically, and it might not have a closed-form solution. However, noting that the objective function of problem (\ref{prob:1}) aims at minimizing $|| \bm{\theta} - \bm{\theta}_{t_{k-1}}||$, we may replace $\mathcal{L}^{l}_{t_k}(\bm{\theta})$ with its first order Taylor expansion at $\bm{\theta}_{t_{k-1}}$. We adopt the resulting problem for online learning of the hyper-parameters: \begin{align} \label{prob:2} \begin{cases} \bm{\theta}_{t_k} = \underset{\bm{\theta}}{\text{argmin}} ~|| \bm{\theta} - \bm{\theta}_{t_{k-1}}||^2 + c_k \xi^2\\ \text{ s.t. } \max\left(-\epsilon -\hat{\mathcal{L}}^{l}_{t_k}(\bm{\theta}), 0\right) \leq \xi \end{cases}, \end{align} with $\hat{\mathcal{L}}^{l}_{t_k}(\bm{\theta}) = \mathcal{L}^{l}_{t_k}(\bm{\theta}_{t_{k-1}}) + \nabla \mathcal{L}^{l}_{t_k}(\bm{\theta}_{t_{k-1}})^T (\bm{\theta} - \bm{\theta}_{t_{k-1}})$. The above optimization problem is convex and has closed-form solution (see \ref{sct:optim_solu} for the derivation, and \ref{sct:optim_upd} for the derivation of $\nabla \mathcal{L}^{l}_{t_k}(\bm{\theta})$): \begin{align} \label{prob:solu} \bm{\theta}_{t_k} = \bm{\theta}_{t_{k-1}} + c_k\frac{\max\left(-\epsilon -\mathcal{L}^{l}_{t_k}\left(\bm{\theta}_{t_{k-1}}\right), 0 \right)}{1 + c_k \| \nabla \mathcal{L}^{l}_{t_k}(\bm{\theta}_{t_{k-1}}) \|^2}\nabla \mathcal{L}^{l}_{t_k}(\bm{\theta}_{t_{k-1}}). \end{align} Noting that the parameters do not change when $\mathcal{L}^{l}_{t_k}(\bm{\theta}_{t_{k-1}}) > -\epsilon$, it follows that our learning scheme can be regarded as a \textit{passive-aggressive} algorithm with \textit{stochastic gradient descent} update (\cite{bottou98}). Moreover, as \begin{align} &\hat{\mathcal{L}}^{l}_{t_k}(\bm{\theta}_{t_k}) - \hat{\mathcal{L}}^{l}_{t_k}(\bm{\theta}_{t_{k-1}}) \nonumber \\ &= \max\left(-\epsilon-\mathcal{L}^{l}_{t_k}\left(\bm{\theta}_{t_{k-1}}\right), 0 \right) \frac{c_k \| \nabla \mathcal{L}^{l}_{t_k}(\bm{\theta}_{t_{k-1}}) \|^2}{1 + c_k \| \nabla \mathcal{L}^{l}_{t_k}(\bm{\theta}_{t_{k-1}}) \|^2} \geq 0, \nonumber \end{align} each change in the hyper-parameters increases the approximate conditional log likelihood $\hat{\mathcal{L}}^{l}_{t_k}$. Moreover, the one-side $\epsilon$-insensitive loss term $\max\left(-\epsilon-\mathcal{L}^{l}_{t_k}\left(\bm{\theta}_{t_{k-1}}\right), 0 \right)$ guards against overfitting when an update is required. Algorithm \ref{alg:fore_pmgp} summarizes online forecasting and hyper-parameters learning under the $p$M-GP filter. \begin{algorithm}[ht] \caption{Forecasting with the $p$M-GP filter.} \label{alg:fore_pmgp} \begin{algorithmic} \STATE {\bfseries In:} $\{ y_{t_k}\}_{k \geq 0}$, average sampling frequency $F_s$. \STATE {\bfseries Out:} Predictive probability density functions $\{ \dots, p(z_t \vert y_{t_0}:y_{t_k}), \dots \}, t \geq t_k$. \\\hrulefill \STATE Set $\omega_i = \frac{1+i}{1+n}\pi F_s$, set the other parameters to $0$. \FORALL{$(t_k, y_{t_k})$} \STATE Evaluate $\bm{\theta}_{t_k}$ using Eq. (\ref{prob:solu}). \STATE Run the prediction step Eqs. (\ref{eq:pred}) with $t=t_k$. \STATE Run the update step Eqs. (\ref{eq:update}) with $t=t_k$. \STATE For $t \geq t_k$ get $p(z_t \vert y_{t_0}:y_{t_k})$ from Eqs. (\ref{eq:solu2}, \ref{eq:pred}). \ENDFOR \end{algorithmic} \end{algorithm} In order to control the aggressiveness of our learning algorithm in a manner that is consistent across datasets, we rewrite the aggressiveness parameter in the form $c_k := c \frac{\| \bm{\theta}_{t_{k-1}}\|^2}{\left(\epsilon +\hat{\mathcal{L}}^{l}_{t_k}(\bm{\theta}_{t_{k-1}})\right)^2}, c>0$, that arises by normalizing the term $|| \bm{\theta} - \bm{\theta}_{t_{k-1}}||^2$ (resp. $\xi^2$) in the objective function of problem (\ref{prob:2}) by $\| \bm{\theta}_{t_{k-1}}\|^2$ (resp. $\left(\epsilon +\hat{\mathcal{L}}^{l}_{t_k}(\bm{\theta}_{t_{k-1}})\right)^2$). \section{EXPERIMENTS} \label{sct:exp} \begin{figure*}[t!] \centering \includegraphics[width=0.4\textwidth]{K_2_trend_lin_eps_0_0_c_100_p_10-eps-converted-to.pdf} \includegraphics[width=0.29\textwidth]{K_5_sensitivity_to_c-eps-converted-to.pdf} \includegraphics[width=0.29\textwidth]{c_100_sensitivity_to_K-eps-converted-to.pdf} \caption{CO2 experiment: (left) running normalized mean absolute error (NMAE), (middle) effect of the $p$M-GP aggressiveness parameter $c$ on NMAE, and (right) effect of the number of spectral components $K$ on NMAE.} \label{fig:co2} \end{figure*} \begin{table}[t!] \caption{Mean $\pm$ 1 standard deviation of normalized absolute errors in the experiments of section \ref{sct:exp}.} \label{tab:bench} \begin{center} \begin{tabular}{lcccc} \toprule & CO2 & Airline \\ \midrule $p$M-GP & \textbf{0.52} $\pm$\textbf{0.70} & \textbf{0.52} $\pm$ \textbf{0.52} \\ PAs AR(2) & 0.77 $\pm$ 0.57 & 0.94 $\pm$ 0.84 \\ BLR AR(2) & 1.31 $\pm$ 0.58 & 0.77 $\pm$ 0.62 \\ PAs AR(10) & 1.02 $\pm$ 0.57 & 0.92 $\pm$ 0.81 \\ BLR AR(10) & 1.77 $\pm$ 1.00 & 0.77 $\pm$ 0.53 \\ \bottomrule \end{tabular} \end{center} \end{table} In this section we start by demonstrating that the $p$M-GP filter provides considerably more accurate forecasts than competing fully-online approaches on standard real-life time series datasets. We then experimentally illustrate the sensitivity of the forecasting errors to the normalized aggressiveness parameter $c$, the trend, and the number of spectral components $K$. \textbf{Benchmarking}: We compare our model to competing fully-online alternatives on the CO2 dataset of \cite{rasswill}, and the airline passengers dataset of \cite{box2015}. We select as competing benchmarks two autoregressive (AR) models, namely AR(2) and AR(10), and we use four different algorithms to learn the autoregressive coefficients online, namely the PA, PA-I and PA-II algorithms of \cite{pajmlr}, and Bayesian online linear regression with i.i.d. standard normal priors on the weights (BLR). For consistency with the $p$M-GP filter, we initialize the autoregressive weights in all four algorithms to $0$. We choose $c=100.0, \epsilon=0.0$ for $p$M-GP models and PA algorithms. We choose a linear trend and twice differentiability ($p=2$) for the $p$M-GP filter. BLR is run with a noise standard deviation of $5\%$ of the sample standard deviation of the corresponding time series. For each model we perform one-step ahead forecast. Mean absolute errors\footnote{Excluding the first forecast, which is the same for all models.} normalized by the standard deviation of increments (NMAE) are reported in Tab. \ref{tab:bench}, where we refer to all three PA algorithms as PAs because they perform identically up to two decimal points. The evolution of the running NMAE as a function of time is illustrated in Fig. \ref{fig:co2} (left) for the CO2 dataset. We note that our approach provides more accurate forecasts than competing alternatives. \textbf{Sensitivity to Parameters}: The sensitivity of the accuracy of the $p$M-GP filter to the trend, the aggressiveness parameter $c$ and the number of spectral components is illustrated in Fig. \ref{fig:co2} (middle and right) for the CO2 dataset. Overall, for our data size (607 points), it can be seen that the error decreases as a function of $c$. This is in line with the empirical observation of \cite{pajmlr} (Fig. 5) that, for small datasets large $c$ perform better, and $c=0.001$ begins to outperform $c=100.0$ for the PA classification problems the author considered when the data size is in the thousands. Moreover, we note from Fig. \ref{fig:co2} (right) that for small datasets, a large $K$ should not necessarily be preferred to a smaller one, as more samples will typically be required to learn the model hyper-parameters. \section{DISCUSSION} \label{sct:con} We propose an exact state space representation of Gaussian process regression for time series forecasting, under a family of kernels that is dense in the family of all stationary kernels, and that allows decoupling the flexibility of the covariance structure from the differentiability requirement of the latent time series. When hyper-parameters are known, exact GP predictive inference can be performed in constant time and with constant memory requirement. Critically, we propose a novel \textit{passive-aggressive} algorithm for online learning of the model hyper-parameters. The overall approach we refer to as the $p$M-GP filter has constant complexity and memory requirement, and provides more accurate forecasts than fully-online competing alternatives on the standard CO2 and airline passengers datasets. The approach may easily be extended to structured time series prediction, thus allowing for online learning of correlations between time series. Techniques that extend the Kalman filter to leptokurtic measurement noise models (e.g. \cite{aga11}) may also be used out-of-the box to robustify our approach. \clearpage

proofpile-arXiv_069-5125

\section{Introduction} The chemical composition of regions where stars are forming influences the thermal balance and the evolution of interstellar clouds on their way to form stars (\citealt{hocuk2014}). Recent observations of cold environments (i.e. pre-stellar cores) have shown unexpected amount of some molecules in the gas phase (\citealt{Bacmann2012}), while these species should be depleted from the gas phase and adsorbed on cold dust grain (T$<$20 K) in physisorbed states (\citealt{collings2004}). In order to understand these observations, the transition between solid and gas phases has been studied in laboratory under physical-chemical conditions present in the ISM. In particular, many studies have been focussing on thermal desorption (\citealt{bisschop2006, noble15}) and non-thermal processes, like photo-desorption (\citealt{desimone2013}; \citealt{bertin2013}), sputtering (\citealt{Johnson2013}; \citealt{cassidy2013}), cosmic ray (CR) desorption (\citealt{leger1985}; \citealt{hasegawa1993}; \citealt{ivlev2015}) and chemical desorption (\citealt{cazaux2010}; \citealt{dulieu2013}; \citealt{minissale2014}). Differently from the other processes, chemical desorption links solid and gas phases without the intervention of any external agents like photons, electrons or other energetic particles. In other words, it could be efficient in UV shielded and low CR environments where photo-desorption, CR (thermal) desorption or sputtering cannot be efficient. The chemical desorption process starts from the energy excess of some radical-radical reactions. As described in \cite{minissaledulieu2014}, its efficiency depends essentially on four parameters: enthalpy of formation, degrees of freedom, binding energy and mass of newly formed molecules. The aim of this article is to quantify the chemical desorption mechanism (part I) and its impact on interstellar gas (part II). The paper is organized as follow: In Sec. II, the experimental protocol and results are presented. In Sec.~III, we derive a law to quantify the chemical desorption mechanism. In Sec.~IV we discuss the main results of this work. \section{Experiments} The experiments have been performed using the FORMOLISM set-up (\citealt{amiaud2006}; \citealt{congiu2012}). In an ultra high vacuum chamber a gold mirror covered with amorphous silicates, oxidized graphite or compact amorphous water substrate is held at very low temperatures ($\sim$ 10~K). All the experiments are performed in sub-monolayer regime, to be sure that the interaction concerns bare surfaces, and is not perturbed by any other layer properties. The surface coverage is known using the specific desorption properties of the second layer (\citealt{noble2012}). A typical experiment proceeds as follow: one layer (or less) of non self-reactive molecules is deposited at low temperature. Just after its deposition another beam is aimed at the solid sample and the desorption flux is monitored simultaneously with a movable quadrupole mass spectrometer, placed 3 mm away from the surface. These experiments are refereed to as \rm{During Exposure Desorption}\rm\ (DED). To deal with radical-radical systems we sometimes use two simultaneous beams (i.e. [O+H] system ) because each separate beams will lead to a complete recombination (i.e. H+H$\longrightarrow$ H$_2$) and so a sequential deposition is meaningless. The DED technique collect species in the gas phase during the reactive phase (beam exposure). Due to poor angular collection, signals are weak but are unambiguously due to the reactive systems. For example, if mass 20 is measured during the irradiation of an O$_2$ layer ( O$_2$ with mass of 32 a.m.u.) by D atoms (mass 2), it is a direct evidence that D$_2$O is produced and undergoes chemical desorption. Such a signal is presented in the Figure 6 of \cite{chaabouni2012}. Mass confusions with cracking patterns of the QMS are still possible but can be estimated by careful calibration. DED measurements allow to determine the coarse efficiency of the chemical desorption process for many considered reactions. Even though it is a direct measurement, most of the reactive systems are actually composed of many reactions. In the former example, it is clear that the O$_2$ + D reaction cannot directly produce D$_2$O, even if this signal appears as soon as the D exposure begins. For the case of water formation, more than 10 reactions are taking place simultaneously (see the Figure 7 of \cite{chaabouni12} for a graphical description). DED is a direct measurement of products, but the link between one product and one reaction is not necessary direct. The DED technique can be completed by indirect measurements of the chemical desorption. In a second phase, after the exposure, the surface temperature is increased and the desorption flux is monitored. This second phase, called temperature programmed desorption (TPD) experiments, allows us to determine the amount of formed products that did not desorb upon formation. TPD experiment is therefore an indirect process to measure the loss of surface species due to chemical desorption. \subsection{How to determine the chemical desorption efficiency: the test case of the O+O system} In a recent experiment set of experiments \citep{minissale2014,congiu2014} we studied the formation of ozone and O$_2$ on different surfaces obtained by deposition of a unique beam composed of 85$\%$ of O and 15$\%$ of O$_2$. We choose this example because contrarily to most of the other reactive systems, there are here only 2 efficient reactions O+O $\longrightarrow$ O$_2$ and O+O$_2$ $\longrightarrow$ O$_3$. Therefore, there is no ambiguity or multiple possible assignment in the chemical reaction network. Each product corresponds to one reaction. At the end of the exposure, the temperature of the surface has been increased and the amount of O$_2$ and O$_3$ (TPD) that were present on the surface could be measured. \figurename~\ref{O2O3} reports the results for experiments performed on water ice (squares), silicates (triangles) and graphitic (circles) surfaces. Keeping the same total exposure beam, the experiments can be repeated with various substrate temperature (so called ``vs T$_s$ experiments''), or keeping the same surface temperature (usually 10K), the exposure time can be varied, so the effect of the deposited dose can be studied (so called``vs dose''). In all the experiments, all the atoms are consumed during the exposure phase and we collect the O$_2$ and O$_3$ signals. The numbers reported in the X and Y axes are the fraction of O$_2$ and O$_3$ measured in the TPD. The point A on the \figurename~\ref{O2O3} (coordinate 0, 1) represents an experiment where almost all the O/ O$_2$ deposited has been transformed into Ozone. This happens in high surface temperature conditions where the mobility of O atoms is high and thus any atoms has the possibility to react with an O$_2$ molecule. The point B (0.8, 0.2), corresponds to the case where 80$\%$ of the products have desorbed under the form of O$_2$. This is the case of very low temperature and coverage. In these conditions, the O+O reaction dominate over the O$_2$+O reaction. If the total fraction of O$_2$ and O$_3$ is equal to 1 (the sum of coordinates), then all the oxygen species sent on the surface are desorbing from the substrate during the TPD. This case is represented by the thick line which crosses both O$_2$ and O$_3$ fraction origin axis at coordinate 1. On the contrary, the point C (0.25,0.25) on the figure represents a case where the amount of species desorbing from the surface after the deposition is about equal to half of the total amount of species that were deposited. The oxygen species missing from the TPD experiments are attributed to the chemical desorption process occurring during the exposure phase. The general conclusion drawn from \figurename~\ref{O2O3} is that the chemical desorption process can be efficient under some specific conditions. We notice that experiments made on water substrates (blue points) all lie around the thick line representing the absence of chemical desorption. The reactions occurring on water ice surfaces lead to products remaining on the surface, and the chemical desorption process is almost negligible, or inside the error bars. Experiments made on silicates and graphite surfaces, however, show a large deficit in oxygen species. The experimental results show that almost 50$\%$ of the oxygen is missing from the surface for reactions occurring on graphite surfaces, while this fraction is of 30$\%$ for reaction on silicates and is negligible on water ices. For graphite and silicates surfaces, the amount of oxygen missing can vary with the coverage of the species on the surface (open triangles for silicates and open circles for graphite), which indicates that the chemical desorption process strongly depends on the surface coverage (\citealt{minissaledulieu2014}). Actually, the presence on the surface of a neighbor of similar mass increases strongly the probability of energy transfer. The energy detained by the molecule upon its formation is quenched very efficiently. In this case, the chemical desorption process is minimized. \begin{figure} \centering \includegraphics[width=8.8cm]{FullO2O3b.png} \caption{Normalized fraction of O$_2$ and O$_3$ for the [O+O] reactive system obtained after TPD. The filled symbols represent the surface temperature variation ( between 6.5~K and 25~K ) while the empty symbols represent the dose variation (between 0.05~ML, and 1.0~ML). The black dots represent the data of \cite{minissaledulieu2014} where the method is detailed. Other points are coming from \cite{minissale2013}, \cite{minissale2014}, \cite{congiu2014} or are original data. The solid black line represent the zone of the plot where no chemical desorption is measured.} \label{O2O3} \end{figure} The oxygen species missing from the TPD experiments are attributed to the chemical desorption process. In order to confirm that the chemical desorption process is at the origin at the loss observed in the TPD, DED measurements are performed. This technique consists in measuring directly the increase of the partial pressure of the gas during the exposure of the reactants. For these type of measurements, the quadrupole mass spectrometer (QMS) head is close to the surface but still allows the deposition of species on the surface. This also implies that the position of the QMS is not optimized for the detection of the desorbing species. Furthermore, molecules directly formed may have a high kinetic energy, which reduces their interaction time passing through the QMS head. This mostly explains why the DED signals are low. On the left panel of the \figurename~\ref{DED}, we show the raw data measured during the exposure of O atoms on silicates and water ice surfaces. Before exposure (for time $\le$1.5 min), and after exposure (for time $\ge$4.5 min), the signal at mass 32 (a.m.u.) is very low. The signal during exposure, located between 1.5 and 4.5 min, is stronger for O$_2$ desorbing from a graphite surface than from a water ice surface. This is exactly the opposite situation to what we found in the TPD. This shows that the missing oxygen from our TPD are released into the gas during exposure, through the chemical desorption process, and that this process strongly depends on the surface on which reactions occur. The chemical desorption is found to be less efficient on water substrate than on oxidized graphite, via direct measurement (DED) or indirect measurement (TPD). Note that for the specific case of DED of a beam of O, a part of the O$_2$ beam is not dissociated and is detected in the QMS, and therefore does not come from the formation and desorption of O$_2$ on the surface. The DED technique provides good qualitative information, but is not suitable for quantitative analyses. It is however possible for us to sort qualitatively desorbing species. The species that we easily detect by DED should have an efficiency of few tens of \%, whereas those close to our detection limit, should be in the 5\%- 15\% range, depending on the level of background noise. \begin{figure} \centering \includegraphics[width=8.8cm]{Fig2.png} \caption{Left panel: Raw DED measurements of mass 32 (O$_2$) during O/O$_2$ deposition (between 1.5 and 4.5 min) on graphite (green) and water ice (red) held at 10 K. Right panel: mean DED measurements over the 3 min exposure with associated error bars. } \label{DED} \end{figure} By combining TPD and DED experiments for O$_2$ on several surfaces, we could obtain an accurate measurement of the chemical desorption efficiency for the [O+O] system. This is possible because of the limited number of reactions occurring in this system. We studied many other systems. Some are quite simple like [N+N] (or [H+H]), where it is reasonable to think that there is only one reaction N+N $\longrightarrow$ N$_2$. But most other systems are more complex. \begin{table*}[t] \centering \caption{List of experimental chemical desorption efficiencies for different reactions on three different surfaces (np-ASW, amorphous silicate, and oxidized graphite). For the cases in which we were not able to measure quantitatively the CD efficiency, we provide upper or lower limits. We list theoretical CD efficiencies calculated for an oxidized graphite.}\label{table1} \resizebox{\textwidth}{!}{% \begin{tabular}{c c c||cc||cc||c||ccc||c} \rowcolor[gray]{0.85} Experiments & Surface & Coverage & \multicolumn{2}{>{\columncolor[gray]{.85}}c}{Method} & Reaction&Desorbing&Delta H$_R$&\multicolumn{3}{>{\columncolor[gray]{.85}}c}{Experimental CD efficiency} &Theoretical CD efficiency\\ \rowcolor[gray]{0.85} & Temperature & Range & DED & TPD & & product & & & & & \\ \rowcolor[gray]{0.9} & & & & & & & &np-ASW & Amorphous & Oxidized & \\ \rowcolor[gray]{0.9} & & & & & & & & &Silicate & HOPG & \\ \rowcolor[gray]{0.85} & (K) & (ML) & & & & &eV&\%&\%&\%&\% \\ \rowcolor[gray]{0.96} & & & & & O+H & OH &4.44& 25$\pm$15{\tiny$^{*}$} & -- & 50$\pm$25{\tiny$^{*}$} & 39\\ \rowcolor[gray]{0.96} \multirow{-2}{*}{[O+H]} & \multirow{-2}{*}{10} & \multirow{-2}{*}{$<$1} & \multirow{-2}{*}{\textit{$\surd$}} & \multirow{-2}{*}{\textit{$\surd$}} & OH+H & H$_2$O & 5.17& 30$\pm$15{\tiny$^{*}$} & -- & 50$\pm$25{\tiny$^{*}$} & 27\\ \rowcolor[gray]{0.94} & & & & & OH+H & H$_2$O &5.17& $<$40$\pm$20{\tiny$^{a}$} & $<$70$\pm$20{\tiny$^{a}$} & $<$80$\pm$20{\tiny$^{b}$} &27 \\ \rowcolor[gray]{0.94} & & & & & O$_2$+H & O$_2$H & 2.24& $<$8{\tiny$^{*}$} & 10$\pm$10{\tiny$^{b}$}& -- &1.4\\ \rowcolor[gray]{0.94} & & & & & O$_2$H+H & H$_2$O$_2$ & 3.69& $<$8{\tiny$^{*}$} & $<$5{\tiny$^{b}$} & -- &0.5\\ \rowcolor[gray]{0.94} & & & & & O$_2$H+H & 2 OH & 1.47& $<$8{\tiny$^{*}$} &$<$5{\tiny$^{b}$} & --&0.3\\ \rowcolor[gray]{0.94} \multirow{-5}{*}{[O$_2$+H]} & \multirow{-5}{*}{10} & \multirow{-5}{*}{0.5-1} & \multirow{-5}{*}{\textit{$\surd$}} &\multirow{-5}{*}{\textit{$\surd$}} & H$_2$O$_2$+H& H$_2$O+OH& 2.95& $<$5{\tiny$^{*}$} & $<$5{\tiny$^{b}$} & --&2.1\\ \rowcolor[gray]{0.96} & & & & & O$_3$+H & O$_2$+OH &3.33& -- & $<$10{\tiny$^{b}$} & $<$8{\tiny$^{*}$} &8 \\ \rowcolor[gray]{0.96} \multirow{-2}{*}{[O$_3$+H]} & \multirow{-2}{*}{10 and 45} & \multirow{-2}{*}{0.5-1}& \multirow{-2}{*}{\textit{$\surd$}} &\multirow{-2}{*}{\textit{$\surd$}} & OH+H & H$_2$O & 5.17& -- & 80$\pm$20{\tiny$^{b}$} & -- &27\\ \rowcolor[gray]{0.94} & & & & & O+O & O$_2$ & 5.16&$<$5{\tiny$^{c}$} & 40$\pm$10$^{\#}${\tiny$^{d}$} & 80$\pm$10$^{\#}${\tiny$^{e}$}&68\\ \rowcolor[gray]{0.94} \multirow{-2}{*}{[O+O]} & \multirow{-2}{*}{8 to 25} &\multirow{-2}{*}{$<$1}& \multirow{-2}{*}{\textit{$\surd$}} & \multirow{-2}{*}{\textit{$\surd$}} & O$_2$+O & O$_3$ & 1.1&$<$5{\tiny$^{c}$} & $<$5{\tiny$^{d}$} & $<$5{\tiny$^{e}$}& 0\\ \rowcolor[gray]{0.96} \multirow{-2}{*}{[N+N]} & \multirow{-2}{*}{10} & \multirow{-2}{*}{$<$1}& \textit{$\surd$} & \textit{$\surd$} & N+N & N$_2$ & 9.79&$>$50$^{\&}${\tiny$^{*}$} & $>$70$^{\&}${\tiny$^{*}$} & $>$70$^{\&}${\tiny$^{*}$}& 89\\ \rowcolor[gray]{0.94} & & & & & CO+H & HCO &0.66& -- & -- & 10$\pm$8{\tiny$^{*}$}& 0.7\\ \rowcolor[gray]{0.94} & & & & & HCO+H& CO+H$_2$ &3.85& -- & -- & 40$\pm$20{\tiny$^{*}$} & 47\\ \rowcolor[gray]{0.94} \multirow{-3}{*}{[CO+H]} & \multirow{-3}{*}{10}& \multirow{-3}{*}{0.8-2.5} & \multirow{-3}{*}{\textit{$\surd$}} & \multirow{-3}{*}{\textit{$\surd$}} & HCO+H& H$_2$CO & 3.91&-- & -- & $<$8{\tiny$^{*}$} &7\\ \rowcolor[gray]{0.96} & & & & & H$_2$CO+H & CH$_3$O & 0.88& -- & -- & $<$8{\tiny$^{*}$}&0\\ \rowcolor[gray]{0.96} & & & & & H$_2$CO+H& HCO+H$_2$ & 0.61& -- & -- & 10$\pm$5{\tiny$^{*}$}& 0\\ \rowcolor[gray]{0.96} & & & & & HCO+H& CO+H$_2$&3.85& -- & -- & 40$\pm$20{\tiny$^{*}$} &47\\ \rowcolor[gray]{0.96} & & & & & HCO+H& H$_2$CO &3.91& -- & -- & 10$\pm$5{\tiny$^{*}$} &7\\ \rowcolor[gray]{0.96} \multirow{-5}{*}{[H$_2$CO+H]}&\multirow{-5}{*}{10 and 55} & \multirow{-5}{*}{0.8-2} & \multirow{-5}{*}{\textit{$\surd$}} &\multirow{-5}{*}{\textit{$\surd$}} & CH$_3$O+H& CH$_3$OH &4.56& $<$8{\tiny$^{*}$} & --& $<$8{\tiny$^{*}$} &2.3\\ \rowcolor[gray]{0.94} \multirow{-1}{*}{[Ar+H]} & \multirow{-1}{*}{10} & \multirow{-1}{*}{1} & \textit{$\surd$} &\textit{$\surd$} & Ar+H & Ar && -- & $<$5{\tiny$^{*}$} & -- &\\ \rowcolor[gray]{0.96} \multirow{-1}{*}{[NO+H/O/N]} & \multirow{-1}{*}{10} & \multirow{-1}{*}{0.5-5} & \textit{X} &\textit{$\surd$} & many & many && $<$8{\tiny$^{*}$} & $<$8{\tiny$^{*}$} & $<$8{\tiny$^{*}$} &\\ \rowcolor[gray]{0.94} \multirow{-1}{*}{[CO+O]} & \multirow{-1}{*}{10} & \multirow{-1}{*}{0.5-4} & \textit{$\surd$} &\textit{$\surd$} & CO+O& CO$_2$ & 5.51& $<$5{\tiny$^{*}$} & -- & $<$5{\tiny$^{*}$}&22\\ \rowcolor[gray]{0.96} \multirow{-1}{*}{[H$_2$CO+O]} & \multirow{-1}{*}{10 and 55} & \multirow{-1}{*}{0.8-2} & \textit{X} &\textit{$\surd$} & H$_2$CO+O& CO$_2$+H$_2$ &5.45& $<$10{\tiny$^{*}$} & -- & $<$10{\tiny$^{*}$}&8\\ \rowcolor[gray]{0.94} \multirow{-1}{*}{[CH$_3$OH+H]} &\multirow{-1}{*}{10} & \multirow{-1}{*}{0.5-2} & \textit{$\surd$} &\textit{$\surd$} & CH$_3$OH+H& CH$_3$OH & &$<$8{\tiny$^{*}$} & -- & $<$8{\tiny$^{*}$}& \\ \rowcolor[gray]{0.96} \multirow{-1}{*}{[CH$_3$OH+O]} & \multirow{-1}{*}{10} & \multirow{-1}{*}{0.5-2} & \textit{$\surd$} &\textit{$\surd$} & CH$_3$OH+O& CH$_3$OH & &$<$8{\tiny$^{*}$} & -- & $<$8{\tiny$^{*}$}&\\ \end{tabular}} {\textit{Surface temperature} indicates deposition temperature of both reactants. \textit{Coverage range} indicates the coverage of molecular ice growth before atoms irradiation; in the case of experiments performed only with atomic species, \textit{coverage range} indicates the total amount of species sent on the surface.} \textit{$\surd$} and \textit{X} are used to indicate if the experimental procedure (DED or TPD) have been used or not, respectively; $^{\#}$ CD decreases as a function of coverage; $^{\&}$ experiments performed with excited particles (see ~\citealt{minissalethesis2014}); $^{a}$ \citealt{chaabouni2012}; $^{b}$ \citealt{dulieu2013}; $^{c}$ \citealt{minissale2013}; $^{d}$ \citealt{minissale2014}; $^{e}$ \citealt{minissaledulieu2014}; $^{*}$ This work. \end{table*} \subsection{How to determine the chemical desorption efficiency: more complex systems} Table~\ref{table1} lists the different experiments performed as well as the main reactions involved in each experiment. In the case of the [O+H] experiment, there are 2 main reactions occurring in the experiment, O+H $\longrightarrow$ OH and OH+H $\longrightarrow$ H$_2$O that we can measure with TPD and DED. To derive the chemical desorption efficiency for the reaction OH+H $\longrightarrow$ H$_2$O, we assume that the fraction of H$_2$O detected is due to the OH molecules that stayed on the surface upon formation. The amount of chemical desorbed H$_2$O observed in this experiment indirectly gives information on the amount of OH staying on the surface or desorbed upon formation. We positively detected OH for the [O+H] system, but not for the [O$_2$+H] nor [O$_3$+H] systems. By looking at the possible reaction network of water, we can attribute the high chemical desorption efficiency of OH to the O+H reaction. It is not trivial since other routes can produce OH, such as H$_2$O$_2$+H$\longrightarrow$ H$_2$O+OH or O$_2$H+H$\longrightarrow$ 2 OH. To summarize, by combining TPD and DED experiments, one can identify which are the ``missing species'' by using a small reaction network. This allows us to derive some estimates of the chemical desorption efficiency for the formation of different species on surfaces. Here we present several experiments which are listed in Table~\ref{table1}. For each experiment, as mentioned previously, more than one reaction can occur leading to different products with distinct chemical desorption efficiencies. These reactions, as well as the enthalpy of reaction, the chemical desorption efficiency on water ice, amorphous silicates and oxidized graphite are reported in Table~\ref{table1}. \section{Theoretical estimate of the desorption efficiency} We propose to use a simple assumption of equipartition of energy to reproduce the chemical desorption observed experimentally. We suppose that the total energy budget $\Delta H_R$ (see Table\ref{table1}) is shared between all the degrees of freedom $N$=3$\times$ n$_{atoms}$. We use the generic 3 degree of freedom per atom. For a diatomic molecule, it would be decomposed in 3 degrees for translation, 1 degree for vibration, and 2 for rotation axis. Of course, the symmetry can affect this number slightly, however this would be a small correction. Among these degrees of freedom, only the kinetic energy perpendicular to the surface will allow the newly formed species to desorb, with the condition that this kinetic energy is more important than the binding energy ($E_{binding}$). The binding energies of the species considered in our network are reported in Table~\ref{bind}. Initially, the translational momentum is negligible since we believe that in most of our experiments the species admitted on the surface thermalize rapidly and the diffusion mechanism is the dominant process. To desorb from the surface, the newly formed products have to bounce against the surface to gain velocity in the direction perpendicular to the surface. The simplest way to treat this process is to assume an elastic collision, \begin{equation} \epsilon=\frac{(M-m)^2}{(M+m)^2}. \end{equation} Here $\epsilon$ is the fraction of kinetic energy retained by the product $m$ colliding with the surface which has an effective mass $M$. For the specific case of hyper thermal scattering of O$_2$ on graphite surface, \cite{hayes2012} have proposed an equivalent mass of 130 a.m.u. The effective mass is much higher than one single C atom, or even a carbon ring, because the rigidity of the surface induces a collective behavior of the atom and a larger effective mass. \begin{table}[h] \caption{Binding energies on bare and icy surfaces. \label{bind}} \begin{tabular}{lll} Species & E$_{bare}$ &E$_{ice}$\\ \hline\hline H&550$^{a,b}$&650$^c$\\ H$_2$&300$^{a,d}$&500$^e$\\ O&1500$^b$&1400$^f$\\ OH&4600$^g$&4600$^g$\\ H$_2$O&4800$^g$&4800$^h$\\ O$_2$&1250$^{i}$&1200$^i$\\ O$_3$&2100$^j$&2100$^j$\\ HO$_2$&4000$^g$&4000$^g$\\ H$_2$O$_2$&6000$^g$&6000$^g$\\ CO&1100$^i$&1300$^{i,k}$\\ CO$_2$&2300$^i$&2300$^{i,k}$\\ HCO&1600$^l$&1600$^l$\\ H$_2$CO&3700$^m$&3200$^{m}$\\ CH$_3$O&3700$^l$&3700$^l$\\ CH$_3$OH&3700$^n$&3700$^{n}$\\ N&720&720$^{f}$\\ N$_2$&790$^o$&1140$^p$\\ \hline\hline \end{tabular} \\ \rm{For HCO, CH$_3$O and CH$3$OH we assume E$_{ice}$=E$_{bare}$.}\rm\ $^a$\citealt{cazaux2004}, $^b$\citealt{bergeron2008}, $^c$\citealt{Al-Halabi2007}, \citealt{amiaud2007}, $^d$\citealt{Pirronello1997}, $^e$\citealt{amiaud2006} for low coverage $^f$~Minissale et al. submitted, $^g$\citealt{dulieu2013}, $^h$\citealt{speedy1996}, \citealt{fraser2001} energy derived of 5800~K with pre-factor of 10$^{15}$ s$^{-1}$ here corrected to have pre-factor of 10$^{12}$ s$^{-1}$, $^i$\citealt{noble2012b}, $^j$\citealt{borget2001}, \citealt{minissalethesis2014}, $^k$\citealt{Karssemeijer2014},$^l$\citealt{garrod2006}, $^m$\citealt{noble2012}, $^n$\citealt{collings2004}, $^o$\citealt{fuchs2006}, $^p$~extrapolated from \citealt{kimmel2001}. \end{table} \rm{The chemical energy is certainly splitted in many different ro-vibronic states, not at all in thermal equilibrium. We assume that an equal share of energy is spread over all the degrees of freedom, so that the total chemical energy available for the kinetic energy perpendicular to the surface is $\epsilon \Delta H_R/N$. We assume that the only velocity perpendicular to the surface corresponds to a distribution of speed/temperature. In this sense, kT = $\epsilon \Delta H_R/N$.}\rm\ The probability for the product to have an energy (or liberation velocity) larger than the binding energy becomes: \begin{equation} CD = e^{-\frac{E_{binding}}{\epsilon \Delta H_R /N}}. \end{equation} In figure ~\ref{FigCD}, we compare the results obtained experimentally (blue triangles) to our predictions. We use first a constant $\epsilon$ of 0.4, which implies that we assume that 40$\%$ of the total energy is retained by the product (red hexagons). In this case, our predictions do not reproduce well the experimental results. In a second step, we consider a fraction $\epsilon$ which depends on the mass of the products, as described in eq. 1. These predictions are represented as green squares and do match very well the chemical desorption efficiencies observed experimentally. This clearly indicates that the chemical desorption process is sensitive to the binding energy, the energy distributed in the products (depending on their mass), and the degrees of freedom. For some of the reactions such as OH+H, O+H, our predictions slightly underestimate the experimental results. \begin{figure} \centering \includegraphics[width=8.8cm]{FIG1.png} \caption{Measured CD efficiencies in blue with error bars. In green, the result of the formulae (eq. 2) which include division among the degree of freedom and fraction of kinetic energy after bounce ($\epsilon$ is mass dependent). In red, equal share of energy and constant value of $\epsilon$.} \label{FigCD} \end{figure} The formula (eq. 2) can be translated to the case of silicates surfaces. The efficiency is slightly smaller, and thus reducing the effective mass from 130 to 110 gives a very good match for the most constrained reaction (O+O). Taking into account the degree of inaccuracy of our experimental determination of CD, adopting a common value of 120 amu is appropriate to treat the trend of CD for bare grains. The case of water substrate (np-ASW), which is even more important from the perspective of explaining the return in the gas phase of complex organic molecules (COMs), appears to be less simple. We have two contradictory results. The chemical desorption of O+O is at least 8 times less, but the OH+H reaction seems to be still efficient, even if lowered by a factor of $\sim$ 2. The mass argument does no longer stand. In the part II of our work, we assume that the chemical desorption efficiency on ices is our CD coefficient divided by 10 for the reactions that have not been constrained (CD$_{ice}$=CD$_{bare}$/10 while CD$_{ice}$(OH+H) = 25\%, CD$_{ice}$(O+H) = 30\% and CD$_{ice}$(N+N) = 50\%). \section{Discussion} Previously \cite{garrod2007} have proposed in their models a formula to describe the chemical desorption (or reactive desorption). They derive the probability to undergo desorption in two steps. The first one is based upon RRKM estimation and takes into account energetic aspects (degree of freedom, enthalpy of reaction and binding energy). In most of the cases this coefficient is found to be high. The second step, interaction with the surface, is parametrized with an analytic formula. We first tried to use this approach, but unsuccessfully. The main problem, already mentioned by the authors, is that the first step (energetic aspects) is not really discriminating in case of exothermic reactions, whereas the second one, which is unknown, is the most important. The net result is that when this formula is used, there is no really discrimination among the reactions. However, we clearly demonstrated that O+O has a high probability to desorb and O$_2$+O has not, while both of these reactions are exothermic. This a strong argument to reject the former approach. Our simple hypothesis of equipartition of energy, coupled with elastic collision with the surface, reproduces the observations well. Although it is an empirical approach, it allows to sort among the products that should undergo the chemical desorption. The basic idea underlying our proposition is inspired by dynamic calculations in specific systems such as H+H on graphite \citep{morisset2004, bachellerie2007} or O+H on graphite \citep{bergeron2008}. We have illustrated the mechanism of this scenario in \figurename~\ref{Figbounce}. The first step consists in a collinear approach parallel to the surface. When the molecule is formed, the trajectories of both atoms adopt a more disordered motion symptomatic of the high internal energy in vibration and rotation of the newly formed molecules, up to a point where the molecule interacts with the surface and that a part of the motion is transferred perpendicularly to the surface provoking the desorption. \begin{figure} \centering \includegraphics[width=8.8cm]{figSabine.png} \caption{Illustration of the chemical desorption process. After an initial meeting (1), the molecule is highly excited (2) and converts a fraction of its energy into vertical motion, thanks to the interaction with the surface (3)} \label{Figbounce} \end{figure} We point out that the chemical desorption process seems to favor small molecules with low binding energies. Therefore, this process may be part of the reason COMS are present in dark clouds. We notice that our prediction of around one percent of chemical desorption for methanol would be sufficient to sustain the observed value, as modeled by \cite{vasyunin2013}. The second part of our paper will examine the impact of the derivation of our formulae in an astrophysical context. The reduction of the chemical desorption efficiency on water ice substrates is somehow to be expected. Similar behavior has been observed in photodesorption of CO. Pure CO ices exhibit an efficient wavelength dependent photodesorption (up to one percent per incident photon, \citealt{bertin2013}) or may be a less by a factor 3-11 \citep{hemert2015}. On water substrate some authors have estimated the CO photodesorption to be around 10$^{-3}$ \citep{oberg2009} or one order of magnitude less (\citealt{desimone2013}). \rm{However, even if the CO photodesorption process is not very well constrained, it is clear that the water substrate prevents or dramatically reduces its efficiency.}\rm\ This probably is due to the very fast transportation of energy in the water network and to the reduction of the indirect photodesorption. In the case of the chemical desorption on water substrate, the weakness of our coarse model lies probably in the assumption of elastic collisions, which are not adapted to the case of water. Many more experimental investigations, as well as detailed computations would help to solve this problem. Finally, we note that even if water ice substrate seems to be a good trapping environment, and can dramatically reduce the chemical desorption efficiency, in the central part of dark clouds, CO layers are expected to cover an inner water layer around the solid seeds of dust. Therefore, another set of experiments should be performed using a CO ice substrate \section{Conclusions} In this study we present a collection of experiments in order to quantify which fraction of species forming on surfaces is ejected into the gas phase upon formation. This process, called chemical desorption, has been observed experimentally (\citealt{dulieu2013}). With this work we derive an analytical expression to quantify the chemical desorption process on different substrates such as oxidized graphite and amorphous silicate. This formula, which depends on the equipartition of the energy of newly formed products and the binding energy of the products, reproduces well the experimental results on bare surfaces. However, on icy surfaces the chemical desorption process is strongly reduced and cannot be well quantified by our experimental results. In part II of our study, we address the importance of the chemical desorption process on the composition of interstellar gas, by combining gas phase chemistry as well as surface and bulk chemistry, and using our analytical formula to account for the chemical desorption process. \begin{acknowledgements} We acknowledge the support of the French National PCMI program funded by the CNRS and the support of the DIM ACAV, a funding program of the R\'egion Ile de France. S. C. is supported by the Netherlands Organization for Scientific Research (NWO; VIDI project 639.042.017) and by the European Research Concil (ERC; project PALs 320620). We would like to thank the anonymous referee for his/her comments and suggestions that greatly improved our manuscript. \end{acknowledgements}

proofpile-arXiv_069-5389

\section{Summary} In this paper we applied a variational quasiparticle theory to study the ground state of a Bose system exposed to a constant external velocity field. We have shown that at small velocities the energy minimum for the variational ansatz occurs at zero quasiparticle excitation, meaning the persistence of a pure superfluid state at zero temperature. Crossing the Landau critical velocity a quasiparticle condensate is formed with spectacular consequences. The condensation takes place into a one-particle state of momentum $\k_\v$ which is parallel to the velocity $\v$ and whose magnitude is determined by the energy minimum. The quasiparticle condensation deeply influences the distribution of real particles. Apart from the original BEC in the $\k=\bold0$ one-particle state, two more condensates appear, a dominant one in the plane wave state $\k_\v$ and another one of a smaller density at $-\k_\v$. The coexistence of these condensates leads to density modulations characterized by the wave vectors $\k_\v$ and $ 2\k_\v$. In the present model, the density of the condensate at $\k_\v$ attains the full density at a finite velocity $v_1$; necessarily, the condensates at $\k=\bold0$ and $\k=-\k_\v$ and the two density modulations together with them vanish here, if not already at smaller velocities. At $v_1$ our model exhibits a bifurcation of $\k_\v$, with one solution increasing and the other one decaying as $v$ tends to infinity. The bifurcation is due to the fact that $\k_\v$ is determined by density saturation, when $v>v_1$. A general ground state is then a superposition of the two pure condensates, corresponding to the two solutions for $\k_\v$. There is no density modulation in these states. \section*{Acknowledgement} We thank Gergely Szirmai for numerical assistance. This work was supported by the Hungarian Science Foundation through OTKA Grant No. 77629.

proofpile-arXiv_069-5429

\section{Introduction} The isovector part of the equation of state (asy-EoS) of asymmetric nuclear matter, known as symmetry energy (SE), represents one of the remaining open questions of nuclear physics. It comprises the lesser known part of the nuclear matter equation of state (EoS) that can be approximately described by the expansion \begin{eqnarray} E(\rho,\beta)&=&E_0(\rho,\beta=0)+E_{sym}(\rho)\,\beta^2\,, \end{eqnarray} where $\beta$=($\rho_n$-$\rho_p$)/($\rho_n$+$\rho_p$) with $\rho_n$, $\rho_p$ and $\rho$ denoting the neutron, proton and total nucleon densities, respectively. The coefficient $E_{\rm sym}(\rho)$ of the asymmetry-dependent term is the symmetry energy. Knowledge of its precise density dependence is mandatory for the proper understanding of the structure of rare isotopes, dynamics and spectra of heavy-ion collisions and most importantly for certain astrophysical processes such as neutron star cooling and supernovae explosions~ \cite{Baran:2004ih,Li:2008gp}. Intermediate energy nuclear reactions involving stable and radioactive beams have allowed by studying the thickness of neutron skins, deformation, binding energies and isospin diffusion, the extraction of constaints on the density dependence of SE at densities below saturation ($\rho_0$)~\cite{Li:1997px,Chen:2004si,Chen:2005ti,Tsang:2012se}. Existing theoretical models describing its density dependence generally agree with each other in this density regime, but their predictions start to diverge well before regions with densities $\rho\geq2\rho_0$ are reached~\cite{Li:2008gp}. Nuclear matter at suprasaturation densities is created in the laboratory in the processes of collisions of heavy nuclei. Several observables that can be measured in such reactions have been determined to bear information on the behavior of the SE above $\rho_0$: the ratio of high transverse momentum neutron/proton yields~\cite{Yong:2007tx}, light cluster emission~\cite{Chen:2003qj}, $\pi^-/\pi^{+}$ multiplicity ratio in central collisions~ \cite{Xiao:2009zza,Feng:2009am,Xie:2013np}, double neutron to proton ratios of nucleon emission from isospin-asymmetric but mass-symmetric reactions~\cite{Li:2006wc} and others. The FOPI experimental data for the $\pi^-/\pi^{+}$ ratio~\cite{Reisdorf:2006ie} have been used to set constraints on the suprasaturation density behavior of SE by various authors with contradicting results: Xiao $et\,al.$~\cite{Xiao:2009zza} made use of the IBUU transport model supplemented by the isovector momentum dependent Gogny inspired parametrization of symmetry potential~\cite{Das:2002fr} to point toward a soft asy-EoS, the study of Feng and Jin~\cite{Feng:2009am}, which employed the isospin-dependent quantum molecular dynamics (IQMD) model and a power-law parametrization of the symmetry energy, \begin{eqnarray} S(\rho)&=&S_0\,(\rho/\rho_0)^{\gamma}, \eqlab{sepowerlaw} \end{eqnarray} favors a stiff SE. Most recently Xie $et\,al.$~\cite{Xie:2013np} addressed the same issue within the Boltzmann-Langevin approach and a power-law parametrization of asy-EoS presenting support for a super-soft scenario for the symmetry energy. Constraints on the high density dependence of $S(\rho)$ extracted from elliptic flow ratios of neutrons and protons (npEFR) $v_2^{n/p}=v_2^n/v_2^p$ and of neutrons and hydrogen $v_2^{n/H}$ have been presented by Russotto $et\,al.$~\cite{Russotto:2011hq}. The experimental data taken by the FOPI-LAND Collaboration for $^{197}$Au+$^{197}$Au collisions at 400 MeV/nucleon incident energy~ \cite{Leifels:1993ir,Lambrecht:1994cp} have been reanalyzed allowing for a reduction of systematical and statistical uncertainties. To model heavy-ion collisions a version of the UrQMD model~\cite{Li:2005gfa,Li:2005zza} and the power-law parametrization of SE mentioned above have been employed. A comparison of the theoretical and experimental elliptic flow ratios of neutrons vs. protons ($v_2^n/v_2^p$) and neutrons vs. hydrogen ($v_2^n/v_2^H$) has led to a constraint compatible with a linear density dependence for the potential part $S(\rho)$: $\gamma$=0.9$\pm$0.4~\cite{Russotto:2011hq}. In an independent study~\cite{Cozma:2011nr} the neutron-proton elliptic flow difference (npEFD) $v_2^{n-p}=v_2^n-v_2^p$ has been proposed as a viable observable for constraining the suprasaturation density dependence of SE. Its dependence on model parameters like in-medium microscopic nucleon-nucleon cross-sections, compressibility of symmetric nuclear matter and width of the gaussian wave packet of nucleons is a rather small fraction of the sensitivity to the changes between a stiff and a soft asy-EoS for kinematical acceptances close to those of the FOPI experiment. A comparison with published FOPI-LAND impact parameter dependent data~\cite{Leifels:1993ir,Lambrecht:1994cp} for $v_2^{n-p}$ was found problematic due to a highly non-monotonous dependence of the experimental data on that variable. Still, the experimental $v_2^{n-H}$, viewed as an upper bound of $v_2^{n-p}$, allowed the exclusion of the super-soft asy-EoS from the list of possible scenarios. The present Article aims at an update of the results of References~\cite{Cozma:2011nr,Russotto:2011hq} by extending the analysis of the former to both neutron-proton elliptic flow differences $v_2^{n-p}$ and ratios $v_2^{n/p}$ and by addressing the model dependence due to the momentum dependent part of the EoS and of the momentum dependence of the symmetry potential. A recent overview addressing the relevance of elliptic flow in the study of the SE at supra-saturation density can be found in Ref.~\cite{Trautmann:2012nk}. \section{The framework} \subsection{The model} In the present study, heavy-ion collisions have been simulated by using the QMD transport model developed in T\"{u}bingen~\cite{Khoa:1992zz,UmaMaheswari:1997ig} and expanded to accommodate density-dependent nucleon-nucleon cross-sections and an isospin dependent EoS. The same model has been previously used to study dilepton emission in heavy-ion collisions~\cite{Shekhter:2003xd,Cozma:2006vp,Santini:2008pk}, stiffness of the equation of state of symmetric nuclear matter~\cite{Fuchs:2000kp} and various in-medium effects relevant for the dynamics of heavy-ion collisions~\cite{Fuchs:1997we,UmaMaheswari:1997ig}. Most of the results of the following Section have been obtained by making use of the Gogny inspired momentum dependent parametrization of the isovector part of the equation of state~\cite{Das:2002fr}. It contains a parameter denoted $x$ which has been introduced to allow adjustments in the stiffness of asy-EoS, negative and positive values corresponding to a stiff and a soft density dependence of the symmetry energy, respectively (see Sect. IIIA). To assess the importance of the momentum dependent part of asy-EoS, the momentum-independent power-law parametrization (\eqref{sepowerlaw}) is used where indicated. Further details of the model, relevant for the current study, can be found in~\cite{Cozma:2011nr}. \subsection{Experimental data} \begin{table*}[t] \centering \begin{tabular}{|c|c|c|c|c|c|} \hline\hline Data set&Centrality&b (fm)&$ v_2^n$ & $v_2^p$ &$v_2^H$\\ \hline\hline &E2&7.2&-0.0939$\pm$0.0059& -0.0966$\pm$0.0052&-0.1045$\pm$0.0040\\ B&E3&4.7&-0.0711$\pm$0.0057&-0.0705$\pm$0.0054&-0.0758$\pm$0.0040\\ $0.25\leq y/y_P\leq 0.75$&E4&3.4&-0.0615$\pm$0.0066&-0.0324$\pm$0.0066&-0.0501$\pm$0.0047\\ $0.3\leq p_T\leq 1.0$&E5&1.9&-0.0245$\pm$0.0068&-0.0201$\pm$0.0072&-0.0222$\pm$0.0049\\ &E2-E5&$<$7.5&-0.0655$\pm$0.0031&-0.0627$\pm$0.0030&-0.0681$\pm$0.0022\\ \hline &E2&7.2&-0.1065$\pm$0.0111&-0.1008$\pm$0.0101&-0.1249$\pm$0.0079\\ C&E3&4.7&-0.0681$\pm$0.0108&-0.0583$\pm$0.0110&-0.0841$\pm$0.0080\\ $0.45\leq y/y_P\leq 0.55$&E4&3.4&-0.0552$\pm$0.0125&-0.0356$\pm$0.0129&-0.0593$\pm$0.0090\\ $0.3\leq p_T\leq 1.0$&E5&1.9&-0.0259$\pm$0.0126&0.0007$\pm$0.0148&-0.0178$\pm$0.0095\\ &E2-E5&$<$7.5&-0.0668$\pm$0.0058&-0.0586$\pm$0.0059&-0.0771$\pm$0.0043\\ \hline\hline \end{tabular} \caption{Experimental FOPI-LAND values for elliptic flow of neutrons, protons and hydrogen for two choices of kinematical conditions referred to in the text as data sets B and C. The applied kinematical cuts are shown in the first column, the values of the transverse momentum $p_T$ are in units of GeV/c. Changes in the elliptic flow values are minute (fourth digit) when the transverse momentum cut is relaxed to $0.3 \leq p_{T} \leq 1.3$.} \tablab{expdata} \end{table*} In the original release of the FOPI-LAND data~\cite{Leifels:1993ir,Lambrecht:1994cp} the extraction of proton spectra required, due to insufficient calorimeter resolution, narrow constraints to be applied in order to minimize the contamination with deuterium and tritium events. As a result proton elliptic flow values show a non-monotonous dependence on the impact parameter, in contrast to neutrons, making a comparison with predicted values troublesome. The data have been recently reanalyzed ~\cite{Russotto:2011hq} in order to determine the optimum conditions for the new ASY-EOS experiment~\cite{Russotto:2012jb} and to extract constraints on the density dependence of the symmetry energy. This effort has also resulted in a smoother impact-parameter dependence of the elliptic flow results for protons. Several data sets, corresponding to different ranges in rapidity $0.25<y/y_P<0.75$ (B) and $0.45<y/y_P<0.55$ (C) and transverse momentum \footnote{Units of momenta, GeV/c, are not displayed explicitly.} ($0.3<p_T<1.0$ and $0.3<p_T<1.3$) are available. The experimental values for the elliptic flow of neutrons and protons, as well as hydrogen, used to obtain the results in this work are displayed in~\tabref{expdata}. In the original FOPI-LAND data~\cite{Leifels:1993ir,Lambrecht:1994cp}, only the rapidity window $0.40 < y/y_p < 0.60$ (A) is covered for which the kinematical acceptance of the FOPI-LAND detector produces constraints for the transverse momentum as well: $0.27 < p_T < 1.06$. The values of the elliptic flow of neutrons and protons for this data set have been derived using the experimental values for the squeeze-out factor $R_N$ presented in Ref.~\cite{Lambrecht:1994cp}. Constraints on the stiffness of asy-EoS extracted by a comparison of the data sets corresponding to different rapidity windows, integrated over impact parameter, with model predictions of EFD $v_2^{n-p}$ agree with each other. One obtains for the $x$ parameter the following values: $x=-2.5\pm1.5$, $x=-1.5\pm0.75$, and $x=-2.0\pm0.75$ for the data sets A, B and C, respectively. Differences between the choices $p_T<1.0$ and $p_T<1.3$ are negligible. The theoretical estimates were obtained using the set of model parameters employed in Section III to generate the central solution, the uncertainty in the values of the $x$ parameter originating solely from the error bars of experimental elliptic flow values. \section{Model dependence} \subsection{Parametrizations of the potential} The nucleon optical potential is an important ingredient of transport models, the sensitivity of heavy-ion observables in general~\cite{Aichelin:1987ti,Peilert:1989kr,Greco:1999zz} and collective flows~\cite{Zhang:1994hpa} in particular to its momentum dependence being well documented. In a recent study~\cite{Zhang:2012fc} the effects of the momentum dependence of the symmetry potential on transverse and elliptic flows have been investigated with the conclusion that the neutron-proton elliptic flow difference exhibits a small sensitivity to the momentum dependent part of the isovector nucleon potential within the constraint of an asy-soft EoS. This is an important finding since the momentum dependence of the isovector part of the nucleon potential is still an open question. Parametrizations of it with various momentum dependences, or none at all, are commonly employed. On the theoretical side, the optical potential has been extracted from first principles~\cite{Baldo:1989zz}, and similar approaches have later been extended to also extract the symmetry potential~\cite{Zuo:2001bd,Zuo:2005hw,Xu:2012fu}. Alternatively, it has been possible to extract the momentum dependence of the bare nucleon interaction within an effective model~\cite{Hartnack:1994zz} starting from the optical potential of Refs.~\cite{Arnold:1982rf,Hama:1990vr} obtained within a relativistic Dirac-equation description of experimental data of proton scattering on Ca and heavier nuclei. The results of the two approaches are somewhat different, the Brueckner-Hartree-Fock approach and its relativistic counterpart favor a potential that is attractive at all values of the momentum, while the relativistic Dirac approach delivers a potential that becomes repulsive above a certain momentum threshold depending on which experimental data sets are considered. Additionally, the Brueckner-Hartree-Fock approach predicts an optical potential that is almost momentum independent at moderate values of the momentum. To account for this model dependence we have simulated heavy-ion collisions by considering three different parametrizations of the optical potential. The first one stems from the isoscalar part of the Gogny interaction~\cite{Das:2002fr} while the last two mimic the parametrizations presented in Ref.~\cite{Hartnack:1994zz} \begin{eqnarray} V_{opt}^{(MDI)}(\vec{p_i},\vec{p_j})&=&(C_l+C_u)\,\frac{1}{1+(\vec{p_i}-\vec{p_j})^2/\Lambda^2}\,\frac{\rho_{ij}}{\rho_0} \nonumber\\ V_{opt}^{(HA)}(\vec{p_i},\vec{p_j})&=&\{V_0+v\,\mathrm{ln}^2[\,a\,(\vec{p_i}-\vec{p_j})^2+1]\}\,\frac{\rho_{ij}}{\rho_0}. \eqlab{optpotparam} \end{eqnarray} The parameters present in $V_{opt}^{(MDI)}(\vec{p_i},\vec{p_j})$ can be found in Ref.~\cite{Das:2002fr}, while for $V_{opt}^{(HA)}(\vec{p_i},\vec{p_j})$ they read: $V_0$=-0.054 GeV, $v$=0.00158 GeV and $a$=500 GeV$^{-2}$, $V_0$=-0.0753 GeV, $v$=0.002526 GeV and $a$=500 GeV$^{-2}$ for the old and new parametrization in Ref.~\cite{Hartnack:1994zz}, respectively; $\rho_{ij}$ is the contribution to the density at the location of nucleon $j$ due to nucleon $i$, recovering in the infinite nuclear matter limit the parametrization of Ref.~\cite{Das:2002fr} and the right EoS. The $V_0$ parameter is absorbed in the linearly density dependent term of the single nucleon potential $V=\alpha\,\frac{\rho}{\rho_0} +\beta\,(\frac{\rho}{\rho_0})^\gamma+V_{opt}$. For completeness, the values of the remaining parameters, producing a soft (K=210 MeV) isoscalar EoS, read: $\alpha$=-0.3901 GeV, $\beta$=0.3203 GeV, $\gamma$=1.14 and $\alpha$=-0.2017 GeV, $\beta$=0.1861 GeV, $\gamma$=1.2104 for the two $HA$ parametrizations. The momentum dependence of the symmetry potential is currently an unsettled issue and consequently various parametrizations have been employed in the literature. To estimate the impact of this unknown on elliptic flow observables, we have selected two of the most widely employed parametrizations for the current study: the Gogny interaction inspired one (Refs.~\cite{Chen:2004si, Das:2002fr}), producing a momentum dependent symmetry potential, \begin{eqnarray} &&V(\rho,\beta)=\frac{A_1}{2\rho_0}\rho^2+\frac{A_2(x)}{2\rho_0}\rho^2\beta^2 +\frac{B}{\sigma+1}\frac{\rho^{\sigma+1}}{\rho_0^\sigma}(1-x\beta^2) \nonumber\\ &&\phantom{aaaa} +\frac{1}{\rho_0}\sum_{\tau,\tau'}\,C_{\tau\tau'}\int\int d^3 p d^3 p'\,\frac{f_{\tau}(\vec{r},\vec{p})\,f_{\tau'}(\vec{r},\vec{p'})} {1+(\vec{p}-\vec{p}')^2/\Lambda^2} \eqlab{sympot} \end{eqnarray} and the power-law parametrization, that leads to a momentum independent potential \begin{eqnarray} S(\rho)=\left\{ \begin{array}{l} S_0\,(\rho/\rho_0)^\gamma \hspace{0.25cm} \text{- linear, stiff} \\ a+(S_0-a)(\rho/\rho_0)^\gamma \hspace{0.25cm} \text{- soft, supersoft}. \end{array} \right. \eqlab{sympotpowerlaw} \end{eqnarray} with $S_0$=18.5 MeV. To reproduce different density dependencies of the symmetry energy predicted by various ab-initio theoretical calculations the original Gogny interaction has been generalized in Ref.~\cite{Das:2002fr} by introducing a real parameter $x$ that can be adjusted to generate an asy-EoS with the desired saturation density magnitude and high density behavior. Values of the parameters present in~\eqref{sympot} for the choices $x=1$ (soft) and $x=0$ (close to linear) can be found in Ref.~\cite{Das:2002fr} for the case of a soft iso-scalar EoS ($K$=210 MeV). Reproduction of the saturation value of the symmetry energy requires that the parameter $A_2$ bears a linear dependence on $x$. Consequently, for a given density value, the symmetry energy's dependence on $x$ is linear. This implies that the coefficients of the Taylor expansion of the SE around saturation density, in particular $L_{sym}$ and $K_{sym}$ defined in~\eqref{lsymksym}, bear a linear dependence on $x$. Stiff and soft SE density dependencies can be simulated by choosing negative and positive values for $x$, respectively. In the case of the power-law parametrization of SE,~\eqref{sympotpowerlaw}, density dependencies close to those provided by the Gogny interaction for $x$=-2.0,-1.0, 0.0 are obtained for values of the parameter $\gamma$=2.0, 1.5, 0.5, respectively. To mimic the soft and super-soft symmetry energy provided by the Gogny interaction with $x$=1 and respectively $x$=2 modified power-law parametrizations, as presented in ~\eqref{sympotpowerlaw}, are employed above the saturation point. The sets of parameters $a=23.0$ MeV, $\gamma$=1.0 and $a=31.0$ MeV, $\gamma$=2.0 reproduce a soft and super-soft density dependence respectively. Below saturation, the standard power-law parametrization with $\gamma$=1.0 is employed in these cases. The soft power-law parametrizations, while producing discontinuous force terms at saturation density, have the advantage of having an identical functional density dependence of the force term as its stiff counterpart and generating a SE stiffness below saturation point compatible with the experimental result for that density region. \subsection{Model dependence of observables} Elliptic flows of protons and neutrons cannot be used separately to constrain the isovector part of the equation of state above the saturation point due to their sizable dependence on particular values of transport model parameters, that are either inaccurately determined or do not represent measurable quantities, like in-medium nucleon-nucleon ($NN$) cross-sections, compressibility modulus of nuclear matter, width of the nucleon wave function and strength of the optical potential. A precise knowledge of the dependence on these parameters would allow the elimination of the most uncertain ones leaving one with a set of observables that bear no or almost no model dependence. In practice, one is forced to make assumptions that are either verified or disproved by employing a definite transport model. Neutron proton elliptic flow ratios require a scenario in which elliptic flows scale linearly with model parameters, while for the differences the variation of elliptic flows of neutrons and protons with respect to model parameters should be equal in order to be able to extract a model independent constraint on the density dependence of SE. \begin{figure}[t] \includegraphics[width=20pc]{flow_v2avip_dif_rat_optpotse_moddep_v2.eps} \caption{\figlab{npefdoptpotsens}(Color online) Variations in the values of the impact parameter integrated ( b$\leq$7.5 fm ) npEFD (a) and npEFR (b) due to different choices for the optical potential ($V_{opt}$), parametrization of symmetry-energy ($S$) as well as the combined, quadratically added, uncertainty.} \end{figure} To extract constraints on the stiffness of the SE the following procedure is employed. We require that the experimental elliptic flow values are reproduced as closely as possible for a particular asy-EoS scenario, while keeping model parameters within limits commonly found in the literature. As a consequence, npEFD and npEDR cannot be treated as independent observables anymore. Differences in the constraints extracted by using each of them independently are related to how close the experimental elliptic flow values are reproduced for the favored asy-EoS scenario. In the following the sensitivity of npEFD and npEFR with respect to model parameters will be presented. For the central estimates, a reproduction of the experimental elliptic flow data for a value of the asy-EoS stiffness parameter $x$=-1 within 10-15$\%$ was possible with the following set of parameter values: stiffness of the isoscalar EoS set to $K$=210 MeV, width of the nucleon wave function $L$=4.33 fm$^2$, the new version of the $V_{opt}^{(HA)}$ optical potential parametrization and Cugnon nucleon-nucleon cross-sections. We would like to note that very few of the possible combinations of model parameters produce values for the elliptic flow compatible with the experimental values, many combinations underestimate its strength severely, sometimes up to a factor of 2. Once the stiffness of the scalar part of the EoS is set to a soft one ($K$=210 MeV) the choices to enhance elliptic flow towards realistic values are to decrease the nucleon wave function width and/or choose an optical potential that is as repulsive as possible in its higher energy region. The Cugnon nucleon-nucleon cross-section parametrization has been used in the collision term. It should be noted that Cugnon neutron-proton cross-sections are lower than the experimental vacuum ones below an incident kinetic energy of $T$=100 MeV, but significantly higher than the in-medium theoretical predictions at saturation density. They can, therefore, be thought of as effectively simulating some in-medium effects. Results for the sensitivity of npEFD and npEFR to both the momentum dependent part of the isospin symmetric EoS and various parametrizations of the symmetry energy are displayed in \figref{npefdoptpotsens} as a function of the stiffness of the asy-EoS. Collisions of Au+Au at 400 MeV/nucleon have been simulated and the kinematical cuts labeled 'B' above have been applied. The widths of the bands represent the variations of npEFD or npEFR when switching between parametrizations of the optical potential while keeping the SE parametrization fixed and vice-versa. The results presented correspond to averages over different choices of the quantity that was kept fixed. Conclusions as $e.g.$ the sensitivity/insensitivity of the studied observables to the momentum dependence of symmetry potential for an asy-soft scenario (as was presented in Ref.~\cite{Zhang:2012fc}) can thus not be drawn from this figure. Each of the possible combinations of optical potential parametrization and symmetry energy parametrization (6 in total) usually yields a different outcome in this respect. The result of~\figref{npefdoptpotsens} should therefore be considered as an estimate. Nevertheless, it can be concluded that the uncertainties in the optical potential and the momentum dependence or independence of the symmetry potential have an important impact on elliptic flow observables like npEFD and npEFR. For precisely constraining the symmetry energy at high density from elliptic flow data, an accurate knowledge of the optical potential will, therefore, be required and the problem of the momentum dependence or independence of the iso-vector potential will have to be resolved. \begin{figure}[tb] \begin{center} \begin{minipage}{0.49\textwidth} \includegraphics[width=18pc]{flow_a2_auau_400_nmp_cv_difeos_moddep_v2_5b.eps} \end{minipage} \begin{minipage}{0.49\textwidth} \includegraphics[width=17.5pc]{flow_a2_auau_400_ndp_cv_difeos_moddep_v2_5b.eps} \end{minipage} \end{center} \caption{\figlab{efdefrmoddep}(Color online) Model dependence of npEFD (a) and npEFR (b) and comparison with FOPI-LAND experimental data, integrated over impact parameter b$\leq$7.5 fm. Sensitivity to the different model parameters, compressibility modulus ($K$), width of nucleon wave function ($L$), optical potential ($V_{opt}$) and parametrization of the symmetry energy ($S$) are displayed. The total model dependence is obtained by adding, in quadrature, individual sensitivities.} \end{figure} In~\figref{efdefrmoddep} the model dependence of impact parameter integrated npEFR and npEFD to variations of the compressibility modulus ($K$), width of the nucleon wave function($L$), optical potential ($V_{opt}$) and SE parametrization ($S$) are presented. The central curve (full line) was obtained employing the parameter values mentioned above. By using the vacuum Li-Machleidt nucleon-nucleon cross-sections~\cite{Li:1993ef,Li:1993rwa} instead of the Cugnon ones, while keeping the other model parameters unchanged, both npEFD and npEFR remain practically the same irrespective of the value of $x$. As such the sensitivity to cross-section parametrizations has not been included in the total sensitivity bands. The commonly employed value for the compressibility modulus, a soft $K$=210 MeV, has been extracted from the multiplicity ratio of $K^+$ production in heavy (Au+Au) over light (C+C) nuclei at incident energies close to 1 AGeV by the KaoS Collaboration~\cite{Sturm:2000dm,Fuchs:2000kp,Hartnack:2005tr} but at lower incident energies the situation is not as clear: the KaoS result points to an even softer EoS while the study of sidewards flow or stopping by the FOPI collaboration~\cite{Andronic:2003dc,FOPI:2011aa} does not exclude a somewhat stiffer isoscalar EoS. For the case of the nucleon wave function width, the value $L$=4.33 fm$^2$ is commonly employed in transport models simulations of collisions of lighter nuclei while for the simulation of heavier nuclei an increase to the value $L$=8.66 fm$^2$ is found necessary to prevent nucleon evaporation. The optical model dependence and SE parametrization dependence are the same as presented in~\figref{npefdoptpotsens}. The variation of the $K$ and $L$ model parameters has been performed within ranges that take into account the facts mentioned in the previous paragraph together with the requirement that the simulated values for the elliptic flow of neutrons and protons for $x$=-1 be within 25$\%$ from the experimental ones. This limit was chosen because it represents about 3 standard deviations of the combined experimental and numerical uncertainties. A search for the allowed values for $K$ and $L$ that obey this constraint has been performed with the following outcome. Increasing the compressibility modulus results in higher elliptic flow absolute values, reaching the upper 25$\%$ off the experimental value boundary in the region $K$=270$\div$280 MeV. By decreasing the compressibility modulus below $K$=210 MeV a saturation region is reached with values well within the 25$\%$ off region just below $K$=190 MeV. The dependence of the elliptic flow values on the nucleon wave function width $L$ proves to be approximately parabolic with a maximum absolute value for the elliptic flows, compatible with the imposed constraint, reached close to $L$=3.5 fm$^2$ and crossing the lower 25$\%$ boundary for the values $L$=2.5 fm$^2$ and $L$=7.0 fm$^2$. Consequently, to produce the results presented in~\figref{efdefrmoddep} the following variation ranges for the K and L model parameters have been adopted: $K$=190$\div$280 MeV and $L$=2.5$\div$7.0 fm$^2$. Increasing the compressibility modulus to $K$=300 MeV or the nucleon wave function width to $L$=8.66 fm$^2$ produces elliptic flow values that can deviate from the experimental ones with up to 40-50$\%$ but with marginal impact on the allowed values for the asy-EoS stiffness parameter $x$. The sensitivities of npEFD and npEFR to model parameter values are similar; both show a model dependence on the optical potential that is almost independent of the stiffness of asy-EoS while the averaged dependence on the parametrization of SE is more pronounced for asy-soft scenarios. The $L$ dependence is slightly more important for an asy-stiff than for an asy-soft EoS for both npEFD and npEFR. The $K$ dependence is important for both npEFD and npEFR irrespective of the value chosen for the asy-EoS stiffness with the exception of the asy-stiff region for npEFD where it represents the most important source of uncertainty. The model dependence due to these last two parameters makes up most of the sensitivity of npEFD, especially for the case of an asy-stiff scenario. The same holds true for npEFR in the asy-stiff region while for the case of an asy-soft scenario the importance of the optical potential and SE parametrization is equally or slightly more important. The total model dependence $K+L+V_{opt}+S$ of both npEFD and npEFR is almost insensitive to the stiffness of asy-EoS and is comparable in absolute magnitude with the experimental uncertainty of the respective quantity. For each observable the experimental value and its uncertainty are depicted in~\figref{efdefrmoddep} by the horizontal line and hatched band, respectively. A clear separation of the theoretical and experimental bands, amounting to about one standard deviation, exists in the super-soft scenario region. \subsection{Constraints on asy-EoS} The comparison with the experimental results (\figref{efdefrmoddep}) permits the extraction of estimates for the stiffness of asy-EoS: x=-1.50$^{+1.75}_{-1.00}$ from npEFD and x=-1.25$^{+1.25}_{-1.00}$ from npEFR. These constraints translate, using the parabolic expansion of the asy-EoS around the saturation point \begin{eqnarray} E_{sym}(\rho)=E_{sym}(\rho_0)+\frac{L_{sym}}{3}\frac{\rho-\rho_0}{\rho_0}+\frac{K_{sym}}{18} \frac{(\rho-\rho_0)^2}{\rho_0^2}, \eqlab{lsymksym} \end{eqnarray} into the following estimates for the slope and curvature parameters of the symmetry energy: $L_{sym}$=129$^{+46}_{-80}$ MeV, $K_{sym}$=272$^{+291}_{-508}$ MeV (npEFD) and $L_{sym}$=118$^{+45}_{-57}$ MeV, $K_{sym}$=199$^{+291}_{-362}$ MeV (npEFR). The obtained values for $L_{sym}$ are larger by a factor of 2 and by 50$\%$ than the ones extracted from an analysis of neutron skin thickness and isospin diffusion at lower energies~\cite{Tsang:2012se}, respectively. The central values of the npEFD and npEFR based constraints favor therefore a density dependence of the symmetry energy above the saturation point close to mildly stiff or linear. They are consistent with each other, the difference between the central values is a consequence of the imperfect theoretical description of the experimental elliptic flow data at the favored value for the $x$ parameter. This difference can in principle be eliminated by a finer than here attempted tuning of model parameters. In~\figref{symenconstraints} the explicit constraints on the density dependence of SE obtained in this study from the comparison of theoretical and experimental values of npEFD and npEFR are presented. As npEFD and npEFR are not independent observables, due to the constraint that experimental elliptical flow data be reproduced at a value of the stiffness parameter $x$ close to the extracted one, only one band, obtained from averaging the npEFD and npEFR constraints, is advanced for the allowed values for the asy-EoS stiffness. The result of Ref.~\cite{Russotto:2011hq} is added for comparison. The two studies employ independent flavors of the QMD transport model (T\"ubingen QMD vs. UrQMD) and parametrizations of isovector EoS that differ: Gogny inspired (momentum dependent potential) vs. power law (momentum independent potential). Abandoning the requirement of a close description of the experimental elliptic flow values for the central estimates allows one to treat npEFD and npEFR as independent observables. The extracted constraints for the values of the $x$ parameter are wider if extracted from npEFD and npEFR independently, but the overlapping region of the two differs only slightly from the one presented in~\figref{symenconstraints}. This brings strong support to the conclusion that the obtained constraint for the symmetry energy stiffness is model independent by providing evidence that an asy-EoS stiffness close to one corresponding to the $x$=-1 scenario is favored The constraints on the density dependence of SE obtained with these different ingredients are in agreement with each other. This contrasts with the current status of the effort to constrain the SE from $\pi^-/\pi^{+}$ ratios: a study employing IBUU transport model and the Gogny inspired asy-EoS~\cite{Xiao:2009zza} favors a soft asy-EoS, a second study which uses IQMD and the power-law parametrization of SE~\cite{Feng:2009am} points towards a stiff asy-EoS while the work of~\cite{Xie:2013np} within the Boltzmann-Langevin approach supplemented by a power-law parametrization of SE concludes that a super-soft scenario is the realistic one. The reason for this strong disagreement has most likely nothing to do with the parametrization used for the EoS or its momentum (in)dependence but may originate in medium effects on both $\Delta$ resonance and pion production cross-sections (including their isospin dependent energy thresholds) or other related topics~\cite{DiToro:2010ku,Ferrini:2005jw}. The result presented in this Article is robust, the model dependence of the presented observables, while important, is well understood and constraints obtained by employing different parametrizations of the asy-EoS are compatible with each other. An improvement of the current theoretical model is mandatory to allow, together with more accurate experimental data of elliptic flow of neutrons and protons as expected to be delivered by the ASY-EOS Collaboration~\cite{Russotto:2012jb}, a tighter constraint on the high density dependence of the symmetry energy. \begin{figure}[t] \includegraphics[width=20pc]{symen_eosdensdep_4d.eps} \caption{\figlab{symenconstraints} (Color online) Constraints on the density dependence of the symmetry energy obtained from comparing theoretical predictions for npEFD and npEFR to FOPI-LAND experimental data. The result of Russotto~{\it et al.} ~\cite{Russotto:2011hq} is also shown together with the Gogny inspired SE parametrization for three values of the stiffness parameter: x=-1 (stiff), x=0 and x=1 (soft).} \end{figure} \section{Conclusions} Constraints on the high density dependence of the symmetry energy (SE) have been extracted by comparing theoretical predictions of neutron-proton elliptic flow differences (npEFD) and neutron-proton elliptic flow ratios (npEFR) with experimental results obtained from a recent analysis of the FOPI-LAND experimental data for Au+Au collisions at 400 MeV/nucleon. The T\"ubingen QMD model supplemented with the Hartnack-Aichelin and Gogny parametrizations of the isoscalar EoS and asy-EoS respectively, for the central values, has been employed. A thorough study of the model dependence of npEFD and npEFR has been performed concluding that, while the sensitivity to uncertainties in the model parameters is important, the two observables offer the opportunity to extract information about the SE above the saturation point. Furthermore, the results of the present study supplemented with those of Ref.~\cite{Russotto:2011hq} allow one to conclude that constraints on symmetry energy extracted from elliptic flow data are independent on its parametrization, suggesting that an almost model independent extraction can be achieved in this case. This contrasts with the case of $\pi^-/\pi^{+}$ ratios where the stiffnesses of asy-EoS extracted using different parametrizations for SE or transport models can be extremely different. We have imposed that the experimental elliptic flow values of neutrons and protons are reproduced as closely as possible and accomplished that to within 10-15$\%$ by changing model parameters within limits commonly employed in the literature. Averaging the constraints extracted independently from npEFD and npEFR one obtains the following allowed values for the parameters describing the stiffness of the symmetry energy: $L_{sym}$=122$\pm$57 MeV and $K_{sym}$=229$\pm$363 MeV. Together with the estimates obtained in Ref.~\cite{Russotto:2011hq} we advance the following constraint, obtained from averaging these results, on the stiffness of asy-EoS: $L_{sym}$=106$\pm$46 MeV and $K_{sym}$=127$\pm$290 MeV. It corresponds to a moderately stiff to linear density dependence and excludes the super-soft and, with a lesser degree of confidence, the soft asy-EoS scenarios from the list of possibilities. An improvement of the current theoretical model, in the sense of reducing theoretical uncertainties, is mandatory to allow, together with more accurate experimental data of elliptic flow of neutrons and protons, a tighter constraint on the high density-dependence of the symmetry energy. \begin{acknowledgments} The research of M.D.C. has been financially supported by the Romanian Ministry of Education and Research through contract PN09370103. Q.L. acknowledges financial support from the National Natural Science Foundation of China (Grant No. 11375062) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y6090210). \end{acknowledgments}

proofpile-arXiv_069-5543

\section{Introduction} Adiabatic quantum computation\cite{Farhi2001} has been shown to be equivalent to the usual circuit model\cite{Equivalent}. This is only known to hold for ideal systems without noise. While there are effective techniques for fault-tolerance in the circuit model\cite{ShorFaultTolerance}, it remains unknown whether adiabatic quantum computation can be made fault-tolerant. In spite of this, there has been some enthusiasm for implementing restricted forms of adiabatic quantum computation in very noisy hardware based on the hope that it would be naturally robust. Even in the original proposal for quantum adiabatic computation\cite{Farhi2001} it was suggested it might be a useful technique for solving optimization problems. Recent papers about D-Wave hardware have studied a particular sort of optimization problem, namely finding the ground state of a set of Ising spins. These spins are taken to live on a graph. The problem instance is determined by a choice of a graph and either ferromagnetic or antiferromagnetic interactions between each pair of bits connected by an edge of the graph. Finding this ground state is NP-hard if the graph is arbitrary\cite{Barahona} and efficiently approximable when the graph is planar\cite{Bansal2009}. The connectivity of the D-Wave machine is somewhere in between and it is not known whether the associated problem is hard. The D-Wave machine is made of superconducting ``flux'' qubits\cite{Harris2007} (first described in Ref.\cite{Mooij1999}). Because of the high decoherence rates associated with these flux qubits, it has been unclear whether the machine is fundamentally quantum or merely performing a calculation equivalent to that of a classical stochastic computer. References\cite{signature,annealing108} attempt to distinguish between these possibilities by proposing tests for quantumness that the D-Wave machine passes but a purely classical computer should fail. This letter presents a classical model that passes the tests, exhibiting all the supposedly quantum behaviors. \section{The claims} \subsection{Eight-spin signature of quantum annealing} In\cite{signature}, a system of eight spins as shown in Fig.\ \ref{fig:eight} is analyzed. It is shown that the ground state of the system is 17-fold degenerate, comprising the states \begin{equation} \{\ket{\!\!\uparrow\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow\downarrow},\ldots, \ket{\!\!\uparrow\uparrow\uparrow\uparrow\downarrow\uparrow\downarrow\downarrow}, \ldots, \ket{\!\!\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow}\}\ \mathrm{and}\ \ket{\!\!\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow}\ . \end{equation} The probability of finding the isolated (all down) state $p_s$ is compared to the average probability of states from the 16-fold ``cluster'' of states, $p_C$. It is computed that classical simulated annealing finds an enhancement of $p_s$, \emph{i.e.} $p_s > p_C$ while quantum annealing both in simulation and running on the D-Wave machine finds a suppression $p_s < p_C$. \begin{figure}[htbf] \centering{\includegraphics[height=2in]{eight.pdf}} \caption{The eight spin graph from Reference\protect\cite{signature}. Each spin is coupled to its neighbors along the graph with a $+1$ coupling such that the spins like to be aligned. In addition, the inner ``core'' spins have a magnetic field applied in the $+z$ direction while the outer ``ancillae'' spins have a field applied in the $-z$ direction. \label{fig:eight}} \end{figure} \subsection{Annealing with 108 spins} In Ref.\cite{annealing108} 108 spins in the D-Wave computer are employed. Their connectivity is displayed in Figure 1 of that paper. Each connection is given a coupling of $\pm 1$ at random. 1000 random cases are studied, and for each case the machine is run 1000 times.\footnote{They also look at what happens with fewer than 108 spins, so the total number of experiments they performed is actually much larger.} The ground state energies calculated are compared to the correct answer, which can be found for cases of this size\cite{spinglassserver}. A histogram of the probabilities of successfully finding the correct answer is shown. The histogram has a bimodal distribution, with significant peaks at probability zero and probability one. The cases where the machine never find the answer are ``difficult'' cases and the ones where it always finds it are ``easy'' cases. By comparison, classical simulated annealing shows a unimodal distribution with no, or nearly no, cases being either hard or easy. See Figure 2 in Ref.\cite{annealing108} \section{Quantum annealing is not annealing} Is it surprising that results of the D-Wave experiments differ greatly from classical simulated annealing? And should this be considered evidence that the machine is quantum or more powerful than classical computation in some way? We argue here that the answer to these questions is ``no.'' What is called ``quantum annealing'' is often compared to classical simulated annealing\cite{SimulatedAnnealing}, or to physical annealing itself. Though both quantum annealing and simulated annealing are used to find lowest-energy configurations, one is not the quantum generalization of the other. More properly, quantum annealing should be considered quantum adiabatic ground state dragging. Simulated annealing proceeds by choosing a random starting state and/or a high temperature, then evolving the system according to a set of rules (usually respecting detailed balance) while reducing the simulated temperature. This process tends to reduce the energy during its evolution but can, of course, become stuck in local energy minima, rather than finding the global ground state. Quantum annealing, on the other hand, involves no randomness or temperature, at least in the ideal. Rather, it is a particular type of adiabatic evolution. Two Hamiltonians are considered: The one for which one desires to know the ground state $H_f$, and one which is simple enough that cooling to its ground state is easy, $H_i$. At the start of the process, the system is initialized to the ground state of $H_i$ by turning on $H_i$ and turning off $H_f$ and waiting for thermal equilibration. Then $H_i$ is gradually turned off while $H_f$ is gradually turned on. If this is done slowly enough, the system will at all times remain in the ground state of the overall Hamiltonian.\footnote{If the ground state is degenerate at some point during the process then this may not be true.} Then, at the end of this process the system will be in the ground state of $H_f$.\footnote{Of course, ``slowly enough,'' depends on the gap between the ground state and other nearby energy eigenstates. If the system is frustrated, there are many such states and the evolution must proceed exponentially slowly, just as frustration hinders classical simulated annealing. Though it has often been suggested that quantum annealing is a panacea, whether it can outperform classical simulated annealing in such cases is unknown.} It remains to measure the ground state. For the problems considered in Refs.\cite{signature,annealing108}, the ground state is typically known to be diagonal in the $z$ basis. Thus, a measurement need not introduce any randomness. Since classical simulated annealing is intrinsically random and ``quantum annealing'' is not, the differences reported in Refs.\cite{signature,annealing108} are not surprising. For the eight-bit suppression of finding the isolated state, two things could have happened: Either the ideal machine would find the isolated state always, or never. It happens that, due to the structure of the state, the ideal outcome is ``never,'' which is certainly a suppression. The bimodal distribution found for the 108-bit computations is also just what one would expect: In the perfect case of no noise either the calculation gets the correct answer or it does not. The outcome is deterministic so there should be exactly two peaks, at probability of success zero and one. Classical annealing, which begins from a random state on each run, is not expected to succeed with probability one, even for cases where the system is not frustrated. The bimodality of the D-Wave results, in contrast to the unimodality of simulated annealing, can be seen as evidence not of the machine's quantumness, but merely of its greater reproducibility among runs using the same coupling constants, due to its lack of any explicit randomization. The simulated annealing algorithm, by contrast uses different random numbers each time, so naturally exhibits more variablity in behavior when run repeatedly on the same set of coupling constants, leading to a unimodal historgram. Indeed if the same random numbers were used each time for simulated annealing, the histogram would be perfectly bimodal. To remove the confounding influence of explicit randomization, we need to consider more carefully what would be a proper classical analog of quantum annealing. If, as we have argued, classical simulated annealing is not the correct classical analog of quantum annealing, what is? The natural answer is to classically transform a potential landscape slowly enough that the system remains at all times in the lowest energy state. In the next section we give a model classical system and, by running it as an adiabatic lowest-energy configuration finder, demonstrate that it exhibits the same computational behavior interpreted as a quantum signature in Refs.\cite{signature,annealing108}. \section{The model} The flux qubits in the D-Wave machine decohere in a time considerably shorter than the time adiabatic evolution experiment runs. The decoherence times are stated to be on the order of tens of nanoseconds while the adiabatic runtime is 5-20 microseconds\cite{annealing108}. For this reason, one would expect that no quantum coherences should exist. We therefore model the qubits as $n$ classical spins (``compass needles''), each with an angle $\theta_i$ and coupled to each other with coupling $J_{ij}=0,\pm 1$. Each spin may also be acted on by external magnetic fields $h_i$ in the $z$-direction as well as an overall field in the $x$-direction $B_X$. The potential energy function is then given by \begin{equation} V_\mathrm{Ising}=-\sum_i \cos(\theta_i) h_i - \frac{1}{2}\sum_{i\ne j} \cos(\theta_i)\cos(\theta_j) J_{ij} \label{eq:vising} \end{equation} and \begin{equation} V_\mathrm{trans}=-\sum_i \sin(\theta_i) B_x \ . \label{eq:vtrans} \end{equation} Compare these to Equations (1) and (2) in Ref.\cite{signature} The adiabatic computation is performed (or simulated) by running the dynamics while gradually changing these potentials from $V_\mathrm{trans}$ to $V_\mathrm{Ising}$ over a time $T$ according to \begin{equation} V(t)=A(t) V_\mathrm{trans}+ B(t) V_\mathrm{Ising} \label{eq:vtotal} \end{equation} with $A(0)=B(T)=1$ and $A(T)=B(0)=0$. The equations of motion are simply: \begin{equation} \frac{d}{dt}\theta_i=\dot{\theta}_i\ \mathrm{and}\ \frac{d}{dt}\dot{\theta}_i=\frac{dV}{d\theta_i} \ . \label{eq:motion} \end{equation} It is straightforward to integrate this system of ordinary differential equations. \section{Results} \subsection{Eight spin model} The simulated adiabatic dragging time $T$ needs to be long compared to the fundamental timescales of the system. Since units have been omitted in Eqs.\ (\ref{eq:vising}-\ref{eq:motion}), these are of order unity. Using $T=1000$ produces good results as shown in the next section. Fig.\ \ref{fig:8spinplot} shows the results of simulating the classical model of the eight spins from Ref.\cite{signature} When the total adiabatic dragging time $T$ is long enough, the dynamics lead to a stationary state (red lines). The core spin is driven to $\theta=0$, corresponding to $\ket{\!\uparrow}$, and the ancillae spin to $\theta=\frac{\pi}{2}$, corresponding to $\ket{\!\!\rightarrow}$. All other core and ancillae states behave identically. The final state $\ket{\!\!\uparrow\uparrow\uparrow\uparrow\rightarrow\rightarrow\rightarrow\rightarrow}$ lies in the space spanned by the 16 cluster states. Within this space the ancillae spins are free to rotate as they contribute nothing to the energy provided the core states are all $\ket{\!\!\uparrow}$. Since there is nothing to break the symmetry of the initial all $\ket{\!\!\rightarrow}$ configuration, this is the final state they choose. This behavior is robust to added noise (blue lines). However, when the dragging time $T$ is too short, the long-time behavior is not a stationary state (black lines). \begin{figure}[htbf] \centering{\includegraphics[height=2.5in]{8spinsplot.pdf}} \caption{Results for eight spin model. Three runs are shown. In each case the angle $\theta$ of a representative core spin and ancillae spin are shown as a function of time. The core spins are all driven to $\theta=0$ while the ancillae spins are ideally driven to $\theta=\Pi/2$. 1) The red lines show a case with no noise and with adiabatic drag time $T=1000$. 2) The blue lines have added noise again with $T=1000$. The noise was simulated by applying random kicks uniformly distributed between $\pm 0.02$ to the angular velocities $\dot{\theta}$ of the spins at $t=10,20, \ldots$. 3) The black lines have no noise but $T=200$ (the evolution continues after the adiabatic drag is complete. It is easily seen the ancillae spin has been driven too fast and winds up with some kinetic energy (the slope is not zero for $t>T$). \label{fig:8spinplot}} \end{figure} \subsection{108 spins} For this case we programmed 108 spins with the same connectivity as run in Ref.\cite{annealing108} Choosing 500 random cases of $\pm$ for the couplings and running our classical compass model simulation with no noise and a dragging time $T=1000$ results in a perfect bimodal distribution. For 249 cases the optimal answer was found always, and for 251 it was found never (since the machine is deterministic it is only necessary to run each case once). It seems to be a coincidence that for problems of this size the number of easy and hard cases is almost equal. We show in Fig.\ \ref{fig:bimodal} the results of running a simulation of the classical compass model with with noise. For the same 500 sets of couplings, 30 noisy runs were performed and a histogram of success probabilities results. The bimodal distribution is maintained. Compare to Fig.\ 2 in Ref.\cite{annealing108} The qualitative nature of the bimodal signatures found is insensitive to the details of the noise so more realistic noise models would give similar behavior. Note also that the noise seems to help find the ground state, as in 401 of the 500 cases the ground state was found at least once in 30 trials. This suggests that noisy adiabatic dragging picks up some of the benefits of classical simulated annealing, avoiding being ``stuck'' with either always or never finding the ground state. \begin{figure}[htbf] \centering{\includegraphics[height=3in]{plot.pdf}} \caption{Results of 500 spin glass instances, each run 30 times on a noisy simulation of the classical compass model with $T=1000$. A bimodal distribution is observed, with a clear separation between easy and hard instances with high and low success probabilities respectively. The noise applied was random kicks to each $\dot{\theta}$ at $t=10,20,\ldots,1000$ uniformly chosen between $\pm 0.0015$. \label{fig:bimodal}} \end{figure} \section{Conclusions} We have argued that quantum annealing and simulated annealing are very different procedures. The deterministic nature of quantum annealing leads to rather different behaviors than the random processes of simulated annealing. However, other deterministic procedures can also lead to behavior very similar to that observed in the D-Wave device. Our classical model reproduces all the claimed signatures of quantum annealing. We recommend using the term ``ground-state adiabatic dragging'' or simply ``adiabatic computation'' for such nonrandom processes. Note that in Ref.\cite{manufactured}, under highly diabatic conditions, evidence was shown that the D-Wave device exhibits quantum tunneling. We emphasize that this does not constitute evidence for an essential use of quantum effects during adiabatic dragging and indeed an effectively classical model may well capture the physics and computational behavior seen in Refs.\cite{signature,annealing108}. Tunneling is a possible additional path for a system to anneal into a lower energy state without needing enough energy to cross a potential barrier. If the decoherence is such that each spin is projected into a local energy eigenstate, then in the adiabatic limit, where each spin is at all times in its local ground state, tunneling locally can be of no assistance. The tunneling in Ref.\cite{manufactured} is indeed only shown locally. Furthermore, there is nothing preventing the implementation of our simulated compass model in hardware. It would be possible to build an analog classical machine that could simulate it very quickly, but it would be simpler use digital programmable array with one processing core per simulated spin. Since each spin requires knowledge of at most six of its neighbors along the connectivity graph, the algorithm can be easily parallelized. A 108 core computer specialized to running our algorithm could easily run hundreds or thousands of times faster than simulating it on a desktop computer, and could be built for modest cost using off-the-shelf components. Similarly, \emph{any} classical physics can be efficiently simulated on a classical machine. This is not to suggest that simulating classical physics directly on a classical computer is a good way to solve optimization problems. Classical simulated annealing\cite{SimulatedAnnealing} and other heuristic techniques have been extremely successful, and can also be very fast on special-purpose hardware. It is shown in Ref.\cite{annealing108} that simulated annealing is competitive with the D-Wave One machine even on desktop CPUs. Stronger evidence of both quantumness and a resulting algorithmic speedup is needed before quantum adiabatic computers will have proved their worth. \vspace{1 in} \bibliographystyle{unsrt}

proofpile-arXiv_069-5637

\section{Introduction} In recent years, the Lattice Boltzmann (LB) method has emerged as an attractive computational approach for complex physical system\cit {SS,BSV,XGL1}. The lattice Bhatnagar-Gross-Krook (BGK) model, based on a single-relaxation-time approximation, is the simplest and the most popular form. However, this simplicity also leads to some deficiencies, such as the numerical stability problem, and fixed Prandtl number. To overcome these deficiencies of BGK model, the Multiple-Relaxation-Time (MRT) lattice Boltzmann method\cite{HSB,13} has been developed, and successfully used in simulating various fluid flow problems\cit {Lallemand1,Lallemand2,PRE036701,JCP539,CF957,CMA1392,JCP453,JCP7774,PRE016705,JPA055501 . Most of the existing MRT models work only for isothermal system. To simulate system with temperature field, many attempts have been made\cite{pre036706,pre026705,Mezrhab2010}. Besides the models mentioned above, we proposed a MRT Finite Difference lattice Boltzmann model for compressible flows with arbitrary specific heat ratio and Prandtl number in previous work\cite{chenepl}. In the model, the kinetic moment space and the equilibria of nonconserved moments are constructed according to the seven-moment relations associated with the local equilibrium distribution function. Numerical experiments showed that compressible flows with strong shocks can be well simulated by this model. In the paper we extend the MRT LB model so that it is suitable also for incompressible flows. In order to efficiently decrease the unphysical oscillations, the flux limiter scheme\cite{Sofonea1,Sofonea3,chenctp} with splitting technique is incorporated into the new model. When the system deviates more from equilibrium, the LB simulation can give more physical information\cite{xufop,yan,JCP}, such as the non-equilibrium characteristics of system. Here, in the new MRT LB model, the non-equilibrium characteristics of system are solved through a dynamic procedure where a shock wave propagates from a heavy medium to a light one. The rest of the paper is organized as follows. Section II presents the extended MRT LB model. Section III describes the finite difference schemes. section IV is for the validation and verification of the new LB model. Non-equilibrium characteristics are shown and analyzed in section V. Section VI makes the conclusion for the present paper. \section{Model description} According to the main strategy of MRT LB method, the MRT LB equation can be described as: \begin{equation} \frac{\partial f_{i}}{\partial t}+v_{i\alpha }\frac{\partial f_{i}}{\partial x_{\alpha }}=-\mathbf{M}_{il}^{-1}\hat{\mathbf{S}}_{lk}(\hat{f}_{k}-\hat{f _{k}^{eq})\text{,} \label{3} \end{equation where $f_{i}$ and $\hat{f}_{i}$ are the particle distribution function in the velocity space and the kinetic moment space respectively, $\mathbf{v _{i} $ is the discrete particle velocity, $i=1$,$\ldots$ ,$N$, $N$\ is the number of discrete velocities, the subscript $\alpha $\ indicates $x$\ or $y . The matrix $\hat{\mathbf{S}}=\mathbf{MSM}^{-1}=diag(s_{1},s_{2},\cdots ,s_{N})$ is the diagonal relaxation matrix. $\mathbf{M}$ is the transformation matrix between the velocity space and the kinetic moment space. $\hat{f}_{i}=m_{ij}f_{j}$, $m_{ij}$ is an element of the transformation matrix. $\hat{f}_{i}^{eq}$ is the equilibrium value of distribution function $\hat{f}_{i}$ in the kinetic moment space. \begin{figure}[tbp] \center\includegraphics*[width=0.40\textwidth]{Fig1.eps} \caption{ Schematics of $\mathbf{v}_{i}$ for the discrete velocity model.} \end{figure} In the previous work, we constructed a two-dimensional MRT LB model based on the model by Kataoka and Tsutahara\cite{Kataoka} (see Fig. 1): \begin{equation*} \left( v_{i1,}v_{i2}\right) =\left\{ \begin{array}{cc} \mathbf{cyc}:\left( \pm 1,0\right) , & \text{for }1\leq i\leq 4, \\ \mathbf{cyc}:\left( \pm 6,0\right) , & \text{for }5\leq i\leq 8, \\ \sqrt{2}\left( \pm 1,\pm 1\right) , & \text{for }9\leq i\leq 12, \\ \frac{3}{\sqrt{2}}\left( \pm 1,\pm 1\right) , & \text{for }13\leq i\leq 16 \end{array \right. \end{equation* where \textbf{cyc} indicates the cyclic permutation. Transformation matrix \mathbf{M}$ and the equilibrium distribution function $\hat{f}_{i}^{eq}$ in the moment space are chosen according to the seven-moment relations (see Appendix for details). At the continuous limit, the above formulation recovers the following Navier-Stokes (NS) equations: \begin{subequations} \begin{equation} \frac{\partial \rho }{\partial t}+\frac{\partial (\rho u_{x})}{\partial x} \frac{\partial (\rho u_{y})}{\partial y}=0, \end{equation \begin{eqnarray} &&\frac{\partial (\rho u_{x})}{\partial t}+\frac{\partial }{\partial x}(\rho u_{x}^{2})+\frac{\partial }{\partial y}(\rho u_{x}u_{y}) \notag \\ &=&-\frac{\partial P}{\partial x}+\frac{\partial }{\partial y}[\frac{\rho R }{s_{7}}(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y )] \notag \\ &&+\frac{\partial }{\partial x}[\frac{\rho RT}{s_{5}}(1-\frac{2}{b})(\frac \partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y})+\frac{\rho R }{s_{6}}(\frac{\partial u_{x}}{\partial x}-\frac{\partial u_{y}}{\partial y )]\text{,} \end{eqnarray \begin{eqnarray} &&\frac{\partial (\rho u_{y})}{\partial t}+\frac{\partial }{\partial x}(\rho u_{x}u_{y})+\frac{\partial }{\partial y}(\rho u_{y}^{2}) \notag \\ &=&-\frac{\partial P}{\partial y}+\frac{\partial }{\partial x}[\frac{\rho R }{s_{7}}(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y )] \notag \\ &&+\frac{\partial }{\partial y}[\frac{\rho RT}{s_{5}}(1-\frac{2}{b})(\frac \partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y})-\frac{\rho R }{s_{6}}(\frac{\partial u_{x}}{\partial x}-\frac{\partial u_{y}}{\partial y )]\text{,} \end{eqnarray \begin{eqnarray} &&\frac{\partial e}{\partial t}+\frac{\partial }{\partial x}[(e+2P)u_{x}] \frac{\partial }{\partial y}[(e+2P)u_{y}] \notag \\ &=&2\frac{\partial }{\partial x}\{\frac{\rho RT}{s_{8}}[(\frac{b}{2}+1) \frac{\partial T}{\partial x}+(2\frac{\partial u_{x}}{\partial x}-\frac{2}{b \frac{\partial u_{x}}{\partial x}-\frac{2}{b}\frac{\partial u_{y}}{\partial })u_{x}+(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y )u_{y}]\} \notag \\ &&+2\frac{\partial }{\partial y}\{\frac{\rho RT}{s_{9}}[(\frac{b}{2}+1) \frac{\partial T}{\partial y}+(\frac{\partial u_{y}}{\partial x}+\frac \partial u_{x}}{\partial y})u_{x}+(2\frac{\partial u_{y}}{\partial y}-\frac{ }{b}\frac{\partial u_{x}}{\partial x}-\frac{2}{b}\frac{\partial u_{y}} \partial y})u_{y}]\}\text{.} \end{eqnarray where $P=\rho RT$, $e=b\rho RT+\rho u_{\alpha }^{2}$ is twice of the total energy, and $b$\ is a constant related to the specific-heat-ratio $\gamma =(b+2)/b$. In order to maintain the isotropy constraint of viscous stress tensor and heat conductivity, some of the relaxation parameters should be equal to one another, namely $s_{5}=s_{6}=s_{7}$, $s_{8}=s_{9}$. The above NS equations reduce to \end{subequations} \begin{subequations} \begin{equation} \frac{\partial \rho }{\partial t}+\frac{\partial (\rho u_{\alpha })} \partial x_{\alpha }}=0\text{,} \end{equation \begin{equation} \frac{\partial (\rho u_{\alpha })}{\partial t}+\frac{\partial \left( \rho u_{\alpha }u_{\beta }\right) }{\partial x_{\beta }}=-\frac{\partial P} \partial x_{\alpha }}+\frac{\partial }{\partial x_{\beta }}[(\mu \frac \partial u_{\alpha }}{\partial x_{\beta }}+\frac{\partial u_{\beta }} \partial x_{\alpha }}-\frac{2}{3}\frac{\partial u_{\chi }}{\partial x_{\chi }\delta _{\alpha \beta })+\mu _{B}\frac{\partial u_{\chi }}{\partial x_{\chi }}\delta _{\alpha \beta }]\text{,} \label{3b} \end{equation \begin{equation} \frac{\partial e}{\partial t}+\frac{\partial }{\partial x_{\alpha }}\left[ (e+2P)u_{\alpha }\right] =2\frac{\partial }{\partial x_{\beta }}[(\frac{b}{2 +1)\lambda^{\prime } R\frac{\partial T}{\partial x_{\beta }}+\lambda^{\prime } (\frac{\partial u_{\alpha }}{\partial x_{\beta }}+\frac{\partial u_{\beta }{\partial x_{\alpha }}-\frac{2}{b}\frac{\partial u_{\chi }}{\partial x_{\chi }}\delta _{\alpha \beta })u_{\alpha }]\text{,} \label{3c} \end{equation} where the viscosity $\mu =\rho RT/s_{5}$, the bulk viscosity $\mu _{B}=(2/3-2/b) \rho RT/s_{5}$, $\lambda^{\prime } =\rho RT/s_{8}$, $\left( \alpha, \beta, \chi=x,y\right)$. However, the viscous coefficient in the energy equation \eqref{3c} is not consistent with that in the momentum equation \eqref{3b}. By modifying the collision operators of the moments related to energy flux: \end{subequations} \begin{subequations} \begin{align} & \hat{\mathbf{S}}_{88}(\hat{f}_{8}-\hat{f}_{8}^{eq})\Rightarrow \hat \mathbf{S}}_{88}(\hat{f}_{8}-\hat{f}_{8}^{eq})+(s_{8}/s_{5}-1)\rho Tu_{x} \notag \\ & \qquad \times (4\frac{\partial u_{x}}{\partial x}-\frac{4}{b}\frac \partial u_{x}}{\partial x}-\frac{4}{b}\frac{\partial u_{y}}{\partial y )+(s_{8}/s_{5}-1)\rho Tu_{y}(2\frac{\partial u_{y}}{\partial x}+2\frac \partial u_{x}}{\partial y})\text{,} \end{align \begin{align} & \hat{\mathbf{S}}_{99}(\hat{f}_{9}-\hat{f}_{9}^{eq})\Rightarrow \hat \mathbf{S}}_{99}(\hat{f}_{9}-\hat{f}_{9}^{eq})+(s_{9}/s_{5}-1)\rho Tu_{x} \notag \\ & \qquad \times (2\frac{\partial u_{y}}{\partial x}+2\frac{\partial u_{x}} \partial y})+(s_{9}/s_{5}-1)\rho Tu_{y}(4\frac{\partial u_{y}}{\partial y} \frac{4}{b}\frac{\partial u_{x}}{\partial x}-\frac{4}{b}\frac{\partial u_{y }{\partial y})\text{,} \end{align we get the following energy equation: \end{subequations} \begin{equation} \frac{\partial e}{\partial t}+\frac{\partial }{\partial x_{\alpha }}\left[ (e+2P)u_{\alpha }\right] =2\frac{\partial }{\partial x_{\beta }}[\lambda \frac{\partial T}{\partial x_{\beta }}+\mu (\frac{\partial u_{\alpha }} \partial x_{\beta }}+\frac{\partial u_{\beta }}{\partial x_{\alpha }}-\frac{ }{b}\frac{\partial u_{\chi }}{\partial x_{\chi }}\delta _{\alpha \beta })u_{\alpha }]\text{,} \end{equation where the thermal conductivity $\lambda =(\frac{b}{2}+1)R\lambda^{\prime }$. \section{Finite Difference Scheme} In the original LB model\cite{chenepl}, the time evolution is based on the usual first-order forward Euler scheme, while space discretization is performed through a Lax-Wendroff scheme. In this work, the flux limiter scheme with splitting technique corresponding to the MRT model is adopted. The proposed flux limiter scheme can efficiently decrease the unphysical oscillations around the interfaces. \begin{figure}[tbp] \center\includegraphics*[width=0.65\textwidth]{Fig2.eps} \caption{Characteristic lines and corresponding projections in the $x$ and $y $ directions. (a): $f_{1}(\mathbf{x},t)$; (b): $f_{9}(\mathbf{x},t)$.} \end{figure} Figure 2 shows the characteristic lines in the flux limiter scheme and corresponding projections in $x$ and $y$ directions. \left( J-1)\right\vert _{x}$ and $\left( J-1)\right\vert _{y}$ are corresponding projections of node $J-1$ in the $x$ and $y$ directions. Let f_{i,J}^{n}$ be the value of distribution function at time $t$ in the node $J$ along the direction $i$, we rewrite the evolution of $f_{i}$ in node $J$ at time step $t+dt$ as follows, \begin{equation} f_{i,J}^{n+1}=f_{i,J}^{n}-\frac{dt}{A_{i}dx}[\left. F_{i,J+1/2}^{n}\right\vert _{x}-\left. F_{i,J-1/2}^{n}\right\vert _{x}] \frac{dt}{A_{i}dy}[\left. F_{i,J+1/2}^{n}\right\vert _{y}-\left. F_{i,J-1/2}^{n}\right\vert _{y}]-dt\mathbf{M}_{il}^{-1}\hat{\mathbf{S}}_{lk} \hat{f}_{k,J}^{n}-\hat{f}_{k,J}^{n,eq})\text{,} \end{equation where \begin{equation} A_{i}=\left\{ \begin{array}{cc} 1, & \text{for }1\leq i\leq 4, \\ 1/6, & \text{for }5\leq i\leq 8, \\ 1/\sqrt{2}, & \text{for }9\leq i\leq 12, \\ \sqrt{2}/3, & \text{for }13\leq i\leq 16 \end{array \right. \end{equation $\left. F_{i,J+1/2}^{n}\right\vert _{x}$ ($\left. F_{i,J-1/2}^{n}\right\vert _{x}$) and $\left. F_{i,J+1/2}^{n}\right\vert _{y}$ ($\left. F_{i,J-1/2}^{n}\right\vert _{y}$) are $x$ and $y$ components of the outgoing (incoming) flux in node $J$ along the direction $i$, \begin{subequations} \begin{equation} \left. F_{i,J+1/2}^{n}\right\vert _{x}=f_{i}^{n}(ix,iy)+\frac{1}{2}(1-\frac dt}{A_{i}dx})[f_{i}^{n}(ix+A_{i}v_{ix},iy)-f_{i}^{n}(ix,iy)]\psi _{x}(ix,iy \text{,} \end{equation \begin{equation} \left. F_{i,J-1/2}^{n}\right\vert _{x}=f_{i}^{n}(ix-A_{i}v_{ix},iy)+\frac{1} 2}(1-\frac{dt}{A_{i}dx})[f_{i}^{n}(ix,iy)-f_{i}^{n}(ix-A_{i}v_{ix},iy)]\psi _{x}(ix-A_{i}v_{ix},iy)\text{,} \end{equation \begin{equation} \left. F_{i,J+1/2}^{n}\right\vert _{y}=f_{i}^{n}(ix,iy)+\frac{1}{2}(1-\frac dt}{A_{i}dy})[f_{i}^{n}(ix,iy+A_{i}v_{iy})-f_{i}^{n}(ix,iy)]\psi _{y}(ix,iy \text{,} \end{equation \begin{equation} \left. F_{i,J-1/2}^{n}\right\vert _{y}=f_{i}^{n}(ix,iy-A_{i}v_{iy})+\frac{1} 2}(1-\frac{dt}{A_{i}dy})[f_{i}^{n}(ix,iy)-f_{i}^{n}(ix,iy-A_{i}v_{iy})]\psi _{y}(ix,iy-A_{i}v_{iy})\text{.} \end{equation The flux limiter is expressed as \end{subequations} \begin{equation} \psi _{\alpha }(ix,iy)=\left\{ \begin{array}{ccc} 0 & \text{,} & \left. \theta _{i}^{n}(ix,iy)\right\vert _{\alpha }\leq 0 \\ 2\left. \theta _{i}^{n}(ix,iy)\right\vert _{\alpha } & \text{,} & 0\leq \left. \theta _{i}^{n}(ix,iy)\right\vert _{\alpha }\leq \frac{1}{3} \\ (1+\left. \theta _{i}^{n}(ix,iy)\right\vert _{\alpha })/2 & \text{,} & \frac 1}{3}\leq \left. \theta _{i}^{n}(ix,iy)\right\vert _{\alpha }\leq 3 \\ 2 & \text{,} & 3\leq \left. \theta _{i}^{n}(ix,iy)\right\vert _{\alpha \end{array \right. \end{equation where the smoothness functions are \begin{subequations} \begin{equation} \left. \theta _{i}^{n}(ix,iy)\right\vert _{x}=\frac f_{i}^{n}(ix,iy)-f_{i}^{n}(ix-A_{i}v_{ix},iy)} f_{i}^{n}(ix+A_{i}v_{ix},iy)-f_{i}^{n}(ix,iy)}\text{,} \end{equation \begin{equation} \left. \theta _{i}^{n}(ix,iy)\right\vert _{y}=\frac f_{i}^{n}(ix,iy)-f_{i}^{n}(ix,iy-A_{i}v_{iy})} f_{i}^{n}(ix,iy+A_{i}v_{iy})-f_{i}^{n}(ix,iy)}\text{.} \end{equation \end{subequations} The Lax-Wendroff scheme is recovered for the flux limiter $\psi _{x}=\psi _{y}=1$, and the first order upwind scheme is recovered when $\psi _{x}=\psi _{y}=0$. \section{validation and verification} \subsection{Performance on discontinuity} \begin{figure}[tbp] \center\includegraphics*[bbllx=15pt,bblly=15pt,bburx=315pt,bbury=230pt,width=0.65\textwidth]{Fig3.eps} \caption{Simulation results with various difference schemes at $t=0.06$.} \end{figure} In order to check the performance of flux limiter scheme on discontinuity, we construct the following problem \begin{equation} \left\{ \begin{array}{cc} (\rho ,u_{1},u_{2},T)=(1.5,0.666667,0.0,1.55556), & x\leq L/2. \\ (\rho ,u_{1},u_{2},T)=(1.0,0.0,0.0,1.0), & L/2<x<L \end{array \right. \end{equation} $L$ is the length of computational domain. In the $x$ direction, $f_{i}=\mathbf{M}_{ij}^{-1}\hat{f}_{j}^{eq}$ is set, where the macroscopic quantities adopt the initial values. In the $y$ direction, the periodic boundary condition is adopted. The physical quantities on the two sides satisfy the Hugoniot relations. Fig. 3 shows the simulation results of density, pressure, $x-$ component of velocity, and temperature at time $t=0.06$ using different space discretization schemes. The parameters are $\gamma=2$, $dx=dy=0.001$, $dt=10^{-5}$, $s_{5}=s_{6}=s_{7}=5\times 10^{4}$, and other collision parameters are $10^{5}$. The simulations with Lax-Wendroff scheme have strong unphysical oscillations in the shocked region. The second order upwind scheme results in unphysical `overshoot' phenomena at the shock front. The simulation results with flux limiter scheme are much more accurate, and this scheme has the ability to decrease the unphysical oscillations at the discontinuity. \subsection{Lax shock tube problem} \begin{figure}[tbp] \center\includegraphics*[bbllx=15pt,bblly=15pt,bburx=320pt,bbury=230pt,width=0.65\textwidth]{Fig4.eps} \caption{LB simulation results and exact solutions for Lax shock tube at $t=0.45$.} \end{figure} The initial condition of the problem is: \begin{equation} \left\{ \begin{array}{cc} (\rho ,u_{1},u_{2},T)=(0.445,0.698,0.0,7.928), & x\leq L/2. \\ (\rho ,u_{1},u_{2},T)=(0.5,0.0,0.0,1.142), & L/2<x<L \end{array \right. \end{equation} The profiles of density, pressure, $x-$ component of velocity, and temperature at $t=0.45$ are shown in Fig. 4, where the exact solutions are presented with solid lines for comparison. The parameters are $\gamma =1.4$, $dx=dy=0.003$, $dt=10^{-5}$, $s_{5}=s_{6}=s_{7}=2\times 10^{3}$, s_{8}=s_{9}=10^{3}$, and other collision parameters are $10^{5}$. Obviously, the simulation results agree well with the exact solutions. The above simulations show that compressible flows, especially those with discontinuity and shock waves, can be well simulated by the present model. \subsection{Couette flow} \begin{figure}[tbp] \begin{center} \includegraphics[bbllx=10pt,bblly=15pt,bburx=340pt,bbury=190pt, scale=0.9]{Fig5.eps} \end{center} \caption{Temperature profiles of Couette flow. (a) $\gamma=2$, $Pr=0.5$ corresponds to $s_{5}=10^3,s_{8}=5\times10^2$, $Pr=5$ corresponds to $s_{5}=2\times10^2,s_{8}=10^3$, and (b) $\gamma=1.4$, $Pr=0.1$ corresponds to $s_{5}=10^3,s_{8}=10^2$, $Pr=5$ corresponds to $s_{5}=2\times10^2,s_{8}=10^3$ (other collision parameters are $10^3$).} \end{figure} Here we conduct a series of numerical simulations of Couette flow. In the simulation, the left wall is fixed and the right wall moves at speed $u_{x}=0$, $u_{y}=0.1$. The initial state of the fluid is $\rho =1$, $T=1$, $u_{x}=0$, $u_{y}=0$. The simulation results are compared with the analytical solution: \begin{align} T=T_{1}+(T_{2}-T_{1})\frac{x}{H}+\frac{\mu }{2\lambda }u_{y}^{2}\frac{x}{H}(1 \frac{x}{H})\text{,} \end{align} where $T_{1}$ and $T_{2}$ are temperatures of the left and right walls ( T_{1}=1$, $T_{2}=1.005$), $H$ is the width of the channel. Periodic boundary conditions are applied to the bottom and top boundaries, and the left and right walls adopt the nonequilibrium extrapolation method. Fig. 5 shows the comparison of LB results with analytical solutions for thermal Couette Flows. (a) corresponds to $\gamma=2$, and (b) corresponds to $\gamma=1.4$. It is clearly shown that the simulation results of new model are in agreement with the analytical solutions, and the Prandtl number effects are successfully captured. New model is suitable for incompressible flows. \section{Non-equilibrium characteristic} To show the merit of LB method over traditional ones, in this section we study the non-equilibrium characteristics using the new model. Among the moment relations required by each LB model, only for the first three (density, momentum and energy), the equilibrium distribution function f_{i}^{eq}$ can be replaced by the distribution function $f_{i}$. If we replace $f_{i}^{eq}$ by $f_{i}$ in the left hand of other moment relations, the value of left side will have a difference from that of the right side. This difference represents the deviation of system from its thermodynamic equilibrium\cite{xufop,yan,JCP}. In this MRT LB model, the kinetic moment space and the corresponding equilibria of nonconserved moments are constructed according to the seven-moment relations. So, the deviation from equilibrium in this model can be defined as $\Delta _{i}=\hat{f}_{i}-\hat{f}_{i}^{eq}=\textbf{M}_{ij}(f_{j}-f_{j}^{eq})$. $\Delta _{i}$ contains the information of macroscopic flow velocity $u_{\alpha}$. Furthermore, we replace $v_{i\alpha}$ by $v_{i\alpha}-u_{\alpha}$ in the transformation matrix $\textbf{M}$, named $\textbf{M}^{\ast}$ (see Appendix for details). $\Delta _{i}^{\ast}=\textbf{M}_{ij}^{\ast}(f_{j}-f_{j}^{eq})$ is only the manifestation of molecular thermalmotion and does not contain the information of macroscopic flow. \begin{figure}[tbp] \center\includegraphics*[bbllx=10pt,bblly=15pt,bburx=325pt,bbury=230pt,width=0.9\textwidth]{Fig6.eps} \caption{LB numerical results and non-equilibrium characteristics at $t=0.3$.} \end{figure} Now, we study the following dynamic procedure. An incident shock wave with Mach number $1.414$ travels from a heavy medium and hits a light one, where the two different fluids are separated by an unperturbed interface. The initial macroscopic quantities are as follows: \begin{equation*} \left\{ \begin{array}{cc} (\rho,u_{1},u_{2},p)_{s}=(1.5,0.666667,0,2.33334) \text{,} & \\ (\rho,u_{1},u_{2},p)_{h}=(1,0,0,1) \text{,} & \\ (\rho,u_{1},u_{2},p)_{l}=(0.5,0,0,1) \text{,} & \end{array \right. \end{equation* where the subscripts $s$, $h$, $l$ indicate the shock wave region, the heavy medium region, and the light medium region. In our simulations, the computational domain is $[0, 1.2]\times[0, 0.01]$, and divided into 1200\times10$ mesh-cells. The initial position of shock wave is $x=0.24$, the unperturbed interface lies at the position $x=0.4$. Inflow boundary is applied at the left side, outflow boundary is applied at the right side, and periodic boundary conditions are applied at the top and bottom boundaries. \gamma =2$ in the whole domain. The density, pressure, $x-$ component of velocity and temperature profiles and $\Delta _{i}^{\ast}$ ($i=5,6,7,10,11$) on the center line $y=0.005$ at time $t=0.3$ are shown in Fig. 6. The parameters are $dt=10^{-5}$, $s_{5}=s_{6}=s_{7}=5\times 10^{4}$, and other collision parameters are $10^{5}$. In the figures, the system shows three different interfaces, rarefaction wave, material interface, and shock wave. Physical quantities change significantly at the three interfaces, and vertical lines indicate the positions of interfaces. The system starts to deviate from equilibrium once the physical quantities starts to change. When the physical quantities arrives at its steady-state required by the Hugoniot relations, the system goes back to its equilibrium state. The peak values of deviations $\Delta _{i}^{\ast}$ at shock wave interface are larger than the others. This is because the shock dynamic procedure is faster than the other two processes, and the system has less time to relax to its thermodynamic equilibrium. At the interfaces, $\Delta _{5}^{\ast}$, $\Delta _{7}^{\ast}$ and $\Delta _{11}^{\ast}$ have small amplitudes. $\Delta _{5}^{\ast}$ contains two parts, $x$ and $y$ components of internal translational kinetic energy. This indicates that the two parts deviate from equilibrium in opposite directions with the same amplitude. $\Delta _{6}^{\ast}$ shows an opposite deviation for the rarefaction wave interface and the shock interface. The physical reason is as below. The temperature gradient first initiates variance of the internal kinetic energy in the direction of temperature gradient. (Here, the temperature shows gradient in the $x$ direction.) Then, part of internal kinetic energy variance is transferred to other degrees of freedoms via collisions of molecules. The internal kinetic energy in the temperature gradient direction further varies, and so on. The shock wave increases density, pressure and temperature, while the rarefaction wave decreases those quantities. So, $\Delta _{6}^{\ast}$ shows a negative deviation for the rarefaction wave interface, while shows a positive deviation for the shock interface. The values of $\Delta _{10}^{\ast}$ at material interface and shock wave interface have the same order, and are much larger than that at rarefaction wave. This is because the sizes of temperature variation near the material interface and shock wave differ little, and larger than that near the rarefaction wave. When the temperature gradient vanishes, the system attains its thermodynamic equilibrium. \section{Conclusions} In the paper a MRT LB model for compressible flows is extended so that it is suitable also for incompressible flows. In order to efficiently decrease the unphysical oscillations, space discretization adopts flux limiter scheme with splitting technique. It is validated and verified via same well-known benchmark tests, including Riemann problem and Couette flow, and satisfying agreements are obtained between the new model results and analytical ones. In order to show the merit of LB model over traditional methods, we studied the behaviors of system deviating from its equilibrium through a dynamic procedure where shock wave propagates from a heavy material to a light one. The simulation results are consistent with the physical analysis. \section*{Acknowledgements} The authors would like to sincerely thank S. Succi and C. Lin for many instructive discussions. We acknowledge support of National Natural Science Foundation of China[under Grant Nos. 11075021 and 11047020]. AX and GZ acknowledge support of the Science Foundation of CAEP [under Grant Nos. 2012B0101014 and 2011A0201002] and the Foundation of State Key Laboratory of Explosion Science and Technology[under Grant No.KFJJ14-1M]. \begin{appendix} \section{Transformation matrix and equilibria of the nonconserved moments} In the model by Kataoka and Tsutahara, the local equilibrium distribution function $f_{i}^{eq}$ satisfies the following relations: \begin{subequations} \begin{equation} \rho =\sum f_{i}^{eq}, \end{equation \begin{equation} \rho u_{\alpha }=\sum f_{i}^{eq}v_{i\alpha }, \end{equation \begin{equation} \rho \left( bRT+u_{\alpha }^{2}\right) =\sum f_{i}^{eq}\left( v_{i\alpha }^{2}+\eta _{i}^{2}\right) , \end{equation \begin{equation} P\delta _{\alpha \beta }+\rho u_{\alpha }u_{\beta }=\sum f_{i}^{eq}v_{i\alpha }v_{i\beta }, \end{equation \begin{equation} \rho \left[ \left( b+2\right) RT+u_{\beta }^{2}\right] u_{\alpha }=\sum f_{i}^{eq}\left( v_{i\beta }^{2}+\eta _{i}^{2}\right) v_{i\alpha }, \end{equation \begin{equation} \rho \left[ RT\left( u_{\alpha }\delta _{\beta \chi }+u_{\beta }\delta _{\alpha \chi }+u_{\chi }\delta _{\alpha \beta }\right) +u_{\alpha }u_{\beta }u_{\chi }\right] =\sum f_{i}^{eq}v_{i\alpha }v_{i\beta }v_{i\chi }, \end{equation \begin{equation} \rho \left\{ \left( b+2\right) R^{2}T^{2}\delta _{\alpha \beta }+\left[ \left( b+4\right) u_{\alpha }u_{\beta }+u_{\chi }^{2}\delta _{\alpha \beta \right] RT+u_{\chi }^{2}u_{\alpha }u_{\beta }\right\} =\sum f_{i}^{eq}\left( v_{i\chi }^{2}+\eta _{i}^{2}\right) v_{i\alpha }v_{i\beta }, \end{equation \end{subequations} where a parameter $\eta _{i}$ is introduced, in order to describe the $(b-2)$ extra-degrees of freedom corresponding to molecular rotation and/or vibration, where $\eta _{i}=5/2$ for $i=1$, $\cdots $,$4$, and $\eta _{i}=0$ for $i=5$, $\cdots $, $16$. The transformation matrix $\mathbf{M}$ in the MRT model is composed as below: $\mathbf{M}=(m_{1},m_{2},\cdots ,m_{16})^{T}$, \begin{subequations} \begin{equation} m_{1i}=1\text{,} \end{equation \begin{equation} m_{2i}=v_{ix}\text{,} \end{equation \begin{equation} m_{3i}=v_{iy}\text{,} \end{equation \begin{equation} m_{4i}=v_{ix}^{2}+v_{iy}^{2}+\eta _{i}^{2}\text{,} \end{equation \begin{equation} m_{5i}=v_{ix}^{2}+v_{iy}^{2}\text{,} \end{equation \begin{equation} m_{6i}=v_{ix}^{2}-v_{iy}^{2}\text{,} \end{equation \begin{equation} m_{7i}=v_{ix}v_{iy}\text{,} \end{equation \begin{equation} m_{8i}=v_{ix}(v_{ix}^{2}+v_{iy}^{2}+\eta _{i}^{2})\text{,} \end{equation \begin{equation} m_{9i}=v_{iy}(v_{ix}^{2}+v_{iy}^{2}+\eta _{i}^{2})\text{,} \end{equation \begin{equation} m_{10i}=v_{ix}(v_{ix}^{2}+v_{iy}^{2})\text{,} \end{equation \begin{equation} m_{11i}=v_{iy}(v_{ix}^{2}+v_{iy}^{2})\text{,} \end{equation \begin{equation} m_{12i}=v_{ix}(v_{ix}^{2}-v_{iy}^{2})\text{,} \end{equation \begin{equation} m_{13i}=v_{iy}(v_{ix}^{2}-v_{iy}^{2})\text{,} \end{equation \begin{equation} m_{14i}=(v_{ix}^{2}+v_{iy}^{2})(v_{ix}^{2}+v_{iy}^{2}+\eta _{i}^{2})\text{,} \end{equation \begin{equation} m_{15i}=v_{ix}v_{iy}(v_{ix}^{2}+v_{iy}^{2}+\eta _{i}^{2})\text{,} \end{equation \begin{equation} m_{16i}=(v_{ix}^{2}-v_{iy}^{2})(v_{ix}^{2}+v_{iy}^{2}+\eta _{i}^{2})\text{,} \end{equation \end{subequations} where $i=1,\cdots ,16$. Replacing $v_{i\alpha}$ by $v_{i\alpha}-u_{\alpha}$ in the transformation matrix $\textbf{M}$, matrix $\mathbf{M}^{\ast}$ is expressed as follows: $\mathbf{M}^{\ast}=(m_{1}^{\ast},m_{2}^{\ast},\cdots ,m_{16}^{\ast})^{T}$, \begin{subequations} \begin{equation} m_{1i}^{\ast}=1\text{,} \end{equation \begin{equation} m_{2i}^{\ast}=v_{ix}-u_{x}\text{,} \end{equation \begin{equation} m_{3i}^{\ast}=v_{iy}-u_{y}\text{,} \end{equation \begin{equation} m_{4i}^{\ast}=(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}+\eta _{i}^{2}\text{,} \end{equation \begin{equation} m_{5i}^{\ast}=(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}\text{,} \end{equation \begin{equation} m_{6i}^{\ast}=(v_{ix}-u_{x})^{2}-(v_{iy}-u_{y})^{2}\text{,} \end{equation \begin{equation} m_{7i}^{\ast}=(v_{ix}-u_{x})(v_{iy}-u_{y})\text{,} \end{equation \begin{equation} m_{8i}^{\ast}=(v_{ix}-u_{x})[(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}+\eta _{i}^{2}]\text{,} \end{equation \begin{equation} m_{9i}^{\ast}=(v_{iy}-u_{y})[(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}+\eta _{i}^{2}]\text{,} \end{equation \begin{equation} m_{10i}^{\ast}=(v_{ix}-u_{x})[(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}]\text{,} \end{equation \begin{equation} m_{11i}^{\ast}=(v_{iy}-u_{y})[(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}])\text{,} \end{equation \begin{equation} m_{12i}^{\ast}=(v_{ix}-u_{x})[(v_{ix}-u_{x})^{2}-(v_{iy}-u_{y})^{2}]\text{,} \end{equation \begin{equation} m_{13i}^{\ast}=(v_{iy}-u_{y})[(v_{ix}-u_{x})^{2}-(v_{iy}-u_{y})^{2}]\text{,} \end{equation \begin{equation} m_{14i}^{\ast}=[(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}][(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}+\eta _{i}^{2}]\text{,} \end{equation \begin{equation} m_{15i}^{\ast}=(v_{ix}-u_{x})(v_{iy}-u_{y})[(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}+\eta _{i}^{2}]\text{,} \end{equation \begin{equation} m_{16i}^{\ast}=[(v_{ix}-u_{x})^{2}-(v_{iy}-u_{y})^{2}][(v_{ix}-u_{x})^{2}+(v_{iy}-u_{y})^{2}+\eta _{i}^{2}]\text{,} \end{equation \end{subequations} where $i=1,\cdots ,16$. The equilibria of nonconserved moments are as follows: \begin{subequations} \begin{equation} \hat{f}_{5}^{eq}=2P+(j_{x}^{2}+j_{y}^{2})/\rho \text{,} \end{equation \begin{equation} \hat{f}_{6}^{eq}=(j_{x}^{2}-j_{y}^{2})/\rho \text{,} \end{equation \begin{equation} \hat{f}_{7}^{eq}=j_{x}j_{y}/\rho \text{,} \end{equation \begin{equation} \hat{f}_{8}^{eq}=(e+2P)j_{x}/\rho \text{,} \end{equation \begin{equation} \hat{f}_{9}^{eq}=(e+2P)j_{y}/\rho \text{,} \end{equation \begin{equation} \hat{f}_{10}^{eq}=(4P+j_{x}^{2}/\rho +j_{y}^{2}/\rho )j_{x}/\rho \text{,} \end{equation \begin{equation} \hat{f}_{11}^{eq}=(4P+j_{x}^{2}/\rho +j_{y}^{2}/\rho )j_{y}/\rho \text{,} \end{equation \begin{equation} \hat{f}_{12}^{eq}=(2P+j_{x}^{2}/\rho -j_{y}^{2}/\rho )j_{x}/\rho \text{,} \end{equation \begin{equation} \hat{f}_{13}^{eq}=(-2P+j_{x}^{2}/\rho -j_{y}^{2}/\rho )j_{y}/\rho \text{,} \end{equation \begin{equation} \hat{f}_{14}^{eq}=2(b+2)\rho R^{2}T^{2}+(6+b)RT(j_{x}^{2}+j_{y}^{2})/\rho +(j_{x}^{2}+j_{y}^{2})^{2}/\rho ^{3}\text{,} \end{equation \begin{equation} \hat{f}_{15}^{eq}=[(b+4)P+(j_{x}^{2}+j_{y}^{2})/\rho ]j_{x}j_{y}/\rho ^{2 \text{,} \end{equation \begin{equation} \hat{f}_{16}^{eq}=[(b+4)P+(j_{x}^{2}+j_{y}^{2})/\rho ](j_{x}^{2}-j_{y}^{2})/\rho ^{2}\text{.} \end{equation \end{subequations} \end{appendix}

proofpile-arXiv_069-5713

\section{Introduction} Let $\Gamma =(V,E)$ be a finite digraph, where $V=\{v_1,v_2,\ldots ,v_n\}$. Set $$D^-(v_i)=\{v_j\in V|(v_j,v_i)\in E\},\quad D^+(v_i)=\{v_j\in V|(v_i,v_j)\in E\}.$$ Two vertices $v_i,v_j\in V$ are \textit{mutually neighbored} if $$D^-(v_i)\cap D^-(v_j)\neq \emptyset \quad \text{or} \quad D^+(v_i)\cap D^+(v_j)\neq \emptyset.$$ Let $$T(\Gamma)=E\circ E^{\text{op}}\cup E^{\text{op}}\circ E=\{(v_i,v_j)|v_i,v_j \text{ are mutually neighbored}\}.$$ For simplicity, we sometimes abuse notation and consider $$T(\Gamma)=\{(i,j)|v_i,v_j \text{ are mutually neighbored}\}.$$ \begin{Theorem A} For any finite digraph without parallel edges $\Gamma =(V,E)$, $$|T(\Gamma)|\geq|E|.$$ \end{Theorem A} A consequence of Theorem A is an estimation of the dimension of the homogeneous components in simple group-graded algebras. A \textit{grading} of an algebra $\Lambda$ by a group $G$ is a vector space decomposition \begin{equation}\label{eq:algebragrading} \Lambda=\bigoplus _{g\in G} \Lambda_g \end{equation} such that $\Lambda_g\Lambda_h\subseteq \Lambda_{gh}$. An algebra $\Lambda$ is \textit{$G$-simple} with respect to a group grading~\eqref{eq:algebragrading} if it admits no non-trivial graded ideals. We prove the following theorem, which is implicit in \cite{giambruno}. \begin{Theorem B} Let $\Lambda$ be a complex $G$-simple algebra with respect to the grading~\eqref{eq:algebragrading}. Denote the trivial element of $G$ by $e$. Then dim$_{\mathbb{C}}(\Lambda_e)\geq$dim$_{\mathbb{C}}(\Lambda_g)$ for any $g\in G$. \end{Theorem B} Theorem A is proven in \S\ref{graph results}. In \S\ref{gradings} we associate a natural digraph to any simple grading of a complex algebra using a classification theorem due to Bahturin, Sehgal and Zaicev. Then, using Theorem A we prove Theorem B. \section{Proof of Theorem A}\label{graph results} Let $V=\{v_1,v_2,\ldots ,v_n\}$. Obviously, we may assume that $\Gamma$ has no isolated vertices. In particular, we may assume that $(i,i)\in T(\Gamma)$ for any $1\leq i\leq n$. Another assumption we can adopt is that $\Gamma$ admits no loops. Indeed, for every loop $(v_i,v_i)\in E$, the condition that either $(v_i,v_j)\in E$ or $(v_j,v_i)\in E$ says that $v_i\in D^-(v_i)\cap D^-(v_j)$ or $v_i\in D^+(v_i)\cap D^+(v_j)$ respectively, which is the same as saying that both $(i,j),(j,i)\in T(\Gamma)$. Hence the number of ordered pairs that $v_i$ contributes to $T(\Gamma)$ is at least as the number of edges that this vertex contributes to $E$.\\ The theorem is clear for $|V|=1,2$. We proceed by induction on the number of vertices in $\Gamma $. Assume that the theorem holds for digraphs with $|V|\leq n-1$. Let $\Gamma=(V,E)$ where $|V|=n$. We distinguish between the Eulerian and the non-Eulerian cases:\\ \underline{\textbf{$\Gamma$ is Eulerian.}}\\ For every $1\leq i\leq n$ denote by $r_i=|D^-(v_i)|=|D^+(v_i)|$. Let $k$ be the length of a shortest directed cycle in $\Gamma $. Without loss of generality we may assume that the vertices in the cycle are $\{v_i\}_{1\leq i\leq k}$, such that $v_i \in D^-(v_{i+1})$ for $1\leq i< k$ and $v_k\in D^-(v_1)$. By minimality of the length of the cycle, if $i\not \equiv j+1($mod $k)$, then $v_j\not \in D^-(v_i)$. Again we distinguish between two different cases.\\ \textbf{Case 1:} Assume that for any $1\leq i\neq j\leq k$ the ordered pair $(i,j)\not \in T(\Gamma)$. By removing the vertices $v_1,v_2,\ldots ,v_k$ as well as their corresponding edges we get a new graph $\Gamma ^{\shortmid}=(V^{\shortmid},E^{\shortmid})$ with $n-k$ vertices, which satisfies the theorem by the induction assumption. That is, \begin{equation}\label{eq:ind} |T(\Gamma ^{\shortmid})|\geq |E^{\shortmid}|. \end{equation} The number of edges that were removed is \begin{equation}\label{eq:edges} |E|-|E^{\shortmid}|=\sum _{i=1}^k 2r_i-k. \end{equation} We count the number of ordered pairs that were removed from $T(\Gamma)$. For $1\leq i<k$, let $$ C_i=\{(v_i,v)|v\in D^-(v_{i+1})\}\cup \{(v,v_i)|v\in D^-(v_{i+1})\}(\subseteq T(\Gamma)),$$ and also $$ C_k=\{(v_k,v)|v\in D^-(v_1)\}\cup \{(v,v_k)|v\in D^-(v_1)\}(\subseteq T(\Gamma)).$$ In order to compute the cardinality of $C_i$, for $1\leq i<k$, every $v\in D^-(v_{i+1})$ is counted twice, except $v_i$ itself which is counted only once. This argument, as well as a similar argument for $C_k$ yields \begin{equation}\label{eq:C_i} |C_i|= 2r_{i+1}-1 , 1\leq i< k ,\quad |C_k|= 2r_1-1. \end{equation} The condition that for $1\leq i\neq l\leq k$ the ordered pair $(i,l)\not \in T(\Gamma)$ ensures that for $i\neq l$, $C_i\cap C_l=\emptyset$. Therefore, the number of distinct ordered pairs that were removed from $T(\Gamma)$ can be bounded as follows. \begin{equation}\label{eq:pairs} |T(\Gamma)|-|T(\Gamma ^{\shortmid})|\geq |\bigcup _{i=1}^kC_i|=\sum _{i=1}^k|C_i|= \sum _{i=1}^k 2r_i-k. \end{equation} By~\eqref{eq:ind}, ~\eqref{eq:edges}, ~\eqref{eq:pairs} we get \begin{equation} |T(\Gamma)|\geq |T(\Gamma ^{\shortmid})|+ \sum _{i=1}^k 2r_i-k\geq |E^{\shortmid}|+(|E|-|E^{\shortmid}|)=|E|. \end{equation} \textbf{Case 2:} Suppose that there exist $1\leq i,j\leq k$ such that $(i,j)\in T(\Gamma)$. We may assume that the path from $v_i$ to $v_j$ is of minimal length such that $(i,j)\in T(\Gamma)$, and relabel the indices such that $i=1$. Let $v_m$ be a vertex such that $$v_m \in D^-(v_1)\cap D^-(v_j)\quad \text{or}\quad v_m \in D^+(v_1)\cap D^+(v_j).$$ By the minimality of $k$, $m>k$. Assume that $v_m \in D^-(v_1)\cap D^-(v_j)$ (The proof for $v_m \in D^+(v_i)\cap D^+(v_j)$ is similar). Let $\textrm{F}:=\{1,2,\ldots ,j\}\cup \{m\}$. By removing the vertices $\{v_s\}_{s \in \textrm{F}}$ as well as their corresponding edges we get a new graph $\Gamma ^{\shortmid}=(V^{\shortmid},E^{\shortmid})$ with $n-j-1$ vertices, which satisfies the theorem by the induction assumption. That is, \begin{equation}\label{eq:ind2} |T(\Gamma ^{\shortmid})|\geq |E^{\shortmid}|. \end{equation} The number of edges that were removed is \begin{equation}\label{eq:edges2} |E|-|E^{\shortmid}|=\sum _{t=1}^j 2r_t-(j-1)+2r_m-2. \end{equation} We subtract $(j-1)$, since any edge in the path from $v_1$ to $v_j$ is counted twice. Also, we subtract $2$ since the edges $(v_m,v_1),(v_m,v_j)$ are counted twice. Next, we count the number of ordered pairs that were removed from $T(\Gamma)$. Again, for every $i \in \textrm{F}$ we define the sets $C_i$. For $1\leq i< j$ $$ C_i=\{(v_i,v)|v\in D^-(v_{i+1})\}\cup \{(v,v_i)|v\in D^-(v_{i+1})\}(\subseteq T(\Gamma)),$$ for $m$ we define $$ C_m=\{(v_m,v)|v\in D^-(v_1)\}\cup \{(v,v_m)|v\in D^-(v_1)\}(\subseteq T(\Gamma)),$$ and for $j$ $$ C_j=\{(v_j,v)|v\in D^+(v_m)\}\cup \{(v,v_j)|v\in D^+(v_m)\}(\subseteq T(\Gamma)).$$ Similarly to~\eqref{eq:C_i} we obtain $$|C_i|= 2r_{i+1}-1,1\leq i< k,\quad |C_m|= 2r_1-1, \quad C_j=2r_m-1.$$ By the minimality property the above sets are distinct, that is $C_{l_1}\cap C_{l_2}=\emptyset$ for any $l_1 \neq l_2 \in \textrm{F}.$ Therefore, the number of distinct ordered pairs that were removed from $T(\Gamma)$ can be bounded as follows. \begin{equation}\label{eq:pairs2} |T(\Gamma)|-|T(\Gamma ^{\shortmid})|\geq |\bigcup _{i\in F}C_i|=\sum _{i\in F}|C_i|=\sum _{t=1}^j 2r_t+2r_m-(j+1). \end{equation} By~\eqref{eq:ind2}, ~\eqref{eq:edges2}, ~\eqref{eq:pairs2} we get \begin{equation} |T(\Gamma)|\geq |T(\Gamma ^{\shortmid})|+ \sum _{t=i}^j 2r_t+2r_m -(j+1)\geq |E^{\shortmid}|+(|E|-|E^{\shortmid}|)=|E|. \end{equation} \underline{\textbf{$\Gamma$ is non-Eulerian}.}\\ First, we show that in this case there exists $(v_j,v_i)\in E$ such that $$|D^-(v_i)|> |D^+(v_i)|\quad \text{and} \quad |D^-(v_j)|\leq |D^+(v_j)|.$$ We write $V$ as a disjoint union, $V=V_1\cup V_2$ where $$V_1=\{v\in V||D^-(v)|> |D^+(v)|\}$$ $$V_2=\{v\in V||D^-(v)|\leq |D^+(v)|\}.$$ By the non-Eulerian property $V_1$ is not empty. Obviously, there must be $v_j\in V_2 ,v_i\in V_1$ such that $(v_j,v_i)\in E$. Let $v_i,v_j$ be as above and set $$|D^-(v_i)|=i_1,|D^+(v_i)|=i_2,|D^-(v_j)|=j_1,|D^+(v_j)|=j_2.$$ Then \begin{equation}\label{eq:notE} i_1>i_2 \quad ,\quad j_1\leq j_2. \end{equation} By removing the vertices $v_i,v_j$ as well as their corresponding edges we get a new graph $\Gamma ^{\shortmid}=(V^{\shortmid},E^{\shortmid})$ with $n-2$ vertices, which satisfies the theorem by the induction assumption. That is, \begin{equation}\label{eq:ind3} |T(\Gamma ^{\shortmid})|\geq |E^{\shortmid}|. \end{equation} Again, let $$ C_j=\{(v_j,v)|v\in D^-(v_i)\}\cup \{(v,v_j)|v\in D^-(v_i)\}(\subseteq T(\Gamma)),$$ and $$ C_i=\{(v_i,v)|v\in D^+(v_j)\}\cup \{(v,v_i)|v\in D^+(v_j)\}(\subseteq T(\Gamma)).$$ The cardinality of $C_i$ is $2i_1-1$, and the cardinality of $C_j$ is $2j_2-1$. Since there are no loops in $\Gamma$, then clearly $C_i\cap C_j=\emptyset$. Therefore, the number of distinct ordered pairs that were removed from $T(\Gamma)$ can be bounded as follows, \begin{equation}\label{eq:ind4} |T(\Gamma)|-|T(\Gamma^{\shortmid})|\geq |C_i\cup C_j|=|C_i|+|C_j|=2i_1+2j_2-2. \end{equation} On the other hand, by~\eqref{eq:notE} the number of edges that were removed is bounded as follows, \begin{equation}\label{eq:ind5} |E|-|E^{\shortmid}|= i_1+i_2+j_1+j_2-1\leq 2i_1+2j_2-2. \end{equation} By~\eqref{eq:ind3},~\eqref{eq:ind4} and~\eqref{eq:ind5} we get \begin{flalign*} &&|T(\Gamma)|\geq |T({\Gamma^{\shortmid}})|+2i_1+2j_2-2\geq |E^{\shortmid}|+(|E|-|E^{\shortmid}|)=|E|.&& \qed \end{flalign*} For an undirected graph $\Gamma$, let $$\gamma _k=\{(i,j)|\text{there exists a path of length } k \text{ between } v_i \text{ and } v_j \}.$$ By interpreting an undirected graph as a digraph, Theorem A yields the following corollary. \begin{corollary} Let $\Gamma$ be a finite undirected graph without parallel edges. Then, with the above notations, $$|\gamma _2|\geq |\gamma _1|.$$ \end{corollary} Another consequence of Theorem A is as follows. Denote the non-negative real numbers by $\mathbb{R}^+$. For $A\in M_n(\mathbb{R}^+)$, let $$\text{Supp}(A)=\{(i,j)\in A|a_{ij}\neq 0\}.$$ Any $A\in M_n(\mathbb{R}^+)$ determines a digraph $\Gamma=(V,E)$ in the following way: $$V=\{v_1,v_2,\ldots ,v_n\}$$ $$(v_i,v_j)\in E \Leftrightarrow a_{ij}\neq 0.$$ Clearly $|\text{supp}(A)|=|E|$. Notice that $(AA^t+A^tA)_{ij}\neq 0 $ if and only if the vertices $v_i,v_j$ are mutually neighbored. \begin{corollary} For any $A\in M_n(\mathbb{R}^+)$ $$|\text{Supp}(AA^t+A^tA)|\geq |\text{Supp}(A)|.$$ \end{corollary} \section{gradings}\label{gradings} In order to deduce Theorem B from Theorem A we need the following classification theorem. \begin{theorem}\cite[see Theorem 3]{MR2488221}\label{th:zaicev2} Let~\eqref{eq:algebragrading} be a $G$-simple grading of a complex algebra $\Lambda$. Then there exists an $n$-tuple $(g_1,g_2,\ldots ,g_n)\in G^n$ and a subgroup $H\leq G$, where dim$_{\mathbb{C}}(\Lambda)=n^2\cdot |H|$ such that for any $g\in G$ \begin{equation}\label{eq:induce} \text{dim}_{\mathbb{C}}(\Lambda_g)=|\{(g_i,h,g_j)|h\in H, g_i^{-1}hg_j=g\}|. \end{equation} \end{theorem} Simply-graded algebras are vastly investigated, e.g. see \cite{giambruno, aljadeff2011simple, MR2226177, MR1941224, MR2488221, das1999, MR2928456, MR1863551}. Let~\eqref{eq:algebragrading} be a $G$-simple grading. Let $(g_1,g_2,\ldots ,g_n)\in G^n$ and $H\leq G$ be the corresponding $n$-tuple and the corresponding subgroup provided by Theorem~\ref{th:zaicev2}. For any $g\in G$ we associate a digraph $\Gamma _g=(V_g,E_g)$ in the following way. $$V_g=\{v_1,v_2,\ldots ,v_n\},$$ $$E_g=\{(v_i,v_j)|\exists h\in H,\quad g_i^{-1}hg_j=g\}.$$ Notice that for any $1\leq i,j\leq n$, if $$g_i^{-1}h_1g_j=g_i^{-1}h_2g_j$$ then $h_1=h_2$. Therefore, by~\eqref{eq:induce} \begin{equation}\label{eq:grad} \text{dim}_{\mathbb{C}}(\Lambda_g)=|E_g|. \end{equation} \begin{remark} \cite[Theorem 3]{MR2488221} describes a way to decompose any simple $G$-graded complex algebra to \textit{fine} and \textit{elementary} gradings. This decomposition is not unique. However, when given a $G$-simple grading~\eqref{eq:algebragrading}, one can show by \cite[Proposition 3.1]{aljadeff2011simple} and by using the ``moves" described in \cite[Lemma 1.3]{aljadeff2011simple} that for any $g\in G$ the associated digraph $\Gamma _g$ is determined up to a graph isomorphism. \end{remark} \begin{center} Proof of Theorem B. \end{center} By~\eqref{eq:grad} we need to show that $|E_e|\geq |E_g|$ for any $g\in G$. Applying Theorem A on the graph $\Gamma_g$ we obtain \begin{equation}\label{eq:grad2} |T({\Gamma _g})|\geq |E_g|. \end{equation} Now, we show that if a pair $(i,j)\in T(\Gamma _g)$ then $(v_i,v_j)\in E_e$. Indeed, if there exists $v_k\in D^+(v_i)\cap D^+(v_j)$ (the case where $v_k\in D^-(v_i)\cap D^-(v_j)$ is similar), then there exist $h_1,h_2$ such that \begin{equation}\label{eq:distinct} g_i^{-1}h_1g_k=g=g_j^{-1}h_2g_k. \end{equation} By~\eqref{eq:distinct} $$(g_i^{-1}h_1g_k)(g_j^{-1}h_2g_k)^{-1}=e$$ and hence $(v_i,v_j)\in E_e.$ As a consequence we get that \begin{equation}\label{eq:grad1} |E_e|\geq |T(\Gamma _g)|. \end{equation} Therefore, by~\eqref{eq:grad2} and~\eqref{eq:grad1} we get that for any $g\in G$ \begin{flalign*} && |E_e|\geq |E_g|. && \qed \end{flalign*} \begin{remark} The dimension of the trivial component is not necessary maximal when the algebra is not simply-graded. For example, consider the natural $\mathbb{Z}$-grading of a polynomial ring with more the one indeterminate. In this case, the trivial component is one dimensional, whereas the other components have strictly larger dimensions. \end{remark} \addcontentsline{toc}{chapter}{Bibliography}

proofpile-arXiv_069-5813

\section{Introduction} The human body is inhabited by complex microbial communities, called microbiomes. It is estimated that the number of microbial cells associated with the human body is 10 times the total number of human cells. The collective genomes of these microbes constitute an extended human genome that provides us with genetic and metabolic capabilities that we do not inherently possess [\citet{Backhed2005}]. With the development of next generation sequencing technology such as the 454 pyrosequencing and Illumina Solexa sequencing, microbiome composition can now be determined by direct DNA sequencing without laborious cultivation. Typically, instead of sequencing all bacterial genomic DNA as in a shotgun metagenomic approach, only the 16S rRNA gene, which is ubiquitous in the bacteria kingdom and has variable regions, is sequenced. Since each bacterial cell is assumed to have the same number of copies of this gene, the basic idea is to isolate from all the bacteria the DNA strands corresponding to some variable region of the gene, to count different versions of the sequences, and then to identify to which bacteria the versions correspond. The types and abundances of different bacteria in a sample can therefore be determined. After preprocessing of the raw sequences, the 16S sequences are either mapped to an existing phylogenetic tree in a taxonomic dependent way [e.g., \citet{Matsen2010}] or clustered into operational taxonomic units (OTUs) at a certain similarity level in a taxonomic independent way [e.g., \citet{Caporaso2010a,Schloss2009a}]. At 97\% similarity level, these OTUs are used to approximate the taxonomic rank \textit{species}. The OTU based approach is most commonly used in 16S based microbiome studies. Each OTU is characterized by a representative DNA sequence and can be assigned a taxonomic lineage by comparing to a known bacterial 16S rRNA database. Most OTUs are in extremely low abundances, with a large proportion being simply singletons (possibly due to sequencing error). We can further aggregate OTUs from the same genus and perform analysis on the abundances at the genus level, which is more robust to sequencing error and can reduce the number of variables significantly. Either way we finally obtain the taxa counts for each sample. Recent studies have linked the microbiome with human diseases including \mbox{obesity} and inflammatory bowel disease [\citet{Virgin2011}]. It is therefore important to understand how genetic or environmental factors shape the human microbiome in order to gain insight into etiology of many microbiome-related diseases and to develop therapeutic measures to modulate the microbiome composition. \citet{Benson2010} demonstrated that genetic variants are associated with the mouse gut microbiome. \citet{Wu2011a} showed that dietary nutrients are associated with the human gut microbiome. Both studies have considered a large number of genetic loci or nutrients and aimed to identify the genetic variants or nutrients that are associated with the gut microbiome. When there are numerous possible covariates affecting the microbiome composition, variable selection becomes necessary. Variable selection cannot only increase biological interpretability but also provide researchers with a short list of top candidates for biological validation. The methods we\vadjust{\goodbreak} develop in this paper are particularly motivated by an ongoing study at the University of Pennsylvania to link the nutrient intake to the human gut microbiome. In this study, gut microbiome data were collected on 98 normal volunteers. In addition, food frequency questionnaire (FFQ) were filled out by these individuals. The questionnaires were scored and the quantitative measurements of 214~micronutrients were obtained. Details of the study and the data set can be found in Section~\ref{secreal} and in \citet{Wu2011a}. Our goal is to identify the nutrients that are associated with the gut microbiome and also their associated bacterial taxa. Most of the microbiome studies used distance-based methods to link the microbiome and environmental covariates, where a distance metric was defined between two microbiome samples and statistical analysis was then performed using the distances. However, the choice of distance metric is sometimes subjective and different distances vary in their power of identifying relevant environmental factors. Another limitation of distance-based methods is its inefficiency for detecting subtle changes since distances summarize the overall relationship. In addition, such distance-based approaches do not provide information on how covariates affect the microbiome compositions and which taxa are affected. Therefore, it is desirable to model the counts directly instead of summarizing the data as distances. One way of testing for covariate effects is by performing a multivariate multiple regression (called redundancy analysis in ecology) after appropriate transformation of the count data such as converting into proportions [\citet{PierreLegendre2002}]. A pseudo-$F$ statistic is then calculated and the significance is then evaluated by permutation test. Alternatively, one can define a distance between the samples and then use a PERMANOVA procedure to test for covariate effects [\citet{McArdle2001}]. It is easy to show that when the distance is Euclidean, these two procedures are equivalent. In this paper, we consider the sparse Dirichlet-multinomial (DM) regression [\citet{MOSIMANN1962}] to link high-dimensional covariates to bacterial taxa counts from microbiome data. The DM regression model is chosen to model the overdispersed taxa counts. The observed taxa count variance is much larger than that predicted by a multinomial model that assumes fixed underlying taxa proportions, an assumption that is hardly met for real microbiome data. Uncontrollable sources of variation such as individual-to-individual variability, day-to-day variability, sampling location variability or even technical variability such as sample preparation lead to enormous variability in the underlying proportions. In contrast, the DM model assumes that the underlying taxa proportions come from a Dirichlet distribution. We use a log-linear link function to associate the mean taxa proportions with covariates. In this DM modeling framework, the effects of the covariates on taxa proportions can be tested using the likelihood ratio test. When the number of the covariates is large, we propose a sparse group $\ell_1$ penalized likelihood approach for variable selection and parameter estimation. The\vadjust{\goodbreak} sparse group $\ell_1$ penalty function [\citet{Friedman2010}] consists of a group $\ell_1$ penalty and an overall $\ell_1$ penalty, which induce both group-level sparsity and within-group sparsity. This is particularly relevant in our setting. For the nutrient-microbiome association example, we have $p$ nutrients and $q$ taxa, so the fully parameterized model has $(p+1)\times q$ coefficients including the intercepts, since each nutrient-taxon association is characterized by one coefficient. The $q$ coefficients for each nutrient constitute a group. If we assume many nutrients have no or ignorable effects on the microbiome composition, the groups of coefficients associated with these irrelevant nutrients should be zero altogether, which is a group-level sparsity that is achieved by imposing a group $\ell_1$ penalty. However, the group $\ell_1$ penalty does not perform within-group selection, wherein if one group is selected, all the coefficients in that group are nonzeros. In the case of nutrient-microbiome association, we are also interested in knowing which taxa are associated with a selected nutrient. By imposing an overall $\ell_1$ penalty, within-group selection becomes possible. Therefore, we impose a sparse group $\ell_1$ penalty not only to select these important nutrients but also to recover relevant nutrient-taxon associations. Section~\ref{secDM} reviews the Dirichlet-multinomial model for count data. Section~\ref{secLR} introduces the Dirichlet-multinomial regression framework for incorporating covariate effects and proposes a likelihood ratio statistic for testing the covariate effect. Section \ref{secgrp} proposes a sparse group $\ell_1$ penalized likelihood procedure for variable selection for the DM models followed by a detailed description of a block-coordinate descent algorithm in Section \ref{secalg}. Section~\ref{secsim} shows simulation results and Section~\ref{secreal} demonstrates the proposed method on a real human gut microbiome data set to associate the nutrient intake with the human gut microbiome composition. \section{Dirichlet-multinomial model for microbiome composition data} \label{secDM} Suppose we have $q$ bacterial taxa and their counts $Y=(Y_1, Y_2,\ldots, Y_q)$ are random variables. Denote ${\mathbf{y}}=(y_1,y_2,\ldots,y_q)$ as the observed counts. The simplest model for count data is the multinomial model and its probability function is given as \[ f_M(y_1, y_2, \ldots, y_q; \bphi) = \pmatrix{y_+ \cr{\mathbf{y}}}\prod_{j=1}^{q} \phi_j^{y_j}, \] where $y_+=\sum_{j=1}^q y_j$ and $\bphi=(\phi_1, \phi_2, \ldots, \phi_q)$ are underlying species proportions with $\sum_{j=1}^q\phi_j=1$. Here the total taxa count $y_+$ is determined by the sequencing depth and is treated as an ancillary statistic since its distribution does not depend on the parameters in the model. The mean and variance of the multinomial component~$Y_j$ $(j=1 ,\ldots, q)$ are \begin{equation} \label{eqM2} \E(Y_j) = y_+\phi_j,\qquad \var(Y_j) = y_+ \phi_j(1-\phi_j). \end{equation} For microbiome composition data, the actual variation is usually larger than what would be predicted by the multinomial model,\vadjust{\goodbreak} which assumes fixed underlying proportions. This increased variation is due to the heterogeneity of the microbiome samples and the underlying proportions vary among samples. To account for the extra variation or overdispersion, we assume the underlying proportions $(\phi_1, \phi_2, \ldots, \phi_q)$ are themselves positive random variables $(\Phi_1, \Phi_2, \ldots, \Phi_q)$ subject to the constraint $\sum_{j=1}^q \Phi_j =1$. One commonly used distribution is the Dirichlet distribution [\citet{MOSIMANN1962}] with the probability function given by \[ f_D(\phi_1, \phi_2, \ldots, \phi_q; \bgamma) = \frac{\Gamma (\gamma_+)}{\prod_{j=1}^q \Gamma(\gamma_j)} \prod _{j=1}^q \phi_j^{\gamma _j - 1}, \] where $\bgamma=(\gamma_1,\gamma_2,\ldots,\gamma_q)$ are positive parameters, $\gamma_+=\sum_{j=1}^q \gamma_j$ and $\Gamma(\cdot)$ is the Gamma function. The mean and variance of the Dirichlet component $\Phi_j$ $(j=1 ,\ldots, q)$ are \[ \E(\Phi_j) = \frac{\gamma_j}{\gamma_+},\qquad \var(\Phi_j)= \frac {\gamma_j(\gamma_+ - \gamma_j)}{(1+\gamma_+)\gamma_+^2}. \] The mean is proportional to $\gamma_j$ and the variance is controlled by $\gamma_+$, which can be regarded as a ``precision parameter.'' As $\gamma_+$ becomes larger, the proportions are more concentrated around the means. The Dirichlet-multinomial (DM) distribution [\citet{MOSIMANN1962}] results from a compound multinomial distribution with weights from the Dirichlet distribution (parametrization I): \begin{eqnarray} \label{eqDM1} f_{\mathrm{DM}}(y_1, y_2, \ldots, y_q;\bgamma) & = & \int f_M(y_1, y_2, \ldots, y_q; \bphi)f_D(\bphi;\bgamma)\,d \bphi \nonumber\\[-8pt]\\[-8pt] & = & \pmatrix{y_+ \cr{\mathbf{y}}}\frac{\Gamma(y_+ + 1) \Gamma(\gamma_+)}{\Gamma(y_+ + \gamma_+)} \prod _{j=1}^{q} \frac{\Gamma(y_j + \gamma_j)}{\Gamma (\gamma_j)\Gamma(y_j+1)}.\nonumber \end{eqnarray} The mean and variance of the DM distribution for each component $Y_j$ $(j=1,\ldots,q)$ is given by \begin{equation} \label{eqDM2} \E(Y_j) = y_+\E(\Phi_j),\qquad \var(Y_j) = y_+\E(\Phi_j)\bigl\{1 - \E ( \Phi_j)\bigr\} \biggl( \frac{y_+ + \gamma_+}{1 + \gamma_+} \biggr). \end{equation} Comparing (\ref{eqDM2}) with (\ref{eqM2}), we see that the variation of the DM component is increased by a factor of $(y_+ + \gamma_+)/(1 + \gamma_+)$, where $\gamma_+$ controls the degree of overdispersion with a larger value indicating less overdispersion. Using an alternative parameterization, the probability function can be written as (parameterization~II) \begin{equation} \label{eqDM3} f_{\mathrm{DM}}^*(y_1, y_2,\ldots, y_q; \bphi, \theta) = \pmatrix{y_+ \cr{\mathbf {y}}}\frac {\prod_{j=1}^q\prod_{k=1}^{y_j}\{\phi_j(1-\theta) + (k-1)\theta\} }{\prod_{k=1}^{y_+}\{1 - \theta+ (k-1)\theta\}}, \end{equation} where $\phi_j= \gamma_j/\gamma_+$ is the mean and $\theta =1/(1+\gamma_+)$ is the dispersion parameter. When $\theta= 0$, it is easy to verify (\ref{eqDM3}) is reduced to the multinomial distribution. \section{Dirichlet-multinomial regression for incorporating the covariate effects} \label{secLR} When there is no covariate effect, the DM model can be used to produce more accurate estimates of taxa proportions of a given microbiome sample than the simple multinomial model, due to its ability to model the overdispersion. Beyond proportion estimation, microbial ecologists are more interested in associating the microbiome composition with some environmental covariates. Suppose we have $n$ microbiome samples and $q$ species. Let $\Y =(y_{ij})_{n \times q}$ be the observed count matrix for the $n$ samples. Let $\X=(x_{ij})_{n \times p}$ be the design matrix of $p$ covariates for $n$ samples. We assume the parameters $\gamma_j$ $(j=1,\ldots,q)$ in the DM model (parametrization I) depend on the covariate via the following log-linear model, \begin{equation} \label{eqlink1} \gamma_j\bigl({\mathbf{x}}^i\bigr) = \exp\Biggl(\alpha_j + \sum_{k=1}^p \beta_{jk}x_{ik}\Biggr), \end{equation} where ${\mathbf{x}}^i$ is the $i$th row vector of $\X$ and $\beta_{jk}$ is the coefficient for the $j$th taxon with respect to $k$th covariate, whose sign and magnitude measure the effect of the $k$th covariate on the $j$th taxon. From (\ref{eqDM2}), we see that $\E(Y_{ij}) \propto\exp(\alpha_j)\prod_{k=1}^p\exp(\beta_{jk}x_{ik})$, where $\exp(\alpha_j)$ can be interpreted as the baseline abundance level for species $j$ and the coefficient $\beta_{jk}$ indicates the magnitude of the $k$th covariate effect on species $j$. Though the log-linear link is assumed mainly for ease of computation, it is biologically consistent, in that microorganisms usually exhibit exponential growth in a favorable environment.\looseness=-1 For notational simplicity, we denote $\beta_{j0}$ as $\alpha_j$ and augment $\X$ with an $n$-vector of $1$'s as its first column. We number the columns from $0$ to $p$. The link function becomes \begin{equation} \label{eqlink2} \gamma_j\bigl({\mathbf{x}}^i\bigr) = \exp\Biggl(\sum_{k=0}^p \beta_{jk}x_{ik}\Biggr). \end{equation} Let $\bbeta$ be the $q\times(p+1)$ regression coefficient matrix, $\bbeta^{j}=(\beta_{j0},\ldots,\beta_{jp})^T$ be the vector of coefficients for the $j$th taxon ($j=1,\ldots,q$) and $\bbeta_{k}=(\beta_{1k},\ldots,\break\beta_{qk})^T$ be the vector of coefficients for the $k$th covariate ($k=0,\ldots,p$). We also use $\bbeta$ to denote the $q(p+1)$ vector that contains all the coefficients. Substituting (\ref {eqlink1}) into DM probability function (\ref{eqDM1}) and ignoring the part that does not involve the parameters, the log-likelihood function given the covariates is given by \begin{eqnarray} \label{eqloglik2} l(\bbeta; \Y,\X) &=& \sum_{i=1}^n \Biggl[ \tilde{\Gamma} \Biggl(\sum_{j=1}^q \gamma_j\bigl({\mathbf{x}}^i;\bbeta^j\bigr) \Biggr) - \tilde{\Gamma } \Biggl(\sum_{j=1}^q y_{ij} + \sum_{j=1}^q \gamma_j\bigl({\mathbf{x}}^i;\bbeta^j\bigr) \Biggr) \nonumber\\[-9pt]\\[-9pt] &&\hspace*{45pt}{} + \sum_{j=1}^q \bigl\{\tilde{ \Gamma} \bigl(y_{ij}+\gamma_j\bigl({\mathbf{x}}^i; \bbeta^j\bigr) \bigr) - \tilde{\Gamma } \bigl(\gamma_j \bigl({\mathbf{x}}^i;\bbeta^j\bigr) \bigr) \bigr\} \Biggr],\nonumber \end{eqnarray} where $\tilde{\Gamma}(\cdot)$ is the log-gamma function.\vadjust{\goodbreak} Based on the likelihood function (\ref{eqloglik2}), one can test the effect of a given covariate or the joint effects of all covariates on the microbiome composition using the standard likelihood ratio test (LRT). To solve the maximization problem, we implemented the Newton--Raphson algorithm, since the gradient and Hessian matrix of the log-likelihood can be calculated analytically. Alternatively, we can use the general-purpose optimization algorithm such as $nlm$ in R, which computes the gradient and Hessian numerically. By selecting an appropriate starting point (e.g., $\balpha=\bbeta=\nulll$), for moderate-size problems in the dimensions $p$ and $q$, the algorithm converges to a stationary point sufficiently fast. With a large number of covariates in the DM regression model, direct maximization of the likelihood function becomes infeasible or unstable. When each covariate is tested separately using the LRT, adjustment for multiple testing is required. In addition, when the number of taxa $q$ is large, the null distribution of the LRT has large degrees of freedom and therefore reduced power. It is also desirable to select the relevant covariates that are associated with the microbiome composition. Although one can test the null hypothesis $H_0\dvtx\beta_{jk}=0$ for each $(j,k)$ pair by the LRT, adjustment of multiple comparisons can lead to a loss of power. In the next section we present a sparse group $\ell_1$ penalized estimation for variable selection and parameter estimation for sparse DM regression models. \section{Variable selection for sparse Dirichlet-multinomial regression} \label{secgrp} To perform variable selection, we estimate the regression coefficient vector $\bbeta$ in model (\ref{eqlink2}) by minimizing the following sparse group $\ell_1$ penalized negative log-likelihood function, \begin{equation} \label{eqploglik} \mathit{pl}(\bbeta; \Y,\X, \lambda_1, \lambda_2) = - l(\bbeta; \Y,\X) + \lambda_1 \sum _{k=1}^p \llVert \bbeta_{k} \rrVert_2 + \lambda_2 \sum_{k=1}^p \llVert \bbeta_{k} \rrVert_1, \end{equation} where $l(\bbeta; \Y,\X)$ is the log-likelihood function defined as in (\ref{eqloglik2}), $\lambda_1$ and $\lambda_2$ are the tuning parameters and $\llVert \bbeta_k \rrVert_1=\sum_{j=1}^q|\beta_{ik}|$ is the $\ell_1$ norm and $\llVert \bbeta_k \rrVert_2=\sqrt{\sum_{j=1}^q \beta_{ik}^2}$ is the group $\ell_1$ norm of the coefficient vector $\bbeta_k$, respectively. We do not penalize the intercept vector $\bbeta_0$. The first part of the sparse group $\ell_1$ penalty is the group $\ell_1$ penalty that induces group-level sparsity, which facilitates selection of the covariates that are associated with taxa proportions. The second $\ell_1$ penalty on all the coefficients facilitates the within-group selection, which is important for interpretability of the resulting model. A similar penalty involving both group $\ell_1$ and $\ell_1$ terms is discussed in \citet{Peng2009} and \citet{Friedman2010} for regularized multivariate linear regression. When $\lambda_2=0$, criterion (\ref{eqploglik}) reduces to the group lasso. \subsection{A block-coordinate gradient descent algorithm for sparse group $\ell_1$ penalized DM regression} \label{secalg} The sparse group $\ell_1$ estimates of $\bbeta$ can be obtained by minimizing the penalized negative log-likelihood function (\ref{eqploglik}): \[ \hat{\bbeta}_{\lambda_1,\lambda_2} = \argmin_{\bbeta} \Biggl\{- l(\bbeta; \Y,\X) + \lambda_1 \sum_{k=1}^p \llVert \bbeta_k \rrVert_2 + \lambda_2 \sum _{k=1}^p \llVert \bbeta_k \rrVert_1 \Biggr\}. \] Using the general block coordinate gradient descent algorithm of \citet{Tseng2007}, we develop in the following an efficient algorithm to solve this optimization problem. \citet{Meier2008} present a block coordinate gradient descent algorithm for group lasso for logistic regression that includes only the group $\ell_1$ penalty (i.e., $\lambda_2=0$). In contrast, our optimization problem (\ref{eqploglik}) has two nondifferentiable parts, both at the individual $\beta_{jk}$ and at the group $\bbeta_k$ levels. The key idea of the algorithm is to combine a quadratic approximation of the log-likelihood function with an additional line search. First we expand (\ref{eqloglik2}) at current estimate $\hat{\bbeta}^{(t)}$ to a second-order Taylor series. The Hessian matrix is then replaced by a suitable matrix $\H^{(t)}$. We define \begin{equation} \label{Taylor} l_Q^{(t)}(\d) = l\bigl(\hat{ \bbeta}{}^{(t)}\bigr) + \d^T\nabla l\bigl(\hat { \bbeta}{}^{(t)}\bigr) + \tfrac{1}{2}\d^T\H^{(t)} \d, \end{equation} where $\d\in\reals^{q(p+1)}$. Also denote $\nabla l(\hat{\bbeta }{}^{(t)})_k$ and $\d_k$ the gradient and increment with respect to $\hat {\bbeta}{}^{(t)}_k$ for the $k$th group, and $\nabla l(\hat{\bbeta }{}^{(t)})_{sk}$ and $\d_{sk}$ with respect to $\hat{\beta}{}^{(t)}_{sk}$. We then minimize the following function $\mathit{pl}_{Q}^{(t)}(\d)$ with respect to the $k$th penalized parameter group: \begin{eqnarray} \label{eqpqlik} \mathit{pl}_{Q}^{(t)}(\d) & = & - l_Q^{(t)}( \d) + \lambda_1\sum_{k=1}^{p} \bigl\llVert \hat{\bbeta}{}^{(t)}_k+\d_k \bigr \rrVert_2 +\lambda_2\sum_{k=1}^{p} \bigl\llVert \hat{\bbeta}{}^{(t)}_k+\d_k \bigr \rrVert_1 \nonumber\\[-8pt]\\[-8pt] &\approx& \mathit{pl}\bigl(\hat{\bbeta}{}^{(t)}+\d; \Y, \X, \lambda_1, \lambda_2\bigr). \nonumber \end{eqnarray} We restrict ourselves to vectors $\d$ with $\d_j = \nulll$ for $j \neq k$ and the corresponding $q \times q$ submatrix $\H_{kk}^{(t)}$ for the $k$th group is a diagonal matrix of the form $\H_{kk}^{(t)}=h_k^{(t)}\I_q$ for some scalar $h_k^{(t)} \in\reals$. The solution\vspace*{1pt} to the general optimization problem of the form (\ref {eqpqlik}) is given by Theorem~\ref{theorem1} and its corollary in the \hyperref[app]{Appendix}. Let $S = \{s \vert \vert\nabla l(\hat{\bbeta }{}^{(t)})_{sk} - h_k^{(t)}\hat{\beta}{}^{(t)}_{sk} \vert< \lambda_2 \}$ and $\bar{S}$ be the set $\{1,\ldots,q\}\setminus S$. Denote $\d_{Sk}$ the subvector of $\d_k$ with indices in $S$ and $\d\bsk$ in $\bar{S}$. The minimizer of (\ref{eqpqlik}) can be decomposed into two parts: The first part $\d_{Sk}^{(t)}$ can be obtained by \[ \d_{Sk}^{(t)}= - \hat{\bbeta}{}^{(t)}_{Sk}. \] The second part $\d\bsk^{(t)}$ can be computed by minimizing \begin{equation} \label{eqpqlik2} f^{(t)}(\d_k) = - \bigl\{ \d_k^T \u_k^{(t)} + \tfrac{1}{2}\d_k^T\H_{kk}^{(t)} \d_k \bigr\} + \lambda_1 \bigl\llVert \hat{ \bbeta}{}^{(t)}_k+\d_k \bigr\rrVert_2 \end{equation} with respect to $\d\bsk$ (set components other than $\d\bsk$ to be $0$), where \[ \u^{(t)}_k = \bigl[\nabla l\bigl(\hat{ \bbeta}{}^{(t)}\bigr)_k - \lambda_2 \operatorname{sgn} \bigl\{\nabla l\bigl(\hat{\bbeta}{}^{(t)} \bigr)_k- h_k^{(t)}\hat{\bbeta }{}^{(t)}_k \bigr\} \bigr] \] and sgn($\cdot$) is the sign function. Minimization of (\ref{eqpqlik2}) with respective to $\d\bsk$ can be performed in a similar fashion as in \citet{Meier2008} for the group $\ell_1$ penalty. Specifically, if $\llVert \u^{(t)}\bsk-h_k^{(t)}\bbeta^{(t)}\bsk\rrVert_2 < \lambda_1$, the minimizer of equation (\ref{eqpqlik2}) for $\d\bsk$ is \[ \d\bsk^{(t)}= -\hat{\bbeta}{}^{(t)}\bsk. \] Otherwise \[ \d\bsk^{(t)}= -\frac{1}{h_k^{(t)}} \biggl\{\u^{(t)}\bsk- \lambda_1 \frac{\u^{(t)} \bsk-h_k^{(t)}\hat{\bbeta}{}^{(t)}\bsk}{ \llVert \u^{(t)}\bsk-h_k^{(t)}\hat{\bbeta}{}^{(t)}\bsk\rrVert_2} \biggr\}. \] For the unpenalized intercept, the solution can be directly computed: \[ \d_0^{(t)} = -\frac{1}{h_0^{(t)}}\nabla l\bigl(\hat{ \bbeta}{}^{(t)}\bigr)_0. \] If $\d^{(t)}\ne\nulll$, an inexact line search using the Armijo rule will be performed. Let $\alpha^{(t)}$ be the largest value in $\{\alpha_0\delta^l\}_{l \ge0}$ such that \[ \mathit{pl}\bigl(\hat{\bbeta}{}^{(t)}+\alpha^{(t)}\d^{(t)} \bigr)-\mathit{pl}\bigl(\hat{\bbeta }{}^{(t)}\bigr) \le \alpha^{(t)}\sigma \Delta^{(t)}, \] where $ 0 < \delta<1, 0 < \sigma< 1, \alpha_0 > 0$, and $\Delta^{(t)}$ is the improvement in the objective function $\mathit{pl}(\bbeta)$ using a linear approximation, that is, \begin{eqnarray*} \Delta^{(t)}&=&-\d^{(t)T}\nabla l\bigl(\hat{\bbeta}{}^{(t)} \bigr) + \lambda_1 \Biggl\{ \sum_{k=1}^p \bigl\llVert \hat{\bbeta}{}^{(t)}_k + \d_k^{(t)} \bigr\rrVert_2 - \sum_{k=1}^p \bigl\llVert \hat{\bbeta}{}^{(t)}_k \bigr\rrVert_2 \Biggr\} \\ &&{}+\lambda_2 \Biggl\{\sum_{k=1}^p \bigl\llVert \hat{\bbeta}{}^{(t)}_k + \d_k^{(t)} \bigr\rrVert_1 - \sum_{k=1}^p \bigl\llVert \hat{\bbeta}{}^{(t)}_k \bigr\rrVert_1 \Biggr\}. \end{eqnarray*} Finally, we update the current estimate by \[ \hat{\bbeta}{}^{(t+1)}=\hat{\bbeta}{}^{(t)} + \alpha^{(t)} \d^{(t)}. \] For $\H_{kk}^{(t)}$, we use the same choice as in \citet{Meier2008}, that is, \[ h_k^{(t)} = -\max \bigl[\diag\bigl\{-\nabla^2l \bigl(\hat{\bbeta }{}^{(t)}\bigr)_{kk}\bigr\}, c^* \bigr], \] where $c^*>0$ is a lower bound to ensure convergence. In this paper, we use the standard choices for the parameters, $\alpha_0=1,\delta =0.5,\sigma=0.1$ and $c^*=0.001$ [\citet{Tseng2007}], in the block coordinate descent algorithm to ensure the convergence of the algorithm. \begin{Remark*} In each iteration of the algorithm detailed above, when estimating the $k$th column of the $q\times p$ coefficient matrix $\bbeta$ with all other columns fixed, the algorithm first identifies the coefficients with zero estimates, denoted by set $S$ in the algorithm. For the coefficients in set $S$, $d_{Sk}^{(t)}=-\hat{\bbeta}{}^{(t)}_{Sk}$ and, therefore, when \mbox{$\alpha^{t}=1$}, $\hat{\bbeta}{}^{(t+1)}_{Sk}=\hat{\bbeta}{}^{(t)}_{Sk}+\alpha^{t}d_{Sk}^{(t)}=0$ and the coefficients\vspace*{1pt} in $S$ are shrunk to zero. Based on its definition, the set $S$ depends on the turning parameter $\lambda_2$ and a larger value of $\lambda_2$ leads to fewer nonzero coefficients. The algorithm then performs a group shrinkage of the nonzero estimates of the coefficients in the complementary set $\bar{S}$. These nonzero coefficients can further be shrunk to zero as a group if the condition $\|\u^{(t)}\bsk-h_k^{(t)}\bbeta^{(t)}\|_2 < \lambda_1$ is met, in which case $d_{\bar{S}k}^{(t)}=-\hat{\bbeta}_{\bar {S}k}^{(t)}$ and, therefore, $\hat{\bbeta}_{\bar{S}k}^{(t+1)}=\hat{\bbeta}_{{\bar {S}k}}^{(t)}+d_{\bar {S}k}^{(t)}=0$. Clearly,\vspace*{2pt} this group shrinkage depends on the tuning parameter $\lambda_1$. Thus, with careful choice of the tuning parameters $\lambda_1$ and $\lambda_2$, some column group coefficients are set to zero and the within-group sparsity is achieved by the plain $\ell_1$ penalty. \end{Remark*} \subsection{Tuning parameter selection} Two tuning parameters $\lambda_1$ and $\lambda_2$ in the penalized likelihood estimation need to be tuned with data by $v$-fold cross-validation or a BIC criterion. To facilitate computation, we reparameterize $\lambda_1$ and $\lambda_2$ as $\lambda_1=c\lambda\sqrt{q}$ and $\lambda_2=(1-c)\lambda$. The multiplier $\sqrt{q}$ in the group penalty is used so that the group $\ell_1$ penalty and overall $\ell_1$ penalty are on a similar scale. Here we use $\lambda$ to control the overall sparsity level and use $c \in[0,1]$ to control the proportion of group $\ell_1$ in the composite sparse group penalty. When $c=0$, the penalty is reduced to the lasso; when $c=1$, it is reduced to a group lasso. We consider the tuning parameter $c$ from the set $\{0, 0.05, 0.1, 0.2, 0.4\}$. For each $c$, to search for the best tuning parameter value, we run the algorithm from $\lambda_{\mathrm{max}}$ so that it produces the sparsest model with the intercepts $\bbeta_0$ only. The value $\lambda_{\mathrm{max}}$ can be roughly determined by using the starting value $\bbeta^{(0)}$ with components $\bbeta_j^{(0)}=\nulll$ $(j \neq0)$ and $\bbeta_0^{(0)}$ the MLE of (\ref{eqloglik2}) without covariates, and choosing the smallest value of $\lambda$ so that the iteration converges in the first iteration, that is, $\bbeta^{(0)}$ is a stationary point. We then decrease the $\lambda$ value and use the estimate of $\bbeta$ from the last $\lambda$ as a warm start. The grid of $\lambda$ can be chosen to be equally spaced on a log-scale, for example, $\lambda_j=0.96^j\lambda_{\mathrm{max}}$ $(j=1,\ldots, m)$, where $m$ is set so that $\lambda_{\mathrm{min}}=0.2\lambda_{\mathrm{max}}$ or, alternatively, we could terminate the loop until the model receives more than the maximum number of nonzero coefficients allowed. \section{Simulation studies} \label{secsim} \subsection{Simulation strategies} We simulate $n$ microbiome samples, $p$ nutrients and $q$ bacterial taxa to mimic the real data set that we analyze in Section \ref {secreal}. The nutrient intake vector is simulated using a multivariate normal distribution with mean $\nulll$ and a covariance matrix $\Sigma_{i,j}=\rho^{\vert i-j\vert}$. We simulate $p_r$ relevant nutrients with each nutrient being associated with $q_r$ taxa. For each nutrient, the association coefficients $\beta_{ij}$ for the $q_r$ taxa are equally spaced over the interval $[0.6f, 0.9f]$ with alternative signs, where $f$ controls the association strength. We consider two growth models to relate the taxa abundances to the covariates. In the exponential growth model, the proportion of the $j$th taxon of the $i$th sample is determined as \begin{equation} \label{eqlinkmult} \phi_{ij} = \frac{ \exp (\beta_{j0} +\sum_{k=1}^p\beta_{jk}x_{ik} )}{ \sum_{j=1}^{q} \exp (\beta_{j0} +\sum_{k=1}^p\beta_{jk}x_{ik} )}. \end{equation} The intercepts $\bbeta_0$, which determine the base abundances of the taxa, are taken from a uniform distribution over $(-2.3, 2.3)$ so that the base taxa abundances can differ up to 100 folds. The exponential growth model is a common model for bacteria growth in response to environmental stimuli. We also consider a linear growth model, in which the proportion of the $j$th taxon of the $i$th sample is determined as \[ \phi_{ij} = \frac{\beta_{j0} + \sum_{k=1}^p\beta_{jk}x_{ik}}{ \sum_{j=1}^{q} (\beta_{j0} + \sum_{k=1}^p \beta_{jk}x_{ik})}. \] The intercepts $\bbeta_0$ are now drawn from a uniform distribution over $(0.02, 2)$ so that the base taxa abundances can also differ up to 100 folds. To deal with possible negative $\sum_{k=0}^p\beta_{jk}x_{ik}$, we add a small constant to make it positive. We then generate the count data using the DM model of parametrization II (\ref{eqDM3}) with a common dispersion $\theta$. The number of individuals (sequence reads) for the $i$th sample $m_i$ is generated from a uniform distribution over $(m, 2m)$. Note that the data are not generated exactly according to our model assumptions, which are based on parametrization I (\ref{eqDM1}) and link (\ref{eqlink2}). This can further demonstrate the robustness of our proposed model. \begin{figure}[b] \vspace*{-3pt} \includegraphics{592f01.eps} \caption{{Effect of the tuning parameter $c$ on variable selection.} The tuning parameter $c$ is varied from $0$ to $0.4$. Under each value of $c$, the best $\lambda$ value, which maximizes the likelihood of the test data set, is selected to generate the sparse model. Group (left) and within-group (right) selection performances are then evaluated using measures of recall, precision and $F_1$ based on 100 replications. Simulation setting: $n = 100$, $p =100$, $ p_r = 4$, $q = 40$, $q_r = 4$, $m = 500$, $\theta = 0.025$, $\rho = 0.4$.} \label{figceffect} \end{figure} \subsection{Evaluation of the penalized likelihood approach for selecting covariates affecting the microbiome composition} To evaluate the variable selection performance of the proposed sparse penalized likelihood approach with group $\ell_1$ penalty, we first simulate the count data using the exponential growth model with $n = 100$, $p = 100$, $ p_r = 4$, $q = 40$, $q_r = 4$, $m = 1000$, $\theta = 0.025$, and $\rho = 0.4$, totaling $4000$ variables. We compare the results to the corresponding penalized estimation of the DM model using only the $\ell_1$ penalty function and two other sparse group $\ell_1$ estimations based on multinomial or Dirichlet regression. In sparse multinomial regression, we use the multinomial model for count data and\vadjust{\goodbreak} the link function is given by (\ref{eqlinkmult}). We set $\beta_{10}=0$ to make the coefficients identifiable. In sparse Dirichlet regression, instead of modeling the counts directly, we model the proportions using the Dirichlet distribution and the link function is the same as that of the DM regression. Since the count data contain zeros, we add 0.5 to the cells with 0 counts. We also include results from the LRT based univariate testing procedure for group selection controlling the false discovery rate (FDR) at 0.05.\looseness=-1 We measure the selection performance using \[ \mathrm{recall} = \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}, \qquad \mathrm{precision} = \frac {\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}},\qquad F_1 = 2 \cdot\frac{\mathrm{precision} \cdot \mathrm{recall}}{ \mathrm{precision} + \mathrm{recall}}, \] where $\mathrm{TP}$, $\mathrm{FN}$ and $\mathrm{FP}$ are true positives, false negatives and false positives, respectively, and $F_1$ is an overall measure, which weights the precision and recall equally. The averages of these measures are reported based on $100$ replications. To select the best tuning parameter values, we simulate an independent test data set of $n/2$ samples. We then run the penalized procedure over the training data set and re-estimate the selected coefficients using an unpenalized procedure (``nlm'' function in R). The log-likelihood of the test data set is calculated based on the re-estimated coefficients and the tuning parameter is selected to maximize the log-likelihood over the test data set. We choose the tuning parameter $c$ from the set $\{0, 0.05, 0.1, 0.2, 0.4\}$. Figure~\ref{figceffect} shows that a small $c$ is sufficient to identify the groups efficiently, while further increase of $c$ only improves the group selection marginally. On the other hand, within-group selection exhibits a unimode pattern indicating slight grouping could lead to better identification of within-group elements. In the following simulations, we tune both $c$ and $\lambda$ to achieve the maximum likelihood values in the test data sets. Table~\ref{tabsgl0} shows the simulation results. The sparse group $\ell_1$ penalized DM regression has a much higher precision rate in group selection than the corresponding $\ell_1$ penalized procedure, while both achieve similar recall rates, demonstrating the gain from including the group $\ell_1$ penalty in the regularization. Interestingly, the sparse group penalized DM regression also performs better in within-group selection, as shown by a higher recall rate and $F_1$, indicating better group selection could also facilitate better overall variable selection. Compared to models based on the sparse Dirichlet regression and multinomial regression, the DM model performs \begin{sidewaystable} \textwidth=\textheight \tablewidth=\textwidth \caption{Comparison of sparse group $\ell_1$ and $\ell_1$ penalized procedures for variable selection under Dirichlet-multinomial (DM),\break Dirichlet (D) and multinomial (M) regression models. The selection performance, both group selection and\break within-group selection, is evaluated using recall rate ({R}), precision rate ({P}) and $F_1$ ({F}), all averaged over 100 runs\break (standard deviation in parenthesis). The selection based on a univariate likelihood ratio test ({LRT}) at FDR${}={}$0.05 is also indicated}\label{tabsgl0} \begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lcccccccccccc@{}} \hline & \multicolumn{6}{c}{\textbf{Sparse group} $\bolds{\ell_1}$ \textbf{penalization}} & \multicolumn{6}{c@{}}{$\bolds{\ell_1}$ \textbf{penalization}} \\[-4pt] & \multicolumn{6}{c}{\hrulefill} & \multicolumn{6}{c@{}}{\hrulefill}\\ & \multicolumn{3}{c}{\textbf{Within-group}} & \multicolumn{3}{c}{\textbf{Group}} & \multicolumn{3}{c}{\textbf{Within-group}} & \multicolumn{3}{c@{}}{\textbf{Group}}\\[-4pt] & \multicolumn{3}{c}{\hrulefill} & \multicolumn{3}{c}{\hrulefill} & \multicolumn{3}{c}{\hrulefill} & \multicolumn{3}{c@{}}{\hrulefill}\\ \textbf{Model} & \textbf{R} & \textbf{P} & \textbf{F} & \textbf{R} & \textbf{P} & \textbf{F} & \textbf{R} & \textbf{P} & \textbf{F} & \textbf{R} & \textbf{P} & \textbf{F}\\ \hline\\[-8pt] & \multicolumn{12}{c}{Exponential growth, $p = 100,q_r = 4, \theta = 0.025$}\\[4pt] DM & 0.59 & 0.70 & 0.59 & 0.86 & 0.92 & 0.87 & 0.42 & 0.76 & 0.48 & 0.88 & 0.68 & 0.70 \\ & (0.23) & (0.23) & (0.18) & (0.23) & (0.16) & (0.18) & (0.21) & (0.23) & (0.18) & (0.22) & (0.29) & (0.22) \\ D & 0.48 & 0.73 & 0.52 & 0.83 & 0.89 & 0.82 & 0.36 & 0.82 & 0.45 & 0.82 & 0.77 & 0.72 \\ & (0.23) & (0.23) & (0.20) & (0.26) & (0.18) & (0.21) & (0.20) & (0.21) & (0.19) & (0.26) & (0.27) & (0.23) \\ M & 0.46 & 0.72 & 0.50 & 0.82 & 0.85 & 0.79 & 0.36 & 0.76 & 0.44 & 0.84 & 0.70 & 0.69 \\ & (0.23) & (0.26) & (0.21) & (0.27) & (0.24) & (0.25) & (0.19) & (0.24) & (0.18) & (0.26) & (0.28) & (0.24) \\ LRT & -- & -- & -- & 0.96 & 0.54 & 0.66 & -- & -- & -- & 0.96 & 0.54 & 0.66 \\ & -- & -- & -- & (0.11) & (0.21) & (0.16) & -- & -- & -- & (0.11) & (0.21) & (0.16) \\ \hline \end{tabular*} \end{sidewaystable} better in variable selection, especially for within-group selection, suggesting the DM model is more appropriate than multinomial or Dirichlet models when the counts exhibit overdispersion. The Dirichlet model performs slightly better than the multinomial model. At 5\% FDR, the LRT based univariate testing procedure selects far more variables than these penalized procedures, yielding a higher recall rate but a much worse precision rate.\looseness=1 \subsection{Effects of overdispersion and model misspecification} We further investigate the effect of overdispersion and simulate the count data with different degrees of overdispersion and present the results in Figure~\ref{figsgl1}. We observe that larger overdispersion makes the selection more difficult for all three models, as shown by smaller $F_1$ values. When the data have slight overdispersion ($\theta = 0.005$), the selection performances of the three models are similar. On the other hand, when the data have much overdispersion ($\theta = 0.1$), DM performs much better than the other two models in terms of both group selection and within-group selection. Therefore, modeling overdispersion can lead to power gains in identifying relevant variables if the data are overdispersed. \begin{figure} \includegraphics{592f02.eps} \caption{Effects of overdispersion (top panel) and model-misspecification (bottom panel) on the performance of three different models and methods. DM-SGL: sparse group $\ell_1$ penalized Dirichlet-multinomial model; DM-L: $\ell_1$ penalized Dirichlet-multinomial model; M-SGL: sparse group $\ell_1$ penalized multinomial model; M-L: $\ell_1$ penalized multinomial model; D-SGL: sparse group $\ell_1$ penalized Dirichlet model; D-L: $\ell_1$ penalized Dirichlet model. For each bar, mean${}\pm{}$standard error is presented based on 100 replications.} \label{figsgl1}\vspace*{6pt} \end{figure} To assess the sensitivity to model misspecification, we simulate the counts using the linear growth model instead and compare the results with the exponential growth model (see Figure~\ref{figsgl1}). Interestingly, both the Dirichlet and DM model are very robust to model misspecification and their selection performances do not decrease significantly. On the other hand, the multinomial model suffers a large performance loss with the $F_1$ measure for group selection decreasing from $0.79$ to $0.56$. We also study the effect of the total counts for each sample (data not shown). Even increasing the total count by 10 folds, the DM model is still better than the proportion based Dirichlet model. Therefore, even though we have much deeper sequencing of the microbiome that results in larger counts for each sample, using the DM model can still lead to improved performance over the model that considers only the proportions. \begin{figure}[b] \includegraphics{592f03.eps} \caption{Effects of the number of relevant taxa (top panel) and the number of the covariates (bottom panel) on the performances of several models and methods. DM-SGL: sparse group $\ell_1$ penalized Dirichlet-multinomial model; DM-L: $\ell_1$ penalized Dirichlet-multinomial model; M-SGL: sparse group $\ell_1$ penalized multinomial model; M-L: $\ell_1$ penalized multinomial model; D-SGL: sparse group $\ell_1$ penalized Dirichlet model; D-L: $\ell_1$ penalized Dirichlet model. For each bar, mean${}\pm{}$standard error is presented based on 100 replications.} \label{figsgl3} \end{figure} \subsection{Effects of the number of the covariates and the relevant taxa} We next study the effect of the number of relevant taxa in each group on the performance of different models and present the results in Figure~\ref{figsgl3}. When each relevant group contains only one relevant taxon, the grouping is not very helpful, so the sparse group regularized DM model and $\ell_1$ regularized DM model do not differ much in selecting the relevant groups. When the relevant group contains $8$ relevant taxa, variable grouping becomes much more important and the sparse group regularized DM model performs much better than the $\ell_1$ penalized DM. The group penalized multinomial and Dirichlet regression models, on the other hand, select groups as well as the DM regression model, since the grouping effect is much stronger. Figure~\ref{figsgl3} also shows the results when we increase the dimension of covariates to 400 ($16\mbox{,}000$ variables in total). Increase of the dimension does not deteriorate the variable selection performance, demonstrating the efficiency of our method in handling high-dimensional data. \section{Associating nutrient intake with the human gut microbiome composition} \label{secreal} Diet strongly affects the human health, partly by modulating gut microbial community composition. \citet{Wu2011a} studied the habitual diet effect on the human gut microbiome, where a cross-section of 98 healthy volunteers were enrolled in the study. Diet information was collected using a food frequency questionnaire (FFQ) and was then converted to nutrient intake values of 214 micronutrients. Nutrient intake was further normalized using the residual method to adjust for caloric intake and was standardized to have mean 0 and standard deviation 1. Since some nutrient measurements were almost identical, we used only one representative for these highly correlated nutrients (correlation $\rho> 0.9$), resulting in 118 representative nutrients. Stool samples were collected and DNA samples were analyzed by the 454$/$Roche pyrosequencing of 16S rDNA gene segments of the V1--V2 region. The pyrosequences were denoised prior to taxonomic assignment, yielding an average of $9265 \pm3864$ (SD) reads per sample. The denoised sequences were then analyzed by the QIIME pipeline [\citet{Caporaso2010a}] with the default parameter settings. The OTU table contained 3068 OTUs (excluding the singletons) and these OTUs can be further combined into 127 genera. We studied 30 relatively common genera that appeared in at least 25 subjects. Finally, we had the count matrix $\Y_{98\times30}$ and covariate matrix $X_{98\times118}$. Our goal is to identify the micronutrients that are associated with the gut microbiomes and the specific genera that the selected nutrients affect. \begin{figure} \includegraphics{592f04.eps} \caption{Model fit using the variables selected by the sparse group $l_1$ penalized DM model. Top plot: square root of the fitted counts versus square root of the observed counts based on the DM model with the selected nutrients; bottom plots: observed counts and simulated counts produced by the fitted sparse DM model and multinomial model.} \label{figfit} \end{figure} \begin{sidewaystable} \tabcolsep=0pt \textwidth=\textheight \tablewidth=\textwidth \caption{Estimated regression coefficients from the sparse group $\ell_1$ penalized DM regression for the diet-gut microbiome data.\break The exponentiation of a given coefficient can be interpreted as the factor of change in proportion of a taxon when\break a given nutrient changes by one unit while other nutrients remain constant. Columns 1--11 represent the selected nutrients: Polyunsaturated fat, Methionine, Sucrose, Animal Protein, Vitamin E-Food Fortification, Maltose, Added Germ from wheats, Choline-Phosphatidylcholine, Taurine, Naringenin-flavanone and Eriodictyol-flavonone. Rows 1--13 represent the selected\break bacteria taxa: Bacteroides, Barnesiella, Odoribacter, Parabacteroides, Prevotella, Alistipes, Coprococcus, Faecalibacterium,\break Oscillibacter, Ruminococcus, Subdoligranulum, Phascolarctobacterium and Parasutterella. The marginal $p$-value\break based on the LRT and the bootstrap selection probability of each of the selected nutrients are also shown}\label{estcoeff} \begin{tabular*}{\tablewidth}{@{\extracolsep{4in minus 4in}}ld{2.2}d{2.2}cd{2.2}d{2.2}d{2.2} d{2.2}d{2.2}d{2.2}d{2.2}@{}} \hline \multicolumn{11}{c}{\textbf{Row: taxon; column: nutrient}}\\ \hline -- &-0.03&-0.08 &0.09 &-0.08 &-0.10 &-0.02 &0.02 &0.10 & \multicolumn{1}{c}{--} &-0.03 \\ $-$0.32 & \multicolumn{1}{c}{--} &-0.33 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &0.22 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c@{}}{--} \\ $-$0.38 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &-0.29 & \multicolumn{1}{c@{}}{--} \\ -- &-0.01 &-0.08 &0.13 &-0.07 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &0.02 &-0.23 & \multicolumn{1}{c@{}}{--} \\ -- & \multicolumn{1}{c}{--} &0.23 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &0.36 &0.63 &-0.72 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c@{}}{--} \\ $-$0.19 &-0.04 & \multicolumn{1}{c}{--} &0.16 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &0.05 \\ -- & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &0.16 \\ -- & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &-0.08 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &0.07 & \multicolumn{1}{c@{}}{--} \\ -- &-0.02 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &-0.10 & \multicolumn{1}{c}{--} & \multicolumn{1}{c@{}}{--} \\ -- & \multicolumn{1}{c}{--} &0.19 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c@{}}{--} \\ -- &0.02 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &-0.12 &-0.12 &0.14 \\ -- & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} &-0.35 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c@{}}{--} \\ $-$0.26 & \multicolumn{1}{c}{--} &-0.29 & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c}{--} & \multicolumn{1}{c@{}}{--} \\ \hline \multicolumn{11}{c}{\textbf{Marginal} $\bolds{p}$\textbf{-value}}\\ \hline $4.5\times10^{-3}$ & \multicolumn{1}{c}{$2.2\times10^{-4}$} & \multicolumn{1}{c}{$8.4\times10^{-4}$} & \multicolumn{1}{c}{$3.6\times10^{-4}$} & \multicolumn{1}{c}{$1.1\times10^{-1}$} & \multicolumn{1}{c}{$6.0\times10^{-3}$} & \multicolumn{1}{c}{$9.5\times10^{-6}$}& \multicolumn{1}{c}{$2.7\times10^{-3}$} & \multicolumn{1}{c}{$5.9\times10^{-3}$} & \multicolumn{1}{c}{$5.8\times10^{-2}$} & \multicolumn{1}{c@{}}{$5.2\times10^{-3}$}\\ \hline \multicolumn{11}{c}{\textbf{Bootstrap selection probability}}\\ \hline 0.50 & 0.93 & 0.72 &0.94&0.35&0.67 & 0.43& 0.58& 0.92& 0.61&0.60\\ \hline \end{tabular*} \end{sidewaystable} We applied the sparse group $\ell_1$ penalized DM regression to this data set. We used the BIC to select the tuning parameters. The final DM model selected 11~nutrients and 13 associated genera. We refit the DM regression model using the selected variables and obtained the maximum likelihood estimates of the coefficients. We compared the fitted counts (total count${}\times{}$fitted proportion) against the observed counts in Figure~\ref{figfit} (top panel). The model fits the data quite well with $r^2=0.79$. Table~\ref{estcoeff} shows the MLEs of the regression coefficients for the selected nutrients and genera. Except for Methionine (second column), the coefficients are not too small.\vadjust{\goodbreak} Since the nutrient measurements are standardized, the exponentiation of a given coefficient can be interpreted as the factor of change in proportion of a taxon when a given nutrient changes by one unit while other nutrients remain constant. The marginal $p$-value based on the LRT for each of the selected nutrients is also shown in this table. Except for Vitamin E and Eriodictyol, these selected nutrients all showed a significant marginal association with the gut microbiome. To further assess the relevance of the nutrients selected, we used the bootstrap to analyze the stability of the selected nutrients [\citet{bach08}]. Specifically, we took 100 bootstrap samples and for each sample we ran our algorithm to select the nutrients. Since some nutrients are highly correlated, we expect that highly correlated nutrients (if the correlation is greater than 0.75) can be selected in different bootstrap samples; we define the bootstrap selection probability of a given nutrient as the number of times that this nutrient or its correlated nutrients were selected. Table \ref {estcoeff} shows the bootstrap probabilities of the nutrients that were selected by the sparse DM regression, indicating quite stable selection of most of the selected microbiome-associated nutrients. Vitamin E had the least stable selection over the 100 bootstrap samples. The identified nutrient-taxon associations are visualized in a bipartite graph shown in Figure~\ref{figcombo}, where the genera and nutrients are depicted with circles and hexagons, respectively. These results further confirmed the findings of \citet{Wu2011a}, where they found the human gut microbiome can be clustered into two enterotypes characterized by Prevotella and Bacteroides, respectively, and the Prevotella enterotype is associated with a high carbohydrate diet while the Bacteroides enterotype is associated with a high protein/fat/choline diet. Figure~\ref{figcombo} shows that two carbohydrates, Maltose and Sucrose, are positively associated with Prevotella and negatively associated with Bacteroides, while animal proteins are positively associated with Bacteroides, Parabacteroides and Alistipes, the three genera mostly enriched in the Bacteroides enterotype. Choline is positively associated with Bacteroides and negatively associated with Prevotella. Polyunsaturated fat is strongly associated with Alistipes, Odoribacter, Barnesiella and Parasutterella, indicating the large effect of fat on the human microbiome. \begin{figure} \includegraphics{592f05.eps} \caption{Association of nutrients with human gut microbial taxa identified by the sparse group $\ell_1$ regularized DM model. We use a bipartite graph to visualize the selected nutrients and their associated genera based on sparse group $\ell_1$ penalized DM regression. Circle: genus; hexagon: nutrient; solid line: positive correlation; dashed line: negative correlation. The thickness of the line represents the association strength.} \label{figcombo} \end{figure} The DM model also identified several other associations that are worth further investigation. For example, we found that Naringenin (flavanone) was positively associated with Faecalibacterium, an anti-inflammatory commensal bacterium identified by gut microbiota analysis of Crohn's disease patients [\citet{Sokol2008}]. If the association is validated, diet with high Naringenin (e.g., Orange, Grapefruit) can be beneficial for patients with Crohn's disease. As a comparison, we also ran the sparse group $\ell_1$ penalized multinomial or Dirichlet regression models and the identified nutrient-genus associations showed significant overlap with those from the DM regression model. However, the interpretability of the DM regression model was the best. To further demonstrate the advantage of the DM model, we simulated taxa counts for each individual based on the fitted models and the observed total taxa counts. The bottom plot of Figure~\ref{figfit} shows that the simulated counts produced by the fitted sparse DM model resemble the observed counts better than those from the sparse multinomial model, where the simulated counts are apparently over-smoothed. This indicates the importance of considering the overdispersion in modeling the gut microbiome data. We also performed the LRT based univariate testing procedure. At FDR${}={}$0.05, the LRT identified 13 nutrients, 8 of which are also identified or highly correlated with the nutrients identified by the sparse group $\ell_1$ penalized DM model. \section{Discussion} \label{secdis} We have proposed a sparse group $\ell_1$ penalized estimation for the DM regression in order to select covariates associated with the microbiome composition. The sparse group $\ell_1$ penalty encourages both group-level and within-group sparsity, with which we can select the relevant taxa associated with the selected covariates. We have performed extensive simulations to evaluate our proposed penalized estimation procedure for both group and within-group selections. We demonstrated the procedure with a real data set on associating nutrient intakes with gut microbiome composition and confirmed the major findings in \citet{Wu2011a}. In our penalized likelihood estimation of the DM model, we use a combination of group $\ell_1$ and individual $\ell_1$ penalties, which result in a convex and separable (in groups of parameters) penalty function. This property facilitates the application of the general coordinate gradient descent method of \citet{Tseng2007} to implement an efficient optimization algorithm. In each iteration, we have a closed form solution for a block update. For a given set of the sparsity tuning parameters, our algorithm is fully automatic and does not require the specification of an algorithmic tuning parameter to ensure convergence. For example, it took about 3 minutes on a standard laptop (Core i5, 2G memory) to finish the analysis of the real data set using an R implementation of the algorithm (available at \url{http://statgene.med.upenn.edu/}). Besides the sparse $l_1$ group penalty, other group penalty functions such as the sup-norm penalty in \citet{zhang-sup} and the composite absolute penalties in \citet{zhao09} can also be used in the setup of the Dirichlet multinomial regression. However, efficient implementation of the optimization problems with these penalty functions is challenging. In microbiome data analysis literature, one commonly used approach is to normalize the counts into proportions and perform statistical analysis using the proportions. However, by converting into the proportions, the variation associated with the multinomial sampling process is lost. In 16S rRNA sequencing, the sequencing depths (total counts) for samples can vary up to 10-fold. Obviously, the accuracy of the proportion estimates under sequencing depth of $500$ reads is very different from that of $10\mbox{,}000$ reads. As shown in our simulations, modeling counts directly can result in gain of power in selecting relevant variables even when the number of sequence reads is very large. Another problem associated with proportions is the existence of numerous zeros in the taxa count data. Many proportion based approaches require taking logarithms of the proportions, which is problematic for the zero proportions. To circumvent this problem, either a pseudo count (e.g., 0.5) is added to these zero counts before converting into proportions or an arbitrary small proportion is substituted for these zero proportions. The effects of creating pseudo counts have not been evaluated thoroughly when the data contain excessive zeros. Besides overdispersion, the taxa count data can also exhibit zero-inflation [\citet{BARRY2002}], where the count data contain more zeros than expected from the DM model. How to model the microbiome count data that allows overdispersion, zero-inflation and possibly the phylogenetic correlations among the taxa is an important future research topic. The multilevel zero-inflated DM regression model for overdispersed count data with extra zeros [\citet{Moghimbeigi2008,Lee2006}] can potentially provide a solution to this problem. Another problem associated with the DM model is its inflexibility in modeling the covariance structure among the taxa counts. The multinomial model for counts compounded by a logistic normal model [\citet{aitchison82}] for proportions provides a possible solution. This needs to be investigated further. \begin{appendix}\label{app} \section*{Appendix} \begin{theorem} \label{theorem1} Letting $\b,{\mathbf{x}}\in\reals^n$, $\lambda_1,\lambda_2,c$ are nonnegative constants and ${\mathbf{x}}^0$ is the minimizer of the following function: \begin{equation} \label{min1} f({\mathbf{x}}) = \tfrac{1}{2}{\mathbf{x}}^T{ \mathbf{x}}+ \b^T{\mathbf{x}}+ c + \lambda_1 \llVert { \mathbf{x}} \rrVert_2 + \lambda_2 \llVert {\mathbf{x}} \rrVert_1, \end{equation} then ${\mathbf{x}}^0_S = \nulll$ and \[ {\mathbf{x}}^0_{\bar{S}} = \arg\min_{{\mathbf{x}}_{\bar{S}}} \bigl\{ \tfrac{1}{2}{\mathbf{x}}^T_{\bar {S}}{\mathbf{x}}_{\bar{S}} + \bigl(\b_{\bar{S}} - \lambda_2 \operatorname{sgn}( \b_{\bar {S}}) \bigr)^T{\mathbf{x}}_{\bar{S}} + c + \lambda_1 \llVert {\mathbf{x}}_{\bar{S}} \rrVert_2 \bigr\}, \] where $S=\{i \in\{1, \ldots,n \} \vert \vert b_i \vert< \lambda_2 \}$ and $\bar{S} = \{1, \ldots,n \} \setminus S$ and $\operatorname{sgn}(\cdot)$ is the sign function. \end{theorem} \begin{pf} We prove ${\mathbf{x}}^0_S = \nulll$ by contradiction. If $x^0_i \neq0$ $(i \in S)$, then we can construct a new ${\mathbf{x}}^1$ with $x^1_i=0$ and other components being the same as ${\mathbf{x}}^0$. Clearly, $\frac{1}{2}{{\mathbf {x}}^1}^T{\mathbf{x}}^1 + \b^T{\mathbf{x}}^1 + c + \lambda_2 \llVert {\mathbf{x}}^1 \rrVert_1 < \frac {1}{2}{{\mathbf {x}}^0}^T{\mathbf{x}}^0 + \b^T{\mathbf{x}}^0 + c + \lambda_2 \llVert {\mathbf{x}}^0 \rrVert_1$ and $\lambda_1 \llVert {\mathbf{x}}^1 \rrVert_2 < \lambda_1 \llVert {\mathbf{x}}^0 \rrVert_2$. The former is due to the fact that $\frac{1}{2}(x_i^0)^2 + b_ix_i^0 + \lambda_2 \vert x_i^0 \vert> 0$ for $\vert b_i \vert< \lambda_2$. Hence, ${\mathbf{x}}^0$ is not the minimizer of $f({\mathbf{x}})$, which is contradictory. Therefore, ${\mathbf {x}}^0_S = \nulll$. To prove the second part, we note that $x_i^0$ must be either $0$ or have an opposite sign of $b_i$ for $i \in\{1,\ldots, n\}$. So the minimization of $\f({\mathbf{x}})$ is equivalent to minimizing \[ f^*({\mathbf{x}}) = \tfrac{1}{2}{\mathbf{x}}^T{\mathbf{x}}+ \bigl(\b- \lambda_2\operatorname{sgn}(\b)\bigr)^T{ \mathbf{x}}+ c + \lambda_1 \llVert {\mathbf{x}} \rrVert_2, \] subject to \[ \operatorname{sgn}(x_i) = -\operatorname{sgn}(b_i) \quad\mbox{or}\quad x_i=0. \] Since ${\mathbf{x}}^0_S = \nulll$, we can restrict the minimization over only ${\mathbf{x}}_{\bar{S}}$, \begin{equation} \label{min2} f^*({\mathbf{x}}_{\bar{S}}) = \tfrac{1}{2}{ \mathbf{x}}_{\bar {S}}^T{\mathbf{x}}_{\bar{S}} + \bigl( \b_{\bar {S}}- \lambda_2\operatorname{sgn}(\b_{\bar{S}}) \bigr)^T{\mathbf{x}}_{\bar {S}} + c + \lambda_1 \llVert {\mathbf{x}}_{\bar{S}} \rrVert_2, \end{equation} subject to \[ \operatorname{sgn}(x_i) = -\operatorname{sgn}(b_i) \quad\mbox{or}\quad x_i=0\qquad (i \in \bar{S}). \] Since ${\mathbf{x}}^0_{\bar{S}}$ is the minimizer of $f^*({\mathbf {x}}_{\bar{S}})$ without the constraint, the sign of ${\mathbf{x}}^0_{\bar{S}}$ should be the opposite of the sign of $(\b_{\bar{S}}- \lambda_2\operatorname{sgn}(\b_{\bar{S}}))$. Because $\vert b_i \vert\ge\lambda_2$ for $i \in\bar{S}$, the sign of $(\b_{\bar{S}}- \lambda_2\operatorname{sgn}(\b_{\bar{S}}))$ is the same as $\b_{\bar{S}}$. So the sign of ${\mathbf{x}}^0_{\bar{S}}$ is the opposite of that of $\b_{\bar{S}}$. Therefore, ${\mathbf{x}}^0_{\bar{S}}$ satisfies the constraint.\vspace*{-2pt} \end{pf} Using simple variable substitution, we have the following corollary. \begin{corollary} \label{cor1} Letting $\b,\bbeta,\d\in\reals^n$, $\lambda_1,\lambda_2,c$ are nonnegative constants and $\d^0$ is the minimizer of the following function, \begin{equation} f(\d) = \tfrac{1}{2}\d^T\d+ \b^T\d+ c + \lambda_1 \llVert \bbeta+\d\rrVert_2 + \lambda_2 \llVert \bbeta+\d\rrVert_1, \end{equation} then $\d^0_S = -\bbeta_S$ and \[ \d^0_{\bar{S}} = \arg\min_{\d_{\bar{S}}} \biggl\{ \frac{1}{2}\d^T_{\bar {S}}\d_{\bar{S}} + \bigl( \b_{\bar{S}} - \lambda_2 \operatorname{sgn}(\b_{\bar {S}}- \bbeta_{\bar{S}}) \bigr)^T\d_{\bar{S}} + c + \lambda_1 \llVert \d_{\bar{S}}+\bbeta_{\bar{S}} \rrVert_2 \biggr\}, \] where $S=\{i \in\{1, \ldots,n \} \vert \vert b_i - \beta_i\vert< \lambda_2 \}$, $\bar{S} = \{1, \ldots,n \} \setminus S$ and $\operatorname{sgn}(\cdot)$ is the sign function.\vspace*{-2pt} \end{corollary} \end{appendix} \section*{Acknowledgments} We thank Doctors Rick Bushman, James Lewis and Gary Wu for providing the data and for many insightful discussions. We also thank Professor Karen Kafadar, an Associate Editor and two reviewers for many helpful comments.

proofpile-arXiv_069-5818

\section*{Introduction} \label{sec:introduction} \subsection*{General settings and requirements} \label{sec:gener-sett-requ} In this paper we introduce a probabilistic framework for a simple asynchronous system distributed over two sites, based on the trace semantics of concurrency. Consider a communicating system consisting of two subsystems, called site $1$ and site~$2$, that need to synchronize with one another from time to time, for example for message exchange. Intended applications are, for instance, simple client-server situations, device-device driver interactions, communication bridge between two asynchronous networks. The synchronization is modeled, for each site, by the fact that the concerned subsystem is entering some \emph{synchronizing state}, corresponding to a synchronization task---there shall be several synchronization states corresponding to different tasks. It is natural to consider that the synchronization states are shared: both subsystems are supposed to enter together into a shared synchronization state. Beside synchronization states, we assume that each subsystem may evolve between other states that concern the local activity of each subsystem, and seen as \emph{private states}. Hence we consider for each site $i=1,2$ some finite set of states~$S^i$, with the intended feature that $Q=S^1\cap S^2$ is a nonempty set of synchronization states. Whenever the two subsystems enter some of their private states, the corresponding events are said to be \emph{concurrent}. It is natural to consider that the private activity of a given site should not influence the private activity of the other site. This ought to be reflected by some kind of statistical independence in the probabilistic modeling. Another feature that we are seeking is that the \emph{local} time scales of private activities do not need to be synchronous. Indeed, the local time scale of each subsystem might be driven for instance by the input of a user, by the arrival of network events, or by its internal chipset clock; therefore, it is realistic \emph{not} to assume any correlation between local time scales, but for synchronization. In particular, in a discrete time setting, the synchronization instants counted on the two different local time scales shall not need to be equal, making the two subsystems \emph{asynchronous}. \subsection*{Sequential probabilistic systems and concurrency} \label{sec:limit-sequ-prob} Classically, Markov chains in either discrete or continuous time are a popular model for adding a probabilistic layer to describe the evolution of a transition system. Since the Markov chain formalism is intrinsically sequential, its straightforward application to a concurrent system brings the issue of translating a concurrent system into a sequential one. A solution to this issue, found in the Probabilistic Automata literature for instance \cite{segala95,lynch03}, is the introduction of a non deterministic scheduler in charge of deciding which subsystem is about to run at each time instant. This defines a Markov Decision Process, a model introduced earlier for control issues in~\cite{bellman57}. Other ways of composing probabilistic systems to form a Markov process, with or without non determinism, are usually based on Milner's CCS \cite{milner89} or Hoare's CSP~\cite{hoare85}, where the synchronization policy for possibly synchronizing processes is either to allow or to force synchronization. In \cite{argenio99} for instance, where both synchronization methods \`a la CSS and \`a la CSP are encoded in the model of bundle probabilistic transition systems, renormalization occurs at each step to take into account the selected synchronization paradigm. \subsection*{Probabilistic trace semantics. Lattice of trajectories} \label{sec:prob-trace-semant} We introduce another way of randomizing our simple concurrent system. We first accept as a basic fact that modeling the evolution of a system as ordered paths of events jeopardizes the concurrency feature of the model. Adopting instead the so-called trace semantics for concurrency (or partial order semantics)~\cite{nie80,nie95}, lattices replace ordered paths to model trajectories. Unordered events of a trajectory are then intrinsically concurrent. This raises a question on the probabilistic side: which part of Markov chain theory can we rebuild on this new basis? The aim of this paper is to provide an answer to the question. Our work is thus largely inspired by Markov chain theory; but we try to adapt the theory to the partial order semantics of concurrency, instead of directly turning a concurrent system into a Markov chain (or a variant of it) as in~\cite{lynch03,argenio99}. Let us be precise about what we mean in this paper by a partial order semantics for concurrency, referring to the two sets of local states $S^1$ and~$S^2$ with synchronization constraint $Q=S^1\cap S^2$. We will then explain how probability concepts apply in this setting. If two sequences of states in $S^1\cup S^2$ only differ by the interleaving order of private states of different sites, such as $a\cdot e$ and $e\cdot a$ with $a\in S^1\setminus Q$ and $e\in S^2\setminus Q$, the trace semantics suggests to simply identify them: $a\cdot e\equiv e\cdot a$. Propagating this identification to sequences of events of arbitrary length, we obtain an equivalence relation on sequences. Sequences that cannot be permuted are those of the form $x\cdot y$ with $x,y\in S^1$ or $x,y\in S^2$, which include those of the form $x\cdot \c$ with $\c\in Q$ and any $x\in S^1\cup S^2$. We adopt a simple representation for equivalence classes of sequences by mapping each equivalence class to a \emph{pair} of sequences, where each coordinate is reserved for a given site; synchronization states appear in both coordinates. Hence the equivalence class of $a\cdot e\equiv e\cdot a$ is mapped to $(a,e)$, the equivalence class of $a\cdot e\cdot\c\equiv e\cdot a\cdot \c$ is mapped to $(a\cdot\c,e\cdot \c)$. We define thus a \emph{trajectory} as a pair $(s^1,s^2)$, where $s^i$ is a sequence of elements in~$S^i$\,, and such that the sequences of synchronization states extracted from $s^1$ and from~$s^2$, and taken in their order of appearance, shall be \emph{equal}. An \emph{infinite trajectory} is defined as a trajectory $\omega=(\omega^1,\omega^2)$ where both sequences $\omega^1$ and $\omega^2$ are infinite. So for example, if $S^1=\{a, b, \bm c, \bm d\}$ and $S^2=\{\bm c,\bm d,e,f\}$, and thus \mbox{$Q=\{\bm c,\bm d\}$}, an infinite trajectory could be $\omega=(\omega^1,\omega^2)$ with $\omega^1$ and $\omega^2$ starting as follows: \mbox{$\omega^1=a\cdot \bm c\cdot b\cdot a\cdot b\cdot b\cdot \bm d\cdot(\cdots)$} and \mbox{$\omega^2=e\cdot f\cdot \bm c\cdot f\cdot \bm d\cdot(\cdots)$}. The common extracted sequence of synchronization states starts in this example with $\bm c\cdot\bm d$. Note the important feature that each local trajectory $\omega^i$ is permitted to have a free evolution between synchronizations: synchronizations occur at instants $2$ and $7$ for~$\omega^1$, while they occur at instants $3$ and $5$ for~$\omega^2$; here, the instants of synchronization are relative to the \emph{local} time scales. The set $\Omega$ of infinite trajectories is the natural sample space to put a probability measure on. There is a natural notion of \emph{subtrajectory}: in the previous example, $v=(a\cdot\c,e\cdot f\cdot\c)$ is a finite subtrajectory of $\omega=(\omega^1,\omega^2)$. ``Being a subtrajectory'' defines a binary relation that equips subtrajectories of a given trajectory with a \emph{lattice} structure. For instance, and denoting by $\epsilon$ the empty word, the subtrajectories of $v$ are: $(\epsilon,\epsilon)$, $(a,\epsilon)$, $(\epsilon,e)$, $(\epsilon,e\cdot f)$, $(a,e)$, $(a,e\cdot f)$ and $(a\cdot\c,e\cdot f\cdot\c)$. Their lattice is depicted in Figure~\ref{fig:qsdqsjpa}. Observe that, for a given trajectory, its subtrajectories are naturally identified with two-components ``time instants''. In case of~$v$, these time instants are $(0,0)$, $(1,0)$, $(0,1)$, $(0,2)$, $(1,1)$, $(1,2)$ and~$(2,3)$, and they form a sublattice of the lattice $\mathbb{N}\times \mathbb{N}$. However, even if one considers an infinite trajectory~$\omega$, the associated lattice of two-components time instants is only a \emph{sub}lattice of $\mathbb{N}\times\mathbb{N}$ in general. For instance, if $\zeta$ is any infinite trajectory that has $v$ as subtrajectory, then $(2,2)$ is a time instant that does not correspond to any subtrajectory of~$\zeta$, because of the synchronization on state~$\c$. \begin{figure} \centerline{ \xymatrix{&(a,\epsilon)\ar[dr]\\ (\epsilon,\epsilon)\ar[ur]\ar[dr]&&(a,e)\ar[r]&(a,e\cdot f)\ar[r]&(a\cdot\c,e\cdot f\cdot\c) \\ &(\epsilon,e)\ar[ur]\ar[r]&(\epsilon,e\cdot f)\ar[ur] }} \caption{\textsl{Lattice of subtrajectories of $v=(a\cdot\c,e\cdot f\cdot\c)$.}} \label{fig:qsdqsjpa} \end{figure} Obviously, considering another trajectory $\omega'$ would lead to another lattice of subtrajectories, not necessarily isomorphic to the one associated with~$\omega$. We sum up the previous observations by saying that time is \emph{partially ordered} on the one hand, since time instants form a lattice and not a total order, and \emph{random} on the other hand, since the lattice structure depends on the trajectory considered, that is, on the execution of the system. \bigskip\bigskip \subsection*{Defining M2CP: absence of transition matrix} \label{sec:absence-trans-matr} This has consequences for the way one may construct a probability measure on the space $\Omega$ of infinite trajectories. Consider again the finite trajectory encountered above, $v=(a\cdot\c,e\cdot f\cdot\c)$. The occurrences of $a$ on site~$1$, and of $e$ on site~$2$, are \emph{concurrent}. Trying to determine the precise interleaving of $a$ and $e$ is irrelevant for us. This desired feature prevents us from applying the standard recursive construction to assign a probability to trajectory~$v$ (that is: the probability that $v$ occurs as a subtrajectory of a sample infinite trajectory~$\omega$): starting from the initial state, there is no obvious choice between $a$ and~$e$; which one should be first plugged in the probability computation? Therefore the lattice structure of trajectories implies to give up, at least temporarily, the familiar inductive computation of probabilities based on transition matrices. Nevertheless, two important notions can be defined in the asynchronous framework by analogy with Markov chain theory: first, the notion of state reached by (``after'') a finite trajectory (\S~\ref{sec:general-framework-1}); second, the probabilistic evolution of the system ``after'' execution of a finite trajectory (Definition~\ref{def:2} in \S~\ref{sec:prob-two-comp}). We define a \emph{Markov two-components process} ({\normalfont M2CP}) as a random system where the probabilistic future after execution of a finite trajectory $v$ only depends on the state reached by~$v$. \bigskip\bigskip \subsection*{Stopping times for M2CP} \label{sec:stopping-times-m2cp} Recall that a stopping time in Markov chain theory identifies with a random halting procedure that does not need anticipation: an observer can decide whether the stopping time has been reached based on the only knowledge of the process history at each step. Stopping times are a basic tool in Markov chain theory. Important notions such as the first return time to a state, recurrent and transient states are defined by means of stopping times. Stopping times are manipulated with the help of the Strong Markov property, a central result in Markov chain theory. We show that several aspects of the Markovian language carry over to the asynchronous framework. Once an adequate notion of stopping time for asynchronous probabilistic processes has been defined (Definition~\ref{def:6} in~\S~\ref{sec:stopping-times}), derived notions such as the first hitting time to a state, and the notions of recurrent and transient states follow by almost literally translating the original ones into the asynchronous language. We show that the Strong Markov property also has an equivalent, called the Asynchronous Strong Markov property, which serves as a basic tool for probabilistic reasoning. Some other notions translate in a more subtle way: the first reaching time of a \emph{set} of states needs some additional care, since the lattice structure of trajectories prevents a straightforward generalization of the analogous notion from Markov chain theory, providing an interesting difference with Markov chain theory. Irreducible processes have an equivalent counterpart in the asynchronous framework, and we detail the decomposition of a {\normalfont M2CP}\ into irreducible components. \subsection*{The Local Independence Property} \label{sec:local-indep-prop} Therefore, we have on the one hand these notions obtained as a generalization of analogous notions from Markov chain theory to the asynchronous framework. But on the other hand, we also have other notions specific to the asynchronous framework, and that would not make sense for Markov chains. In particular the way the two local components behave with respect to one another is a question specific to the asynchronous framework. Since the two local components synchronize with one another, they cannot be fully independent in the probabilistic sense. There is however a weaker notion of independence in probability theory, adapted to our purpose, which is \emph{conditional} independence. We call Local Independence Property (LIP) the property that the two components are independent conditionally to their synchronization constraint. Informally, the LIP says that the local components have the maximal independence that they can have, up to their synchronization constraint. We characterize {\normalfont M2CP}\ with the LIP by a finite family of transition matrices; and we show how to construct a {\normalfont M2CP}\ from an adapted family of such transition matrices. The finite collection of numbers this family of matrices defines is an equivalent, in the asynchronous framework, of the transition matrix for a Markov chain. \subsection*{Synchronization of systems} \label{sec:synchr-syst} The composition of probabilistic systems has always been a challenge, with multiple applications in the theory of network analysis \cite{argenio99,lynch03,baccelli92:_synch_linear}. The main limitation of the theory of probabilistic event structures as it has been developed so far by the author together with A.~Benveniste in~\cite{abbes08,abbes06a} (another probabilistic model with trace semantics targeting applications to probabilistic $1$-safe Petri nets), and by other authors in \cite{varacca04:_probab} is the non ability to define a suitable synchronization product. This very limitation has motivated the development of the present framework, by starting with the definition of a synchronization product for two Markov chains. By recursively ``forcing'' their synchronization, it is shown in this paper how the synchronization of two Markov chains on shared common states naturally leads to a {\normalfont M2CP}. Even if one was interested in this construction only (the author is aware of current work on this kind of \textit{a priori} model, simply because it was the only one people could think of), including it inside a more general picture as it is done in this paper is useful to better understand its properties. \subsection*{Organization of the paper} \label{sec:organization-paper} We describe the model in~\S~\ref{sec:prob-proc-mark}, defining a general notion of probabilistic two-components process, and then specializing to \emph{Markov} two-components process es. In~\S~\ref{sec:synchr-two-mark} we introduce the synchronization product of two Markov chains. This construction provides an example of {\normalfont M2CP}, intended to support the intuition for {\normalfont M2CP}\ in general. Next section, \S~\ref{sec:stopp-times-strong}, is devoted to Markovian concepts in the asynchronous framework, centered around the Asynchronous Strong Markov property. We introduce recurrence and transience of states and the decomposition of {\normalfont M2CP}\ into irreducible components. The new notions of closed and open processes are studied in this section, as well as the definition of stopping times for asynchronous processes. The Local Independence Property (LIP) is the topic of~\S~\ref{sec:prop-mark-two}, and it is shown that the synchronization of Markov chains introduced in \S~\ref{sec:synchr-two-mark} satisfies the LIP. Finally, \S~\ref{sec:gener-constr-mark}~is devoted to the construction and characterization of general {\normalfont M2CP}\ with the LIP. A concluding section introduces directions for future work. It discusses limitations imposed by the two-components hypothesis, and possible ways to remove this limitating hypothesis. \section{Probabilistic Processes and Markov Processes on Two Sites} \label{sec:prob-proc-mark} \subsection{General Framework} \label{sec:general-framework-1} A \tdefine{distributed system} is given by a pair $(S^1,S^2)$, where $S^i$ for $i=1,2$ is a finite set, called the set of \tdefine{local states} of site~$i$. A \tdefine{local trajectory} attached to site $i$ is a sequence of local states of this site. For $i=1,2$, we denote by $\Omega^i$ the set of infinite local trajectories attached to site~$i$. The two local state sets $S^1$ and $S^2$ are intended to have a non empty intersection, otherwise the theory has little interest. We put $Q=S^1\cap S^2$. Elements of $Q$ are called \tdefine{common states} or \tdefine{shared states}. In contrast, states in $S^i\setminus Q$ are said to be \tdefine{private to site~$\bm i$}, for $i=1,2$. From now on, \textbf{we will always assume that $\bm S^{\bm i}\bm\setminus\bm Q\bm\neq\bm\emptyset$} for $i=1,2$: each site has at least one private state. This is a convenient technical assumption; removing it would not harm if needed. Given a sequence $(x_j)_j$ of elements in a set~$S$, either finite or infinite, and given a subset $A\subseteq S$, the \tdefine{$\bm A$-sequence induced by $\bm(\bm x_{\bm j}\bm)_{\bm j}$} is defined as the sequence of elements of $A$ encountered by the sequence~$(x_j)_j$\,, in their order of appearance. Given two local trajectories $\seq {x^1}n$ and $\seq{x^2}n$ on sites $1$ and $2$ respectively, we will say that they \tdefine{synchronize} if the two $Q$-sequences they induce are \emph{equal}. A pair of two synchronizing local trajectories will be called \tdefine{a global trajectory}, or simply a \tdefine{trajectory} for brevity. Among them, \tdefine{finite trajectories} are those whose components are both finite sequences of states. Trajectories are ordered component by component: if $s=(s^1,s^2)$ and $t=(t^1,t^2)$ are two trajectories, we define $s\leq t$ if $s^1\leq t^1$ and $s^2\leq t^2$, where the order on sequences is the usual prefix order. The resulting binary relation on trajectories is a partial order, the maximal elements of which are exactly those whose components are both infinite: this relies on the fact that $S^i\setminus Q\neq\emptyset$ for $i=1,2$ (for instance, if $S^1=\{a,\bm b\}$ and $S^2=\{\bm b\}$ so that $Q=\{\bm b\}$ and $S^2\setminus Q=\emptyset$, then $(\bm baaa\cdots,\bm b)$ is maximal, but the second component is finite). The set of maximal trajectories is denoted by~$\Omega$, and we have that $\Omega\subseteq\Omega^1\times\Omega^2$. For $s$ a \emph{finite} trajectory, the subset of $\Omega$ defined by \begin{equation} \label{eq:11} \uparrow s=\{\omega\in\Omega\;|\; s\leq\omega\} \end{equation} is called the \tdefine{elementary cylinder} of base~$s$---adapting a standard notion from Measure theory to our framework. Given any trajectory $s=(s^1,s^2)$, the \tdefine{subtrajectories of~$\bm s$} are those trajectories $t$ such that $t\leq s$. Observe that not any prefix $t$ of $s$ is a subtrajectory; since $t$ could very well not be a trajectory itself. Given a trajectory $(s^1,s^2)$, we denote by $(y_j)_j$ the $Q$-sequence induced by both sequences $s^1$ and~$s^2$\,. It can be finite or infinite, even empty. We refer to $(y_j)_j$ as to the \tdefine{$\bm Q$-sequence induced} by $ (s^1,s^2)$. A \tdefine{global state} is any pair $\alpha=(x^1,x^2)\in S^1\times S^2$. We reserve the letters $\alpha$ and $\beta$ to denote global states. Observe that trajectories are \emph{not} defined as sequences of global states; since the length of the two components may very well differ. Let $\alpha=(x,y)$ be some fixed global state, thought of as the \emph{initial} state of the system. If $s=(s^1,s^2)$ is a finite trajectory, we define \begin{equation} \label{eq:2} \gamma_\alpha(s)=(x^1,x^2)\in S^1\times S^2 \end{equation} as the pair of last states of the two sequences $x\cdot s^1$ and~$y\cdot s^2$\,. We understand $\gamma_\alpha(s)$ as the current global state after the execution of finite trajectory~$s$, starting from~$\alpha$. Note that, with this definition, $\gamma_\alpha$~is well defined on the empty sequence and $\gamma_\alpha(\emptyset)=\alpha$. By an abuse of notation, we will omit $\alpha$ and write $\gamma$ instead of~$\gamma_\alpha$\,, the context making clear which initial state $\alpha$ we refer to. We introduce a notion of length for trajectories. We denote by $\mathcal{T}$ the set \begin{equation*} \mathcal{T}=\bigl(\mathbb{N}\times\mathbb{N})\cup\{\infty\}\,. \end{equation*} The set $\mathcal{T}$ is partially ordered component by component, with the natural order on each component, and $(m,n)\leq\infty$ for all $(m,n)\in\mathbb{N}\times\mathbb{N}$. If $s=(s^1,s^2)$ is any trajectory, the \textbf{length of~$\bm s$} is defined by \begin{equation*} |s|= \begin{cases} (|s^1|,|s^2|)\in\mathcal{T},&\text{if $s$ is finite,}\\ \infty,&\text{otherwise,} \end{cases} \end{equation*} where $|s^1|$ and $|s^2|$ denote the length of sequences. Roughly speaking, lengths can be thought of as time instants; it becomes then clear that time is only partially ordered, and not totally ordered---see random times in \S~\ref{sec:stopping-times} for a finer notion. There is a \tdefine{concatenation} operation partially defined on trajectories. If $s=(s^1,s^2)$ is a \emph{finite trajectory}, and $t=(t^1,t^2)$ is any trajectory, then the concatenation denoted by $s\cdot t$ and defined by $s\cdot t=(s^1\cdot t^1,s^2\cdot t^2)$ is obviously a trajectory. If $t\in\Omega$, then $s\cdot t\in\Omega$ as well. There is an obvious \tdefine{addition} on lengths, compatible with concatenation of finite trajectories, in the sense that $|s\cdot t|=|s|+|t|$. If we fix~$s$, the concatenation defines a bijection onto the cylinder of base~$s$: \begin{equation} \label{eq:1} \Phi_s: \begin{cases} \Omega\to\uparrow s\\ \omega\mapsto\Phi_s(\omega)= s\cdot \omega\,. \end{cases} \end{equation} \subsection{Trajectory Structure} \label{sec:struct-traj} The fact that we consider only two sites allows to precisely describe the structure of trajectories. \begin{defi} \label{def:5} \ \begin{enumerate}[(1)] \item An \tdefine{elementary trajectory} is a finite trajectory with a unique synchronization, that occurs at its end. Equivalently, a finite trajectory $s$ is elementary if $\gamma(s)=(x,x)$ for some $x\in Q$, and $(x)$ is the $Q$-sequence induced by~$s$. \item We say that a trajectory is \tdefine{synchronization free} if its associated $Q$-sequence is empty. \end{enumerate} \end{defi} \noindent We omit the proof of the following proposition, which is elementary, but fundamental for some constructions introduced later in~\S~\ref{sec:synchr-two-mark} and in~\S~\ref{sec:gener-constr-mark}. \begin{prop} \label{prop:1} \ \begin{enumerate}[\em(1)] \item\label{item:10} Any finite trajectory has a unique decomposition as a concatenation of elementary trajectories, followed by a synchronization free trajectory. \item\label{item:11} Any maximal trajectory is either, according to its $Q$-sequence being infinite or finite: \begin{enumerate}[\em(a)] \item\label{item:1} A countable infinite concatenation of elementary trajectories, and the decomposition as such a concatenation is unique; or \item\label{item:2} A finite concatenation of elementary trajectories, followed by a synchronization free trajectory, infinite on both sides. This decomposition is unique. \end{enumerate} \end{enumerate} \end{prop} \begin{figure} \centering \centering \mbox{\textbf{Case~2a}\quad\xymatrix@1@C=2cm@M=1ex@R=3em{ {\rule{0em}{2em}}\POS!U(2)\drop{\bullet}\ar@/^2em/[r]!D(1.5) \POS!D(20)\drop{\bullet} \ar@/_2em/[r]!U(1) & \fbox{\makebox[.5em]{$\begin{array}{c} \bullet\\[-.7em]\bullet\end{array}$}} \ar@/^2em/[r]!D(.5) \ar@/_3em/[r]!U(.5) & \fbox{\makebox[.5em]{$\begin{array}{c} \bullet\\[-.7em]\bullet\end{array}$}} \ar@/^3em/[r]!D(.5) \ar@/_2em/[r]!U(.5) & \fbox{\makebox[.5em]{$\begin{array}{c} \bullet\\[-.7em]\bullet\end{array}$}} &\rule{1em}{0em}\POS!L(4)\drop{\makebox[2cm]{\dotfill}} }}\par\bigskip\bigskip\bigskip \mbox{\textbf{Case~2b}\quad\xymatrix@1@C=2cm@M=1ex@R=3em{ {\rule{0em}{2em}}\POS!U(1)\drop{\bullet}\ar@/^2em/[r]!D(1.5) \POS!D(15)\drop{\bullet} \ar@/_2em/[r]!U(1) & \fbox{\makebox[.5em]{$\begin{array}{c} \bullet\\[-.7em]\bullet\end{array}$}} \ar@/^2em/[r]!D(.5) \ar@/_3em/[r]!U(.5) & \fbox{\makebox[.5em]{$\begin{array}{c} \bullet\\[-.7em]\bullet\end{array}$}} \ar@/^3em/[r]!D(.5) \ar@/_2em/[r]!U(.5) & \fbox{\makebox[.5em]{$\begin{array}{c} \bullet\\[-.7em]\bullet\end{array}$}}\POS!U(.5)\ar@/_1em/[r]!U(4.5) \POS!D(.5)\ar@/^1em/[r]!D(4.5) &\rule{1em}{0em} }}% \par\vspace{2em} \caption{\textsl{Illustration of the decomposition of a maximal trajectory, according to Proposition~{\normalfont\ref{prop:1}}, Cases~{\normalfont\ref{item:1}} and~{\normalfont\ref{item:2}}. The framed boxes represent the synchronizations, the arrows represent the private paths. In Case~{\normalfont\ref{item:1}}, the synchronization pattern keeps repeating on the right.}} \label{fig:gt} \end{figure} \noindent Figure~\ref{fig:gt} depicts the decomposition of global trajectories in cases~\ref{item:1} and~\ref{item:2}. Finally, the following lemma will be useful. \begin{lem} \label{lem:5} For any trajectory $v$, the set of subtrajectories of $v$ is a well founded and complete lattice. Lower and upper bounds are taken component by component. \end{lem} \proof Let $v=(s^1,s^2)$, and let $\mathcal{I}(s^i)$ denote, for $i=1,2$, the set of initial subsequences of~$s^i$\,. It is well known that $\mathcal{I}(s^i)$ is a total and well-founded order with arbitrary \emph{lub}s (least upper bounds). Therefore the component-wise order on $\mathcal{I}(s^1)\times\mathcal{I}(s^2)$ is a complete lattice, with lower and upper bounds taken component by component. To prove the lemma, it suffices thus to check that the component-wise upper and lower bounds of subtrajectories of $v$ yield again subtrajectories of~$v$, and this is obvious, hence we are done.\qed \subsection{Probabilistic Two-Components Processes} \label{sec:prob-two-comp} Although time has been abstracted from the framework, the notion of trajectory is still present; this is all we need to introduce a probabilistic layer. We consider the \mbox{$\sigma$-alge}\-bra\ $\mathfrak{F}$ on $\Omega$ generated by the countable family of elementary cylinders, defined above in Eq.~(\ref{eq:11}). The \mbox{$\sigma$-alge}\-bra\ $\mathfrak{F}$ coincides with the trace on $\Omega$ of the product \mbox{$\sigma$-alge}\-bra\ on the infinite product $\Omega^1\times\Omega^2=(S^1\times S^2)^\mathbb{N}$\,, where of course~$S^i$, as a finite set for $i=1,2$, is equipped with the discrete \mbox{$\sigma$-alge}\-bra. Unless stated otherwise, the set $\Omega$ will be equipped with the \mbox{$\sigma$-alge}\-bra~$\mathfrak{F}$. Assume thus that $\mathbf{P}$ is a probability defined on~$\Omega$. By an abuse of notation, if $s$ is a finite trajectory we simply denote by $\mathbf{P}(s)$ the probability of the elementary cylinder of base~$s$, so that $\mathbf{P}(s)=\mathbf{P}(\uparrow s)$. We say that a global state $\alpha$ is \tdefine{reachable} w.r.t.~$\mathbf{P}$ if there exists a finite trajectory $s$ such that $\mathbf{P}(s)>0$ and $\alpha=\gamma(s)$. A probabilistic two-components process\ on a distributed system is defined as follows. \begin{defi} \label{def:1}\ \begin{enumerate}[(1)] \item A \tdefine{probabilistic two-components process}, or \tdefine{probabilistic process} for brevity, is a family $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ of probability measures on $\Omega$ indexed by a set $X_0$ of global states, and satisfying the following property: for all \mbox{$\alpha\in X_0$}, if $\beta$ is reachable with respect to~$\mathbf{P}_\alpha$\,, then \mbox{$\beta\in X_0$}\,. \item If $\beta$ is reachable w.r.t.~$\mathbf{P}_\alpha$\,, we say that $\beta$ is \tdefine{reachable from~$\bm \alpha$}. \item A \tdefine{subprocess} of a probabilistic process $\mathbb{P}$ is a subfamily $(\mathbf{P}_\alpha)_{\alpha\in X_1}$\,, with $X_1\subset X_0$, that forms a probabilistic process. \end{enumerate} \end{defi} \noindent The probability $\mathbf{P}_\alpha$ is intended to describe the probabilistic behavior of the system starting from~$\alpha$. However, for technical reasons that will appear later, we consider the evolution of the system \emph{after}~$\alpha$. In other words, we assume that $\alpha$ has already been reached, and we put ourselves just after it. In particular, we do \emph{not} assume that $\mathbf{P}_\alpha(\uparrow\alpha)=1$, contrary to the usual convention adopted in Markov chain theory. \begin{defi} \label{def:17} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a probabilistic two-components process. Let also $*$ be an arbitrary specified value not in $S^1\cup S^2$. \begin{enumerate}[(1)] \item For $\omega\in\Omega$, we denote by $Y(\omega)=\bigl(Y_n(\omega)\bigr)_{n\geq0}$ the $Q$-sequence induced by~$\omega$, followed by the constant value $*$ if the $Q$-sequence is finite. In all cases, we also put $Y_{-1}=*$. We refer to $Y$ as to the \tdefine{(random) synchronization sequence.} \item We say that $\omega\in\Omega$ \tdefine{synchronizes infinitely often} if $Y_n(\omega)\neq *$ for all $n\geq0$. \item We say that $\mathbb{P}$ is \tdefine{closed} if for all $\alpha\in X_0$\,, $Y_n\neq*$ for all $n\geq0$ and\/ $\mathbf{P}_\alpha$-\text{a.s.} \item We say that\/ $\mathbb{P}$ is \tdefine{open} if for all $\alpha\in X_0$\,, $Y_n=*$ for all $n\geq0$ and\/ $\mathbf{P}_\alpha$-\text{a.s.} \end{enumerate} \end{defi} \medskip\noindent Consider any probability measure $\mathbf{P}$ on~$\Omega$, and let $s$ be a finite trajectory. Observe that $\Phi_s:\Omega\to\uparrow s$ is not only a bijection, it is also bi-measurable. Considering the action of $\Phi_s^{-1}$ on measures is thus meaningful. In particular, if $\mathbf{P}(s)>0$, we define the probability $\mathbf{P}_s$ on $\Omega$ as the image of the conditional probability $\mathbf{P}(\,\cdot\,|\uparrow s)$. It satisfies, and is characterized by the relations $\mathbf{P}_s(t)=\frac1{\mathbf{P}(s)}\mathbf{P}(s\cdot t)$, for $t$ ranging over the set of finite trajectories. \begin{defi} \label{def:2} If\/ $\mathbf{P}$ is a probability measure on\/~$\Omega$, and if $s$ is a finite trajectory such that\/ $\mathbf{P}(s)>0$, we define the probability measure $\mathbf{P}_s$ on\/ $\Omega$ characterized by: \begin{equation} \label{eq:3} \mathbf{P}_s(t)=\frac1{\mathbf{P}(s)}\mathbf{P}(s\cdot t), \end{equation} for $t$ ranging over the set of finite trajectories, as the \tdefine{probabilistic future} of $s$ w.r.t. probability\/~$\mathbf{P}$. \end{defi} Markov two-components process es can now be defined as follows, without reference to any explicit notion of time. \begin{defi} \label{def:3} Given a distributed system, a \tdefine{Markov two-components process}, abbreviated\/ \tdefine{{\normalfont M2CP}}, is defined as a probabilistic process\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ over this system, satisfying the following property: for $\alpha$ ranging over $X_0$ and $s$ ranging over the set of finite trajectories such that\/ $\mathbf{P}_\alpha(s)>0$, the probabilistic future of trajectory $s$ w.r.t.~$\mathbf{P}_\alpha$ only depends on~$\gamma(s)$. This is equivalent to saying: \begin{equation} \label{eq:4} \forall\alpha\in X_0\quad \forall s\quad \mathbf{P}_\alpha(s)>0\Rightarrow \bigl(\mathbf{P}_\alpha\bigr)_s=\mathbf{P}_{\gamma(s)}\,. \end{equation} \end{defi} Equation~\eqref{eq:4} formalizes the intuition that ``the probabilistic future only depends on the present state''; we shall refer to it as to the \emph{Markov property}. Some additional comments about Definition~\ref{def:3}: \begin{enumerate}[1.] \item Markov chains are usually defined by their transition matrix, from which a probability measure on the space of trajectories is derived. Here, on the contrary, the lack of a totally ordered time index leads us to first consider a measure on the space of trajectories with the Markov property already encoded in it. It will be our task to find an equivalent for the transition matrix, that would characterize the probability measure through a finite number of real parameters with adequate normalization conditions. This is the topic of~\S~\ref{sec:gener-constr-mark}. \item Considering the same definition for a probability measure on a space of trajectories with only one component---for instance, taking $S^2=\{*\}$ a singleton disjoint from~$S^1$---, would exactly bring us back to the definition of a homogeneous Markov chain on~$S^1$. The transition matrix $P_{i,j}$ would then be given by $P_{i,j}=\mathbf{P}_{(i,*)}\bigl(\uparrow(j,*)\bigr)$. \item Contrast this definition with an alternative, naive model consisting of a Markov chain on the state of global states. Note that the Markov property stated in Eq.~\eqref{eq:4} is relative to any ``cut'' $\gamma(s)$ of the trajectory. However, for a Markov chain, the property would only hold for particular cuts, namely those such that $|s|$ has the form $(n,n)$ for some integer~$n$. \end{enumerate} \noindent Checking that a probabilistic process $\mathbb{P}$ satisfies the Markov property amounts to verifying the equality: \begin{equation} \label{eq:7} \frac1{\mathbf{P}_\alpha(s)}{\mathbf{P}_\alpha(s\cdot t)}=\mathbf{P}_{\gamma(s)}(t) \end{equation} for all finite trajectories $s$ and $t$ such that $\mathbf{P}_\alpha(s)>0$. The following lemma however shows that, for closed processes, it suffices to verify Eq.~\eqref{eq:7} for elementary trajectories~$t$. \begin{lem} \label{lem:1} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a closed two-components process, such that: \begin{equation} \label{eq:10} \forall\alpha\in X_0\qquad \bigl(\mathbf{P}_\alpha\bigr)_s(t)=\mathbf{P}_{\gamma(s)}(t), \end{equation} for every elementary trajectory $t$ and finite trajectory $s$ with\/ $\mathbf{P}_\alpha(s)>0$. Then\/ $\mathbb{P}$ is a Markov two-components process. \end{lem} \proof Let $\mathcal{E}$ denote the set of elementary trajectories (Definition~\ref{def:5}). We also denote by $\mathcal{E}^+$ the set of trajectories that are finite concatenations of elementary trajectories, and by $\mathcal{V}$ the set of finite trajectories. We proceed in two steps to show that Eq.~\eqref{eq:10} is valid for $s,t\in\mathcal{V}$. \par\medskip \textsl{Step~1: Equation~\eqref{eq:10} is true for $s\in\mathcal{V}$ and $t\in\mathcal{E}^+$.}\quad By induction, we show that Eq.~\eqref{eq:10} is true for $s\in\mathcal{V}$ and $t=t_1\cdot\ldots\cdot t_n$ with $t_i\in\mathcal{E}$\,. The case $n=1$ is given by the hypothesis of the lemma, assume it is true for all $k<n$. Assume moreover that $\mathbf{P}_\alpha(s\cdot t_1\cdot\ldots\cdot t_k)>0$ for all $k=1,\ldots,n-1$. We calculate as follows, using the hypothesis of the lemma and the induction hypothesis: \begin{align} \notag \bigl(\mathbf{P}_\alpha\bigr)_s(t_1\cdot\ldots\cdot t_n)&= \frac{\mathbf{P}_\alpha(s\cdot t_1\cdot\ldots\cdot t_n)} {\mathbf{P}_\alpha(s)}\\ \notag &=\bigl(\mathbf{P}_\alpha\bigr)_s(t_1\cdot\ldots\cdot t_{n-1})\cdot \bigl(\mathbf{P}_\alpha\bigr)_{s\cdot t_1\cdot\ldots\cdot t_{n-1}}(t_n)\\ \label{eq:38}&=\mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_{n-1})\cdot \mathbf{P}_{\gamma(t_{n-1})}(t_n). \end{align} We also have, using again the hypothesis of the lemma: \begin{align} \notag \mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_n)&=\mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_{n-1})\cdot\mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_n|t_1\cdot\ldots\cdot t_{n-1})\\ \notag &=\mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_{n-1})\cdot \bigl(\mathbf{P}_{\gamma(s)}\bigr)_{t_1\cdot\ldots\cdot t_{n-1}}(t_n)\\ \label{eq:32} &= \mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_{n-1})\cdot\mathbf{P}_{\gamma(t_{n-1})}(t_n)\,. \end{align} Comparing \eqref{eq:38} and \eqref{eq:32}, we get $ \bigl(\mathbf{P}_\alpha\bigr)_s(t_1\cdot\ldots\cdot t_n)= \mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_n)$\,, completing the induction in this case. To be complete, we examine the case where $\mathbf{P}_\alpha(s\cdot t_1\cdot\ldots\cdot t_k)=0$ for some integer $k\in\{1,\ldots,n-1\}$. Then, on the one hand, this implies $\mathbf{P}_\alpha(s\cdot t_1\cdot\ldots\cdot t_n)=0$ and thus $\bigl(\mathbf{P}_\alpha\bigr)_s(t_1\cdot\ldots\cdot t_n)=0$. On the other hand, let $i$ be the smallest integer $1\leq i<n$ such that $\mathbf{P}_\alpha(s\cdot t_1\cdot\ldots\cdot t_i)=0$. Then the minimality of $i$ yields $\bigl(\mathbf{P}_\alpha\bigr)_{s\cdot t_1\cdot\ldots\cdot t_{i-1}}(t_i)=0$, and by the hypothesis of the lemma this is $\mathbf{P}_{\gamma(t_{i-1})}(t_i)=0$\,. Applying again the hypothesis of the lemma: $\mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_i)=\mathbf{P}_{\gamma(s)}(t_1\cdot\ldots\cdot t_{i-1})\cdot\mathbf{P}_{\gamma(t_{i-1})}(t_i)=0$, which implies $\mathbf{P}_{\gamma(s)}(t_1\cdot\ldots t_n)=0$. The induction is complete. \par\medskip \textsl{Step~2: Equation~\eqref{eq:10} is true for $s,t\in\mathcal{V}$.}\quad Let $s$ and $t$ be any finite trajectories. For $\omega\in\,\uparrow(s\cdot t)$, we put \begin{equation} \label{eq:12} \notag E_\omega=\{v\in\mathcal{E}^+\;|\; s\cdot t\leq v\leq\omega\}\,,\qquad \omega_T=\inf E_\omega\,. \end{equation} On the one hand, $E_\omega\neq\emptyset$ $\mathbf{P}_\alpha$-\text{a.s.}\ since $\mathbb{P}$ is assumed to be closed. On the other hand, the trajectories of $E_\omega$ form a chain, which is well founded by Lemma~\ref{lem:5}; hence $\omega_T=\min E_\omega$ is $\mathbf{P}_\alpha$-\text{a.s.}\ well defined and $\omega_T\in E_\omega$\,. It is easy to observe that, for $v=\omega_T$\,, we have: \begin{equation} \label{eq:13} \{\omega'\in\Omega\;|\;\omega'_T=v\}=\uparrow v. \end{equation} (Later, we will interpret this by saying that $\omega\mapsto\omega_T$ is a stopping time). Since $\omega_T$ ranges over finite trajectories, the set of values it can take is countable. Therefore, decomposing with respect to the possible values: \begin{align*} \label{eq:14} \notag \bigl(\mathbf{P}_\alpha\bigr)_s(t)&=\sum_v\bigl(\mathbf{P}_\alpha\bigr)_s(\uparrow t\cap\{\omega_T=v\})\\ &=\sum_{v}\bigl(\mathbf{P}_\alpha\bigr)_s(\omega_T=v)&&\text{since $\uparrow t\subset\{\omega_T=v\}$}\\ &=\sum_{v}\bigl(\mathbf{P}_\alpha\bigr)_s(\uparrow v)&&\text{by Eq.~\eqref{eq:13}}\\ &=\sum_{v}\mathbf{P}_{\gamma(s)}(\uparrow v)&&\text{by Step~1 since $v\in\mathcal{E}^+$}\\ &=\mathbf{P}_{\gamma(s)}(t)&&\text{recomposing.} \end{align*} The proof is complete.\qed \section{Synchronization of Two Markov Chains} \label{sec:synchr-two-mark} In this section we introduce a way of constructing {\normalfont M2CP} s. It first shows that our object of study is not empty. It also provides a bridge between {\normalfont M2CP} s and usual Markov chains---another, maybe deeper link is developed in~\S~\ref{sec:stopp-times-strong}. Consider two Markov chains $\seq{X^1}n$ and $\seq {X^2}n$ on $S^1$ and $S^2$ respectively. We denote by $M^i_x$ the probability measure on $\Omega^i$ associated with the chain $X^i$ starting from state $x\in S^i$, for $i=1,2$. We assume for simplicity that both transition matrices have all their coefficients positive. The construction consists of recursively forcing the next synchronization of the chains on a shared state. The formal construction is given in Definition~\ref{def:4} below, after an informal explanation. The case where there is only one synchronization state is trivial, in the sense that it reduces to the independent product of the two Markov chains as shown by Proposition~\ref{prop:8}, point~\ref{item:5}. A numerical example with two synchronization states is analyzed in~\S~\ref{sec:few-examples}. Denoting as above by $Q$ the set $S^1\cap S^2$ of shared states, let $\tau^i$ be the first hitting time of $Q$ for the chain~$X^i$, defined on $\Omega^i$ by \begin{equation} \label{eq:5} \notag \tau^i=\inf\{n>0 \;|\; X^i_n\in Q\},\quad \text{noting that $\tau^i<\infty$ $M^i_x$-a.s.} \end{equation} We consider the subset $X_0$ of global states given by \begin{equation} \notag \label{eq:6} X_0=\{(x,z)\in S^1\times S^2\;|\; x\in Q\wedge z\in Q\Rightarrow x=z\}. \end{equation} Introduce also $\Delta=\bigl\{(\tau^1<\infty)\wedge(\tau^2<\infty) \wedge(X^1_{\tau^1}=X^2_{\tau^2})\bigr\}$, a measurable subset of $\Omega^1\times\Omega^2$. Since the transition matrices we consider have all their coefficients positive, we have $M_x^1\otimes M_z^2(\Delta)>0$ for any global state~$(x,z)$. We therefore equip the random pair of sequences $\sigma_0=\bigl(X^1_1X^1_2\dots X^1_{\tau^1},X^2_1X^2_2\dots X^2_{\tau^2}\bigr)$ with the conditional law \begin{equation} \notag\label{eq:9} U_{(x,z)}(\,\cdot\,)=M^1_x\otimes M^2_z(\,\cdot\;|\Delta). \end{equation} Starting now from the global state $(X^1_{\tau^1},X^2_{\tau^2})$, we consider a fresh copy $\sigma_1$ of the same random pair of sequences, now equipped with the law $U_{(X^1_{\tau^1},X^2_{\tau^2})}$\, (observe that, by construction, $X^1_{\tau^1}=X^2_{\tau^2}$). We construct inductively in this way a sequence $\seq\sigma n$ of random trajectories, for which the concatenation $\omega=\sigma_0\cdot\sigma_1\cdot\ldots$ is an element of~$\Omega$ since $|\sigma_k|\geq(1,1)$ for all $k\geq0$. Denoting by $\mathbf{P}_{(x,z)}$ the law of $\omega$ thus constructed, we obtain a probabilistic two-components process\ (Definition~\ref{def:1}), which is a closed process by construction. \medskip \begin{defi} \label{def:4} The \tdefine{synchronization of the two Markov chains} $\seq {X^1}n$ and $\seq{X^2}n$ is the probabilistic process $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$, where: \begin{enumerate}[(1)] \item $\alpha$ ranges over the set $X_0=\{(x,z)\in S^1\times S^2\;|\; x\in Q\wedge z\in Q\Rightarrow x=z\}$\,. \item $\mathbf{P}_{(x,z)}$ is defined as the law of the infinite concatenation $\sigma_1\cdot\sigma_2\cdot\ldots$, where $\seq\sigma n$ is the countable Markov chain on the set $\mathcal{E}$ of elementary trajectories with $U_{(x,z)}=M^1_x\otimes M^2_z(\,\cdot\;|\Delta)$ as initial law, and transition kernel $K$ given by: \begin{equation} \label{eq:8} \notag \forall\sigma,\sigma'\in\mathcal{E},\qquad K(\sigma,\sigma')=U_{\gamma(\sigma)}(\sigma'). \end{equation} \end{enumerate}\medskip \end{defi} \noindent Translating the above definition in the two-components processes language consists of determining the value of $\mathbf{P}_\alpha(\uparrow v)$ for any finite trajectory~$v$. This can be easily done only for $v$ of the following special form: \begin{equation*} v=\sigma_1\cdot\ldots\cdot\sigma_n\,,\quad\sigma_i\in\mathcal{E}, \qquad \mathbf{P}_\alpha(\uparrow v)= U_\alpha(\sigma_1)\cdot K(\sigma_1,\sigma_2)\cdot\ldots \cdot K(\sigma_{n-1},\sigma_n)\,. \end{equation*} Note that this entirely determines the probability~$\mathbf{P}_\alpha$\,; since $\mathbf{P}_\alpha(v)$ for any finite trajectory $v$ will be computed as the sum of all $\mathbf{P}_\alpha(w)$, for $w$ of the form $w=\sigma_1\cdot\ldots\cdot\sigma_n$ and $v\leq w$, very much as we did in Step~$2$ in the proof of Lemma~\ref{lem:1}. \medskip \begin{thm} \label{thr:1} The synchronization of two Markov chains is a Markov two-components process. \end{thm} \proof Let $\alpha=(x,z)$ be an initial state. Let $t\in\mathcal{E}$ be any elementary trajectory, and let $s$ be any finite trajectory. Denote the coordinates of trajectories on each site by \mbox{$s=(s^1,s^2)$} and \mbox{$t=(t^1,t^2)$}, and put $\gamma(s)=(x',z')$. Applying Lemma~\ref{lem:1}, we have to show that \mbox{$\bigl(\mathbf{P}_\alpha\bigr)_s(t)=\mathbf{P}_{\gamma(s)}(t)$}. We proceed in two steps. \begin{enumerate}[(1)] \item \textsl{Step~$1$: $s$ is synchronization free.} Then $s\cdot t$ is an elementary trajectory, and by construction of $\sigma_0$ we have $\uparrow(s\cdot t)=\{\sigma_0=s\cdot t\}$ and thus $\mathbf{P}_\alpha(s\cdot t)=U_\alpha(s\cdot t)$ by construction. From this we compute: \begin{equation} \label{eq:15} \bigl(\mathbf{P}_\alpha\bigr)_s(t)= \frac{M_x\otimes M_z\bigl(\uparrow(s\cdot t)\cap\Delta\bigr)} {M_x\otimes M_z(\uparrow s\cap\Delta)}\,. \end{equation} On the one hand, noting that $\uparrow(s\cdot t)\subset\Delta$, we have $M_x\otimes M_z\bigl(\uparrow(s\cdot t)\cap\Delta\bigr)=M_x(s^1\cdot t^1)M_z(s^2\cdot t^2)$. On the other hand, we have $M_x\otimes M_z(\uparrow s\cap\Delta)=M_x(s^1)M_z(s^2) M_x\otimes M_z(\Delta|\uparrow s)$ and, since $M_x$ and $M_z$ are Markov chains, $M_x\otimes M_z(\Delta|\uparrow s)=M_{x'}\otimes M_{z'}(\Delta)$. Going back to Eq.~\eqref{eq:15} we get: \begin{align*} \bigl(\mathbf{P}_\alpha\bigr)_s(t) &=\frac{M_x(s^1\cdot t^1)}{M_x(s^1)}\times \frac{M_z(s^2\cdot t^2)}{M_z(s^2)}\times \frac1{M_{x'}\otimes M_{z'}(\Delta)}\\ &=M_{x'}(t^1)M_{z'}(t^2)\frac1{M_{x'}\otimes M_{z'}(\Delta)}\\ &=M_{x'}\otimes M_{z'}(t|\Delta)=\mathbf{P}_{\gamma(s)}(t). \end{align*} \item \textsl{Step $2$: $s$ is any finite trajectory.} Let $s=\sigma_0\cdot\sigma_1\cdot\ldots\cdot\sigma_p\cdot s'$ be the decomposition of $s$ according to Proposition~\ref{prop:1}, case~\ref{item:10}, so that $\sigma_0,\dots,\sigma_p$ are elementary trajectories, and $s'$ is a synchronization free trajectory (the case $s'=\emptyset$ is admissible). We compute: \begin{align*} \bigl(\mathbf{P}_\alpha\bigr)_s(t)&= \frac{\mathbf{P}_\alpha(\sigma_0\cdot\ldots\cdot\sigma_p\cdot s'\cdot t)} {\mathbf{P}_\alpha(\sigma_0\cdot\ldots\cdot\sigma_p\cdot s')} \\ &=\frac{U_{(x,z)}(\sigma_0)K(\sigma_0,\sigma_1)\ldots K(\sigma_{p-1},\sigma_p) K(\sigma_p,s'\cdot t)} {U_{(x,z)}(\sigma_0)K(\sigma_0,\sigma_1)\ldots K(\sigma_{p-1},\sigma_p) U_{\gamma(\sigma_p)}(\uparrow s')}\\ &=U_{\gamma(\sigma_p)}(s'\cdot t|\uparrow s')\\ &=\bigl(\mathbf{P}_{\gamma(\sigma_p)}\bigr)_{s'}( t)\\ &=\mathbf{P}_{\gamma(s)}(t), \end{align*} the last equality following from Step~$1$ together with $\gamma(s')=\gamma(s)$. \end{enumerate} Conclusion: Lemma~\ref{lem:1} applies, and $(\mathbf{P}_\alpha)_{\alpha\in X_0}$ is a {\normalfont M2CP}. \qed \section{Stopping Times and the Asynchronous Strong Markov Property} \label{sec:stopp-times-strong} All the notions and results of this section do not depend on the particular structure of trajectories, and in particular they do not rest on Proposition~\ref{prop:1}. It follows that they have straightforward generalizations to an asynchronous model with an arbitrary number $n\geq2$ of sites. \subsection{Stopping Times} \label{sec:stopping-times} Stopping times are a fundamental tool in the theory of probabilistic processes in general, and in the theory of Markov chain in particular. Recall that a stopping time associated to a Markov chain is a random integer, maybe infinite and seen as a random time instant, with the following property: an observer aware of the successive values of the chain can decide at each instant whether the stopping time has already been reached or not. Standard examples of stopping times in Markov chain theory are: constant times (trivial since non random); the \emph{first} instant where the chain hits a given state; more generally the first instant where a chain reaches a given set of states. A standard example of a random time which is \emph{not} a stopping time is the \emph{last} instant where the chain hits a given state. It is natural to introduce an equivalent notion for two-components process es, and this is the topic of this subsection. We will see that the first instant a process hits a global state defines a stopping time; but, and contrasting with Markov chains, the first instant of reaching a given \emph{set} of global states does not define a stopping time in general, unless one considers special kinds of sets. Recall that $\mathcal{T}=\bigl(\mathbb{N}\times\mathbb{N}\bigr)\cup\{\infty\}$ denotes the partially ordered set of ``two-components time instants''. \begin{defi}[Random times and stopping times] \label{def:6} Let $T:\Omega\to\mathcal{T}$ be an arbitrary mapping. For any $\omega\in\Omega$, we denote by $\omega_T$ the prefix of $\omega$ of length $T(\omega)$ if $T(\omega)<\infty$, and we put $\omega_T=\omega$ if $T(\omega)=\infty$. \begin{enumerate}[(1)] \item We say that $T$ is a \tdefine{random time} if $\omega_T$ is a subtrajectory of $\omega$ for all $\omega\in\Omega$. \item If $T$ is a random time we say that $T$ is a \tdefine{stopping time} if furthermore the following property holds: \begin{equation} \label{eq:45} \forall\omega,\omega'\in\Omega\quad \omega'\geq \omega_T\Rightarrow \omega_T=\omega'_T\,. \end{equation} \end{enumerate} \end{defi} \noindent Actually since the space $\Omega$ is always implicitly equipped with an initial state~$\alpha$, a more general notion of stopping times would be as for probabilistic processes a \emph{family} of random times~$(T_\alpha)_{\alpha\in X_0}$\,, each one satisfying condition~(\ref{eq:45}). But, since we will only be concerned with stopping times independent of~$\alpha$, we prefer limiting ourselves to Definition~\ref{def:6} as it is formulated. \medskip Since $\mathcal{T}$ is a countable set, it is naturally equipped with its discrete \mbox{$\sigma$-alge}\-bra. It turns out that a stopping time $T:\Omega\to\mathcal{T}$ is always measurable; and so is the mapping $\omega\in\Omega\mapsto\omega_T$\,, provided we equip the set of trajectories (either finite or infinite) with the \mbox{$\sigma$-alge}\-bra\ generated by the sets of the form $\{v\;|\; s\leq v\}$, for $s$ ranging over finite trajectories, and $v$ ranging over trajectories. If the set of trajectories is seen as a DCPO (Directed Complete Partial Order~\cite{gierz03}), this is the Borel \mbox{$\sigma$-alge}\-bra\ associated with the Scott topology on the DCPO. Obviously, it induces by restriction the \mbox{$\sigma$-alge}\-bra\ $\mathfrak{F}$ on the subset~$\Omega$. \begin{prop} \label{prop:4} Let $T:\Omega\to\mathcal{T}$ be a stopping time. We equip $\mathcal{T}$ with its discrete $\mbox{$\sigma$-alge}\-bra$, and we equip the set of trajectories with its Borel \mbox{$\sigma$-alge}\-bra\ described above. \begin{enumerate}[\em(1)] \item Then $T$ and $\omega_T$ are two measurable mappings. \item Let $\mathfrak{F}_T$ denote the \mbox{$\sigma$-alge}\-bra\ generated by~$\omega_T$\,. Then $\mathfrak{F}_T$ is finer than the \mbox{$\sigma$-alge}\-bra\ generated by~$T$, and it is characterized as follows: \begin{equation*} \forall A\in \mathfrak{F}\quad A\in\mathfrak{F}_T\iff \forall\omega,\omega'\in\Omega\quad \omega\in A\wedge\omega'\geq\omega_T\Rightarrow\omega'\in A. \end{equation*} \end{enumerate} \end{prop} \proof If $Y:(\Omega,\mathfrak{F})\to (A,\mathfrak{G})$ is a measurable mapping, we denote by $\langle Y\rangle$ the sub-\mbox{$\sigma$-alge}\-bra\ of $\mathfrak{F}$ generated by~$Y$, and given by $\langle Y\rangle =\{Y^{-1}(U)\;|\; U\in\mathfrak{G}\}$. For $\omega\in\Omega$, let $\zeta(\omega)=\omega_T$. For any finite trajectory~$v$, we put $$ S_v=\{\text{$w$ trajectory}\;|\; v\leq w\}\,. $$ Since $T$ is a stopping time, $\zeta^{-1}\bigl(\{v\}\bigr)$~is either empty or equal to~$\uparrow v$, so $\zeta^{-1}\bigl(\{v\}\bigr)$ is measurable in either cases. Let us denote by $\mathcal{V}$ the set of finite trajectories. Since $\mathcal{V}$ is countable, it follows that $\zeta^{-1}(S_v\cap\mathcal{V})=\bigcup_{w\in S_v\cap\mathcal{V}}\zeta^{-1}\bigl(\{w\}\bigr)$ is measurable for any $v\in\mathcal{V}$, as well as $\zeta^{-1}(\mathcal{V})$. By definition of $\zeta=\omega_T$, we have that $\zeta(\omega)$ is either finite or maximal. From this, it follows first that $\zeta^{-1}(\Omega)=\Omega\setminus\zeta^{-1}(\mathcal{V})$ is measurable; and second: \begin{equation*} \zeta^{-1}(S_v)=\zeta^{-1}(S_v\cap\mathcal{V})\cup\zeta^{-1}(\uparrow v). \end{equation*} But $\zeta^{-1}(\uparrow v)=\zeta^{-1}(\Omega)\cap\uparrow v$, hence $\zeta^{-1}(S_v)$ is the union of two measurable subsets of~$\Omega$, and is thus measurable. This shows that $\zeta$ is a measurable mapping. To prove that $T$ is measurable, observe that: \begin{equation} \label{eq:20} \notag \forall (m,n)\in\mathcal{T}\quad \bigl\{T=(m,n)\bigr\}=\bigcup_{|v|=(m,n)}\zeta^{-1}(v)\,. \end{equation} Since the union is finite, it follows that $\{T=(m,n)\}$ is a $\langle\zeta\rangle$-measurable subset, from which we deduce that $\{T=\infty\}=\bigcup_{(m,n)\in\mathbb{N}\times\mathbb{N}}\{T\neq(m,n)\}$ is also a $\langle\zeta\rangle$-measurable subset. Therefore $\langle T\rangle\subset\langle\zeta\rangle$. By the property of stopping times $\omega'\geq\omega_T$ is equivalent to $\omega_T=\omega'_T$\,, from which follows the characterization of $\mathfrak{F}_T=\langle\zeta\rangle$. \qed Note that any function $f:(\Omega,\mathfrak{F})\to(A,\mathfrak{G})$ with value in some measurable space is measurable with respect to $\mathfrak{F}_T$ if and only if it is constant on elementary cylinders of the form $\uparrow v=\{\omega_T=v\}$ with $v$ ranging over the values of~$\omega_T$---since it is well known that $f$ is $\mathfrak{F}_T$-measurable if and only if it can be written as $f(\omega)=g(\omega_T)$ with $g$ some measurable mapping. \subsection{Shift Operators} In Markov chain theory, the ``universal'' shift operator $\theta$ is classically defined on the space of trajectories of a Markov chain by $\theta(x_0x_1\ldots)=(x_1x_2\ldots)$. Its iterations $\theta_n$ are defined for $n\geq0$ by $\theta_0=\text{Id}$ and $\theta_{n+1}=\theta\circ\theta_n$\,. Allowing the time index $n$ to be random, one defines~$\theta_T$, for $T:\Omega\to \mathbb{N}$ any random variable, by $\theta_T(\omega)=\theta_{T(\omega)}(\omega)$. In our framework, there is no such ``universal'' shift operator~$\theta$. Yet, each stopping time $T:\Omega\to\mathcal{T}$ induces a shift operator~$\theta_T:\Omega\to\Omega$. Informally $\theta_T(\omega)$ is the queue of trajectory $\omega$ that remains ``after'' the prefix trajectory~$\omega_T$\,. \begin{defi} \label{def:7} Let $T:\Omega\to\mathcal{T}$ be a stopping time. The \tdefine{shift operator} associated with $T$ is the mapping $\theta_T:\Omega\to\Omega$ , which is only partially defined; if $T(\omega)<\infty$, then $\theta_T(\omega)$ is defined as the unique element of\/ $\Omega$ such that \begin{equation*} \omega=\omega_T\cdot \theta_T(\omega), \end{equation*} and $\theta_T(\omega)$ is undefined otherwise. \end{defi} The shift operator allows to define an addition on stopping times, as shown by the following result which mimics an equivalent result widely used in Markov chain theory. \begin{lem} \label{lem:8} Let $S,T:\Omega\to\mathcal{T}$ be two stopping times. Then $U=S+T\circ\theta_S$ is a stopping time (it is understood that $U=\infty$ if $S=\infty$). \end{lem} \proof It is clear that $U$ is a random time. Let $\omega,\omega'\in\Omega$ such that $\omega'\geq\omega_U$, we have to show that $\omega'_U=\omega_U$. If $S(\omega)=\infty$ then $U(\omega)=\infty$ and then $\omega'=\omega$ and $\omega_U=\omega'_U$, trivially. Hence we assume without loss of generality that $S(\omega)<\infty$, and we put $\zeta=\theta_S(\omega)$ and $\zeta'=\theta_S(\omega')$. We have $\omega_U=\omega_S\cdot \zeta_T$. Hence $\omega'\geq\omega_S$ and thus $\omega'_S=\omega_S$ since $S$ is a stopping time. Therefore: $\omega'=\omega_S\cdot\zeta'\geq\omega_U=\omega_S\cdot \zeta_T$\,, hence $\zeta'\geq\zeta_T$\,. Thus $\zeta'_T=\zeta_T$ since $T$ is a stopping time. We have finally $\omega'_U=\omega'_S\cdot\zeta'_T=\omega_S\cdot\zeta_T=\omega_U$, proving that $U$ is a stopping time. \qed Starting from a given stopping time~$T$, we use Lemma~\ref{lem:8} above to iterate the ``addition'' of~$T$ to itself. \begin{defi} \label{def:15} Let $T:\Omega\to\mathcal{T}$ be a stopping time, and let $\theta_T$ be the associated shift operator. The sequence $(T^n)_{n\geq0}$ of mappings $\Omega\to\mathcal{T}$ defined as follows: \begin{align*} T^0&=(0,0)& \forall n\geq0\quad T^{n+1}&=T^n+T\circ\theta_{T^n} \end{align*} with the convention that $T^{n+1}=\infty$ on $\{T^n=\infty\}$, is a sequence of stopping times, called \tdefine{iterated stopping times associated with~$T$}. \end{defi} Remark that $\theta_{T^0}=\text{Id}_\Omega$, and $T^1=T$. \subsection{Examples of Stopping Times} \label{sec:exampl-stopp-times} In this subsection we review some examples of random times, and analyze whether they are stopping times or not. Some of the examples introduced here will be used later in~\S\S~\ref{sec:recurr-transc-glob}--\ref{sec:open-closed-tcp}. \subsubsection{Constant Times are not Random Times in General.} In general, if $(m,n)\in\mathbb{N}\times\mathbb{N}$, then the random variable constant and equal to $(m,n)$ is not a random time. For instance, take $(m,n)=(2,2)$ and consider as in the Introduction a maximal trajectory $\omega$ starting with $(a\cdot\c,e\cdot f\cdot\c)$ with $\c$ as synchronization state. Then the prefix of length $(2,2)$ of $\omega$ is $(a\cdot\c,e\cdot f)$, which is not a trajectory. Hence the constant $T=(2,2)$ is not a random time. This contrasts with Markov chain theory, where constant times are a basic example of stopping times. However note that any constant time is indeed a random time if the process is open (Definition~\ref{def:17}). And in this case, it is also a stopping time. \subsubsection{A Random Time which is not a Stopping Time.} For $\omega$ a maximal trajectory, let $v$ be the first elementary trajectory in the decomposition of $\omega$ as in Proposition~\ref{prop:1}, which is defined if $\omega$ has at least one synchronization. Then $v$ has the form $v=u\cdot(y,y)$ for some unique finite trajectory $u$ and state $y\in Q$. Put $\omega_T=u$ in this case, and $\omega_T=\omega$ if $v$ is not defined. Time $T(\omega)$ represents the ``last instant before first synchronization''. By construction, $T$~is a random time since $\omega_T$~is a subtrajectory of~$\omega$. However $T$ is not a stopping time in general. For example, consider $S^1=\{a,b,\c,\d\}$ and $S^2=\{\c,\d,e,f\}$, if $\omega$ starts with $(a\cdot\c,f\cdot f\cdot e\cdot\c)$, then $T(\omega)=(1,3)$ and $\omega_T=(a,f\cdot f\cdot e)$, corresponding to the last private states $a$ and $e$ before synchronization on $(\c,\c)$. And if $\omega'$ starts with $(a\cdot b\cdot\c,f\cdot f\cdot e\cdot f\cdot\c)$, then $T(\omega')=(2,3)\neq T(\omega)$ although $\omega'\geq \omega_T$\,. This shows that $T$ is not a stopping time. \subsubsection{The First Return Time of a Global State.} \label{sec:first-hitting-time} Let $\alpha\in X_0$ be a given global state. For $\omega\in\Omega$, consider the following set of finite subtrajectories of~$\omega$: \begin{equation*} N_\alpha(\omega)=\{v\leq\omega\;|\; \text{$v$ finite subtrajectory of $\omega$} \wedge \gamma(v)=\alpha\wedge |v|\geq(1,1)\}. \end{equation*} If nonempty, $N_\alpha$ is a sublattice of the lattice of subtrajectories of~$\omega$ since, by Lemma~\ref{lem:5}, lower bounds are taken component by component so that $\gamma(v\wedge v')=\alpha$ whenever $v,v'\in N_\alpha(\omega)$ and $|v\wedge v'|\geq(1,1)$. In particular, if we put $v=\min N_\alpha(\omega)$, which exists whenever \mbox{$N_\alpha(\omega)\neq\emptyset$}, then $\gamma(v)=\alpha$ and $|v|\geq(1,1)$. We define thus the first return time to $\alpha$ as follows. \begin{defi} \label{def:8} For any $\alpha\in X_0$\,, the \tdefine{first return time to~$\bm\alpha$} is the stopping time $R_\alpha:\Omega\to\mathcal{T}$ defined by: \begin{equation*} \forall\omega\in\Omega\quad \omega_{R_\alpha}= \begin{cases} \omega,&\text{if $N_\alpha(\omega)=\emptyset$, and thus $R_\alpha(\omega)=\infty$,}\\ \min N_\alpha(\omega),&\text{otherwise, and thus $R_\alpha(\omega)=\bigl|\min N_\alpha(\omega)\bigr|$.} \end{cases} \end{equation*} The \tdefine{successive return times to~$\bm\alpha$} are the iterated stopping times $(R^n_\alpha)_{n\geq1}$ associated with~$R_\alpha$ as in Definition~{\normalfont\ref{def:15}}. \end{defi} For any finite subtrajectory $v$ of~$\omega$, we have: \[ \bigl(\gamma(v)=\alpha\wedge |v|\geq(1,1)\bigr)\Longrightarrow \omega_{R_\alpha}\leq v\,, \] which is consistent with the intuition of what a ``first return time'' should be. To show that $R_\alpha$ is indeed a stopping time, observe first that $\omega_{R_\alpha}$ is clearly a subtrajectory of~$\omega$. And second, if $\omega'\in\Omega$ is such that $\omega'\geq\omega_{R_\alpha}$, that implies that $\omega_{R_\alpha}\in N_\alpha(\omega')$, and thus $\omega'_{R_\alpha}\leq\omega_{R_\alpha}$ by minimality of~$\omega'_{R_\alpha}$\,. But then $\omega'_{R_\alpha}\in N_\alpha(\omega)$, and thus $\omega_{R_\alpha}\leq\omega'_{R_\alpha}$ by minimality of~$\omega_{R_\alpha}$\,. Hence $\omega_{R_\alpha}=\omega'_{R_\alpha}$\,, and this shows that $R_\alpha$ is a stopping time. \begin{figure} \setbox1=\hbox{$\bullet$} \newcommand{\rlap{\raisebox{-2.5 pt}{$\bullet$}}\raisebox{2.5 pt}{$\bullet$}}{\rlap{\raisebox{-2.5 pt}{$\bullet$}}\raisebox{2.5 pt}{$\bullet$}} \newcommand{\lu}[2]{\POS[]!U(#1)\drop{#2}} \newcommand{\ld}[2]{\POS[]!D(#1)\drop{#2}} \renewcommand{\ll}[1]{\POS[]!L(6)\drop{#1}} \centering \[ \xymatrix@M=0pt{ \bullet\ar@{-}[rr]!R\lu5{a} &&\bullet\ar@{-}[dr]\lu5{b} &&\ar@{-}[r] &\bullet\ar@{-}[r]\lu5{a} &\ar@{-}[dr]&&\\ &&&\rlap{\raisebox{-2.5 pt}{$\bullet$}}\raisebox{2.5 pt}{$\bullet$}\ar@{-}[ur]\ar@{-}[dr]\lu3{\c}\ld{2.666}{\c} &&&&\rlap{\raisebox{-2.5 pt}{$\bullet$}}\raisebox{2.5 pt}{$\bullet$}\ar@{-}[ur]\ar@{-}[dr]\lu3{\d}\ld{2.666}{\d}&\\ &\bullet\ar@{-}[r]!R\ld4e &\ar@{-}[ur] &&\bullet\ar@{-}[rr]\ld4e &&\bullet\ar@{-}[ur]\ld4f&& } \] \caption{\textsl{A finite trajectory synchronizing on shared states $\c$ and~$\d$.}} \label{fig:mkjsdfqqs} \end{figure} \medskip As an example, consider $S^1=\{a,b,\c,\d\}$ and $S^2=\{\c,\d,e,f\}$, and a maximal trajectory $\omega$ starting with \mbox{$(a\cdot b\cdot\c\cdot a\cdot \d,e\cdot\c\cdot e\cdot f\cdot\d)$}, which is depicted in Figure~\ref{fig:mkjsdfqqs}. Consider the global state $\alpha=(a,e)$. Then $R_\alpha(\omega)=(1,1)$, and $\omega_{R_\alpha}=(a,e)$. Note that, since $R_\alpha$ is indeed a stopping time, we do not need to know the queue of $\omega$ to already have information on $R(\omega)$. Let us determine the value of next return $R^2_\alpha(\omega)$ to~$\alpha$. The shifted trajectory $\theta_{R_\alpha}(\omega)$ starts with $(b\cdot\c\cdot a\cdot\d,\c\cdot e\cdot f\cdot\d)$. Therefore $R_\alpha\bigl(\theta_{R_\alpha}(\omega)\bigr)=(3,2)$, and $R^2_\alpha(\omega)=(1,1)+(3,2)=(4,3)$. Note that $R^3(\omega)$ is undetermined at this stage. If $\zeta$ is the trajectory \mbox{$(b\cdot b\cdot\ldots,e\cdot\cdot e\cdot \ldots)$}, with only $b$ on the first component and only $e$ on the second component, then $R_\alpha(\zeta)=\infty$ and $\zeta_{R_\alpha}=\zeta$. \subsubsection{Supremum of Stopping Times.} If $S$ and $T$ are two stopping times, then the random time $S\vee T$ defined by $\omega_{S\vee T}=\omega_S\vee \omega_T$ is a stopping time. For, if $\omega'\geq\omega_S\vee\omega_T$, then $\omega'\geq\omega_S$ and $\omega'\geq\omega_T$, therefore $\omega'_S=\omega_S$ and $\omega'_T=\omega_T$, hence $\omega'_{S\vee T}=\omega_{S\vee T}$\,. The same line of proof shows that the supremum of any family of stopping times is a stopping time. \subsubsection{The Infimum of Stopping Times may not be a Stopping Time.} Contrasting with stopping times from Markov chain theory however, the infimum of two stopping times $S$ and~$T$, defined by $\omega_{S\wedge T}=\omega_S\wedge\omega_T$\,, may not be a stopping time. Let us consider an example. Let $S^1=\{a,b,\c\}$ and $S^2=\{\c,e,f\}$. Let $\alpha=(a,e)$ and $\beta=(b,f)$, and let $S=T_\alpha$ and $T=T_\beta$ be the first return times to $\alpha$ and to $\beta$ respectively. Consider a trajectory $\omega$ starting with \mbox{$(a\cdot b,f\cdot e)$}. Then \mbox{$\omega_S=(a,f\cdot e)$} and \mbox{$\omega_T=(a\cdot b,f)$}, and thus $\omega_{S\wedge T}=(a,f)$. However, if $\omega'$ is the maximal trajectory defined by \mbox{$\omega'=(a\cdot a\cdots,f\cdot f\cdots)$} we have $\omega'\geq\omega_{S\wedge T}$ on the one hand, and $\omega'_S=\omega'$ and $\omega'_T=\omega'$ on the other hand, so that $\omega'_{S\wedge T}=\omega'\neq\omega_{S\wedge T}$. This show that $S\wedge T$ is not a stopping time. This example is specific to the asynchronous structure we consider, since it makes use of the \emph{partially} ordered structure of trajectories. \subsubsection{First Return Time to a Square Set of Global States.} \label{sec:first-hitting-time-1} Since the infimum of stopping times is not a stopping time in general, there is an issue for defining the first return time to a set of global states. There is actually no obvious way of defining such a thing in general, as the analysis of the above example reveals. The situation however becomes favorable if one considers a set of states satisfying the following property. \begin{defi} \label{def:9} We say a subset $A\subset X_0$ of global sets is a \tdefine{square set} if it has the form $A=X_0\cap(S'_1\times S'_2)$ where $S'_1\subset S_1$ and $S'_2\subset S_2$\,. \end{defi} A first example of a square set is $X_0$ itself. We will also encounter the square set $(Q\times Q)\cap X_0$\,. If $\alpha=(x,z)$ and $\beta=(x',z')$, the smallest square set containing $\alpha$ and $\beta$ is $\{\alpha,\beta,(x,z'),(x',z)\}$. Assume that $A$ is a square set of global states. Define then, for any $\omega\in\Omega$: \begin{align*} N_A(\omega)&=\{v\leq\omega\;|\; \text{$v$ finite subtrajectory of $\omega$}\wedge \gamma(v)\in A\wedge|v|\geq(1,1)\}\,. \end{align*} Then $N_A(\omega)$ is a sublattice of the lattice of finite subtrajectories of $\omega$ whenever it is nonempty. Indeed, since $A$ is a square set. The random time $R_A$ defined by \begin{equation*} \omega_{R_A}=\min N_A(\omega)\,, \end{equation*} and by $R_A=\infty$ as usual when $N_A(\omega)$ is empty, is a stopping time that satisfies $\gamma(\omega_{R_A})\in A$ whenever or $R_A<\infty$. We define $R_A$ as the \emph{first return time} to the square set~$A$. One furthermore checks that $\omega_{R_A}=\bigwedge_{\alpha\in A}\omega_{R_\alpha}$\,, providing an example of infimum of stopping times the result of which is indeed a stopping time. \medskip Let us examine the first return times associated with the square sets $X_0$ and \mbox{$(Q\times Q)\cap X_0$}. In Markov chain theory, $R_{X_0}$~would correspond to the constant time~$1$. But in the asynchronous framework its action is less simple. Stopping time $R_{X_0}$ can be described as follows: $\omega_{R_{X_0}}$~is the smallest subtrajectory of $\omega$ with length $\geq(1,1)$. In particular, $R_{X_0}(\omega)$ is always \emph{finite}. We detail the action of $R_{X_0}$ on an example. Consider $S^1=\{a,b,\c,\d\}$ and $S^2=\{\c,\d,e,f\}$, and let $\omega$ be some maximal trajectory starting with \mbox{$(a\cdot b\cdot\c\cdot a\cdot \d,e\cdot\c\cdot e\cdot f\cdot\d)$}, as depicted in Figure~\ref{fig:mkjsdfqqs} above. The exercise consists in finding the values of $R^n_{X_0}(\omega)$ for the first integers~$n$, where $R^n_{X_0}$ denote the iterated stopping times associated with $R_{X_0}$ as in Definition~\ref{def:15}. Obviously $R_{X_0}(\omega)=(1,1)$. The shifted trajectory $\theta_{R_{X_0}}(\omega)$ starts with \mbox{$(b\cdot\c\cdot a\cdot\d,\c\cdot e\cdot f\cdot\d)$}. The smallest subtrajectory of $\theta_{R_{X_0}}(\omega)$ of length at least $(1,1)$ is \mbox{$(b\cdot\c,\c)$}, and thus $R_{X_0}\bigl(\theta_{R_{X_0}}(\omega)\bigr)=(2,1)$. Hence $R^2_{X_0}(\omega)=(1,1)+(2,1)=(3,2)$. The finite trajectories $$\omega_{R_{X_0}}=(a,e) \quad\text{and}\quad \bigl(\theta_{R_{X_0}}(\omega)\bigr)_{R_{X_0}}=(b\cdot\c,\c) $$ yield the following initial decomposition of~$\omega$: $\omega=(a,e)\cdot(b\cdot\c,\c)\cdot \theta_{R^2_{X_0}} (\omega)$. For the next values $n=3,4$ we find $R^3_{X_0}(\omega)=(4,3)$ and $R^4_{X_0}(\omega)=(5,5)$, corresponding to the initial decomposition $\omega=(a,e)\cdot(b\cdot\c,\c)\cdot(a,e)\cdot(\d,f\cdot\d)\cdots$. \medskip Coming now to the square set $(Q\times Q)\cap X_0$\,, and denoting by $R_Q$ the first return time associated with it, we may rephrase the definition of infinite synchronization of trajectories (Definition~\ref{def:17}) as follows: a maximal trajectory $\omega$ synchronizes infinitely often if \mbox{$\omega\in\bigcap_{n\geq1}\{R_Q^n<\infty\}$}. A probabilistic process $\mathbb{P}$ is closed if $R_Q^n<\infty$ for all $n\geq1$ and $\mathbf{P}_\alpha$-almost surely, for all $\alpha\in X_0$\,. It is open if $R_Q=\infty$, $\mathbf{P}_\alpha$-almost surely and for all $\alpha\in X_0$\,. \medskip We end this series of examples with the following result which will be useful in the study of recurrence of global states. It makes use of the finitary assumption on the set of global states. \begin{lem} \label{lem:6} Let $A$ be a square set. Denoting by $(R^n_A)_{n\geq1}$ the successive returns to~$A$, \textsl{i.e.}, the iterated stopping times associated with the first return time~$R_A$, and by $(R^n_\alpha)_{n\geq1}$ the successive return times to~$\alpha$ for any $\alpha\in A$, we have the following equality of sets: \[ \bigcap_{n\geq1}\{R_A^n<\infty\}= \bigcup_{\alpha\in A}\bigcap_{n\geq1} \{R^n_\alpha<\infty\}\,. \] \end{lem} \proof The $\supset$ inclusion is obvious. For the converse inclusion, let $\omega\in\Omega$ be such that $R^n_A(\omega)<\infty$ for all $n\geq1$. Since $A$ is a finite set, there exists some state $\alpha\in A$ and a strictly increasing sequence of integers $(n_k)_{k\geq1}$ such that $\gamma(R^{n_k}_A)=\alpha$ for all~$k$. By induction on~$k$, we show that $R^k_\alpha(\omega)\leq R_A^{n_k}(\omega)$ for all integers $k\geq1$. The finite trajectory $v=R_A(\omega)$ is a subtrajectory of $\omega$ satisfying $\gamma(v)=\alpha$ and $|v|\geq(1,1)$, and therefore $\omega_{R_\alpha}\leq v$. Since the sequence $\bigl(R^n_A(\omega)\bigr)_{n\geq1}$ is increasing, as shown by the formula in Definition~\ref{def:15} that defines it, we have $\omega_{R_\alpha}\leq v=\omega_{R^1_A}\leq\omega_{R^{n_1}_A}$\,. Assume for the induction that $R^k_\alpha(\omega)\leq R^{n_k}_A(\omega)$. Then there is some finite trajectory $v$ such that $\omega_{R^{n_k}_A}=\omega_{R^k_\alpha}\cdot v$. Since $n_k<n_{k+1}$, there is also some finite trajectory $v'$ such that $\gamma(v')=\alpha$, $|v'|\geq(1,1)$ and $\omega_{R_A^{n_{k+1}}}= \omega_{R_A^{n_{k}}}\cdot v'$\,. We obtain thus: \begin{equation*} \omega_{R^{n_{k+1}}_A}=\omega_{R^k_\alpha}\cdot v\cdot v',\quad |v\cdot v'|\geq(1,1),\quad\gamma(v\cdot v')=\alpha. \end{equation*} This implies that $R_\alpha\bigl(\theta_{R^k_\alpha}(\omega)\bigr)\leq|v\cdot v'|$. By definition, we have $R^{k+1}_\alpha=R^k_\alpha+R_\alpha\circ\theta_{R^k_\alpha}$, whence: \begin{equation*} \bigl|R^{k+1}_\alpha(\omega)\bigr|\leq \bigl|R^{k}_\alpha(\omega)\bigr|+|v|+| v'| = \bigl|R^{n_k}_A(\omega)\bigr|+| v'|=\bigl| R^{n_{k+1}}_A(\omega)\bigr|, \end{equation*} completing the induction. This implies in particular that $R^{k}_\alpha(\omega)<\infty$ for all $k\geq1$, as expected. \qed \subsection{The Asynchronous Strong Markov Property} \label{sec:strong-mark-prop} The Asynchronous Strong Markov Property that we state below has the exact same formulation than the Strong Markov property for Markov chains found in classical references~\cite[Theorem~3.5 p.23]{revuz75}. The syntactical identity underlines the parallel with Markov chain theory, although the interpretation of symbols must be changed of course: stopping times must be understood in the sense of Definition~\ref{def:6}, the associated \mbox{$\sigma$-alge}\-bra\ in the sense given in Proposition~\ref{prop:4}, and of course {\normalfont M2CP} s replace Markov chains. Nevertheless, once the Asynchronous Strong Markov property has been established, it is possible to transfer verbatim some pieces of Markov chain theory. Examples of such transfers are Lemma~\ref{lem:7} given just after Theorem~\ref{thr:2} and the \mbox{$0$-$1$} law for the infinite return to a given global state, given in point~\ref{item:14} of Proposition-definition~\ref{prop:3} below. \medskip \begin{thm}[Asynchronous Strong Markov property] \label{thr:2} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}. For any measurable and non negative function $h:\Omega\to\mathbb{R}$ and for any stopping time $T:\Omega\to\mathcal{T}$, we have \begin{equation} \label{eq:17} \forall\alpha\in X_0\qquad \mathbf{E}_\alpha(h\circ\theta_T\,|\,\mathfrak{F}_T)=\mathbf{E}_{\gamma(\omega_T)}(h)\,,\quad \text{$\mathbf{P}_\alpha$-\text{a.s.},} \end{equation} where $\mathbf{E}_\alpha(\cdot|\mathfrak{F}_T)$ denotes the conditional expectation with respect to probability\/~$\mathbf{P}_\alpha$\, and \mbox{$\sigma$-alge}\-bra~$\mathfrak{F}_T$\,. By convention, both sides of Eq.~\eqref{eq:17} vanish outside $\{T<\infty\}$. \end{thm} Note that, as for the Strong Markov Property for Markov chains, both sides of Eq.~\eqref{eq:17} are random variables: the left side, since it is a conditional expectation with respect to \mbox{$\sigma$-alge}\-bra~$\mathfrak{F}_T$; and the right side, since it depends on the random variable~$\gamma(\omega_T)$. \medskip \proof Let $Z$ denote the random variable $Z=\mathbf{E}_{\gamma(\omega_T)}(h)$, which is obviously $\mathfrak{F}_T$-measurable since $\gamma(\omega_T)$ is. Let $\phi$ be any non negative, bounded and $\mathfrak{F}_T$-measurable function. Denote by $R$ the set of finite trajectories taken by~$\omega_T$. Then, since $R$ is at most countable: \begin{equation} \label{eq:22} \mathbf{E}_\alpha\bigl(\phi\cdot h\circ\theta_T\bigr)=\sum_{\begin{substack}v\in R\end{substack}} \mathbf{E}_\alpha\bigl(\un{\omega_T=v}\cdot\phi\cdot h\circ\theta_T\bigr). \end{equation} Since $T$ is a stopping time, and since $T^{-1}(v)\neq\emptyset$ if $v\in R$, we have $\{\omega_T=v\}=\uparrow v$. Furthermore, $\phi$~is constant on $\{\omega_T=v\}$, so that if $\phi(v)$ denote this constant, we get: \begin{align} \label{eq:23} \notag \mathbf{E}_\alpha\bigl(\un{\omega_T= v}\cdot\phi\cdot h\circ\theta_T\bigr)&= \phi(v)\mathbf{P}_\alpha\bigl(\uparrow v\bigr) \frac{\mathbf{E}_\alpha\bigl(\un{\uparrow v}h\circ\theta_T\bigr)} {\mathbf{P}_\alpha\bigl(\uparrow v\bigr)} \end{align} Recognizing the conditional expectation defined as the future of $v$ w.t.r.\ to probability~$\mathbf{P}_\alpha$, we use the Markov property~\eqref{eq:4} of Definition~\ref{def:3} to get: \begin{equation} \label{eq:24} \notag \mathbf{E}_\alpha\bigl(\un{\omega_T= v}\cdot\phi\cdot h\circ\theta_T\bigr)= \mathbf{P}_\alpha(\uparrow v)\phi(v)\mathbf{E}_{\gamma(v)}(h). \end{equation} Going back to Eq.~\eqref{eq:22} we obtain: \begin{align} \notag \mathbf{E}_\alpha\bigl(\phi\cdot h\circ\theta_T\bigr)&=\sum_{\begin{substack}v\in R\end{substack}} \mathbf{P}_\alpha(\uparrow v)\phi(v)\mathbf{E}_{\gamma(v)}(h)=\mathbf{E}_\alpha \bigl(\phi Z\bigr). \end{align} This shows that $Z=\mathbf{E}_\alpha(h\circ\theta_T|\mathfrak{F}_T)$. \qed The following result is a typical application of the Strong Markov property in Markov chain theory that applies here too. It intuitively says this: the probability of returning infinitely often to a state~$\beta$, starting from~$\alpha$, is the product of the probability of hitting $\beta$ once starting from~$\alpha$, by the probability of returning to $\beta$ infinitely often, starting from~$\beta$. \vfill \begin{lem} \label{lem:7} Let $\alpha,\beta$ be two global states, and let $(R^n_\beta)_{n\geq1}$ be the successive return times to~$\beta$. Let $h=\un{\bigcap_{n\geq1}\{R^n_\beta<\infty\}}$\,. Then: \begin{equation} \label{eq:40} \mathbf{E}_\alpha(h)=\mathbf{P}_\alpha(R_\beta<\infty)\cdot\mathbf{E}_\beta(h)\,. \end{equation} \end{lem} \vfill \proof Applying the Asynchronous Strong Markov property (Theorem~\ref{thr:2}) with stopping time $R_\beta$ and function~$h$, we get: $\mathbf{E}_\alpha\bigl(h\circ\theta_{R_\beta}|\mathfrak{F}_{R_\beta}\bigr)= \mathbf{E}_{\gamma(\omega_{R_\beta})}(h)$\,. The right side of this equality is simply the constant $\mathbf{E}_\beta(h)$ on $\{R_\beta<\infty\}$. We multiply both sides by $\un{R_\beta<\infty}$, which is $\mathfrak{F}_{R_\beta}$-measurable by definition of $\mathfrak{F}_{R_\beta}$ and can therefore be put inside the $\mathbf{E}_\alpha(\cdot|\mathfrak{F}_{R_\beta})$ sign, to obtain: \begin{equation*} \label{eq:39} \mathbf{E}_\alpha(\un{R_\beta<\infty} h\circ\theta_{R_\beta}|\mathfrak{F}_{R_\beta})= \un{R_\beta<\infty}\mathbf{E}_\beta(h). \end{equation*} We observe that $\un{R_\beta<\infty}h\circ\theta_{R_\beta}=h$, and therefore $\mathbf{E}_\alpha(h|\mathfrak{F}_{R_\beta})=\un{R_\beta<\infty}\mathbf{E}_\beta(h)$. Taking the $\mathbf{E}_\alpha$-expectations yields identity~(\ref{eq:40}). \qed \vfill \subsection{Recurrent and Transient Global States} \label{sec:recurr-transc-glob} In Markov chain theory, the Strong Markov property is a fundamental tool for studying so-called recurrent states, those states to which the chain returns infinitely often almost surely. There is a strong parallel between Markov chain theory and this part of {\normalfont M2CP}\ theory: recurrence concerns global states, and the infinite return is defined through the successive return times defined in~\S~\ref{sec:exampl-stopp-times}. And the Asynchronous Strong Markov property is the fundamental tool in this study. Denoting as in Definition~\ref{def:8} by $(R^n_\alpha)_{n\geq1}$ the successive returns to $\alpha\in X_0$\,, we say that a global trajectory $\omega\in\Omega$ \tdefine{returns infinitely often} to~$\alpha$ if $R^n_\alpha(\omega)<\infty$ for all integers $n\geq1$. \vfill \begin{propdef} \label{prop:3} Let $(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}. \begin{enumerate}[\em(1)] \item\label{item:14} For any $\alpha\in X_0$, the set of trajectories that return infinitely often to $\alpha$ has\/ $\mathbf{P}_\alpha$-probability either $0$ or~$1$. Following Markov chain terminology, we will say that: \begin{itemize} \item $\alpha$ is \tdefine{recurrent} if\/ $\mathbf{P}_\alpha\bigl(\bigcap_{n\geq1}\{R^n_\alpha<\infty\}\bigr)=1$, which is equivalent to: \begin{equation*} \mathbf{P}_\alpha(R_\alpha<\infty)=1\,. \end{equation*} \item $\alpha$ is \tdefine{transient} if\/ $\mathbf{P}_\alpha\bigl(\bigcap_{n\geq0}\{R^n_\alpha<\infty\}\bigr)=0$, which is equivalent to: \begin{equation*} \mathbf{P}_\alpha(R_\alpha<\infty)<1. \end{equation*} \end{itemize} \item\label{item:15} There is at least one recurrent state in~$X_0$\,. \item\label{item:16} If $\alpha$ is a recurrent state, then the successive returning trajectory to $\alpha$ defined by $\rho_n=\bigl(\theta_{R^{n-1}_\alpha}(\omega)\bigr)_{R_\alpha}$ for $n\geq1$, form a sequence of independent and identically distributed finite trajectories w.r.t.\ probability\/~$\mathbf{P}_\alpha$. \item\label{item:17} If $\alpha$ is a recurrent state, and if $\beta$ is reachable from~$\alpha$, then $\beta$ is recurrent and $\alpha$ is reachable from~$\beta$. \end{enumerate} \end{propdef} \bigskip \ \pagebreak \proof \ \begin{enumerate}[(1)] \item The proof is adapted from~\cite[Proposition~1.2 p.65]{revuz75}. Recall the usual transformation, for a measurable subset $A$ and some sub-\mbox{$\sigma$-alge}\-bra\ $\mathfrak{G}$ of a probability space $(\Omega,\mathfrak{F},\mathbf{P})$: $\mathbf{P}(A)=\mathbf{E}(\mathbf{1}_A)=\mathbf{E}\bigl(\mathbf{E}(\mathbf{1}_A|\mathfrak{G})\bigr)$. Putting $R=R_\alpha$ and $R^n=R^n_\alpha$, we apply this transformation to $(\Omega,\mathfrak{F},\mathbf{P}_\alpha)$ with $A=\{R^n<\infty\}$ and $\mathfrak{G}=\mathfrak{F}_{R^{n-1}}$: \begin{align*} \mathbf{P}_\alpha(R^n<\infty) &=\mathbf{E}_\alpha\bigl( \mathbf{E}_\alpha(\un{R^n<\infty}|\mathfrak{F}_{R^{n-1}})\bigr). \end{align*} Since $R^n=R^{n-1}+R\circ\theta_{R^{n-1}}$ we have: $\un{R^n<\infty}=\un{R^{n-1}<\infty}\cdot \un{R\circ\theta_{R^{n-1}}<\infty}$\,. Since $\un{R^{n-1}<\infty}$ is $\mathfrak{F}_{R^{n-1}}$-measurable, the usual property of conditional expectation yields: \begin{align} \label{eq:30} \mathbf{P}_\alpha(R^n<\infty) &=\mathbf{E}_\alpha\bigl( \un{R^{n-1}<\infty}\mathbf{E}_{\alpha}(\un{R\circ\theta_{R^{n-1}}<\infty}|\mathfrak{F}_{R^{n-1}}) \bigr). \end{align} Applying the Asynchronous Strong Markov property (Theorem~\ref{thr:2}) with stopping time $R^{n-1}$ and function $\un{R<\infty}$ we have: \begin{equation} \label{eq:41} \mathbf{E}_\alpha(\un{R\circ\theta_{R^{n-1}}<\infty}|\mathfrak{F}_{R^{n-1}})= \mathbf{E}_{\gamma(\omega_{R^{n-1}})}(\un{R<\infty})\,. \end{equation} Since $\gamma(\omega_{R^{n-1}})=\alpha$ on $\{R^{n-1}<\infty\}$, multiplying both sides of~(\ref{eq:41}) by $\un{R^{n-1}<\infty}$ brings: \begin{equation} \label{eq:42} \un{R^{n-1}<\infty}\mathbf{E}_\alpha(\un{R\circ\theta_{R^{n-1}}<\infty}|\mathfrak{F}_{R^{n-1}})= \un{R^{n-1}<\infty}\mathbf{E}_\alpha(\un{R<\infty})\,. \end{equation} We take the $\mathbf{P}_\alpha$-expectation of both sides of~(\ref{eq:42}) and report the result in~(\ref{eq:30}) to obtain: \begin{equation} \label{eq:43} \mathbf{P}_\alpha(R^n<\infty)=\mathbf{P}_\alpha(R<\infty)\cdot\mathbf{P}_\alpha(R^{n-1}<\infty)\,. \end{equation} It follows from Borel-Cantelli Lemma that $\alpha$ is recurrent if $\mathbf{P}_\alpha(R<\infty)=1$, and transient if $\mathbf{P}_\alpha(R<\infty)<1$. \item Pick any $\alpha\in X_0$\,, and let $(R^n)_{n\geq1}$ be the successive return times to the square set~$X_0$ (cf.~\S~\ref{sec:first-hitting-time-1}). With $\mathbf{P}_\alpha$-probability~$1$, $R^n<\infty$ for all $n\geq1$. It follows from Lemma~\ref{lem:6} applied with $A=X_0$ that, for some $\beta\in X_0$: \begin{equation} \label{eq:16} \mathbf{P}_\alpha\Bigl(\bigcap_{n\geq1}\{ R^n_{\beta}<\infty\}\Bigr) >0. \end{equation} It remains to show that~(\ref{eq:16}) is still valid with $\mathbf{P}_\beta$ in place of~$\mathbf{P}_\alpha$\,. Let $h$ be the non negative function $h=\un{\bigcap_{n\geq1}\{R^n_\beta<\infty\}}$. Then $\mathbf{E}_\alpha(h)=\mathbf{P}_\alpha(R_\beta<\infty)\cdot \mathbf{E}_\beta(h)$ by Lemma~\ref{lem:7}. Since $\mathbf{E}_\alpha(h)>0$ by Eq.~(\ref{eq:16}), it follows that $\mathbf{E}_\beta(h)>0$, showing that $\beta$ is recurrent. \item\label{itemiid} Observe that the $\rho_n$ are related to $R^n_\alpha$ through the identity: $\omega_{R^{n+1}_\alpha}=\omega_{R^n_\alpha}\cdot\rho_n$\,. Now, let $v_1,\dots,v_n$ be $n$ finite trajectories in the range of~$R_\alpha$\,. Since the $R^i_\alpha$ are stopping times, we have the equality $\{\rho_1=v_1,\dots,\rho_{n}=v_{n}\}= \{\omega_{R^{n}_\alpha}=v_1\cdot\ldots\cdot v_{n}\}=\uparrow (v_1\cdot\ldots\cdot v_{n})$\,. The chain rule yields: \begin{equation} \label{eq:25} \mathbf{P}_\alpha(\rho_1=v_1,\dots,\rho_n=v_n)= \mathbf{P}_\alpha(\uparrow v_1\cdot\ldots\cdot v_{n-1})\times\\ \mathbf{P}_\alpha\bigl(\uparrow v_1\cdot\ldots\cdot v_n|\uparrow v_1\cdot\ldots\cdot v_{n-1}\bigr)\,. \end{equation} The Markov property~(\ref{eq:4}) combines with $\gamma(v_1\cdot\ldots\cdot v_{n-1})=\alpha$ to rewrite the conditional probability in Eq.~(\ref{eq:25}) as follows: \[ \mathbf{P}_\alpha\bigl(\uparrow v_1\cdot\ldots\cdot v_n|\uparrow v_1\cdot\ldots\cdot v_{n-1}\bigr)=\mathbf{P}_\alpha(\uparrow v_n)\,. \] Since $v_n$ is in the range of~$R_\alpha$\,, and since $R_\alpha$ is a stopping time, $\uparrow v_n=\{R_\alpha=v_n\}$. We replace thus the conditional probability in Eq.~(\ref{eq:25}) by $\mathbf{P}_\alpha(R_\alpha=v_n)$, and apply $n$ times the same transformation to finally obtain the identity: \begin{equation*} \label{eq:18} \mathbf{P}_\alpha(\rho_1=v_1,\dots,\rho_n=v_n)=\mathbf{P}_\alpha (R_\alpha= v_1) \cdot\ldots\cdot \mathbf{P}_\alpha(R_\alpha= v_n)\,, \end{equation*} showing that the $\rho_n$ are i.i.d.\ random variables, with the law of~$R_\alpha$\,. \item Consider the two measurable and non negative functions: \begin{equation*} h_\alpha=\un{\bigcap_{n\geq1}\{R^n_\alpha<\infty\}}\,,\qquad h_\beta=\un{\bigcap_{n\geq1}\{R^n_\beta<\infty\}}\,. \end{equation*} Since the successive returns to $\alpha$ are i.i.d. by virtue of point~\ref{itemiid} above, each $\rho_n$ one has positive $\mathbf{P}_\alpha$-probability of hitting~$\beta$, otherwise the $\mathbf{P}_\alpha$-probability of ever hitting $\beta$ would be zero, contradicting the assumption that $\beta$ is reachable from~$\alpha$. Hence, by Borel-Cantelli Lemma, $\mathbf{E}_\alpha(h_\beta)>0$. Since $\mathbf{E}_\alpha(h_\beta)=\mathbf{P}_\alpha(R_\beta<\infty)\cdot\mathbf{E}_\beta(h_\beta)$ by Lemma~\ref{lem:7}, this implies that $\mathbf{E}_\beta(h_\beta)>0$ and thus $\beta$ is recurrent. To prove that $\alpha$ is reachable from~$\beta$, we apply the Asynchronous Strong Markov property (Theorem~\ref{thr:2}) with stopping time $\mathfrak{F}_{R_\beta}$ and function~$h_\alpha$\,. We then multiply the resulting identity by $\un{R_\beta<\infty}$, and take into account that $\un{R_\beta<\infty}$ is $\mathfrak{F}_{R_\beta}$-measurable on the one hand, and that $\gamma(\omega_{R_\beta})=\beta$ on $\{R_\beta<\infty\}$ on the other hand to obtain: \begin{equation} \label{eq:21} \mathbf{E}_\alpha\bigl( \un{R_\beta<\infty}h_\alpha\circ\theta_{R_\beta}|\mathfrak{F}_{R_\beta}\bigr) =\un{R_\beta<\infty}\cdot \mathbf{E}_\beta(h_\alpha)\,. \end{equation} Observe that $h_\alpha\circ\theta_{R_\beta}=h_\alpha$ on $\{R_\beta<\infty\}$, therefore the following identity is valid everywhere: \mbox{$\un{R_\beta<\infty}h_\alpha\circ\theta_{R_\beta}=\un{R_\beta<\infty}h_\alpha$}\,. By assumption, $\alpha$~is recurrent, hence $h_\alpha=1$ $\mathbf{P}_\alpha$-almost surely, and finally $\un{R_\beta<\infty}h_\alpha\circ\theta_{R_\beta}=\un{R_\beta<\infty}$ $\mathbf{P}_\alpha$-almost surely. Replacing thus \mbox{$\un{R_\beta<\infty}h_\alpha\circ\theta_{R_\beta}$} by $\un{R_\beta<\infty}$ in Eq.~(\ref{eq:21}), and taking the $\mathbf{P}_\alpha$-expectations of both sides yields: \[ \mathbf{P}_\alpha(R_\beta<\infty)=\mathbf{P}_\alpha(R_\beta<\infty)\cdot\mathbf{E}_\beta(h_\alpha)\,. \] But $\beta$ is assumed to be reachable from~$\alpha$, hence $\mathbf{P}_\alpha(R_\beta<\infty)>0$, and thus $\mathbf{E}_\beta(h_\alpha)=1$, implying in particular that $\alpha$ is reachable from~$\beta$. \qed \end{enumerate} \subsection{Irreducible Components} \label{sec:irred-comp} With the notion of recurrent state at hand, it is now possible to introduce the notions of irreducible process and irreducible components of a {\normalfont M2CP}. \begin{defi} \label{def:11} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}. We say that\/ $\mathbb{P}$ is \tdefine{irreducible} if every $\alpha\in X_0$ is reachable from every $\beta\in X_0$\,. \end{defi} \begin{prop} \label{prop:5} If a {\normalfont M2CP}\ is irreducible, then every global state is recurrent. \end{prop} \proof By Proposition~\ref{prop:3}, point~\ref{item:15}, there is some recurrent state $\alpha\in X_0$. But then, since any $\beta\in X_0$ is reachable from~$\alpha$, $\beta$~is recurrent by point~\ref{item:17} of the same proposition. \qed The result in Proposition~\ref{prop:9} below says that the study of Markov two-components process es essentially reduces to the study of irreducible processes, especially if one is interested in asymptotic properties (so-called limit theorems from probability theory such as the Law of Large Numbers or the Central Limit Theorem). For this we use the notion of subprocess introduced in Definition~\ref{def:1}, and introduce irreducible components for {\normalfont M2CP} s. \pagebreak \begin{defi} \label{def:16} An \tdefine{irreducible component} of a probabilistic two-components process\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ is any subset $X_1\subset X_0$ such that, for all $\alpha\in X_1$~: \begin{enumerate}[(1)] \item\label{item:3} any $\beta\in X_1$ is reachable from~$\alpha$; and \item\label{item:4} if $\beta\in X_0$ is reachable from~$\alpha$, then $\beta\in X_1$\,. \end{enumerate} \end{defi} Point~\ref{item:4} in Definition~\ref{def:16} ensures that $(\mathbf{P}_\alpha)_{\alpha\in X_1}$ is indeed a probabilistic process (Definition~\ref{def:1}). Therefore, if $X_1$ is an irreducible component of {\normalfont M2CP}\ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$, then the family $(\mathbf{P}_\alpha)_{\alpha\in X_1}$ forms a subprocess of~$\mathbb{P}$, which is obviously an irreducible {\normalfont M2CP}. It follows from Proposition~\ref{prop:5} that any element $\alpha$ of an irreducible component is recurrent. Any two irreducible components are disjoint. Finally, if $\alpha$ is recurrent, then $\alpha$ belongs to a unique irreducible component, namely the set $X_1$ of those $\beta$ which are reachable from~$\alpha$ (the fact that $X_1$ is indeed an irreducible component follows from Proposition~\ref{prop:3}). Since recurrent states exist by Proposition~\ref{prop:3}, this implies that any {\normalfont M2CP}\ has at least one irreducible component. \begin{prop} \label{prop:9} If\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ is a {\normalfont M2CP}, there exists a stopping time $T:\Omega\to\mathcal{T}$ such that $T$ is almost surely finite and $\gamma(\omega_T)$ belongs to some irreducible component of\/~$\mathbb{P}$. \end{prop} \proof We fix an initial state $\alpha\in X_0$\,. Let $(R^n)_{n\geq1}$ denote the successive return times to the square set~$X_0$ (cf.~\S~\ref{sec:first-hitting-time-1}). As already observed several times, $\mathbf{P}_\alpha(R^n<\infty)=1$ for all $n\geq1$, and therefore, if we put $B=\bigl\{\beta\in X_0\;|\; \mathbf{P}_\alpha\bigl(\bigcap_{n\geq1}R^n_\beta<\infty\}\bigr)>0\bigr\}$\,, it follows from Lemma~\ref{lem:6} that: \begin{equation} \label{eq:46} \Omega=\bigcup_{\beta\in B}\bigcap_{n\geq1}\{R^n_\beta<\infty\}\,. \end{equation} Pick exactly one global state $\alpha_i$ for each irreducible component. Let $T_i$ be the first hitting time of~$\alpha_i$, and put: \begin{equation} \label{eq:36} \notag \forall\omega\in\Omega,\quad\omega_T=\inf_i\{\omega_{T_i}\}. \end{equation} For each $\beta\in B$, let $\widetilde{\beta}$ be the unique recurrent state $\alpha_i$ of the same irreducible component. Then $\widetilde{\beta}$ is reachable from~$\beta$, and therefore: \begin{equation} \label{eq:47} \omega\in\bigcap_{n\geq1}\{R^n_\beta<\infty\}\Rightarrow R_{\widetilde{\beta}}(\omega)<\infty\quad\text{$\mathbf{P}_\alpha$-a.s.} \end{equation} >From Eqs.~(\ref{eq:46})(\ref{eq:47}) we deduce that $\omega_T<\infty$ $\mathbf{P}_\alpha$-almost surely. Hence, on the one hand, at least one $T_i$ is finite $\mathbf{P}_\alpha$-almost surely. On the other hand, only one of them is finite, since the $\alpha_i$ have been chosen in different irreducible components. Therefore $\omega_T=\omega_{T_i}$ for some~$T_i$, and thus $\gamma(\omega_T)$ does belong to some irreducible component, as claimed. \qed \subsection{Open and Closed Markov Two-Components Processes} \label{sec:open-closed-tcp} Besides the classical application of the Strong Markov Property to recurrence and transience, it also applies to the notion of open and closed processes which is specific to the two-components framework. Open and closed processes have been defined in Definition~\ref{def:17}. \begin{prop} \label{prop:6} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}. \begin{enumerate}[\em(1)] \item\label{item:18} Let $\alpha\in X_0$ be a recurrent state. Then a global trajectory $\omega$ synchronizes infinitely often with\/ $\mathbf{P}_\alpha$-probability\/ $1$ if at least some synchronization state is reachable from~$\alpha$, and with\/ $\mathbf{P}_\alpha$-probability\/ $0$ otherwise. \item If\/ $\mathbb{P}$ is irreducible, then\/ $\mathbb{P}$ is closed or open. \end{enumerate} \end{prop} \proof\ \begin{enumerate}[(1)] \item\label{item:9} Let $R_Q$ be first return time to the square set $X_0\cap(Q\times Q)$ (see~\S~\ref{sec:first-hitting-time-1}), and consider the stopping time $U=R_Q+R_\alpha\circ\theta_{R_Q}$\,, corresponding to reaching $\alpha$ after having reached~$Q$. This is indeed a stopping time by virtue of Lemma~\ref{lem:8}. Let $(U_n)_{n\geq1}$ be the iterated stopping times associated with $U$ as in Definition~\ref{def:15}. If $h_n=\un{U_n<\infty}$, the same technique involving the Asynchronous Strong Markov property (Theorem~\ref{thr:2}) than in the proof of Proposition~\ref{prop:3}, point~\ref{item:14}, shows that: $\mathbf{E}_\alpha(h_n|\mathfrak{F}_{U_{n-1}})=h_{n-1}\mathbf{E}_\alpha(\un{U<\infty})$. Therefore, if $a=\mathbf{P}_\alpha(U<\infty)$ one has $\mathbf{P}_\alpha(U_n<\infty)=a^n$\,. Since $\alpha$ is recurrent, Proposition~\ref{prop:3}, point~\ref{item:14} implies that $R_\alpha$ is $\mathbf{P}_\alpha$-almost surely finite, hence \mbox{$\mathbf{P}_\alpha(U<\infty)=\mathbf{P}_\alpha(R_Q<\infty)$}. Since a trajectory $\omega$ synchronizes infinitely often if and only if $U_n<\infty$ for all $n\geq1$, Borel-Cantelli Lemma implies that $\omega$ has $\mathbf{P}_\alpha$-probability $1$ of synchronizing infinitely often if $\mathbf{P}_\alpha(R_Q<\infty)=1$, and $0$ otherwise. It remains to show that $\mathbf{P}_\alpha(R_Q<\infty)=1$ if and only if some state of the form $(x,x)$ with $x\in Q$ is reachable from~$\alpha$. Since $R_Q=\bigwedge_{x\in Q}R_{(x,x)}$\,, obviously if no $(x,x)$ is reachable from $\alpha$ then $\mathbf{P}_\alpha(R_Q<\infty)=0$. Conversely, assume that some $(x,x)$ with $x\in Q$ is reachable from~$\alpha$. Then $(x,x)$ is recurrent, by point~\ref{item:17} of Proposition~\ref{prop:3}, and Lemma~\ref{lem:7} implies that $R_{(x,x)}<\infty$ $\mathbf{P}_\alpha$-almost surely. But $R_Q\leq R_{(x,x)}$, hence $\mathbf{P}_\alpha(R_Q<\infty)=1$, as claimed. \item If $\mathbb{P}$ is irreducible, then by Proposition~\ref{prop:3}, every $\alpha\in X_0$ is recurrent, therefore point~\ref{item:9} above applies to any $\alpha\in X_0$. Assume that the $\mathbf{P}_\alpha$-probability of synchronizing infinitely often is~$0$ for some $\alpha\in X_0$\,, and let $\beta\in X_0$\,. Consider a finite trajectory $v$ such that $\mathbf{P}_\alpha(v)>0$ and $\gamma(v)=\beta$; such a $v$ exists since any $\beta$ is reachable from~$\alpha$. Then $\mathbf{P}_\alpha$-\text{a.s.}\ every trajectory $\omega\in\uparrow v$ has no synchronization. But the $\mathbf{P}_\alpha$ probability measure on $\uparrow v$ coincides, up to the factor $\mathbf{P}_\alpha(v)\neq0$, with $\mathbf{P}_\beta$ on~$\Omega$. Hence $\mathbf{P}_\beta$-\text{a.s.}\ every $\omega\in\Omega$ has no synchronization, and since this is true for every $\beta\in X_0$, the process $\mathbb{P}$ is open. The same method applies to show that $\mathbb{P}$ is closed if the probability of synchronizing infinitely often is $1$ for some $\alpha\in X_0$. This concludes the proof.\qed \end{enumerate} \section{The Local Independence Property} \label{sec:prop-mark-two} Having adapted Markovian concepts from Markov chain theory, we now focus on a topic specific to the asynchronous framework, without equivalent in Markov chain theory: the probabilistic correlation between private behaviors of local components. It is desirable to have a kind of probabilistic independence between private parts of trajectories: otherwise, hidden synchronization constraints would be encoded in the probabilistic structure, while we expect synchronization to occur only on explicit synchronization states. Probabilistic independence of random variables $\omega^1$ and $\omega^2$ however is too much to ask; their synchronization is an obstacle to their mere probabilistic independence. This is easy to understand from an information theoretic viewpoint: the knowledge of $\omega^1$ gives indeed information on~$\omega^2$, since it precisely determines the $Q$-sequence of~$\omega^2$. The weaker notion of conditional independence proves to be adapted to our purpose. The Local Independence Property that we introduce informally states that the two local components have the maximal probabilistic independence they can have, considering their natural synchronization constraints. \medskip Recall that $Y=\seq Yn$ has been defined in Definition~\ref{def:17} as the $Q$-sequence induced by some trajectory $\omega\in\Omega$, to which we have added $Y_{-1}=*$ and $Y_n=*$ for large $n$ if the $Q$-sequence is finite, for some fixed specified value~$*$. We proceed in a similar way to define the sequence $\seq\sigma n$ of random elementary trajectories, referring to the decomposition of a trajectory $\omega$ as a concatenation of elementary trajectories from Proposition~\ref{prop:1}. If $\sigma_n$ is defined only until some integer~$N$ (that is, in case \ref{item:2} of Proposition~\ref{prop:1}) we define $\sigma_{N+1}$ as the synchronization free trajectory such that $\omega=\sigma_1\cdot\ldots\sigma_N\cdot\sigma_{N+1}$ and $\sigma_n=*$ for $n>N+1$. Then we observe the following property: \begin{prop} \label{prop:7} Let\/ $\mathbb{P}$ be the synchronization product of two Markov chains. Decomposing $\sigma_n$ as $\sigma_n=(\sigma^1_n,\sigma^2_n)$ we have: for all $\alpha\in X_0$ and for every integer $n\geq0$, $\sigma^1_n$~and $\sigma^2_n$ are two random variables independent conditionally on the pair $(Y_{n-1},Y_n)$ with respect to\/~$\mathbf{P}_\alpha$\,. \end{prop} \proof Since $\mathbb{P}$ satisfies the Markov property, the statement is equivalent to the independence of $\sigma_n^1$ and~$\sigma^2_n$, conditionally on~$Y_n$, and with respect to~$\mathbf{P}_{Y_{n-1}}$\,. But this follows from the construction of the law of $\sigma_n=(\sigma^1_n,\sigma^2_n)$ given in~\S~\ref{sec:synchr-two-mark}. \qed In order to generalize the above property to processes which may not be closed, and at the cost of a little more abstraction, we introduce the following definition. \begin{defi} \label{def:12} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}, let\/ $Y$ be the associated random synchronization sequence. Let $\omega^1$ and $\omega^2$ denote the local components of global trajectories, so that $\omega=(\omega^1,\omega^2)$ for $\omega\in\Omega$. We say that\/ $\mathbb{P}$ has the \tdefine{local independence property} (abbreviated {\normalfont\tdefine{LIP}}) if\/ $\omega^1$ and\/ $\omega^2$ are independent conditionally\footnote{Recall that two random variables $X_1$ and $X_2$ are independent w.r.t.\ a \mbox{$\sigma$-alge}\-bra~$\mathfrak{G}$ if $\mathbf{E}(\varphi_1\cdot\varphi_2|\mathfrak{G})=\mathbf{E}(\varphi_1|\mathfrak{G}) \cdot\mathbf{E}(\varphi_2|\mathfrak{G})$, for all non negative and bounded variables $\varphi_1$ and~$\varphi_2$, measurable with respect to $X_1$ and to $X_2$ respectively. See~\textsl{e.g.}~\cite[Chapter~IV]{neveu65}. Here, the independence conditionally to $Y$ means the independence w.r.t.\ the \mbox{$\sigma$-alge}\-bra\ $\langle Y\rangle$ generated by~$Y$.} to~$Y$ with respect to\/~$\mathbf{P}_\alpha$\,, for all \mbox{$\alpha\in X_0$}\,. \end{defi} The following theorem relates this definition with the previous property stated in Proposition~\ref{prop:7} for the synchronization of Markov chains. \begin{thm} \label{thr:3} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}. Then\/ $\mathbb{P}$ satisfies the {\normalfont LIP} if and only if the random variables $\sigma_n^1$ and $\sigma_n^2$ are independent conditionally on the pair $(Y_{n-1},Y_n)$, with respect to\/ $\mathbf{P}_\alpha$ for all $n\geq0$ and for all $\alpha\in X_0$\,. \end{thm} \proof Let $(a)$ be the property that $\omega^1$ and $\omega^2$ are independent conditionally on~$Y$, and let $(b)$ be the property stated in the theorem. \medskip \textsl{Proof of $(a)\Rightarrow(b)$.}\quad Thanks to the Markov property, it is enough to consider \mbox{$n=1$}. We denote $\sigma^1_1$ and $\sigma^2_1$ by $\sigma^1$ and~$\sigma^2$\,, and we put: $Z^1=\mathbf{E}_\alpha\bigl(\un{\sigma^1=z^1}\,|\,Y\bigr)$\,, $Z^2=\mathbf{E}_\alpha\bigl(\un{\sigma^2=z^2}\,|\,Y\bigr)$\,, and $ Z=\mathbf{E}_\alpha\bigl(\un{\sigma^1=z^1,\,\sigma^2=z^2}\,|\,Y\bigr)$\,. These three random variables are constant on $\{Y_1=b\}$ and, by~$(a)$, satisfy $Z=Z^1\cdot Z^2$, whence: \begin{align*} \mathbf{P}_\alpha(\sigma^1=z^1,\,\sigma^2=z^2|Y_1=b)&= Z\big|_{\{Y_1=b\}}\\ &= Z^1\big|_{\{Y_1=b\}}\cdot Z^2\big|_{\{Y_1=b\}}\\ &=\mathbf{P}_\alpha(\sigma^1=z^1\,|Y_1=b)\times \mathbf{P}_\alpha(\sigma^2=z^2|Y_1=b)\,, \end{align*} as expected. \medskip \textsl{Proof of $(b)\Rightarrow(a)$.}\quad From $(b)$ used in conjunction with the Markov property and the chain rule, we get for integers $m\geq n$ and with short notations: \begin{multline} \label{eq:33} \notag \mathbf{P}_\alpha\bigl(\sigma^1_1,\ldots,\sigma^1_m,\sigma^2_1,\ldots,\sigma^2_m\big| Y_1,\ldots,Y_n\bigr)= \mathbf{P}_\alpha\bigl(\sigma^1_1,\ldots,\sigma^1_m\big| Y_1,\ldots,Y_n\bigr)\times\\ \mathbf{P}_\alpha\bigl(\sigma^2_1,\ldots,\sigma^2_m\big| Y_1,\ldots,Y_n\bigr). \end{multline} The \mbox{$\sigma$-alge}\-bra\ generated by the random trajectories $(\sigma^i_k,\,k\geq 1)$ for $i=1,2$ coincides with the \mbox{$\sigma$-alge}\-bra\ generated by~$\omega^i$, since $\omega^i$ is obtained as the concatenation of these---the concatenation being finite or infinite. Hence, for any bounded non negative and measurable functions $h^1$ and~$h^2$: \begin{equation*} \mathbf{E}_\alpha\bigl( h^1(\omega^1)\cdot h^2(\omega^2)\big| Y_1,\ldots,Y_n\bigr)= \mathbf{E}_\alpha\bigl(h^1(\omega^1)\big| Y_1,\ldots,Y_n\bigr) \cdot \mathbf{E}_\alpha\bigl(h^2(\omega^2)\big| Y_1,\ldots,Y_n\bigr). \end{equation*} The sequence of \mbox{$\sigma$-alge}\-bra s $\langle Y_1,\ldots,Y_n\rangle$ is increasing, and converges to~$\langle Y\rangle$. Therefore by the special case \cite[Theorem~35.6 p.470]{billingsley95} of the Martingale convergence theorem, we get by taking the limit $n\to\infty$: \begin{equation} \label{eq:35} \notag \mathbf{E}_\alpha\bigl( h^1(\omega^1)\cdot h^2(\omega^2)\big|Y\bigr)= \mathbf{E}_\alpha\bigl(h^1(\omega^1)\big|Y\bigr) \cdot \mathbf{E}_\alpha\bigl(h^2(\omega^2)\big|Y\bigr), \end{equation} completing the proof. \qed \begin{cor} \label{cor:1} The synchronization product of Markov chains satisfies the {\normalfont LIP}. \end{cor} Having the specified value $*$ assigned to some $Y_{k}$ and $\sigma_k$ described above has the following effect with regard to Theorem~\ref{thr:3}: the statement is trivial if both $\sigma_k$, $Y_{k-1}$ and $Y_k$ assume their constant values~$*$; but it implies the probabilistic independence of $\sigma^1_{N+1}$ and $\sigma^2_{N+1}$ with respect to~$\mathbf{P}_{Y_{N}}$\,, where $N$ is the last synchronization index. In other words, the local trajectories are independent after their last synchronization. It is useful to examine a degenerated case of Definition~\ref{def:12}, where the conditional independence reduces to probabilistic independence. \begin{lem} \label{lem:2} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}. Let\/ $\omega^1$ and\/ $\omega^2$ denote the local components of global trajectories. Assume that, with respect to\/ $\mathbf{P}_\alpha$ for some state $\alpha\in X_0\,$, the two components\/ $\omega^1$ and\/ $\omega^2$ are independent. Then $\omega^1$ and\/ $\omega^2$ are the sample paths of two independent Markov chains. \end{lem} \proof Fix $\alpha\in X_0$, and for each $i=1,2$ let $P^i$ denote the law of~$\omega^i$, characterized by $P^i(\omega^i\geq s^i)=\mathbf{P}_\alpha(\omega^i\geq s^i)$, with $s^i$ ranging over the finite local trajectories on site~$i$. We show that the conditional law $P^i(s^i\cdot\bullet|\uparrow s^i)$ only depends on the last state of~$s^i$, which is enough to obtain that $\omega^i$ follows the law of a homogeneous Markov chain. Consider $i=1$, the case $i=2$ is identical. Consider $s^1$ a finite sequence in~$S^1$ such that $P^1(\omega^1\geq s^1)>0$. It implies that there exists some sequence in~$S^2$, say~$s^2$, such that $\mathbf{P}_\alpha\bigl(\uparrow(s^1,s^2)\bigr)>0$. Put $s=(s^1,s^2)$ and let $(x^1,x^2)=\gamma(s)$. For any finite sequence $\sigma$ in~$S^1$, we have: \begin{align} \label{eq:29} \notag P^1(\omega^1\geq s^1\cdot\sigma|\omega^1\geq s^1)&= \frac{\mathbf{P}_\alpha(\omega^1\geq s^1\cdot\sigma)}{\mathbf{P}_\alpha(\omega^1\geq s^1)}\\ \notag &=\frac{\mathbf{P}_\alpha(\omega^1\geq s^1\cdot\sigma,\,\omega^2\geq s^2)}{\mathbf{P}_\alpha(\omega^1\geq s^1,\,\omega^2\geq s^2)}&\text{by independence}\\ &=\mathbf{P}_{(x^1,x^2)}\bigl(\omega^1\geq\sigma \bigr). \end{align} Obviously, the expression $P^1(\omega^1\geq s^1\cdot\sigma|\omega^1\geq s^1)$ does not depend on~$x^2$, since $x^2$ is the last state of the arbitrary chosen sequence~$s^2$. Therefore, the right member of~\eqref{eq:29} does not depend on $x^2$ neither, hence it only depends on $x^1$ and~$\sigma$, which was to be proved. \qed \begin{prop} \label{prop:8} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}\ with the {\normalfont LIP}. Let\/ $\omega^1$ and\/ $\omega^2$ denote the local components of global trajectories. \begin{enumerate}[\em(1)] \item If $Q=\emptyset$, then $\omega^1$ and $\omega^2$ are two independent Markov chains, with respect to\/~$\mathbf{P}_\alpha$ for any $\alpha\in X_0$. \item\label{item:5} If $Q$ is a singleton, and if $\alpha$ is a recurrent state, then $\omega^1$ and $\omega^2$ are two independent Markov chains, with respect to\/~$\mathbf{P}_\alpha$. \end{enumerate} \end{prop} \proof\ \begin{enumerate}[(1)] \item\label{item:19} Since $Q=\emptyset$, the synchronization sequence $Y$ is constant, $Y=(*,*,*,\ldots)$. The conditional independence in the definition of the LIP reduces to probabilistic independence. The result follows then by Lemma~\ref{lem:2}. \item Let $Q=\{\beta\}$. By Proposition~\ref{prop:6}, point~\ref{item:18}, $\omega$~synchronizes infinitely often with $\mathbf{P}_\alpha$-probability either $0$ or~$1$. If it is with probability~$0$, then $Y=(*,*,*,\ldots)$ $\mathbf{P}_\alpha$-\text{a.s.}, and the same method than in point~\ref{item:19} above applies. If it is with probability~$1$, then $Y$ is still constant, now $Y=(*,\beta,\beta,\beta,\ldots)$. The same method applies again.\qed \end{enumerate} \begin{cor} \label{cor:2} An open {\normalfont M2CP}\ with the {\normalfont LIP} identifies with two independent homogeneous Markov chains. \end{cor} \section{Characterization of Markov Two-Components Processes with the LIP} \label{sec:gener-constr-mark} The topic of this section is to characterize a {\normalfont M2CP}\ with the LIP by means of a finite family of real numbers, very much as the transition matrix of a Markov chain does. It turns out that the law of a {\normalfont M2CP}\ with the LIP is entirely specified by a finite family of transition matrices. We will also investigate, conversely, if such a family of transition matrices always induces a {\normalfont M2CP}\ with the LIP, providing a more general way of constructing {\normalfont M2CP} s than the synchronization product of Markov chains. We show through a numerical example at the end of the section that not any {\normalfont M2CP}\ can be obtained as the synchronization product of two Markov chains. \subsection{Technical Preliminaries} We begin with two lemmas. \begin{lem} \label{lem:3} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a closed {\normalfont M2CP}, and let $Y$ denote the associated synchronization sequence. Then for any $\alpha\in X_0$, $Y$~is a homogeneous Markov chain with respect to\/~$\mathbf{P}_\alpha$. \end{lem} \proof The formulation of Definition~\ref{def:3} applies to $Y$ as follows: for any two finite sequences $s$ and $u$ in~$Q$, the conditional probability $\mathbf{P}_\alpha(Y\geq s\cdot u|Y\geq s)$ only depends on $u$ and on the last state of~$s$. This shows that $Y$ is a homogeneous Markov chain. \qed \begin{lem} \label{lem:4} Let\/ $\mathbb{P}=(\mathbf{P}_\alpha)_{\alpha\in X_0}$ be a {\normalfont M2CP}\ with the {\normalfont LIP}, let $Y$ denote the associated synchronization sequence, and let $\seq \sigma n$ denote the sequence of elementary trajectories that decompose global trajectories (see~\S~{\normalfont\ref{sec:prop-mark-two}}). Then for every $n\geq0$ and for $i=1,2$, the sequence of states that appear in $\sigma^i_n$ is a stopped Markov chain with respect to the conditional probability $\mathbf{P}_\alpha(\,\cdot\,|Y_{n-1},Y_{n})$. \end{lem} \proof By the Markov property, there is no loss of generality in assuming that $n=0$. Using the notation $\sigma^i=\sigma^i_0$ for short, we thus have to prove that $\sigma^i$ is a stopped Markov chain with respect to $\mathbf{P}_{\alpha}(\,\cdot\,|Y_{0}=y)$, for any value $y\in Q$. We consider $i=1$ only, the case $i=2$ is similar. Let $(X_1,\dots,X_\tau)$ be the sequence of states in~$\sigma^1$, and let $\mathbf{Q}$ denote the conditional probability $\mathbf{Q}=\mathbf{P}_\alpha(\,\cdot\,|Y_0=y)$. Let $x_1,\dots,x_n$ be values in $S^1\setminus Q$, let $x_{n+1}\in Q\cup\{y\}$, and put $\delta=\mathbf{Q}(X_{n+1}=x_{n+1}|X_1=x_1,\dots,X_n=x_n)$. We claim that $\delta$ only depends on $x_n$ and~$x_{n+1}$. Put $\alpha=(x,z)$. We calculate: \begin{align*} \delta&=\frac{\mathbf{Q}(X_1=x_1,\dots, X_{n+1}=x_{n+1})} {\mathbf{Q}(X_1=x_1,\dots, X_{n}=x_{n})}\\ &=\frac{\mathbf{P}_\alpha(X_1=x_1,\dots, X_{n+1}=x_{n+1},X_\tau=y)} {\mathbf{P}_\alpha(X_1=x_1,\dots, X_{n}=x_{n},X_\tau=y)}\,. \end{align*} We can rephrase $\{X_1=x_1,\dots, X_{n+1}=x_{n+1}\}$ in the two-components framework as \mbox{$\{\omega^1\geq (x_1\cdot\ldots\cdot x_{n+1})\}=\uparrow (x_1\cdot\ldots\cdot x_{n+1},\epsilon)$}, observing that $(x_1\cdot\ldots\cdot x_{n+1},\epsilon)$ is indeed a trajectory. The same applies to $\{X_1=x_1,\ldots,X_n=x_n\}=\uparrow(x_1\cdot\ldots\cdot x_n,\epsilon)$. Therefore the calculation continues as follows: \begin{align*} \delta&=\frac{\mathbf{P}_\alpha\bigl(\omega^1\geq(x_1\cdot\ldots\cdot x_{n+1}),\,X_\tau=y\bigr)}{\mathbf{P}_\alpha\bigl(\uparrow(x_1\cdot\ldots\cdot x_{n},\epsilon),\,X_\tau=y\bigr)}\\ &=\frac{\mathbf{P}_\alpha\bigl(\omega^1\geq(x_1\cdot\ldots\cdot x_{n+1}),\;X_\tau=y \big|\uparrow(x_1\cdot\ldots\cdot x_n,\epsilon)\bigr)} {\mathbf{P}_\alpha\bigl(\uparrow(x_1\cdot\ldots\cdot x_n,\epsilon),\, X_\tau=y \big|\uparrow(x_1,\cdot\ldots\cdot x_n,\epsilon)\bigr)}\\ &=\frac{\mathbf{P}_{(x_n,z)}(\omega^1\geq x_{n+1},\,X_\tau=y)} {\mathbf{P}_{(x_n,z)}(X_\tau=y)}=\mathbf{P}_{(x_n,z)}(\omega^1\geq x_{n+1}\big|X_\tau=y). \end{align*} On the last expression, it is clear that $\delta$ only depends on $x_n$ and~$x_{n+1}$, and not on $x_1,\dots,x_n$, showing our claim. This is enough to imply that $X_1,\dots,X_\tau$ are the terms of a homogeneous Markov chain. \qed \subsection{Adapted Family of Transition Matrices} \label{sec:adpat-family-trans} The two above lemmas suggest the following construction for {\normalfont M2CP}\ with the LIP. First consider a Markov chain $Y$ on the set of shared states; then for any two consecutive values $y_{n-1}$ and $y_n$ of~$Y$, consider two independent stopped Markov chains $\sigma^1_n$ and~$\sigma^2_n$, with $\sigma^i_n$ taking values in $\{y_n\}\cup(S^i\setminus Q)$, that reaches $y_n$ with probability one and which is stopped at the first hitting time of~$y_n$. This description is formalized in Theorem~\ref{thr:4} below. It is first convenient to introduce the following definition. \begin{defi} \label{def:13} An \tdefine{adapted family of transition matrices} is given by two families $(R^i_y)_{y\in Q_0}$\,, one for each $i=1,2$ and with $Q_0$ some subset of~$Q$, such that: \begin{enumerate}[(1)] \item For each $y\in Q_0$ and $i=1,2$, $R^i_y$ is a stochastic matrix on $\{y\}\cup(S^i\setminus Q)$; \item With respect to the transition matrix~$R^i_y$\,, the state $y$ is reachable from any state in $S^i\setminus Q$. \end{enumerate} \end{defi} \noindent Using this definition, the existence and uniqueness result concerning {\normalfont M2CP}\ with the LIP states as follows. We focus on closed processes only, as suggested by Proposition~\ref{prop:9}, Proposition~\ref{prop:6} and Corollary~\ref{cor:2}. \newpage \begin{thm} \label{thr:4} Any closed {\normalfont M2CP}\/\ $\mathbb{P}$ with the {\normalfont LIP} induces the following elements, that entirely characterize\/~$\mathbb{P}$: \begin{enumerate}[\em(1)] \item A transition matrix $R$ on the set $Q$ of shared states, defined as the transition matrix of the synchronization sequence $Y$ from Definition~{\normalfont\ref{def:17}}; \item An adapted family of transition matrices $(R^i_y)_{y\in Q_0}$, for $i=1,2$, where $Q_0$ is the essential set of values of\/~$Y$. For each $i=1,2$, and for $y\in Q_0$\,, $R^i_y$~is the transition matrix of the Markov chain~$\sigma^i_n$ with respect to the conditional probability $\mathbf{P}_\alpha(\,\cdot\,|Y_{n-1},Y_n=y)$, which is independent of the integer $n$ and of $\alpha\in X_0$, provided it is defined for these values. \end{enumerate} \noindent Conversely, given a set of global states \begin{equation} \label{eq:19} \notag X_0\subset \bigl\{(x,z)\in S^1\times S^2\;|\; (x\in Q)\wedge(z\in Q)\Rightarrow x=z\bigr\}, \end{equation} such that the set \begin{equation} \label{eq:37} \notag Q_0=\{y\in Q\;|\;(y,y)\in X_0\}. \end{equation} is nonempty; and considering: \begin{enumerate}[\em(1)] \item a transition matrix $R$ on the set~$Q_0$; and \item an adapted family of transition matrices $(R_y^i)_{y\in Q_0}$\,, \end{enumerate} then there exists a unique {\normalfont M2CP}\ with the {\normalfont LIP}, defined on $X_0$ and inducing $R$ and~$(R_y^i)_{y\in Q_0}$\,. This {\normalfont M2CP}\ is closed. \end{thm} \proof The first part of the theorem follows from Lemmas~\ref{lem:3} and~\ref{lem:4}. For the second part, assume that the considered data are given. The construction of the process $\mathbb{P}$ is essentially the same as the construction of the synchronization product of Markov chains, therefore we omit the routine arguments showing the existence and uniqueness of~$\mathbb{P}$. What we need to show is that the two-components process\ obtained is indeed a {\normalfont M2CP}\ with the LIP. The LIP is obvious from the construction of $\mathbb{P}$ combined with Theorem~\ref{thr:3}, hence we focus on the Markov property. Since the process is closed by construction, we rely on Lemma~\ref{lem:1} for this. Hence, let $\alpha\in X_0$, let $t$ be any elementary trajectory and let $s$ be any finite trajectory. The proof then follows the same steps than the proof of Theorem~\ref{thr:1}: \begin{enumerate}[(1)] \item \textsl{Step~$1$: $s$ synchronization free.}\quad Then $s\cdot t$ is an elementary trajectory. Put $\alpha=(x_0,z_0)$, $\gamma(s)=(x_1,z_1)$ and $\gamma(t)=(y,y)$. We have: $\uparrow(s\cdot t)=\{\sigma_1^1=s^1\cdot t^1,\,\sigma^2_1=s^2\cdot t^2\}$. Let $\mathbf{Q}^i_b$ denote the probability associated with the Markov chain starting from $b$ and with transition matrix~$R^i_y$, for $i=1,2$ and $b\in Q$. We compute using the independence conditionally on~$Y_1$: \begin{align*} \bigl(\mathbf{P}_\alpha\bigr)_s\bigl(\uparrow t\bigr)&= \bigl(\mathbf{P}_\alpha\bigr)_s\bigl(\uparrow t\wedge Y_1=y\bigr)\\ &= \frac{\mathbf{P}_\alpha\bigl(\uparrow(s\cdot t)\big|Y_1=y \bigr)}{\mathbf{P}_\alpha\bigl(\uparrow s\big|Y_1=y \bigr)}\\ &=\frac{\mathbf{Q}^1_{x_0}\bigl(\uparrow(s^1\cdot t^1)\bigr)} {\mathbf{Q}^1_{x_0}\bigl(\uparrow s^1\bigr)} \cdot \frac{\mathbf{Q}^2_{z_0}\bigl(\uparrow(s^2\cdot t^2)\bigr)} {\mathbf{Q}^2_{z_0}\bigl(\uparrow s^2\bigr)}\\ &=\mathbf{Q}^1_{x_1}(\uparrow t^1)\cdot \mathbf{Q}^2_{z_1}(\uparrow t^2). \end{align*} The last quantity only depends on $(x_1,z_1)=\gamma(s)$ and~$t$. In particular, as expected, we have $\bigl(\mathbf{P}_\alpha\bigr)_s\bigl(\uparrow t\bigr)=\mathbf{P}_{\gamma(s)}(\uparrow t)$. \item \textsl{Step~$2$: $s$ is any finite trajectory.}\quad Using Step~$1$, as in the proof of Theorem~\ref{thr:1}.\qed \end{enumerate} \subsection{A Numerical Example} \label{sec:few-examples} In this subsection, we show on an example how the synchronization product of Markov chains is to be interprated in terms of an adapted family of transition matrices. We show that not any {\normalfont M2CP}\ can be obtained from the synchronization of two Markov chains. \medskip Let $S^1=\{a,b,\c,\d\}$ and $S^2=\{\c,\d,e,f\}$, and let two transition matrices $M^1$ and $M^2$ on $S^1$ and $S^2$ respectively. Take for instance: \begin{align*} M^1&=\begin{matrix} a\\[\smalldim]b\\[\smalldim]\c\\[\smalldim]\d \end{matrix} \begin{pmatrix} \frac13&\frac13&\frac13&0\\[\smalldim] \frac12&\frac18&\frac18&\frac14\\[\smalldim] \frac12&0&\frac14&\frac14\\[\smalldim] 0&\frac12&\frac14&\frac14 \end{pmatrix} & M^2&=M^1=\begin{matrix} e\\[\smalldim]f\\[\smalldim]\c\\[\smalldim]\d \end{matrix} \begin{pmatrix} \frac13&\frac13&\frac13&0\\[\smalldim] \frac12&\frac18&\frac18&\frac14\\[\smalldim] \frac12&0&\frac14&\frac14\\[\smalldim] 0&\frac12&\frac14&\frac14 \end{pmatrix}\,. \end{align*} The matrices contain $0$ in some places, but that will not harm. \subsubsection*{Computation of the adapted family of transition matrices.} \label{sec:comp-adapt-family} We need to compute the matrices $R^1_{\c}=R^2_{\c}$ and $R^1_{\d}=R^2_{\d}$\,. Matrix $R^1_{\c}$ is a stochastic matrix on $\{a,b,\c\}$, and drives the subsystem on site~$1$, conditionally on ``next synchronization is~$\c$''. Referring to the construction detailed in~\S~\ref{sec:synchr-two-mark}, $R^1_{\c}$~is simply obtained as follows: starting from matrix~$M^1$, suppress all lines and columns attached to states in $Q$ different from~$\c$, here, this is only state~$\d$. Finally, renormalize each line to obtain a stochastic matrix. The same process is applied to obtain~$R^1_{\d}$\,: \begin{align*} R^1_{\c}&= \begin{matrix} a\\[\smalldim]b\\[\smalldim]\c \end{matrix} \begin{pmatrix} \frac13&\frac13&\frac13\\[\smalldim] \frac23&\frac16&\frac16\\[\smalldim] \frac23&0&\frac13 \end{pmatrix}& R^1_{\d}&= \begin{matrix} a\\[\smalldim]b\\[\smalldim]\d \end{matrix} \begin{pmatrix} \frac12&\frac12&0\\[\smalldim] \frac47&\frac17&\frac27\\[\smalldim] 0&\frac23&\frac13 \end{pmatrix} \end{align*} This construction implies that the lines obtained from matrices $R^1_{\c}$ and $R^1_{\d}$ by deleting the lines and columns relative to shared states are \emph{proportional}: $ \begin{pmatrix} \frac13&\frac13 \end{pmatrix}$ is proportional to $ \begin{pmatrix} \frac12&\frac12 \end{pmatrix}$, and $ \begin{pmatrix} \frac23&\frac16 \end{pmatrix}$ is proportional to $ \begin{pmatrix} \frac47&\frac17 \end{pmatrix}$. Indeed, the lines of $R_{\c}^1$ and $R_{\d}^1$ are obtained by renormalization after extraction from the same transition matrix~$M^1$\,. We deduce from this observation a way to construct a {\normalfont M2CP}\ with the LIP not obtained as a synchronization product of Markov chains. Replace for example the $b$ line of $R^1_{\c}$ by $ \begin{pmatrix} 0&0&1 \end{pmatrix}$ and leave $R^1_{\d}$ unchanged. This corresponds to some closed {\normalfont M2CP}\ with LIP according to Theorem~\ref{thr:4}, which cannot be a synchronization product of Markov chains. We have obtained: \emph{not every {\normalfont M2CP}\ with the {\normalfont LIP} can be obtained as the synchronization product of two Markov chains}. \subsubsection*{Computation of the matrix of the synchronization chain.} \label{sec:comp-matr-synchr} It remains to compute the transition matrix of the chain $Y=(Y_n)_{n\geq1}$\,, which involves the law of $X^1_{\tau^1}$ and~$X^2_{\tau^2}$\,, where $\tau^i$ are the first hitting times to $Q$ of chains $X^1$ and $X^2$ respectively, which we do here ``by hand''. For a general theory, see for instance~\cite[Ch.~XII \S\S58--59 \emph{Entrance and exit laws}, p.262\textit{ff}]{dellacherie88}. Denoting by $M^1_{x}$ the law of chain $X^1$ starting from~$x$, one has: $M^1_{x}(X^1_{\tau^1}=\c)=\sum_w M^1_{x}(w)$\,, where $w$ ranges over words of the form $w=v\cdot \c$, and $v$ is any word on $\{a,b\}$. Therefore, if $q_k(x)$ denotes, for any integer $k\geq0$: \[ q_k(x)=\sum_{l_1,\ldots,l_k\in\{a,b\}}M^1_{x}(l_1\cdot\ldots\cdot l_k\cdot \c)\,, \] one has $M^1_{x}(X^1_{\tau^1}=\c)=\sum_{k\geq0}q_k(x)$. Decomposing over the two possible values of $l_1$ yields: \[ q_k(x)=M^1(x,a)q_{k-1}(a)+M^1(x,b)q_{k-1}(b)\,. \] Therefore the vector $\begin{pmatrix} q_k(a)&q_k(b) \end{pmatrix}$ satisfies the following recurrence relation: \[ \begin{pmatrix} q_k(a)\\[\smalldim]q_k(b) \end{pmatrix} =N \begin{pmatrix} q_{k-1}(a)\\[\smalldim]q_{k-1}(b) \end{pmatrix}\,,\qquad\text{with } N= \begin{pmatrix} M^1(a,a)&M^1(a,b)\\[\smalldim] M^1(b,a)&M^1(b,b) \end{pmatrix} \,.\] We observe that $\begin{pmatrix} q_0(a)\\q_0(b) \end{pmatrix}=\begin{pmatrix} M^1(a,\c)\\M^1(b,\c) \end{pmatrix}$ and therefore: \[ \begin{pmatrix} M^1_{a}(X^1_{\tau^1}=\c)\\[\smalldim] M^1_{b}(X^1_{\tau^1}=\c) \end{pmatrix}= (I-N)^{-1}\begin{pmatrix} M^1(a,\c)\\[\smalldim] M^1(b,\c) \end{pmatrix}\,. \] We find in a similar fashion: \[ \begin{pmatrix} M^1_{a}(X^1_{\tau^1}=\d)\\[\smalldim] M^1_{b}(X^1_{\tau^1}=\d) \end{pmatrix}= (I-N)^{-1}\begin{pmatrix} M^1(a,\d)\\[\smalldim] M^1(b,\d) \end{pmatrix}\,, \] with same matrix~$N$. Finally we have: \begin{equation} \label{eq:31} \begin{split} M^1_{\c}(X^1_{\tau^1}=\c)&=M^1(\c,\c)+M^1(\c,a)M^1_a(X^1_{\tau^1}=\c)+M^1(\c,b)M^1_b(X^1_{\tau^1}=\c)\\ M^1_{\c}(X^1_{\tau^1}=\d)&=M^1(\c,\d)+M^1(\c,a)M^1_a(X^1_{\tau^1}=\d)+M^1(\c,b)M^1_b(X^1_{\tau^1}=\d)\,. \end{split} \end{equation} And in a similar fashion: \begin{equation} \label{eq:44} \begin{split} M^1_{\d}(X^1_{\tau^1}=\c)&=M^1(\d,\c)+M^1(\d,a)M^1_a(X^1_{\tau^1}=\c)+M^1(\d,b)M^1_b(X^1_{\tau^1}=\c)\\ M^1_{\d}(X^1_{\tau^1}=\d)&=M^1(\d,\d)+M^1(\d,a)M^1_a(X^1_{\tau^1}=\d)+M^1(\d,b)M^1_b(X^1_{\tau^1}=\d)\,. \end{split} \end{equation} Applying these calculations to our numerical example, we find: \begin{align*} N&= \begin{pmatrix} \frac13&\frac13\\[\smalldim] \frac12&\frac18 \end{pmatrix}& (I-N)^{-1}&=\frac{12}5 \begin{pmatrix} \frac78&\frac13\\[\smalldim] \frac12&\frac23 \end{pmatrix}\\ \begin{pmatrix} M^1_a(X^1_{\tau^1}=\c)\\[\smalldim] M^1_b(X^1_{\tau^1}=\c) \end{pmatrix}&= \frac{12}5\begin{pmatrix} \frac 13\\[\smalldim] \frac14 \end{pmatrix} & \begin{pmatrix} M^1_a(X^1_{\tau^1}=\d)\\[\smalldim] M^1_b(X^1_{\tau^1}=\d) \end{pmatrix}&=\frac{1}5 \begin{pmatrix} 1\\[\smalldim] 2 \end{pmatrix}\,. \end{align*} We obtain thus, using Eqs.~(\ref{eq:31})(\ref{eq:44}): \begin{align*} M^1_{\c}(X^1_{\tau^1}=\c)&=\frac{13}{20}&M^1_{\c}(X^1_{\tau^1}=\d)&=\frac7{20}\\ M^1_{\d}(X^1_{\tau^1}=\c)&=\frac{11}{20}&M^1_{\d}(X^1_{\tau^1}=\d)&=\frac9{20} \end{align*} Since we have taken $M^2=M^1$, we obtain the same laws depending on the initial state $\c$ or $\d$ for $X^2_{\tau^2}$\,. The $2\times2$ transition matrix of $Y$ is now obtained by conditioning the free product $(X^1_{\tau^1},X^2_{\tau^2})$ on $X^1_{\tau^1}=X^2_{\tau^2}$\,, which yields the following transition matrix: \[ \begin{matrix} \c \\[\smalldim] \d \end{matrix} \begin{pmatrix} \frac{169}{218}&\frac{49}{218}\\[\smalldim] \frac{121}{202}&\frac{81}{202} \end{pmatrix}\,. \] \section*{Conclusion} \label{sec:conclusion} \subsection*{Summary of results} \label{sec:summary-results} Following the idea that, in a network, the knowledge a node has about time is related to its local clock, and to its local clock only, we have introduced a probabilistic model based on a simple trace model, that allows private changes of states and synchronizations between two sites. We have focused on a Markov model where local components are independent up to the synchronization constraints, which brought us to the formulation of a Markov property without reference to any time index on the one hand, and to the Local Independence Property on the other hand. Triples $(\Omega,\mathfrak{F},\mathbf{P})$ where $(\Omega,\mathfrak{F})$ is the space of trajectories and $\mathbf{P}$ is a probability measure satisfying both properties have been constructed and entirely characterized by a finite family of transition matrices, extending the familiar transition matrix from discrete time Markov chain theory. A singular feature of the model is the absence of \emph{constant times}; instead, only random times may be considered, and among them stopping times play a distinguished role. Note that despite the absence of a totally ordered time index, we can conduct probabilistic reasoning about our two-components models at the level of stopping times. \subsection*{Potential applications} Open research fields involving asynchronous systems are numerous. In some cases, trace models have proved to be more relevant than interleaving models: distributed observation, supervision and diagnosis of concurrent systems, distributed optimization and planning \cite{benveniste03:_diagn} provide examples. In the formal verification community, people have considered interleaving models for composing probabilistic systems (cf.\ the discussion in the Introduction). Although product of Probabilistic Automata for instance has shown to be efficient for developing proving techniques based on bisimulation relations, it is worth trying other ways for modeling network system where asynchrony plays an important role. One can therefore expect new advances in the theory of networked systems through the development of a probabilistic layer for trace models. In this respect, asymptotic analysis of probabilistic trace models may have applications in network dimensioning. \subsection*{Limitations and extensions} Although the model of Markov concurrent process adopted in this paper is limited to \emph{two} components only, it is important to notice that it has a straightforward generalization to an arbitrary number $n\geq2$ of components. In this generalized framework, the notion of stopping time, the Asynchronous Strong Markov Property and all the results developed in \S~\ref{sec:stopp-times-strong} carry over without additional difficulty. The LIP may also be expressed for $n\geq2$ components in a similar way than we did for two components only. However, the mere existence of Markov processes with $n\geq2$ components is not trivial to prove. This relies on the additional combinatorial complexity that appears when at least four components are involved, since then different synchronization events can occur concurrently. Therefore the simple structure of trajectories given by Proposition~\ref{prop:1} is no longer valid, making in turn the constructions of this paper found in Sections~\ref{sec:synchr-two-mark} and~\ref{sec:gener-constr-mark} ineffective. Nevertheless, the task of proving the existence of Markov processes with the LIP has been tackled in~\cite{abbes11}, generalizing the synchronization product of Markov chains. However, this construction is not very natural, and its main advantage is to encourage further study in this direction, since at least it ensures that the object of study is not empty. Regarding a general theory of Markov multi-components processes, one may retain the following elements from the present paper: firstly, stopping times and the Asynchronous Strong Markov Property have a straightforward extension to $n\geq2$ components. These are basic tools that remain unchanged. Secondly, the generalized LIP allows to focus on the synchronization process only, since it implies a conditional decorrelation between the synchronization process on the one hand, and the private parts of each component on the other hand. The core of the remaining challenge is thus the construction and characterization of the synchronization process---we have shown above that, for two components, the synchronization process identifies with a homogeneous Markov chain, a drastic simplification compared to the general case of an arbitrary number of components. Recent work by G.~Winskel~\cite{winskel13} on probabilistic event structures has shown to be promising in this respect. {\small \section*{Acknowledgments} \label{sec:aknowledgment} Many thanks go to Albert Benveniste from IRISA in Rennes (France) for his support, his help and his friendship. I would like also to thank the anonymous referees for their many comments and suggestions, and the Editor Prakash Panangaden, to whom I am profoundly grateful.}

proofpile-arXiv_069-5907

\section{Introduction} Conversational AI in the form of dialogue agents (DA) and chatbots are nowadays commonly applied in various areas helping with a diverse set of tasks. For example, Amazon Alexa enables people to manage their grocery shopping using natural language, whereas chatbots are used for digital customer-service. Although such agents are constantly evolving and are often described as intelligent, humans still perceive them in the role of a butler \cite{mcmillan2021leaving}. Thus, they are still only trusted fulfilling simple, less risky tasks, and are not applied for tasks that require a higher degree of cooperation, e.g. assisting in stock exchanges, or as a business advisor. A reason for this lack of trust stems from the low degree of conversational intelligence of contemporary systems. Chaves and Gerosa \cite{chaves2021should} describe a chatbot's conversational intelligence as ``to actively participate in the conversation and to demonstrate awareness of the topic discussed, the evolving conversational context, and the flow of the dialog''. However, current systems still mostly act in a reactive manner, i.e. they just react on commands, or express low-level proactive behavior for simple tasks \cite{sarikaya2017technology}, e.g. reminding the user of a specific event which the user explicitly set in the calendar or providing movie recommendations based on user habits or routines. For being trusted and effective in assisting with more complex tasks, the conversational intelligence of DAs needs to be improved. For achieving this, we deem sophisticated and sound proactive dialog strategies inevitable. Proactive behavior of conversational systems can be defined as self-initiated and anticipatory behavior \cite{nothdurft2015finding}. Providing systems with appropriate proactivity during cooperative tasks has shown to benefit a system's helpfulness \cite{peng2019design}, and user satisfaction \cite{baraglia2016initiative}. However, proactivity is a double-edged sword, where irrelevant or untimely interventions may have highly negative consequences. The probably best example in this regard forms Microsoft's infamous office assistant \textsc{Clippit}~\cite{horvitz1998lumiere}. The agent did not act according to the user's expectations and was perceived as distractive and not trustworthy, which resulted in a negative system reception \cite{bickmore2005establishing}. Therefore, the design, modelling, and implementation of effective and trusted proactive dialog is a delicate task. We tackle this issue by presenting the first principled approach to proactive dialog modelling by extending the Reinforcement Learning (RL) based dialog management approach. In modular task-oriented dialog systems, the dialog management component is responsible for deciding on appropriate system responses taking into account a semantic representation of the user's input, as well as the dialog history, and other contextual information with the aim of fulfilling a specific user goal (e.g. see McTear \cite{mctear2020conversational}). For reactive assistance, RL-based dialog management approaches have shown to provide optimal results considering task success and efficiency \cite{young2013pomdp,lemon2008adaptive, rieser2011reinforcement}, as well as user satisfaction \cite{ultes2015quality,ultes2019improving}. Therefore, we adopt a RL approach for optimizing proactive dialog strategies with the goal of improving the conversational intelligence of a system and thus enhancing the human-machine cooperation. Unlike user-initiated reactive dialog systems, that usually focus on optimising solely task-related metrics, proactive dialog system also require a certain social awareness for acting in favour of the user expectations. Recent research \cite{kraus2022design} provides evidence that the level of perceived trust in the system reflects whether a proactive system acts in accordance with the user's expectations. Therefore, we trained a proactive DA in a simulated environment, in which the agent assisted the user with a sequential decision-making task, using both trust estimate and usability measures for providing adequate behavior. In a standardized user simulation evaluation, we compared the RL-based proactive DA with agents using only static and rule-based proactive dialog strategies. The results demonstrate that including trust in the dialog model (and in the reward function) for enabling trust-adaptive dialog proved to be successful for creating socially (i.e. ``human-likeness'') and task efficient agents, where they were to achieve the best compromise of contributing to task completion effectively, but also acting in a trustworthy manner. \section{Related Work} \label{sec:rel} \subsection{Proactive Conversational AI} Proactivity has been an extensive research field both in human-computer interaction (HCI) and AI for the last decades \cite{meurisch2020exploring, yorke2012design, chaves2021should, horvitz1999principles, sarikaya2017technology}. However, the definitions for the concept of proactivity differ greatly between the individual research areas. In recommendation systems \cite{christakopoulou2016towards}, for example, proactive behavior is understood as suggesting specific items for simplifying a user's navigation in large product or information spaces. In non-task-oriented open-domain dialog systems, proactivity is understood as actively leading the dialog and changing the topic from a start to a goal topic (e.g. see Wu et al.; Zhu et al.; Tang et al.; Xu et al. \cite{wu2019proactive,zhu2021proactive, tang2019target, xu2020knowledge}). In both areas, proactive conversation is often modelled using various combinations of knowledge graphs and deep learning models for providing personalized suggestions (recommender systems) or achieving a natural, coherent and engaging dialog flow (open-domain dialog). In personal intelligent assistants \cite{sarikaya2017technology, yorke2012design}, proactive behavior is closely related to concept of mixed-initiative interaction \cite{horvitz1999principles} and may be realized in multiple different ways. In such interactions, a user and an autonomous agent capable of acting on its own collaborate to solve tasks. To provide support, the agent must keep track of the user's activities and goals and weigh the costs and benefits of automated actions. Here, task-oriented proactive dialog is used to communicate and negotiate a system's decision-making process to minimize the risk of system failure and to solve task efficiently. We understand proactive dialog under this definition in the scope of this paper. Here, the essential challenges are the timing and the level of proactive actions. Usually, the degree of proactive behavior in personal assistants can be modelled using different levels of autonomy (LoA). In this regard, Isbell and Pierce \cite{isbell2005ip} adopted the LoA from Sheridan and Verplank \cite{sheridan1978human} for defining the Interface-Proactivity (IP) continuum. The IP continuum consists of five levels and ranges from reactive behavior (``users do it themselves'') to completely autonomous system behavior (``system does it by itself''). Typically, proactive behavior of intelligent personal assistants may span several levels on that continuum. Due to the complexity of the task domains, usually rule-based approaches have been applied to guarantee reliable and predictable behavior. For example, \textsc{Radar} was a personal assistant that could help office employees to solve their tasks more efficiently \cite{faulring2010agent, garlan2007radar}. Similarly, \textsc{Calo} \cite{yorke2012design} was a proactive assistant that helped users with task management in an office environment. It could assist with organizing meetings or reminding of important activities. For deciding the level and timing of proactive dialog, e.g. whether a meeting was scheduled automatically or an interaction was initiated, both examples made use of task-specific dialog managers. They contained knowledge about a specific user's attention and interruption policies. Based on this information, the system could weigh the cost-benefit ratio of proactive behavior and adjust its level of proactivity accordingly. For example, if a high workload was detected, the system could suggest automatically preparing background material for the meeting or offering a reminder. The cost-benefit was calculated using system-related metrics such as the urgency of a proactive behavior or the cost of an error or interruption. The decision on which level of autonomy to choose was based on for the specific task, and fixed rules. Thus, the generated rules for proactive behavior may be not transferable to different scenarios and task domains. Further, they are quite costly to develop, due to the large set of necessary rules for reproducing adequate behavior. In addition, the rules were only designed with a limited set of users. Hence, they may not meet the expectations of all users and could be perceived as inflexible. For these reasons, we deem statistical methods for proactive dialog management beneficial. Especially, the RL paradigm seems to be predestined for proactive dialog management due to its nature of long-term planning and optimal decision-making under uncertainty. Therefore, we apply RL for deciding on optimal proactive system actions during mixed-initiative cooperation. For modelling proactive system behavior, we use several proactive dialog actions defined in previous work \cite{kraus2020effects}. They reflect the degree of interference of autonomous systems in human actions and will be explained in more detail in the Section 3. Contrary to reactive RL-based dialog management, it may not be beneficial to focus only on rewarding task-efficient behavior but also to include a certain social awareness. Due to a shift of control towards the agent. a potential loss of self-autonomy may occur. Therefore, a formation of trust for the user is required, otherwise the assistance possibly will be rejected and becomes obsolete \cite{schaefer2016meta}. For this reason, including a trust measure in the reward function needs to be considered. We shortly describe the concept of human-computer trust (HCT) in the following. \subsection{Human-Computer Trust} Among the various definitions of trust in relationships between artificial agents and humans, we utilize the notion provided by Lee and See \cite{lee2004trust} who describe trust as ``the attitude that an agent will help achieve an individual's goal in a situation characterized by uncertainty and vulnerability''. This often used definition was deemed to be the most suitable for explaining the trust relationship during cooperation. According to Lee and See \cite{lee2004trust}, three factors are proposed to be considered when modeling trust: the human, the autonomous partner, and the environment. Each factor has specific properties that influence the HCT relationship, e.g., gender of the user/agent, personality of the user/agent, degree of automation, anthropomorphism of the agent, type/difficulty of the task. For a more detailed information on the impact of the individual factors, we refer the reader to Schaefer et al. \cite{schaefer2016meta}. In this work, we utilize trust-related properties as features for estimating the user's trust in the proactive DA during cooperation. Details of the trust estimation module are described in Section 3. In human-computer studies, the user's trust in a system is usually measured using subjective measurements based on self-reported questionnaires. For example, Madsen and Gregor \cite{madsen2000measuring} developed a questionnaire based on a hierarchical model of trust where subjects can agree or disagree with statements about the system's trustworthiness. Here, the authors differentiate between affect-based and cognition-based trust. While affect-based trust, mostly refers to attributes relevant for a long-term relationships, cognition-based trust encompasses features relevant for short-term interaction, such as perceived understandability, perceived technical competence, and perceived reliability. In short-term interactions, mostly the functionality and usability of a system are of importance. Especially the system's competence and reliability are proven to have a large impact on the HCT relationship~\cite{muir1996trust,lee1994trust}. In the scope of this paper, we refer to trust by means of cognition-based trust as we observe rather short-term interactions with the proactive DA. For designing conversational strategies to achieve both task success and socially effective interaction between user and system, \cite{pecune2020framework} studied an RL-based approach. For this, the authors made use of a user simulator that included a rapport estimation module \cite{jain2018user} and equipped a conversational system with task-oriented and rapport building behavior, e.g. small talk, self-disclosure. Further, they included the estimated rapport besides task metrics in the reward function for optimising both task and social dialog policies. Training and testing the agent with the social user simulator showed the usefulness of their approach. In summary, we study how to include the concept of trust for RL-based proactive dialog management. For this, we contribute a simulation-based framework for equipping proactive DAs with social awareness concerning the HCT relationship between user and agent. For integrating socially awareness into conversational systems, Pecune and Marsella \cite{pecune2020framework} also studied an RL-based approach. Contrary to our work, the authors made use of a user simulator that included a rapport estimation module \cite{jain2018user} and equipped a conversational system with task-oriented and rapport building behavior, e.g. small talk, self-disclosure. Further, they included the estimated rapport besides task metrics in the reward function for optimising both task and social dialog policies. Training and testing the agent with the social user simulator showed the usefulness of their approach. In this paper, we contribute an evaluation of our proposed approach for testing whether a socially-aware proactive DA enhances cooperation by measuring task success, duration and the user's trust in the agent's actions. \label{sec:sim} \begin{figure*} \centering \includegraphics[scale=0.5]{OverviewSociallyAware.png} \caption{Components and information flows of the simulated RL-based proactive dialog environment.} \label{img:arch} \end{figure*} Typically, reactive RL-based DAs are trained using some kind of bootstrapping approach \cite{rieser2011reinforcement}. According to this approach, firstly, dialog data is collected using a Wizard-of-Oz (WoZ) approach. Based on this data set, different components of the simulated dialog environment can be build, e.g. user simulation, or noise model. Dialog policies can then be trained and evaluated in interaction with this simulated environment. For training proactive dialog policies with an RL-based DA, we used a similar approach. However, data was not collected in a WoZ fashion but with a simplistic proactive DA. Moreover, for enabling socially-aware RL, we integrated a trust estimation module in the simulated environment for measuring the simulated user's trust level. The trust estimate was included both in the state space and the reward function of the RL-based proactive DA. In the following, we describe the bootstrap approach: First, we shortly review the data collection process based on previous work, and then detail the novel components that we developed for training a socially-aware RL-based proactive DA in a simulated environment. A depiction of the environment and interaction flows is depicted in Fig. \ref{img:arch}. \section{Collecting Proactive Dialog Data for Socially-Aware RL-based Dialog Management} In previous work \cite{kraus2022prodial}, a proactive dialog corpus consisting of 308 dialogs (3696 system-user exchanges) was created. For this, 308 users participated in an online data collection via the clickworker\footnote{www.clickworker.de} framework. During the data collection, users had to cooperate with a proactive DA in a serious dialog game, which was modelled as a sequential decision-making problem. The game consisted of 12 task steps, which were noted as system-user exchanges. The main difference between proactive dialog systems assisting with decision-making in comparison to reactive information-seeking systems is that the dialog goal is both known to the agent and the user. Thus, its task is not to identify the user's intention but to provide timely and appropriate assistance during task execution. Each exchange was annotated with self-reported measures on the system's trustworthiness and its related concepts user's perceived competence, predictability, reliability to represent the user's cognition-based trust. These variables were labeled on a 5-point Likert scale and ranged from 1="very low" to 5="very high". Further, we captured objective features such as task-related properties (complexity, exchange duration, user actions) as well as the system's actions, and static user information provided in a pre-test questionnaire, e.g. age, gender, personality, domain expertise. This allowed to developed a model for estimating the user's trust state for each exchange during the dialog game. In the following, we describe the main components of the data collection. \subsection{Dialog Game} The dialog game was implemented as a servlet-application based on a client-server model. On the client-side a user played the game and interacted with the proactive assistant using a clickable graphical user interface (GUI). On the server-side a self-implemented dialog control logic received user input from the GUI and provided task-related content to the interface by accessing a database. The user's goal was to successfully manage the company through skilful strategic action and to maximize profits. In doing so, a user had to make step-by-step decisions and plan undertakings in the interest of the company, such as location planning or personnel management. Individual decisions had consequences and affected the success of the management. The game was designed as a turn-based planning task, in which the system sequentially presented a task step and the available choices. The user could take different actions and cooperatively solve the task with the DA. Thus, the structure of the game can be describes as follows: the system takes an action providing task step relevant information, upon the user takes an action solving the respective task step after which the cycle is repeated until the game ends after a total of 12 task steps. The order of the tasks was fixed and could not be changed by the user. For each step, several options were presented from which the user had to choose. The number of options changed from task to task and ranged from a minimum of three to a maximum of five options. At each task step, the user could perform four actions: select an option, ask for help, explicitly ask the DA for a suggestion, and continue with the game. When asking for help, general information about the game is given, such as what previous decisions need to be considered in the current task step. To add cause and effect to the user's decision making, options were linked to numerical scores. This allowed previous decisions to directly influence the value of future actions. The concept of a game state was to create a vulnerable, yet engaging environment for the user. Performance mattered, which should lead to the need to trust the assistant. For illustration, consider the following example: User Alice is currently required to make a decision on task ``Research'', where a plausible research direction with regard to the built up company needs to be chosen. This task is influenced by Alice selections in previous tasks ``Management'' and ``Banking''. Depending on the combination of selections in respective tasks, one of the four options (Hydrogen Drive, Autonomous Driving, Battery Research, Climate Neutral Production) would yield the most points, whereas in the worst case scenario a user would yield zero points. The game score was based on an artificial scoring model, particularly developed for this application. \subsection{Proactive Dialog Agent} An agent system was able to choose from four different dialog actions to serve as a personal advisor to the user. The agent used natural language based on dynamically created templates. With regard to data collection purposes, it was designed as an expert system to avoid unintended side effects of incompetent system behavior on its trustworthiness. This allowed only the effects of proactive levels on HCT to be considered. To select the best option per task step, the agent used a simple reasoning mechanism by having knowledge about previous user decisions and accordingly querying the game's evaluation model. The proactive assistance was modeled according to the proactive dialog actions defined by Kraus et al. \cite{kraus2021role}: \textit{None}, \textit{Notification}, \textit{Suggestion}, and \textit{Intervention}. These actions are tiered and range from no intervention (non-intrusive) to full intervention (very intrusive). Since the scenario was a planning task and the user was asked to select what they thought was the best option for each task step, the purpose of the system behavior was to use natural language to provide helpful information and suggestions for the selection process. With the reactive \textit{None} action, the system waited for the user to explicitly ask for suggestions. The more conservative proactive actions, \textit{Notification} and \textit{Suggestion}, let the user confirm the wizard's suggestions and differ only in the degree of directness. The \textit{Intervention} actions took the responsibility completely out of the user's hands and autonomously chose an option. Assisting with sequential decision-making, proactive actions can be considered as the initiation of sub-dialogs, where the assistant influences a user action. For data collection purposes, the agent utilized a combined rule-based and randomized strategy for creating a diverse data set. In the following a typical example dialog is presented (note, that the user was only able to select pre-defined natural language templates): \begin{compactitem} \item[]\textbf{(Task:``Management''; \#Options = 3; Agent's Action: Notification)} \item[\textbf{Agent (A):}] I have a suggestion for you! \item[\textbf{User (U):}] Ok, tell me more? \item[\textbf{A:}] I would suggest to take Ben as your Manager as he is the best fit according to the brand image you want to create! \item[\textbf{U:}] Ok, thanks. I'll take him. \item[]\textbf{(Task:``Banking''; \#Options = 4; Agent's Action: None)} \item[\textbf{U:}] I need a suggestion! \item[\textbf{A:}] I would suggest to take the Wallace-bank, as you have taken Ben as your manager who suits the communication style of the bank best! \item[\textbf{U:}] Ok, but I'll take a different bank. \item[]\textbf{(Task:``Research''; \#Options = 4; Agent's Action: Intervention)} \item[\textbf{A:}] Based on your previous selections regarding your management and banking, I select autonomous driving as future research direction! \textit{Automatically makes decision and game moves on to next task...} \end{compactitem} \section{Simulated Proactive Dialog Environment} Based on the data corpus, we created a simulated environment for training and evaluating the RL-based proactive DA. Therefore, we replaced the user with corpus-based user simulation and implemented an RL-based proactive DA. For adding social awareness to the agent, we developed and evaluated a supervised learning-based trust state model that allows to estimate the simulated user's current trust in the agent's action. The estimate was then included in the DA's state and its reward model. \subsection{Simulated User} For user simulation, the main objective was to replicate realistic user characteristics as well as task and trusting behavior for training and testing proactive dialog policies. For this purpose, the simulation relied on relevant personal and dialog data gathered from the previously described corpus collection. For creating distinct user types, the corpus' user-dependent information was used: age, gender, technical affinity, propensity to trust, domain expertise, and the Big 5 personality traits \cite{mccrae1992introduction}. Noise was added to the variables using truncated Gaussian distributions and likelihood of appearance based on corpus data distributions. Task-related behavior was simulated by generating values for the game score (task success dependent on the user's option selection), whether a user initiated a suggestion or help request, task step duration, and perceived difficulty. The probabilities for each specific user behavior were based on structured data distributions depending on the specific user type and the current dialog state. Consequently, this enabled to reproduce user behavior as a reaction to proactive system actions. For evaluating the quality of the developed user simulator, we used the Kullback-Leiber distance \cite{kullback1951information} between distributions of behavior generated by the user simulator and real users. Distances range from 0, i.e. distributions are equal, to 1, i.e distributions are completely different. The lower the distance between the distributions, the more realistic is the respective simulation. Our user simulator achieved a score of 0.172 (.142) showing realistic performance. The simulated user-specific and task-related features as well as proactive system actions were then fed as input to the trust state model which is described in the following. \subsection{Trust State Model} In previous work \cite{kraus2021modelling}, we developed a novel user model for predicting trust during interaction with proactive DAs based on the collected data. As we modelled trust and its related concepts on a discrete, ordinal scale, the prediction problem was formulated as a multi-class classification task. The target classes were the distinct trust values ranging from 1 to 5. For prediction, the simulated user-specific as well as task-related features and the proactive dialog action type were quantified and concatenated into a feature vector at each system-user exchange (task step). The feature vector was then fed as input to a classifier. In previous work, we compared the performance of a Support Vector Machine (SVM), Gated Recurrent Unit (GRU) Networks, and Extreme Gradient Boosting. Typically for smaller data sets, the SVM outperformed the other approaches achieving an $F_{1}$-score of 0.533, Cohen's $\kappa$ of 0.363, a Spearman's $\rho$ of 0.426, and an extended accuracy $eA$ \cite{rach2017interaction} of 0.895 by evaluating the different algorithms using cross-validation on the data corpus. Even though the scores seem to be rather low at first sight, we deem the the proposed trust prediction model to be useful for including trust as metric for training a socially-aware agent, as trust measurement is complicated, even for human beings. This is due to trust being multi-faceted and also a latent variable that cannot be observed directly. Further, the SVM clearly outperformed the random baseline ($F_{1}$~=~0.2, $\kappa = 0.0$). Therefore, we used the trained SVM as model for predicting the user's trust state on exchange-level basis for enabling socially-aware proactive dialog modeling. \subsection{Socially-Aware RL-Based Proactive Dialog Agent} \label{sec:imp} RL allows an agent to learn strategies for solving complex problems by maximising a reward, e.g. see Sutton and Barto \cite{sutton2018reinforcement} for more information. This reward is a feedback signal from the agent's environment for determining the goodness of agent behavior. To apply an RL-based approach for the adaptation of proactive dialog behavior, it was required to model the interaction between the simulated user and the agent as a Markov Decision Process (MDP). Thus, dialog states, actions, and rewards needed to be defined. For modelling the dialog state, we included the current task step $s_{step}$ of the serious dialog game and the respective complexity $s_{complexity}$. These states represented the agent's static knowledge of the task. Dynamic knowledge was represented by integrating the last known estimated user trust value $s_{trust}$ (1-5), task success $s_{success}$ (0-10-20-30-40) and the duration $s_{duration}$ of the last task step in seconds. For modelling the action space, we relied on the taxonomy of proactive dialog acts $a \in \{a_{none}, a_{notification}, a_{suggestion}, a_{intervention}\}$. The reward function was modelled to promote task and also socially effective proactive behavior by the means of trustworthy interaction. As trustworthy proactive dialog behavior does not necessarily imply task effective or efficient behavior and vice versa, several aspects were taken into account for designing the reward function. To enable both trustworthy, successful, and efficient proactive dialog behavior, the reward was modelled as the sum of the rewards for estimated trust, task success, and task duration \begin{equation} r_{t} = r_{trust} + r_{success} + r_{duration} \end{equation} The reward function $r_{trust}$ for each task step was modelled to reward high levels and to punish low levels of trust: \begin{equation} r_{trust} = \left\{ \begin{array}{ll} \, \, \, \, 20 & \textrm{, if $s_{trust} = 5$}\\ \, \, \, \, 10 & \textrm{, if $s_{trust} = 4$} \\ \, \, \, \, 0 & \textrm{, if $s_{trust} = 3$} \\ - 10 & \textrm{, if $s_{trust} = 2$} \\ -20 & \textrm{, if $s_{trust} = 1$} \end{array} \right. \end{equation} Further, we rewarded task step success depending on the average task success score for the respective task: \begin{equation} r_{success} = \left\{ \begin{array}{ll} \, \, \, \, 15 & \textrm{, if $s_{success} > mean$}\\ \, \, \, \, 10 & \textrm{, if $s_{success} = mean$} \\ \, \, \, \, 5 & \textrm{, if $s_{success} < mean$} \\ \, \, \, \, 0 & \textrm{, if $s_{success} = min$} \\ \end{array} \right. \end{equation} Similarly, we rewarded task efficient behavior with regard to the duration per task step: \begin{equation} r_{duration} = \left\{ \begin{array}{ll} \, \, \, \, 10 & \textrm{, if $s_{duration} \leq mean$}\\ \, \, \, \, 0 & \textrm{, if $s_{duration} > mean$} \\ \end{array} \right. \end{equation} Note that we weighted the individual reward functions regarding their importance. Foremost, we aimed for trustworthy behavior. Therefore, high trust received the highest possible reward amongst all functions, whereas low trustworthy behavior even resulted in negative numerical scores. To not to neglect usability, we also rewarded successful and efficient actions. However, unsuccessful and not efficient behavior did not receive a negative reward for balancing the training more towards benefiting trustworthy dialog actions. The proactive dialog policy was then trained in numerous interactions with simulated users. As the state space was quite large ($\approx$ 90,000 states), conventional model-free Q-leaning was not feasible. For this reason, we implemented a Deep-Q-Network (DQN) \cite{mnih2015human} approach with a stacked Multi-Layer Perceptron (MLP) for function approximation. For implementing the DQN, we utilized the stable-baseline implementation \footnote{https://stable-baselines.readthedocs.io/en/master/modules/dqn.html}. The architecture of the DQN consisted of two MLP-layers with 256 neurons, an input layer sized in the dimension of the state space, and an output player for producing the Q-values of the dialog actions. For creating the output a softmax layer was used. Using heuristic search, we applied the following hyper parameters to the DQN: For discounting future rewards, we set $\gamma =0.99$. Further, we trained the network using the RMSProp-algorithm with ADAM optimisation~\cite{kingma2014adam}, a learning rate of 0.00005, and a mini-batch size of 64. We used $\epsilon$-greedy for training the behavior policy with $\epsilon$ annealed linearly from 1 to 0.1 over 15 \% of the training sample, and fixed at 0.1 thereafter. The DQN was trained on a total of 300000 training samples (task steps) or 25000 dialog games with different simulated users. For speeding up the training process, we normalized the state space using min-max scaling. The trained RL-based strategy was then evaluated against the rule-based, and the static proactive dialog strategies. \section{Experiments and Results} \label{sec:exp} We conducted an empirical evaluation with simulated users. For comparison, we tested the RL-based proactive DA against agents using four static baseline strategies, i.e only one proactive dialog act type, e.g. None, Notification, ..., was used by the respective agent during dialog games. Further, we tested against a rule-based strategy that made decisions on proactive behavior depending on the user's current trust level and task complexity (see appendix for a detailed description of the rule-based strategy). For evaluation, we simulated 500 dialog games per strategy. For each dialog game a different user type was simulated. The set of simulated users was kept constant for each strategy to ensure comparability of the results. The amount of dialog games was selected to produce normally distributed data sets, that allowed the usage of parametric statistical significance tests. As evaluation metrics, we used the average overall trust ratings, overall task success score, and overall task duration. Significance tests for the differences between the strategies were conducted using t-tests with Bonferroni correction regarding multiple testing. \begin{figure} \centering \includegraphics[width=\columnwidth]{colingresultstrust.png} \caption{Average trust ratings per DA (\emph{max} indicates maximum value).} \label{img:rltrust} \end{figure} Considering the impact on HCT, the RL-based proactive DA achieved the third highest average trust value among all agents (see Fig. \ref{img:rltrust}). The differences between the RL-based agent and the agents achieving higher trust values using the \textit{None}-strategy ($p=0.315$) respectively the \textit{Rule-based}-strategy ($p=1.000$) were not significant, however. In comparison with the agents using medium-levels proactive strategies, the RL-based agent achieved significantly higher average trust values than the agent using only suggestions ($p=0.002$), but not significantly higher values than the agent using only notifications ($p=0.174$). Further, the RL-based agent scored significantly higher average trust values than the agents using the \textit{Intervention}-strategy ($p<0.001$) \begin{figure} \centering \includegraphics[width=\columnwidth]{colingresultstd.png} \caption{Average task duration per DA.} \label{img:rltd} \end{figure} Considering the impact on task duration, the agent utilising the \textit{RL-based} strategy followed by the agent applying the \textit{Intervention}-strategy led to the fastest dialog game completion (see Fig. \ref{img:rltd}). The difference between those strategies was not significant ($p=0.113$). However, the average task duration using the RL-based agent was significantly lower in comparison to all other agents (all $p<0.001$). Considering the impact on task success, the RL-based proactive DA produced significantly higher task success than all other agents (all $p<0.001$) except the agent using the \textit{Intervention-strategy} (see Fig. \ref{img:rlts}). Conducting the dialog game with an agent utilising the \textit{Intervention}-strategy resulted in a significantly higher task success than interacting with the RL-based proactive DA ($p<0.001$). \section{Discussion and Conclusion} \label{sec:dis} \begin{figure} \centering \includegraphics[width=\columnwidth]{colingsresultsts.png} \caption{Average task success per DA (\emph{max} indicates maximum value).} \label{img:rlts} \end{figure} \begin{table} \centering \small \begin{tabular}{c|cc||c} \emph{Policy} & \emph{Task Efficiency} & \emph{Trust} & \emph{Cooperation}\\ \hline RL-based & 0.359 & 3.10 & \textbf{1.14} \\ None & 0.292 & \textbf{3.18} & 0.96 \\ Notification & 0.296 & 3.01 & 0.91 \\ Suggestion & 0.302 & 2.97 & 0.92 \\ Intervention & \textbf{0.389} & 2.46 & 0.97 \\ Rule-based & 0.318 & 3.14 & 1.05 \\ \end{tabular} \caption{Task efficiency, trust, and cooperation scores per dialog policy.} \label{tab:my_label} \end{table} The results show that the more reactive a DA acts, the more trustworthy it is perceived. This is also in line with findings from studies with real users \cite{rau2013effects,kraus2020effects} and seems to be primarily connected with the level of control the user possesses on making the final decisions. Simultaneously, the more proactive an agent acts, the higher the task success becomes due to the agent being an expert system on the given task. Thus, a DA needs to appropriately balance reactive and different kinds of proactive behavior for enhancing cooperation. An agent needs to find the fine line on improving the user's task success, but also not to lose the user's trust which may lead to the system's disusage \cite{muir1994trust}. For making these kinds of decisions, including a degree of social awareness in the dialog policy seems to be beneficial based on our results. Observing the metrics task efficiency $\frac{success}{duration}$ and trust it becomes evident that the RL-based agent provides the best compromise between task efficient and trustworthy behavior for improving cooperation (see Table \ref{tab:my_label}). The RL-based proactive dialog policy outperforms all others except the \textit{Intervention}-strategy in task efficiency while achieving near optimal performance regarding trust which is provided by the \textit{None}-strategy. Calculating the a cooperation measure $cooperation = efficiency \cdot trust$ ultimately shows the benefit of including trust in the dialog state and/or reward function for improving cooperation (the differences between RL-based strategy and all other strategies are significant with $p < 0.05$). Observing the proactive dialog act types the RL-based agent selected, a predominant use of the \textit{Notification}-action becomes evident (38 \% of all actions), while the extreme levels \textit{None} and \textit{Intervention} were selected well-balanced (23 \% and 25 \%). In an experiment with real users, \cite{kraus2020effects} found that notifying behavior was perceived as most appropriate with regard to a proactive system's trustworthiness. Therefore, the perception of this proactive act type maybe more invariant to the respective situation and user, while purely reactive and autonomous system behavior needs to be adapted to user and the situation. As this study is the first of its kind utilising RL for proactive dialog management, further validation of the results are necessary. Especially, the learned policy needs to be evaluated with real users. Another limitation was the application of hand-crafted reward modelling. Here, different reward shaping methods need to be explored, which may lead to improved results. Also, other approaches need to be investigated. For example, proactive dialog policies may be learned directly from human-behavior using end-to-end approaches, e.g. by applying transformers on human proactive dialog data. Finally, including trust is a beginning for creating socially-aware AI, but what is next? Trust is an important concept that should be considered when developing AI agents, also other important aspects need to be studied. For example, Ritschel et al. \cite{ritschel2017adapting} considered measurements of user engagement to be included in an RL-based framework for learning a robot's appropriate linguistic style. Besides, other factors of conversational intelligence, namely conscientiousness and communicability (a system's ability to self-disclose internal features and interactive principles) need to be included in sophisticated AI agents \cite{chaves2021should}. In conclusion, we developed an RL-based proactive DA with the aim of achieving a socially and task effective interaction. For this, the agent was trained in simulated dialog environment using both trust estimate and usability measures for providing adequate proactive dialog behavior. For evaluating the approaches, we compared the RL-based proactive DA with agents using only static and rule-based proactive dialog strategies. Including trust in the dialog model for enabling adaptive dialog proved to be successful for creating socially and task effective agents. Our approach achieved the best compromise of contributing to task efficiency, but also acting in a trustworthy manner. In future work, we plan to validate the results in a study with real users. \bibliographystyle{IEEEtran}

proofpile-arXiv_069-5929

\section{Introduction} Image manipulation aims to modify some aspects of a given image, and the manipulation content includes low-level color or texture to high-level semantics to match the user's preference. It has a wide range of applications in education, image editing, and video game development. In recent years, with the rise of artificial intelligence, automated image manipulation research, including image inpainting\cite{Wang_2021_CVPR}\cite{Li_2020_CVPR}, image colorization\cite{Bahng_2018_ECCV}, style transfer\cite{Kalischek_2021_CVPR}\cite{liu2021adaattn}, and domain or attribute transformation \cite{Chen_2019_CVPR}, has gained widespread attention and made significant progress. Despite achieving stunning results, most studies focus on specific problems and require specialized knowledge, resulting in poor practicality. To address this issue, By using text information that is closely related to people's lives to manipulate the content of the corresponding image region, text-guided image manipulation implements a user-friendly way of image manipulation. With the rapid development of GAN \cite{goodfellow2014generative} technology, text-guided image manipulation has achieved encouraging results. SISGAN \cite{dong2017semantic} and TAGAN \cite{nam2018text} have achieved decent image manipulation results using a GAN-based encoder-decoder structure and a text-adaptive network, respectively. However, the image results generated by these methods didn't perform well in detail synthesis. In contrast, ManiGAN \cite{li2020manigan} achieves better detail manipulation and makes the final image manipulation result more realistic. Nevertheless, on the one hand, ManiGAN will not only modify the text-relevant content during the manipulation process but also modify the text-irrelevant content, which is not expected. On the other hand, its overall manipulation performance still has room for further improvement. In summary, these existing text-guided image manipulation methods mainly suffer from the following problems: (1) the content not related to the text description is randomly changed, (2) the performance of image manipulation still needs to be further improved, (3) only can manipulate descriptive attributes, lacking specific actions such as removing, enlarging, or dwindling an object. The root of these problems is that it is difficult to accurately recognize the target content. To solve these problems, we divide the image manipulation task into three phases: pre-processing phase, manipulation phase, and combination phase. In the pre-processing phase, we propose a segmentation network to obtain the segmentation map (as shown in Fig. \ref{fig:model}). The white area in the segmentation map represents the content that needs to be manipulated, and the black area represents the content that does not need to be modified. Based on the segmentation map, the text-relevant content and the text-irrelevant content can be accurately determined. In the manipulation phase, the text-relevant content is manipulated according to the semantic information of the input text, and the text-irrelevant content remains unchanged. In the combination phase, the content of the text-relevant region after manipulation is fused with the text-irrelevant region that keeps the content unchanged to form the final image manipulation result. The whole process is first to determine the text-relevant content and the text-irrelevant content, then only modify the text-relevant content, and then put the modified result back to the original position. The above process can solve the first problem mentioned earlier. At the same time, since only the text-relevant content is modified and the text-irrelevant content is retained, the manipulation network can pay more attention to the modification of the text-relevant content, thereby further improving the quality of manipulation. In addition, since the text-irrelevant content is inherently high-quality, the image manipulation results after fusion will have higher quality due to the high quality of text-relevant content and text-irrelevant content. This solves the aforementioned second problem. However, there are still two issues in the above process. One is that it cannot be guaranteed that the pre-processing phase can obtain an accurate segmentation map. If the segmentation map is inaccurate, the quality of the final manipulation result will be poor. Another is that for the third problem, the above process cannot be solved. For these two issues, we propose a user-interactive segmentation map editing method and use a super-resolution network to solve them. The user-interactive segmentation map editing method allows the user to manually edit the segmentation map to obtain more accurate segmentation map results to ensure the high quality of the final manipulation results. The super-resolution network is used before the image manipulation phase. It can enlarge and dwindle the text-relevant content so that the ways of our image manipulation are more flexible and diverse. In addition to the above content, to further improve the practicability of our manipulation method and the flexibility of manipulation results, we allow the user to input two original images, select the text-relevant content from one, and retain the text-irrelevant content from the other. However, since the shapes of the text-relevant regions of the two images are mostly different, this leads to holes in the combination stage. Therefore, we adopt the image inpainting methods \cite{yi2020contextual}\cite{ulyanov2018deep} to fill these holes to synthesize high-quality image manipulation results. \begin{figure*}[t] \begin{center} \includegraphics[width=0.8\linewidth]{img/model.pdf} \end{center} \caption{The overview of our proposed model. It allows users to edit the segmentation map automatically.} \label{fig:model} \end{figure*} Our contributions are as follows: \begin{itemize} \item We propose a novel and effective text-guided image manipulation method. By accurately detecting and manipulating text-relevant content, we achieve better image manipulation results. \item In order to improve the detection accuracy of the text-relevant region and the user-friendly of the whole method, we propose a user-interactive segmentation map editing method, which allows users to manually edit the detected segmentation map to achieve a more accurate detection effect to make the final image manipulation result more conform to the user's expectations. \item To make the proposed method more flexible and diverse, we introduce a super-resolution network and an image inpainting network to achieve richer manipulation operation forms such as enlarge, dwindle, background replacement, and so on. \item Experiments on the CUB \cite{wah2011caltech} and MS COCO \cite{DBLP:journals/corr/LinMBHPRDZ14} datasets demonstrate the effectiveness and superiority of our proposed method. \end{itemize} \section{Related Works} \textbf{Text-to-Image Synthesis.} The success of text-to-image synthesis by generative adversarial networks (GAN) \cite{goodfellow2014generative} attracted a lot of attention. Reed et al. \cite{reed2016generative} pioneered the end-to-end text-to-image research base on GAN. Following that, the Generative Adversarial What-Where Network (GAWWN) \cite{reed2016learning} was shown to synthesize content at specified locations, resulting in superior quality images. Moreover, StackGAN \cite{zhang2017stackgan} and StackGAN++ \cite{zhang2018stackgan++} architecture aim to generate realistic images using multi-stage structures with generators and discriminators at each stage. Based on this, various improvements have been made with the aim of improving realism \cite{xu2018attngan} \cite{zhu2019dm} \cite{zhang2021draw}, semantics \cite{qiao2019mirrorgan} and diversity \cite{bodla2018semi}\cite{dash2017tac}. \textbf{Text-guided Image Manipulation} aims to use text to modify text-relevant content in the input image and maintain text-irrelevant content. Dong et al.\cite{dong2017semantic} proposed an encoder-decoder architecture to edit an image according to a given text. Then, TAGAN \cite{nam2018text} aimed to preserve the text-relevant content by introducing a text-adaptive discriminator, which can give the generator finer training feedback, to achieve better manipulation results. More recently proposed ManiGAN \cite{li2020manigan} introduced the multi-stage structures with generators and discriminators at each stage for high-quality manipulation results. In addition, StyleCLIP \cite{StyleCLIP} achieved the style transfer effect of images based on text-driven. However, this work is more applied to style transfer of facial content, which leads to its general practicability. Besides, the common problem with the aforementioned methods is that they do not work well in preserving text-irrelevant content. To solve this problem, Tomoki et al. \cite{haruyama2021segmentation} proposed the approach to introducing a segmentation function to reduce the loss of text-irrelevant content. This work is close to our proposed method. However, this method can only reduce the modification degree of the background as much as possible and cannot avoid the modification of the background content. Different from \cite{haruyama2021segmentation}, we design an enhanced architecture that divided an image into text-relevant content and text-irrelevant content and achieved zero loss for text-irrelevant content. Furthermore, we achieve challenging tasks such as enlarging, dwindling, and removing an object, which other existing methods can not do. Additionally, the proposed method introduces an interface that can edit the segmentation interactively so that our method has excellent flexibility and practicality. \section{Methods} \label{sec:pagestyle} As shown in Fig. \ref{fig:model}, the inputs of the proposed system are an input image $I$ and a text instruction $S$. The target is to edit the content associated with $S$, preserve the content not associated with $S$, and generate a corresponding image $I^{\prime}$. Firstly, the segmentation network is applied to divide the input image into two regions: text-relevant content $I_{tr}$ and text-irrelevant content $I_{ti}$. Secondly, we input text-relevant content in the super-resolution network \cite{zhang2021designing} to obtain the refined text-relevant content $I_{tr\_SR}$. Then, we adopt the ManiGAN \cite{li2020manigan} as the basic framework to edit the refined text-relevant content according to the text instruction. Finally, the output image $I'$ is obtained by combining the edited text-relevant and the original text-irrelevant content into the combination network. Moreover, in order to more accurately detect the content of text-relevant regions, we propose a user interface that allows users to edit the segmentation map detected before so that the final manipulation result is more conform to the user's expectation. Fig. \ref{fig:model} shows our entire system divided into three phases: pre-processing phase, manipulation phase, and combination phase. Next, we introduce the specific content of each phase one by one. \subsection{Pre-processing phase} \label{ssec:segmentaiton} \textbf{Segmentation network.} Existing works are not able to recognize the target object in the text description, causing current text-guided image manipulation methods to suffer problems such as randomly changing the content of the text-irrelevant region. To solve this issue, we introduce a segmentation network. This network allows linking the text with the corresponding region information in the image. It takes the input image $I$ and the text instruction $S$ as the input, then links text-region information by performing two tasks simultaneously. The first task is to obtain the segmentation mask $M_{seg}$. We detect all objects with re-trained Deeplabv3 \cite{chen2018encoder} from scratch on the CUB and COCO datasets, respectively, to achieve this task. The second task is to acquire the candidate class label $W$ of objects, we employ pre-trained YOLOv4 \cite{DBLP:journals/corr/abs-2004-10934} to achieve this task. When finishing the above two tasks, the cosine similarity between the words in the obtained candidate class label $W$ and the noun words in the text instructions $S$ is calculated to obtain the class information $c$ of the most relevant object. \begin{equation} c = \text{max}(\text{Similarity}(W, S)) \end{equation} The region corresponds to text-relevant class $c$ is selected from $W$ and then multiplied with $M_{seg}$ to obtain the text-relevant mask $M_{tr}$. \begin{equation} M_{tr} = M_{seg} * W(c) \end{equation} We mask the input image $I$ by using text-relevant mask $M_{tr}$ to obtain the text-relevant and text-irrelevant content: \begin{equation} I_{tr}, I_{ti} = \text{Mask}(M_{tr}, I) \end{equation} \textbf{Super-Resolution network.} When the text-relevant content is in a small region or when modifying the detailed information (e.g., eyes), the existing works are unable to achieve sufficient accurate positioning detection resulting in a poor manipulation effect. Besides, the existing methods can only modify the manipulation at the pixel level and cannot achieve manipulation effects such as object enlarging and dwindling. To solve the above issues, we adopt the super-resolution network. First, we crop the object region that needs to be manipulated in the text-relevant content. Then, it is up-sampled by an image super-resolution network (pre-trained BSRGAN \cite{zhang2021designing}) to obtain the super-resolution text-relevant content $I_{tr\_SR}$. The super-resolution network can zoom in on the small region and detailed information to provide more effective information for the subsequent manipulation phase and achieve better manipulation results. Besides, it allows enlarging and dwindling of the object so as to achieve more manipulation effects. Furthermore, this network is first used to make the object in the largest size that would fit in the image. And then, in the manipulation phase, the size of the object is modified by receiving text instructions and matched to that size to generate the object content. \begin{figure}[t] \begin{center} \includegraphics[width=\linewidth]{img/detail.pdf} \end{center} \caption{The detailed content of TRDCM and Combination network is shown above.} \label{fig:detail} \end{figure} \begin{figure*}[tb] \begin{center} \includegraphics[width=0.8\linewidth]{img/result_bird.pdf} \end{center} \caption{The qualitative comparison of the proposed ours(auto seg.) and related works on the CUB dataset. Existing methods lack details (e.g., eyes, patterns, etc.), and fail to preserve the text-irrelevant content. In contrast, by adopting semantic map information and the super-resolution method, this method successfully keeps text-irrelevant content and generates better text-relevant content.} \label{fig:qualitative_wo_interface} \end{figure*} \subsection{Manipulation phase} \label{ssec:manipulation} In the manipulation phase, we refer to the affine combination module (ACM) and the detail correction module (DCM) in ManiGAN\cite{li2020manigan}. Unlike ManiGAN, we propose a novel Text-Relevant DCM (TRDCM) to replace DCM. The contents of ACM and TRDCM in our method are as follows. \textbf{ACM.} Different from ACM in ManiGAN, ACM in our method processes the features by the super-resolution network. Since the features of the super-resolution network belong to the text-relevant region, our manipulation results can more conform to the semantic information of the text and do not change the content of the text-irrelevant region content. In contrast, ManiGAN's ACM processes the features of the entire image, which leads to the problem that the content of the text-irrelevant region also changes. \textbf{TRDCM.} We do not introduce the user interface during the training processing, so our TRDCM content is consistent with ManiGAN's DCM. However, our TRDCM is quite different from ManiGAN due to the introduction of user interaction in the testing. Its basic structure is shown on the left side of Fig. \ref{fig:detail}. Specifically, it includes four inputs: (1) the last hidden feature $h_{last}$ from the ACM, (2) the text features $t$, (3) the super-resolution text-relevant content $I_{tr\_SR}$ and, (4) the segmentation map $I_{seg}$ from the pre-processing network. The text features consist of the word-visual vector $t_{v}$ and the word-instruction vector $t_{i}$ encoded by a pre-trained RNN \cite{xu2018attngan}. First, the word-visual vector $t_{v}$ is input to the spatial and channel-wise attention \cite{NEURIPS2019_1d72310e} to create the attention features, which are then combined with $h_{last}$ to obtain the intermediate feature $a$. Secondly, to obtain the detail visual vector $v$ as the same size as $a$, we upsample the image feature encoded $I_{tr\_SR}$ by the pre-trained VGG-16 \cite{simonyan2014very}. Then, by using the ACM, $v$ and $a$ fused to produce the feature $\tilde{a}$. we refine $\tilde{a}$ with residual block according to \cite{li2020manigan} to obtain the image feature which edit the descriptive information $\tilde{I_{tr\_{SR}}}$. Finally, we concatenate the image feature $\tilde{I_{tr\_{SR}}}$ with word-instruction vector $t_{i}$ to produce the final modified text-relevant image $I^{\prime}_{tr}$. Since the text-relevant content obtained after the super-resolution network processing is the largest size, during the generation process in TRDCM, the segmentation map is used to guide the generation of text-relevant region manipulation result that is consistent with the size of the segmentation map. Furthermore, in order to introduce more manipulation operations (such as object enlarging, dwindling, and removal), we detect keywords in the text in the TRDCM of the testing phase. For example, if the word ``2x large'' or ``4x small'' appears, we will scale the segmentation map accordingly to obtain $I_{seg}^{'}$ and then use it to guide the generated size of the text-relevant region content. If the word ``remove'' appears, we will remove the content of the corresponding region according to the current input segmentation map. \begin{figure*}[tb] \begin{center} \includegraphics[width=0.8\linewidth]{img/result_COCO-min.pdf} \end{center} \caption{The qualitative comparison of the proposed ours (auto seg.) and related works on the COCO dataset. We can see that ManiGAN manipulates all regions of the image. In contrast, our model is capable of editing images only for a specific object so that it can generate better results.} \label{fig:qualitative_wo_interface_COCO} \end{figure*} \subsection{Combination phase} \label{ssec:com} \textbf{Combine network.} We introduce this network (Fig. \ref{fig:detail}) to combine the edited text-relevant content and the original text-irrelevant content to synthesize a plausible image. It takes three required inputs and one optional input. (1) the edited text-relevant content $I_{tr}^{'}$ from the manipulation phase, (2) the original text-irrelevant content $I_{ti}$, (3) the segmentation map $I_{seg}$ from the pre-processing phase, (option) the reference background image $I_{back}$. First, according to \cite{yi2020contextual}, $I_{ti}$ is inpainted based on the segmentation map to remove the text-relevant content to obtain the inpainted text-irrelevant content $I_{ti\_inpaint}$. In the background change task, the reference background image $I_{back}$ is input to Deeplabv3 \cite{chen2018encoder} to obtain the pure background image $I_{back\_pure}$ and segmentation map $I_{back\_seg}$. Then, based on the segmentation map, the inpainted background image $I_{back\_inpaint}$ can be obtained. And then, to obtain the output image $I'$, based on the segmentation map $I_{seg}^{'}$ obtained by the manipulation network, we combine the modified text-relevant content $I_{tr}^{'}$ and the inpainted text-irrelevant content $I_{ti\_inpaint}$, or the inpainted background image $I_{back\_inpaint}$ for the background change task. Finally, we introduce a function to absorb the color differences in these contents that are caused by processing them separately. Specifically, the outline of the segmentation map $I_{seg}^{'}$ is extracted to make the mask. By masking this outline for the combined image, we produce the image without the contour region. By inpainting the image without the contour region based on \cite{ulyanov2018deep}, the color difference can be eliminated. \subsection{Interactive user interface} \label{ssec:interactive_editing} As shown in Fig. \ref{fig:mis_seg1}, for some cases, the output segmentation map automatically generated by the network is not perfect. To solve this issue, we adopt the user interface to edit the segmentation map in real-time. With the user interface, when the initial image manipulation does not work well, accurate image manipulation can be performed by revising the segmentation result. As shown in Fig. \ref{fig:n_times_modify}, the editing can be done recursively, and it can be corrected and redone many times. \subsection{Loss function and Training} \label{ssec:function} The above parts that need to be trained only include the manipulation phase, and the baseline used in the training process is ManiGAN \cite{li2020manigan}. The pre-processing phase, combination phase, and user interface proposed by our are to solve the problem that the existing methods cannot accurately manipulate the required objects and simultaneously achieve more manipulation operations so that the entire model has better applicability. In the training process, the overall structure is based on GAN. Following ManiGAN \cite{li2020manigan}, it includes a main module and a TRDCM, where the main module includes three ACM, corresponding to three generators and three discriminators. The TRDCM module includes a generator and a discriminator. The main module and TRDCM are trained individually. \textbf{Generator objective.} Based on \cite{li2020manigan}\cite{NEURIPS2019_1d72310e}, the generator loss $L_{G}$ consists of an adversarial loss $L_{adv}$, a perceptual loss $L_{per}$, a text-image matching loss $L_{DAMSM}$, and a regularization loss $L_{reg}$. We show the definition: \begin{equation} L_{reg} = 1-\frac{1}{CHW}||I^{'}_{tr}-I_{tr}|| \end{equation} \begin{equation} \fontsize{8pt}{0cm}\selectfont \begin{split} L_{G} = L_{adv} + L_{per} + {1 - L_{cor}(I_{tr}^{'}, S)} \\ + L_{DAMSM}(I', S) + L_{reg} \end{split} \end{equation} where $C$, $H$, and $W$ are the number of color channels, the height and width of the input image $I_{tr}$, respectively. \textbf{Discriminator objective.} Based on \cite{li2020manigan}, the discriminator loss $L_{D}$ consists of an adversarial loss $L_{adv}$ and the text-image correlation loss $L_{cor}$. The discriminator loss $L_{D}$ is defined as follows: \begin{equation} \fontsize{8pt}{0cm}\selectfont \begin{split} L_{D} = L_{adv} + {1 - L_{cor}(I_{tr}^{'},S)} + L_{cor}(I_{tr}^{'},S^{'}) \end{split} \end{equation} where $S^{'}$ is a randomly chosen mismatched textual description from the dataset. \section{Experiments} \label{sec:experiments} We evaluate our method on the CUB and COCO datasets. The CUB dataset \cite{wah2011caltech} contains 11,788 bird images of 200 categories. 8,855 and 2,933 images are used for training and testing, respectively. The COCO dataset \cite{DBLP:journals/corr/LinMBHPRDZ14} contains 123,287 wide range of genre images. 82,783 and 40,504 images are used for training and testing, respectively. Using a user interface does not ensure reproducibility and results in unbiased evaluations. For fair evaluation, we compare our model (auto seg.) with state-of-the-art models in qualitative and quantitative performance. Our model (auto seg.) means a model that performs image manipulation using a segmentation map automatically detected by the segmentation network, without the user interface to allow segmentation editing. Based on the Adam optimizer\cite{kingma2014adam}, we train the main module 600 epochs on the CUB dataset and 120 epochs on the COCO dataset, and train the TRDCM module 100 epochs for both datasets. \subsection{Comparison with state-of-the-art approaches} \label{ssec:exp_comparison} \textbf{Quantitative comparison.} In the quantitative experiment, we evaluate the IS \cite{salimans2016improved}, NIMA \cite{talebi2018nima}, and FID \cite{heusel2017gans} on randomly selected images from the CUB and COCO datasets with a randomly chosen text description. Following the ManiGAN, 30,000 images are generated for quantitative evaluation. The results are shown in Table \ref{table:qualitative}. On the CUB and COCO datasets, our method (auto seg.) outperforms on almost all metrics to compare state-of-the-art models, except for the NIMA on the COCO dataset. In the IS, this means that our segmentation network produces highly discriminative images by considering the text-relevant and text-irrelevant content. Furthermore, the NIMA of our method is also high, indicating that our method achieves generating high-quality manipulated images. Furthermore, the excellent FID results indicate that our results have the best fidelity. \begin{table}[tb] \caption{Quantitative comparison results between our method and other existing methods are shown below. We use IS, NIMA, and FID for quantitative comparisons.} \label{table:qualitative} \centering \setlength{\tabcolsep}{3mm} \begin{tabular}{l|ccc} % \hline & IS $\uparrow$ & NIMA $\uparrow$ & FID $\downarrow$ \\ \hline \hline \multicolumn{4}{c}{CUB}\\ \hline \hline SISGAN \cite{dong2017semantic} & 2.33 & 4.51 & 258.56 \\ TAGAN \cite{nam2018text} & 3.05 & 4.70 & 184.79 \\ ManiGAN \cite{li2020manigan} & 4.60 & 5.17 & 11.97 \\ SEGMani \cite{haruyama2021segmentation} & 4.55 & 4.26 & - \\ Ours w/o SR & 6.30 & 5.25 & 9.56 \\ Ours (auto seg.) & {\bf6.79} & {\bf5.32} & {\bf7.15} \\ \hline \hline \multicolumn{4}{c}{COCO}\\ \hline \hline ManiGAN \cite{li2020manigan} & 18.20 & {\bf5.19} & 39.74 \\ Ours w/o SR & 21.47 & 5.15 & 28.48 \\ Ours (auto seg.) & {\bf21.67} & 5.16 & {\bf25.60} \\ \hline \end{tabular} \end{table} \textbf{Qualitative comparison.} Fig. \ref{fig:qualitative_wo_interface} and Fig. \ref{fig:qualitative_wo_interface_COCO} show the comparison between the previous models and ours (auto seg.). The comparison results indicate that our model is able to manipulate the image to match the text description. Moreover, the edited results produced by previous models sometimes are less satisfactory, such as the text-relevant content is not edited and the text-irrelevant content is changed. In contrast, because the text-relevant and text-irrelevant region is separated by segmentation, only the text-relevant content is edited in our results. A comparison of the details of each image shows that our method is able to preserve the details. Specifically, for the background image, our method is able to retain detailed information such as the name of the book and the surface information of the tree. Moreover, previous models often fail to maintain complex information and fail to manipulate the target objects. This problem can be solved by our model because our model is capable of recognizing the target objects automatically, and then only the recognized region is manipulated. Furthermore, previous models only can accurately manipulate large objects. In contrast, ours can accurately manipulate both large objects and small objects due to the use of a super-resolution network. \begin{figure}[tb] \begin{center} \includegraphics[width=8cm]{img/ablation.pdf} \end{center} \caption{The qualitative comparison of ours w/o super-resolution, w super-resolution, and ManiGAN.} \label{fig:ablation} \end{figure} \begin{figure}[tb] \begin{center} \includegraphics[width=8cm]{img/change_size.pdf} \end{center} \caption{The result of changing the size and the background.} \label{fig:change_size} \end{figure} \subsection{Component Analysis} \label{ssec:exp_SR} \textbf{Improvement from super-resolution.} To better understand what has the benefit of our super-resolution network, we visualize the generated images without the super-resolution in Fig. \ref{fig:ablation}. In the model without super-resolution, detailed information about the bird is missing, while in the model with super-resolution are made detailed information is retained, such as textures and colors. It implies that super-resolution helps retain detailed information in the image. As shown in Table \ref{table:qualitative}, compared with w/o super-resolution, our method with super-resolution performs better, which proves the effectiveness of adding the super-resolution network. \begin{figure}[tb] \begin{center} \includegraphics[width=7cm]{img/extra_inpint.pdf} \end{center} \caption{Comparison of the results before and after inpainting.} \label{fig:inpaint} \end{figure} Furthermore, this architecture can also be used to resize the objects in the input image. As shown in Fig. \ref{fig:change_size}, the size of the object can now be changed to any size. Whether making a small object larger or making a large object smaller, both processes can be achieved in our model. It's noted that the above operations are not possible to achieve in existing models due to the limitation of only being able to modify the entire image at once. \textbf{Improvement from Inpaint network.} During the combination phase, holes appear when objects dwindle and background replacement operations are performed. To fill in the hole information and make the final manipulation result more realistic, we employ two image inpainting networks \cite{yi2020contextual}\cite{ulyanov2018deep}. The networks are specifically used for content repair at two levels. The first one \cite{yi2020contextual} is to repair the hole information left in the original image after segmenting the text-relevant content. As shown in Fig. \ref{fig:change_size}, due to effective content repair, our method is able to synthesize satisfactory results when encountering the operations of object enlarging, dwindling, and background replacement. The second \cite{ulyanov2018deep} is to fix the color difference phenomenon. We find that when the object is divided, manipulated, and put back to its original position, there will be color differences. As shown in Fig. \ref{fig:inpaint}, there are some tiny holes in the combination junction, which makes the overall smoothness of the result poor. To this issue, we use \cite{ulyanov2018deep} to repair these tiny holes so that the result has good smoothness. \begin{figure}[tb] \begin{center} \includegraphics[width=7cm]{img/mis_seg1.pdf} \end{center} \caption{The qualitative result of before and after modifying the segmentation map.} \label{fig:mis_seg1} \end{figure} \begin{figure}[tb] \begin{center} \includegraphics[width=7cm]{img/n_times_modify.pdf} \end{center} \caption{The results of modifying the $n$ times segmentation using the interface.} \label{fig:n_times_modify} \end{figure} \textbf{Improvement from editing segmentation.} Without the user interface to edit segmentation, there are two cases where the text-irrelevant content has penetrated text-relevant content and vice versa. As shown in Fig. \ref{fig:mis_seg1}, in both cases mentioned above, the image is edited more correctly after the correction than before. This indicates that the user interface makes it possible to improve the accuracy of image manipulation. In addition, Fig. \ref{fig:overview} shows an example the image has more than one bird in the image, but the text only describes one bird. In existing works, two birds are forced to change when the edit to match the text description. However, by using a user interface that interactively modifies the segmentation, we can focus on the specific bird to edit. Fig. \ref{fig:n_times_modify} shows the improvement of accuracy to modify the segmentation map recursively. In our model, it is possible to modify the segmentation map again even after making the segmentation fail. Therefore, if the generated image does not match the user's desires, we can generate a more accurate image by redoing it many times. \section{Conclusion} \label{sec:page} In this paper, we propose a novel image manipulation method that interactively edits an image using complex text instructions. By introducing semantic segmentation, it segments the text-relevant and text-irrelevant content of the input image so that the model can only modify the text-relevant content and maintain the text-irrelevant content. Furthermore, we introduce a super-resolution network and an inpainting network to achieve more operations (such as object enlarging and dwindling, background replacement, etc.) and generate more realistic manipulation results. Moreover, we propose a user interface that can edit the segmentation map interactively so that the user can obtain satisfactory manipulation results. Experimental results show that our method outperforms the state-of-the-art models both quantitatively and qualitatively, and also demonstrate that our work has good application value. \section{Acknowledgement} \label{sec:acknowledgement} This Research is supported by the Joint Research Project of Young Researchers of Hosei University in 2021. {\small \bibliographystyle{ieee_fullname} \section{Introduction} We thank all the reviewers for their constructive feedback. We have received comments that the differences between the existing model and our model are not apparent, and the content of each new network did not fully explain. Additionally, we have received comments that grammatical and vocabulary errors make the paper difficult to understand. Currently, we have modified the content and shown more details to illuminate each part of our proposed method. Besides, we have corrected the grammar and vocabulary errors, and we have used a grammar tool (Grammarly) to ensure that our content is basically grammatically correct. The parts of the paper with major changes are marked in red. Besides, we summarize and respond to the significant questions raised by the reviewers, and the specific contents are as follows: \color{blue}\textbf{[R3, R4] How does the segmentation network extract text-relevant regions from input images and text?} \color{black}Our segmentation network first processes the text information to extract the class labels from it, and then detects text-relevant region content in the input image based on the class labels. The specific content is in lines 339 to 370 in the paper. \color{blue}\textbf{[R3] The inpainting component is not described with enough detail and clarity for the reader to understand exactly what is done.} \color{black}The inpainting network is used in the combination phase. In the combination phase, there are two processes: the normal process and the background replacement process. In the normal process, the text-irrelevant content is inpainted. Then, according to the segmentation map, the manipulated text-relevant content combines with the inpainted text-irrelevant content to obtain the final result. In the background replacement process, the reference background image is input into the segmentation network to obtain the background image without text-relevant region and corresponding segmentation map. The background image without text-relevant region is inpainted at first, and then it will combine the manipulated text-relevant content to form the final result. Besides, in the combination phase, there may be color difference phenomena. To solve this problem, we use the inpainting network to repair it so as to synthesize more realistic results. The specific content is in lines 519 to 570 in the paper. \color{blue}\textbf{[R1, R3] What is the difference in the text encoder, ACM, and DCM between the ManiGAN and our model?} \color{black}For the text encoder and the ACM, we use the same architecture as ManiGAN [14]. However, the input information is different for the ACM. The input of ManiGAN's ACM is the whole image features, while the input of our ACM is the text-relevant image content features dealt with by the super-resolution network. For the DCM, the DCM structure in our training processing is consistent with ManiGAN. But in the testing processing, we propose Text-Relevant DCM (TRDCM), which additionally adds the input of the segmentation map. TRDCM can realize more manipulation operations, such as object enlarging, dwindling, and removal operations, by processing and utilizing the segmentation map. The specific content is in lines 411 to 517 in the paper. Besides, we only refer to the structure of ManiGAN in our manipulation phase, the pre-processing phase, combination phase, and user interface proposed by our are to solve the problem that the existing methods cannot accurately manipulate the required objects and simultaneously achieve more manipulation operations so that the entire model has better applicability. \color{blue}\textbf{[R4] It is not clear to me which part of the proposed model is trainable and some technical details of the proposed method are not clear.} \color{black} In Sec3.5, we describe the part that needs to be trained. The only part that needs to be trained is the manipulation phase, which borrows from ManiGAN. The differences between our model and ManiGAN are explained in lines 411 to 517 in the paper. Furthermore, the specific details of each part of our proposed method are presented in Sec 3.1 to 3.4. We have made detailed revisions to the previous Sec 3.1 and 3.4, not only revising the existing grammar and logic errors but also making the overall details more clear. \color{blue}\textbf{[R3] Some of the diagrams are not precise enough and create confusion.} \color{black} We have carefully revised the details in some diagrams so that they can be more refined while matching the content of the main text. \color{blue}\textbf{[R1] Does the interactive user interface only apply at the segmentation editing steps or the whole image manipulation workflow?} \color{black}This model uses the user interface only at the segmentation editing step. If the synthesized results are not satisfactory, the user can re-edit the segmentation map (as shown in Fig. 10) to obtain satisfactory results. \color{blue}\textbf{[R4] Missing the important literature StyleCLIP} \color{black}Thanks for your suggestion. We have added it to related work. \color{blue}\textbf{[R1] Will this work be available for everyone?} \color{black}Yes, we will release the code when the paper is accepted. \color{blue}\textbf{[R1] Is the generated result stable according to the text, or will there be slight changes anytime the method is run for the same text?} \color{black}If the same text is entered, the generated image is stable. \color{blue}\textbf{[R1] In Fig. 1, would it be possible for the method to remove the other bird instead?} \color{black}Yes, all objects that recognize can be manipulated. {\small

proofpile-arXiv_069-5975

\section{Introduction} \label{sec:intro} In the last decade formal techniques have received a renewed attention as the basis of a methodology for increasing the reliability of software artifacts and reducing the cost of software production~\cite{Mi&10}. In particular, a massive effort has been made to devise automatic verification techniques, such as {\em software model checking}~\cite{JhM09}, for proving the correctness of programs with respect to their specifications. In many software model checking techniques, the notion of a {\em constraint} has been shown to be very effective, both for constructing models of programs and for reasoning about them~\cite{Be&07,CoH78,De&13a,DeP99,Fla04,Gr&12,Ja&11b,Pe&98,PoR07}. Several types of constraints have been considered, such as equalities and inequalities over the booleans, the integers, the reals, the finite or infinite trees. By using constraints we can represent in a symbolic, compact way (possibly infinite) sets of values computed by programs and, more in general, sets of states reached during program executions. Then, in order to reason about program properties in an efficient way, we can use solvers specifically designed for the various classes of constraints we have mentioned above. In this paper we consider a simple imperative programming language with integer and array variables, and we adopt Constraint Logic Programming (CLP)~\cite{JaM94} as a metalanguage for representing imperative programs, their executions, and their properties. We use constraints consisting of equalities and inequalities over the integers, but the method presented here is parametric with respect to the constraint domain which is used. By following an approach first presented in~\cite{Pe&98}, a given imperative program~\textit{prog} and its interpreter are encoded as a CLP program. Then, the proofs of the properties of interest about the program~\textit{prog} are sought by analyzing the derived CLP program. In order to improve the efficiency of analysis, it is advisable to first {\em compile-away} the CLP interpreter of the language in which~\textit{prog} is written. This is done by specializing the interpreter with respect to the given program~\textit{prog} using well-known {\em program specialization} (also known as {\em partial evaluation}) techniques~\cite{Jo&93,Pe&98}. We have shown in previous work~\cite{De&13b,De&13a,Fi&11b} that program specialization can be used not only as a preprocessing step to improve the efficiency of program analysis, but also as a means of analysis on its own. In this paper, we extend that approach and we propose a verification methodology based on more general, semantics preserving {\em unfold/fold transformation rules} for CLP programs~\cite{BuD77,EtG96,TaS84}. Transformation-based verification techniques are very appealing because they are parametric with respect to both the programming languages in which programs are written, and the logics in which the properties of interest are specified. Moreover, since the output of a transformation-based verification of a program is an \textit{equivalent} program with respect to the properties of interest, we can apply a \textit{sequence} of transformations, thereby refining the analysis to the desired degree of precision. Our approach can be summarized as follows. Suppose we are given: (i) an imperative program~\textit{prog}, (ii)~a predicate \texttt{initConf} expressing the property, called the {\em initial property}, holding in the configuration from which the execution of~\textit{prog} starts, and (iii)~a predicate \texttt{errorConf} expressing the property, called the {\em error property}, holding in the configuration which should {\em not} be reached at the end of the execution of~\textit{prog}. The partial correctness of \textit{prog} is defined as the negation of a predicate $\mathtt{incorrect}$ specified by the following CLP program $T$:\nopagebreak \smallskip $\mathtt{incorrect}$ {\tt{:-}} $\mathtt{initConf(X) ,\ reach(X).}$\nopagebreak $\mathtt{reach(X)}$ {\tt{:-}} $\mathtt{tr(X,X1) ,\ reach(X1).}$\nopagebreak $\mathtt{reach(X)}$ {\tt{:-}} $\mathtt{errorConf(X).}$ \smallskip \noindent where: (i)~$\texttt{reach(X)}$ holds iff an error configuration can be reached from the configuration $\texttt{X}$, and (ii)~$\mathtt{tr(X,X1)}$ holds iff the configuration $\texttt{X1}$ can be reached in one step from the configuration $\texttt{X}$ via the transition relation that defines the operational semantics of the imperative language. Thus, $\mathtt{incorrect}$ holds iff there exists an error configuration that can be reached from an initial configuration. The verification method we propose in this paper is made out of the following two steps. \noindent {\it Step}~(A). This step, called the {\em removal of the interpreter}, consists in specializing the program $T$ (which includes the clauses defining the predicate $\mathtt{tr}$) with respect to the given program \textit{prog} and properties \texttt{initConf} and \texttt{errorConf}. After this specialization step we derive from program~$T$ a new program~$T_A$, where there is no reference to the predicate~$\mathtt{tr}$ (and in this sense we say that during this Step~(A) the interpreter is removed or `compiled-away'). \noindent {\it Step}~(B). This step, called the {\em propagation of the initial and error properties}, consists in applying a sequence of unfold/fold transformation rules, and deriving from the CLP program~$T_A$ a new CLP program $T_B$ such that ${\tt incorrect}$ holds in $T_B$ if and only if \textit{prog} is {\em not} correct with respect to the given initial and error properties. The objective of this Step~(B) is to derive a program $T_B$ where the predicate {\tt incorrect} is defined by: either (i)~the fact `{\tt incorrect.}' (and in this case \textit{prog} is not correct), or (ii)~the empty set of clauses (and in this case \textit{prog} is correct). If neither Case~(i) nor Case~(ii) holds, that is, in program $T_B$ the predicate {\tt incorrect} is defined by a non-empty set of clauses which does not contain the fact `{\tt incorrect.}', we can conclude neither the correctness nor the incorrectness of~\textit{prog}. Thus, similarly to what has been proposed in~\cite{De&13a}, we continue our verification method by iterating this Step~(B) in the hope of deriving a program where either Case~(i) or Case~(ii) holds. Obviously, due to undecidability limitations, it may be the case that we never derive such a program. \smallskip During Step (B) we apply transformation rules that are more powerful than those needed for program specialization and, in particular, for the removal of the interpreter done during Step~(A). Indeed, the rules used during Step~(B) include the {\em conjunctive definition} and the {\em conjunctive folding} rules and they allow us to introduce and transform new predicate definitions that correspond to \textit{conjunctions} of old predicates, while program specialization can deal only with new predicates that correspond to specialized versions of \textit{one} old predicate. During Step~(B) we use also the {\em goal replacement} rule, which allows us to replace conjunctions of constraints and predicates by applying equivalences that hold in the least model of $T$, while program specialization can only replace conjunctions of constraints. These more powerful rules extend the specialization-based verification techniques in two ways. First, we can verify programs with respect to complex initial and error properties defined by sets of CLP clauses (for instance, recursively defined relations among program variables), whereas program specialization can only deal with initial and error properties specified by (sets of) constraints. Second, we can verify programs which manipulate arrays and other data structures by applying equivalences between predicates that express suitable properties of those structures. In particular, in this paper we will apply equivalences which follow from the axiomatization of the theory of arrays~\cite{Br&06}. \smallskip The paper is organized as follows. In Section~\ref{sec:syntax_semantics} we present the imperative language and the definition of its interpreter as a CLP program. In Section~\ref{sec:translating-correctness-into-CLP} we describe how partial correctness properties of imperative programs can be translated to predicates defined by CLP programs. In Section~\ref{sec:verification_method} we present our transformation-based verification method, and a general strategy to apply the transformation rules. Next, we present two examples of application of our verification method. In particular, in Section~\ref{sec:conjunctivefolding} we show how we deal with specifications provided by recursive CLP clauses, and in Section~\ref{sec:array_properties} we show how we deal with programs that manipulate arrays. Finally, in Section~\ref{sec:conclusions} we discuss related work in the area of program verification. \section{Translating Imperative Programs into CLP} \label{sec:syntax_semantics} We consider a simple imperative language with arrays. We are given: (i)~the set {\it{Vars}} of {\rm{integer variable identifiers}}, (ii)~the set~{\it{AVars}} of {\rm{integer array identifiers}}, and (iii)~the set~$\mathbb{Z}$ of the integer constants. Our language has the following syntax: \vspace{1mm} \hspace{-7mm} {\rm{ \begin{tabular}{l@{\hspace{0.5mm}}c@{\hspace{1mm}}lllll} $x,y,z,i,j,m,n,\ldots~ $ & $\in$ & $\mathit{Vars}$ (integer variable identifiers) & \\ $a,b,\ldots$& $\in$ & $\mathit{AVars}$ (integer array identifiers) & \\ $\mathit{k,\ldots}$ & $\in$ & $\mathbb{Z}$ (integer constants) & \\ $\mathit{\ell}, \mathit{\ell}_{0}, \mathit{\ell}_{1}, \ldots $ & $\in$ & Labels ($\subseteq \mathbb{Z}$) & \\ $\mathit{uop}$, $\mathit{bop}$ & $\in$ & Unary and binary operators ({\tt{-}}, {\tt{+}}, {\tt{<}}, $\ldots$) & \\[1.5mm] \end{tabular} }} \hspace{-7mm} {\rm{ \begin{tabular}{l@{\hspace{1.5mm}}c@{\hspace{2mm}}lllll} $ {\mathrm{prog}}$ & ::= & $ {\mathrm{lab\!\_\!cmd}} ^{+}$ & &\\ \makebox[13mm][l]{$ {\mathrm{lab\!\_\!cmd}}$} & ::= & $\ell\! :\! {\mathrm{cmd}} $ \\ \end{tabular} }} \vspace{-0.5mm} \hspace{-7mm} {\rm{ \begin{tabular}{l@{\hspace{1.5mm}}c@{\hspace{2mm}}lll} \makebox[13mm][l]{${\mathrm{cmd}}$}& ::= & ${\mathtt{halt}} ~~|~~ x\! =\! {\mathrm{expr}} ~~|~~ a${\tt[}$\mathrm{expr}${\tt]}$=\!\mathrm{expr}$ ~~$|$~~ ${\mathtt{if~}}${\tt(}$ {\mathrm{expr}}${\tt)}$~\ell_{1}~{\mathtt{else}}$ $\ell_{2}$ ~~$|$~~ ${\mathtt{goto}}$ $\ell$ \end{tabular} }} \vspace{-0.5mm} \hspace{-7mm} {\rm{ \begin{tabular}{l@{\hspace{1.5mm}}c@{\hspace{2mm}}ll} \makebox[13mm][l]{${\mathrm{expr}}$}& ::= & $\emph{k} ~~|~~ \emph{x} ~~|~~ \emph{uop} ~ {\mathrm{expr}} ~~|~~ {\mathrm{expr}} $ \emph{bop} $ {\mathrm {expr}} ~~|~~ a${\tt[}$\mathrm{expr}${\tt]}$ $ \\ \end{tabular} }} \smallskip \noindent A program is a non-empty sequence of labeled commands (also called commands, for brevity). The elements of a sequence are separated by semicolons. We assume that in every program each label occurs at most once. Note that in our language we can deal with conditional and iterative commands, such as `${\mathtt{if~}}${\tt(}$ {\mathrm{expr}}${\tt)}~{cmd}', `${\mathtt{if~}}${\tt(}$ {\mathrm{expr}}${\tt)}~{cmd}~${\mathtt{else}}$ {cmd}', and `${\mathtt{while~}}${\tt(}$ {\mathrm{expr}}${\tt)}~\{{cmd}\}', by translating them using ${\mathtt{if}}$-${\mathtt{else}}$ and ${\mathtt{goto}}$ commands. In order to focus our attention on the verification issues, we do not consider in our imperative language other features such as: (i)~type and variable declarations, (ii)~function declarations and function calls, and (iii) multidimensional arrays. Those features can be added in a straightforward way (see, for instance,~\cite{De&13a}). \smallskip Now we give the semantics of our language by defining a binary relation $\Longrightarrow$ which will be encoded by a CLP predicate {\tt tr}. For that purpose let us first introduce the following auxiliary notions. An {\it{environment}} $\delta$ is a function that maps: (i)~every integer variable identifier $x$ to its value $v\in \mathbb Z$, and (ii)~every array identifier $a$ to a finite function from the set $\{0,\ldots,{\textit{dim}}(a)\!-\!1\}$ to $\mathbb Z$, where ${\mathit{dim}}(a)$ is the dimension of $a$. For any expression~$\mathit{e}$, array identifier $a$, and environment~$\delta$, (i) $\llbracket \mathit{e}\rrbracket\delta$ is the integer value of $\mathit{e}$ defined by induction on the structure of $\mathit{e}$ (in particular, for any integer variable identifier~$x$, $\llbracket x\rrbracket\delta =_\textit{def} \delta(x)$), and (ii) $\llbracket a[\mathit{e}]\rrbracket\delta=_\textit{def}\delta(a)(\llbracket \mathit{e}\rrbracket\delta)$. We assume that the evaluation of expressions has no side effects. A {\it{configuration}} is a pair of the form $\langle{\hspace*{-2.5pt}}\langle c, \delta\rangle{\hspace*{-2.5pt}}\rangle$ where: (i)~$c$~is a labeled command, and (ii)~$\delta$ is an environment. By $\mathit{update}(f,x,v)$ we denote the function $f'$ such that, for every $y$, if $y\!=\!x$ then $f'\!(y)\!=\!v$ else $f'\!(y)\!=\!f(y)$. By $\mathit{write}(f,x,v)$ we denote the function $\mathit{update}(f,x,v)$ in the case where $f$ is a finite function denoting an array, $x$ is an integer in the domain of $f$, and $v$ is an integer value. For any program~{\textit{prog}}, for any label $\ell$, (i)~${\mathit{at}}(\ell)$ denotes the command in~{\textit{prog}} with label~$\ell$, and (ii)~{\it{nextlab}}$(\ell)$ denotes the label of the command in~{\textit{prog}} which is written {\it{immediately after}} the command with label~$\ell$. \smallskip The operational semantics (that is, the interpreter) of our language is given as a transition relation $\Longrightarrow$ between configurations according to the following rules $R1$--$R3$. Notice that no rules are given for the command $\ell\!:\! {\mathtt{halt}}$. Thus, no configuration $\gamma$ exists such that $\ell\!:\! \mathtt{halt}\, \Longrightarrow\, \gamma$. \smallskip \noindent $(R1)$. {\it{Assignment}}. \smallskip \noindent If \emph{$x$} is an integer variable identifier: \noindent \makebox[6mm][l]{}$\langle{\hspace*{-2.5pt}}\langle \ell\!:\!x\!=\!e,\ \delta\rangle{\hspace*{-2.5pt}}\rangle \Longrightarrow \langle{\hspace*{-2.5pt}}\langle{\mathit{at}}(\!{\mathit{nextlab}}(\ell)\!),\ {\mathit{update}}(\delta,x,\llbracket e\rrbracket\delta)\rangle{\hspace*{-2.5pt}}\rangle$ \smallskip \noindent If \emph{$a$} is an integer array identifier: \noindent \makebox[6mm][l]{}$\langle{\hspace*{-2.5pt}}\langle \ell\!:\!a[\mathit{ie}]\!=\!e,\ \delta\rangle{\hspace*{-2.5pt}}\rangle \Longrightarrow \langle{\hspace*{-2.5pt}}\langle{\mathit{at}}(\!{\mathit{nextlab}}(\ell)\!),\ {\mathit{update}}(\delta, a,\mathit{write}(\delta(a),\llbracket \mathit{ie}\rrbracket\delta,\llbracket e\rrbracket\delta))\rangle{\hspace*{-2.5pt}}\rangle$ \smallskip \noindent \smallskip \noindent \makebox[22mm][l]{$(R2)$. {\it{Goto}}.} $\langle{\hspace*{-2.5pt}}\langle \ell\!:\!{\mathit{goto}}~\ell',\ \delta\rangle{\hspace*{-2.5pt}}\rangle\Longrightarrow\langle{\hspace*{-2.5pt}}\langle {\mathit{at}}(\ell'),\ \delta\rangle{\hspace*{-2.5pt}}\rangle$ \smallskip \noindent $(R3)$. {\it{If-else}}. \smallskip \noindent \makebox[22mm][l]{If $\llbracket e\rrbracket\delta\!\not =\!0$:} $\langle{\hspace*{-2.5pt}}\langle \ell\!: {\mathtt{if}}~${\tt(}$e${\tt)}$~ \ell_{1}~{\mathtt{else}}~\ell_{2}, \ \delta\rangle{\hspace*{-2.5pt}}\rangle \Longrightarrow\langle{\hspace*{-2.5pt}}\langle{\mathit{at}}(\ell_{1}),\ \delta\rangle{\hspace*{-2.5pt}}\rangle$ \smallskip \noindent \makebox[22mm][l]{If $\llbracket e\rrbracket\delta\!=\!0$:} $\langle{\hspace*{-2.5pt}}\langle \ell\!: {\mathtt{if}}~${\tt(}$e${\tt)}$~ \ell_{1}~{\mathtt{else}}~\ell_{2}, \ \delta\rangle{\hspace*{-2.5pt}}\rangle \Longrightarrow\langle{\hspace*{-2.5pt}}\langle{\mathit{at}}(\ell_{2}),\ \delta\rangle{\hspace*{-2.5pt}}\rangle$ \smallskip Let us now recall some notions and terminology concerning constraint logic programming (CLP). For more details the reader may refer to~\cite{JaM94}. If $\mathtt{p_1}$ and $\mathtt{p_2}$ are linear polynomials with integer coefficients and integer variables, then $\mathtt{p_1\!=\!p_2}$, $\mathtt{p_1\!\not =\!p_2}$, $\mathtt{p_1\!<\! p_2}$, $\mathtt{p_1\!\leq\! p_2}$, $\mathtt{p_1\!\geq\! p_2}$, and $\mathtt{p_1\!>\!p_2}$ are {\em atomic constraints}. A {\em constraint} is either $\tt{true}$, or $\tt{false}$, or an atomic constraint, or a {\it{conjunction}} of constraints. A CLP program is a finite set of clauses of the form $\texttt{A\,:-\,c,B}$, where $\texttt{A}$ is an atom, $\texttt{c}$ is a constraint, and $\texttt{B}$ is a (possibly empty) conjunction of atoms. A clause of the form: $\texttt{A\,:-\,c}$ is called a {\em constrained fact} or simply a {\em fact} if~$\texttt{c}$ is $\tt{true}$. The semantics of a CLP program $P$ is defined to be the {\em least model} of $P$ which extends the standard interpretation on the integers. This model is denoted by $M(P)$. The CLP interpreter for our imperative language is given by the following predicate $\mathtt{tr}$ which encodes the transition relation $\Longrightarrow$. \smallskip \noindent \makebox[6mm][r]{1.~}~$\mathtt{tr(cf(cmd(L,asgn(int(X),E)),D),\ cf(cmd(L1,C),D1))}$ {\tt{:-}}\nopagebreak ~~~~$\mathtt{eval(E,D,V), ~~update(D,int(X),V,D1), ~~nextlab(L,L1), ~at(L1,C). }$ \noindent \makebox[6mm][r]{2.~}~$\mathtt{tr(cf(cmd(L,asgn(arrayelem(A,IE),E)),D),\ cf(cmd(L1,C),D1))}$ {\tt{:-}}\nopagebreak ~~~~$\mathtt{eval(IE,D,I), ~~eval(E,D,V), ~~lookup(D,array(A),FA), ~~write(FA,I,V,FA1),}$ ~~~~$\mathtt{update(D,array(A),FA1,D1), ~~nextlab(L,L1), ~~at(L1,C). }$ \noindent \makebox[6mm][r]{3.~}~$\mathtt{tr(cf(cmd(L,ite(E,L1,L2)),D),\ cf(cmd(L1,C),D))}$ {\tt{:-}} $\mathtt{V\neq0, ~~eval(E,D,V), ~~at(L1,C). }$ \noindent \makebox[6mm][r]{4.~}~$\mathtt{tr(cf(cmd(L,ite(E,L1,L2)),D),\ cf(cmd(L2,C),D))}$ {\tt{:-}} $\mathtt{V=0, ~~eval(E,D,V), ~~at(L2,C). }$ \noindent \makebox[6mm][r]{5.~}~$\mathtt{tr(cf(cmd(L,goto(L1)),D),\ cf(cmd(L1,C),D))}$ {\tt{:-}} $\mathtt{at(L1,C).}$ \smallskip \noindent In the above clauses the term $\mathtt{asgn(int(X),E)}$ encodes a variable assignment of the form $x\!=\!e$, while $\mathtt{asgn(arrayelem(A,IE),E)}$ encodes an array assignment of the form $a${\tt [}$\mathit{ie}${\tt ]}$=\!e$. Similarly, the terms $\mathtt{ite(E,L1,L2)}$ and $\mathtt{goto(L)}$ encode the conditional ${\mathtt{if}}\,${\tt(}$e${\tt)}$\ \ell_{1}\ {\mathtt{else}}\ \ell_{2}$ and the jump ${\mathtt{goto}}\ \ell$, respectively. The term $\mathtt{cmd(L,C)}$ encodes the command~$\mathtt{C}$ with label $\mathtt{L}$. The predicate $\mathtt{at(L,C)}$ binds to $\mathtt{C}$ the command with label~$\mathtt{L}$. The predicate $\mathtt{nextlab(L,L1)}$ binds to~$\mathtt{L1}$ the label of the command which is written immediately after the command with label~$\mathtt{L}$. The predicate $\mathtt{eval(E,D,V)}$ computes the value $\mathtt{V}$ of the expression~$\mathtt{E}$ in the environment~$\mathtt{D}$. Below we list a subset of the clauses that define $\mathtt{eval(E,D,V)}$ by induction on the structure of~$\mathtt{E}$. The others are similar and we shall omit them. \smallskip \noindent \makebox[6mm][r]{6.~}~$\mathtt{eval(int(X),D,V)}$ {\tt{:-}} $\mathtt{lookup(D,int(X),V).}$ \noindent \makebox[6mm][r]{7.~}~$\mathtt{eval(not(E),D,V)}$ {\tt{:-}} $\mathtt{{V\!\not =\!0}, ~~V1\!=\!0, ~~eval(E,D,V1).}$ \noindent \makebox[6mm][r]{8.~}~$\mathtt{eval(not(E),D,V)}$ {\tt{:-}} $\mathtt{V\!=\!0, ~~V1\!\neq\!0, ~~eval(E,D,V1).}$ \noindent \makebox[6mm][r]{9.~}~$\mathtt{eval(plus(E1,E2),D,V)}$ {\tt{:-}} $\mathtt{V=V1\!+\!V2, ~~eval(E1,D,V1), ~~eval(E2,D,V2).}$ \noindent \makebox[6mm][r]{10.~}~$\mathtt{eval(arrayelem(A,IE),D,V)}$ {\tt{:-}} $\mathtt{eval(IE,D,I), ~~lookup(D,array(A),FA), ~~read(FA,I,V).}$ \smallskip \noindent The predicate $\mathtt{update(D,Id,B,D1)}$ updates the environment~$\mathtt{D}$, thereby constructing the new environment~$\mathtt{D1}$ by binding the (integer or array) identifier~$\mathtt{Id}$ to the (integer or {array}) value~$\mathtt{B}$. The predicate $\mathtt{lookup(D,Id,B)}$ retrieves from the environment $\mathtt{D}$ the (integer or {array}) value $\mathtt{B}$ bound to the identifier~$\mathtt{Id}$. The predicate {\tt read} gets the value of an element of an array, and the predicate {\tt write} sets the value of an element of an array. As already mentioned, the environment maps an array to a finite function. We represent this function as a pair $\texttt{(A,N)}$, where $\texttt{N}$ is the dimension of the array and $\texttt{A}$ is a list of integers of length $\texttt{N}$. The predicate ${\texttt{read((A,N),I,X)}}$ holds iff the~{\tt I}-th element of~{\tt A} is~{\tt X}, and the predicate ${\texttt{write((A,N),I,X,(A1,N))}}$ holds iff the list~{\tt A1} is equal to~{\tt A} except that the {\tt I}-th element of~{\tt A1} is~{\tt X}. For reasons of space, we do not list the clauses defining $\mathtt{update}$, $\mathtt{lookup}$, $\mathtt{read}$, and $\mathtt{write}$. \section{Translating Partial Correctness into CLP} \label{sec:translating-correctness-into-CLP} The problem of verifying the partial {\rm correctness} of a {\rm{program}} {\textit{prog}} is the problem of checking whether or not, starting from an initial configuration, the execution of {\textit{prog}} leads to an {\rm{error configuration}}. This problem is formalized by defining an {\it{incorrectness triple}} of the form $\{\hspace{-3pt}\{\varphi_{\mathit{init}}\}\hspace{-3pt}\} $ {\textit{prog}} $ \{\hspace{-3pt}\{\varphi_{\mathit{error}}\}\hspace{-3pt}\}$, where: \hangindent=7mm \noindent \makebox[7mm][l]{(i)~}{\textit{prog}} is a program which acts on the variables $z_1,\ldots,z_r$, each of which is either an integer variable or an integer array, \hangindent=7mm \noindent \makebox[7mm][l]{(ii)~}$\varphi_{\mathit{init}}$ is a first-order formula with free variables in $\{z_1,\ldots,z_r\}$ characterizing the initial configurations, and \hangindent=7mm \noindent \makebox[7mm][l]{(iii)~}$\varphi_{\mathit{error}}$ is a first-order formula with free variables in $\{z_1,\ldots,z_r\}$ characterizing the error configurations. \noindent We assume that: (i)~the formulas $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$ can be encoded using CLP predicates, and (ii)~the domain of every environment $\delta$ is the set $\{z_1,\ldots,z_r\}$. We also assume, without loss of generality, that the last command of {\textit{prog}} is $\ell_{h}\!:\!{\mathtt{halt}}$ and no other {\tt halt} command occurs in {\textit{prog}}. We say that a program {\textit{prog}} is {\it{incorrect}} with respect to a set of initial configurations satisfying $\varphi_{\mathit{init}}$ and a set of error configurations satisfying $\varphi_{\mathit{error}}$, or simply, ${\textit{prog}}$ is {\it{incorrect}} with respect to $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$, if there exist environments $\delta_{\mathit{init}}$ and $\delta_{\mathit{h}}$ such that \smallskip \noindent \makebox[7mm][l]{(i)~}$\varphi_{\mathit{init}}(\delta_{\mathit{init}}(z_{1}),\ldots,\delta_{\mathit{init}}(z_{r}))$ holds, \smallskip \noindent \makebox[7mm][l]{(ii)~}$\langle{\hspace*{-2.5pt}}\langle \ell_{0}\!:\!c_{0}, \ \delta_{\mathit{init}}\rangle{\hspace*{-2.5pt}}\rangle$ $\Longrightarrow ^*$ $\langle{\hspace*{-2.5pt}}\langle \ell_{h}\!:\!{\mathtt{halt}},\ \delta_{\mathit{h}}\rangle{\hspace*{-2.5pt}}\rangle$ and \smallskip \noindent \makebox[7mm][l]{(iii)~}$\varphi_{\mathit{error}}(\delta_{\mathit{h}}(z_{1}),\ldots,\delta_{\mathit{h}}(z_{r}))$ holds, \smallskip \noindent where $\ell_{0}\!:\!c_{0}$ is the first command in~{\textit{prog}}. Obviously, in $\varphi_{\mathit{init}}$ or in $\varphi_{\mathit{error}}$ some of the variables $z_{1},\ldots,z_{r}$ may be absent. A program is said to be {\it{correct}} with respect to $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$ iff it is not incorrect with respect to $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$. We now show, by means of an example, how to encode an incorrectness triple as a CLP program. The reader will have no difficulty to see how this encoding can be done in general. Let us consider the incorrectness triple $\{\hspace{-3pt}\{\varphi_{\mathit{init}}(i,j)\}\hspace{-3pt}\}$ {\it{increase}} $\{\hspace{-3pt}\{\varphi_{\mathit{error}}(i,j)\}\hspace{-3pt}\}$ where:\nopagebreak \noindent -- $\varphi_{\mathit{init}}(i,j)$ is $ i\!=\!0 \andw j\!=\!0$,\nopagebreak \noindent \makebox[40mm][l]{-- {\it{increase}} is the program} \noindent \hspace{-3mm}\makebox[6mm][l]{}$\ell_{0}$:~ $\mathtt{while}~(i\!<\!2n)~\{ \mathtt{if}~(i\!<\!n)~i=i\!+\!1;~\mathtt{else}~j=i\!+\!1;$\hspace{4mm}$i=i\!+\!1\}$\,;\nopagebreak \noindent \hspace{-3mm}\makebox[6mm][l]{}$\ell_{h}$:~ $\mathtt{halt}$\nopagebreak \noindent -- $\varphi_{\mathit{error}}(i,j)$ is $ i\!<\!j$. \smallskip \noindent First, we replace the given program {\textit{increase}} by the following sequence of commands: \smallskip \hspace{-3mm}$\ell_{0}$:~ $\mathtt{if}~(i\!<\!2n)~ \ell_{1} \mathtt{~else~} \ell_{h};$\nopagebreak \hspace{-3mm}$\ell_{1}$:~ $\mathtt{if}~(i\!<\!n)~ \ell_{2} \mathtt{~else~} \ell_{4};$ \hspace{-3mm}$\ell_{2}$:~ $i=i\!+\!1;$\nopagebreak \hspace{-3mm}$\ell_{3}$:~ $\mathtt{goto}~ \ell_{5};$\nopagebreak \hspace{-3mm}$\ell_{4}$:~ $j=i\!+\!1;$\nopagebreak \hspace{-3mm}$\ell_{5}$:~ $i=i\!+\!1;$\nopagebreak \hspace{-3mm}$\ell_{6}$:~ $\mathtt{goto}~ \ell_{0};$\nopagebreak \hspace{-3mm}$\ell_{h}$:~ $\mathtt{halt}$ \smallskip \noindent Then, this sequence of commands is translated into the following CLP facts of the form: ${\mathtt{at(}}\ell,{\textit{cmd}}{\mathtt{)}}$ meaning that at label $\ell$ there is the command $\mathit{cmd}$: \smallskip \noindent \makebox[6mm][r]{1.~}~$\mathtt{at(0,ite(less(int(i),mult(int(2),int(n))),1,h))}$. \noindent \makebox[6mm][r]{2.~}~$\mathtt{at(1,ite(less(int(i),int(n)),2,4))}$. \noindent \makebox[6mm][r]{3.~}~$\mathtt{at(2,asgn(int(i),plus(int(i),int(1))))}$. \noindent \makebox[6mm][r]{4.~}~$\mathtt{at(3,goto(5))}$. \noindent \makebox[6mm][r]{5.~}~$\mathtt{at(4,asgn(int(j),plus(int(i),int(1))))}$. \noindent \makebox[6mm][r]{6.~}~$\mathtt{at(5,asgn(int(i),plus(int(i),int(1))))}$. \noindent \makebox[6mm][r]{7.~}~$\mathtt{at(6,goto(0))}$. \noindent \makebox[6mm][r]{8.~}~$\mathtt{at(h,halt)}$. \smallskip \noindent We also have the following clauses that specify that an error configuration can be reached from an initial configuration, in which case the atom {\tt incorrect} holds: \smallskip \noindent \makebox[6mm][r]{9.~}~$\mathtt{incorrect}$ {\tt{:-}} $\mathtt{initConf(X) ,\ reach(X).}$ \noindent \makebox[6mm][r]{10.~}~$\mathtt{reach(X)}$ {\tt{:-}} $\mathtt{tr(X,X1) ,\ reach(X1).}$ \noindent \makebox[6mm][r]{11.~}~$\mathtt{reach(X)}$ {\tt{:-}} $\mathtt{errorConf(X).}$ \smallskip \noindent In our case the predicates $\mathtt{initConf}$ and $\mathtt{errorConf}$ specifying the initial and the error configurations, respectively, are defined by the following clauses: \smallskip \noindent \makebox[6mm][r]{12.~}~$\mathtt{initConf(cf(cmd(0,ite(less(int(i),mult(int(2),int(n))),1,h)),}$ \hspace{12mm}$\mathtt{[[int(i),I],[int(j),J],[int(n),N]]))}$ {\tt{:-}} $\mathtt{phiInit(I,J)}$ \noindent \makebox[6mm][r]{13.~}~$\mathtt{errorConf(cf(cmd(h,halt),}$ \hspace{12mm}$\mathtt{[[int(i),I],[int(j),J],[int(n),N]]))}$ {\tt{:-}} $\mathtt{phiError(I,J)}$ \noindent \makebox[6mm][r]{14.~}~$\mathtt{phiInit(I,J)}$ {\tt{:-}} $\mathtt{I\!=\!0},\ \mathtt{J\!=\!0}.$ \noindent \makebox[6mm][r]{15.~}~$\mathtt{phiError(I,J)}$ {\tt{:-}} $\mathtt{I\!<\!J}.$ \smallskip \noindent In the initial configuration (see clause~12) the command (labeled by~{\tt 0}) is: \smallskip $\mathtt{cmd(0,ite(less(int(i),mult(int(2),int(n))),1,h))}$ \smallskip \noindent In clauses~12 and~13 the environment $\delta$ has been encoded by the list: \smallskip $\mathtt{[ [int(i),I],[int(j),J],[int(n),N]]}$ \smallskip \noindent which provides the bindings for the integer variables~$i,j$, and $n$, respectively. The initial environment is any environment which binds $i$ to 0 and $j$ to 0. The CLP program consisting of clauses 1--15 above, together with the clauses that define the predicate {\tt tr} (see clauses {1--5} of Section~\ref{sec:syntax_semantics}), is called the {\em CLP encoding} of the given incorrectness triple \mbox{$\{\hspace{-3pt}\{\varphi_{\mathit{init}}(i,j)\}\hspace{-3pt}\}$ {\textit{prog}} $ \{\hspace{-3pt}\{\varphi_{\mathit{error}}(i,j)\}\hspace{-3pt}\}$.} \vspace*{-1mm} \begin{theorem}{\em (Correctness of CLP Encoding)} \label{thm:encoding} Let $\mathit{T}$ be the CLP encoding of the incorrectness triple $\{\hspace{-3pt}\{\varphi_{\mathit{init}}\}\hspace{-3pt}\} $ {\textit{prog}} $ \{\hspace{-3pt}\{\varphi_{\mathit{error}}\}\hspace{-3pt}\}$. The program {\textit{prog}}~is correct with respect to $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$ iff $\,{\tt incorrect}\notin\!M(\textit{T\/})$. \end{theorem} \section{The Verification Method} \label{sec:verification_method} In this section we present a method for verifying whether or not a program \textit{prog} is correct with respect to any given pair of $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$ formulas. By Theorem~\ref{thm:encoding}, this verification task is equivalent to checking whether or not $\,{\tt incorrect}\notin\!M(\textit{T\/})$, where $\mathit{T}$ is the CLP encoding of the incorrectness triple \mbox{$\{\hspace{-3pt}\{\varphi_{\mathit{init}}\}\hspace{-3pt}\} $ {\textit{prog}} $ \{\hspace{-3pt}\{\varphi_{\mathit{error}}\}\hspace{-3pt}\}$.} Our verification method is based on transformations of CLP programs that preserve the least model semantics~\cite{EtG96,Fi&04a}. It makes use of the following {\em transformation rules}, collectively called {\em unfold/fold rules}: {\em unfolding}, {\em goal replacement}, {\em clause removal}, {\em definition}, and {\em folding}. The verification method is an extension of the method for proving safety of imperative programs by specialization of CLP programs presented in~\cite{De&13a}. Actually, the transformations presented in this paper are much more powerful than program specialization and, as we will show in the next sections, they enable the verification of correctness properties which are more complex than those considered in~\cite{De&13a}. \smallskip Our verification method consists of the following two steps. \noindent {\it Step}~(A): {\em Removal of the Interpreter.} The CLP program $T$ which encodes the incorrectness triple \mbox{$\{\hspace{-3pt}\{\varphi_{\mathit{init}}\}\hspace{-3pt}\} $ {\textit{prog}} $ \{\hspace{-3pt}\{\varphi_{\mathit{error}}\}\hspace{-3pt}\}$} is specialized with respect to the clauses encoding $\varphi_{\mathit{init}}$, \textit{prog}, and $\varphi_{\mathit{error}}$. The result of this first transformation step is a new CLP program~$T_A$ such that ${\tt incorrect} \in M(T)$ iff ${\tt incorrect} \in M(T_A)$. Program~$T_A$ incorporates the operational semantics of the imperative program~${\mathit{prog}}$, but the clauses defining the predicate {\tt tr}, that is, the interpreter of the imperative language we use, do not occur in $T_A$, hence the name of this transformation step. Step~(A) is common to other verification techniques which are based on program specialization~\cite{De&13b,Pe&98}. \smallskip \noindent {\it Step}~(B):~{\em Propagation of the initial and error properties.} By applying the unfold/fold transformation rules, program~$ T_A$ is transformed into a new CLP program $T_B$ such that ${\tt incorrect} \in M(T_A)$ iff ${\tt incorrect} \in M(T_B)$. This transformation exploits the interactions among the predicates encoding the initial property $\varphi_{\mathit{init}}$, the operational semantics of \textit{prog}, and the error property $\varphi_{\mathit{error}}$, with the objective of deriving a program $T_B$ where the predicate {\tt incorrect} is defined either by (i) the fact `{\tt incorrect.}', or by (ii) the empty set of clauses (that is, no clauses for {\tt incorrect} occur in~$T_B$). In Case (i) the imperative program \textit{prog} is incorrect with respect to the given $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$ properties, while in Case (ii) \textit{prog} is correct with respect to these properties. There is a third case where neither (i) nor (ii) holds, that is, in program $T_B$ the predicate {\tt incorrect} is defined by a non-empty set of clauses not containing the fact `{\tt incorrect.}'. In this third case we cannot conclude anything about the correctness of~\textit{prog} by a simple inspection of~$T_B$ and, similarly to what has been proposed in~\cite{De&13a}, we iterate Step~(B) by propagating at each iteration the initial and error properties (that, in general, could have been modified during the previous iterations), in the hope of deriving a program where either Case~(i) or Case~(ii) holds. Obviously, due to undecidability limitations, we may never get to one of these two cases. Note that either Case~(i) or Case~(ii) may hold for the program~$T_A$ that we have derived at the end of Step~(A) and, if this happens, we need not perform Step~(B) at all. \smallskip Step~(A) and Step~(B) are both instances of the {\it Transform} strategy presented in Figure~\ref{fig:transf_proc}. These instances are obtained by suitable choices of the {\em unfolding}, {\em generalization}, and {\em goal replacement} auxiliary strategies. Step~(A) has been fully automated using the MAP system~\cite{MAP} and always returns a program with no residual call to the predicate {\tt tr} (this is a consequence of the fact that {\tt tr} has no recursive calls, and hence all calls to {\tt tr} can be fully unfolded). A detailed description of Step~(A) for a simple \mbox{C-like} imperative language without arrays can be found in~\cite{De&13a}. As discussed below, we can also design the unfolding, generalization, and goal replacement auxiliary strategies in such a way that Step~(B) always terminates (see, in particular, Theorem~\ref{thm:term_corr_Transform}). However, the design of auxiliary strategies that are effective in practice is a very hard task. Some of those strategies have been automated and, at the moment, we are performing experiments for their evaluation. \medskip \begin{figure}[!ht] \noindent\hrulefill \noindent \emph{Input\/}: Program~$P$.\\ \noindent \emph{Output\/}: Program $\textit{TransfP}$ such that $\texttt{incorrect}\!\in\! M(P)$ iff $\texttt{incorrect}\!\in\! M(\textit{TransfP})$. \vspace*{-2mm} \rule{30mm}{0.1mm} \noindent \textsc{Initialization}: $\textit{TransfP}:=\emptyset$; ~ ~~$\textit{InDefs}:=\{\texttt{incorrect:-\,c\!,\,G}\}$; ~ ~~$\textit{Defs}:= \textit{InDefs}$; \medskip \noindent \textit{while}~in \textit{InDefs} there is a clause~$C$ ~\textit{do} \smallskip \hspace*{3mm}\begin{minipage}{154mm} \noindent \textsc{Unfolding}: \noindent $\textit{TransfC} :=\textit{Unf\/}(C,A)$,~ where $A$ is the leftmost atom with high predicate in the body of~$C$; \smallskip \noindent \makebox[9mm][l]{\textit{while}}in $\textit{TransfC}$ there is a clause $D$ whose body contains an occurrence of an unfoldable atom $B$ ~\textit{do} ~~$\textit{TransfC} := (\textit{TransfC}- \{ D\}) \cup \textit{Unf\/}(D,B)$ \hangindent=8mm \noindent \textit{end-while}; \smallskip \noindent \textsc{Goal Replacement}: \noindent select a subset $R $ of $ \textit{TransfC\/}$; \noindent {\it{for every}} $D\! \in\! R$ {\it do} \hspace*{8mm}\makebox[4mm][l]{\it if} \makebox[5mm][r]{(i)~} the constrained goal $\mathtt{c_1, G_1}$ occurs in the body of $D$, \hspace*{13mm}\makebox[5mm][r]{(ii)~} all predicates in $\mathtt{G_1}$ are low, ~and~ \hspace*{13mm}\makebox[5mm][r]{(iii)~} $M(P)\models \forall\, ({\mathtt{c_1, G_{1}}}\!\leftrightarrow\!{\mathtt{c_2, G_{2}}})$, \hspace*{8mm}{\it then} ~replace $\mathtt{c_1, G_1}$ by $\mathtt{c_2, G_2}$ in the body of $D$; \smallskip \noindent \textsc{Clause Removal}: \noindent \makebox[8mm][l]{\textit{while}}~in $\textit{TransfC}$ there is a clause $F$ whose body contains an unsatisfiable constraint ~\textit{do}\\ $\textit{TransfC} := \textit{TransfC}- \{ F \}$ \hangindent=8mm \noindent \textit{end-while}; \smallskip \noindent {\textsc{Definition} \& \textsc{Folding}:} \noindent \makebox[8mm][l]{\textit{while}}~in $\textit{TransfC}$ there is a clause $E$ of the form: ~$\texttt{H\,:-\,e,\,L,\,Q,\,R}$~ such that at least one high predicate occurs in $\mathtt{Q}$~ \textit{do}\hangindent=0mm \smallskip \hspace*{4mm} \begin{minipage}{150mm} \noindent \hangindent=9mm\hspace*{4mm}\makebox[4mm][l]{\textit{if}}~in $\textit{Defs}$ there is a clause $D$ of the form: $\texttt{Newd\,:-\,d,\,D}$, where:\\{\hspace*{1mm}}(i) for some substitution $\vartheta$, $\mathtt{Q=D\,\vartheta}$, ~and~ \\ (ii) $\texttt{e}\sqsubseteq\,\texttt{d}\,\vartheta$ \noindent \hspace*{4mm}\textit{then}~~$\textit{TransfC} :=(\textit{TransfC} -\{E \})\cup \{\texttt{H\,:-\,e,\,L,\,Newd} \,\vartheta\texttt{,\,R}\}$; \noindent \hspace*{4mm}\makebox[7mm][l]{\textit{else}}~~let $\textit{Gen}(E,\textit{Defs})$ be ~$\texttt{Newg\,:-\,g,\,G}$, ~~where: \hspace*{13mm}\makebox[7mm][r]{(i)~}~\texttt{Newg} is an atom with high predicate symbol not occurring in\,$P\cup \textit{Defs}$, \hspace*{13mm}\makebox[7mm][r]{(ii)~}~for some substitution $\vartheta$, $\mathtt{Q} = \mathtt{G}\,\vartheta$, ~and~ \hspace*{13mm}\makebox[7mm][r]{(iii)~}~${\tt e} \sqsubseteq{\tt g}\,\vartheta$; \makebox[13mm]{~}$\textit{Defs}:=\textit{Defs}\cup \{\textit{Gen}(E,\textit{Defs})\}$; \makebox[5mm]{~}$\textit{InDefs}:=\textit{InDefs} \cup \{\textit{Gen}(E,\textit{Defs})\}$; \\ \makebox[13mm]{~}$\textit{TransfC} :=(\textit{TransfC} -\{E \})\cup \{\texttt{H\,:-\,e,\,L,\,Newg}\,\vartheta\texttt{,\,R}\}$ \end{minipage} \smallskip \noindent \textit{end-while}; \smallskip \noindent $\textit{InDefs}:=\textit{InDefs}-\{C\}$; ~~~ $\textit{TransfP}:=\textit{TransfP}\cup \textit{TransfC}$; \end{minipage} \smallskip \noindent \textit{end-while}; \smallskip \noindent {\textsc{Removal of Useless Clauses}:} Remove from $\textit{TransfP}$ all clauses whose head predicate is useless. \vspace*{-1mm} \noindent\hrulefill \vspace*{-1mm} \caption {The \textit{Transform} strategy.} \label{fig:transf_proc} \vspace*{-4mm} \end{figure} Let us briefly illustrate the \textit{Transform} strategy. We assume that the CLP program $P$ taken as input by the strategy contains a single clause defining the predicate \texttt{incorrect} of the form: $\texttt{incorrect\,:-\,c\!,\,G}$, where $\texttt{c}$ is a constraint (possibly $\texttt{true}$) and $\texttt{G}$ is a non-empty conjunction of atoms. (The strategy can easily be extended to the case where \texttt{incorrect} is defined by more than one clause.) In particular, we will consider the program~$P$ made out of: (i)~the clauses defining the predicates \texttt{incorrect} and \texttt{reach} (see {clauses~9--11} of Section~\ref{sec:translating-correctness-into-CLP}), (ii)~the clauses defining the predicate~{\tt tr} (see {clauses~1--5} of Section~\ref{sec:syntax_semantics}) which is the interpreter of our imperative language, (iii)~the clauses for the predicates \texttt{initConf} and \texttt{errorConf}, and (iv) the clauses defining the predicates on which~{\tt tr} depends (such as {\tt at} and {\tt eval}). In~$P$ the predicate \texttt{incorrect} is defined by the single clause $\texttt{incorrect\,:-\,\texttt{initConf(X),\,reach(X)}}$. The set of predicate symbols is partitioned into two subsets, called the {\em high} and {\em low} predicates, respectively, such that no low predicate depends on a high predicate. Recall that a predicate~{\tt p} {\em immediately depends on} a predicate~{\tt q} if in the program there is a clause of the form $\mathtt{p(\ldots)\,\texttt {:-}\,\ldots,q(\ldots),\ldots}$ The {\em depends on} relation is the transitive closure of the {\em immediately depends on} relation. The predicates \texttt{incorrect}, \texttt{initConf}, \texttt{reach}, and \texttt{errorConf} are high predicates and, in general, the body of the clause $\texttt{incorrect\,:-\,c\!,\,G}$ has at least one occurrence of a high predicate. Moreover, every new predicate introduced during the {\textsc{Def\-inition}} \& \textsc{Folding} phase of the \textit{Transform} strategy is a high predicate. This partition is needed for guaranteeing the correctness of the {\it{Transform}} strategy (see Theorem~\ref{thm:term_corr_Transform} below). The \textit{Transform} strategy makes use of two functions, \textit{Unf} and \textit{Gen}, which are used for performing unfolding and generalization steps, respectively. Let us first consider the function~\textit{Unf}.\nopagebreak Given a clause $C$ of the form $\tt H\,\texttt{:-\,c,L,A,R}$, where $\texttt{H}$ and $\texttt{A}$ are atoms, $\texttt{c}$ is a constraint, and $\texttt{L}$ and~$\texttt{R}$ are (possibly empty) conjunctions of atoms, let $\{\texttt{K}_i\, \texttt{:-}\, \texttt{c}_i\texttt{,B}_i \mid i=1, \ldots, m\}$ be the set of the (renamed apart) clauses of program~$P$ such that, for $i\!=\!1,\ldots ,m,$ $\texttt{A}$ is unifiable with $\texttt{K}_i$ via the most general unifier~$\vartheta_i$ and $(\texttt{c,c}_i)\, \vartheta_i$ is satisfiable (thus, the unfolding function performs also some constraint solving operations). We define the following function:\nopagebreak \smallskip $\textit{Unf\/}(C,\texttt{A}) = \{(\texttt{H}\, \texttt{:-}\, \texttt{c,c}_i\texttt{,L,B}_i\texttt{,R})\,\vartheta_i \mid i=1,\ldots, m\}$ \smallskip\noindent Each clause in $\textit{Unf\/}(C,\texttt{A})$ is said to be derived by {\em unfolding $C$ w.r.t.}~\texttt{A}. In order to perform unfolding steps during the execution of the {\em Transform} strategy, we assume that it is given a {\em selection function} that determines which atoms occurring in the body of clause~$C$ are {\em unfoldable}. This selection function, which depends on the sequence of applications of $\textit{Unf\/}$ through which we have derived~$C$, will ensure that any sequence of clauses constructed by unfolding w.r.t.~unfoldable atoms is finite. A survey of techniques which ensure the finiteness of unfolding can be found in~\cite{LeB02}. The function {\it{Gen}}, called the {\em generalization operator}, is used for introducing new predicate definitions. Indeed, given a clause $E$ and the set \textit{Defs} of clauses defining the predicates introduced so far, $\textit{Gen}(E,\textit{Defs})$ returns a new predicate definition~$G$ such that $E$ can be {\em folded} by using clause~$G$. The generalization operator guarantees that during the execution of the \textit{Transform} strategy, a {\em finite} number of new predicates is introduced (otherwise, the strategy does not terminate). One can define several generalization operators based on the notions of {\em widening}, {\em convex hull}, {\em most specific generalization}, and {\em well-quasi orderings} which have been introduced for analyzing and transforming constraint logic programs (see, for instance,~\cite{CoC77,CoH78,De&99,Fi&13a,PeG02}). For lack of space we do not present here the definitions of those generalization operators, and we refer the reader to the relevant literature. The equivalences which are considered when performing the goal replacements are called {\em laws} and their validity in the least model~$M(P)$ can be proved once and for all before applying the \textit{Transform} strategy. During the application of the \textit{Transform} strategy we also need an auxiliary strategy for performing the goal replacement steps (this auxiliary strategy has to select the clauses where the replacements should take place and the law to be used). In Section~\ref{sec:array_properties} we will consider programs acting on arrays and we will see an application of the goal replacement rule which makes use of a law that holds for arrays. However, we leave it for future work the study of general strategies for performing goal replacement. In the \textit{Transform} strategy we also use the following notions. We say that a constraint \texttt{c} {\em entails} a constraint \texttt{d}, denoted $c \sqsubseteq d$, if $\forall\, (\mathtt{c}\! \rightarrow\! \mathtt{d})$ holds, where, as usual, by $\forall(\varphi)$ we denote the universal closure of a formula~$\varphi$. The set of \emph{useless predicates} in a program \( P \) is the maximal set \( U \) of predicates occurring in \( P \) such that \texttt{p} is in \( U \) iff every clause with head predicated \texttt{p} is of the form \(\mathtt{ p(\ldots)\ \texttt{:-}\ c, G_{1}, q(\ldots ), G_{2} }\), for some \( \texttt q \) in $U$. A clause in a program~$P$ is {\em useless} if the predicate of its head is useless in $P$. We have the following result. \vspace{-2mm} \begin{theorem}{\em (Termination and Correctness of the \textit{Transform} strategy)} \label{thm:term_corr_Transform} {\em (i)}~The Transform strategy terminates. {\em (ii)}~Let program $\textit{TransfP}$ be the output of the Transform strategy applied on the input program $P$. Then, $\mathtt{incorrect}\! \in \!M(\textit{P\/})$ \mbox{iff} $\mathtt{incorrect}\! \in\! M(\textit{TransfP})$. \end{theorem} \vspace{-1mm} \noindent The termination of the \textit{Transform} strategy is guaranteed by the assumption that one can make only a finite number of unfolding and generalization steps. The correctness of the strategy with respect to the least model semantics directly follows from the correctness of the transformation rules~\cite{EtG96,TaS86}. Indeed, the conditions on high and low predicates ensure that the \textit{Transform} strategy complies with the conditions on predicate {\it levels} given in~\cite{TaS86}. \medskip Let us briefly explain how the \textit{Transform} strategy may achieve the objective of deriving a program $\textit{TransfP}$ with either the fact `$\texttt{incorrect.}$' (hence proving that \textit{prog} is incorrect) or the empty set of clauses for the predicate $\texttt{incorrect}$ (hence proving that \textit{prog} is correct). If \textit{prog} is incorrect, the fact `$\texttt{incorrect.}$' can, in principle, be derived by unfolding the clause $\texttt{incorrect\,:-\,c\!,\,G}$. Indeed, observe that if $\texttt{incorrect}\in M(P)$ then, by the completeness of the {\em top-down} evaluation strategy of CLP programs~\cite{JaM94}, there exists a sequence of unfolding steps that leads to the fact `$\texttt{incorrect.}$'. In practice, however, the ability to derive such a fact depends on the specific strategy used for selecting the atoms for performing unfolding steps during the \textsc{Unfolding} phase. A program $\textit{TransfP}$ where the predicate $\texttt{incorrect}$ is defined by the empty set of clauses can be obtained by first deriving a program where the set of clauses defining \texttt{incorrect} and the high predicates upon which \texttt{incorrect} depends, contains no constrained facts. Indeed, in this case the predicate \texttt{incorrect} is useless and all the clauses of its definition are removed by the last step of the \textit{Transform} strategy. A set of clauses without constrained facts may be derived as follows. We perform the \textsc{Unfolding}, \textsc{Goal Replacement}, and \textsc{Clause Removal} phases as indicated in the \textit{Transform} strategy. If we get a set $\textit{TransfC}$ of clauses with constrained facts, then the strategy will necessarily derive a final program $\textit{TransfP}$ with constrained facts. Otherwise, all clauses in $\textit{TransfC}$ are folded by (possibly) introducing new predicate definitions and the strategy continues by processing the newly introduced predicates (if any). If the strategy exits the while-loop without producing any constrained fact, we derive a program $\textit{TransfP}$ defining a set of mutually recursive predicates without any constrained fact. \medskip Let us consider the incorrectness triple $\{\hspace{-3pt}\{ \varphi_{\mathit{init}}(i,j) \}\hspace{-3pt}\}$ {\it{increase}} $\{\hspace{-3pt}\{ \varphi_{\mathit{error}}(i,j)\}\hspace{-3pt}\}$ of Section~\ref{sec:translating-correctness-into-CLP}. In order to show that the program \textit{increase} is correct with respect to $\varphi_{\mathit{init}}=_{\mathit{def}}i\!=\!0 \andw j\!=\!0$ and $\varphi_{\mathit{error}}=_{\mathit{def}}i\!<\!j$, we start off from clauses~1--15 associated with the program \textit{increase} (see Section~\ref{sec:translating-correctness-into-CLP}), together with the clauses for the predicate~$\mathtt{tr}$ ({clauses~1--5} of Section~\ref{sec:syntax_semantics} and the clauses on which~$\mathtt{tr}$ depends) which, as already mentioned, define the interpreter of our imperative language. We perform Step~(A) of the verification method by applying the {\it{Transform}} strategy. \smallskip \noindent \textsc{Unfolding.} First we unfold clause~9 with respect to the leftmost atom with high predicate, that is, $\mathtt{initConf(X)}$, and we get: \smallskip \noindent 16.~$\mathtt{incorrect}$ {\tt{:-}} $\mathtt{I\!=\!0},\ \mathtt{J\!=\!0},\ \mathtt{reach(cf(cmd(0,ite(less(int(i),mult(int(2),int(n))),1,h)),}$\nopagebreak \hspace{60mm}$\mathtt{[[int(i),I],[int(j),J],[int(n),N]])).}$ \smallskip \noindent \textsc{Definition \& Folding.} We introduce the new predicate definition: \smallskip \noindent 17.~$\mathtt{new1(I,J,N)}$ {\tt{:-}} $\mathtt{reach(cf(cmd(0,ite(less(int(i),mult(int(2),int(n))),1,h)),}$\nopagebreak \hspace{60mm}$\mathtt{[[int(i),I],[int(j),J],[int(n),N]])).}$ \smallskip \noindent We fold clause 16 using clause 17 and we get: \smallskip \noindent 18.~$\mathtt{incorrect}$ {\tt{:-}} $\mathtt{I\!=\!0,\ J\!=\!0,\ new1(I,J,N).}$ \smallskip \noindent Then, we continue the execution of the \textit{Transform} strategy, starting from the last definition we have introduced, that is, clause~17 (indeed, we have ${\mathit{InDefs}}\!=\!\{\mbox{clause 17}\}$). Eventually we get the following program~$T_A$: \smallskip \noindent 18.~$\mathtt{incorrect}$ {\tt{:-}} $\mathtt{I\!=\!0,~~ J\!=\!0,~~ new1(I,J,N).}$ \noindent 19.~$\mathtt{new1(I,J,N)}$ {\tt{:-}} $\mathtt{I\!<\!2\!*\!N,~~I\!<\!N,~~ I1\!=\!I\!+\!2,~~ \mathtt{new1(I1,J,N)}.}$ \noindent 20.~$\mathtt{new1(I,J,N)}$ {\tt{:-}} $\mathtt{I\!<\!2\!*\!N,~~I\!\geq\!N,~~ I1\!=\!I\!+\!1,~~ J1\!=\!I\!+\!1,~~ \mathtt{new1(I1,J1,N)}.}$ \noindent 21.~$\mathtt{new1(I,J,N)}$ {\tt{:-}} $\mathtt{I\!\geq\!2\!*\!N,~~ I\!<\!J.}$ \smallskip \noindent Now we have completed Step~(A). Unfortunately, it is not possible to prove by direct evaluation that \texttt{incorrect} is not a consequence of the above CLP clauses. Indeed, the evaluation of the query \texttt{incorrect} using the standard top-down strategy enters into an infinite loop. Tabled evaluation~\cite{CuW00} does not terminate either, as infinitely many tabled atoms are generated. Analogously, bottom-up evaluation is unable to return an answer. Indeed, the presence of a constrained fact for $\mathtt{new1}$ in program $T_A$ (see clause~21) generates in the least model $M(T_A)$, by repeatedly using the recursive clauses~19 and 20, infinitely many new atoms with predicate ${\mathtt{new1}}$, and thus we cannot show that ${\mathtt{new1}}$ does not hold for $\mathtt{I\!=\!0 \andw J\!=\!0}$. Hence, in this way we cannot show that $\mathtt{incorrect}$ does not hold in $M(T_A)$ and we cannot conclude that the program {\it{increase}} is correct. Our verification method, instead of directly evaluating the query \texttt{incorrect} in $T_A$, proceeds to Step~(B). In order to perform this transformation step we may choose to propagate either the initial property or the error property. Let us opt for the second choice. (However, verification would succeed also by propagating the initial property.) In order to do so, we have first to `reverse' program~$T_A$, as indicated in~\cite{De&13a}. We proceed as follows. First, program~$T_A$ is transformed into a program of the form: \smallskip \noindent s1.~\makebox[18mm][l]{$\mathtt{incorrect}$} {\tt{:-}} $\mathtt{a(U), ~r1(U).}$\nopagebreak \noindent s2.~\makebox[9mm][l]{$\mathtt{r1(U)}$} {\tt{:-}} $\mathtt{trans(U,V), ~r1(V).}$\nopagebreak \noindent s3.~\makebox[9mm][l]{$\mathtt{r1(U)}$} {\tt{:-}} $\mathtt{b(U).}$ \smallskip \noindent where the predicates $\mathtt{a(U)}$, $\mathtt{trans(U,V)}$, and $\mathtt{b(U)}$ are defined by the following clauses: \smallskip \noindent s4.~$\mathtt{a([new1,I,J,N])}$ {\tt{:-}} $\mathtt{I\!=\!0,~~ J\!=\!0.}$\nopagebreak \noindent s5.~$\mathtt{trans([new1,I,J,N],[new1,I1,J,N])}${\tt{~:-}} $\mathtt{I\!<\!2\!*\!N,~~I\!<\!N,~~I1\!=\!I\!+\!2.}$ \noindent s6.~$\mathtt{trans([new1,I,J,N],[new1,I1,J1,N])}${\tt{:-}} $\mathtt{I\!<\!2\!*\!N,~~I\!\geq\!N,~~I1\!=\!I\!+\!1,~~J1\!=\!I\!+\!1.}$ \noindent s7.~$\mathtt{b([new1,I,J,N])}$ {\tt{:-}} $\mathtt{I\!\geq\!2\!*\!N,~~ I\!<\!J.}$ \smallskip \noindent This transformation is correct because program~$T_A$ can be obtained from clauses~s1--\,s7 by: (i)~unfolding clauses~s1--\,s3 with respect to~$\tt a(U)$, $\tt trans(U,V)$, and $\tt b(U)$, and then (ii)~rewriting the atoms of the form $\mathtt{r1([new1,X,Y,N])}$ as $\mathtt{new1(X,Y,N)}$. (The occurrences of the predicate symbol $\mathtt{new1}$ in the arguments of $\mathtt{r1}$ should be considered as an individual constant.) Then, the {\rm{reversed program}}~$T_A^{\textit{rev}}$ is given by the following clauses (with the same definitions of $\mathtt{a(U)}$, $\mathtt{trans(U,V)}$, and $\mathtt{b(U)}$): \smallskip \noindent rev1.~~\makebox[17mm][l]{$\mathtt{incorrect}$} {\tt{:-}} $\mathtt{b(U), ~r2(U).}$\nopagebreak \noindent rev2.~~\makebox[9mm][l]{$\mathtt{r2(V)}$} {\tt{:-}} $\mathtt{trans(U,V), ~r2(U).}$\nopagebreak \noindent rev3.~~\makebox[9mm][l]{$\mathtt{r2(U)}$} {\tt{:-}} $\mathtt{a(U).}$ \smallskip \noindent One can show that program reversal is correct in the sense that $\mathtt{incorrect}\in M(T_A)$ iff $\mathtt{incorrect} \in M(T_A^{\textit{rev}})$~\cite{De&13a}. Now, we perform Step~(B) of the verification method. Let us apply the \textit{Transform} strategy taking as input program $T_A^{\textit{rev}}$ (clauses rev1--rev3) and clauses s4--\,s7. We assume that the high predicates are: {\tt{incorrect}}, {\tt{b}}, {\tt{r2}}, {\tt{trans}}, and {\tt{a}}. \smallskip \noindent \textsc{Unfolding.} First we unfold clause rev1 w.r.t.~the leftmost atom with high predicate, that is, {\tt b(U)}, and we get: \smallskip \noindent 22.~$\mathtt{incorrect}$ {\tt{:-}} $\mathtt{I\!\geq\!2\!*\!N,~~ I\!<\!J,~~ r2([new1,I,J,N]).}$ \smallskip \noindent The {\sc goal replacement} and the {\sc clause removal} phases leave unchanged the set of clauses we have derived so far. \smallskip \noindent \textsc{Definition \& Folding.} In order to fold clause 22 we introduce the clause: \smallskip \noindent 23.~$\mathtt{new2(I,J,N)}$ {\tt{:-}} $\mathtt{I\!\geq\!2\!*\!N,~~ I\!<\!J,~~ r2([new1,I,J,N]).}$ \smallskip \noindent and we fold clause 22 using clause 23. Thus, we get: \smallskip \noindent 24.~$\mathtt{incorrect}$ {\tt{:-}} $\mathtt{I\!\geq\!2\!*\!N,~~ I\!<\!J,~~ new2(I,J,N).}$ \smallskip \noindent At this point we execute again the outermost body of the while-loop of the \textit{Transform} strategy because {\it{InDefs}} contains clause~23, which is not a constrained fact. \smallskip \noindent \textsc{Unfolding.} By unfolding clause~23 w.r.t.~the atom $\mathtt{r2([new1,I,J,N])}$, we get the following two clauses: \smallskip \noindent 25.~$\mathtt{new2(I,J,N)}$ {\tt{:-}} $\mathtt{I\!\geq\!2\!*\!N,~~ I\!<\!J,~~ trans(U,[new1,I,J,N]),~~ r2(U).}$ \noindent 26.~$\mathtt{new2(I,J,N)}$ {\tt{:-}} $\mathtt{I\!\geq\!2\!*\!N,~~ I\!<\!J,~~ a([new1,I,J,N]).}$ \smallskip \noindent Then, by unfolding clause~25 w.r.t.~the atom~$\mathtt{trans(U,[new1,I,J,N])}$, we get: \smallskip \noindent 27.~$\mathtt{new2(I1,J,N)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!2,~~ I\!<\!2\!*\!N,~~ I\!<\!N,~~ I\!+\!2\!\geq\!2\!*\!N,~~ I\!+\!2\!<\!J,~~ r2([new1,I,J,N]).}$ \smallskip \noindent By unfolding clause~26 w.r.t.~$\mathtt{a([new1,I,J,N])}$, we get an empty set of clauses (indeed, the constraint `$\mathtt{I\!<\!J, I\!=\!0, J\!=\!0}$' is unsatisfiable). \smallskip \noindent \textsc{Definition \& Folding.} In order to fold clause~27 we perform a generalization operation as follows. Clause~27 can be folded introducing the clause: \smallskip\noindent 28.~$\mathtt{new3(I,J,N)}$ {\tt{:-}} $\mathtt{I\!<\!2\!*\!N,~~ I\!<\!N,~~ I\!+\!2\!\geq\!2\!*\!N,~~ I\!+\!2\!<\!J,~~ r2([new1,I,J,N]).}$ \smallskip\noindent However, the comparison between clause~23 introduced in a previous step and clause~28 shows the risk of introducing an unlimited number of definitions whose body contains the atom $\mathtt{r2([new1,I,J,N])}$ (see, in particular, the constraint `$\mathtt{I\!\geq\!2\!*\!N,\ I\!<\!J}$' in clause~23 and the constraint `$\mathtt{I\!+\!2\!\geq\!2\!*\!N,\ I\!+\!2\!<\!J}$' in clause~28), thereby making the {\it{Transform}} strategy diverge. To avoid this divergent behaviour, we apply {\em widening}~\cite{CoH78} to clauses~23 and~28, and we introduce the following clause~29, instead of clause~28: \smallskip \noindent 29.~$\mathtt{new3(I,J,N)}$ {\tt{:-}} $\mathtt{I\!<\!J,~~ r2([new1,I,J,N]).}$ \smallskip \noindent Recall that the widening operator applied to two clauses $c1$ and $c2$ (in this order) behaves, in general, as follows. After replacing every equality constraint $\mathtt{A\!=\!B}$ by the equivalent conjunction `$\mathtt{A\!\leq\! B,~ A\!\geq\! B}$', the widening operator returns a clause which is like~$c1$, except that the atomic constraints are only those of~$c1$ which are implied by the constraint of~$c2$. In our case, the widening operator drops the atomic constraint $\mathtt{I\!\geq\!2\!*\!N}$ and keeps $\mathtt{I\!<\!J}$ only. \smallskip \noindent By folding clause 27 w.r.t.~the atom $\mathtt{r2([new1,I,J,N])}$, we get: \smallskip \noindent 30.~$\mathtt{new2(I1,J,N)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!2,~~ I\!<\!2\!*\!N,~~ I\!<\!N,~~ I\!+\!2\!\geq\!2\!*\!N,~~ I\!+\!2\!<\!J,~~ new3(I,J,N).}$ \smallskip \noindent Now, we process the newly introduced definition clause~29 and we perform a new iteration of the body of the outermost while-loop of the \textit{Transform} strategy. \smallskip \noindent \textsc{Unfolding.} After a few unfolding steps from clause~29, we get: \smallskip \noindent 31.~$\mathtt{new3(I1,J,N)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!2,~~ I\!<\!2\!*\!N,~~ I\!<\!N,~~ I\!+\!2\!<\!J,~~ r2([new1,I,J,N])}$. \smallskip \noindent \textsc{Definition \& Folding.} In order to fold clause~31 we do not need to introduce any new definition. Indeed, it is possible to fold clause~31 by using clause 29 and we get: \smallskip \noindent 32.~$\mathtt{new3(I1,J,N)}${\tt{:-}}$\mathtt{I1\!=\!I\!+\!2,~~ I\!<\!2\!*\!N,~~ I\!<\!N,~~ I\!+\!2\!<\!J,~~ new3(I,J,N)}$. \smallskip \noindent Now, \textit{InDefs} is empty and we exit the outermost while-loop. We get the following program~$T_B$: \smallskip \noindent 24.~$\mathtt{incorrect}$ {\tt{:-}} $\mathtt{I\!\geq\!2\!*\!N,~~ I\!<\!J,~~ new2(I,J,N).}$ \noindent 30.~$\mathtt{new2(I1,J,N)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!2,~~ I\!<\!2\!*\!N,~~ I\!<\!N,~~ I\!+\!2\!\geq\!2\!*\!N,~~ I\!+\!2\!<\!J,~~ new3(I,J,N).}$ \noindent 32.~$\mathtt{new3(I1,J,N)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!2,~~ I\!<\!2\!*\!N,~~ I\!<\!N,~~ I\!+\!2\!<\!J,~~ new3(I,J,N)}$. \smallskip \noindent Since program~$T_B$ contains no constrained facts, all its clauses are useless and can be removed from~$T_B$. Thus, at the end of Step~(B) we get the final empty program~$T_B$. Hence, $M(T_B)$ is the empty set and the atom $\mathtt{incorrect}$ does not belong to it. We conclude that the given program {\it{increase}} is correct with respect to the given properties $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$. The various transformation steps which lead to program $T_B$, including the removal of the interpreter, the program reversal, and the generalization steps performed during Step~(B), have been automatically performed by the MAP system~\cite{MAP} (see~\cite{De&13a} for some experimental results). \section{Verifying Complex Correctness Properties by Conjunctive Folding} \label{sec:conjunctivefolding} In this section we show how our verification method can be used also in the case when the error properties are specified by sets of CLP clauses, rather than by constraints only. In order to deal with this class of error properties we make use of transformation rules which are more powerful than the ones used in the verification example of the previous section. Indeed, during the \textsc{Definition\&Folding} phase of the \textit{Transform} strategy, we allow ourselves to introduce new predicates by using definition clauses of the form: $\texttt{Newp}\, \texttt{:-}\, \mathtt{c,\, G}$ \noindent where \texttt{Newp} is an atom with a new predicate symbol, \texttt{c} is a constraint, and \texttt{G} is a {\em conjunction of one or more atoms}. Clauses of that form will then be used for performing folding steps. (Note that the new predicate definitions introduced during the verification example of the previous section are all of the form:~ $\texttt{Newp}\, \texttt{:-}\, \mathtt{c,\, A}$, where \texttt{A} is a single atom.) The more powerful definition and folding rules, called {\em conjunctive definition} and {\em conjunctive folding}, respectively, allow us to perform verification tasks that cannot be handled by the technique presented in~\cite{De&13a}, which is based on {\em atomic} definition and {\em atomic} folding. \smallskip Let us consider the following program \textit{gcd} for computing the {\em greatest common divisor}~$z$ of two positive integers $m$ and $n$: \hspace{-4mm}$\ell_{0}$: $x=m\,;$ \hspace{-4mm}$\ell_{1}$: $y=n\,;$ \hspace{-4mm}$\ell_{2}$: $\mathtt{while}~(x\neq y)\ \{\,\mathtt{if}~(x>y)~x\!=\!x\!-\!y\,;~\mathtt{else}~y\!=\!y\!-\!x\,\}$\,; \hspace{-4mm}$\ell_{3}$: $z=x\,;$ \noindent \makebox[5mm][l]{} \hspace{-4mm}$\ell_{h}$: $\mathtt{halt}$ \noindent and the incorrectness triple $\{\hspace{-3pt}\{\varphi_{\mathit{init}}(m,n)\}\hspace{-3pt}\} \ \textit{gcd\/}\ \{\hspace{-3pt}\{\varphi_{\mathit{error}}(m,n,z)\}\hspace{-3pt}\}$, where: \noindent -- $\varphi_{\mathit{init}}(m,n)$ is $ m\!\geq\!1 \andw n\!\geq\!1$, and \nopagebreak \noindent -- $\varphi_{\mathit{error}}(m,n,z)$ is $\exists\, d\,(\textit{gcd\/}(m,n,d\/) \andw d\!\neq\! z)$. The~property $\varphi_{\mathit{error}}$ uses the ternary predicate~\textit{gcd} defined by the following CLP clauses: \smallskip \noindent \makebox[5mm][l]{1.~}$\mathtt{gcd(X,Y,D)~ \texttt{:-} ~X\!>\!Y,~~ X1\!=\!X\!-\!Y,~~ gcd(X1,Y,D).}$ \noindent \makebox[5mm][l]{2.~}$\mathtt{gcd(X,Y,D)~ \texttt{:-} ~X\!<\!Y,~~ Y1\!=\!Y\!-\!X,~~ gcd(X,Y1,D).}$ \noindent \makebox[5mm][l]{3.~}$\mathtt{gcd(X,Y,D)~ \texttt{:-} ~X\!=\!Y,~~ Y\!=\!D.}$ \smallskip \noindent Thus, the incorrectness triple holds if and only if, for some positive integers $m$ and $n$, the program \textit{gcd} computes a value of $z$ that is different from the greatest common divisor of $m$ and $n$. As indicated in Section~\ref{sec:verification_method}, the program \textit{gcd} can be translated into a set of CLP facts defining the predicate~\texttt{at}. We will not show them here. The predicates $\mathtt{initConf}$ and $\mathtt{errorConf}$ specifying the initial and the error configurations, respectively, are defined by the following clauses: \smallskip \noindent \makebox[5mm][l]{4.~}$\mathtt{initConf(cf(cmd(0,asgn(int(x),int(m))),}$ \hspace{8mm}$\mathtt{[[int(m),M],[int(n),N],[int(x),X],[int(y),Y],[int(z),Z]]))}$ {\tt{:-}}~ $\mathtt{phiInit(M,N).}$ \noindent \makebox[5mm][l]{5.~}$\mathtt{errorConf(cf(cmd(h,halt),}$ \hspace{8mm}$\mathtt{[[int(m),M],[int(n),N],[int(x),X],[int(y),Y],[int(z),Z]]))}$ {\tt{:-}} ~$\mathtt{phiError(M,N,Z).}$ \noindent \makebox[5mm][l]{6.~}$\mathtt{phiInit(M,N)}$ {\tt{:-}} $\mathtt{M\!\geq\!1},~~ \mathtt{N\!\geq\!1}.$ \noindent \makebox[5mm][l]{7.~}$\mathtt{phiError(M,N,Z)}$ {\tt{:-}} $\mathtt{gcd(M,N,D),~~ D \!\neq\! Z}.$ \smallskip \noindent The CLP program encoding the given incorrectness triple consists of clauses~1--7 above, together with the clauses defining the predicate \texttt{\,at\,} that encode the program~\textit{gcd}, and the clauses that define the predicates {\tt incorrect}, {\tt reach}, and {\tt tr} (that is, clauses~9--11 of Section~\ref{sec:translating-correctness-into-CLP} and clauses~1--5 of Section~\ref{sec:syntax_semantics}). Now we perform Step~(A) of our verification method, which consists in the removal of the interpreter, and we derive the following CLP program: \smallskip \noindent \makebox[6mm][r]{8.\hspace{1.5mm}}$\mathtt{incorrect\ \texttt{:-}\ M\!\geq\!1,~~ N\!\geq\! 1,~~ new1(M,N,M,N,Z).}$ \noindent \makebox[6mm][r]{9.\hspace{1.5mm}}$\mathtt{new1(M,N,X,Y,Z)\ \texttt{:-}\ X\!>\!Y,~~ X1\!=\!X\!-\!Y,~~ new1(M,N,X1,Y,Z).}$ \noindent \makebox[6mm][l]{10.}$\mathtt{new1(M,N,X,Y,Z)\ \texttt{:-}\ X\!<\!Y,~~ Y1\!=\!Y\!-\!X,~~ new1(M,N,X,Y1,Z).}$ \noindent \makebox[6mm][l]{11.}$\mathtt{new1(M,N,X,Y,Z)\ \texttt{:-}\ X\!=\!Y,~~ Z\!=\!X,~~ gcd(M,N,D),~~ Z\!\neq\!D.}$ \smallskip \noindent Clauses 8 and 11 can be rewritten, respectively, as: \smallskip \noindent \makebox[7mm][r]{8r.~}$\mathtt{incorrect\ \texttt{:-}~ M\!\geq\!1,~~ N\!\geq\! 1,~~ Z\!\neq\!D,~~ new1(M,N,M,N,Z),~~ gcd(M,N,D).}$ \noindent \makebox[7mm][r]{11r.~}$\mathtt{new1(M,N,X,Y,Z)\ \texttt{:-}\ X\!=\!Y,~~ Z\!=\!X.}$ \smallskip \noindent This rewriting is correct because {\tt{new1}} modifies the values of neither {\tt M} nor {\tt N}. Note that we could avoid performing the above rewriting and automatically derive a similar program where the constraints characterizing the initial and the error properties occur in the same clause, by starting our derivation from a more general definition of the reachability relation. However, an in-depth analysis of this variant of our verification method is beyond the scope of this paper (see also \cite{Pe&98} for a discussion about different styles of encoding the reachability relation and the semantics of imperative languages in CLP). Now we perform Step (B) of the verification method by applying the \textit{Transform} strategy to the program consisting of clauses $\{1,2,3,8{\rm r},9,10,11{\rm r}\}$. We assume that the only high predicates are {\tt{incorrect}} and {\tt new1}. \smallskip \noindent \textsc{Unfolding.} We start off by unfolding clause 8r w.r.t.~the atom $\mathtt{new1(M,N,M,N,Z)}$, and we get: \smallskip \noindent \makebox[6mm][r]{12.~}$\mathtt{incorrect \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ M\!>\!N,~~ X1\!=\!M\!-\!N,~~ Z\!\neq\!D,~}$ $\mathtt{new1(M,N,X1,N,Z),~~ gcd(M,N,D).}$ \noindent \makebox[6mm][r]{13.~}$\mathtt{incorrect\ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ M\!<\!N,~~ Y1\!=\!N\!-\!M,~~ Z\!\neq\!D,~}$ $\mathtt{new1(M,N,M,Y1,Z),~~ gcd(M,N,D).}$ \noindent \makebox[6mm][r]{14.~}$\mathtt{incorrect \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ M\!=\!N,~~ Z\!=\!M,~~ Z\!\neq\!D,~~ gcd(M,N,D).}$ \smallskip \noindent By unfolding clauses 12--14 w.r.t.~the atom $\mathtt{gcd(M,N,D)}$ we derive: \smallskip \noindent \makebox[6mm][r]{15.~}$\mathtt{incorrect \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ M\!>\!N,~~ X1\!=\!M\!-\!N,~~ Z\!\neq\!D,~~ new1(M,N,X1,N,Z),~~ gcd(X1,N,D).}$ \noindent \makebox[6mm][r]{16.~}$\mathtt{incorrect \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ M\!<\!N,~~ Y1\!=\!N\!-\!M,~~ Z\!\neq\!D,~~ new1(M,N,M,Y1,Z),~~ gcd(M,Y1,D).}$ \smallskip \noindent (Note that by unfolding clause~14 we get an empty set of clauses because the constraints derived in this step are all unsatisfiable.) \smallskip \noindent The \textsc{Goal Replacement} and \textsc{Clause Removal} phases do not modify the set of clauses derived after the \textsc{Unfolding} phase because no laws are available for the predicate \texttt{gcd}. \smallskip \noindent \textsc{Definition \& Folding.} In order to fold clauses 15 and 16, we perform a generalization step and we introduce a new predicate defined by the following clause: \smallskip \noindent \makebox[6mm][r]{17.~}$\mathtt{new2(M,N,X,Y,Z,D) \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ Z\!\neq\!D,~~ new1(M,N,X,Y,Z),~~ gcd(X,Y,D).}$ \smallskip\noindent The body of this clause is the most specific generalization of the bodies of clauses~8r,~15 and~16. Here, we define a conjunction $G$ to be a generalization of a conjunction $C$ if there exists a substitution $\vartheta$ such that $G\vartheta$ can be obtained by deleting some of the conjuncts of $C$. Now, clauses 15 and 16 can be folded by using clause~17, thereby deriving:\nopagebreak \smallskip \noindent \makebox[6mm][r]{18.~}$\mathtt{incorrect \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ M\!>\!N,~~ X1\!=\!M\!-\!N,~~ Z\!\neq\!D,~~ new2(M,N,X1,N,Z,D).}$\nopagebreak \noindent \makebox[6mm][r]{19.~}$\mathtt{incorrect \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ M\!<\!N,~~ Y1\!=\!N\!-\!M,~~ Z\!\neq\!D,~~ new2(M,N,M,Y1,Z,D).}$ \medskip\noindent Clause~17 defining the new predicate \texttt{new2}, is added to \textit{Defs} and \textit{InDefs} and, since the latter set is not empty, we perform a new iteration of the while-loop body of the \textit{Transform} strategy. \smallskip \noindent\textsc{Unfolding.} By unfolding clause 17 w.r.t.~\texttt{new1(M,N,X,Y,Z)} and then unfolding the resulting clauses w.r.t.~\texttt{gcd(X,Y,Z)}, we derive: \smallskip \noindent \makebox[6mm][r]{20.~}$\mathtt{new2(M,N,X,Y,Z,D) \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ X\!>\!Y,~~ X1\!=\!X\!-\!Y,~~ Z\!\neq\!D,~~ }$ $\mathtt{new1(M,N,X1,Y,Z),~~ gcd(X1,Y,D).}$ \noindent \makebox[6mm][r]{21.~}$\mathtt{new2(M,N,X,Y,Z,D) \ \texttt{:-}\ M\!\geq\! 1,~~ N\!\geq\! 1,~~ X\!<\!Y,~~ Y1\!=\!Y\!-\!X,~~ Z\!\neq\!D,~~ }$ $\mathtt{new1(M,N,X,Y1,Z),~~ gcd(X,Y1,D).}$ \smallskip \noindent \textsc{Definition \& Folding.} Clauses 20 and 21 can be folded by using clause~17, thereby deriving: \smallskip \noindent \makebox[6mm][r]{22.~}$\mathtt{new2(M,N,X,Y,Z,D)\, \texttt{:-}\, M\!\geq\! 1,~~ N\!\geq\! 1,~~ X\!>\!Y,~~ X1\!=\!X\!-\!Y,~~ Z\!\neq\!D,~~ new2(M,N,X1,Y,Z).}$ \noindent \makebox[6mm][r]{23.~}$\mathtt{new2(M,N,X,Y,Z,D)\, \texttt{:-}\, M\!\geq\! 1,~~ N\!\geq\! 1,~~ X\!<\!Y,~~ Y1\!=\!Y\!-\!X,~~ Z\!\neq\!D,~~ new2(M,N,X,Y1,Z).}$ \smallskip \noindent No new predicate definition is introduced, and the \textit{Transform} strategy exits the while-loop. The final program~$\textit{TransfP}$ is the set $\{18,19,22,23\}$ of clauses that contains no constrained facts. Hence both predicates $\mathtt{incorrect}$ and $\mathtt{new2}$ are useless and all clauses of~$\textit{TransfP}$ can be deleted. Thus, the \textit{Transform} strategy terminates with $\textit{TransfP}=\emptyset$ and we conclude that the imperative program \textit{gcd} is correct with respect to the given properties $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$. \section{Verifying Correctness of Array Programs} \label{sec:array_properties} In this section we apply our verification method of Section~\ref{sec:verification_method} for proving properties of a program, called {\it{arraymax}}, which computes the maximal element of an array. Let us consider the following incorrectness triple $\{\hspace{-3pt}\{\varphi_{\mathit{init}}(i,n,a,\textit{max})\}\hspace{-3pt}\} ~\mathit{arraymax}~ \{\hspace{-3pt}\{\varphi_{\mathit{error}}(n,a,\textit{max})\}\hspace{-3pt}\}$, where: \noindent -- $\varphi_{\mathit{init}}(i,n,a,\textit{max})$ is~ $ i\!=\!0~ \andw ~n\!=\!{\mathit{dim}}(a)~ \andw ~n\!\geq\! 1~ \andw ~max\!=\!a[i]$,\nopagebreak \noindent \makebox[45mm][l]{-- $\mathit{arraymax}$ is the program} \noindent \makebox[6mm][l]{} $\ell_{0}\!:~\mathtt{while}~(i\!<\!n)~\{$\nopagebreak $\mathtt{if}~(a[i] > max)~~max=a[i];$~~~ $i=i\!+\!1\,\};$\nopagebreak \noindent \makebox[6mm][l]{} $\ell_{h}\!:~\mathtt{halt}$ \noindent -- $\varphi_{\mathit{error}}(n,a,\textit{max})$ is~ $\exists k ~(0\!\leq\! k\! <\! n \andw a[k]\!>\!\mathit{max})$. \smallskip \noindent If this triple holds, then the value of ${\mathit{max}}$ computed by the program $\mathit{arraymax}$ is not the maximal element of the given array $a$ with $n~(\geq 1)$ elements. We start off by constructing a CLP program $T$ which encodes the incorrectness triple. This construction is done as indicated in Section~\ref{sec:translating-correctness-into-CLP} and, in particular, the clauses for the predicates~${\mathtt{phiInit}}$ and ${\mathtt{phiError}}$ are generated from the formulas~$\varphi_{\mathit{init}}$ and~$\varphi_{\mathit{error}}$. \smallskip As indicated in Section~\ref{sec:verification_method}, the program \textit{arraymax} is translated into a set of CLP facts defining the predicate \texttt{at}. The predicates $\mathtt{initConf}$ and $\mathtt{errorConf}$ specifying the initial and the error configurations, respectively, are defined by the following clauses: \smallskip \noindent 1.~$\mathtt{initConf(cf(cmd(0,asgn(int(x),int(0))),}$ \hspace{8mm}$\mathtt{[[int(i),I],[int(n),N],[array(a),(A,N)],[int(max),Max]]))}$ {\tt{:-}} $\mathtt{phiInit(I,N,A,Max).}$ \noindent 2.~$\mathtt{errorConf(cf(cmd(h,halt),}$ \hspace{8mm}$\mathtt{[[int(i),I],[int(n),N],[array(a),(A,N)],[int(max),Max]]))}$ {\tt{:-}} $\mathtt{phiError(N,A,Max).}$ \noindent 3.~$\mathtt{phiInit(I,N,A,Max)}$ {\tt{:-}} $\mathtt{I\!=\!0, ~~N\!\geq\!1, ~~read((A,N),I,Max)}.$ \noindent 4.~$\mathtt{phiError(N,A,Max)}$ {\tt{:-}} $\mathtt{K\!\geq\! 0, ~~N\!>\!K, ~~Z\!>\!Max, ~~read((A,N),K,Z).}$ \smallskip \noindent Next, we apply Step~(A) of our verification method which consists in removing the interpreter from program $T$. By applying the \textit{Transform} strategy as indicated in Section~\ref{sec:verification_method}, we obtain the following program $\mathit{T_A}$: \smallskip \noindent 5.~$\texttt{incorrect}$ {\tt{:-}} $\mathtt{I\!=\!0, ~~N\!\geq\!1, ~~read((A,N),I,Max),~~ new1(I,N,A,Max).}$ \noindent 6.~$\mathtt{new1(I,N,A,Max)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~I\!<\!N, ~~I\geq 0, ~~M\!>\!Max,}$ $\mathtt{~read((A,N),I,M), ~~new1(I1,N,A,M).}$ \noindent 7.~$\mathtt{new1(I,N,A,Max)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~I\!<\!N, ~~I\!\geq\!0, ~~M\!\leq\!Max,}$ $\mathtt{~read((A,N),I,M), ~~new1(I1,N,A,Max).}$\nopagebreak \noindent 8.~$\mathtt{new1(I,N,A,Max)}$ {\tt{:-}} $\mathtt{I\!\geq\! N,~~ K\!\geq\! 0, ~~N\!>\!K, ~~Z\!>\!Max, ~~read((A,N),K,Z).}$ \smallskip \noindent We have that $\mathtt{new1(I,N,A,Max)}$ encodes the reachability of the error configuration from a configuration where the program variables $\mathit{i, n, a, max}$ are bound to $\mathtt{I,N,A,Max}$, respectively. In order to propagate the error property, similarly to the example of Section~\ref{sec:verification_method}, we first `reverse' program $\mathit{T_A}$ and we get the following program~$T_A^{\textit{rev}}$: \smallskip \noindent rev1.~~\makebox[17mm][l]{$\mathtt{incorrect}$} {\tt{:-}} $\mathtt{b(U), ~~r2(U)}$.\nopagebreak \noindent rev2.~~\makebox[8mm][l]{$\mathtt{r2(V)}$} {\tt{:-}} $\mathtt{trans(U,V), ~~r2(U)}$.\nopagebreak \noindent rev3.~~\makebox[8mm][l]{$\mathtt{r2(U)}$} {\tt{:-}} $\mathtt{a(U)}$. \smallskip \noindent where predicates {\tt a(U)}, {\tt b(U)}, and {\tt trans(U,V)} are defined as follows: \smallskip \noindent s4.~$\mathtt{a([new1,I,N,A,Max])}$ {\tt{:-}} $\mathtt{I\!=\!0,~~ N\!\geq\!1, ~~read((A,N),I,Max)}$ \noindent s5.~$\mathtt{trans([new1,I,N,A,Max],[new1,I1,N,A,M])}$\, {\tt{:-}}\, \hspace{40mm}$\mathtt{I1\!=\!I\!+\!1, ~~I\!<\!N, ~~I\!\geq\! 0, ~~M\!>\!Max, ~~read((A,N),I,M)}$. \noindent s6.~$\mathtt{trans([new1,I,N,A,Max],[new1,I1,N,A,Max])}$\, {\tt{:-}}\, \hspace{40mm}$\mathtt{I1\!=\!I\!+\!1,~~ I\!<\!N,~~ I\!\geq\! 0,~~ M\!\leq\! Max, ~~read((A,N),I,M)}$. \noindent s7.~$\mathtt{b([new1,I,N,A,Max])}$ {\tt{:-}} $\mathtt{I\!\geq\! N, ~~K\!\geq\! 0, ~~K\!<\!N, ~~Z\!>\!Max, ~~read((A,N),K,Z)}$. \smallskip \noindent For the application of Step (B) of the verification method we assume that the predicates \texttt{incorrect}, \texttt{b}, \texttt{r2}, \texttt{trans}, and \texttt{a} are high predicates, while the predicate \texttt{read} is a low predicate. For the \textsc{Goal Replacement} phase we use the following law, which is a consequence of the fact that an array is a finite function: \smallskip \noindent \makebox[16mm][l]{(Law~L1)~} $\mathtt{read((A,N)\!,K,Z), ~~read((A,N),I,M)} ~\leftrightarrow$ \hspace{29mm}$(\mathtt{K\!=\!I,~~ Z\!=\!M,~~ read((A,N),K,Z)}) ~~~~\vee~~ $ \hspace{29mm}$(\mathtt{K\!\neq\! I,~~ read((A,N),K,Z),~~ read((A,N),I,M))}$ \smallskip \noindent In general, when applying the \textit{Transform} strategy in the case of array programs, some additional laws may be required (see, for instance,~\cite{Br&06}). For the \textsc{Definition}~\&~\textsc{Folding} phase we use a particular generalization operator, called \mbox{\it{WidenSum}}~\cite{Fi&13a}, which is a variant of the classical widening operator~\cite{CoC77} and behaves as follows. Given any atomic constraint $a$, let us denote by $\textit{sumcoeff}(a)$ the sum of the absolute values of the coefficients of $a$. Given any two constraints~$c$ and~$d$, ${\mathit{WidenSum}}(c,d)$ returns the conjunction of: (i) all atomic constraints $a$ in $c$ such that $d\sqsubseteq a$, and (ii) all atomic constraints~$b$ in $d$ such that $\textit{sumcoeff}(b)\!\leq\! \textit{sumcoeff}(e)$ for some atomic constraint $e$ in $c$. \smallskip \noindent Let us now apply Step (B) of the verification method. \smallskip \noindent \textsc{Unfolding}. We start off by unfolding clause~rev1 w.r.t.~the atom $\mathtt{b(U)}$, and we get: \smallskip \noindent 9.~$\mathtt{incorrect}$\,{\tt{:-}}\, $\mathtt{I\!\geq\!N,~~ K\!\geq\! 0,~~ K\!<\!N, ~~Z\!>\!Max,~~ read((A,N),K,Z),~~ r2([new1,I,N,A,Max])}$. \smallskip \noindent The \textsc{Goal Replacement} and \textsc{Clause Removal} phases leave unchanged the set of clauses we have derived so far. \smallskip \noindent \textsc{Definition \& Folding.} In order to fold clause~9 we introduce the following clause: \smallskip \noindent 10.~$\mathtt{new2(I,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{I\!\geq\!N, ~~K\!\geq\! 0,~~ K\!<\!N,~~ Z\!>\!Max,~ read((A,N),K,Z),~~ r2([new1,I,N,A,Max])}$. \smallskip \noindent By folding clause 9 using clause~10, we get: \smallskip \noindent 11. $\mathtt{incorrect}$ {\tt{:-}} $\mathtt{I\!\geq\!N, ~~K\!\geq\! 0, ~~K\!<\!N, ~~Z\!>\!Max, ~~new2(I,N,A,Max,K,Z)}$. \smallskip \noindent Now we proceed by performing a second iteration of the body of the while-loop of the \textit{Transform} strategy because clause~10 is in {\it{InDefs}}. \smallskip \noindent \textsc{Unfolding}. By unfolding clause~10 w.r.t.~the atom $\mathtt{r2([new1,I,N,A,Max])}$, we get the following clauses: \smallskip \noindent 12.~$\mathtt{new2(I,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{I\!\geq\!N, ~~K\!\geq\! 0,~~ K\!<\!N,~~ Z\!>\!Max,~~}$ \hspace{22mm}$\mathtt{read((A,N),K,Z), ~~trans(U,[new1,I,N,A,Max]), ~~r2(U).}$ \noindent 13.~$\mathtt{new2(I,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{I\!\geq\!N, ~~K\!\geq\! 0, ~~K\!<\!N,~~ Z\!>\!Max,~~ read((A,N),K,Z),~~ a([new1,I,N,A,Max])}$. \smallskip \noindent By unfolding clause 12 w.r.t.~$\mathtt{trans(U,[new1,I,N,A,Max])}$, we get: \smallskip \noindent 14.~$\mathtt{new2(I1,N,A,M,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~N\!=\!I1, ~~K\!\geq\! 0, ~~K\!<\!I1, ~~M\!>\!Max, ~~Z\!>\!M,}$ \hspace{22mm}$\mathtt{read((A,N),K,Z), ~~read((A,N),I,M), ~~r2([new1,I,N,A,Max]).}$ \smallskip \noindent 15.~$\mathtt{new2(I1,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1,~~ N\!=\!I1,~~ K\!\geq\! 0,~~ K\!<\!I1,~~ M\!\leq\!Max,~~ Z\!>\!Max,}$\nopagebreak \hspace{22mm}$\mathtt{read((A,N),K,Z), ~~read((A,N),I,M), ~~r2([new1,I,N,A,Max]).}$ \smallskip \noindent By unfolding clause 13 w.r.t.~$\mathtt{a([new1,I,N,A,Max])}$, we get an empty set of clauses (indeed, the constraint $\mathtt{I\!\geq\!N,~ I\!=\!0,~ N\!\geq\!1}$ is unsatisfiable). \smallskip \noindent \textsc{Goal Replacement}. This phase performs the following two steps: \noindent (i)~it replaces the conjunction of atoms `$\mathtt{read((A,N),K,Z)},~\mathtt{read((A,N),I,M)}$' occurring in the body of clause~14 by the right hand side of Law~L1, and then (ii)~it splits the derived clause into the following two clauses, each of which corresponds to a disjunct of that right hand side. \smallskip \noindent 14.1~$\mathtt{new2(I1,N,A,M,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~N\!=\!I1, ~~K\!\geq\! 0, ~~K\!<\!I1, ~~M\!>\!Max, ~~Z\!>\!M,}$ \hspace{22mm}$\mathtt{{K\!=\!I, ~~Z\!=\!M}}$,~~ $\mathtt{read((A,N),K,Z)}$,~~ $\mathtt{r2([new1,I,N,A,Max])}$. \noindent 14.2~$\mathtt{new2(I1,N,A,M,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~N\!=\!I1, ~~K\!\geq\! 0, ~~K\!<\!I1, ~~M\!>\!Max, ~~Z\!>\!M,}$ \hspace{22mm}$\mathtt{{K\!\neq\!I}}$,~~ $\mathtt{read((A,N),K,Z)}$,~~ $\mathtt{read((A,N),I,M)}$,~~ $\mathtt{r2([new1,I,N,A,Max])}$. \smallskip \noindent \textsc{Clause Removal}. The constraint ~`$\mathtt{Z\!>\!M, ~Z\!=\!M}$'~ in the body of clause 14.1 is unsatisfiable. Therefore, this clause is removed from \textit{TranfP}. From clause~14.2, by replacing `$\mathtt{K\!\not = \!I}$' by `$\mathtt{K\!<\!I ~\orv~ K\!>\!I}$' and simplifying the constraints, we get: \smallskip \noindent 16.~$\mathtt{new2(I1,N,A,M,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~N\!=\!I1, ~~K\!\geq\! 0, ~~K\!<\!I, ~~M\!>\!Max, ~~Z\!>\!M,}$ \hspace{22mm}$\mathtt{read((A,N),K,Z)}$,~~ $\mathtt{read((A,N),I,M)}$,~~ $\mathtt{r2([new1,I,N,A,Max])}$. \smallskip \noindent By performing from clause~15 a sequence of goal replacement and clause removal transformations similar to that we have performed from clause~14, we get the following clause: \smallskip \noindent 17.~$\mathtt{new2(I1,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1,~~ \,N\!=\!I1,~~ \,K\!\geq\! 0,~~ K\!<\!I,~~ M\!\leq\!Max,~~ Z\!>\!Max,}$ \hspace{22mm}$\mathtt{read((A,N),K,Z)}$,~~ $\mathtt{read((A,N),I,M)}$,~~ $\mathtt{r2([new1,I,N,A,Max])}$. \smallskip \noindent \textsc{Definition \& Fold}. The comparison between the definition clause~10 we have introduced above, and clauses 16 and~17 which we should fold, shows the risk of introducing an unlimited number of definitions whose body contains the atoms $\mathtt{read((A,N),K,Z)}$ and $\mathtt{r2([new1,I,N,A,Max])}$. Thus, in order to fold clauses 16 and 17, we introduce the following new definition: \smallskip \noindent 18.~$\mathtt{new3(I,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{K\!\geq\! 0, ~K\!<\!N, ~K\!<\!I, ~Z\!>\!Max,~ read((A,N),K,Z)), ~r2([new1,I,N,A,Max])}$. \smallskip \noindent The constraint in the body of this clause is obtained by generalizing: (i) the projection of the constraint in the body of clause~16 on the variables $\mathtt{I,N,A,Max,K,Z}$ (which are the variables of clause 16 that occur in the atoms $\mathtt{read((A,N),K,Z)}$ and $\mathtt{r2([new1,I,N,A,Max])}$), and (ii) the constraint occurring in the body of clause~10. This generalization step can be seen as an application of the above mentioned \mbox{\textit{WidenSum}} generalization operator. The same definition clause 18 could also be derived by generalizing the projection of the constraint in the body of clause 16 (instead of 17) and the constraint occurring in the body of clause~10. Thus, by folding clause 16 and clause 17 using clause 18 we get:\nopagebreak \smallskip \noindent 19.~$\mathtt{new2(I1,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~N\!=\! I1, ~~K\!\geq\!0, ~~K\!<\!I, ~~M\!>\!Max, ~~Z\!>\!M,}$\nopagebreak \hspace{22mm}$\mathtt{read((A,N),I,M),\ new3(I,N,A,Max,K,Z)}$. \noindent 20.~$\mathtt{new2(I1,N,A,M,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~N\!=\! I1, ~~K\!\geq\!0, ~~K\!<\!I, ~~M\!\leq\!Max, ~~Z\!>\!Max,}$\nopagebreak \hspace{22mm}$\mathtt{read((A,N),I,M),\ new3(I,N,A,Max,K,Z)}$. \smallskip \noindent Now we perform the third iteration of the body of the while-loop of the strategy. \smallskip \noindent \textsc{Unfolding}, \textsc{goal replacement}, and \textsc{clause removal}. By unfolding, goal replacement, and clause removal, from clause 18 we get: \smallskip \noindent 21.~$\mathtt{new3(I1,N,A,M,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~K\!\geq\!0, ~K\!\!<\!I, ~~N\!\geq\!I1, ~~M\!>\!Max, ~~Z\!>\!M,}$ \hspace{22mm}$\mathtt{read((A,N),K,Z),~~ read((A,N),I,M),~~ r2([new1,I,N,A,Max]).}$ \noindent 22.~$\mathtt{new3(I1,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~K\!\geq\!0, ~~K\!\!<\!I, ~~N\!\geq\!I1, ~~M\!\leq\!Max, ~~Z\!>\!Max,}$ \hspace{22mm}$\mathtt{read((A,N),K,Z), ~~read((A,N),I,M), ~~r2([new1,I,N,A,Max]).}$ \smallskip \noindent \textsc{Definition \& Folding}. In order to fold clauses 21 and 22, we do not need to introduce any new definition. Indeed, it is possible to fold these clauses by using clause 18, thereby obtaining: \smallskip \noindent 23.~$\mathtt{new3(I1,N,A,M,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1, ~~K\!\geq\!0,~~ K\!\!<\!I,~~ N\!\geq\!I1,~~ M\!>\!Max,~~ Z\!>\!M,}$ \hspace{22mm}$\mathtt{read((A,N),I,M),~~ new3(I,N,A,Max,K,Z)}$. \noindent 24.~$\mathtt{new3(I1,N,A,Max,K,Z)}$ {\tt{:-}} $\mathtt{I1\!=\!I\!+\!1,~~ K\!\geq\!0,~~ K\!\!<\!I, ~~N\!\geq\!I1,~~ M\!\leq\!Max,~~ Z\!>\!Max,\ }$ \hspace{22mm}$\mathtt{read((A,N),I,M),~~ new3(I,N,A,Max,K,Z)}$. \smallskip \noindent Since no clause to be processed is left (because $\textit{InDefs}\!=\!\emptyset$), the \textit{Transform} strategy exits the outermost while-loop, and the program derived is the set $\{11, 19, 20, 23, 24\}$ of clauses. No clause in this set is a constrained fact, and hence by \textsc{removal of useless clauses} we get the final program~$\mathit{T_B}$ consisting of the empty set of clauses. Thus, $\mathit{arraymax}$ is correct with respect to the given properties $\varphi_{\mathit{init}}$ and $\varphi_{\mathit{error}}$. \section{Related Work and Conclusions} \label{sec:conclusions} The verification framework introduced in this paper is an extension of the framework presented in~\cite{De&13a}, where CLP and \textrm{iterated specialization} have been used to define a general verification framework which is parametric with respect to the programming language and the logic used for specifying the properties of interest. The main novelties we have introduced in this paper are the following: (i)~we consider imperative programs acting also on array variables, and (ii)~we consider a more expressive specification language, which allows us to write properties involving elements of arrays and, in general, fields of complex data structures. In order to deal with these additional features (i)~we have defined the operational semantics for array manipulation, and (ii)~we have considered powerful transformation rules, such as conjunctive definition, conjunctive folding, and goal replacement. The transformation rules and some strategies for their application have been implemented in the MAP transformation system~\cite{MAP}, so as to perform proofs of program correctness in a semi-automated way. Our approach has many connections to various techniques developed in the field of static program analysis, to which David Schmidt has given an outstanding contribution, specially in clarifying its relationships with methodologies like denotational semantics, abstract interpretation, and model checking (see, for example,~\cite{Sch98}). Now we briefly overview the verification techniques that are particularly relevant to the method described in the present paper, and make use of logic programming, constraints, abstract interpretation, and automated theorem proving. The use of logic programming techniques for program analysis is not novel. For instance, Datalog (the function-free fragment of logic programming) has been proposed for performing various types of program analysis such as {\em dataflow analysis}, {\em shape analysis}, and {\em points-to analysis} \cite{BrS09,Rep98,WhL04}. In order to encode the properties of interest into Datalog, all these analysis techniques make an abstraction of the program semantics. In contrast, our transformational method manipulates a CLP program which encodes the full semantics (up to a suitable level of detail) of the program to be analyzed. An advantage of our approach is that the output of a transformation-based analysis is a new program which is {\em equivalent} to the initial one, and thus the analysis can be iterated to the desired degree of precision. Constraint logic programming has been successfully applied to perform model checking of both finite~\cite{NiL00} and infinite state systems~\cite{DeP99,Fi&01a,Fi&13a}. Indeed, CLP turns out to be suitable for expressing both (i)~the symbolic executions and (ii)~the invariants of imperative programs~\cite{Ja&11b}. Moreover, there are powerful CLP-based tools, such as ARMC~\cite{PoR07}, TRACER~\cite{Ja&11c}, and HSF~\cite{HSF} that can be used for performing model checking of imperative programs. These tools are fully automatic, but they are applicable to classes of programs and properties that are much more limited than those considered in this paper. We have shown in~\cite{De&13a} that, by focusing on verification tasks similar to those considered by ARMC, TRACER, and HSF, we can design a fully automatic, transformation-based verification technique whose effectiveness is competitive with respect to the one of the above mentioned tools. The connection between imperative programs and constraint logic programming has been investigated in~\cite{Fla04,Pe&98}. The verification method presented in~\cite{Fla04} is based on a semantics preserving translation from an imperative language with dynamic data structures and recursive functions into CLP. This translation reduces the verification of the (in)correctness of imperative programs to a problem of constraint satisfiability within standard CLP systems. A method based on specialization of CLP programs for performing the analysis of imperative programs has been presented in~\cite{Pe&98}. In this work the authors first present a CLP interpreter which defines the operational semantics of a simple imperative language. Then, given an imperative program, say $\textit{prog}$, they specialize that interpreter with respect to a CLP translation of~$\textit{prog}$ thereby getting a residual, specialized CLP program~$P_{\!\mathit{sp}}$. Finally, a static analyzer for CLP programs is applied to~$P_{\!\mathit{sp}}$ and its results are used to annotate the original imperative program with invariants. In~\cite{HeG06} a very similar methodology is applied for the verification of low-level programs for PIC microcontrollers. The program specialization which is done during Step~(A) of our verification method (that is, the removal of the interpreter) is very similar to the specialization proposed in~\cite{Pe&98}. However, having removed the interpreter, in order to verify the correctness of the given imperative program, in Step (B) we apply again program transformation and we not use static analyzers. CLP approaches to the verification of program correctness have recently received a growing attention because of the development of very efficient \textrm{constraint solving} tools~\cite{Ryb10}. These approaches include: (i)~the template-based invariant synthesis~\cite{Be&07}, and (ii)~the interpolant generation~\cite{Ry&10}. Related to this line of work, we would like to mention the paper~\cite{Gr&12} where the authors propose a method for constructing verification tools starting from a given set of proof rules, called Horn-like rules, specified by using constraints. Our transformational method for verifying properties of array programs is related to several methods based on abstract interpretation and predicate abstraction. In~\cite{HaP08}, which builds upon~\cite{Go&05}, relational properties among array elements are discovered by partitioning the arrays into symbolic slices and associating an abstract variable with each slice. This approach offers a compromise between the efficiency of {\em array smashing} (where the whole array is represented by a single abstract variable) and the precision of {\em array expansion}~\cite{Bl&02} (where every array element is associated with a distinct abstract variable). In~\cite{Co&11} the authors present a scalable, parameterized abstract interpretation framework for the automatic analysis of array programs based on slicing. In~\cite{Gu&08} a powerful technique using template-based quantified abstract domains, is applied to successfully generate quantified invariants. Other authors (see~\cite{FlQ02,LaB07}) use indexed predicate abstraction for inferring universally quantified properties about array elements. Also theorem provers have been used for discovering invariants in programs which manipulate arrays. In particular, in~\cite{Br&06} a satisfiability decision procedure for a decidable fragment of a theory of arrays is presented. That fragment is expressive enough to prove properties such as sortedness of arrays. In~\cite{JhM07,KoV09,McM08} the authors present some theorem provers which may generate array invariants and interpolants. In~\cite{Se&09} a backward reachability analysis based on predicate abstraction and the CEGAR technique is used for deriving invariants which are universally quantified formulas over array indexes, and existing techniques for quantifier-free assertions are adapted to the verification of array properties. Finally, we would like to mention two more papers which deal with the problem of verifying properties of array programs by using Satisfiability Modulo Theory~(SMT) techniques. Paper~\cite{La&13} presents a constraint-based invariant generation method for generating universally quantified loop invariants over arrays, and paper~\cite{Al&12} proposes a model checker for verifying universally quantified safety properties of arrays. As future work, we plan to investigate the issue of designing a general, fully automated strategy for guiding the application of the program transformation rules we have considered in this paper. In particular, we want to study the problem of devising unfolding strategies, generalization operators, and goal-replacement techniques which are tailored to the specific task of verifying program correctness. We also intend to extend the implementation described in~\cite{De&13a} and to perform more experiments for validating our transformation-based verification approach by using some benchmark suites which include array programs. As a further line of future research, we plan to enrich our framework by considering some more theories, besides the theory of the arrays, so that one can prove properties of programs acting on dynamic data structures such as lists and heaps. \section{Acknowledgments} We thank the anonymous referees for their constructive comments. We would like to thank Anindya Banerjee, Olivier Danvy, Kyung-Goo Doh, and John Hatcliff for their kind invitation to contribute to this symposium in honor of Dave Schmidt. Alberto recalls with great joy and gratitude the time together with Dave in Edinburgh and in Rome.

proofpile-arXiv_069-6496

\section{Introduction} \input{introduction} \section{iDataCool architecture} \label{sec:idc-arch} \input{idc-arch} \section{Infrastructure} \label{sec:installation} \input{installation} \section{Measurements} \label{sec:benchmarks} \input{benchmarks} \section{Conclusions} \input{conclusions} \section*{Acknowledgments} \input{acknowledgment} \bibliographystyle{splncs}

proofpile-arXiv_069-6689

\section{Introduction, terminologies and notation} \label{sec:1} As a higher dimensional versions of Morse functions, {\it fold maps} have been fundamental tools in the studies of smooth manifolds by using generic maps. A {\it fold map} is defined as a smooth map such that each singular point is of the form $$(x_1, \cdots, x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m-i}{x_k}^2-\sum_{k=m-i+1}^{m}{x_k}^2)$$ for two positive integers $m \geq n$ and an integer $0 \leq i \leq m-n+1$. A Morse function is naturally regarded as a fold map ($n=1$). For a fold map from a closed smooth manifold of dimension $m$ into a smooth manifold of dimension $n$ (without boundary), the following two hold ($m \geq n \geq 1$). \begin{enumerate} \item The {\it singular set}, defined as the set of all the singular points, is a closed smooth submanifold of dimension $n-1$ of the source manifold. \item The restriction map to the singular set is a smooth immersion of codimension $1$. \end{enumerate} Studies of such maps were started by Whitney (\cite{whitney}) and Thom (\cite{thom}) in the 1950s. We also note that if the restriction map to the singular set is an immersion with normal crossings, it is {\it stable} (stable maps are important in the theory of global singularity; see \cite{golubitskyguillemin} for example). \\ \ \ \ Since around the 1990s, fold maps with additional conditions have been actively studied. For example, in \cite{burletderham}, \cite{furuyaporto}, \cite{saeki2}, \cite{saekisakuma} and \cite{sakuma}, {\it special generic} maps, which are defined as smooth maps such that singular points are always of the form $$(x_1, \cdots, x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m}{x_k}^2)$$ for two positive integers $m \geq n$, were studied. {\it Simple} fold maps are defined as fold maps such that fibers of singular values do not have any connected component with more than one singular points (see cite \cite{saeki} and \cite{sakuma2}) and special generic maps are simple. In \cite{kobayashisaeki}, Kobayashi and Saeki investigated topology of {\it stable} maps including fold maps which are stable from closed manifolds of dimensions larger than $2$ into the plane. maps including fold maps which are stable from closed manifolds of dimensions larger than $2$ into the plane. \ \ \ Later, in \cite{saekisuzuoka}, Saeki and Suzuoka found good properties of manifolds admitting stable maps such that the inverse images of regular values are disjoint unions of spheres. The main theorem \cite[Theorem 4.1]{saekisuzuoka} states that a closed smooth manifold of dimension $4$ admitting such a stable map into a surface without boundary bounds a nice compact smooth manifold of dimension $5$ we can construct by observing the inverse images of the maps. As \cite[Lemma 1]{kitazawa}, the theorem is generalized as a proposition for a simple fold map such that the inverse images of regular values are disjoint unions of spheres from a closed smooth manifold of dimension $m$ into a smooth manifold of dimension $n$ without boundary (see also \cite[Proposition 3.12]{saeki} and its proof). In this paper, we consider extensions of these works. We consider simple fold maps such that the inverse images of regular values are more general and show that under appropriate differential topological conditions, the source manifolds are bounded by compact PL or smooth manifolds. \ \ \ This paper is organized as follows. \ \ \ Section \ref{sec:2} is for preliminaries. We review the {\it Reeb space} of a smooth map, which is the space consisting of all the connected components of all the fibers of the map. We review fundamental facts on simple fold maps in Proposition \ref{prop:1}. \ \ \ In section \ref{sec:3}, as main theorems, under some appropriate situations, we prove facts similar to ones in \cite{saekisuzuoka} introduced before for closed smooth manifolds admitting simple fold maps such that the inverse images of regular values are not always disjoint unions of spheres by analogy of the proofs of the original statements with extra new technique on algebraic and differential topology. More precisely, we first decompose the Reeb space $W_f$ of the given simple fold map $f:M \rightarrow N$ into some pieces by using its simplicial structure, construct manifolds piece by piece and then we glue them together to obtain the desired manifold $W$ such that $\partial W=M$. As a result, we also obtain other objects such as a polyhedron simple homotopy equivalent to $W$, which is not obtained in the proofs of known results. \ \ \ In this paper, smooth manifolds and smooth maps between them are of class $C^{\infty}$ unless otherwise stated. Throughout this paper, we assume that $M$ is a closed smooth manifold of dimension $m$ ($m \geq 1$) and that $N$ is a smooth manifold of dimension $n$ without boundary ($m \geq n \geq 1$). We set $f$ as a smooth map from $M$ into $N$ and denote the {\it singular set} of the map, defined as the set of all the singular points of the map, by $S(f)$. \\ \ \ \ We also note about homeomorphisms. In this paper, we often consider ${\rm PL}$ homeomorphisms between two polyhedra including ones between two ${\rm PL}$ manifolds. In this paper, for two polyhedra $X_1$ and $X_2$, a homeomorphism $\phi:X_1 \rightarrow X_2$ which is isotopic (in the topology category) to a homeomorphism from $X_1$ onto $X_2$ which gives an isomorphism between underlying simplicial complexes is said to be a {\it ${\rm PL}$ homeomorphism}. Note that any diffeomorphism between two diffeomorphic manifolds is regarded as a ${\rm PL}$ homeomorphism, where we consider the canonical ${\rm PL}$ structures.\\ \section{Preliminaries} \label{sec:2} \subsection{Reeb spaces} We introduce the {\it Reeb space} of a continuous map. \begin{Def} \label{def:3} Let $X$, $Y$ be topological spaces. For $p_1, p_2 \in X$ and for a map $c:X \rightarrow Y$, we define as $p_1 {\sim}_c p_2$ if and only if $p_1$ and $p_2$ are in the same connected component of $c^{-1}(p)$ for some $p \in Y$. ${\sim}_{c}$ is an equivalence relation. \ \ \ We set the quotient space $W_c:=X/{\sim}_c$. we call $W_c$ the {\it Reeb space} of $c$. \end{Def} We denote the induced quotient map from $X$ into $W_c$ by $q_c$. We define $\bar{c}:W_c \rightarrow Y$ so that $c=\bar{c} \circ q_c$. For example, for a Morse function, the Reeb space is a graph and for a simple fold map, the Reeb space is homeomorphic to a polyhedron which is not so complex (see Proposition \ref{prop:1} later). For a special generic map, the Reeb space is homeomorphic to a smooth manifold (see section 2 of \cite{saeki2}). \ \ \ Here, we introduce terms on spheres, fiber bundles and polyhedra which are important in this paper. \ \ \ In this paper, an {\it almost-sphere} means a smooth homotopy sphere given by glueing same dimensional two standard discs together by a diffeomorphism between the boundaries. We note that the underlying ${\rm PL}$ manifold of an almost-sphere is a standard sphere. \ \ \ We often use terminologies on (fiber) bundles in this paper (see also \cite{steenrod} for example). For a topological space $X$, an {\it $X$-bundle} is a bundle whose fiber is $X$. A bundle whose structure group is $G$ is a {\it trivia bundle} if it is equivalent to the product bundle as a bundle whose structure group is $G$. In this paper, a {\it PL {\rm (}smooth{\rm )} bundle} means a bundle whose fiber is a polyhedron (resp. smooth manifold) and whose structure group is a group consisting of some PL homeomorphisms (resp. diffeomorphisms) of the fiber. A {\it linear} bundle is a bundle whose fiber is a standard sphere or a standard disc and whose structure group consist of linear transformations on the fiber. \ \ \ In this paper, we also use terminologies on polyhedra. In this paper, for a polyhedron of dimension $k \geq 1$, a {\it branched} point means a point such that every open neighborhood of the point is not homeomorphic to any open set of ${\mathbb{R}}^k$ or ${{\mathbb{R}}^k}_{+}:=\{(x_1,\cdots,x_k) \in {\mathbb{R}}^k \mid x_k \geq 0\}$. If a polyhedron $X$ of dimension $k$ does not have branched points, then it is a manifold with triangulation and we can define the {\it interior} ${\rm Int} X$ and the {\it boundary} $\partial X$. \ \ \ The following proposition is well-known and we omit the proof. \begin{Prop} \label{prop:1} Let $f:M \rightarrow N$ be a special generic map, a simple fold map, or a stable fold map. Then, $W_f$ is regarded as a polyhedron and the following statements hold. \begin{enumerate} \item \label{prop:1.1} $W_f-q_f(S(f))$ is uniquely regarded as a smooth manifold of dimension $n$ such that $q_f:M-S(f) \rightarrow W_f-q_f(S(f))$ is a smooth submersion. For any compact subset $P$ of any connected component of $W_f-q_f(S(f))$, is regarded as the total space of a bundle given by $q_f {\mid}_{{q_f}^{-1}(P)}:{q_f}^{-1}(P) \rightarrow P$. \item \label{prop:1.2} Suppose that $f$ is simple. Then, for any connected component $C$ of $S(f)$, there exists a small regular neighborhood $N(q_f(C))$ of $q_f(C)$ in $W_f$ such that $N(q_f(C))$ is regarded as the total space of a PL bundle over $C$. Furthermore, ${q_f}^{-1}(N(q_f(C)))$ is regarded as the total space of a PL bundle given by the composition of the map ${q_f} {\mid}_{{q_f}^{-1}(N(q_f(C)))}:{q_f}^{-1}(N(q_f(C))) \rightarrow N(q_f(C))$ and the projection of the bundle $N(q_f(C))$ over $C$. \item \label{prop:1.3} If $C$ is a connected component consisting of singular points of index $0$, then the bundle $N(q_f(C))$ before is a trivial $[0,1]$-bundle over $q_f(C)$ and $q_f(C)$ is the image of a section of the bundle over $N(q_f(C))$ corresponding to the point $0 \in [0,1]$. Furthermore, the bundle ${q_f}^{-1}(N(q_f(C)))$ is a linear $D^{m-n+1}$-bundle over $q_f(C)$. \item \label{prop:1.4} If in the situation of Proposition \ref{prop:1} (\ref{prop:1.2}), $C$ is a connected component of the singular set consisting of singular points of index larger than $0$, then the bundle $N(q_f(C))$ before is a PL $[-1,1]$-bundle over $q_f(C)$ and $q_f(C)$ is the image of a section of the bundle over $N(q_f(C))$ corresponding to $0 \in [-1,1]$. This bundle is trivial if and only if $N(q_f(C))-q_f(C)$ is not connected. \item \label{prop:1.5} Ifin the situation of Proposition \ref{prop:1} (\ref{prop:1.2}), $C$ is a connected component of the singular set such that $q_f(C)$ consists of branched points, then the bundle $N(q_f(C))$ before is a PL bundle over $q_f(C)$ whose fiber is $K:=\{re^{i \theta} \mid 0 \leq r \leq 1, \theta=0, \frac{2}{3}\pi, \frac{4}{3}\pi \}$ and whose structure group consists of only the identity map and the conjugation $z \mapsto \bar{z}$. Furthermore, $q_f(C)$ is the image of a section of the bundle over $N(q_f(C))$ corresponding to $0 \in K$. \end{enumerate} \end{Prop} \section{Simple fold maps and compact manifolds bounded by the source manifolds of the maps} \label{sec:3} As main works, in this section, we consider simple fold maps satisfying appropriate conditions on regular fibers and extra algebraic and differential topological conditions and construct manifolds bounded by their source manifolds. We review results on simple fold maps and stable maps whose regular fibers are disjoint unions of spheres and then, we prove Theorems \ref{thm:1}-\ref{thm:3}. \subsection{Simple fold maps whose regular fibers are disjoint unions of spheres} \label{subsec:3.1} We review \cite[Theorem 4.1]{saekisuzuoka}. \begin{Prop}[\cite{saekisuzuoka}] \label{prop:2} Let $f:M \rightarrow N$ be a simple fold map from a closed manifold of dimension $4$ into a manifold of dimension $2$ without boundary. For each regular value $p$, let $f^{-1}(p)$ be a disjoint union of finite copies of $S^2$. \ \ \ Then, there exist a compact smooth manifold $W$ of dimension $5$ such that $\partial W = M$ and a continuous map $r: W \rightarrow W_f$ such that $r \mid_{\partial W}$ coincides with $q_f:M \rightarrow W_f$ and the following five hold. \begin{enumerate} \item For each $p \in W_f-q_f(S(f))$, $r^{-1}(p)$ is diffeomorphic to $D^3$. \item $\bar{f} \circ r$ is a smooth submersion. \item There exist a smooth triangulation of $W$ and a triangulation of $W_f$ such that $r$ is a simplicial map. \item For each $p \in W_f$, $r^{-1}(p)$ collapses to a point and $r$ is a homotopy equivalence. \item $W$ collapses to a subpolyhedron ${W_f}^{\prime}$ such that $r {\mid}_{{W_f}^{\prime}}:{W_f}^{\prime} \rightarrow W_f$ is a ${\rm PL}$ homeomorphism. \end{enumerate} If $M$ is orientable, then we can construct $W$ as an orientable manifold. \end{Prop} As a corollary to the proposition, we can prove the following. \begin{Cor}[\cite{saekisuzuoka}] \label{cor:1} In the situation of Proposition \ref{prop:2}, let $M$ be connected and $i:M \rightarrow W$ be the natural inclusion. Then, $${q_f}_{\ast}=r_{\ast} \circ i_{\ast}:{\pi}_k(M) \rightarrow {\pi}_k(W_f)$$ gives an isomorphism for $k=0,1$. \end{Cor} See also \cite[Lemma 1]{kitazawa2}, which is a similar statement for simple fold maps whose regular fibers are disjoint unions of spheres between manifolds of arbitary dimensions. \subsection{More general simple fold maps and compact manifolds bounded by their source manifolds} \label{subsec:3.2} We study more general simple fold maps. Before precise studies, we define several technical conditions on the maps, which we often pose in the present paper, to obtain theorems. \begin{Def} \label{def:4} Let $f:M \rightarrow N$ be a simple fold map sastisfying $m>n \geq 1$. \begin{enumerate} \item For each connected component $C$ of the set $q_f(q_f(S(f)))$, let $N(C)$ be a small regular neighborhood such that ${q_f}^{-1}(N(C))$ is regarded as the total space of a bundle over $C$. We call the bundle ${q_f}^{-1}(N(C))$ a {\it monodromy representation} on $C$ and if we can take this as a topologically trivial bundle, then $f$ is said to {\it have a topologically trivial monodromy} on $C$ and the bundle is said to be a {\it topologically trivial monodromy bundle} on $C$. If $f$ has a topologically trivial monodromy on $C$ for each component $C$, then $f$ is said to {\it have topologically trivial monodromies on the singular part}. We can replace "topologically" by "PL" and "smooth" and define similar terminologies. \item Let $N(q_f(S(f)))$ be a small regular neighborhood of the set $q_f(S(f))$. For any connected component $R$ of the set $W_f-{\rm Int} N(q_f(S(f)))$, ${q_f}^{-1}(R)$ is regarded as the total space of a bundle over $R$. We call the bundle ${q_f}^{-1}(R)$ a {\it monodromy representation} on $R$ and if we can choose the bundle ${q_f}^{-1}(R)$ as a topologically trivial bundle, then $f$ is said to {\it have a topologically trivial monodromy} on $R$ and the bundle is said to be a {\it topologically trivial monodromy bundle} on $R$. If $f$ has a topologically trivial monodromy on $R$ for each component $R$, then $f$ is said to {\it have topologically trivial monodromies on the regular part}. We can replace "topologically" by "PL" and "smooth" and define similar terminologies. \end{enumerate} \end{Def} \begin{Thm} \label{thm:1} Let $M$ be a closed and connected manifold of dimension $m$, $f:M \rightarrow {\mathbb{R}}^n$ be a simple fold map and $m>n \geq 1$ hold. Suppose that $f$ has topologically trivial monodromies on the singular part and the regular part and that the following conditions hold. \begin{enumerate} \item For each connected component $C$ of the set $q_f(S(f))$, we can take a topologocally trivial monodromy bundle on $C$ as a bundle whose fiber is a compact topological manifold ${F^{\prime}}_C$ with non-empty boundary. \item Let $N(C)$ be a small regular neighborhood of each connected component $C$ as in Definition \ref{def:4} and let $N(q_f(S(f)))$ be the disjoint union of $N(C)$ for every $C$. For any connected component $R$ of the set $W_f-{\rm Int} N(q_f(S(f)))$, we can take a topologically trivial monodromy bundle on $R$ as a bundle whose fiber is a topological manifold $F_R$ bounding a compact topological manifold $E_R$. \item For each connected component $C$ of the set $q_f(S(f))$, let us denote the family of all the connected components of the set $W_f-{\rm Int} N(q_f(S(f)))$ intersecting $N(C)$ by $\{R_{C_{\lambda}}\}$. Then the boundary $\partial {F^{\prime}}_C$ of the manifold ${F^{\prime}}_C$ before is represented as the disjoint union of closed topological manifolds $\{F_{C_{\lambda}}\}$ {\rm (}$\lambda \in \Lambda${\rm )}, where $F_{C_{\lambda}}$ is homeomorphic to $F_{R_{C_{\lambda}}}$. Furthermore, the subbundle corresponding to the boundary $\partial {F^{\prime}}_C \subset F^{\prime}$ of a topologically trivial monodromy bundle on $C$ is trivial. \end{enumerate} Then, we can construct a topologically trivial bundle $W_R$ over $R$ whose fiber is a compact topological manifold $E_R$ satisfying so that the topologically trivial monodromy bundle over $R$ before is a subbundle of this bundle correspondig to the boundary. Moreover as a result, we can construct a topologically trivial bundle over $C$ whose fiber is a closed topological manifold $F_C$ obtained by attaching the manifolds $\{F_{C_{\lambda}}\}$ {\rm (}$\lambda \in \Lambda${\rm )} on the boundary of ${F^{\prime}}_C$. Furthermore, assume also that $F_C$ bounds a compact topological manifold $E_C$. Then, there exist a compact topological manifold $W$ of dimension $m+1$ bounded by $M$. \ \ \ Especially, let similar conditions hold even if we replace "topologically" by "PL". Then, we can construct $W$ as a PL manifold and we can obtain a polyhedron $V$ whose dimension is smaller than $W$ and continuous maps $r:W \rightarrow V$ and $s:V \rightarrow W_f$ and triangulations of $W$, $V$ and $W_f$ so that the following conditions hold. \begin{enumerate} \item $r$ and $s$ are simplicial. \item If $p$ is in a connected component in $W_f-{\rm Int} N(q_f(S(f)))$, then $s^{-1}(p)$ is a subpolyhedron of $V$, $(s \circ r)^{-1}(p)$ collapses to $s^{-1}(p)$ and the dimension of $s^{-1}(p)$ is smaller than that of $(s \circ r)^{-1}(p)$. \item For each connected component $C$ of $N(q_f(S(f)))$, there exists a PL subbudle $V_C$ of the trivial PL $E_C$-bundle over $C$ which is trivial and whose fiber is a subpolyhedron of $E_C$, simple homotopy equivalent to $E_C$ and of dimension smaller than $E_C$. \item $W$ collapses to a subpolyhedron ${V}^{\prime}$ such that $r {\mid}_{{V}^{\prime}}:{V}^{\prime} \rightarrow V$ is a PL homeomorphism. \end{enumerate} In addition, let similar conditions hold even if we replace "PL" by "smooth". Then, we can take the manifold $W$ as a smooth manifold and we can obtain similar polyhedron, continuous maps and smooth triangulations. \end{Thm} \begin{proof} We can easily obtain a topological manifold $W$ by gluing the following ($m+1$)-dimensional manifolds. \begin{itemize} \item The total space $W_R$ of the topologically trivial $E_R$-bundle over $R$ explained in the assumption. \item The total space $W_C$ of a topologically trivial $E_C$-bundle over $C$ such that the trivial $F_C$-bundle over $C$ explained in the assumption is a subbundle of this bundle satisfying $\partial E_C=F_C$. \end{itemize} We can obtain $W$ as a PL (smooth) manifold in the PL (resp. smooth) case. We show additional statements in the PL case and by using a similar method we can show them in the smooth case. For any connected component $R$ of $W_f-{\rm Int} N(q_f(S(f)))$, by considering a handle decomposition of $E_R$ in the PL category, we obtain a polyhedron which $E_R$ collapses to and whose dimension is smaller than that of $E_R$ and thus, we obtain a trivial PL subbundle $V_R$ of the trivial PL bundle $W_R$ over $R$ whose fiber is a PL manifold $E_R$. $V_R$ is a polyhedron whose dimension is smaller than that of $W_R$ and we obtain a polyhedron ${{V}^{\prime}}_R$, continuous maps $r_R:W_R \rightarrow V_R$ and $s_R:V_R \rightarrow R$ and triangulations of $W_R$, $V_R$, ${{V}^{\prime}}_R$ and $R$ so that they satisfy desired conditions. For any connected component $C$ of the set $q_f(S(f))$, by considering a handle decomposition of $E_C$ in the PL category, we obtain a polyhedron which $E_C$ collapses to and whose dimension is smaller than that of $E_C$ and thus, we obtain a trivial PL subbundle $V_C$ of the trivial PL bundle $W_C$ over $R$ whose fiber is a PL manifold $E_C$. We can take $V_C$ so that we can attach $V_C \bigcap \partial W_C$ with the disjoint union of $V_{R_{C_{\lambda}}} \bigcap \partial W_{R_{C_{\lambda}}}$ for every $C_{\lambda}$ compatiably together. This completes the proof. \end{proof} We have the following corollary similar to Corollary \ref{cor:1}, which we prove later with Corollary \ref{cor:3}. \begin{Cor} \label{cor:2} In the situation of Theorem \ref{thm:1}, let $i:M \rightarrow W$ be the natural inclusion. Then, the homomorphism $r_{\ast} \circ i_{\ast}:{\pi}_j(M) \rightarrow {\pi}_j(V)$ is an isomorphism for $0 \leq j \leq m-\dim{V}-1$. \end{Cor} As specific cases, we have the following two statements. \begin{Thm} \label{thm:2} Let $m,n \in \mathbb{N}$ and let $m>n \geq 1$ hold. Let $M$ be a closed and connected manifold of dimension $m$ and $f:M \rightarrow {\mathbb{R}}^n$ be a simple fold map. Suppose that $f$ has topologically trivial monodromies on the singular part. \begin{enumerate} \item \label{thm:2.1} Suppose that $f$ has topologically trivial monodromies on the regular part. We also assume that for each closed manifold of dimension $m-n$ or $m-n+1$, there exists a compact topological manifold bounded by it. Then, we can construct a compact topological manifold $W$ of dimension $m+1$ bounded by $M$ in Theorem \ref{thm:1}. In the PL case, we can construct $W$ as a PL manifold and we can obtain a polyhedron $V$ whose dimension is smaller than $W$ and continuous maps $r:W \rightarrow V$ and $s:V \rightarrow W_f$ and triangulations of $W$, $V$ and $W_f$ so that the four conditions listed in Theorem \ref{thm:1} hold. In the smooth case, in addition, we can construct $W$ as a smooth manifold. \item \label{thm:2.2} Suppose that the fiber of the $F_C$-bundle over $C$ in Theorem \ref{thm:1} is a sphere. Then, we can construct a compact topological manifold $W$ of dimension $m+1$ bounded by $M$ in Theorem \ref{thm:1}. In the PL and smooth cases, we obtain objects similar to those of Theorem \ref{thm:1} and (\ref{thm:2.1}) of this theorem. \end{enumerate} \end{Thm} \begin{proof} The first statement follows immediately from Theorem \ref{thm:1}. We can prove the second statement by noticing that a sphere bundle is naturally a subbundle of a disc bundle whose dimension is larger than that of the sphere bundle by one. \end{proof} We introduce and prove Theorem \ref{thm:3}, which is regarded as an extension of a specific case of Theorem \ref{thm:2} (\ref{thm:2.2}). We define the {\it mapping cylinder} of a continuous map $c:X \rightarrow Y$ as the quotient space of $(X \times [0,1]) \sqcup Y$ obtained by identifying $(x,1)$ and $c(x)$ ($x \in X$). \begin{Thm} \label{thm:3} Let $M$ be a closed manifold of dimension $m$, $N$ be a manifold of dimension $n$ without boundary and $f:M \rightarrow N$ be a simple fold map. Let $n \geq 1$ and $m-n \geq 2$. \begin{enumerate} \item For any point $p \in W_f-q_f(S(f))$, ${q_f}^{-1}(p)$ is an almost-sphere of dimension $m-n$ or PL homeomorphic to the product of two standard spheres whose dimensions do not coincide. If a connected component $R$ of $W_f-q_f(S(f))$ contains a point $p$ such that ${q_f}^{-1}(p)$ is an almost-sphere, then $R$ is said to be an {\rm AS-region}. Furthermore, if ${q_f}^{-1}(p)$ is not an almost-sphere, then each connected component of the boundary of the closure of the connected component of $W_f-q_f(S(f))$ containing $p$ contains no branched point and it is also the boundary of the closure of an AS-region. \item Let $C$ be a connected component of the set $S(f)$ such that $q_f(C)$ consists of branched points, then every connected component of $W_f-q_f(S(f))$ such that the boundary of its closure contains $C$ as a connected component is an AS-region. \item Let $C$ be a connected component of the set $S(f)$ and let $N(C)$ be a small tubular neighborhood in Proposition \ref{prop:1}(\ref{prop:1.3}). Let $0<k<m-n$ be an integer and let the fiber of the bundle ${q_f}^{-1}(N(C))$ be PL homeomorphic to an ($m-n+1$)-dimensional manifold simple homotopy equivalent to $S^{m-n-k}$ with the interior of a smoothly embedded ($m-n+1$)-dimensional stadard closed disc removed. Then, for any point $p$ in the connected component $R_C$ of $W_f-q_f(S(f))$ such that the boundary of its closure contains $C$ as a connected component, ${q_f}^{-1}(p)$ is PL homeomorphic to $S^k \times S^{m-n-k}$ and we can take the monodromy representation on $R_C$ as a PL $S^k \times S^{m-n-k}$-bundle over $R_C$ whose structure group consists of PL homeomorphisms regarded as bundle isomorphisms on this trivial $S^{m-n-k}$-bundle $S^k \times S^{m-n-k}$ over $S^{m-n-k}$. In this situation, $R_C$ is said to be a {\rm $k$ S-region}. Furthermore, each connected component of the boundary of the closure of a $k$ S-region satisfies the condition before which $C$ satisfies. Such connected components of the boundary of the closure of a $k$ S-region are said to be {\rm $k$ S-loci}. \end{enumerate} \ \ \ Then, there exist a compact PL manifold $W$ of dimension $m+1$ such that $\partial W = M$, a polyhedron $V$ and continuous maps $r: W \rightarrow V$ and $s:V \rightarrow W_f$ and the following two hold. \begin{enumerate} \item There exist a triangulation of $W$, a triangulation of $V$ and a triangulation of $W_f$ such that $r$ is a simplicial map and that $s$ is a simplicial map and the following three hold. \begin{enumerate} \item For each $p \in V$, $r^{-1}(p)$ collapses to a point and $r$ is a homotopy equivalence. \item If $p$ is in the closure of an AS-region in $W_f$, then $s^{-1}(p)$ is a point. If $p$ is in an AS-region in $W_f$ and $q \in s^{-1}(p)$, then $r^{-1}(q)$ is ${\rm PL}$ homeomorphic to $D^{m-n+1}$. \item If $p$ is in a $k$ S-region in $W_f$ whose closure is bounded by a disjoint union of $k$ S-loci, then $s^{-1}(p)$ is PL homeomorphic to $S^k$. If $p$ is in a $k$ S-region in $W_f$ whose closure is bounded by a disjoint union of $k$ S-loci and $q \in s^{-1}(p)$, then $r^{-1}(q)$ is PL homeomorphic to $D^{m-n-k+1}$. \end{enumerate} \item $W$ collapses to a subpolyhedron ${V}^{\prime}$ such that $r {\mid}_{{V}^{\prime}}:{V}^{\prime} \rightarrow V$ is a PL homeomorphism. \end{enumerate} If $M$ is orientable, then we can construct $W$ as an orientable manifold. \end{Thm} \begin{proof} We construct a compact PL manifold of dimension $m+1$ bounded by $M$. We denote the set of all the singular points of index $i$ by $F_i(f)$. \\ \\ Step 1 \ \ \ Around a regular neighborhood of $q_f(F_0(f))$. $q_f(F_0(f))$ is the image of the set $F_0(f)$ of all the definite fold points of $f$. Let $N(q_f(F_0(f)))$ be a small regular neighborhood of $q_f(F_0(f))$ as in Proposition \ref{prop:1} (\ref{prop:1.3}). $N(q_f(F_0(f)))$ is regarded as the total space of a trivial PL bundle over $q_f(F_0(f))$ with the fiber $[0,1]$. We may assume that $q_f(F_0(f))$ corresponds to the $0$-section ($0 \in [0,1]$). \ \ \ For each $p \in q_f(F_0(f))$, set $K_p:={q_f}^{-1}(\{p\} \times [0,1])$ for a fiber $\{p\} \times [0,1]$ of the bundle $N(q_f(F_0(f)))$ over $q_f(F_0(f))$. It is diffeomorphic to $D^{m-n+1}$. $q_f^{-1}(N(q_f(F_0(f))))$ is regarded as the total space of a linear bundle over $q_f(F_0(f))$ whose fiber is $D^{m-n+1}$ and $K_p$ is the fiber over $p \in q_f(F_0(f))$. \ \ \ We obtain the following objects. See also the proof of Lemma 1 of \cite{kitazawa}. \begin{enumerate} \item A linear $D^{m-n+2}$-bundle $V_0$ over $q_f(F_0(f))$ (we denote by $V_p$ the fiber over $p \in q_f(F_0(f))$) such that $q_f^{-1}(N(q_f(F_0(f))))$ is regarded as the total space of a subbundle of the bundle $V_0$ with the fiber $D^{m-n+1}$ and that $q_f^{-1}(N(q_f(F_0(f))))$ is in the boundary of $V_0$. \item A simplicial map $r_0:V_0 \rightarrow N(q_f(F_0(f)))$. such that ${r_0}^{-1}(p,t)$ is ${\rm PL}$ homeomorphic to $D^{m-n+1}$ for $p \in q_f(F_0(f))$ and $t \in (0,1]$ and that ${r_0}^{-1}(p,t)$ is a point for $p \in q_f(F_0(f))$ and $t=0$. \item A subbundle $\widetilde{N(q_f(F_0(f)))} \subset V_0$ of dimension $n$ such that the followings hold. \begin{enumerate} \item $\widetilde{N(q_f(F_0(f)))}$ is of dimension $n$. \item ${r_0} {\mid}_{\widetilde{N(q_f(F_0(f)))}}:\widetilde{N(q_f(F_0(f)))} \rightarrow N(q_f(F_0(f)))$ is a ${\rm PL}$ homeomorphism (a bundle isomorphism between the two PL bundles). \item $\widetilde{N(q_f(F_0(f)))} \bigcap \partial V_p$ consists of two points $(p,0),(p,1) \in \{p\} \times [0,1]$. One of the two points is in $q_f^{-1}(N(q_f(F_0(f))))$ and the other point is not. \item $V_0$ collapses to $\widetilde{N(q_f(F_0(f)))}$. \end{enumerate} \end{enumerate} \ \ \ Let $B(f)$ be the set of all the branched points of $W_f$. By considering the attachments of handles, $B(f) \subset q_f(F_1(f))$ follows. \\ \\ Step 2 \ \ \ A regular neighborhood of $B(f)$. Let $N(B(f))$ be a small regular neighborhood of $B(f)$. From Proposition \ref{prop:1} (\ref{prop:1.5}), $N(B(f))$ is regarded as the total space of a PL bundle over $B(f)$ whose fiber is $K:= \{r \exp (i \theta) \in \mathbb{C} \mid 0 \leq r \leq 1, \theta = 0, \frac{2}{3}\pi, \frac{4}{3}\pi \}$. We may assume that $B(f)$ corresponds to the $0$-section ($0 \in K$). For each $p \in B(f)$, set $K_p:={q_f}^{-1}(\{p\} \times K)$ for a fiber $\{p\} \times K$ of the bundle $N(B(f))$ over $B(f)$. It is PL homeomorphic to $S^{m-n+1}$ with the interior of a union of disjoint three {\rm (}$m-n+1${\rm )}-dimensional closed standard discs removed. We may assume that $q_f^{-1}(N(B(f)))$ is regarded as the total space of a smooth bundle over $Q(f)$ with a fiber PL homeomorphic to $S^{m-n+1}$ with the interior of a union of disjoint three standard closed {\rm (}$m-n+1${\rm )}-discs removed and $K_p$ is the fiber over $p \in B(f)$. \\ \ \ \ We obtain the following objects. See also the proof of Lemma 1 of \cite{kitazawa}. \begin{enumerate} \item A PL $D^{m-n+2}$-bundle $V_B$ over $B(f)$ (we denote by $V_p$ the fiber over $p \in B(f)$) such that the bundle $q_f^{-1}(N(B(f)))$ is a subbundle of the bundle $V_B$ and that ${q_f}^{-1}(N(B(f)))$ is in the boundary of $V_B$. \item A simplicial map $r_B:V_B \rightarrow N(B(f))$ such that ${r_B}^{-1}(p,t)$ is PL homeomorphic to $D^{m-n+1}$ for $p \in B(f)$ and $t \in K-\{0\}$ and that ${r_B}^{-1}(p,t)$ collapses to a point for $p \in B(f)$ and $t=0$. \item A subpolyhedron $\widetilde{N(B(f))} \subset V_B$ of dimension $n$ such that the following four hold. \begin{enumerate} \item $\widetilde{N(B(f))}$ is regarded as the total space of a subbundle of the bundle $V_1$. \item ${r_B} {\mid}_{\widetilde{N(B(f))}}:\widetilde{N(B(f))} \rightarrow N(B(f))$ is a PL homeomorphism (a bundle isomorphism between the two PL bundles). \item $\widetilde{N(B(f))} \bigcap \partial V_p$ consists of three points $(p,\exp(\frac{2}{3} k \pi i)) \in \{p\} \times K$ ($k=0,1,2$) and is not in ${q_f}^{-1}(N(B(f)))$. Furthermore, each connected component of $\partial V_p-q_f^{-1}(N(B(f)))$ includes one of these points and $V_B$ collapses to $\widetilde{N(B(f))}$. \item $V_B$ collapses to $\widetilde{N(B(f))}$. \end{enumerate} \end{enumerate} For $2 \leq k \leq m-n$, let $G_k(f) \subset q_f(S(f))$ be the disjoint union of all the $k-1$ S-loci. By the assumption, we can take a regular neighborhood $N(G_k(f))$ so that ${q_f}^{-1}(N(G_k(f)))$ is regarded as the total space of a smooth bundle over $G_k(f)$ whose fiber is PL homeomorphic to an ($m-n+1$)-dimensional manifold simple homotopy equivalent to $S^{m-n-k+1}$ with the interior of a smoothly embedded ($m-n+1$)-dimensional standard closed disc removed. As mentioned in Proposition \ref{prop:1} (\ref{prop:1.4}), $N(G_k(f))$ is regarded as the total space of a trivial PL $[-1,1]$-bundle over $G_k(f)$ and $G_k(f)$ is the image of the section corresponding to the point $0 \in [-1,1]$. \\ \\ Step 3 \ \ \ Around the set $W_f-{\rm Int} ((N(q_f(F_0(f))) \sqcup N(B(f))) \sqcup {\sqcup}_{k} N(G_k(f)))$. \ \ \ This set is a compact manifold of dimension $n$ and let $\{R_{\lambda}\}_{\lambda \in \Lambda}$ be the family of all the connected components of the set. $R_{\lambda}$ is in an AS-region or an S-region. \ \ \ Let $R_{\lambda}$ be in an AS-region. Then, ${q_f}^{-1}(R_{\lambda})$ is regarded as the total space of a smooth bundle over $R_{\lambda}$ whose fiber is ${\rm PL}$ homeomorphic to $S^{m-n}$. We can define a PL $D^{m-n+1}$-bundle $V_{R_{\lambda}}$ over $R_{\lambda}$ which is an associated bundle of the bundle ${q_f}^{-1}(R_{\lambda})$ over $R_{\lambda}$ (we regard $S^{m-n}=\partial D^{m-n+1}$) by a PL map $r_{R_{\lambda}}:V_{R_{\lambda}} \rightarrow {R_{\lambda}}$. In addition, we take the associated bundle so that the structure group is a group consisting of PL homeomorphisms $r$ on $D^{m-n+1}$ such that $r(0)=0$ and that for a PL homeomorphism ${r}^{\prime}$ on $S^{m-n}$, $\frac{r(x)}{|x|}={r}^{\prime}(\frac{x}{|x|})$ ($x \neq 0$). Let $\widetilde{P(R_{\lambda})} \subset V_{R_{\lambda}}$ be the section of the associated bundle corresponding to the point $0 \in D^{m-n+1}$. Finally we set $P(R_{\lambda}):=R_{\lambda}$ and let $s_{R_{\lambda}}:P(R_{\lambda}) \rightarrow R_{\lambda}$ be the identity map. \ \ \ Let $R_{\lambda}$ be in a $k$ S-region whose closure is bounded by a disjoint union of $k$ S-loci. Then, ${q_f}^{-1}(R_{\lambda})$ is regarded as the total space of a smooth bundle over $R_{\lambda}$ whose fiber is ${\rm PL}$ homeomorphic to $S^k \times S^{m-n-k}$. We can take a PL ($S^k \times D^{m-n-k+1}$)-bundle $V_{R_{\lambda}}$ over ${R_{\lambda}}$ which is an associated bundle of the bundle ${q_f}^{-1}(R_{\lambda})$ over $R_{\lambda}$ (we regard $S^k \times S^{m-n-k}=S^k \times \partial D^{m-n-k+1}$). In addition ,we take the associated bundle so that the structure group is a group consisting of some PL homeomorphisms $r$ on $S^k \times D^{m-n-k+1}$ such that $r$ is a bundle isomorphism on a PL bundle $S^k \times D^{m-n-k+1}$ over $S^k$ whose structure group consists of PL homeomorphisms ${r_1}$ on $D^{m-n-k+1}$, where $r_1(0)=0$ and for a PL homeomorphism ${r_2}$ on $S^{m-n-k}$, $\frac{r_1(x)}{|x|}={r_2}(\frac{x}{|x|})$ ($x \neq 0$) and that $r$ induces a PL homeomorphism on the base space $S^k$. Let $\widetilde{P(R_{\lambda})} \subset V_{R_{\lambda}}$ be the subbundle whose fiber is $S^k \times \{0\} \subset S^k \times D^{m-n-k+1}$. The bundle $V_{R_{\lambda}}$ over ${R_{\lambda}}$ is also given by the composition of two PL maps $r_{R_{\lambda}}:V_{R_{\lambda}} \rightarrow P(R_{\lambda})$ and $s_{R_{\lambda}}:P(R_{\lambda}) \rightarrow R_{\lambda}$, where $P(R_{\lambda})$ is a PL manifold and by $s_{R_{\lambda}}$ makes $P(R_{\lambda})$ a bundle over $R_{\lambda}$ equivalent to the bundle $\widetilde{P(R_{\lambda})}$ over $R_{\lambda}$. \ \ \ Now we set disjoint unions $V_R:={\sqcup}_{\lambda \in \Lambda} V_{R_{\lambda}}$, $P_R:={\sqcup}_{\lambda \in \Lambda} P(R_{\lambda})$, $r_R:={\sqcup}_{\lambda \in \Lambda} r_{R_{\lambda}}$, $s_R:={\sqcup}_{\lambda \in \Lambda} s_{R_{\lambda}}$ and $\widetilde{P_R}:={\sqcup}_{\lambda \in \Lambda} \widetilde{P(R_{\lambda})}$. \\ \\ Step 4 \ \ \ Around a regular neighborhood of $G_k(f)$ ($2 \leq k \leq m-n$) \ \ \ After Steps 1, 2 and 3, for any connected component $C_k$ of $G_k(f)$ and a small regular neighborhood $N(C_k)$ of $C_k$ as in Proposition \ref{prop:1} (\ref{prop:1.4}), we obtain a PL bundle over $C_k$ whose fiber is an ($m-n+1$)-dimensional PL manifold $B_{C_k}$ simple homotopy equivalent to $S^{m-n-k+1}$ whose boundary $\partial B_{C_k}$ is $S^{k-1} \times S^{m-n-k+1}$ with the product manifold $D^{m-n-k+2} \times S^{k-1}$ attached by the product of a pair of PL homeomorphisms $({\phi}_{{C_1}_k}:\partial D^{m-n-k+2} \rightarrow S^{m-n-k+1}, {\phi}_{{C_2}_k}:S^{k-1} \rightarrow S^{k-1})$ and that the bundle ${q_f}^{-1}(N(C_k))$ is a subbundle of the $B_{C_k}$-bundle. \ \ \ We obtain the following objects. \begin{enumerate} \item A PL $D^{m-n+2}$-bundle $V_{C_k}$ over $C_K$ (let $V_p$ denote the fiber over $p$ of the base space) such that the subbundle $\partial V_{C_k}$ corresponding to $\partial D^{m-n+2}=B_{C_k} {\bigcup}_{{\phi}_{{C_1}_k} \times {\phi}_{{C_2}_k}} (D^{m-n-k+2} \times S^{k-1})$ is the PL $S^{m-n+1}$-bundle in the previous paragraph. The bundle $q_f^{-1}(N(C_k))$ over $C_k$ is a subbundle of the bundle $V_{C_k}$ and is in the boundary of $V_{C_k}$ (we regard it is also a subbundle of the $B_{C_k}$-bundle). \item A PL bundle $P(C_k)$ whose fiber is the mapping cylinder of the constant map $c_k:S^{k-1} \rightarrow [0,1]$ satisfying $c_k(S^{k-1})=\{0\}$ and whose base space is $C_k$. \item Simplicial maps $r_{C_k}:V_{C_k} \rightarrow P(C_k)$ and $s_{C_k}:P(C_k) \rightarrow N(C_k)$ such that ${s_{C_k}}^{-1}(p)$ is a point for $p$ in the closure of an AS-region, that ${s_{C_k}}^{-1}(p)$ is PL homeomorphic to $S^{k-1}$ for $p$ in a $k-1$ S-region, that ${r_{C_k}}^{-1}(q)$ is PL homeomorphic to $D^{m-n+1}$ for $q \in {s_{C_k}}^{-1}(p)$ for $p$ in an AS-region, that ${r_{C_k}}^{-1}(q)$ is PL homeomorphic to $D^{m-n-k+2}$ for $q \in {s_{C_k}}^{-1}(p)$ for $p$ in a $k-1$ S-region and that ${(s_{C_k} \circ {r_{C_k}})}^{-1}(\{p\} \times [-1,1])=V_p$ for all $p \in C_k$ ($\{p\} \times [-1,1]$ is a fiber of the bundle $N(C_k)$ over $C_k$; $N(C_k)$ is regarded as the total space of a trivial PL $[-1,1]$-bundle over $C_k$ and $C_k$ is the image of the section corresponding to the point $0 \in [-1,1]$. Furthermore, ${r_{C_k}}^{-1}(q)$ collapses to a point for $q \in P(C_k)$. \item A subpolyhedron $\widetilde{P(C_k)} \subset V_{C_k}$ of dimension $n+k-1$ such that the following four hold. \begin{enumerate} \item $\widetilde{P(C_k)}$ is regarded as the total space of a subbundle of the bundle $V_{C_k}$. \item ${r_{C_k}} {\mid}_{\widetilde{P(C_k)}}:\widetilde{P(C_k)} \rightarrow P(C_k)$ is a PL homeomorphism and a bundle isomorphism between the two PL bundles. \item $\widetilde{P(C_k)} \bigcap \partial V_p$ does not contain any points in ${q_f}^{-1}(N(C_k))$ and is the disjoint union of a point in $B_{C_k}$ and a ($k-1$)-dimensional sphere $\{0\} \times S^{k-1} \subset D^{m-n-k+2} \times S^{k-1}$, where $B_{C_k}$ and $D^{m-n-k+2} \times S^{k-1}$ are the naturally embedded submanifolds of the fiber $\partial V_p$ over $p$ of the bundle $\partial V_{C_k} \subset V_{C_k}$, which is PL homeomorphic to $B_{C_k} {\bigcup}_{{\phi}_{{C_1}_k} \times {\phi}_{{C_2}_k}} (D^{m-n-k+2} \times S^{k-1})$. \item $V_{C_k}$ collapses to $\widetilde{P(C_k)}$. \end{enumerate} \end{enumerate} $V_G:={\sqcup}_{C_k} V_{C_k}$, $P_G:={\sqcup}_{C_k} P(C_k)$, $r_G:={\sqcup}_{C_k} r_{C_k}$, $s_G:={\sqcup}_{C_k} s_{C_k}$ and $\widetilde{P_G}:={\sqcup}_{C_k} \widetilde{P(C_k)}$. Thus, by gluing the manifolds $V_0$, $V_B$, $V_R$ and $V_G$ together, we obtain the desired manifold $V$. We can obtain other desired objects. This completes the proof. \end{proof} We have the following corollary. \begin{Cor} \label{cor:3} In the situation of Theorem \ref{thm:3}, let $M$ be connected and $i:M \rightarrow W$ be the natural inclusion. Then \\ $$r_{\ast} \circ i_{\ast}:{\pi}_k(M) \rightarrow {\pi}_k(V)$$ gives an isomorphism for $0 \leq k \leq m-\dim{V}-1$. \end{Cor} \begin{proof}[The proof of Corollaries \ref{cor:2} and \ref{cor:3}] Since $r$ is a homotopy equivalence, we have only to show that $i_{\ast}:{\pi}_{k}(M) \rightarrow {\pi}_{k}(W)$ ($0 \leq k \leq m-\dim{V}-1$) is an isomorphism. Since $W$ collapses to a polyhedron of dimension $\dim{V}$, it is regarded as a polyhedron consisting of handles whose indices are not larger than $\dim{V}$. By dualizing the handles, $V$ is regarded as a PL manifold obtained by attaching handles whose indices are not smaller than $m-\dim{V}+1$ along the component $M \times \{1\}$ of $M \times [0,1]$. Hence, the homomorphism $i_{\ast}:{\pi}_{k}(M) \rightarrow {\pi}_{k}(W)$ ($0 \leq k \leq m-\dim{V}-1$) is an isomorphism and this completes the proof. \end{proof}

proofpile-arXiv_069-6751

\section{Introduction} The (adjacency) eigenvalues of a graph $G$ on $m$ vertices are defined as the eigenvalues of its adjacency matrix $A$. Since $A$ is a real symmetric matrix, its eigenvalues $\lambda_i(A)$ are real; we arrange them as follows \[ \lambda_1(A) \leqslant \lambda_2(A) \leqslant \cdots \leqslant \lambda_m(A). \] For convenience we will sometimes also refer to each $\lambda_i(A)$ as $\lambda_i(G)$. Much attention has been directed towards the study of graphs with smallest eigenvalue at least $-2$ \cite{BusseNeum,Doob:LineGraphsEigen73,DoobCvetkovic:LAA79,McKee:IntSymCyc07,ArsComb1993}. Most of this attention has centred around the beautiful theorem of Cameron, Goethals, Shult, and Seidel \cite{CGSS}, which classifies graphs having smallest eigenvalue at least $-2$. In the late 1970s Hoffman \cite{Hoffman:lt-2} studied graphs $G$ with $\lambda_1(G) \geqslant -1 - \sqrt{2}$ and later Woo and Neumaier \cite{Woo:ltHoff} furthered Hoffman's work, introducing the so-called Hoffman graph. Recently, Jang, Koolen, Munemasa, and Taniguchi \cite{-3} proposed a programme to classify fat Hoffman graphs with smallest eigenvalue at least $-3$. The present work fills a part of this programme, and includes the results of \cite{gt-3}. In this article we classify, up to switching equivalence, edge-signed graphs with smallest eigenvalue greater than $-2$. (See Section~\ref{sec:pre} for the definition of the eigenvalues of an edge-signed graph.) In particular, we recover as a special case the classification of graphs with smallest eigenvalue greater than $-2$ given earlier by Doob and Cvetkovi\'c \cite{DoobCvetkovic:LAA79}. As an application, we classify the special graphs of fat Hoffman graphs with smallest eigenvalue greater than $-3$. Some of such graphs are related to the modified adjacency matrix that appeared in a paper of Hoffman \cite{Hoffman:LimitLeastEig1977}. Below we describe the conjecture Hoffman made about these modified adjacency matrices. Let $T$ be a tree on $m \geqslant 2$ vertices with line graph $\operatorname{\mathfrak{L}}(T)$ and let $e$ be an end-edge of $T$ (one of whose vertices has valency $1$). Then $e$ is a vertex of $\operatorname{\mathfrak{L}}(T)$. For a graph $G$ and a vertex $v \in V(G)$, define $\hat{A}(G,v)$ to be the adjacency matrix of $G$, modified by putting a $-1$ in the diagonal position corresponding to $v$. In one of his papers \cite{Hoffman:LimitLeastEig1977}, Hoffman conjectured that $\lambda_{1}(\hat{A}(\operatorname{\mathfrak{L}}(T),e)) < \lambda_1(\operatorname{\mathfrak{L}}(T))$ for all trees $T$ and end-edges $e$. In Section~\ref{sec:Hoffcon} we prove this conjecture, which we record as the following theorem. \begin{theorem}\label{thm:Hoffman} Let $T$ be a tree and let $e$ be an end-edge of $T$. Then $\lambda_1(\hat{A}(\operatorname{\mathfrak{L}}(T),e)) < \lambda_1(\operatorname{\mathfrak{L}}(T))$. \end{theorem} Furthermore, using the classification of edge-signed graphs (see Theorem~\ref{thm:3}) with smallest eigenvalue greater than $-2$, we prove a generalised version of Hoffman's conjecture (see Theorem~\ref{thm:11}). In Section~\ref{sec:pre} we give our preliminaries. In Section~\ref{sec:class} we prove the main part of the classification theorem of edge-signed graphs with smallest eigenvalue greater than $-2$ leaving the exceptional case to Section~\ref{sec:ex.grph}. In Section~\ref{sec:Hoffcon} we prove Theorem~\ref{thm:Hoffman} and its generalised version, and in Section~\ref{sec:Hoffgraph} we comment on the application to Hoffman graphs with smallest eigenvalue greater than $-3$. \section{Edge-signed graphs and representations} \label{sec:pre} In this section we introduce some terminology that we use in our results. An \textbf{edge-signed graph} $G$ is a triple $(V, E^{+}, E^{-})$ of a set $V$ of vertices, a set $E^{+}$ of $2$-subsets of $V$ (called $(+)$-\textbf{edges}), and a set $E^{-}$ of $2$-subsets of $V$ (called $(-)$-\textbf{edges}) such that $E^{+} \cap E^{-} = \emptyset$. Let $G$ be an edge-signed graph. We denote the set of vertices of $G$ by $V(G)$, the set of $(+)$-edges of $G$ by $E^{+}(G)$, and the set of $(-)$-edges of $G$ by $E^{-}(G)$. By a \textbf{subgraph} $G' = (V(G'), E^{+}(G'), E^{-}(G'))$ of $G$ we mean a vertex induced edge-signed subgraph, i.e., $V(G') \subseteq V(G)$ and $E^{\pm}(G')=\{\{x,y\} \in E^{\pm}(G) \mid x,y \in V(G')\}$. The \textbf{underlying graph} $U(G)$ of $G$ is the unsigned graph $(V(G),E^+(G) \cup E^-(G))$. Two edge-signed graphs $G$ and $G'$ are said to be \textbf{isomorphic} if there exists a bijection $\phi: V(G) \to V(G')$ such that $\{u,v\} \in E^+(G)$ if and only if $\{\phi(u), \phi(v) \} \in E^+(G')$ and that $\{u,v\} \in E^-(G)$ if and only if $\{\phi(u), \phi(v) \} \in E^-(G')$. For an edge-signed graph $G$, we define its \textbf{signed adjacency matrix} $A(G)$ by \[ (A(G))_{uv}= \begin{cases} 1 & \text{if } \{u,v\} \in E^+(G), \\ -1 & \text{if } \{u,v\} \in E^-(G), \\ 0 & \text{otherwise.} \end{cases} \] To ease language, we will refer to the signed adjacency matrix simply as the adjacency matrix. The eigenvalues of $G$ are defined to be those of $A(G)$. A \textbf{switching} at vertex $v$ is the process of swapping the signs of each edge incident to $v$. Two edge-signed graphs $G$ and $G'$ are said to be \textbf{switching equivalent} if there exists a subset $W \subset V(G)$ such that $G^\prime$ is isomorphic to the graph obtained by switching at each vertex in $W$. Note that switching equivalence is an equivalence relation that preserves eigenvalues. Let $G$ be an edge-signed graph with smallest eigenvalue at least $-2$. A \textbf{representation} of $G$ is a mapping $\phi$ from $V(G)$ to $\mathbb{R}^n$ for some positive integer $n$, such that $(\phi(u),\phi(v)) = \pm 1$ if $\{u,v\} \in E^\pm (G)$ respectively, and $(\phi(u),\phi(v))=2\delta_{u,v}$ otherwise, where $\delta_{u,v}$ is Kronecker's delta, i.e., $\delta_{u,v}=1$ if $u=v$ and $\delta_{u,v}=0$ if $u \neq v$. Since, for $A$ the adjacency matrix of $G$, the matrix $A + 2I$ is positive semidefinite, $A + 2I$ is the Gram matrix of a set $S$ of vectors $\mathbf{x}_1, \dots, \mathbf{x}_m$. These vectors satisfy $(\mathbf{x}_i, \mathbf{x}_i) = 2$ and $(\mathbf{x}_i, \mathbf{x}_j) = 0,\pm 1$ for $i \ne j$. Sets of vectors satisfying these conditions determine line systems. We denote by $[\mathbf{x}]$ the line determined by a nonzero vector $\mathbf{x}$, in other words, $[\mathbf{x}]$ is the one-dimensional subspace spanned by $\mathbf{x}$. We say that $G$ is \textbf{represented by} the line system $S$ if $G$ has a representation $\phi$ such that $S = \{ [\phi(v)] : v \in V(G) \}$. Below we give descriptions of three line systems, $A_n$, $D_n$ and $E_8$. Let $\mathbf{e}_1, \dots, \mathbf{e}_n$ be an orthonormal basis for $\mathbb{R}^n$. \begin{align*} A_n &= \left \{ [ \mathbf{e}_i - \mathbf{e}_j ] : 1 \leqslant i < j \leqslant n+1 \right \}\quad(n\geqslant1), \\ D_n &= A_{n-1} \cup \left \{ [ \mathbf{e}_i + \mathbf{e}_j ] : 1 \leqslant i < j \leqslant n \right \}\quad(n\geqslant4),\\ E_8 &= D_8 \cup \left \{ [ \frac{1}{2} \sum_{i=1}^8 \epsilon_i\mathbf{e}_i ] : \epsilon_i = \pm 1, \prod_{i=1}^8 \epsilon_i = 1 \right \}. \end{align*} These line systems are used in the following classical result of Cameron, Goethals, Shult, and Seidel. \begin{theorem}[\cite{CGSS}] \label{thm:CGSS} Let $G$ be a connected edge-signed graph with $\lambda_1(G) \geqslant -2$. Then $G$ is represented by a subset of either $D_n$ or $E_8$. \end{theorem} Let $G$ be an edge-signed graph represented by a line system $S$. If $S$ embeds into $\mathbb{Z}^n$ for some $n$, then we say that $G$ is \textbf{integrally represented} or that $G$ has an \textbf{integral representation}. By Theorem~\ref{thm:CGSS}, for an edge-signed graph $G$ with $\lambda_1(G) \geqslant -2$, $G$ has an integral representation if and only if $G$ is represented by a subset of $D_n$ for some $n$, or equivalently, $G$ is represented by $D_\infty$, in the sense of \cite{EJC1987}. We record this observation as the following corollary. \begin{corollary}\label{cor:integralReps} Let $G$ be a connected edge-signed graph with $\lambda_1(G) \geqslant -2$. Then $G$ has no integral representation if and only if $G$ is represented by a subset of $E_8$ but not by a subset of $D_n$ for any $n$. \end{corollary} Corollary~\ref{cor:integralReps} motivates our next definition. Let $G$ be a connected edge-signed graph with $\lambda_1(G) \geqslant -2$. We call $G$ \textbf{exceptional} if it does not have an integral representation. Clearly there are only finitely many exceptional edge-signed graphs. \section{Classification of edge-signed graphs with $\lambda_1 > -2$} \label{sec:class} In this section we classify integrally represented edge-signed graphs with smallest eigenvalue greater than $-2$. We leave the exceptional case until Section~\ref{sec:ex.grph}. \begin{lemma}\label{lem:signed-cycle} Let $G$ be an edge-signed graph whose underlying graph is a cycle. Then $\lambda_{1}(G)>-2$ if and only if the number of $(+)$-edges of $G$ is odd. \end{lemma} \begin{proof} If the number of $(+)$-edge is even, then $G$ is switching equivalent to the edge-signed cycle in which all edges are $(-)$-edges, hence $\lambda_{1}(G)=-2$. Conversely, suppose $G$ has an odd number of $(+)$-edges. If the length of $G$ is odd, then $G$ has an even number of $(-)$-edges, hence $G$ is switching equivalent to an unsigned cycle. Thus $\lambda_{1}(G)>-2$. If the length of $G$ is even, then, up to switching, $A(G) = B^\top B - 2I$ where \[ B = \begin{pmatrix} 1&0&\cdots&0&-1\\ 1&1&\ddots&&0\\ 0&1&1&\ddots&\vdots\\ \vdots&&\ddots&\ddots&0\\ 0&\cdots&\cdots&1&1 \end{pmatrix}. \] Since this matrix is nonsingular, $\lambda_{1}(G)>-2$. \end{proof} Let $G$ be a graph with an orientation assigned to its edges. We define the \textbf{oriented incidence matrix} $B = B(G)$ of $G$ to be the $\{0, \pm 1 \}$-matrix whose rows and columns are indexed by $V(G)$ and $E(G)$ respectively such that the entry $B_{ve}$ is equal to $1$ if $v$ is the head of $e$, $-1$ if $v$ is the tail of $e$, and $0$ otherwise. See \cite{Godsil:AlgGraph} for properties of oriented incidence matrices. Let $G$ be an edge-signed graph with smallest eigenvalue greater than $-2$. Assume that $G$ has an integral representation $\phi$. This means that, with $m=|V(G)|$, there exists an $n\times m$ matrix \[ M=\begin{pmatrix} \mathbf{v}_1&\cdots&\mathbf{v}_m\end{pmatrix}, \] with entries in $\mathbb{Z}$, such that $(\mathbf{v}_i,\mathbf{v}_j) = \pm 1$ if $\{i,j\} \in E^\pm (G)$ respectively, and $(\mathbf{v}_i,\mathbf{v}_j)= 2\delta_{i,j}$ otherwise. We may assume that $M$ has no rows consisting only of zeros. Since $\mathbf{v}_i\in\mathbb{Z}^n$, $\mathbf{v}_i$ has two entries equal to $\pm1$, and all other entries $0$. This means that we can regard $M$ as an oriented incidence matrix of a graph $H$ and the underlying graph of $G$ is the line graph of $H$. More precisely, $H$ is a graph with $n$ vertices, and the vertices $i$ and $j$ are joined by the edge $k$ whenever $\{i,j\}=\supp(\mathbf{v}_k)$. Note that the graph $H$ may have multiple edges. A graph without multiple edges is called \textbf{simple}. We call $H$ the \textbf{representation graph} of $G$ associated with the representation $\phi$. Note that $H$ has no isolated vertex. If $G$ is connected, then so is $H$. \begin{lemma}\label{lem:1} Let $G$ be an $m$-vertex connected edge-signed graph having an integral representation $\phi$ and smallest eigenvalue greater than $-2$. Let $H$ be the $n$-vertex representation graph of $G$ associated with the representation $\phi$. Then $n\in\{m,m+1\}$. Moreover, if $n=m$, then $H$ is a unicyclic graph or a tree with a double edge and if $n=m+1$, then $H$ is a tree. \end{lemma} \begin{proof} Since $M^\top M=A(G)+2I$ is positive definite, $M$ has rank $m$. This implies $n\geqslant m$. If $H$ is disconnected, then so is $G$, which is absurd. Thus $H$ is connected, which forces $n\leqslant m+1$. \end{proof} Let $H$ be a unicyclic graph whose unique cycle $C$ has at least $4$ vertices and let $G = \operatorname{\mathfrak{L}}(H)$. Then for each edge $e$ of $G$ there exists a unique maximal clique that contains $e$. For such a graph $G$, we denote by $\mathfrak C_G(e)$ the unique maximal clique of $G$ containing the edge $e$. Let $uu'$ be an edge of $\operatorname{\mathfrak{L}}(C)$. Define $\operatorname{\mathfrak{L}}^\dag(H,uu^\prime)$ to be the edge-signed graph $(V,E^{+},E^-)$, where $V = V(\operatorname{\mathfrak{L}}(H))$, \[ E^- = \left \{ uv \in E(\operatorname{\mathfrak{L}}(H)) \mid v \in \mathfrak C_G(uu^\prime) \right \} \] and $E^+ = E(\operatorname{\mathfrak{L}}(H)) \backslash E^-$. Observe that, for all edges $uu^\prime$ and $vv^\prime$ of $\operatorname{\mathfrak{L}}(C)$, the graph $\operatorname{\mathfrak{L}}^\dag(H,uu^\prime)$ is switching equivalent to $\operatorname{\mathfrak{L}}^\dag(H,vv^\prime)$. \begin{theorem}\label{thm:3} Let $G$ be a connected integrally represented edge-signed graph having smallest eigenvalue greater than $-2$. Let $H$ be the representation graph of $G$ for some integral representation. Then one of the following statements holds: \begin{enumerate} \item $H$ is a simple tree or $H$ is unicyclic with an odd cycle, and $G$ is switching equivalent to the line graph $\operatorname{\mathfrak{L}}(H)$, \item $H$ is unicyclic with an even cycle $C$, and $G$ is switching equivalent to $\operatorname{\mathfrak{L}}^\dag(H,uu^\prime)$ where $uu'$ is an edge of $\operatorname{\mathfrak{L}}(C)$. \item $H$ is a tree with a double edge, and $G$ is switching equivalent to the edge-signed graph obtained from the line graph $\operatorname{\mathfrak{L}}(H)$, by attaching a new vertex $u'$, and join $u'$ by $(+)$-edges to every vertex of a clique in the neighbourhood of $u$, $(-)$-edges to every vertex of the other clique in the neighbourhood of $u$, where $u$ is a fixed vertex of $\operatorname{\mathfrak{L}}(H)$. \end{enumerate} Conversely, if $G$ is an edge-signed graph described by {\rm(i)}, {\rm(ii)}, or {\rm(iii)} above, then $G$ is integrally represented and has smallest eigenvalue greater than $-2$. \end{theorem} \begin{proof} By Lemma~\ref{lem:1}, we can divide the proof into three cases. \paragraph{Case 1: $H$ is a simple tree} \label{par:case_1_h_is_a_tree} Since $G$ has smallest eigenvalue greater than $-2$, $G$ cannot contain a triangle switching equivalent to one with three $(-)$-edges. By repeatedly applying switching, one can move the locations of $(-)$-edges toward an end block, and eventually end up with an unsigned graph. Therefore, $G$ is switching equivalent to $\operatorname{\mathfrak{L}}(H)$. \paragraph{Case 2: $H$ is unicyclic} \label{par:case_2_h_is_unicyclic} We prove either (i) or (ii) holds by induction on the number of vertices of $H$ minus the length of the cycle in $H$. First suppose that $H$ is a cycle. By Lemma~\ref{lem:signed-cycle}, $G$ is either an odd cycle with an even number of $(-)$-edges, or $G$ is an even cycle with odd number of $(-)$-edges. In the former case, $G$ is switching equivalent to an unsigned odd cycle. In the latter case, $G$ is switching equivalent to an even cycle with one $(-)$-edge and the clique of the underlying graph of $G$ containing the $(-)$-edge is then switching equivalent to the one described in (ii). Now suppose that $H$ is not a cycle. Then $H$ has a vertex $v$ of degree one, adjacent to a vertex $w$. Let $\phi$ be the integral representation of $G$ to which $H$ is associated. Let $H'$ be the graph obtained from $H$ by removing the vertex $v$, and let $G^\prime$ be the graph obtained from $G$ by removing the vertex $x$ corresponding to the edge $vw$ in $H$. Clearly $H^\prime$ is the representation graph of $G^\prime$ associated to $\phi$ restricted to $G^\prime$ and, by induction, $H^\prime$ and $G^\prime$ satisfy either (i) or (ii). If $H^\prime$ and $G^\prime$ satisfy (i) then, since $H$ does not have any double edges, each nonzero entry $M_{w,y}$ (for $y \in V(G^\prime)$) has the same sign. On the other hand, if $H^\prime$ and $G^\prime$ satisfy (ii), that is, $G^\prime$ is switching equivalent to $\operatorname{\mathfrak{L}}^\dag(H^\prime,uu^\prime)$ then, without loss of generality we can assume that $w$ is not incident to the edge $u$ of $H$. Now, again, since $H$ does not have any double edges, we observe that each nonzero entry $M_{w,y}$ (for $y \in V(G^\prime)$) has the same sign. Since $v$ has degree one, we are free to switch the vertex $x$ so that either (i) or (ii) holds for $G$ and $H$. \paragraph{Case 3: $H$ is a tree with a double edge} \label{par:case_3_h_has_a_double_edge} If $H$ has a double edge, then the matrix $M$ has a submatrix \[ \begin{pmatrix} 1&1\\ -1&1\end{pmatrix}. \] Let $u'$ (resp. $u$) denote the vertex of $G$ corresponding to the first (resp. second) column of this matrix (which in turn corresponds to a column of $M$), and let $v^+$ (resp. $v^-$) denote the vertex of $H$ corresponding to the first (resp. second) row of this matrix (which in turn corresponds to a row of $M$). Let $G'=G-u'$. Then the graph $H'$ obtained from $G'$ is a tree. We have already shown that, in this case, we may take $G'$ to be the unsigned line graph of $H'$. Let $K^+$ (resp. $K^-$) be the clique of $G$ in the neighbourhood of $u$ consisting of vertices $u''$ with $M_{v^+,u''}=1$ (resp. $M_{v^-,u''}=1$). Then in the graph $G$, $u'$ is joined to every vertex of $K^+$ by $(+)$-edges, and $u'$ is joined to every vertex of $K^-$ by $(-)$-edges. Therefore (iii) holds. Conversely, suppose $G$ is described by (i), (ii), or (iii). First, we describe how to construct $M$ for each case. For (i), $M$ is the incidence matrix of $H$. For (ii), let $v$ be the vertex incident to both the edges $u$ and $u^\prime$ of $H$. Then $M$ is the incidence matrix of $H$ adjusted so that $M_{v,u}=-1$. For (iii), let $v$ and $w$ be incident to the edge $u$ in $H$. Then $M$ is the incidence matrix of $H$ together with an extra column for $u^\prime$ with $M_{v,u^\prime}=1$, $M_{w,u^\prime}=-1$, and the remaining entries $0$. To prove the converse, it suffices to show that, in each case, $M^\top M$ is positive definite. If $G$ is the line graph of a tree then this is well known. Thus we can immediately restrict our attention to when $n = m$. We will show that, in both remaining cases, $M^\top M$ has determinant $4$. We inductively show that $\det(M^\top M) = 4$ for $H$ unicyclic. Suppose that the underlying graph of $G$ is the line graph of a unicyclic graph. If $H$ is a cycle then, by Lemma~\ref{lem:signed-cycle}, $M$ is nonsingular. Hence the rows of $M$ are a basis for $D_n$, which has discriminant $4$. Thus $M$ has determinant $\pm 2$. Otherwise, $H$ has a vertex $v$ of degree $1$. Let $M^\prime$ be a the matrix obtained by removing $v$. Then $\det(M) = \pm\det(M^\prime)$. Hence $M^\top M$ is positive definite, as required. The same inductive approach can be applied when starting with the double-edge where $M$ is the matrix \[ \begin{pmatrix} 1&1\\ -1&1\end{pmatrix}, \] which has determinant $2$. \end{proof} Note that, if $G$ is represented by the line system $A_n$, then one can relax the assumption of Theorem~\ref{thm:3} to having smallest eigenvalue \emph{at least} $-2$. Ishihara~\cite{Ishihara} shows that, in this case, the underlying graph of $G$ is a claw-free block graph. \section{Hoffman's conjecture} \label{sec:Hoffcon} In this section we settle Hoffman's conjecture, i.e., we prove Theorem~\ref{thm:Hoffman}. Moreover, we prove a stronger version of Hoffman's conjecture extended to edge-signed graphs. \begin{lemma}\label{lem:v1zero} Let $A=(a_{i,j})$ be a real symmetric matrix, and let $A'=(a'_{i,j})$ be the matrix defined by $a'_{i,j}=a_{i,j}-\delta_{i,1}\delta_{j,1}$. Suppose there exists an eigenvector $\mathbf{x}$ of $A$ belonging to the eigenvalue $\lambda_1(A)$ with $x_1\neq0$. Then $\lambda_1(A) > \lambda_1(A^\prime)$. \end{lemma} \begin{proof} We may assume without loss of generality that $\norm{\mathbf{x}} = 1$. Then $\lambda_1(A) = \mathbf{x}^\top A \mathbf{x}= \mathbf{x}^\top A^\prime \mathbf{x} + x_1^2 \geqslant \lambda_1(A^\prime) + x_1^2>\lambda_1(A^\prime)$. \end{proof} \begin{remark} One might wonder if Theorem~\ref{thm:Hoffman} can be proved by showing that the adjacency matrix of $\operatorname{\mathfrak{L}}(T)$ satisfies the assumption of Lemma~\ref{lem:v1zero} when we take the first entry to correspond to an end-edge of a tree $T$. This approach, however, does not work. In fact, let $T$ be the Dynkin diagram $E_6$, and let the first entry of $A$ correspond to the unique end edge attached to the vertex of degree $3$. Then the smallest eigenvalue $-(\sqrt{5}+1)/2$ of $\operatorname{\mathfrak{L}}(T)$ has multiplicity $1$, and the eigenvector has $0$ in its first entry. Hence, $E_6$ is an example of a graph to which we cannot apply Lemma~\ref{lem:v1zero}. \end{remark} \begin{lemma}\label{lem:LGtoL} Let $M$ be an $n \times m$ real matrix. Then $M^\top M$ and $MM^\top$ have the same nonzero eigenvalues (including multiplicities). More explicitly, for any nonzero eigenvalue $\theta$ of $M^\top M$, the multiplication by $M$ gives a linear map from $\operatorname{ker}(M^\top M-\theta I)$ to $\operatorname{ker}(MM^\top-\theta I)$ whose inverse is given by $\mathbf{v}\mapsto\theta^{-1}M^\top\mathbf{v}$. \end{lemma} \begin{proof} Follows from \cite[Lemma 2.9.2]{Brouwer:SpectraGraphs}. \end{proof} It is easy to see that if $G$ is a bipartite graph then there exists an orientation of $G$ such that the oriented incidence matrix $B$ of $G$ satisfies $B^\top B = A(\operatorname{\mathfrak{L}}(G)) + 2I$. \begin{lemma}\label{lem:LT} Let $G$ be a connected bipartite graph on $m$ vertices and $n$ edges, with oriented incidence matrix $B$ satisfying $B^\top B=A+2I$ where $A$ is the adjacency matrix of the line graph of $G$, and let $L$ be the Laplacian matrix of $G$. Then for each $i\in\{2,\dots,m\}$, $B\operatorname{ker}(A-\lambda_{i+n-m}(A)I)=\operatorname{ker}(L-\lambda_{i}(L) I)$ and $\operatorname{ker}(A-\lambda_{i+n-m}(A)I)=B^\top\operatorname{ker}(L-\lambda_{i}(L) I)$. \end{lemma} \begin{proof} Since $G$ is connected, the multiplicity of $0$ as an eigenvalue of $L$ is $1$. Since $B^\top B=A+2I$ and $BB^\top=L$, Lemma~\ref{lem:LGtoL} implies that $\lambda_{i}(L)=\lambda_{i+n-m}(A+2I) =\lambda_{i+n-m}(A)+2$ for $1<i\leq m$. Moreover, Lemma~\ref{lem:LGtoL} implies $B\operatorname{ker}(B^\top B-\lambda_{i+n-m}(B^\top B)I) =\operatorname{ker}(L-\lambda_{i}(L) I)$ and $\operatorname{ker}(B^\top B-\lambda_{i+n-m}(B^\top B)I) =B^\top\operatorname{ker}(L-\lambda_{i}(L) I)$. Since $B^\top B-\lambda_{i+n-m}(B^\top B)I=A-\lambda_{i+n-m}(A)I$, we obtain the desired result. \end{proof} Let $G$ be a graph and let $v$ be a vertex of $G$. Recall that $\hat{A}(G,v)$ is the adjacency matrix of $G$, modified by putting a $-1$ in the diagonal position corresponding to $v$. \begin{lemma}[{\cite[Lemma 2.1]{Hoffman:LimitLeastEig1977}}]\label{lem:gtminus22} Let $T$ be a tree and let $e$ be an end-edge of $T$. Then $\lambda_1(\hat{A}(\operatorname{\mathfrak{L}}(T),e))>-2$. \end{lemma} We are now ready to prove Theorem~\ref{thm:Hoffman}. \begin{proof}[Proof of Theorem~\ref{thm:Hoffman}] Let $T$ be a tree on $n+1$ vertices and $n$ edges, and let $A = (a_{i,j})$ denote the adjacency matrix of the line graph $\operatorname{\mathfrak{L}}(T)$ of $T$. Since $T$ is bipartite, one can orient its edges so that its oriented incidence matrix $B = (b_{i,j})$ satisfies \[ B^\top B = A + 2I. \] We also have $BB^\top = L$, the Laplacian matrix of $T$. Let $r$ and $s$ be the vertices of the end-edge $e$, and assume $r$ has valency $1$ in $T$. We may choose $B$ so that the first row and column correspond to $r$ and $e$ respectively, and the second row corresponds to $s$. Let the column vectors $\mathbf{e}_i$ (resp. $\mathbf{f}_i$) be the canonical bases of the Euclidean spaces $\mathbb{R}^{n}$ (resp. $\mathbb{R}^{n+1}$). Without loss of generality, we assume $B \mathbf{e}_1 = \mathbf{f}_1 - \mathbf{f}_2$ and $B^\top \mathbf{f}_1 = \mathbf{e}_1$. We obtain the matrix $C$ from $B$ by setting $b_{1,1} = 0$. Define matrices $A^\prime$ and $L^\prime$ by \[ C^\top C = A^\prime + 2I, \text{ and } CC^\top = L^\prime. \] Then $A^\prime$ can be obtained from $A$ by setting $a_{1,1}=-1$, that is, $A^\prime=\hat{A}(\operatorname{\mathfrak{L}}(T),e)$. The matrix $L^\prime$ can be obtained from $L$ by setting all entries of the first row and column to zero. By Lemma~\ref{lem:gtminus22}, $C^\top C$ is positive definite. It then follows from Lemma~\ref{lem:LGtoL} that $L'$ is a positive semidefinite $(n+1)\times (n+1)$ matrix with rank $n$. Let $X$ be the principal submatrix of $L'$ obtained by removing the first row and column of $L'$. Since the matrix $L'$ has only zeros in its first row and column, the matrix $X$ is positive definite. Moreover, $X$ is an M-matrix, that is, in addition to being positive definite, its off-diagonal entries are non-positive. By \cite[Theorem 2.5.3]{HornJohnson}, $X^{-1}$ is a non-negative matrix. By the Perron-Frobenius theorem (see, for example, \cite{Godsil:AlgGraph}), any eigenvector corresponding to the smallest eigenvalue of $X$ has no zero entry. By Lemma~\ref{lem:LGtoL}, $\lambda_1(A'+2I)=\lambda_2(L')=\lambda_1(X)$, and $\lambda_1(A+2I)=\lambda_2(L)$. Thus, to prove Theorem~\ref{thm:Hoffman}, it suffices to show that $\lambda_1(X) < \lambda_2(L)$. By Lemma~\ref{lem:v1zero}, we can assume $\operatorname{ker} (A-\lambda_1(A)I) \subset \mathbf{f}_1^\perp$. Let $\mathbf{w}$ be an eigenvector of $L$ belonging to the eigenvalue $\lambda_2(L)$. Then \begin{align*} B^\top\mathbf{w} &\in B^\top\operatorname{ker}(L-\lambda_2(L)I) \\&=\operatorname{ker}(A-\lambda_1(A)I) &&\text{(by Lemma~\ref{lem:LT})} \\&\subset\mathbf{e}_1^\perp. \end{align*} Thus $\mathbf{w}\in (B\mathbf{e}_1)^\perp=(\mathbf{f}_1-\mathbf{f}_2)^\perp$. Therefore $w_1 = w_2$. On the other hand, again by Lemma~\ref{lem:LT}, the eigenvector $\mathbf{w}$ can be written as $B \mathbf{v}$ where $\mathbf{v}$ is in $\operatorname{ker} (A-\lambda_1(A)I) \subset \mathbf{e}_1^\perp$. Then it follows that $w_1=\mathbf{f}_1^\top B\mathbf{v}=\mathbf{e}_1^\top\mathbf{v} =0$. Hence $w_1 = w_2 = 0$. Since the first row of $L$ can be written as $\mathbf{f}_1^\top - \mathbf{f}_2^\top$, we have $$ L \mathbf{w} = \begin{pmatrix} 0 \\ X \mathbf{y} \end{pmatrix},$$ where $\mathbf{w}^\top = (0, \mathbf{y}^\top)$. Hence, $\mathbf{w}$ restricts to an eigenvector $\mathbf{y}$ of $X$. But the first entry of $\mathbf{y}$ is zero. Since $X$ is an M-matrix, $\lambda_2(L)$ is not the smallest eigenvalue of $X$. This implies $\lambda_1(X) < \lambda_2(L)$. \end{proof} \begin{figure}[htbp] \centering \begin{tikzpicture}[yshift=4cm, xshift=0.8cm] \foreach \type/\pos/\name in {{vertex/(0,0.5)/bgn}, {vertex/(1,0.5)/b1}, {vertex/(2,0.5)/e1}, {empty/(2.6,0.5)/b11}, {empty/(3.4,0.5)/c11}, {vertex/(4,0.5)/c1}, {vertex/(5,0.5)/d1}, {vertex/(5.7,1)/f1}, {vertex/(5.7,0)/f2}} \node[\type] (\name) at \pos {}; \foreach \pos/\name in {{(0,0.1)/e_n}, {(1,0.1)/e_{n-1}}, {(2,0.1)/e_{n-2}}, {(4,0.1)/e_4}, {(5,0.1)/e_3}, {(5.7,-0.3)/e_2}, {(5.7,1.3)/e_1}} \node at \pos {$\name$}; \foreach \pos/\name in {{(3,0.5)/\dots}} \node at \pos {$\name$}; \foreach \edgetype/\source/ \dest in {pedge/b1/e1, pedge/e1/b11, pedge/c11/c1, pedge/c1/d1, pedge/d1/f1, pedge/d1/f2} \path[\edgetype] (\source) -- (\dest); \foreach \edgetype/\source/\dest in {pedge/b1/bgn} \path[\edgetype] (\source) -- (\dest); \end{tikzpicture} \caption{The graph $\mathcal X_n^{(1)} (n \geqslant 3)$} \label{fig:x1} \end{figure} \begin{figure}[htbp] \centering \begin{tikzpicture} \begin{scope}[xshift=6.5cm, yshift=0.5cm] \newdimen\rad \rad=1cm \newdimen\radi \radi=1.04cm \newdimen\radii \radii=1.44cm \draw (365.5:\radi) node[empty] {$\vdots$}; \foreach \y/\lab in {70/e_{2k-1},170/e_3,210/e_2,250/e_1,350/e_{2k+1},390/e_{2k}} { \def\y - 120{\y - 120} \draw (\y - 120:\radii) node[empty] {$\lab$}; } \foreach \y in {90,150,210,270,330,390} { \def\y - 120{\y - 120} \draw (\y - 120:\rad) node[vertex] {}; } \foreach \y in {210,270,330,390} { \def\y - 120{\y - 120} \draw[pedge] (\y - 120:\rad) arc (\y - 120:\y - 120+60:\rad); } \draw[pedge] (30:\rad) arc (30:90:\rad); \draw (210:\rad) node[vertex] {}; \draw[pedge] (330:\rad) arc (330:340:\rad); \draw[pedge] (380:\rad) arc (380:390:\rad); \node (x) at (140:\rad) {}; \node (y) at (220:\rad) {}; \end{scope} \begin{scope} \foreach \type/\pos/\name in {{vertex/(0,0.5)/bgn}, {vertex/(1,0.5)/b1}, {vertex/(2,0.5)/e1}, {empty/(2.6,0.5)/b11}, {empty/(3.4,0.5)/c11}, {vertex/(4,0.5)/c1}, {vertex/(5,0.5)/d1}} \node[\type] (\name) at \pos {}; \foreach \pos/\name in {{(0,0.1)/f_l}, {(1,0.1)/f_{l-1}}, {(2,0.1)/f_{l-2}}, {(4,0.1)/f_2}, {(5,0.1)/f_1}} \node at \pos {$\name$}; \foreach \pos/\name in {{(3,0.5)/\dots}} \node at \pos {$\name$}; \foreach \edgetype/\source/ \dest in {pedge/b1/e1, pedge/e1/b11, pedge/c11/c1, pedge/c1/d1, pedge/d1/x, pedge/d1/y} \path[\edgetype] (\source) -- (\dest); \foreach \edgetype/\source/\dest in {pedge/b1/bgn} \path[\edgetype] (\source) -- (\dest); \end{scope} \end{tikzpicture} \caption{The graph $\mathcal X_{k,l}^{(2)} (k,l \geqslant 1)$} \label{fig:x2} \end{figure} \begin{figure}[htbp] \centering \begin{tikzpicture}[yshift=-5cm] \begin{scope}[xshift=6.5cm, yshift=0.5cm] \newdimen\rad \rad=1cm \newdimen\radi \radi=1.04cm \newdimen\radii \radii=1.44cm \draw (365.5:\radi) node[empty] {$\vdots$}; \foreach \y/\lab in {70/e_{2k},170/e_3,210/e_2,250/e_1,350/e_{2k+2},390/e_{2k+1}} { \def\y - 120{\y - 120} \draw (\y - 120:\radii) node[empty] {$\lab$}; } \foreach \y in {90,150,210,270,330,390} { \def\y - 120{\y - 120} \draw (\y - 120:\rad) node[vertex] {}; } \foreach \y in {150,210,330,390} { \def\y - 120{\y - 120} \draw[pedge] (\y - 120:\rad) arc (\y - 120:\y - 120+60:\rad); } \draw[nedge] (150:\rad) arc (150:210:\rad); \draw (210:\rad) node[vertex] {}; \draw[pedge] (330:\rad) arc (330:340:\rad); \draw[pedge] (380:\rad) arc (380:390:\rad); \node (x) at (140:\rad) {}; \node (y) at (220:\rad) {}; \end{scope} \begin{scope} \foreach \type/\pos/\name in {{vertex/(0,0.5)/bgn}, {vertex/(1,0.5)/b1}, {vertex/(2,0.5)/e1}, {empty/(2.6,0.5)/b11}, {empty/(3.4,0.5)/c11}, {vertex/(4,0.5)/c1}, {vertex/(5,0.5)/d1}} \node[\type] (\name) at \pos {}; \foreach \pos/\name in {{(0,0.1)/f_l}, {(1,0.1)/f_{l-1}}, {(2,0.1)/f_{l-2}}, {(4,0.1)/f_2}, {(5,0.1)/f_1}} \node at \pos {$\name$}; \foreach \pos/\name in {{(3,0.5)/\dots}} \node at \pos {$\name$}; \foreach \edgetype/\source/ \dest in {pedge/b1/e1, pedge/e1/b11, pedge/c11/c1, pedge/c1/d1, nedge/d1/x, pedge/d1/y} \path[\edgetype] (\source) -- (\dest); \foreach \edgetype/\source/\dest in {pedge/b1/bgn} \path[\edgetype] (\source) -- (\dest); \end{scope} \end{tikzpicture} \caption{The graph $\mathcal X_{k,l}^{(3)} (k,l \geqslant 1)$} \label{fig:x3} \end{figure} \begin{lemma}\label{lem:-2eig} The matrices $\hat{A}(\mathcal{X}_n^{(1)},e_n)$, $\hat{A}(\mathcal{X}_{k,l}^{(2)},f_l)$, and $\hat{A}(\mathcal{X}_{k,l}^{(3)},f_l)$ (see Figures~\ref{fig:x1}, \ref{fig:x2}, and \ref{fig:x3}) have smallest eigenvalue $-2$. \end{lemma} \begin{proof} We give the eigenvectors corresponding to the eigenvalue $-2$ of each matrix in the statement of the lemma. \begin{itemize} \item $\hat{A}(\mathcal{X}_n^{(1)},e_n)$: set $e_1, e_2 = -1$, and $e_j = (-1)^{j+1}\cdot 2$ for $j \in \{3,\dots, n\}$. \item $\hat{A}(\mathcal{X}_{k,l}^{(2)},f_l)$: set $e_j = (-1)^{j+1}$ for $j \in \{1,\dots, 2k+1\}$ and set $f_j = (-1)^j\cdot 2$ for $j \in \{1,\dots, l\}$. \item $\hat{A}(\mathcal{X}_{k,l}^{(3)},f_l)$: set $e_j = (-1)^{j}$ for $j \in \{1,\dots, 2k+2\}$ and set $f_j = (-1)^j\cdot 2$ for $j \in \{1,\dots, l\}$. \end{itemize} Deleting the row and column containing the $-1$ on the diagonal, we obtain the adjacency matrix of a graph with smallest eigenvalue greater than $-2$. This is immediate for $\mathcal X_n^{(1)}$ since the obtained graph has spectral radius less than $2$. As for $\mathcal{X}_{k,l}^{(2)}$ and $\mathcal{X}_{k,l}^{(3)}$, the result follows from (i) and (ii), respectively, of Theorem~\ref{thm:3}. By interlacing, $\hat{A}(\mathcal{X}_n^{(1)},e_n)$, $\hat{A}(\mathcal{X}_{k,l}^{(2)},f_l)$, and $\hat{A}(\mathcal{X}_{k,l}^{(3)},f_l)$ have at most one eigenvalue less than or equal to $-2$. This implies that $-2$ is indeed the smallest eigenvalue. \end{proof} \begin{theorem}\label{thm:E8} Let $G$ be a connected edge-signed graph and let $v\in V(G)$ such that $\lambda_{1}(\hat{A}(G,v)) \geqslant -2$. Then $G$ is integrally represented. \end{theorem} \begin{proof} Suppose that $\hat{A}(G,v) + 2I$ is positive semidefinite. Then we can write $\hat{A}(G,v) + 2I = U^\top U$ for some matrix $U$. Label the columns of $U$ as $\mathbf{u}_1, \dots, \mathbf{u}_n$ where $\norm{\mathbf{u}_1} = 1$ and $\norm{\mathbf{u}_i}^2 = 2$ for $i \in \{2,\dots, n\}$. Let $\Lambda = \bigoplus_{i=1}^{n}\mathbb{Z} \mathbf{u}_i$ and let $B = \{ \mathbf{v} \in \Lambda \mid \norm{\mathbf{v}} =1 \}$. Clearly $B = \{\pm \mathbf{v}_1, \dots, \pm \mathbf{v}_m\}$ for some $m$ with $(\mathbf{v}_i,\mathbf{v}_j) = \delta_{ij}$. Define $\Lambda^\prime$ as the $\mathbb{Z}$-span of the vectors of $B$ and set $X = \Lambda \cap (\Lambda^\prime)^\perp$. It is easily checked that a vector $\mathbf{v} \in \Lambda$ with $\norm{\mathbf{v}}^2 = 2$ and $\mathbf{v} \not \in \Lambda^\prime$ is orthogonal to $\Lambda^\prime$. Hence we can write $\Lambda$ as the orthogonal sum $\Lambda = \Lambda^\prime \perp X$ and so $\mathbf{v} \in X$. Unless either $\Lambda^\prime = 0$ or $X = 0$, this orthogonal decomposition of $\Lambda$ violates our assumption that $G$ is connected. Since $\mathbf{u}_1 \in \Lambda^\prime$, we must have $X = 0$. Therefore $\Lambda = \Lambda^\prime \cong \mathbb{Z}^m$. Finally, the vectors \[ \begin{pmatrix} 1 \\ \mathbf{u}_1, \end{pmatrix}, \begin{pmatrix} 0 \\ \mathbf{u}_2, \end{pmatrix}, \dots, \begin{pmatrix} 0 \\ \mathbf{u}_n \end{pmatrix} \] all have norm $\sqrt{2}$ and their Gram matrix gives $A + 2I$. Clearly these vectors are contained in a $\mathbb{Z}$-lattice and this lattice represents $G$. \end{proof} \begin{remark} The proof of Theorem~\ref{thm:E8} is essentially the same as that of \cite[Theorem 3.7]{-3}. \end{remark} \begin{theorem}\label{thm:11} Let $G$ be an edge-signed graph with $\lambda_{1}(G) > -2$, and let $v\in V(G)$. Then $\lambda_{1}(G)>\lambda_{1}(\hat{A}(G,v))$. Furthermore, $\lambda_{1}(\hat{A}(G,v))>-2$ if and only if the underlying graph of $G$ is the line graph of a tree $T$ and $v$ corresponds to an end-edge of $T$. Otherwise, $\lambda_{1}(\hat{A}(G,v)) \leqslant -2$. \end{theorem} \begin{proof} By Theorem~\ref{thm:CGSS}, $G$ is represented by $D_n$ or $E_8$. We may assume that $G$ is represented by $D_n$, otherwise by Corollary~\ref{cor:integralReps} and Theorem~\ref{thm:E8} we would have $\lambda_{1}(\hat{A}(G,v)) < -2$ in which case the theorem holds. Therefore the structure of $G$ can be described by Theorem~\ref{thm:3}. First suppose $G$ is of type (i) from Theorem~\ref{thm:3}. Suppose $G$ is the line graph of a tree $T$. If $v$ is not an end-edge of $T$ then $\hat{A}(G,v)$ contains $\hat{A}(\mathcal X_3^{(1)},e_3)$ as a principal submatrix, hence $\lambda_{1}(\hat{A}(G,v))\leqslant-2$. If $v$ is an end-edge of $T$, then $\lambda_{1}(G)>\lambda_{1}(\hat{A}(G,v))$ by Theorem~\ref{thm:Hoffman}, and $\lambda_{1}(\hat{A}(G,v))>-2$ by Lemma~\ref{lem:v1zero}. Next suppose $G$ is of type (i) but not the line graph of a tree $T$. That is, $G$ is the line graph of a unicyclic graph with an odd cycle (and $G$ is not equal to $C_3$). Then $\hat{A}(G,v)$ contains (as a principal submatrix) either $\hat{A}(\mathcal X_3^{(1)},e_3)$ or $\hat{A}(\mathcal X_{k,l}^{(2)},f_l)$ for some $k$ and $l$. Suppose $G$ is of type (ii). Then $\hat{A}(G,v)$ contains (as a principal submatrix) either $\hat{A}(\mathcal X_3^{(1)},e_3)$ or $\hat{A}(\mathcal X_{k,l}^{(3)},f_l)$ for some $k$ and $l$. Suppose $G$ is of type (iii). Then $G$ contains $\mathcal X_n^{(1)}$ for some $n$. Therefore, by Lemma~\ref{lem:gtminus22}, we have $\lambda_{1}(\hat{A}(G,v)) \leqslant -2$ for these cases. \end{proof} \begin{remark} A special case of Theorem~\ref{thm:11} for unsigned graphs is given in \cite[Theorem 5.2]{gt-3}. \end{remark} \section{Exceptional graphs}\label{sec:ex.grph} In this section we enumerate the exceptional edge-signed graphs with smallest eigenvalue greater than $-2$, i.e., those that are not integrally represented. In the tables in the appendix we list (up to switching) every such edge-signed graph. We generalise the following result about graphs with smallest eigenvalue greater than $-2$. \begin{theorem}[\cite{BusseNeum,DoobCvetkovic:LAA79}]\label{thm:CvetDoob} Let $G$ be an exceptional graph having smallest eigenvalue greater than $-2$. Then $G$ is one of \begin{enumerate} \item $20$ graphs on $6$ vertices; \item $110$ graphs on $7$ vertices; \item $443$ graphs on $8$ vertices. \end{enumerate} \end{theorem} \begin{figure} \begin{center} \begin{tikzpicture}[yshift=4cm, xshift=0.8cm] \foreach \type/\pos/\name in {{vertex/(0,0.5)/bgn}, {vertex/(1,0.5)/b1}, {vertex/(2,0.5)/e1}, {vertex/(3,0.5)/b11}, {vertex/(4,0.5)/c11}, {vertex/(4,1.5)/c1}, {vertex/(5,0.5)/d1}, {vertex/(6,0.5)/f1}} \node[\type] (\name) at \pos {}; \foreach \pos/\name in {{(0,0.1)/\alpha_8}, {(1,0.1)/\alpha_7}, {(2,0.1)/\alpha_6}, {(3,0.1)/\alpha_5}, {(4,0.1)/\alpha_4}, {(4,1.8)/\alpha_2}, {(5,0.1)/\alpha_3}, {(6,0.1)/\alpha_1}} \node at \pos {$\name$}; \foreach \edgetype/\source/ \dest in {pedge/bgn/b1, pedge/b1/e1, pedge/e1/b11, pedge/b11/c11, pedge/c11/c1, pedge/c11/d1, pedge/d1/f1} \path[\edgetype] (\source) -- (\dest); \end{tikzpicture} \end{center} \caption{The simple roots of $E_8$} \label{fig:E8} \end{figure} To describe our results, we need a list of $120$ lines of the root system $E_8$. Such a list can be found in \cite[Appendix B]{Gleitz:KNSconj}, and this is also sufficient to describe our results for $E_6$ and $E_7$, since these root systems can be embedded in $E_8$. Each of the $120$ lines are determined by a vector $\beta=\sum_{i=1}^8 b_i\alpha_i$ and the coefficients $(b_1,\dots,b_8)$ for each $\beta$ are given in \cite[Appendix B]{Gleitz:KNSconj}. The inner products among the basis vectors $\alpha_1,\dots,\alpha_8$ are described by Figure~\ref{fig:E8}, where $(\alpha_i,\alpha_i)=2$ for all $i\in\{1,\dots,8\}$, $(\alpha_i,\alpha_j)=-1$ if $\alpha_i$ and $\alpha_j$ are adjacent, $(\alpha_i,\alpha_j)=0$ otherwise. The lines determined by $E_6$ are precisely the $36$ lines with $b_7=b_8=0$, and the lines determined by $E_7$ are precisely the $63$ lines with $b_8=0$. \begin{remark} The numbering of the $120$ lines of $E_8$ in \cite[Appendix B]{Gleitz:KNSconj} is the natural one in the following sense. Every line can be represented by a vector $\alpha=\sum_{i=1}^8 a_i\alpha_i$ with $\|\alpha\|^2=2$ and $a_i\geqslant0$ for all $i$. We assign a total ordering to the lines as follows. Let $\alpha=\sum_{i=1}^8 a_i\alpha_i$, $\beta=\sum_{i=1}^8 b_i\alpha_i$. We define $[\alpha] \prec [\beta]$ if either \begin{itemize} \item[] $\sum_{i=1}^8 a_i < \sum_{i=1}^8 b_i$; \item[or] \item[] $\sum_{i=1}^8 a_i = \sum_{i=1}^8 b_i$ and \item[] $a_1=b_1,\dots,a_i=b_i,a_{i+1}>b_{i+1}$. \end{itemize} Note that this ordering on the set of lines is the default ordering given by \texttt{MAGMA}~\cite{Magma} on the set of positive roots of the root system $E_8$ (which are in one-to-one correspondence with the lines of the line system $E_8$). \end{remark} Let $n=7$ or $8$ and let $G$ be an $n$-vertex exceptional edge-signed graph. By \cite[Theorem 1.10]{ArsComb1993}, $G$ can be obtained from an $(n-1)$-vertex exceptional edge-signed graph $H$ by attaching a vertex to $H$. In the tables in the appendix we describe, up to switching equivalence, the edge-signed graphs that are not integrally represented. Each edge-signed graph is described by referring to the lines used to construct it and each line is referred to by its number in \cite[Appendix B]{Gleitz:KNSconj}, or equivalently, by its position in the total ordering. Clearly, exceptional edge-signed graphs with smallest eigenvalue greater than $-2$ must have at least $6$ vertices and at most $8$ vertices. In Table~\ref{swc:6} the $32$ exceptional switching classes $\mathcal{S}_{6,i}$ $(1\leqslant i\leqslant 32)$ of $6$-vertex edge-signed graphs are described. The first $20$ out of the $32$ consist of those classes which contain an unsigned graph, and these ordered according to \cite[Table A2]{CRS:LMS314}, in which graphs are ordered by the number of edges. The remaining $12$ switching classes are also ordered by the number of edges. In Table~\ref{swc:7} the $233$ exceptional switching classes $\mathcal{S}_{7,i}$ $(1\leqslant i\leqslant 233)$ of $7$-vertex edge-signed graphs are described. Each switching class $\mathcal{S}_{7,i}$ is obtained by adding a line $ l $ to $\mathcal{S}_{6,k} $. In Table~\ref{swc:7}, the triples $i, l, k$ are listed, and the first $110$ switching classes consist of those classes which contain an unsigned graph, and these ordered according to \cite[Table A2]{CRS:LMS314}. Similarly, in Tables~\ref{swc:8_1}, \ref{swc:8_2}, \ref{swc:8_3}, \ref{swc:8_4}, and \ref{swc:8_5}, the $ 1242 $ exceptional switching classes $\mathcal{S}_{8,i}$ $(1\leqslant i\leqslant 1242)$ of $8$-vertex edge-signed graphs are described. Each switching class $\mathcal{S}_{8,i}$ is obtained by adding a line $ l $ to $\mathcal{S}_{7,k}$ and these triple $i,k,l$ are given in the tables. As before, in the tables the $443$ unsigned graphs in \cite[Table A2]{CRS:LMS314} come first. In Tables~\ref{swc:7}--\ref{swc:8_5}, the columns labelled by $i^\prime$ denote the last two digits of $i$, that is, $i^\prime \equiv i \pmod{100}$. We can summarise the tables below in the following theorem. \begin{theorem}\label{thm:E8enum} Let $G$ be an exceptional edge-signed graph having smallest eigenvalue greater than $-2$. Then $G$ is one of \begin{enumerate} \item $32$ edge-signed graphs on $6$ vertices; \item $233$ edge-signed graphs on $7$ vertices; \item $1242$ edge-signed graphs on $8$ vertices. \end{enumerate} \end{theorem} \section{Hoffman graphs} \label{sec:Hoffgraph} A \textbf{Hoffman graph} $\mathfrak H$ is defined as a graph $(V,E)$ with a distinguished coclique $V_f(\mathfrak H) \subset V$ called \textbf{fat} vertices, the remaining vertices $V_s(\mathfrak H) = V\backslash V_f(\mathfrak H)$ are called \textbf{slim} vertices. For more background on Hoffman graphs see some of the authors' previous papers \cite{-3,-1-tau,gt-3}. Let $\mathfrak H$ be a Hoffman graph and suppose its adjacency matrix $A$ has the following form \[ A = \begin{pmatrix} A_s & C \\ C^\top & 0 \end{pmatrix}, \] where the fat vertices come last. Define $B(\mathfrak H) := A_s - C C^\top$. The eigenvalues of $\mathfrak H$ are defined to be the eigenvalues of $B(\mathfrak H)$. A Hoffman graph $\mathfrak H$ is called \textbf{fat} if every slim vertex of $\mathfrak H$ has at least one fat neighbour. In this section we show how our results relate to the classification of fat Hoffman graphs with smallest eigenvalue greater than $-3$. It is shown in \cite{gt-3} that if $\lambda_1(B(\mathfrak H)) > -3$ then every slim vertex is adjacent to at most two fat vertices and at most one slim vertex can be adjacent to more than one fat vertex. It therefore follows that if a fat Hoffman graph $\mathfrak H$ has smallest eigenvalue $\lambda_1(\mathfrak H) > -3$ then the diagonal of $B(\mathfrak H) + I$ consists of at most one $-1$ entry and the remaining entries $0$. In other words, $B(\mathfrak H) + I$ is either the adjacency matrix $A(G)$ of a signed graph $G$, or the modified adjacency matrix $\hat{A}(G,v)$ of $G$ with respect to some vertex $v$. The \textbf{special graph} $\mathcal{S}(\mathfrak H)$ of a Hoffman graph $\mathfrak H$ is the edge-signed graph whose adjacency matrix has the same off-diagonal entries as $B(\mathfrak H)$ and zeros on the diagonal. \begin{theorem} Let $\mathfrak H$ be a fat Hoffman graph in which every slim vertex has exactly one fat neighbour. Then $\mathfrak H$ has smallest eigenvalue greater than $-3$ if and only if $\mathcal{S}(\mathfrak H)$ is switching equivalent to one of the edge-signed graphs in Theorem~\ref{thm:3} or Theorem~\ref{thm:E8enum}. \end{theorem} \begin{proof} Since every slim vertex has exactly one fat neighbour, $B(\mathfrak H)+I$ is the adjacency matrix of $\mathcal{S}(\mathfrak H)$. Thus $\mathfrak H$ has smallest eigenvalue greater than $-3$ if and only if $\mathcal{S}(\mathfrak H)$ has smallest eigenvalue greater than $-2$. The result then follows since Theorems~\ref{thm:3} and \ref{thm:E8enum} give a classification of edge-signed graphs with smallest eigenvalues greater than $-2$. \end{proof} \begin{lemma}\label{lm:012} Let $\mathfrak H$ be a fat Hoffman graph in which every slim vertex has exactly one fat neighbour. Then in the special graph $\mathcal{S}(\mathfrak H)$ of $\mathfrak H$, there are no $(+)$-edges in the neighbourhood of any fat vertex. In particular, $\mathcal{S}(\mathfrak H)$ does not contain a cycle in which all but one edge are $(-)$-edges. \end{lemma} \begin{proof} Let $N$ be the set of neighbours of a fat vertex. Then the off-diagonal entry of $B(\mathfrak H)$ corresponding to two vertices of $N$ cannot be $1$. This shows the first claim. Suppose that $\mathcal{S}(\mathfrak H)$ contains a cycle $v_0,v_1,\dots,v_n,v_0$ such that all but the edge $\{v_n,v_0\}$ are $(-)$-edges. Then $v_i,v_{i+1}$ have a common fat neighbour for $i=0,1,\dots,n-1$. Since every slim vertex has exactly one fat neighbour, it follows that $v_0,\dots,v_n$ have a common fat neighbour. But this contradicts the first claim. \end{proof} Lemma~\ref{lm:012} implies that not every edge-signed graph can be the special graph of a fat Hoffman graph in which every slim vertex has exactly one fat neighbour. For an edge-signed graph $\mathcal{S}=(V,E^+,E^-)$, we denote by $\mathcal{S}^-$ the unsigned graph $(V,E^-)$. Let $\mathcal{S}$ be an edge-signed graph that is switching equivalent to one of the edge-signed graphs in Theorem~\ref{thm:3} or Theorem~\ref{thm:E8enum}, and $\mathcal{S}$ has no cycle in which all but one edge are $(-)$-edges. Then every fat Hoffman graph $\mathfrak H$ in which every slim vertex has exactly one fat neighbour, satisfying $\mathcal{S}(\mathfrak H)=\mathcal{S}$ is obtained as follows. Let \[ V(\mathcal{S})=\bigcup_{i=1}^n V_i\quad\text{(disjoint)} \] be a partition satisfying \begin{enumerate} \item for all $i\in\{1,\dots,n\}$, the subgraph induced on $V_i$ contains no $(+)$-edges, \item every connected component of $\mathcal{S}^-$ is contained in $V_i$ for some $i\in\{1,\dots,n\}$. \end{enumerate} Define a Hoffman graph $\mathfrak H$ as follows. The vertex set of $\mathfrak H$ consists of the set of slim vertices $V(\mathcal{S})$ and the set of fat vertices $\{f_1,\dots,f_n\}$. The edges are $\{f_i,u\}$ with $u\in V_i$, $1\leq i\leq n$; $\{u,v\}$ with $u,v\in V_i$, $1\leq i\leq n$, $\{u,v\}\notin E(\mathcal{S})$; and $\{u,v\}$ with $u\in V_i$, $v\in V_j$, $1\leq i,j\leq n$, $i\neq j$, $\{u,v\}\in E(\mathcal{S})$. Observe that for $u,v\in V_i$, $\{u,v\}\notin E(\mathcal{S})$ if and only if $\{u,v\}\notin E^-(\mathcal{S})$, and for $u\in V_i$, $v\in V_j$ with $i\neq j$, $\{u,v\}\in E(\mathcal{S})$ if and only if $\{u,v\}\in E^+(\mathcal{S})$. Then $\mathcal{S}(\mathfrak H)=\mathcal{S}$ holds, and $\lambda_{1}(\mathfrak H)=\lambda_{1}(\mathcal{S})-1$. In general, given an edge-signed graph $\mathcal{S}$, there are a number of fat Hoffman graphs $\mathfrak H$ with $\mathcal{S}(\mathfrak H)=\mathcal{S}$, satisfying the condition that every slim vertex of $\mathfrak H$ has exactly one fat neighbour. For example, consider the simplest case where $\mathcal{S}$ is a path consisting only of $(+)$-edges. Then such Hoffman graphs are in one-to-one correspondence with partitions of the vertex-set into cocliques. \section{Acknowledgement} \label{sec:acknowledgement} The authors are grateful for the comments of the referees. \input{hoffcon.bbl} \newpage \section*{Appendix} \begin{table}[h] \caption{Switching classes $\mathcal{S}_{6,i}\ (i=1,\ldots,32) $} \input{table6.txt} \label{swc:6} \end{table} \begin{table} \caption{Switching classes $\mathcal{S}_{7,i}\ (1 \leqslant i\leqslant 233)$} {\scriptsize \input{table7.txt} } \label{swc:7} \end{table} \begin{table} \caption{Switching classes $\mathcal{S}_{8,i}\ (1 \leqslant i\leqslant 250) $} {\tiny \input{table8_1_250.txt} } \label{swc:8_1} \end{table} \begin{table} \caption{Switching classes $\mathcal{S}_{8,i}\ (251 \leqslant i\leqslant 500) $} {\tiny \input{table8_251_500.txt} } \label{swc:8_2} \end{table} \begin{table} \caption{Switching classes $\mathcal{S}_{8,i}\ (501 \leqslant i\leqslant 750) $} {\tiny \input{table8_501_750.txt} } \label{swc:8_3} \end{table} \begin{table} \caption{Switching classes $\mathcal{S}_{8,i}\ (751\leqslant i\leqslant 1000) $} {\tiny \input{table8_751_1000.txt} } \label{swc:8_4} \end{table} \begin{table} \caption{Switching classes $\mathcal{S}_{8,1000+i}\ (1 \leqslant i\leqslant 242) $} {\tiny \input{table8_1001_1250.txt} } \label{swc:8_5} \end{table} \end{document}

proofpile-arXiv_069-6770

\section{Introduction} \textit{Introduction.} Floquet engineering, which refers to temporal modulation of system parameters by periodic drives, has been known as an effective approach for controlling the dynamics of a given system. In recent years, it has been implemented in a large variety of systems ranging from cold atoms \cite{2017_RMP_Eckardt,2015_NPhysc_Eisert,2007_PRL_Lignier,2011_PRL_Jiang,2017_PRL_Potirniche,2019_PRA_Li} and quantum dots \cite{2016_PRX_Stehlik,2018_PRL_Koski} to integrated photonics \cite{2019_NPhoton_Zhang} and Josephson junction devices, \cite{2019_PRA_Sameti,2020_NJP_Wang} enabling a diverse variety of novel functionalities. Aside from practical applications, Floquet-driven systems have also significantly advanced fundamental research, leading to the experimental observation of novel non-equilibrium phenomena such as discrete time-crystalline phases \cite{2017_Nature_Zhang,2017_Nature_Choi,2018_PRL_Gong} or Floquet spin-glass phases. \cite{2019_PRL_Raposo} Among all coherent information systems, magnonic systems have been emerging as a highly promising platform because of their unique properties. In magnonic systems, magnons---quasiparticles of spin waves---are used as the information carrier. Their coherent interactions with a broad variety of other systems have been demonstrated recently \cite{2014_PRL_Zhang,2014_PRL_Tabuchi,2014_PRApplied_Goryachev,2015_PRL_Bai,2016_SciAdv_Zhang,2016_PRL_Zhang_Optomagnonics,2016_PRL_Osada,2016_PRL_Haigh,2017_PRB_Sharma,2018_PRB_Graf, 2019_PRL_Hou,2019_PRL_Li}. For instance, magnons, with their frequencies in the gigahertz range, naturally interact with microwave photons through magnetic dipole-dipole interactions. Most importantly, the coupling strength is significantly enhanced by the large spin density in the magnon medium and can reach the strong coupling regime. As of today, such hybrid cavity electromagnonic systems have been experimentally demonstrated in both classical \cite{2014_PRL_Zhang,2014_PRL_Tabuchi,2014_PRApplied_Goryachev,2015_PRL_Bai,2019_PRL_Hou,2019_PRL_Li} and quantum \cite{2015_Science_Tabuchi,2017_SciAdv_Lachance-Quirion,2020_Science_Lachance-Quirion} regimes, becoming the most intensively studied hybrid magnonic systems. With unique magnon properties such as large tunability and time reversal symmetry breaking, novel functionalities can be achieved in these systems \cite{2015_NComm_Zhang,2020_PRAppl_Zhang,2019_PRL_Wang,2019_PRL_Zhang,2020_PRL_Yuan}. However, unlike in systems such as optomechanics \cite{2008_Nature_Li,2010_Science_Weis,2011_Nature_Safavi-Naeini,2011_Nature_Teufel,2014_RMP_Aspelmeyer} that employ parametric coupling, the direct magnon-photon coupling in a given hybrid electromagnonic system is usually difficult to manipulate. Although magnons naturally possess great tunability, fast or on-demand tuning of the coupling is extremely challenging, making the dynamic control of coherent signals practically impossible. This poses a tremendous obstacle in the broad applications of hybrid electromagnonics. In this work, we show that by introducing Floquet engineering into cavity electromagnonics, in-situ tuning of the magnon-photon interaction is, to the best of our knowledge, achieved for the first time. In our Floquet cavity electromagnonic system, a driving field induces mode splittings that are analogous to the Autler-Townes splitting (ATS) in atomic physics, where the coupling strength between two energy levels is determined by the strength of the Floquet driving field. The system response is studied in both the frequency and temporal domains. More interestingly, our system supports a new coupling regime -- Floquet ultrastrong coupling (FUSC), which has not been observed previously in existing Floquet systems. In this regime, the mode splitting exceeds the energy level spacing of the two interacting modes, distinguishing it from the conventional ultrastrong coupling where the coupling strength is a significant fraction of the bare frequencies of the uncoupled systems. We further show that in this regime, the rotating wave approximation no longer holds and the counter rotating terms in the Hamiltonian start to exhibit non-negligible effects on the system response. All these findings point to a new direction for advancing magnon-based coherent information processing. \begin{figure}[tb] \centering \includegraphics[width=0.9\linewidth]{Figure1.pdf} \caption{(a) Device schematics. A spherical YIG magnonic resonator is placed underneath a cylindrical dielectric resonator DR30 inside a copper housing. An external magnetic field $H$ is applied along $x$ direction. The coaxial probe loop is for microwave excitation and readout; the drive loop wrapping around the YIG sphere along $x$ provides the Floquet driving field. (b) Simulated magnetic components of the cavity field for the TE$_{01\delta}$ mode of the dielectric resonator. Color: field intensity; arrows: field direction. (c) Energy level diagram. $m$: magnon mode; $c$: microwave photon mode; $\delta\omega$: magnon detuning; $g_{cm}$: magnon-photon coupling strength; $d_\pm$: upper (lower) hybrid mode; $\Delta\omega$: energy separation between two hybrid modes; $\Delta\omega_\mathrm{AT}$: Autler-Townes splitting (ATS). A driving field is applied to enable the transition between hybrid modes $d_-$ and $d_+$. Under strong drives the system enters the Floquet ultrastrong coupling regime where $\Delta\omega_\mathrm{AT}>\Delta\omega$. (d), Measured cavity reflection spectra at various magnetic fields (controlled by the magnet position $x$). Two avoided crossings are visible, corresponding to the strong coupling between the cavity photon mode ($c$) and two magnon modes ($m_0$ and $m_1$). (e), Cavity reflection spectra measured at $x=21.75$ mm with different DC bias conditions applied to the drive loop.} \label{fig1} \end{figure} \textit{System Description}. Our system consists of a high-quality dielectric resonator and a highly polished yttrium iron garnet (YIG) sphere [Fig.\,\ref{fig1}(a)]. The cylinder-shaped dielectric resonator (DR30) supports a TE$_{01\delta}$ mode [Fig.\,\ref{fig1}(b)] at 8.5 GHz, with its microwave magnetic fields along the axial ($z$) direction. The cavity mode volume is greatly reduced because of the high dielectric permittivity ($\varepsilon=30$), which enhances its coupling with the magnon mode. The cylinder is hosted inside a copper housing to eliminate radiation losses. Together with the low dielectric loss, this leads to quality factors as high as 10,000 for the cavity resonances at room temperature. A loop antenna is placed above the dielectric resonator with its loop along $z$ direction to probe the cavity photon mode. The YIG sphere is placed underneath the dielectric resonator and close to its end surface. With an external bias magnetic field that saturates all the spins, it supports a uniform magnon mode (the Kittel mode) whose frequency is determined by the field strength. When the bias field is not parallel with the microwave magnetic fields of the photon mode, there exists non-zero magnon-photon interaction. In order to maximize such interaction, the bias field is applied perpendicular to $z$ (along $x$ direction). In our experiments, the bias field strength is controlled by the magnet position along $x$ direction. The system is described by the Hamiltonian: \begin{equation}\label{eq_full} \hat{H}=\hbar\omega_c\hat{c}^\dagger\hat{c}+\hbar\omega_m\hat{m}^\dagger\hat{m}+\hbar g_{cm}(\hat{c}^\dagger\hat{m}+\hat{c}\hat{m}^\dagger)+\hat{H}_\mathrm{F}, \end{equation} \noindent where $\hbar$ is the reduced Planck's constant, $\hat{c}^\dagger$ and $\hat{c}$ ($\hat{m}^\dagger$ and $\hat{m}$) are the creation and annihilation operators for the cavity photon (magnon) mode, $g_{cm}$ is the beam-splitter type coupling strength, $\omega_c$ ($\omega_m$) is the resonance frequency of the cavity photon (magnon) mode, and $\hat{H}_\mathrm{F}$ is the Floquet driving term. The magnon frequency is controlled by an external magnetic field: $\omega_m=\gamma H$, where $\gamma=28$ GHz/T is the gyromagnetic ratio, and $H$ is the bias magnetic field. Under strong coupling condition $2g_{cm}>\kappa_c,\kappa_m$ where $\kappa_c$ and $\kappa_m$ represent the dissipation rate of the cavity photon and magnon mode, respectively, the two modes hybridize with each other, allowing coherent information conversion among the magnonic and electromagnetic degrees of freedom. In our experiment, strong coupling is confirmed by the avoided crossing features in the measured cavity reflection spectra when the magnon frequency is swept by varying the magnet position [Fig.\,\ref{fig1}(d)]. The Floquet driving is realized in our system through frequency modulation of the magnon mode. A small coil, which has previously been used only for GHz readout or controls \cite{2019_NJP_Boventer}, is looped tightly around the YIG sphere as a means to modulate the bias magnetic field. The loop is aligned along the bias field ($x$) direction, and has only 3 turns in order to reduce the inductance to allow fast modulation. The effect of the driving coil is confirmed by the shifted magnon resonances when different DC drives are applied [Fig.\,\ref{fig1}(e)]. With a sinusoidal drive, the Floquet term of the Hamiltonian reads: \begin{equation}\label{eq_Floquet} \hat{H}_\mathrm{F}=\hbar\Omega\hat{m}^\dagger\hat{m}\cos(\omega_\mathrm{D} t), \end{equation} \noindent where $\Omega$ and $\omega_\mathrm{D}$ are the strength and frequency of the driving field, respectively. \begin{figure}[t] \centering \includegraphics[width=0.95\linewidth]{Figure2.pdf} \caption{(a)--(c), Measured cavity reflection spectra versus the bias magnetic field at a 10-V peak-to-peak driving amplitude for three frequencies: $\omega_\mathrm{D}=5,15,28$ MHz, respectively. In addition to the avoided crossing caused by the strong coupling between the cavity photon mode ($c$) and magnon mode ($m_0$), sidebands are created by the AC driving fields. (d), Measured cavity reflection spectra as a function driving frequency at $x=21.762$ mm when $\delta\omega=0$. The dashed lines correspond to conditions for (a)--(c). (e), Measured cavity reflection spectra as a function of driving amplitude, with $\delta\omega=0$ and $\omega_\mathrm{D}=28$ MHz. (f), Extracted Autler-Townes splitting for both $d_-$ (square) $d_+$ (circle) modes in (e) as a function of the driving voltage. Solid line is from numerical calculation. SB: sideband.} \label{fig2} \end{figure} \textit{Magnonic Autler-Townes Effect}. When the magnon and photon modes are on resonance ($\delta\omega=\omega_m-\omega_c=0$), two hybrid modes $d_\pm=(\hat{c}\pm\hat{m})/\sqrt{2}$ form at frequencies $\omega_\pm=\omega_c\pm g_{cm}$. Using $d_\pm$ as the new basis and under rotating wave approximation (RWA), the Hamiltonian can be rewritten as \begin{equation}\label{eq_RWA} \hat{H}_\mathrm{RWA}=\frac{\delta\omega_\mathrm{D}}{2}\hat{d}^\dagger_{-}\hat{d}_{-}-\frac{\delta\omega_\mathrm{D}}{2}\hat{d}_{+}^\dagger\hat{d}_{+}+\frac{\Omega}{4}(\hat{d}_{-}^\dagger\hat{d}_{+}+\hat{d}_{-}\hat{d}_{+}^\dagger), \end{equation} \noindent where $\delta\omega_\mathrm{D}=\Delta\omega-\omega_\mathrm{D}$. Here $\Delta\omega=\omega_+-\omega_-=2g_{cm}$ is the level spacing between the two hybrid modes. This Hamiltonian shows that the external Floquet field drives the transition between the two hybrid modes, which is analogous to a two-level system driven by a laser field. As a result, each hybrid mode splits further into two energy levels with a level spacing $\Delta\omega_\mathrm{AT}$ that is determined by the driving strength, resembling the ATS in a laser-driven two-level system [Fig.\,\ref{fig1}(c)]. The cavity responses under a Floquet drive are plotted in Figs.\,\ref{fig2}\,(a)--(c), which are zoomed in to show the avoided-crossing between the fundamental magnon mode ($m_0$) and the cavity mode ($c$). Two hybrid modes ($d_\pm$) resulting from the magnon-photon hybridization are clearly visible. The Floquet drive creates two sidebands for each hybrid mode. As the driving frequency $\omega_\mathrm{D}$ increases, the inner sideband of one hybrid mode moves closer to, and eventually meets with, the other hybrid mode (when $\omega_\mathrm{D}=\Delta\omega=2\pi\times28$ MHz), leading to the ATS discussed above [Fig.\,\ref{fig2}(c)]. \begin{figure}[t] \centering \includegraphics[width=0.95\linewidth]{Figure3.pdf} \caption{(a),(b) Measured cavity temporal response under different driving frequencies after a pulsed excitation (center frequency: 8.52 GHz, width: 20 ns) at time zero when the Floquet drive is on and off, respectively. (c),(d) Calculated cavity temporal response with and without the Floquet drive, respectively. (e),(f) Line plot of the theoretical (solid line) and experimental (greyed area) temporal responses at Floquet driving frequencies as indicated by the vertical dashed lines in (a)--(d) ($\omega_\mathrm{D}=2\pi\times 28$ MHz for the theoretical calculation, while $\omega_\mathrm{D}=2\pi\times 31$ MHz for experiment results which is slightly higher because of magnon non-linearity effects induced by the strong microwave input) with and without the Floquet drive, respectively. Norm. refl.: normalized reflection.} \label{fig3} \end{figure} The on-resonance ($\delta\omega=0$) response of the system as a function of the driving frequency is shown in Fig.\,\ref{fig2}(d). The most prominent feature is that the upper (lower) sideband of $d_-$ ($d_+$) mode moves closer to the other mode $d_+$ ($d_-$) as the driving frequency increases, and eventually crosses that mode with an avoided-crossing feature at $\omega_\mathrm{D}=\Delta\omega=2\pi\times28$ MHz, which corresponds to the ATS. Note that our modulation approach creates sidebands at both $\omega_\pm\pm\omega_\mathrm{D}$ frequencies, therefore there also exists a lower (upper) sideband for $d_-$ ($d_+$), which rapidly disappears as it moves away from the hybrid mode because of the lacking of coupling with the other mode. Analogous to the laser-driven two-level systems, the ATS observed in our system is also determined by the strength of the driving field. The driving strength, characterized by $\Omega$, is controlled by the signal voltage sent into the driving coil in our experiments. Larger driving strengths result in wider ATS, as shown in Fig.\,\ref{fig2}(e), where the on-resonance cavity reflection is plotted as a function of the driving voltage at a driving frequency $\omega_\mathrm{D}=2\pi\times 28$ MHz. According to Eq.\,(\ref{eq_RWA}), the ATS is linearly proportional to the driving strength $\Delta\omega_\mathrm{AT}=\Omega/2$. This is confirmed by Fig.\,\ref{fig2}(f), where the extracted and calculated ATS show an excellent agreement with each other. In addition to modifying the equilibrium spectra, the Floquet drive also provides a new approach for manipulating the electromagnonic dynamics. This has long been a grand challenge such that all previous demonstrations were based on quasi-static controls \cite{2014_PRL_Zhang, 2017_NComm_You, 2019_PRL_Zhang}. Figures\,\ref{fig3}(a) and (c) plot the measured and calculated cavity reflection signals after a 20-ns-wide microwave pulse centered at 8.52 GHz is sent into the cavity, which show a great agreement. At low driving frequencies where the Floquet drive cannot induce coupling between the two hybrid modes $d_\pm$, periodic ``Rabi-like'' oscillations between the magnon and cavity photon modes are observed with a period $T=2\pi/2g_{cm}=36$ ns. This is more clearly visible when the Floquet drive is completely turned off [Figs.\,\ref{fig3}(b) and (d)], and it is evident that the amplitude of the oscillating signal monotonically decreases as a result of dissipation [Figs.\,\ref{fig3}(f)]. When the Floquet drive is turned on and, in particular, when the driving frequency matches the spacing between the two hybrid modes ($\omega_\mathrm{D}=\omega_+-\omega_-= 2\pi\times 28$ MHz), coherent coupling between the hybrid modes is enabled and consequently the temporal response of the system is substantially modified. As shown in Fig.\,\ref{fig3}(e), the oscillation amplitude first rapidly decays into a minimum at around 120 ns and then increases again at around 200 ns, with the time interval matching the ATS ($2\pi/2\Delta\omega_\mathrm{AT}\approx80$ ns for a 10V Floquet drive). The measurement results agree well with our theoretical model. Because such Floquet drive-induced coherent interaction can be controlled by the amplitude of the drive, it provides opportunities for complex real-time manipulations of the magnon-photon coupling using electrical pulses. \textit{Floquet Ultrastrong Coupling}. In the above analysis, the Floquet drive is relatively weak, yielding small ATS $\Delta\omega_\mathrm{AT}<\Delta\omega$. However, in the strong-drive regime where the ATS is comparable with or even larger than the level spacing between $d_\pm$ ($\Delta\omega_\mathrm{AT}\gtrapprox\Delta\omega$), the RWA Hamiltonian in Eq.\,(\ref{eq_RWA}) is no longer sufficient to describe the system. This corresponds to a novel and rarely investigated coupling regime: the FUSC regime. Here, the counter rotating term need to be included and the Hamiltonian becomes: \begin{equation}\label{Eq_Bessel} \begin{aligned} \hat{H}={}& \omega_{-}\hat{d}_{-}^\dagger\hat{d}_{-}+\omega_{+}\hat{d}_{+}^\dagger\hat{d}_{+}+\\ & g_{cm}\sum_{n=\text{odd}}^{\infty}J_n(\frac{\Omega}{\omega_\mathrm{D}})(\hat{d}_{-}^\dagger\hat{d}_{+}e^{i(n\omega_\mathrm{D} t)}+\hat{d}_{-}\hat{d}_{+}^\dagger e^{-i(n\omega_\mathrm{D} t)}). \end{aligned} \end{equation} \noindent where $n$ is the sideband order, and $J_n(x)$ is the $n$-th Bessel function of the first kind. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{Figure4.pdf} \caption{(a)--(c), Cavity reflection spectra when the driving frequency $\omega_\mathrm{D}$ is swept at a fixed driving amplitude (10 V peak-to-peak), obtained from experiment, Floquet model, and rotating wave approximation (RWA) model, respectively. (d)--(f), Cavity reflection spectra when the driving amplitude is swept at a fixed driving frequency ($\omega_\mathrm{D}=2\pi\times 3.85$ MHz), obtained from experiment, Floquet model, and RWA model, respectively.} \label{fig4} \end{figure} The last term in Eq.\,(\ref{Eq_Bessel}) represents the summation of interactions between different sidebands of $d_\pm$ modes. The Floquet drive generates a series of sidebands at frequencies $\omega_\pm\pm n\omega_\mathrm{D}$. Multiple sidebands and their interactions with the other mode are clearly visible in Fig.\,\ref{fig2}(d) when $\omega_\mathrm{D}$ is small. Specifically, the individual coupling strength of the $n$-th sideband of mode $d_\pm$ with mode $d_\mp$ is determined as $g_n=g_{cm}J_n(\frac{\Omega}{\omega_\mathrm{D}})$. Apparently, this coupling strength is determined by the intrinsic magnono-photon coupling strength $g_{cm}$, but can be controlled by the frequency and strength of the Floquet drive. Interestingly, only sidebands with odd $n$ values have non-zero coupling strengths. This is confirmed by the experimental observation in Fig.\,\ref{fig2}(d), where only the first and third sidebands of $d_\pm$ cross $d_\mp$ with an avoided crossing (at 28 and 11 MHz, respectively), while the second sideband does not. When the Floquet drive is relatively small, the ATS can be determined as $\Delta\omega_\mathrm{AT}=2g_n$. But this does not apply for strong drives, where the effects from multiple sidebands need to be considered. In general, experimental investigation of the strong-drive regime is very challenging because the maximally achievable driving strength is limited. Alternatively, it can be achieved by reducing the level spacing $\Delta\omega$ to make it comparable with or smaller than the ATS $\Delta\omega_\mathrm{AT}$. This is realized with a reduced intrinsic magnon-photon coupling strength $g_{cm}$ by increasing the gap between the YIG sphere and the dielectric resonator surface. Figure\,\ref{fig4}(a) shows the measured cavity reflection spectra as a function of the driving frequency with a much reduced magnon-photon coupling strength $g_{cm}=2\pi\times1.825$ MHz, where ATS is observed at $\omega_\mathrm{D}=2g_{cm}=2\pi\times3.650$ MHz for both $d_-$ and $d_+$. This agrees well with the theory results calculated using the Floquet scattering matrix derived from the full Hamiltonian Eq.\,(\ref{eq_full}), as shown in Fig.\,\ref{fig4}(b). As a comparison, the calculated cavity reflection spectra based on the Hamiltonian with RWA in Eq.\,(\ref{eq_RWA}) are plotted in Fig.\,\ref{fig4}(c), which exhibit severe deviation from the experimental results, showing the significant effects of the counter rotating term. Similarly, the measured cavity reflection spectra under different driving strengths are plotted in Fig.\,\ref{fig4}(d), which show excellent agreement with theory calculations based on the full Floquet Hamiltonian [Fig.\,\ref{fig4}(e)], as well as large deviation from the calculations using RWA [Fig.\,\ref{fig4}(f)]. Compared with the RWA results, the RWA breaking has two major effects: first, the outer two branches become much weaker than the inner two branches in both $\omega_\mathrm{D}$-dependent [Figs.\,\ref{fig4}\,(a)--(c)] and $\Omega$-dependent [Figs.\,\ref{fig4}\,(d)--(f)] cavity reflections; second, the inner two branches merge and form a much deeper dip. \textit{Conclusion}. To conclude, this work demonstrates a Floquet cavity electromagnonic system that provides controllable hybridization between magnons and microwave photons. The Floquet engineering technique provides a versatile approach for manipulating hybrid magnon states and enables the observation of on-demand ATS. The measurement results in both the frequency and temporal domains show excellent agreements with our Floquet driving model and the corresponding scattering matrix. More importantly, this approach leads to a new coupling regime---FUSC---for hybrid magnonics, where RWA breaks down and new phenomena are observed. This technique can be readily applied as a general approach to a broad range of magnonic systems that have been lacking the essential controllability, producing not only new functionalities but also novel non-equilibrium magnon dynamics. For instance, more complicated spectral control can be obtained if the driving strength can be further enhanced. Besides, time domain operations such as Rabi or Ramsey pulse sequences will allow on-demand mode swapping or storage. Although our experiments are carried out at room temperatures in the classical regime, the principles demonstrated here also directly apply to quantum operations, where hybrid magnonics has shown great potentials for applications such as quantum transduction. \begin{acknowledgments} This work was performed, in part, at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility, and supported by the U.S. Department of Energy, Office of Science, under Contract No. DE-AC02-06CH11357. L.J. acknowledges support from the ARL-CDQI (W911NF-15-2-0067), ARO (W911NF-18-1-0020, W911NF-18-1-0212), ARO MURI (W911NF-16-1-0349), AFOSR MURI (FA9550-15-1-0015, FA9550-19-1-0399), NSF (EFMA-1640959, OMA-1936118), and the Packard Foundation (2013-39273). \end{acknowledgments} \section*{Appendix I: The Floquet scattering matrix} The system is described by the following periodic Hamiltonian ($ \hbar=1$) \begin{equation}\label{eq1} \begin{split} \hat{H}&=\omega_c\hat{c}^\dagger\hat{c}+\omega_m\hat{m}^\dagger\hat{m}+g_{cm}\hat{c}^\dagger\hat{m}+g_{cm}^\ast\hat{c}\hat{m}^\dagger \\ &+\Omega\hat{m}^\dagger\hat{m}\cos(\omega_\mathrm{D} t+\phi), \end{split} \end{equation} where $\hat{c},\hat{m}$ are the microwave and magnon mode operators, with corresponding mode frequencies $\omega_c$ and $\omega_m$. $g_{cm}$ is the Beam-splitter type coupling strength. $\Omega$ and $\omega_\mathrm{D}$ are the drive strength and drive frequency, respectively. In the Heisenberg picture, the mode operators follow the dynamics $\dot{\textbf{A}}(t)=-i\textbf{H}(t)\textbf{A}(t)$, which is short for \begin{equation} \frac{d}{dt}\begin{pmatrix}\hat{c}(t)\\ \hat{m}(t)\end{pmatrix} =-i\begin{pmatrix} \omega_c&g_{cm}\\g_{cm}^\ast&\omega_m+\Omega\cos(\omega_\mathrm{D}t+\phi) \end{pmatrix}\begin{pmatrix}\hat{c}(t)\\ \hat{m}(t)\end{pmatrix}. \end{equation} It is a periodic time dependent differential equation. According to Floquet theorem, the eigen-mode can be written in the form $\textbf{A}_m(t)=\textbf{a}_m(t)e^{-i{\epsilon_m} t}$, where $\epsilon_m$ is the called the Floquet quasi-energy. $\textbf{a}_m(t)$ is periodic in time with the same period as $\textbf{H}(t)$ and obviously satisfies the eigen equation \begin{equation}\label{eq3} \left[\textbf{H}(t)-i\frac{d}{dt}\right]\textbf{a}_m(t)=\epsilon_m\textbf{a}_m(t). \end{equation} In order to find the Floquet mode, we write the time-dependent mode in its Fourier component $\textbf{a}_m(t)=\sum_{N}\textbf{a}_m^Ne^{-iN\omega_\mathrm{D}t}$. If we further take the Fourier index as a new mode index, the above time-dependent differential equation can be mapped to a time-independent tight-binding Hamiltonian $\textbf{H}_F$ \cite{Shirley1965}, with matrix elements given by \begin{equation} \braket{\alpha n|H_F|\gamma k}=H_{\alpha\gamma}^{n-k}+n\omega_\mathrm{D}\delta_{{\omega_\mathrm{D}}\alpha\gamma}\delta_{{\omega_\mathrm{D}}nk}, \end{equation} where $H_{\alpha\gamma}$ is the matrix element of $\textbf{H}(t)$ and it has the Fouirer expansion $H_{\alpha\gamma}=\sum_nH^n_{\alpha\gamma}e^{-in\omega_\mathrm{D}t}$. The matrix $\textbf{H}_F$ is usually called the Floquet Hamiltonian. The problem of solving the time-dependent system is then reduced to find the eigenvalue of the equation \begin{equation} \textbf{H}_F\textbf{V}=\epsilon_m \textbf{V}, \end{equation} where the vector $\textbf{V}=\{...,\hat{c}_{\omega_c-2\omega_\mathrm{D}},\hat{c}_{\omega_c-\omega_\mathrm{D}},\hat{c}_{\omega_c},\hat{c}_{\omega_c+\omega_\mathrm{D}},\\ \hat{c}_{\omega_c+2\omega_\mathrm{D}},...,\hat{m}_{\omega_m-2\omega_\mathrm{D}},\hat{m}_{\omega_m-\omega_\mathrm{D}},\hat{m}_{\omega_m},\hat{m}_{\omega_m+\omega_\mathrm{D}},\hat{m}_{\omega_m+2\omega_\mathrm{D}},...\}^T$. The equation is infinitely dimensional and is generally numerically solved by truncating the matrix. The above discussion ignores the system losses and the readout. To get the scattering matrix, we need to take them into account. Suppose the magnon mode suffers from loss with rate $\kappa_m$, the cavity has readout rate $\kappa_{c,ex}$, intrinsic loss $\kappa_{c,i}$ and $\kappa_c=\kappa_{c,ex}+\kappa_{c,i}$. Thus each mode follows the dynamical equation \begin{equation}\label{deq} \frac{d}{dt}\textbf{V}=-i\textbf{H}_F\textbf{V}+\textbf{D}\textbf{V}+\textbf{M}\textbf{V}_{in} \end{equation} where $\textbf{D}=\text{Diag}[\kappa_c/2,\kappa_m/2]\otimes \textbf{I}_d$ and $\textbf{I}_d$ is an identity matrix with dimension $d$ determined by the truncation. The matrix $\textbf{M}=\text{Diag}[\textbf{I}_d\otimes\begin{pmatrix}\sqrt{\kappa_{c,ex}}&\sqrt{\kappa_{c,i}}\end{pmatrix},\textbf{I}_d*\sqrt{\kappa_m}]$. $\textbf{V}_{in}$ collects all the input operators. The Floquet scattering matrix can be obtained by combining the input-output theory \begin{equation} \textbf{V}_{out}=\textbf{M}^T\textbf{V}-\textbf{V}_{in}. \end{equation} The Floquet scattering matrix is given \begin{equation}\label{eq8} \textbf{S}_F[\omega]=\textbf{M}^T(-i\omega \textbf{I}_{2d}+i\textbf{H}_F-\textbf{D})^{-1}\textbf{M}-\textbf{I}_{3d}. \end{equation} In the numerical calculation, the element of $d_{th}$ row and $d_{th}$ column gives the reflection from the cavity, which we denoted as $S_{11}$. In general, the analytical expression of $S_{11}$ is very complicated resulting from the matrix inverse in Eq. \ref{eq8}. To get a simplified expression, before calculating the matrix inverse in Eq. \ref{eq8}, we split it into diagonal and anti-diagonal part, then expand it by adoping the Woodbury identity \cite{footnote1}. By collecting the first several terms, we arrive at \begin{equation} \begin{split} &S_{11}=-1+\frac{\kappa_{c,ex}}{\kappa_c/2-i(\omega-\omega_c)}\times\\ &\left(1- \frac{g^2_{cm}}{\kappa_c/2-i(\omega-\omega_c)}\sum_{n=-d}^d\frac{J^2_n(\frac{\Omega}{\omega_\mathrm{D}})}{\kappa_m/2-i(\omega+n\omega_\mathrm{D})+i\omega_m}+... \right), \end{split} \end{equation} where $J_n(\Omega/\omega_\mathrm{D})$ is the $n_{th}$ Bessel function of the first kind. {We see there is no phase $\phi$ dependence in the reflection spectrum and we will set them to be zero in later discussions.} With the dynamical equation Eq.(\ref{deq}), one can further evaluate the response signal in the time domain. In the numerical calculation, we suppose the cavity mode is populated by the input pulse with $\Bar{n}$ photons, which will be flopping back and forth among different modes. Thus, we have \begin{equation} \frac{d}{dt}\textbf{V}=(-i\textbf{H}_F+\textbf{D})\textbf{V}, \end{equation} with an initial condition $\textbf{V}_{t=0}=\{0,0,...\sqrt{\Bar{n}}_{\omega_c},...,0,0\}^T$. This first order differential can be easily solved by diagonalizing the coefficient matrix. The output signal in the time domain is giving by $c_{\omega_c}^\text{out}(t)=\sqrt{\kappa_{c,ex}}c_{\omega_c}(t)$. \section*{Appendix II: The modified coupling strength} In the Floquet picture, we know different side-bands are going to show up due to the periodic drive. The driving field will also modify the coupling strength, which can be seen by analyzing the Hamiltonian \begin{equation} \hat{H}=\omega_c\hat{c}^\dagger\hat{c}+\omega_m\hat{m}^\dagger\hat{m}+g_{cm}(\hat{c}^\dagger\hat{m}+\hat{c}\hat{m}^\dagger) +\Omega\hat{m}^\dagger\hat{m}\cos(\omega_\mathrm{D} t). \end{equation} Define a dynamical phase \begin{equation} \begin{split} \varepsilon(t)=\frac{\Omega}{\omega_\mathrm{D}}\left[\sin{(\omega_\mathrm{D} t)}\right], \end{split} \end{equation} then express the Hamiltonian in the rotating frame $U=\exp[i\varepsilon(t)\hat{m}^\dagger\hat{m}]$, \begin{equation}\label{eq11} \hat{H}=\omega_c\hat{c}^\dagger\hat{c}+\omega_m\hat{m}^\dagger\hat{m}+g_{cm}(\hat{c}^\dagger\hat{m}e^{-i\varepsilon(t)}+\hat{c}\hat{m}^\dagger e^{i\varepsilon(t)}). \end{equation} The phase factor $\varphi(t)=e^{i\varepsilon(t)}$ can be expressed in the following series \begin{equation} \varphi(t)=\sum_{n=-\infty}^\infty J_n(\frac{\Omega}{\omega_\mathrm{D}}) e^{-i\cdot n(\omega_\mathrm{D} t)}, \end{equation} where $J_n(\Omega/\omega_\mathrm{D})$ is the $n_{th}$ Bessel function of the first kind with argument $\Omega/\omega_\mathrm{D}$. Thus we have \begin{equation} \begin{split} \hat{H}&=\omega_c\hat{c}^\dagger\hat{c}+\omega_m\hat{m}^\dagger\hat{m}\\ &+g_{cm}\sum_{n=-\infty}^\infty J_n(\frac{\Omega}{\omega_\mathrm{D}})(\hat{c}^\dagger\hat{m}e^{i\cdot n(\omega_\mathrm{D} t)}+\hat{c}\hat{m}^\dagger e^{-i\cdot n(\omega_\mathrm{D} t)} ). \end{split} \end{equation} From the above Hamiltonian, we see that the coupling strength is modified by the Bessel function. Specifically, if we look at the term with $n=0$, we have \begin{equation} \begin{split} \hat{H}&=\omega_c\hat{c}^\dagger\hat{c}+\omega_m\hat{m}^\dagger\hat{m}+g_{cm}J_0(\frac{\Omega}{\omega_\mathrm{D}})(\hat{c}^\dagger\hat{m}+\hat{c}\hat{m}^\dagger) \\ &+g_{cm}\sum_{n\neq 0} J_n(\frac{\Omega}{\omega_\mathrm{D}})(\hat{c}^\dagger\hat{m}e^{-i\cdot n(\omega_\mathrm{D} t)}+\hat{c}\hat{m}^\dagger e^{i\cdot n(\omega_\mathrm{D} t)} ), \end{split} \end{equation} where a modified coupling strength $g_{cm}J_0(\Omega/\omega_\mathrm{D})$ between the magnon and the cavity mode is obtained. \section*{Appendix III: Crossing and anti-crossing of reflection spectrum} In the main text, we see in the reflection spectrum that some side-bands are coupled and others are not when we tune the drive frequency. Since the magnon and cavity mode are strongly coupled,approximately two hybridized modes can be defined (this is true in the strong coupling limit $g_{cm}\gg\kappa_c,\kappa_m$ and take $\omega_c=\omega_m=\omega_0$) \begin{equation} \begin{split} \hat{d}_+&=\frac{\sqrt{2}}{2}(\hat{c}+\hat{m}),\\ \hat{d}_-&=\frac{\sqrt{2}}{2}(\hat{c}-\hat{m}), \end{split} \end{equation} with which we can rewrite the Hamiltonian Eq. \ref{eq11}, \begin{equation} \hat{H}=\omega_+\hat{d}_+^\dagger\hat{d}_++\omega_-\hat{d}_-^\dagger\hat{d}_-+g_{cm}\sin\varepsilon(t)(i\hat{d}_+^\dagger\hat{d}_--i\hat{d}_+\hat{d}_-^\dagger), \end{equation} where $\omega_+=\omega_0+g_{cm}\cos\varepsilon(t)$, $\omega_-=\omega_0-g_{cm}\cos\varepsilon(t)$. We see the hybridized mode separation is given by $2g_{cm}\cos\varepsilon(t)$. A time integral of the separation will give $2g_{cm}J_0(\Omega/\omega_\mathrm{D})$, where we see the separation is approximated by $2g_{cm}$ when either $\Omega/\omega_D$ approaches zero or infinity. Making use of the identity \begin{equation} \sin(z\sin \theta)=-i\sum_{n=\text{odd}}^{\infty}J_n(z)e^{in\theta}, \end{equation} we have \begin{equation} \begin{split} \hat{H}&=\omega_+\hat{d}_+^\dagger\hat{d}_++\omega_-\hat{d}_-^\dagger\hat{d}_-\\ &+g_{cm}\sum_{n=\text{odd}}^{\infty}J_n(\frac{\Omega}{\omega_\mathrm{D}})(\hat{d}_+^\dagger\hat{d}_-e^{in(\omega_\mathrm{D} t)}+\hat{d}_+\hat{d}_-^\dagger e^{-in(\omega_\mathrm{D} t)}). \end{split} \end{equation} It is important to notice that in the above expression, we only have odd terms in the summation. Based on this Hamiltonian, we can look at the coupled side-bands by taking the corresponding rotating frame transformation. For example, $U=e^{i\omega_\mathrm{D}\hat{d}_+^\dagger\hat{d}_+t}$, we get \begin{equation} \begin{split} \hat{H}&=(\omega_+-\omega_\mathrm{D})\hat{d}_+^\dagger\hat{d}_++\omega_-\hat{d}_-^\dagger\hat{d}_-\\ &+g_{cm}J_1(\frac{\Omega}{\omega_\mathrm{D}})(\hat{d}_+^\dagger\hat{d}_-+\hat{d}_+\hat{d}_-^\dagger )+......, \end{split} \end{equation} where the left out term is non-resonant. Obviously, we see the $\hat{d}_-$ mode is coupled to $\hat{d}_+$ negative one side-band with coupling strength $g_{cm}J_1(\Omega/\omega_\mathrm{D})$, which leads to the anti-crossing in the reflection spectrum. In the experiment, we didn't see any anti-crossing between the $\pm 1$ side-band of each hybridized modes. The reason is obvious if we notice that there is only odd terms in the summation. Specifically, for example, there is no $n=2$ term, which means in whatever rotating frame we can't get a Hamiltonian in the form \begin{equation} \begin{split} \hat{H}&=(\omega_+-\omega_\mathrm{D})\hat{d}_+^\dagger\hat{d}_++(\omega_-+\omega_\mathrm{D})\hat{d}_-^\dagger\hat{d}_-\\&+{g_{cm}J_2(\frac{\Omega}{\omega_\mathrm{D}})(\hat{d}_+^\dagger\hat{d}_-+\hat{d}_+\hat{d}_-^\dagger )}+...... \end{split} \end{equation} Thus there will be {no anti-crossing} between the two $\pm 1$ side bands. In fact, we can generalize the result: denote $k$ ($l$) as the $k_{th}$ ($l_{th}$) side-band of $\hat{d}_+$ ($\hat{d}_-$) mode, then there will be no anti-crossing between the side-bands if $k+l=even$. \section*{Appendix IV: Autler Townes splitting} Similar to a driven two level system, there is mode splitting when we apply a weak drive. This can be easily seen by rewriting Eq. \ref{eq1} in terms of the hybridized mode and express it in the rotating frame $U=\exp(-i\hat{d}_+^\dagger\hat{d}_+\varepsilon(t)/2-i\hat{d}^\dagger_-\hat{d}_-\varepsilon(t)/2)$, thus we have \begin{equation} \hat{H}=\omega_+\hat{d}_+^\dagger\hat{d}_++\omega_-\hat{d}_-^\dagger\hat{d}_-+\frac{\Omega}{2}\cos(\omega_\mathrm{D}t)(\hat{d}_+^\dagger\hat{d}_-+\hat{d}_+\hat{d}_-^\dagger), \end{equation} where the hybridized mode frequency $\omega_+=\omega_0+g_{cm},\omega_-=\omega_0-g_{cm}$. We see that the problem becomes resembling a two level system driven by a laser field. The Autler Townes splitting just follows. A standard method of solving the Hamiltonian is to adopt the rotating wave approximation. First, take the Hamiltonian in the rotating frame, we have $U=\exp({i\omega_+\hat{d}^\dagger_+\hat{d}_+t+i\omega_-\hat{d}^\dagger_-\hat{d}_-t})$, \begin{equation} \begin{split} \hat{H}&=\frac{\Omega}{2} \cos(\omega_\mathrm{D}t)(\hat{d}_+^\dagger\hat{d}_-e^{i(\omega_+-\omega_-)t}+\hat{d}_+\hat{d}_-^\dagger e^{-i(\omega_+-\omega_-)t})\\ &=\frac{\Omega}{2} (\frac{e^{i\omega_\mathrm{D}t}+e^{-i\omega_\mathrm{D}t}}{2})(\hat{d}_+^\dagger\hat{d}_-e^{i\Delta\omega t}+\hat{d}_+\hat{d}_-^\dagger e^{-i\Delta\omega t})\\ &\simeq \frac{\Omega}{4} (\hat{d}_+^\dagger\hat{d}_-e^{i(\Delta\omega-\omega_\mathrm{D}) t}+\hat{d}_+\hat{d}_-^\dagger e^{-i(\Delta\omega-\omega_\mathrm{D}) t}), \end{split} \end{equation} where we define $\Delta\omega=\omega_+-\omega_-$, and the counter rotating term is ignored in the third line. We can further write down an equivalent form \begin{equation} \hat{H}=\frac{\delta\omega_\mathrm{D}}{2}\hat{d}^\dagger_1\hat{d}_+-\frac{\delta\omega_\mathrm{D}}{2}\hat{d}_-^\dagger\hat{d}_-+\frac{\Omega}{4}(\hat{d}_+^\dagger\hat{d}_-+\hat{d}_+\hat{d}_-^\dagger), \end{equation} where $\delta\omega_\mathrm{D}=\Delta\omega-\omega_\mathrm{D}$. We thus obtain a time-independent Hamiltonian, and it can be easily diagonalized. Each mode furthre splits into two branches with the separation given by \begin{equation} \Delta=\sqrt{\frac{\Omega^2}{4}+\delta_{\omega_\mathrm{D}}^2}. \end{equation} \section*{Appendix V: Model comparison} Figure\,\ref{figS1} provides a more detailed comparison of the calculation results using the Floquet model and the RWA model versus the measurement result. It is evident that the Floquet model shows excellent agreement with the experimental results while the RWA model shows significant deviation. These line plots correspond to the maximum driving amplitude condition in the intensity maps of Figs.4 (d)--(f) in the main text. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{FigureS1.pdf} \caption{Line plots comparing the experiment and calculation results for $\omega_\mathrm{D}=2\pi\times 3.85$ MHz and a driving amplitude of 10 V peak-to-peak, which corresponds to $\Omega=2\pi\times8$ MHz.} \label{figS1} \end{figure} \section*{Appendix VI: System configuration and parameters} The dielectric resonator we used has a diameter of 6 mm and height of 4 mm. It is hosted inside a copper housing to eliminate radiation losses. The housing is sufficiently large (diameter: 32 mm; height: 25 mm) to minimize the absorption of the metal wall. A Teflon spacer (not shown in Fig.\,\ref{figS2}a) is used to support the cylinder inside the housing. TE$_{01\delta}$ is the fundamental mode of this dielectric resonator. The loop antennas are made by terminating non-magnetic, semi-rigid coaxial cables with a loop. The innder diameter of the driving loop is around 900 $\mu$m, which is sufficiently small to enhance the driving efficiency but not too small to severely perturb the magnon resonance. The YIG sphere is glued inside the loop of the driving antenna using curable varnish. It has a diameter of 400 $\mu$m with a highly polished surface. The sphere is made of high-quality single-crystal YIG, with a saturation magnetization of 1780 Oe. System parameters extracted from experimental results in Figs.\,1 and 2 of the main text are: cavity resonance frequency $\omega_c=2\pi\times 8.505$ GHz; coupling strength between cavity resonance and fundamental magnon mode ($m_0$) $g_{cm0}=2\pi\times 14.0$ MHz; coupling strength between cavity resonance and the second magnon mode ($m_1$) $g_{cm1}=2\pi\times 6.3$ MHz; dissipation rate of fundamental magnon mode $\kappa_{m0}=2\pi\times4.4$ MHz; dissipation rate of second magnon mode $\kappa_{m1}=2\pi\times3.7$ MHz; dissipation rate of cavity photon mode $\kappa_c=2\pi\times2.0$ MHz. In Fig.\,\ref{figS2}, the cavity resonance is shifted to $\omega_c=2\pi\times 8.41$ GHz with a dissipation rate: $\kappa_c=2\pi\times1.5$ MHz. With the maximum driving amplitude that is available in our measurements (10 V peak-to-peak), the corresponding driving strength is calculated as $\Omega=2\pi\times 8$ MHz. Fundamental magnon mode ($m_0$) is used in Fig.\,2 and Fig.\,3 in the main text. The second magnon mode ($m_1$) is used in Fig.\,\ref{figS2}. \section*{Appendix VII: Magnetic Tunability} One distinct advantage of cavity electromagnonics is that the magnon frequency can be conveniently tuned by the external field. This allows us to study the Floquet driven magnon-photon coupling under various detuning conditions. The most prominent effect of the magnetic field is that it determines the driving frequency for ATS. From the above zero-detuning analysis, it is clear that ATS only occurs when the driving frequency matches the level separation: $\omega_\mathrm{D}=\Delta\omega$. However, the detuning can take non-zero values in our system and it is therefore magnetically tunable, which accordingly affects the level separation $\Delta\omega=\sqrt{4g_{cm}^2+\delta\omega^2}$ and consequently the required driving frequency $\omega_\mathrm{D}$. Figure\,\ref{figS2}a plots the measured cavity reflection spectra with ATS at various bias magnetic fields. The driving frequency needed for observing ATS is summarized in Fig.\,\ref{figS2}b as a function of the bias field. \begin{figure}[tb] \centering \includegraphics[width=\linewidth]{FigureS2.pdf} \caption{\textbf{a}, Summary of the cavity reflection spectra measured with $\omega_\mathrm{D}=\Delta\omega$ for different magnet positions. The driving amplitude is 10 V peak-to-peak. \textbf{b}, Extracted Autler-Townes splitting (ATS) $\Delta\omega_\mathrm{AT}$ for both the lower and upper branches (left y axis), and driving frequency $\omega_\mathrm{D}$ needed to observing ATS (right y axis) as a function of the magnet position.} \label{figS2} \end{figure} In addition, the splitting ($\Delta\omega_\mathrm{AT}$) of each hybrid mode $d_\pm$ is also controlled by the bias magnetic field, as shown in Fig.\,\ref{figS2}a. This is more clearly shown in Fig.\,\ref{figS2}b, where the splitting varies with the bias field for both hybrid modes and reaches the maximum when magnon and cavity photon modes are on resonance (at $x=21.75$ mm). These capability of off-resonance operations demonstrated here enables new approaches for hybridizing magnons and microwave photons. For instance, magnons can be applied off-resonance to have minimal intrinsic coupling with cavity photons, and then the magnon-photon coupling can be achieved by the Floquet drive in a controllable way which allows the coupling to be completely turned off and on. This is different from the on-resonance operations where even if the Floquet drive is turned off, magnons and microwave photons are still hybridized because of the intrinsic coupling.

proofpile-arXiv_069-6846

\section{Introduction}{\label{Intro}} Six dimensional supergravity theories \cite{6DRef1,6DRef2,6DRef3,6DRef4,6DRef5,6DRef6,6DRef7} have been studied from various perspectives as they can be helpful in our understanding of the fundamental properties of the nature. The $\mathcal{N} = (1,0)$ gauged theory, known as the Salam - Sezgin model \cite{SalamSezgin}, spontaneously compactifies on $M_4 \times S^2$ with $\mathcal{N} = 1$ supersymmetry, thus becoming a natural starting point of phenomenological studies with a string theory origin \cite{SalamSezginString}. When extended with certain higher derivative terms, six dimensional $\mathcal{N} = (1,0)$ supergravity provides a useful testbed for checking proposals regarding string duality at higher orders \cite{MinasianLiu1,6DGB,MinasianLiu2}. Six dimensional supergravity theories also found themselves applications in $AdS_3/CFT_2$ correspondence \cite{deBoer}, thus providing a framework for studying the superconformal field theory in two dimensions. Initial works on six-dimensional supergravity were based on the Noether procedure and the construction of $\mathcal{N} = (1,0)$ supergravity as well as its matter couplings was put into a systematic approach using superconformal tensor calculus in \cite{Bergshoeff6D}. This approach, which was originally developed in \cite{SCTC1,SCTC2,SCTC3,SCTC4}, is an off-shell methodology based on enhancing the symmetries as much as possible which restricts the possible couplings of fields within a multiplet and ease the construction of an action principle. In six dimensions, the conformal $\mathcal{N} = (1,0)$ supergravity is based on the superalgebra $OSp(6,2|1)$ with the generators \begin{eqnarray} P_a\,, M_{ab}\,, D\,, K_a \,, U_{ij} \,, Q_\alpha^i\,, S_\alpha^i \,, \end{eqnarray} where $a,b,\ldots $ are the Lorentz indices, $\alpha$ is a spinor index and $i,j = 1, 2$ is an $SU(2)$ index. Here, the first four generators, $\{P_a, M_{ab}, D, K_a \}$ are the generators of conformal algebra. $U_{ij}$ is the generator of $SU(2)$ R-symmetry and $Q_\alpha^i$ and $S_\alpha^i$ are the generators of the $Q-$SUSY and the $S-$SUSY respectively. In the superconformal approach to supergravity, one associates a gauge field to each generator and imposes a set of constraints to relate the gauge theory of $OSp(6,2|1)$ superalgebra to gravity. For six-dimensional $\mathcal{N} = (1,0)$ theory, such a set of constraints separates the gauge fields in a dependent and independent set of fields such that the independent set does not provide an equal number of off-shell bosonic and fermionic degrees of freedom. This can be cured by adding a matter content, which is not unique and lead to two different Weyl multiplets: the standard Weyl multiplet and the dilaton Weyl multiplet \cite{Bergshoeff6D}. It is, however, not possible to obtain a consistent two-derivative supergravity with a standard Weyl multiplet\footnote{A consistent model for the standard Weyl multiplet can be achieved by the addition of higher curvature terms, see \cite{6DHD}.} but a dilaton Weyl multiplet coupled to a scalar (hyper) \cite{Bergshoeff6D,Sokatchev}, or a non-linear \cite{Bergshoeff6D}, or a real $O(2n)$ \cite{O2nMultiplet} or a linear multiplet is necessary \cite{Bergshoeff6D}. If an off-shell supergravity is required, then the linear multiplet is the simplest and the most utilized option whose gauge fixing of redundant superconformal symmetries lead to an off-shell six dimensional $\mathcal{N}=(1,0)$ supergravity. In \cite{6DHD,OzkanThesis}, it was shown that the linear multiplets are also essential in the construction of higher derivative models. Thus, following the previous work on four dimensional tensor multiplets \cite{deWit4D} and five dimensional linear multiplets \cite{Ozkan5D}, our aim in this paper is to provide a detailed investigation of six dimensional $\mathcal{N} = (1,0)$ linear multiplets, their rigid and supergravity couplings and their higher derivative actions. As we will discuss in the next sections, the couplings of $n$-number of linear multiplets are controlled by a function $\mathcal{F}_{IJ}$ which is a function of $SU(2)$ triplet of scalars $L_{ij}$ of the linear multiplet. The function is not completely free but constrained by symmetries of the theory, although the restrictions are quite generic and mild and we provide various possible choices of $\mathcal{F}_{IJ}$. This paper is organized as follows. In Section \ref{Sec2}, we first introduce rigid vector and linear multiplet and their supersymmetric coupling. The vector multiplet is an essential element in the construction of an action principle for the linear multiplets and their coupling gives rise to an a rigid linear multiplet action once we express the elements of the vector multiplet in terms of a proper combination of the elements of the linear multiplet. At this step, we introduce the function $\mathcal{F}_{IJ}$ which completely determines the interaction of linear multiplets. Various choices of $\mathcal{F}_{IJ}$ as well as rigid higher derivative models are discussed. Finally, we provide the dimensional reduction of vector and linear multiplets to 5D. In Section \ref{Sec3}, we introduce local superconformal two-derivative vector and linear multiplet actions. The linear multiplet action can be obtained both in the standard and the Weyl multiplet background. For the vector multiplet, the necessity for a compensating scalar for scaling symmetry implies either the use of dilaton Weyl multiplet or a linear multiplet. From a more minimalist approach, we provide a model in a dilaton Weyl multiplet. Nonetheless, we discussed an ansatz for a conformal vector-linear multiplet action, which we plan to address its details in a future publication. Next, in Section \ref{Sec4} we gauge fix and provide various supergravity models. These models include off-shell Poincar\'e supergravity both in Einstein and Jordan frames, the supersymmetric coupling of linear multiplets to supergravity and an off-diagonal $R Y_{ij}$ invariant which would lead to an on-shell six dimensional $R^2$ supergravity upon coupling to an off-shell vector multiplet and the elimination of auxiliary fields. We give our comments and conclusions in Section \ref{Sec5}. \section{Rigid Linear Multiplet Couplings}{\label{Sec2}} The six dimensional ${\mathcal{N}}=(1,0)$ linear multiplets can be realized off-shell in a general superconformal background. In this section, we focus on its rigid supersymmetric realization on flat Minkowski background. The linear multiplet consists of an $SU(2)$ triplet of scalars $L_{ij}$, a tensor gauge field $E_{\mu\nu}$ and an $SU(2)$ Majorana spinor $\varphi^i$ of negative chirality. The supersymmetry transformation rules $(\epsilon^i)$ can be given as follows\footnote{We use the conventions of \cite{Coomans6D}. In particular, the spacetime signature is $(-, +, +, +, +,+)$ and $\psi^i \gamma_{(n)} \chi^j = t_n \chi^j \gamma_{(n)} \psi^i$ where $t_0 = t_3 = t_4 = -1$ and $t_1 = t_2 = h_5 = t_6 = 1$. When $SU(2)$ indices on spinors are omitted, northwest -southeast contraction is understood.} \cite{Bergshoeff6D,Coomans6D} \begin{eqnarray} \delta L^{ij} &=& \bar{\epsilon}^{(i} \varphi^{j)}\nonumber \,,\\ \delta \varphi^i &=& \frac12 \slashed{\partial} L^{ij} \epsilon_j - \frac14 \slashed{E} \epsilon^i \nonumber\,,\\ \delta E_{\mu\nu} &=& \bar\epsilon \gamma_{\mu\nu} \varphi \,, \label{LinearRigidSUSY} \end{eqnarray} where the $Q$-supersymmetry parameter, $\epsilon^i$, is of positive chirality. Furthermore, $E_\mu$ is a constrained vector, $\partial^\mu E_\mu = 0$, and is related to the tensor gauge field $E_{\mu\nu}$ via \begin{eqnarray} E^\mu = \partial_\nu E^{\mu\nu} \,. \end{eqnarray} \subsection{Composite Vector Multiplets}{\label{SSec21}} Supersymmetric two-derivative actions for linear multiplets can be constructed by using vector multiplets as composite multiplets. The vector multiplet consists of a vector field $W_\mu$, an $SU(2)$ Majorana spinor of positive chirality $\Omega^i$ and a triplet of scalar fields $Y^{ij}$. The supersymmetry transformations are given as \cite{Bergshoeff6D, Coomans6D} \begin{eqnarray} \delta W_\mu &=& - \bar{\epsilon} \gamma_\mu \Omega \,,\nonumber\\ \delta \Omega^i &=& \frac{1}{8} \gamma \cdot F \epsilon^i - \frac12 Y^{ij} \epsilon_j \,,\nonumber\\ \delta Y^{ij} &=& - \bar{\epsilon}^{(i} \slashed\partial \Omega^{j)} \,, \label{VectorRigidSUSY} \end{eqnarray} where $F_{\mu\nu} = \partial_\mu W_\nu - \partial_\nu W_\mu$. The key point in the construction of linear multiplet actions is that the supersymmetric coupling of a linear multiplet to a vector multiplet takes the following form \cite{Bergshoeff6D, Coomans6D} \begin{eqnarray} {\mathcal L} &=& Y_{ij} L^{ij} + 2 \bar\Omega \varphi + \frac14 F_{\mu\nu} E^{\mu\nu} \,. \label{VectorLinearAction} \end{eqnarray} As was already demonstrated in four \cite{deWit4D} and five dimensions \cite{Ozkan5D}, such auxiliary expressions can be used to derive supersymmetric Lagrangians for $n$-linear (or vector) multiplets. Such a construction is based on the observation that it is possible to construct composite expressions from the elements of the linear (or vector) multiplet that precisely transform as a vector (or linear) multiplet. Once such an expression is obtained, it can be used in the auxiliary action (\ref{VectorLinearAction}) to construct a supersymmetric linear (or vector) multiplet action. To construct an interacting $n$-linear multiplets, we follow the footsteps of \cite{deWit4D, Ozkan5D} and introduce a real function ${\mathcal F}_{IJ}(L)$ which is a function of the linear multiplet scalars $L_{ij}$. Note that the index $I,J=1,2,\ldots,n$ label the number of linear multiplets. A reasonable guess would be to start the construction by setting the lowest element of the vector multiplet as $\Omega^{i}_I = \mathcal{F}_{IJ} \slashed\partial \varphi^{iI}$, however, if $\mathcal{F}_{IJ} \neq \delta_{IJ}$, then such a starting point does not have the transformation structure given by (\ref{VectorRigidSUSY}). A starting point that gives rise to the correct structure is \begin{eqnarray} \Omega^i_I &=& {\mathcal{F}}_{IJ}\slashed \partial \varphi^{iJ} + \mathcal{F}_{IJKjk} \slashed\partial L^{ijJ} \varphi^{kK} - \frac12 \mathcal{F}_{IJK}{}{}^{ij} \slashed{E}^J \varphi^{K}_j + \frac16 \mathcal{F}_{IJKL}{}^{ijkl} \gamma^a \varphi_j^J \bar\varphi_k^K \gamma_a \varphi_l^L \,, \label{CompositeOmega} \end{eqnarray} where \begin{eqnarray} \mathcal{F}_{IJKij} = \frac{\partial \mathcal{F}_{IJ}}{\partial L^{ij K}} \,, \qquad \text{and} \qquad \mathcal{F}_{IJKLijkl} = \frac{\partial \mathcal{F}_{IJ}}{\partial L^{ij J} \partial L^{kl K}} \end{eqnarray} The necessity that composite expression for $\Omega_I^i$ transforms like the fermionic component of the vector multiplet implies three important constraints on $\mathcal{F}_{IJ}$ and its descendants \begin{eqnarray} \mathcal{F}_{IJK}{}^{ij} = \mathcal{F}_{I(JK)}{}^{ij} \,, \qquad \mathcal{F}_{IJKi[jk]l} = 0 \,. \label{RigidConstraints} \end{eqnarray} Upon varying the composite expression (\ref{CompositeOmega}) and insisting on the transformation rules (\ref{VectorRigidSUSY}), we obtain the following composite expressions for $Y_{ij}^I$ and $F_{ab}^I$ \begin{eqnarray} Y^{ij}_{I}&=& - \mathcal{F}_{IJ} \Box L^{ijJ} - \mathcal{F}_{IJKkl} \partial_a L^{ik J} \partial^a L^{jl K} - \mathcal{F}_{IJKk}{}^{(i} \partial_a L^{j)k J} E^{a K} - \frac14 \mathcal{F}_{IJK}{}^{ij} E_a^J E^{a K} \,,\nonumber\\ && - \frac14 \mathcal{F}_{IJK}{}^{ij} \bar\varphi^J \slashed\partial \varphi^K - 2 \mathcal{F}_{IJK}{}^{k(i} \bar\varphi_k^J \slashed \partial \varphi^{j) K} - \mathcal{F}_{IJKL}{}^{pqk(i} \partial_a L^{J j)}{}_k \bar\varphi_p^K \gamma^a \varphi_q^L \nonumber\\ && + \frac12 \mathcal{F}_{IJKL}{}^{k(ij)l} \bar\varphi^J_k \gamma^a \varphi_l^K E_a^L - \frac{1}{12} \mathcal{F}_{IJKLM}{}^{ijklmn}\bar\varphi_k^J \gamma_a \varphi_l^K \bar\varphi_m^L \gamma^a \varphi_n^M \,,\nonumber\\ F_{ab I}&=& - 2 \partial_{[a} \left( \mathcal{F}_{IJ} E_{b]}^J \right) + 2 \mathcal{F}_{IJKj}{}^k \partial_{[a} L^{jl J} \partial_{b]} L_{lk}^K + 2 \partial_{[a} \left(\mathcal{F}_{IJK}{}^{ij} \bar\varphi_i^J \gamma_{b]} \varphi_j^K\right)\,. \label{RigidYandF} \end{eqnarray} Before we proceed to the actual construction of the action, a couple of remarks on these composite formulae are in order. First of all, these composite expressions match with the ones in \cite{Bergshoeff6D,Coomans6D} if only a single linear multiplet is chosen. In term of $\mathcal{F}_{IJ}$ such a choice would correspond to setting $\mathcal{F}_{11} = L^{-1}$. In the same spirit, an $n$-number of non-interacting linear multiplets can be obtained by setting $\mathcal{F}_{IJ} = \delta_{IJ} \left(L^I\right)^{-1}$ where $(L^I )^2= L_{ij}^I L^{ij I}$ \cite{deWit4D}. Second, we note that although $F_{ab}$ is closed, i.e. $\partial_{[a}F_{bc]}^I = 0$, it is not exact \cite{deWit4D}. Consequently, the present form of the vector-linear action (\ref{VectorLinearAction}) is essential to us although its last term can be written as $A_\mu E^\mu$ by means of an integration by parts. Finally, it is important to note that the index $I$ is fixed while the index $J$ is being summed over. This is also reflected by the fact that $\mathcal{F}_{IJ}$ has no particular symmetry in the $(IJ)$ indices. This was first noted in \cite{Ozkan5D} and it was shown that a non-symmetric choice plays an important role in the construction of higher derivative superinvariants. It is also possible to construct vector multiplet actions using the auxiliary action (\ref{VectorLinearAction}) as well as a composite linear multiplet that is constructed from the elements of the vector multiplet. \begin{eqnarray} L_{ij} = Y_{ij}\,, \qquad \varphi^i = - \slashed\partial \Omega^i \,,\qquad E^a = \partial_b F^{ba} \,. \label{RigidVectorMap} \end{eqnarray} The vector multiplet action will be of particular use in constructing higher derivative actions for linear multiplets. \subsection{Supersymmetric Linear and Vector Multiplet Actions}{\label{SSec22}} With the composite fields in hand, we now proceed to give the rigid supersymmetric linear multiplet action. Substituting the composite formulae (\ref{CompositeOmega}) and (\ref{RigidYandF}) into the auxiliary action (\ref{VectorLinearAction}) we obtain \begin{eqnarray} \mathcal{L}_L &=& - \mathcal{F}_{IJ} L_{ij}^I \Box L^{ijJ} - \mathcal{F}_{IJKkl} L_{ij}^I \partial_a L^{ik J} \partial^a L^{jl K} - \mathcal{F}_{IJKk}{}^{i} L_{ij}^I \partial_a L^{jk J} E^{a K} - \frac12 \mathcal{F}_{IJ} E^{aI} E_{a}^J \nonumber\\ && - \frac14 \mathcal{F}_{IJK}{}^{ij} L_{ij}^I E_a^J E^{a K} + \frac12 \mathcal{F}_{IJKj}{}^k E^{ab I} \partial_{a} L^{jl J} \partial_{b} L_{lk}^K - \frac14 \mathcal{F}_{IJK}{}^{ij} L_{ij}^I \bar\varphi^J \slashed\partial \varphi^K \nonumber\\ && - 2 \mathcal{F}_{IJK}{}^{ki} L_{ij}^I \bar\varphi_k^J \slashed \partial \varphi^{j K} + 2 {\mathcal{F}}_{IJ}\bar\varphi^I \slashed\partial \varphi^{J} + \frac12 \mathcal{F}_{IJKL}{}^{kijl} L_{ij}^I \bar\varphi^J_k \slashed{E}^L \varphi_l^K + \mathcal{F}_{IJK}{}{}^{ij} \bar\varphi^I_i \slashed{E}^J \varphi^{K}_j \nonumber\\ && + \frac12 \mathcal{F}_{IJK}{}{}^{ij} \bar\varphi^J_i \slashed{E}^I \varphi^{K}_j - 2\mathcal{F}_{IJKjk} \bar\varphi^I_i \slashed\partial L^{ijJ} \varphi^{kK} + \mathcal{F}_{IJKLk}{}^{ipq} L_{ij}^I \bar\varphi_p^K \slashed\partial L^{ jkJ} \varphi_q^L \nonumber\\ && - \frac13 \mathcal{F}_{IJKL}{}^{ijkl} \bar\varphi^I_i \gamma^a \varphi_j^J \bar\varphi_k^K \gamma_a \varphi_l^L - \frac{1}{12} \mathcal{F}_{IJKLM}{}^{ijklmn}L_{ij}^I \bar\varphi_k^J \gamma_a \varphi_l^K \bar\varphi_m^L \gamma^a \varphi_n^M \,, \label{RigidNLinear} \end{eqnarray} up to partial integrations. Once again, we remind the reader that the index $I$ is fixed and not being summed over. However, if we choose to sum over the $I$ index as well, then the linear multiplet action can be written as \begin{eqnarray} \mathcal{L}_L &=& - \frac12 F_{IJ} \partial_a L_{ij}^I \partial^a L^{ijJ} + \frac14 F_{IJ} E_a^I E^{aJ} - \frac12 F_{IJKj}{}^k E^{abI} \partial_a L^{jl J} \partial_b L^K_{kl} \,, \end{eqnarray} up to an overall minus sign and partial integrations. Here we only provide the bosonic part and made use of the following definitions \cite{deWit4D} \begin{eqnarray} F_{IJ} &=& 2 \mathcal{F}_{(IJ)} + \mathcal{F}_{KIJ}{}^{ij} L_{ij}^K \,,\nonumber\\ F_{IJK}{}^{ij} &=& 3 \mathcal{F}_{(IJK)}{}^{ij} + \mathcal{F}_{LIJK}{}^{klij} L_{kl}^L \,, \label{DefFIJ} \end{eqnarray} and heavily used the following $SU(2)$ identities \cite{deWit4D} \begin{eqnarray} K_{ik} L^{jk} + K^{jk} L_{ik} &=& \delta_i^j K_{mn} L^{mn} \,,\nonumber\\ K_{ij} L_{kl} - K_{kl} L_{ij} &=& \varepsilon_{ik} \varepsilon^{mn} \left(K_{lm} L_{nj} + K_{jm} K_{nl}\right)|_{(i,j) (k,l)} \,. \end{eqnarray} At this stage, a brief discussion on various choice of $\mathcal{F}_{IJ}$ is in order \begin{enumerate}[i.] \item {\underline{$\mathcal{F}_{IJ} = \delta_{IJ}$:} For this choice, $\mathcal{F}_{IJ}$ is independent of $L_{ij}$ and all descendants of $\mathcal{F}_{IJ}$ vanish. The map between the vector and the linear multiplets significantly simplifies, i.e. for a single linear multiplet we have \begin{eqnarray} \Omega^i = \slashed\partial \varphi^i \,, \qquad Y^{ij} = - \Box L^{ij} \,, \qquad F_{ab} = - 2 \partial_{[a} E_{b]} \,, \label{SingleLinear1} \end{eqnarray} and the linear multiplet action is given by \begin{eqnarray} \mathcal{L}_L &=& - \partial_a L_{ij} \partial^a L^{ij} + \frac12 E_a E^a - 2 \bar\varphi \slashed\partial \varphi \,. \label{LLinear1} \end{eqnarray} up to an overall minus sign and boundary terms. The map (\ref{SingleLinear1}) and the action (\ref{LLinear1}) will be particularly useful in the construction of higher derivative linear and vector multiplet actions. } \item{{\underline{$\mathcal{F}_{IJ} = \delta_{IJ} (L^I)^{-1}$:} } This choice is particularly relevant to the construction of superconformal linear multiplet actions. In this case, we have \begin{eqnarray} \mathcal{F}_{IJK}{}^{ij} = - (L^I)^{-3} L^{ij I}\,, \qquad \mathcal{F}_{IJKL}{}^{ijkl} = (L^I)^{-5} \left(3 L^{ijI} L^{jkI} + \left(L^I \right)^2 \varepsilon^{i(k} \varepsilon^{l)j} \right) \,. \end{eqnarray} For a single linear multiplet, the bosonic part of Lagrangian that corresponds to this choice is \begin{eqnarray} \mathcal{L}_{CL} &=& - \frac12 L^{-1} \partial_a L_{ij} \partial^a L^{ij} + \frac14 L^{-1} E_a E^a + \frac12 L^{-3} E^{ab} L^l{}_k \partial_a L^{kp} \partial_b L_{lp} \,, \label{ConformalRigid1} \end{eqnarray} up to an overall minus sign and boundary terms. } \end{enumerate} We may also use the composite expressions for the components of the linear multiplet (\ref{RigidVectorMap}) to construct a vector multiplet action \begin{eqnarray} \mathcal{L}_V &=& - \frac14 F_{ab} F^{ab} - 2 \bar\Omega \slashed\partial \Omega + Y^{ij} Y_{ij} \,. \label{RigidVectorAction} \end{eqnarray} As with the linear multiplet action, the composite formulae (\ref{RigidVectorMap}) as well as the vector multiplet action (\ref{RigidVectorAction}) are useful in the construction of higher derivative linear and vector multiplet actions. When coupled to supergravity, $N= (1,0)$ higher derivative vector and linear multiplet actions are known to give rise to higher curvature superinvariants \cite{OzkanThesis,6DHD}. In the case of rigid supersymmetry, we may obtain a higher derivative linear multiplet action by employing the composite vector multiplet fields (\ref{RigidVectorMap}) in the vector multiplet action (\ref{RigidVectorAction}) \begin{eqnarray} \mathcal{L}_L &=& \frac{1}{M^2} \left( \Box L_{ij} \Box L^{ij} - \partial_{[a} E_{b]}\partial^{[a} E^{b]} + 2 \Box \bar\varphi \slashed\partial \varphi \right) \,, \end{eqnarray} where $M$ is some mass parameter. We may follow a similar procedure for the vector multiplets and use the composite linear multiplet (\ref{SingleLinear1}) into the linear multiplet action (\ref{LLinear1}) \begin{eqnarray} \mathcal{L}_V &=& \frac{1}{M^2} \left( - \partial_a Y_{ij} \partial^a Y^{ij} + \frac12 \partial_b F^{ab} \partial^c F_{ac} - 2 \Box \bar\varphi \slashed\partial \varphi \right) \,. \end{eqnarray} One may alternatively wish to use the linear multiplet action (\ref{ConformalRigid1}), however, this Lagrangian involved inverse powers of $L$. To construct a higher derivative vector multiplet action in such a fashion, we first need to consider two distinct linear multiplets represented by $(L_{ij}, \varphi_i, E_a)$ and $(L_{ij}^\prime, \varphi_i^\prime, E_a^\prime)$ and choose $\mathcal{F}_{IJ}$ as \begin{eqnarray} \mathcal{F}_{22} = L^{-1}\,, \qquad \mathcal{F}_{21} = - L^{-3} L^\prime_{ij} L^{ij} \,, \qquad \mathcal{F}_{11} = \mathcal{F}_{12} = 0 \,. \label{NonSymmetricFIJ} \end{eqnarray} As noted in \cite{Ozkan5D}, such a choice is not symmetric in \textit{(1,2)} indices and satisfies all properties associated with $\mathcal{F}_{IJ}$. In this case, we may use the primed multiplet as a composite vector multiplet to obtain a higher derivative vector multiplet action. However, such an action also includes the fields of the unprimed linear multiplets. \subsection{Reduction to Five Dimensions}{\label{SSec23}} The rigid supersymmetric actions for five dimensional $\mathcal{N} = 2$ vector and linear multiplet actions has already been established for single and multiple number of multiplets \cite{Bergshoeff5D1, Bergshoeff5D2,Ozkan1}. As these multiplets and models can be obtained from a six dimensional $\mathcal{N} = (1,0)$ theory by a circle reduction, a brief discussion on this issue would be in order before we end our discussion on the rigid six dimensional vector and linear multiplets. The components of the vector multiplet decomposes according to \begin{eqnarray} \left(W_a, \Omega^i, Y_{ij} \right) \quad \rightarrow \quad \left( W_{\hat a}, W_5, \lambda^i, \mathcal{Y}_{ij} \right) \,, \end{eqnarray} and the linear multiplet decomposes according to \begin{eqnarray} \left( L_{ij}, \varphi^i, E_a \right) \quad \rightarrow \quad \left( {\rm{L}_{ij}}, \hat\varphi^i, {\rm{E}}_{\hat a} , {\rm{E}}_5 \right) \,. \end{eqnarray} where $E^{\hat a}$ is the four-dimensional divergence-free vector field, i.e. $\partial^{\hat a} {\rm{E}}_{\hat a} = 0$. To be more precise, let us introduce our convention for gamma-matrices, supersymmetry parameter $\epsilon$ and Dirac conjugated spinors \cite{BergshoeffSezgin5D} \begin{eqnarray} \gamma_a = {\rm i} \Gamma_{\hat a} \Gamma_5 \,, \qquad \epsilon = \varepsilon\,, \qquad \bar\epsilon = {\rm i} \bar\varepsilon \Gamma_5 \,. \end{eqnarray} We now make the following ansatz for the fields of the five dimensional $\mathcal{N} = 2$ vector multiplet \cite{BergshoeffSezgin5D} \begin{eqnarray} W_a = A_{\hat a}\,, \qquad W_5 = \rho \,, \qquad Y_{ij} = - {\mathcal Y}_{ij} \,, \qquad \Omega = -\frac12 \lambda \end{eqnarray} and obtain the supersymmetry transformation rules as \begin{align} \delta \rho =& \frac12 {\rm i} \bar\varepsilon \lambda \,, & \delta A_{\hat \mu} =& \frac12 \bar\varepsilon \Gamma_{\hat\mu} \lambda \,,\nonumber\\ \delta \lambda^i =& - \frac14 \Gamma \cdot F \varepsilon^i - \frac12 {\rm i} \slashed\partial \rho \varepsilon^i - {\mathcal Y}^{ij} \varepsilon_j\,, & \delta {\mathcal Y}^{ij} =& -\frac12 \varepsilon^{(i} \slashed\partial \lambda^{j)} \end{align} These transformation rules precisely match with the ones given in \cite{Bergshoeff5D1}. For the fields of the linear multiplet, we make the following ansatz \begin{eqnarray} L_{ij} = {\rm L}_{ij} \,, \qquad \varphi^i = \Gamma_5 \hat\varphi^i \,, \qquad E_a = -2 \rm{E}_{\hat a}\,, \qquad E_5 = - 2 N \,, \end{eqnarray} which gives rise to the following supersymmetry transformation rules \begin{align} \delta \rm{L}^{ij} =& {\rm i} \bar\varepsilon^{(i} \hat\varphi^{j)} \,, & \delta \hat{\varphi}^i =& - \frac12 {\rm i} \slashed\partial \rm{L}^{ij} \varepsilon_j - \frac 12 {\rm i} \slashed{\rm{E}} \varepsilon^i \,,\nonumber\\ \delta {\rm{E}}_{\hat a} =& -\frac12 {\rm i} \bar\varepsilon \Gamma_{\hat a \hat b} \partial^{\hat b} \hat\varphi \,, & \delta N =& \frac12 \bar\varepsilon \slashed\partial \hat\varphi \,. \end{align} These transformation rules precisely match with \cite{Bergshoeff5D1} and its application to supersymmetric linear multiplet Lagrangian (\ref{RigidNLinear}) reproduces the five dimensional linear multiplet actions \cite{Ozkan5D}. \subsection{Superconformal Vector and Linear Multiplets}{\label{SSec24}} So far we only considered vector and linear multiplets that are representation of six-dimensional $\mathcal{N}= (1,0)$ super-Poincar\'e algebra. However, these multiplets can be assigned full $\mathcal{N}=(1,0)$ superconformal symmetry in which case the fields pick up additional dilatation $(\Lambda_D)$ and special supersymmetry $(\eta)$ transformations. For the linear multiplet, the transformation rules are given by \begin{eqnarray} \delta L^{ij} &=& \bar{\epsilon}^{(i} \varphi^{j)} + 4 \Lambda_D L^{ij} \nonumber \,,\\ \delta \varphi^i &=& \frac12 \slashed{\partial} L^{ij} \epsilon_j - \frac14 \slashed{E} \epsilon^i - 4 L^{ij} \eta_j + \frac92 \Lambda_D \varphi^i \nonumber\,,\\ \delta E_{a} &=& \bar\epsilon \gamma_{ab}\partial^b \varphi - 5 \bar\eta \gamma_a \varphi + 5 \Lambda_D E_a \,, \label{LinearRigidConformalSUSY} \end{eqnarray} and for the vector multiplet, the transformation rules are \begin{eqnarray} \delta W_\mu &=& - \bar{\epsilon} \gamma_\mu \Omega \,,\nonumber\\ \delta \Omega^i &=& \frac{1}{8} \gamma \cdot F \epsilon^i - \frac12 Y^{ij} \epsilon_j + \frac32 \Lambda_D \Omega^i \,,\nonumber\\ \delta Y^{ij} &=& - \bar{\epsilon}^{(i} \slashed\partial \Omega^{j)} + 2 \bar\eta^{(i} \Omega^{j)} + 2 \Lambda_D Y^{ij} \,. \label{VectorRigidConformalSUSY} \end{eqnarray} These additional symmetries have immediate implications on the function $\mathcal{F}_{IJ}$. First of all, the scaling dimension of $\mathcal{F}_{IJ}$ must be $-4$ in order to match the scaling dimension of both sides in the composite formulae (\ref{CompositeOmega}) and (\ref{RigidYandF}) \begin{eqnarray} \delta_D \mathcal{F}_{IJ}(L) = - 4 \Lambda_D \mathcal{F}_{IJ}(L)\,. \label{ScalingF} \end{eqnarray} As a result, the choice $\mathcal{F}_{IJ} = \delta_{IJ}$ no longer holds in the superconformal case, $\mathcal{F}_{IJ} = \delta_{IJ} (L^I)^{-1}$ or the non-symmetric choice (\ref{NonSymmetricFIJ}) are still valid. Next, we turn to the special supersymmetry transformations of the fields. As $\Omega^i$ is S-SUSY invariant, $\mathcal{F}_{IJ}$ must satisfy \begin{eqnarray} \mathcal{F}_{IJKik}L^{kjK} = - \frac12 \,\delta_i^j \, \mathcal{F}_{IJ} \,. \end{eqnarray} We may multiply this identity with $L^{km J}$ and obtain a useful result \cite{deWit4D} \begin{eqnarray} \mathcal{F}_{IJKij} L_{pq}^{J} L^{pqK} = - \mathcal{F}_{IJ} L^J_{ij} \,, \end{eqnarray} which further implies that \begin{eqnarray} \mathcal{F}_{IJ} L_{ij}^I L^{ijJ} = F_{IJ} L_{ij}^I L^{ijJ} \,. \end{eqnarray} To summarize, the conformal $\mathcal{F}_{IJ}$ needs to satisfy the following four identities \begin{align} \mathcal{F}_{IJK}{}^{ij} &= \mathcal{F}_{I(JK)}{}^{ij} \,, & \mathcal{F}_{IJKi[jk]l} &= 0 \,,\nonumber\\ \delta_D \mathcal{F}_{IJ} = & - 4 \Lambda_D \mathcal{F}_{IJ}(L)\,, & \mathcal{F}_{IJKik}L^{kjK} &= - \frac12 \,\delta_i^j \, \mathcal{F}_{IJ} \,. \label{ConformalF} \end{align} While the composite superconformal vector multiplet can be achieved by the proper choice of $\mathcal{F}_{IJ}$, the composite linear multiplet (\ref{RigidVectorMap}) immediately fails due to the scaling dimension of the fields. However, one may push the idea that a superconformal composite linear multiplet can be obtained by mapping a vector as well as a compensating linear multiplet. For instance, one can start with the following ansatz for the $SU(2)$ scalar of a primed vector multiplet $L_{ij}^\prime$ \begin{eqnarray} L_{ij I}^\prime &=& {\mathcal{M}}_{IJ} Y_{ij}^J + {\mathcal M}_{IJKk(i} \bar\varphi^{kK} \Omega_{j)}^J \,. \end{eqnarray} where $\mathcal{M}_{IJ} $ is a function of $L_{ij}$ i.e. $\mathcal{M}_{IJ}= \mathcal{M}_{IJ}(L)$. By demanding proper dilatation, S- and Q-SUSY transformations, it is possible to push this ansatz to all fields and find the constraints that the function ${\mathcal{M}}_{IJ}$ should satisfy. However, as we will see in the next section, there is a simpler possibility to achieve a superconformal composite linear multiplet when we couple to conformal supergravity. The ``dilaton Weyl multiplet" of $\mathcal{N}=(1,0)$ superconformal theory includes a dilaton field $\sigma$ and a dilatino field $\psi^i$ that we can use to compensate the dilatation and S-SUSY transformations of the superconformal completion of the map (\ref{RigidVectorMap}). \section{Conformal Supergravity}{\label{Sec3}} In the previous section, we discussed the superconformal transformations of linear and vector multiplets that are defined in flat space where we dealt with space-time independent transformation parameters. In a superconformal background, where the rigid parameters are replaced by space-time ones, the transformation rules contain the gauge fields of the superconformal theory. In general, however, the gauge fields do not have the right counting to form a closed background ``Weyl" multiplet but the inclusion of matter fields is necessary. For the six dimensional $\mathcal{N} = (1,0)$ theory there are two sets of possible matter fields, one leading to the so-called standard Weyl multiplet and the other leading to the dilaton Weyl multiplet. We defer the details of these multiplets to the Appendix \ref{Appendix1}. The $Q-$ and $S-$supersymmetry transformation rules for the linear multiplet are given by \cite{Bergshoeff6D,Coomans6D} \begin{eqnarray} \delta L^{ij} &=& \bar{\epsilon}^{(i} \varphi^{j)} \nonumber \,,\\ \delta \varphi^i &=& \frac12 \slashed{\mathcal{D}} L^{ij} \epsilon_j - \frac14 \slashed{E} \epsilon^i - 4 L^{ij} \eta_j \nonumber\,,\\ \delta E_{a} &=& \bar\epsilon \gamma_{ab}\mathcal{D}^b \varphi + \frac{1}{24} \bar\epsilon \gamma_a \cdot T^- \varphi - \frac13 \bar\epsilon^i \gamma_a \chi^j L_{ij} - 5 \bar\eta \gamma_a \varphi \,, \label{LinearLocalConformalSUSY} \end{eqnarray} where the superconformal covariant derivatives are defined as \begin{eqnarray} \mathcal{D}_\mu L^{ij} &=& \partial_\mu L^{ij} - 4 b_\mu L^{ij} + 2 \mathcal{V}_\mu{}^{(i}{}_k L^{j)k} - \bar\psi_\mu^{(i} \, \varphi^{j)}\,,\nonumber\\ \mathcal{D}_{\mu} \varphi^i &=& \partial_\mu \varphi^i - \frac92 b_\mu \varphi^i - \mathcal{V}_\mu^{ij} \varphi_j - \frac12 \slashed\mathcal{D} L^{ij} \psi_{\mu j} + \frac14 \slashed{E} \psi_\mu^i + 4 L^{ij} \phi_j \,. \end{eqnarray} As in the rigid case, the algebra closes if $E_a$ satisfies \cite{Bergshoeff6D,Coomans6D} \begin{eqnarray} \mathcal{D}^a E_a - \frac12 \bar\varphi \chi =0 \,, \label{ConstraintE} \end{eqnarray} where the superconformal covariant derivative of $E_a$ is defined as \begin{eqnarray} \mathcal{D}_\mu E_a &=& \partial_\mu E_a - 5 b_\mu E_a + \omega_{\mu a}{}^b E_b - \bar\psi_\mu \gamma_{ab} \mathcal{D}^b \varphi- \frac{1}{24} \bar\psi_\mu \gamma_a \gamma \cdot T^- \varphi \nonumber\\ &&+ \frac13 \bar\psi^i_\mu \gamma_a \chi^j L_{ij} + 5 \bar\phi_\mu \gamma_a \varphi \,. \end{eqnarray} Note that the constraint equation (\ref{ConstraintE}) requires an additional $\bar\varphi \chi$ term in order to maintain the $Q-$ and $S-$invariance of the constraint. The constraint on $E_a$ allow us to define a four-form gauge field $E_{\mu\nu\rho\sigma}$ which can be dualized to a two-form gauge field $E_{\mu\nu}$ that is defined via \begin{eqnarray} E^a = e_\mu{}^a \mathcal{D}_\nu E^{\mu\nu} \,, \end{eqnarray} where the supersymmetry transformation rule for $E_{\mu\nu}$ is given by \begin{eqnarray} \delta E_{\mu\nu} &=& \bar\epsilon \gamma_{\mu\nu} \varphi + \bar\varphi^{\rho}_i \gamma_{\mu\nu\rho} \epsilon_j L^{ij} \,. \end{eqnarray} The gauge fields of the Weyl multiplets that appear here are the sechsbein $e_\mu{}^a$, the spin-connection $\omega_\mu{}^{ab}$, the dilatation gauge field $b_\mu$, the $SU(2)$ R-symmetry gauge field $\mathcal{V}_\mu^{ij}$, and the $Q-$ and the $S-$ supersymmetry gauge fields $\psi_\mu^i$ and $\phi_\mu^i$. As the matter multiplets are inert under special conformal symmetry, its corresponding gauge field $f_\mu{}^a$ does not appear in the transformation rules. The set of fields $(\omega_\mu{}^{ab},f_\mu{}^a, \phi_\mu^i)$ are not independent but can be expressed in terms of the independent fields $(e_\mu{}^a, b_\mu, \mathcal{V}_\mu^{ij}, \psi_\mu^i)$. Transformation rules also consist of a real scalar field $D$, an antisymmetric tensor of negative duality $T_{abc}^-$ and an $SU(2)$ Majorana-Weyl spinor of negative chirality $\chi^i$. These matter fields are the fundamental fields of the standard Weyl multiplet but can be expressed in terms of a dilaton Weyl multiplet coupled to a tensor multiplet. For the vector multiplet, the $Q-$ and $S-$ transformation rules are given by \cite{Bergshoeff6D,Coomans6D} \begin{eqnarray} \delta W_\mu &=& - \bar{\epsilon} \gamma_\mu \Omega \,,\nonumber\\ \delta \Omega^i &=& \frac{1}{8} \gamma \cdot \widehat F \epsilon^i - \frac12 Y^{ij} \epsilon_j \,,\nonumber\\ \delta Y^{ij} &=& - \bar{\epsilon}^{(i} \slashed\mathcal{D} \Omega^{j)} + 2 \bar\eta^{(i} \Omega^{j)} \,, \label{VectorLocalConformalSUSY} \end{eqnarray} where the superconformal field strength $\widehat F_{\mu\nu}(W)$ and the supercovariant derivative $\mathcal{D}_\mu \Omega^i$ are defined as \begin{eqnarray} \widehat F_{\mu\nu}(W) &=& 2 \partial_{[\mu} W_{\nu]} + 2 \bar\psi_{[\mu} \gamma_{\nu]} \Omega \,,\nonumber\\ \mathcal{D}_\mu \Omega^i &=& \partial_\mu \Omega^i - \frac32 b_\mu \Omega^i + \frac14 \omega_\mu{}^{ab} \gamma_{ab} \Omega^i - \frac12 \mathcal{V}_{\mu}{}^i{}_j \Omega^j - \frac18 \gamma \cdot \widehat F \psi_\mu^i + \frac12 Y^{ij} \psi_{\mu j} \,. \end{eqnarray} With the supersymmetry transformation rules in hand, we are now in a position to generalize rigid composite vector multiplet (\ref{CompositeOmega}) and (\ref{RigidYandF}) and to construct local linear and vector multiplet actions. The starting point of an action principle is the generalization of rigid auxiliary action (\ref{VectorLinearAction}) with the inclusion of the Weyl multiplet fields \cite{Bergshoeff6D,Coomans6D} \begin{eqnarray} e^{-1} \mathcal{L}_{VL} = Y^{ij} L_{ij} + 2 \bar\Omega \varphi - L_{ij} \bar\psi_{\mu}^i \gamma^\mu \Omega_j + \frac14 F^{\mu\nu} E_{\mu\nu} \,. \label{LocalVL} \end{eqnarray} Given the properties (\ref{ScalingF}) and (\ref{ConformalF}), the composite vector multiplet is given by \begin{eqnarray} \Omega^i_I &=& {\mathcal{F}}_{IJ}\slashed \mathcal{D} \varphi^{iJ} + \mathcal{F}_{IJKjk} \slashed\mathcal{D} L^{ijJ} \varphi^{kK} + \frac23 \mathcal{F}_{IJ} L^{ij J} \chi_j + \frac{1}{12} \mathcal{F}_{ij} \gamma \cdot T^- \varphi^{iJ} - \frac12 \mathcal{F}_{IJK}{}{}^{ij} \slashed{E}^J \varphi^{K}_j \nonumber\\ && + \frac16 \mathcal{F}_{IJKL}{}^{ijkl} \gamma^a \varphi_j^J \bar\varphi_k^K \gamma_a \varphi_l^L \,,\nonumber\\ Y^{ij}_{I}&=& - \mathcal{F}_{IJ} \Box^c L^{ijJ} - \mathcal{F}_{IJKkl} \mathcal{D}_a L^{ik J} \mathcal{D}^a L^{jl K} - \mathcal{F}_{IJKk}{}^{(i} \mathcal{D}_a L^{j)k J} E^{a K} - \frac14 \mathcal{F}_{IJK}{}^{ij} E_a^J E^{a K} \,,\nonumber\\ && - \frac13 \mathcal{F}_{IJ} L^{ij J} D + \frac16 \mathcal{F}_{IJ} \bar\chi^{(i} \varphi^{j)J} + \frac43 \mathcal{F}_{IJK}{}^{k(i} L^{j)lJ} \bar\chi_k \varphi_l^K + \frac{1}{12} \mathcal{F}_{IJK}{}^{ij} \bar\varphi \gamma \cdot T^- \varphi \nonumber\\ && - \frac14 \mathcal{F}_{IJK}{}^{ij} \bar\varphi^J \slashed\mathcal{D} \varphi^K - 2 \mathcal{F}_{IJK}{}^{k(i} \bar\varphi_k^J \slashed \mathcal{D} \varphi^{j) K} - \mathcal{F}_{IJKL}{}^{pqk(i} \mathcal{D}_a L^{J j)}{}_k \bar\varphi_p^K \gamma^a \varphi_q^L \nonumber\\ && + \frac12 \mathcal{F}_{IJKL}{}^{k(ij)l} \bar\varphi^J_k \gamma^a \varphi_l^K E_a^L - \frac{1}{12} \mathcal{F}_{IJKLM}{}^{ijklmn}\bar\varphi_k^J \gamma_a \varphi_l^K \bar\varphi_m^L \gamma^a \varphi_n^M \,,\nonumber\\ \widehat F_{ab I}&=& - 2 \mathcal{D}_{[a} \left( \mathcal{F}_{IJ} E_{b]}^J \right) + 2 \mathcal{F}_{IJ} L^{ijJ} \widehat{R}_{abij}(\mathcal{V})+ 2 \mathcal{F}_{IJKj}{}^k \mathcal{D}_{[a} L^{jl J} \mathcal{D}_{b]} L_{lk}^K \nonumber\\ && + \mathcal{F}_{IJ} \bar{\widehat{R}}_{ab}(Q) \varphi^J + 2 \mathcal{D}_{[a} \left(\mathcal{F}_{IJK}{}^{ij} \bar\varphi_i^J \gamma_{b]} \varphi_j^K\right)\,. \label{ConformalCompositeVector} \end{eqnarray} where $\widehat R_{\mu\nu}{}^{ij} (\mathcal{V})$ and $\widehat R_{\mu\nu}{}^i (Q)$ are the superconformal covariant curvatures of the gauge fields $\mathcal{V}_{\mu}{}^{ij}$ and $\psi_\mu^i$ respectively, see Appendix \ref{Appendix1}. The superconformal d'Alembertian $\Box^c L_{ij}^I$ is defined as \begin{eqnarray} \Box^c L^{ij}_I &=& \left(\partial^a - 5 b^a + \omega_b{}^{ba} \right) \mathcal{D}_a L^{ij}_I - 8 f_a{}^a L^{ij}_I + 2 V^{\mu(i}{}_k \mathcal{D}_\mu L^{j)k}_I - \bar\psi_\mu^{(i} \mathcal{D}^\mu \varphi^{j)}_I- \frac16 \bar\psi_\mu \gamma^\mu \chi L^{ij}_I \nonumber\\ && + \frac{1}{6} \bar\psi_\mu^{(i} \gamma^\mu \chi^{k)} L^{j}{}_{kI} + \frac{1}{6} \bar\psi_\mu^{(j} \gamma^\mu \chi^{k)} L^{i}{}_{kI} - \frac1{24} \bar\varphi^{(i}_I \gamma \cdot T^- \gamma^\mu \psi_{\mu}^{j)} - \bar\varphi_I^{(i} \gamma^\mu \phi_\mu^{j)} \,. \end{eqnarray} As mentioned in the previous section, it is not possible to construct a conformal action for the vector multiplet unless we use a compensating multiplet. The compensating multiplet can be chosen as a linear multiplet, however, a local vector multiplet action then contains the fields of vector, linear as well as a Weyl multiplet. From a more minimalist approach, it is possible to use the dilaton Weyl multiplet to compensate the conformal symmetries. In that case, the correct transformation rule for a composite $L_{ij}$ can be obtained by using the scalar field of the dilaton Weyl multiplet $\sigma$ as well as the dilatino field $\psi^i$ as extra fields \cite{Bergshoeff6D} \begin{eqnarray} L^{ij} &=& \sigma Y^{ij} + 2 \bar\psi^{(i} \Omega^{j)} \,. \label{CompositeLocalLinear1} \end{eqnarray} The rest of the composite linear multiplet can then be obtained by the $Q-$variation of the composite $L^{ij}$, which is given by \cite{Bergshoeff6D} \begin{eqnarray} \varphi^i &=& - \sigma \slashed\mathcal{D} \Omega^i - \frac12 \slashed\mathcal{D} \sigma\, \Omega^i + \frac{1}{24} \gamma \cdot H\, \Omega^i + \frac14 \gamma \cdot \widehat F\, \psi^i + Y^{ij} \psi_j \,,\nonumber\\ E_{\mu\nu} &=& - \sigma F_{\mu\nu} - \frac14 \epsilon_{\mu\nu\rho\sigma\lambda\tau} B^{\rho\sigma} F^{\lambda\tau} + 2 \bar\Omega \gamma_{\mu\nu} \psi + \sigma \bar\Omega \gamma^\rho \gamma_{\mu\nu} \psi_\rho \,, \label{CompositeLocalLinear2} \end{eqnarray} where $H_{\mu\nu\rho}$ is the field strength of two-form gauge field $B_{\mu\nu}$, which is one of the matter fields in the dilaton Weyl multiplet \begin{eqnarray} H_{\mu\nu\rho} = 3 \partial_{[\mu} B_{\nu\rho]} + 3 \bar\psi_{[\mu} \gamma_{\nu\rho]} \psi + \frac32 \sigma \bar\psi_{[\mu} \gamma_\nu \psi_{\rho]} \,. \end{eqnarray} With the composite multiplets and the action principle (\ref{LocalVL}) it is possible to construct local superconformal actions for vector and linear multiplets. For the linear multiplet, the action contains a Ricci scalar term via the composite $Y_{ij}$ due to superconformal d'Alembertial of $L_{ij}$. This action can therefore be used to express an off-shell Poincar\'e supergravity after gauge fixing. Noticing that the bosonic part of composite $\widehat F_{ab I}$ can be written as \begin{eqnarray} F_{ab I} &=& - 2 \partial_{[a} \left( \mathcal{F}_{IJ} E_{b]}^J - 2 \mathcal{F}_{IJ} L_{ij}^J \mathcal{V}_{b]}{}^{ij} \right) + 2 \mathcal{F}_{IJKj}{}^k \partial_{[a} L^{jl J} \partial_{b]} L_{lk}^K \,, \end{eqnarray} we give the bosonic part of the linear multiplet action for $n$-number of linear multiplets as \begin{eqnarray} e^{-1} \mathcal{L}_L &=& - \mathcal{F}_{IJ} L_{ij}^I \Box^c L^{ijJ} - \mathcal{F}_{IJKkl} L_{ij}^I D_a L^{ik J} D^a L^{jl K} - \mathcal{F}_{IJKk}{}^{i} L_{ij}^I D_a L^{jk J} E^{a K} \,,\nonumber\\ && - \frac14 \mathcal{F}_{IJK}{}^{ij} L_{ij}^I E_a^J E^{a K} - \frac13 \mathcal{F}_{IJ} L_{ij}^I L^{ij J} D - \frac12 \mathcal{F}_{IJ} E^{aI} E_{a}^J + \mathcal{F}_{IJ} E^{a I} L_{ij}^J \mathcal{V}_{a}^{ij} \nonumber\\ && + \frac12 \mathcal{F}_{IJKj}{}^k E^{ab I} \partial_{a} L^{jl J} \partial_{b} L_{lk}^K \,, \label{LocalLinear} \end{eqnarray} where the $SU(2)$ covariant derivative is defined as \begin{eqnarray} D_\mu L^{ij} &=& \partial_\mu L^{ij} + 2 \mathcal{V}_\mu{}^{(i}{}_k L^{j)k} \,. \end{eqnarray} As before, there is no sum in the $I$ index and no particular symmetry in $(I,J)$ indices. However, if we choose to sum over the index $I$, then the bosonic part of the supersymmetric action considerably simplifies \begin{eqnarray} e^{-1} \mathcal{L}_{L, SW} &=& \frac25 F_{IJ} L_{ij}^I L^{ij J} R - \frac2{15} F_{IJ} L_{ij}^I L^{ij J} D - \frac12 F_{IJ} D_a L_{ij}^I D^a L^{ijJ} + \frac14 F_{IJ} E_a^I E^{aJ} \,,\nonumber\\ && + F_{IJ} E^{a I} L_{ij}^J \mathcal{V}_{a}^{ij} - \frac12 F_{IJKj}{}^k E^{abI} \partial_a L^{jl J} \partial_b L^K_{kl} \,, \end{eqnarray} where $F_{IJ}$ and its descendants are as defined (\ref{DefFIJ}) and $R$ is the Ricci scalar. There are three notable features of this action. First, as the linear multiplet is inert under special conformal transformations, $b_\mu$, the gauge field of dilatations, is the only independent field that transforms non-trivially under special conformal symmetry. As a result, all $b_\mu$ terms cancel each other out and the action does not contain a $b_\mu$ term. Second, the scalar field $D$ imposes a severe constraint on the scalars of the linear multiplet, i.e. $F_{IJ} L_{ij}^I L^{ij J}$. Such a constraint also annihilates the pre-factor of the Ricci scalar, not allowing us to obtain off-shell Einstein-Hilbert supergravity after gauge fixing. Third, as can be seen from the composite formulate (\ref{ConformalCompositeVector}), the field equation for $T_{abc}^-$ imposes a constraint on the fermionic fields. Thus, we use the map between the standard and the dilaton Weyl multiplets, see Appendix \ref{Appendix1}, and establish the linear multiplet action in a dilaton Weyl multiplet background \begin{eqnarray} e^{-1} \mathcal{L}_{L, DW} &=& \frac12 F_{IJ} L_{ij}^I L^{ij J} R - \frac12 \sigma^{-1} F_{IJ} L_{ij}^I L^{ij J}\Box \sigma - \frac1{24} \sigma^{-2} F_{IJ} L_{ij}^I L^{ij J} H_{abc} H^{abc} \nonumber\\ && - \frac12 F_{IJ} D_a L_{ij}^I D^a L^{ijJ} + \frac14 F_{IJ} E_a^I E^{aJ} + F_{IJ} E^{a I} L_{ij}^J \mathcal{V}_{a}^{ij} \nonumber\\ && - \frac12 F_{IJKj}{}^k E^{abI} \partial_a L^{jl J} \partial_b L^K_{kl} \,. \label{ConformalLinDW} \end{eqnarray} As we will discuss in detail, the presence of two scalar fields, $\sigma$ and $L_{ij}$, allow us two distinct gauge fixing possibilities, one giving rise to an off-shell supergravity in the Jordan frame and the other in Einstein frame. Deferring this discussion to the next section, we give the bosonic part of the conformal vector multiplet action \begin{eqnarray} e^{-1} \mathcal{L}_{V} &=& -\frac14 \sigma F^{ab} F_{ab} - \frac1{16} \epsilon^{abcdef} B_{ab} F_{cd} F_{ef} +\sigma Y^{ij} Y_{ij} \,, \end{eqnarray} where we use the composite linear multiplet (\ref{CompositeLocalLinear1}) and (\ref{CompositeLocalLinear2}) as well as the local vector-linear action (\ref{LocalVL}). Finally, for future reference, we take the vector multiplet in the local vector-linear action and the composite linear multiplet to be different, which gives rise to a vector multiplet action for two vector multiplets \begin{eqnarray} e^{-1} \mathcal{L}_{V V^\prime} &=& -\frac14 \sigma F^{ab} F_{ab}^\prime - \frac1{16} \epsilon^{abcdef} B_{ab} F_{cd} F_{ef}^\prime +\sigma Y^{ij} Y_{ij}^\prime \,, \label{VVprime} \end{eqnarray} where the second multiplet is expressed by the primed quantities $(\Omega_i^\prime, W_\mu^\prime, Y_{ij}^\prime)$. \section{Off-Shell Supergravity Models with Multiple Linear Multiplets}{\label{Sec4}} In this section, we take advantage of the conformal linear and vector multiplet actions to construct various off-shell supergravity models. First, we use the superconformal linear multiplet action in the dilaton Weyl multiplet background (\ref{ConformalLinDW}) to obtain an off-shell description of six dimensional $\mathcal{N} = (1,0)$ supergravity. This can be achieved in two ways, one giving rise to Einstein-frame model and the other giving rise to a Jordan-frame model, depending on the gauge fixing condition. Next, we eliminate the auxiliary fields and present the on-shell supergravity coupled to $n$-number of linear multiplets. Finally, we discuss higher derivative models where the leading terms are not higher-order curvature terms but curvature terms coupled to auxiliary fields. When coupled to Poincar\'e supergravity, off-diagonal invariants give rise to on-shell higher curvature supergravity models after the elimination of auxiliary fields. \subsection{Poincar\'e Supergravity}{\label{Sec41}} An off-shell Poincar\'e supergravity can be obtained from the conformal linear multiplet action (\ref{ConformalLinDW}) by gauge fixing the redundant dilatation, special conformal and special supersymmetry transformation, see Table \ref{Table}. It is also possible to break the $SU(2)$ R-symmetry to a $U(1)$ subgroup. \subsubsection*{Einstein-frame Off-Shell Supergravity} If we consider a single linear multiplet coupled to dilaton Weyl multiplet, which corresponds to setting $\mathcal{F}_{11} = L^{-1}$ in (\ref{ConformalLinDW}), we may adopt the following gauge fixing conditions \begin{eqnarray} b_\mu = 0 \,, \qquad L_{ij} = \frac1{\sqrt 2} \delta_{ij} L\,, \qquad L = 1\,, \qquad \varphi^i = 0\,, \label{GaugeFixing1} \end{eqnarray} \begin{table}[h!] \centering \begin{tabular}{|c|c|c|c|} \hline \textbf{Weyl Multiplet} & \textbf{Compensator} & \textbf{Gauge Fixing} & \textbf{Poincar\'e Multiplet } \\ \hline \begin{tabular}[c]{@{}c@{}}Dilaton Weyl Multiplet\\ $(e_\mu{}^a, \psi_\mu^i, b_\mu, {\mathcal V}_\mu{}^{ij}, \sigma, B_{\mu\nu}, \psi^i)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}Linear Multiplet\\ $(L_{ij}, \varphi^i, E_a)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$b_\mu = 0$\\ $L_{ij} = \frac1{\sqrt 2} \delta_{ij} L$\\ $L= 1$\\ $\varphi^i = 0$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$(e_\mu{}^a, \psi_\mu^i, b_\mu, {\mathcal V}_\mu,{\mathcal V}^\prime_\mu{}^{ij}$\\ $\sigma, B_{\mu\nu}, \psi^i, E_a)$\\ \\ (Einstein-frame)\end{tabular} \\ \hline \begin{tabular}[c]{@{}c@{}}Dilaton Weyl Multiplet\\ $(e_\mu{}^a, \psi_\mu^i, b_\mu, {\mathcal V}_\mu{}^{ij}, \sigma, B_{\mu\nu}, \psi^i)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}Linear Multiplet\\ $(L_{ij}, \varphi^i, E_a)$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$b_\mu = 0$\\ $L_{ij} = \frac1{\sqrt 2} \delta_{ij} L$\\ $\sigma= 1$\\ $\psi^i = 0$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$(e_\mu{}^a, \psi_\mu^i, b_\mu, {\mathcal V}_\mu, {\mathcal V}^\prime_\mu{}^{ij}$\\ $L, B_{\mu\nu}, \varphi^i, E_a)$\\ \\ (Jordan-frame)\end{tabular} \\ \hline \end{tabular} \caption{List of possible off-shell supergravity construction via gauge fixing. In a standard Weyl multiplet background, it is not possible to obtain a consistent on-shell Einstein-Hilbert supergravity. In a Weyl multiplet background, with a single linear multiplet coupling, a gauge fixing condition that utilizes the scalar field of dilaton Weyl multiplet leads to a Jordan-frame off-shell supergravity while a gauge choice that uses the fields of linear multiplet leads to Einstein-frame supergravity.} \label{Table} \end{table} where the first choice fixes special conformal symmetry, the second one breaks the $SU(2)$ R-symmetry to $U(1)$, the third breaks the dilatation symmetry and the last one fixes the special supersymmetry. As a result, we obtain an off-shell Poincar\'e supergravity in Einstein frame \cite{Bergshoeff6D,Coomans6D} \begin{eqnarray} e^{-1} \mathcal{L} = \frac{1}{2} R - \frac14 E^a E_a + \mathcal{V}^\prime_{a ij} \mathcal{V}^{\prime ij}_a + \frac{1}{\sqrt2} E^a \mathcal{V}^\prime_a{}^{ij} \delta_{ij} - \frac12 \sigma^{-2} \partial_\mu \sigma \partial^\mu \sigma - \frac{1}{24} \sigma^{-2} H_{\mu\nu\rho} H^{\mu\nu\rho} \,, \label{EHinEF} \end{eqnarray} where we only provide the bosonic part. Due to our gauge choice that breaks the $SU(2)$ R-symmetry, we also decompose the $SU(2)$ R-symmetry gauge field $\mathcal{V}_{a}^{ij}$ into its trace and traceless part \begin{eqnarray} \mathcal{V}_a = \mathcal{V}_a{}^{ij} \delta_{ij} \,, \qquad \mathcal{V}^\prime_a{}^{ij} = \mathcal{V}_a{}^{ij} - \frac12 \delta_{ij} \mathcal{V}_a{}^{kl} \delta_{kl} \,. \end{eqnarray} When $n$-linear multiplet coupling is considered, we may adopt the following gauge fixing conditions to obtain the supergravity coupling of $(n-1)$-linear multiplets to off-shell supergravity in Einstein frame \begin{eqnarray} b_\mu = 0 \,, \qquad F_{IJ} L_{ij}^I L^{ij J} = 1 \,, \qquad F_{IJK}{}^{ij} \varphi_j^K L_{mn}^I L^{mn J} + 2 F_{IJ} L^{ij I} \varphi_j^J = 0 \,. \label{GaugeFixing2} \end{eqnarray} where the second choice fixes the dilatation symmetry while the last one fixes the special supersymmetry. As a result, we obtain an off-shell action for linear multiplets \begin{eqnarray} e^{-1} \mathcal{L} &=& \frac12 R - \frac12 \sigma^{-2} \partial_\mu \sigma \partial^\mu \sigma - \frac{1}{24} \sigma^{-2} H_{\mu\nu\rho} H^{\mu\nu\rho}- \frac12 F_{IJ} D_a L_{ij}^I D^a L^{ijJ} \nonumber\\ && + \frac14 F_{IJ} E_a^I E^{aJ} + F_{IJ} E^{a I} L_{ij}^J \mathcal{V}_{a}^{ij} - \frac12 F_{IJKj}{}^k E^{abI} \partial_a L^{jl J} \partial_b L^K_{kl} \,, \label{EHinJF} \end{eqnarray} where the scalars $L_{ij}^I$ and the fermions $\varphi^I$ are restricted by the equation (\ref{GaugeFixing2}). \subsubsection*{Jordan-frame Off-Shell Supergravity} We may use the scalar of the dilaton Weyl multiplet $\sigma$ to fix the dilatation symmetry. In this case, the consistent set of gauge fixing condition is given by \cite{BergshoeffHD} \begin{eqnarray} b_\mu = 0 \,, \qquad \sigma = 1 \,, \qquad \psi^i = 0 \,, \label{GaugeFixing3} \end{eqnarray} where the first choice fixes special conformal symmetry, the second one fixes the dilatations and the third one fixes special supersymmetry. In this case, the off-shell supersymmetric action is given by \begin{eqnarray} e^{-1} \mathcal{L} &=& g_{{\rm Lin}} \left(\frac12 R - \frac{1}{24} H_{\mu\nu\rho} H^{\mu\nu\rho} \right) - \frac12 F_{IJ} D_a L_{ij}^I D^a L^{ijJ} + \frac14 F_{IJ} E_a^I E^{aJ} \nonumber\\ && + F_{IJ} E^{a I} L_{ij}^J \mathcal{V}_{a}^{ij} - \frac12 F_{IJKj}{}^k E^{abI} \partial_a L^{jl J} \partial_b L^K_{kl} \,, \label{EHinJF2} \end{eqnarray} where the potential $g_{Lin} (L) \equiv g_{\rm Lin}$ is defined by \begin{eqnarray} g_{Lin} = F_{IJ} L_{ij}^I L^{ij J} \,. \end{eqnarray} \subsection{Off-Diagonal $R Y_{ij}$ Invariant and $R^2$ Supergravity}{\label{Sec42}} When higher curvature supergravity models are needed, the most straightforward strategy is to construct off-shell models, if possible, then add the off-shell higher curvature models to the off-shell Poincar\'e supergravity and perturbatively eliminate the auxiliary fields order by order in the small perturbation parameter, see i.e. \cite{6DHD,Cremonini5D,OzkanThesis} for six and five dimensional examples. This approach has been widely used in various dimensions for a various number of supercharges. The supersymmetric higher curvature models usually include coupling between the auxiliary field of the off-shell Poincar\'e supergravity and the curvature terms. Then, the elimination of auxiliary fields leads to gravitational higher derivative terms, spoiling the particular combination one is after. Off-diagonal invariants are supersymmetric off-shell models where the leading terms are not higher-order curvature terms but curvature terms coupled to auxiliary fields. Previously, these models have been used to eliminate such undesired couplings \cite{SuperBHT,BISugra,3DN2}. Here, we aim to construct an off-diagonal invariant that leads to an on-shell $R^2$-supergravity (with a coupling to vector multiplet) which can be compared with \cite{OzkanThesis,6DHD}. We start with a composite superconformal vector multiplet that is achieved according to a single conformal truncation, i.e. \begin{eqnarray} \mathcal{F}_{11} = L^{-1} \,, \qquad \mathcal{F}_{111}{}^{ij} = - L^{-3} L^{ij} \,. \label{SingleMultipletTruncation} \end{eqnarray} Upon gauge fixing (\ref{GaugeFixing3}), the composite off-shell vector multiplet, $(\Omega_i^\prime, W_\mu^\prime, Y_{ij}^\prime)$, is given by \begin{eqnarray} Y^{\prime ij} &=& - L^{-1} D^a D_a L^{ij} + \frac12 L^{-1} L^{ij}R - \frac1{24} L^{-1} L^{ij} H_{abc}H^{abc} + L^{-3} L_{kl} D_a L^{ik} D^a L^{jl}\nonumber\\ &&+ L^{-3} E^a L_k{}^{(i} D_a L^{j)l} + \frac14 L^{-3} L^{ij} E^a E_a \,,\nonumber\\ F_{ab}^\prime &=& - 2 \partial_{[a} \left( L^{-1} E_{b]} - 2 L^{-1} L_{ij} \mathcal{V}_{b]}{}^{ij} \right) - 2 L^{-3} L_{j}{}^k \partial_{[a} L^{jl} \partial_{b]} L_{lk} \,. \end{eqnarray} where \begin{eqnarray} D^a D_a L^{ij} &=& (\partial^a + \omega_b{}^{ba} ) D_a L^{ij} + 2 \mathcal{V}^{a(i}{}_k D_a L^{j)k} \,. \end{eqnarray} Note that we only present the bosonic fields here and the fermionic fields can be read from (\ref{ConformalCompositeVector}) given the single multiplet truncation condition (\ref{SingleMultipletTruncation}). This composite vector multiplet can be used in the two vector multiplet action (\ref{VVprime}), which takes the following form after gauge fixing (\ref{GaugeFixing3}) \begin{eqnarray} e^{-1} \mathcal{L}_{V V^\prime} &=& -\frac14 F^{ab} F_{ab}^\prime - \frac1{16} \epsilon^{abcdef} B_{ab} F_{cd} F_{ef}^\prime +Y^{ij} Y_{ij}^\prime \,, \end{eqnarray} and the resulting off-diagonal action, which we refer to as $RY_{ij}$ action, is given by \begin{eqnarray} e^{-1} \mathcal{L}_{R Y_{ij}} &=& - L^{-1}Y_{ij} D^a D_a L^{ij} + \frac12 L^{-1} Y_{ij} L^{ij}R - \frac1{24} L^{-1} L^{ij} Y_{ij} H_{abc}H^{abc}\nonumber\\ && + L^{-3} Y_{ij} L_{kl} D_a L^{ik} D^a L^{jl} + L^{-3} E^a Y_{ij} L_k{}^{i} D_a L^{jl} + \frac14 L^{-3} Y_{ij} L^{ij} E^a E_a \,,\nonumber\\ && + \frac12 F^{ab} \left(\partial_{a} \left( L^{-1} E_{b} - 2 L^{-1} L_{ij} \mathcal{V}_{b}{}^{ij} \right) + L^{-3} L_{j}{}^k \partial_{a} L^{jl} \partial_{b} L_{lk} \right) \nonumber\\ && + \frac18 \epsilon^{abcdef} B_{ab} F_{cd}\left(\partial_{e} \left( L^{-1} E_{f} - 2 L^{-1} L_{ij} \mathcal{V}_{f}{}^{ij} \right) + L^{-3} L_{j}{}^k \partial_{e} L^{jl} \partial_{f} L_{lk} \right) \,. \label{RYAction} \end{eqnarray} With this off-diagonal action in hand we can obtain an $R^2$ extended Einstein-Maxwell supergravity by considering the following action \begin{eqnarray} \mathcal{L} = \mathcal{L}_{EH} + \mathcal{L}_{V} + \mathcal{L}_{R Y_{ij}} + g \mathcal{L}_{VL} \,, \label{R2SalamSezgin} \end{eqnarray} where $\mathcal{L}_{EH}$ refers to the off-shell Poincar\'e supergravity in Jordan frame (\ref{EHinJF2}), $\mathcal{L}_{VL}$ refers to the vector-linear coupling (\ref{LocalVL}), $\mathcal{L}_V$ refers to the off-shell vector multiplet action \begin{eqnarray} e^{-1} \mathcal{L}_{V} &=& -\frac14 F^{ab} F_{ab} - \frac1{16} \epsilon^{abcdef} B_{ab} F_{cd} F_{ef} +Y^{ij} Y_{ij} \,, \label{VectorAction} \end{eqnarray} and $\mathcal{L}_{R Y_{ij}}$ is the off-diagonal $RY_{ij}$ action (\ref{RYAction}). Upon imposing the field equation for $Y_{ij}$, which solves $Y_{ij}$ as \begin{eqnarray} Y^{ij} &=& \frac12 L^{-1} D^a D_a L^{ij} - \frac14 L^{-1} L^{ij}R + \frac1{48} L^{-1} L^{ij} H_{abc}H^{abc} - \frac12 L^{-3} L_{kl} D_a L^{ik} D^a L^{jl}\nonumber\\ &&-\frac12 L^{-3} E^a L_k{}^{(i} D_a L^{j)l} - \frac18 L^{-3} L^{ij} E^a E_a - \frac12 g L^{ij} \,, \end{eqnarray} we obtain $R^2$ action via $Y^{ij} Y_{ij}$ term in the vector multiplet action. As shown in \cite{BergshoeffHD}, an off-shell version of the Salam - Sezgin model \cite{SalamSezgin} can be obtained by a combination of a single linear multiplet truncation of the off-shell Poincar\'e supergravity in Jordan frame (\ref{EHinJF2}), the off-shell vector multiplet action (\ref{VectorAction}) and a local vector-linear multiplet action (\ref{LocalVL}). With the off-diagonal $RY_{ij}$ action, one may improve off-shell Salam - Sezgin model of \cite{BergshoeffHD} with $RY_{ij}$ action, in which case the elimination of the auxiliary field $Y_{ij}$ would lead to an $R^2$ extension of Salam - Sezgin model. \section{Discussion}\label{Sec5} In this paper, we provide a systematic analysis of linear multiplets of six dimensional $\mathcal{N} = (1,0)$ supergravity, which has been shown to be crucial in the construction of higher curvature models \cite{OzkanThesis,6DHD}. Our analysis start with an investigation of rigid linear multiplets, in which case the couplings of linear multiplets are determined by a function $\mathcal{F}_{IJ}(L)$ that is subject to two mild constraints (\ref{RigidConstraints}). After establishing the relation between the five dimensional $\mathcal{N} = 2$ and six dimensional $\mathcal{N}=(1,0)$ rigid linear multiplets, we repeat our analysis for the case of full $\mathcal{N}=(1,0)$ superconformal symmetry which paves the way for the local supersymmetric couplings of linear multiplets. For the local superconformal models, we work in a dilaton Weyl multiplet background and use superconformal tensor calculus to provide a superconformal linear multiplet action for $n$-number of linear multiplets. For this case, the function $\mathcal{F}_{IJ}(L)$ picks up two more constraints which are imposed due to dilatation and S-supersymmetry invariance (\ref{ConformalF}). These constraints are also mild and we provide various examples of $\mathcal{F}_{IJ}$ for the superconformal scenario. Finally, we discussed various gauge fixing procedures and off-shell supergravity models. In particular, we discussed an off-diagonal invariant, which we refer to as the $RY_{ij}$ invariant (\ref{RYAction}), which leads to an $R^2$-extended Einstein-Maxwell supergravity upon imposing the field equations. There are various directions to pursue following our work. First, it would be interesting to investigate the supersymmetry solutions of an $R^2$-extended Einstein-Maxwell supergravity. The off-shell model that we have in mind here is given by the equation (\ref{R2SalamSezgin}). Note that there is also an off-shell $R^2$ model constructed in \cite{OzkanThesis,6DHD} which can be added to the combination that we discussed in (\ref{R2SalamSezgin}). It remains to be checked whether these two different paths to improve the Salam-Sezgin model with an $R^2$ term lead to the same physical theory. Second, noticing that a composite superconformal linear multiplet contain a constrained vector $E_a^\prime$ given by (\ref{CompositeLocalLinear2}) \begin{eqnarray} E_a^\prime &=& \mathcal{D}^b (\sigma \widehat F_{ba}) + \ldots \,, \label{CompositeEa} \end{eqnarray} where we remind the reader that $E_{\mu\nu}$ is related to $E_a$ via $E^a = e_\mu{}^a \mathcal{D}_\nu E^{\mu\nu}$. Therefore, a linear multiplet action that contains an $E^a E_a$ term would produce a higher derivative vector multiplet action with an $F \Box F$ term. Such a model can easily be obtained with a non-symmetric choice of $\mathcal{F}_{IJ}$ given in (\ref{NonSymmetricFIJ}) where the unprimed multiplet can be used as a compensating multiplet and the primed multiplet can be used as a composite linear multiplet. Note that as $\mathcal{F}_{22} = L^{-1}$ for the non-symmetric choice, an action for such an $\mathcal{F}_{IJ}$ would contain a term $L^{-1} E_a^\prime E^{\prime a}$ due to $\mathcal{F}_{IJ} E_a^I E^{a J}$ in (\ref{LocalLinear}). Thus, upon using the composite expression for $E_a$ in (\ref{CompositeEa}) one would obtain the desired higher derivative vector multiplet action. This result should be compared with \cite{FBoxF1,FBoxF2}. Finally, it would be interesting to see if other off-diagonal invariants can be constructed and if they can have interesting physical implications in higher derivative extended supersymmetric models. \section*{Acknowledgment} The work of U.A is supported by TUBITAK grant 118F091. M.O. is supported in part by TUBITAK grant 118F091.

proofpile-arXiv_069-6875

\section{Introduction}\label{introduction}} The singular value decomposition \citep[SVD;][]{golub_singular_1971} is one of the most important tools in a multivariate toolbox. Conceptually and practically, the SVD is the core technique behind numerous statistical methods; the most common of which is principal component analysis \citep[PCA;][]{jolliffe_principal_2012, abdi_principal_2010, jolliffe_principal_2016}. A lesser known---but far more ubiquitous---tool is the generalized SVD \citep[GSVD;][]{abdi2007singular, greenacre_theory_1984, takane_relationships_2003, holmes_multivariate_2008}. The GSVD is more ubiquitous because it is the technique behind---and generalizes---many more statistical methods. The core concept behind the GSVD is that a data matrix has two companion matrices: one for the rows and one for the columns. These companion matrices are weights, constraints, or metrics imposed on the rows and columns of a data matrix. For examples, via the GSVD we can implement weighted solutions to PCA, or analyses under different metrics, such as correspondence analysis \citep[CA;][]{greenacre_theory_1984, lebart_multivariate_1984, escofier-cordier_analyse_1965, greenacre_correspondence_2010} or canonical correlation analysis \citep[CCA;][]{harold1936relations, abdi2017canonical}. Though the GSVD generalizes---and is more flexible than---the SVD, there are few if any direct implementations of the GSVD in \proglang{R}. Rather, the GSVD is typically part of specific packages, and these packages are typically designed around more general analyses and broad usage. Rarely, if ever, are these GSVD implementations accessible to, or meant for, more direct access by users. Given the frequent and ubiquitous uses of the SVD and GSVD, a more accessible implementation would benefit a wide range of users. Here I introduce a package designed around the GSVD, called \pkg{GSVD}. \pkg{GSVD} is a lightweight implementation of the GSVD and two other generalized decompositions: the generalized eigendecomposition (GEVD) and the generalized partial least squares-SVD (GPLSSVD). \pkg{GSVD} was designed for a wide range of users, from analysts to package developers, all of whom would benefit from more direct access to the GSVD and similar decompositions. More importantly, the \pkg{GSVD} package and the idea of the GSVD provide a basis to unify concepts, nomenclature, and techniques across a wide array of statistical traditions and multivariate analyses approaches. \pkg{GSVD} has three core functions: \code{geigen()}, \code{gsvd()}, and \code{gplsssvd()}. These core functions provide a way for users to implement a wide array of methods including (but not limited to) multidimensional scaling, principal components analysis, correspondence analysis, canonical correlation, partial least squares, and numerous variants and extensions of the aforementioned. \pkg{GSVD} also helps simplify and unify concepts across techniques because, at their core, all of these techniques can be accomplished with the SVD. In this paper, I introduce the core functions and functionality of \pkg{GSVD}. Furthermore, I show how the GSVD can provide a more unified nomenclature across techniques, and I show how various techniques can be implemented through \code{geigen()}, \code{gsvd()}, and \code{gplsssvd()}. Here I provide sufficient detail for the majority of users and readers. However, some readers may want to take a deeper dive into various techniques and literatures. Where possible and relevant, I point the readers to works with much more detail and substance. This paper is outlined as follows. In \emph{Generalized decompositions} I provide background on, notation for, and mathematical explanations of the three decompositions discussed here (GEVD, GSVD, and GPLSSVD), followed by some additional notes and literature. In \emph{Package description}, I provide an overview of the core functions, their uses, and other development notes. In \emph{Examples of multivariate analyses} I provide detailed implementations of numerous multivariate techniques, as well as variations of those techniques or variations of how to implement them with \pkg{GSVD} (e.g., PCA via GEVD vs PCA via GSVD). In \emph{Discussion} I make some comments about the package, its uses, its potential, and possible future development. \hypertarget{generalized-decompositions}{% \section{Generalized decompositions}\label{generalized-decompositions}} The GSVD is probably best known by way of the correspondence analysis literature \citep{greenacre_theory_1984, lebart_multivariate_1984, escofier-cordier_analyse_1965, greenacre_correspondence_2010} and, more generally, the ``French way'' of multivariate data analyses \citep{holmes_discussion_2017, holmes_multivariate_2008}. There is considerable breadth and depth of materials on the GSVD and related techniques since the 1960s (and even well before then). Though such breadth and depth is beyond the scope of this paper, I provide a list of references and resources throughout the paper for any interested readers. This section instead focuses on the formulation---and to a degree the nomenclature---of the three core techniques. I first introduce notation, followed by the GEVD, the GSVD, and then formalize and introduce the GPLSSVD. \hypertarget{notation}{% \subsection{Notation}\label{notation}} Bold uppercase letters denote matrices (e.g., \(\mathbf{X}\)). Upper case italic letters (e.g., \(I\)) denote cardinality, size, or length. Subscripts for matrices denote relationships with certain other matrices, for examples \({\mathbf Z}_{\mathbf X}\) is some matrix derived from or related to the \({\bf X}\) matrix, where something like \({\bf F}_{I}\) is a matrix related to the \(I\) set of elements. When these matrices are introduced, they are also specified. Two matrices side-by-side denotes standard matrix multiplication (e.g., \(\bf{X}\bf{Y}\)). Superscript \(^{T}\) denotes the transpose operation, and superscript \(^{-1}\) denotes standard matrix inversion. The diagonal operator, denoted \(\mathrm{diag\{\}}\), transforms a vector into a diagonal matrix, or extracts the diagonal of a matrix in order to produce a vector. \hypertarget{generalized-eigendecomposition}{% \subsection{Generalized eigendecomposition}\label{generalized-eigendecomposition}} The generalized eigendecomposition (GEVD) requires two matrices: a \(J \times J\) (square) data matrix \({\bf X}\) and a \(J \times J\) constraints matrix \({\bf W}_{\bf X}\). For the GEVD, \({\bf X}\) is typically positive semi-definite and symmetric (e.g., a covariance matrix) and \({\bf W}_{\bf X}\) is required to be positive semi-definite. The GEVD decomposes the data matrix \({\bf X}\)---with respect to its constraints \({\bf W}_{\bf X}\)---into two matrices as \begin{equation} {\bf X} = {\bf Q}{\bf \Lambda}{\bf Q}^{T}, \end{equation} where \({\bf \Lambda}\) is a diagonal matrix that contains the eigenvalues and \({\bf Q}\) are the \emph{generalized} eigenvectors. The GEVD finds orthogonal slices of a data matrix, with respect to its constraints, where each orthogonal slice explains the maximum possible variance. That is, the GEVD maximizes \({\bf \Lambda} = {\bf Q}^{T}{\bf W}_{\bf X}{\bf X}{\bf W}_{\bf X}{\bf Q}\) under the constraint of orthogonality where \({\bf Q}^{T}{\bf W}_{\bf X}{\bf Q} = {\bf I}\). Practically, the GEVD is performed with the standard eigenvalue decomposition (EVD) as \begin{equation} \widetilde{\bf X} = {\bf V}{\bf \Lambda}{\bf V}, \end{equation} where \(\widetilde{\bf X} = {\bf W}_{\bf X}^{\frac{1}{2}}{\bf X}{\bf W}_{\bf X}^{\frac{1}{2}}\) and \({\bf V}\) are the eigenvectors which are orthonormal, such that \({\bf V}^{T}{\bf V} = {\bf I}\). The relationship between the GEVD and EVD can be explained as the relationship between the generalized and standard eigenvectors where \begin{equation} {\bf Q} = {\bf W}_{\bf X}^{-\frac{1}{2}}{\bf V} \Longleftrightarrow {\bf V} = {\bf W}_{\bf X}^{\frac{1}{2}}{\bf Q}. \end{equation} When \({\bf W}_{\bf X} = {\bf I}\), the GEVD produces exactly the same results as the EVD because \(\widetilde{\bf X} = {\bf X}\) and thus \({\bf Q} = {\bf V}\). Analyses with the EVD and GEVD---such as PCA---typically produce component or factor scores. With the GEVD, component scores are defined as \begin{equation} {\bf F}_{J} = {\bf W}_{\bf X}{\bf Q}{\bf \Delta}, \end{equation} where \({\bf \Delta} = {\bf \Lambda}^{\frac{1}{2}}\), which are singular values. The maximization in the GEVD can be reframed as the maximization of the component scores where \({\bf \Lambda} = {\bf F}_{J}^{T}{\bf W}_{\bf X}^{-1}{\bf F}_{J}\), still subject to \({\bf Q}^{T}{\bf W}_{\bf X}{\bf Q} = {\bf I}\). \hypertarget{generalized-singular-value-decomposition}{% \subsection{Generalized singular value decomposition}\label{generalized-singular-value-decomposition}} The generalized singular value decomposition (GSVD) requires three matrices: an \(I \times J\) (rectangular) data matrix \({\bf X}\), an \(I \times I\) row constraints matrix \({\bf M}_{\bf X}\), and a \(J \times J\) columns constraints matrix \({\bf W}_{\bf X}\). For the GSVD \({\bf M}_{\bf X}\) and \({\bf W}_{\bf X}\) are each required to be positive semi-definite. The GSVD decomposes the data matrix \({\bf X}\)---with respect to both of its constraints \({\bf M}_{\bf X}\) and \({\bf W}_{\bf X}\)---into three matrices as \begin{equation} {\bf X} = {\bf P}{\bf \Delta}{\bf Q}^{T}, \end{equation} where \({\bf \Delta}\) is a diagonal matrix that contains the singular values, and where \({\bf P}\) and \({\bf Q}\) are the left and right \emph{generalized} singular vectors, respectively. From the GSVD we can obtain eigenvalues as \({\bf \Lambda}^{2} = {\bf \Delta}\). The GSVD finds orthogonal slices of a data matrix, with respect to its constraints, where each slice explains the maximum possible \emph{square root} of the variance. That is, the GSVD maximizes \({\bf \Delta} = {\bf P}^{T}{\bf M}_{\bf X}{\bf X}{\bf W}_{\bf X}{\bf Q}\) under the constraint of orthogonality where \({\bf P}^{T}{\bf M}_{\bf X}{\bf P} = {\bf I} = {\bf Q}^{T}{\bf W}_{\bf X}{\bf Q}\). Typically, the GSVD is performed with the standard SVD as \begin{equation} \widetilde{\bf X} = {\bf U}{\bf \Delta}{\bf V}, \end{equation} where \(\widetilde{\bf X} = {{\bf M}_{\bf X}^{\frac{1}{2}}}{\bf X}{{\bf W}^{\frac{1}{2}}_{\bf X}}\), and where \({\bf U}\) and \({\bf V}\) are the left and right singular vectors, respectively, which are orthonormal such that \({\bf U}^{T}{\bf U} = {\bf I} = {\bf V}^{T}{\bf V}\). The relationship between the GSVD and SVD can be explained as the relationship between the generalized and standard singular vectors where \begin{equation} \begin{aligned} {\bf P} = {{\bf M}^{-\frac{1}{2}}_{\bf X}}{\bf U} \Longleftrightarrow {\bf U} = {{\bf M}^{\frac{1}{2}}_{\bf X}}{\bf P} \\ {\bf Q} = {{\bf W}^{-\frac{1}{2}}_{\bf X}}{\bf V} \Longleftrightarrow {\bf V} = {{\bf W}^{\frac{1}{2}}_{\bf X}}{\bf Q}. \end{aligned} \end{equation} When \({\bf M}_{\bf X} = {\bf I} = {\bf W}_{\bf X}\), the GSVD produces exactly the same results as the SVD because \(\widetilde{\bf X} = {\bf X}\) and thus \({\bf P} = {\bf U}\) and \({\bf Q} = {\bf V}\). Analyses with the SVD and GSVD---such as PCA or CA---typically produce component or factor scores. With the GSVD, component scores are defined as \begin{equation} {\bf F}_{I} = {\bf M}_{\bf X}{\bf P}{\bf \Delta} \textrm{ and } {\bf F}_{J} = {\bf W}_{\bf X}{\bf Q}{\bf \Delta}, \end{equation} for the left (rows) and right (columns) of \({\bf X}\), respectively. The optimization in the GSVD can be reframed as the maximization of the component scores where \({\bf F}_{I}^{T}{\bf M}_{\bf X}^{-1}{\bf F}_{I} = {\bf \Lambda} = {\bf F}_{J}^{T}{\bf W}_{\bf X}^{-1}{\bf F}_{J}\), still subject to \({\bf P}^{T}{\bf M}_{\bf X}{\bf P} = {\bf I} = {\bf Q}^{T}{\bf W}_{\bf X}{\bf Q}\). Note how the optimization with respect to the component scores shows a maximization for the eigenvalues. \hypertarget{generalized-partial-least-squares-singular-value-decomposition}{% \subsection{Generalized partial least squares singular value decomposition}\label{generalized-partial-least-squares-singular-value-decomposition}} The generalized partial least squares-singular value decomposition (GPLSSVD) is a reformulation of the PLSSVD. The PLSSVD is a specific type of PLS from the broader PLS family \citep{tenenhaus1998regression}. The PLSSVD has various other names---for example, PLS correlation \citep{krishnan_partial_2011}---but canonically comes from the psychology \citep{ketterlinus1989partial} and neuroimaging literatures \citep{mcintosh_spatial_1996}, though it traces its origins to Tucker's interbattery factor analysis \citep{tucker_inter-battery_1958}. Recently, with other colleagues, I introduced the idea of the GPLSSVD as it helped us formalize a new version of PLS for categorical data \citep{beaton_partial_2016, beaton_generalization_2019}. However, the GPLSSVD also allows us to generalize other methods, such as canonical correlation analysis \citep[see Supplemental Material of][]{beaton_generalization_2019}. The GPLSSVD requires six matrices: an \(N \times I\) (rectangular) data matrix \({\bf X}\) with its \(N \times N\) row constraints matrix \({\bf M}_{\bf X}\) and its \(I \times I\) columns constraints matrix \({\bf W}_{\bf X}\), and an \(N \times J\) (rectangular) data matrix \({\bf Y}\) with its \(N \times N\) row constraints matrix \({\bf M}_{\bf Y}\) and its \(J \times J\) columns constraints matrix \({\bf W}_{\bf Y}\). For the GPLSSVD all constraint matrices are required to be positive semi-definite. The GPLSSVD decomposes \emph{the relationship between} the data matrices, with respect to their constraints, and expresses the common information as the relationship between latent variables. The goal of partial least squares-SVD (PLSSVD) is to find a combination of orthogonal latent variables that maximize the relationship between two data matrices. PLS is often presented as \(\mathrm{arg max(} {\bf {l_{\bf X}}_{\ell}^{T}}{{\bf l}_{\bf Y}}_{\ell}\mathrm{)} = \mathrm{arg max}\textrm{ }\mathrm{cov(} {\bf {l_{\bf X}}_{\ell}}, {{\bf l}_{\bf Y}}_{\ell}\mathrm{)}\), under the condition that \({\bf {l_{\bf X}}_{\ell}^{T}}{{\bf l}_{\bf Y}}_{\ell'} = 0\) when \({\ell} \neq {\ell'}\). This maximization can be framed as \begin{equation} {{\bf L}_{\bf X}^{T}}{\bf L}_{\bf Y} = {\bf \Delta}, \end{equation} where \({\bf \Delta}\) is the diagonal matrix of singular values, and so \({\bf \Delta}^{2} = {\bf \Lambda}\) which are eigenvalues. Like with the GSVD, the GPLSSVD decomposes the relationship between two data matrices into three matrices as \begin{equation} [({\bf M}_{\bf X}^{\frac{1}{2}}{\bf X})^{T}({\bf M}_{\bf Y}^{\frac{1}{2}}{\bf Y})] = {\bf P}{\bf \Delta}{\bf Q}^{T}, \end{equation} where \({\bf \Delta}\) is the diagonal matrix of singular values, and where \({\bf P}\) and \({\bf Q}\) are the left and right \emph{generalized} singular vectors, respectively. Like the GSVD and GEVD, the GPLSSVD finds orthogonal slices of \(({\bf M}_{\bf X}^{\frac{1}{2}}{\bf X})^{T}({\bf M}_{\bf Y}^{\frac{1}{2}}{\bf Y})\) with respect to the column constraints. The GPLSSVD maximizes \({\bf \Delta} = {\bf P}^{T}{\bf W}_{\bf X}[({\bf M}_{\bf X}^{\frac{1}{2}}{\bf X})^{T}({\bf M}_{\bf Y}^{\frac{1}{2}}{\bf Y})]{\bf W}_{\bf Y}{\bf Q}\) under the constraint of orthogonality where \({\bf P}^{T}{\bf W}_{\bf X}{\bf P} = {\bf I} = {\bf Q}^{T}{\bf W}_{\bf Y}{\bf Q}\). Typically, the GPLSSVD is performed with the SVD as \begin{equation} \widetilde{\bf X}^{T}\widetilde{\bf Y} = {\bf U}{\bf \Delta}{\bf V}, \end{equation} where \(\widetilde{\bf X} = {{\bf M}_{\bf X}^{\frac{1}{2}}}{\bf X}{{\bf W}^{\frac{1}{2}}_{\bf X}}\) and \(\widetilde{\bf Y} = {{\bf M}_{\bf Y}^{\frac{1}{2}}}{\bf Y}{{\bf W}^{\frac{1}{2}}_{\bf Y}}\), and where \({\bf U}\) and \({\bf V}\) are the left and right singular vectors, respectively, which are orthonormal such that \({\bf U}^{T}{\bf U} = {\bf I} = {\bf V}^{T}{\bf V}\). The relationship between the generalized and standard singular vectors are \begin{equation} \begin{aligned} {\bf P} = {{\bf W}^{-\frac{1}{2}}_{\bf X}}{\bf U} \Longleftrightarrow {\bf U} = {{\bf W}^{\frac{1}{2}}_{\bf X}}{\bf P} \\ {\bf Q} = {{\bf W}^{-\frac{1}{2}}_{\bf Y}}{\bf V} \Longleftrightarrow {\bf V} = {{\bf W}^{\frac{1}{2}}_{\bf Y}}{\bf Q}. \end{aligned} \end{equation} When all constraint matrices are \({\bf I}\), the GPLSSVD produces exactly the same results as the PLSSVD because \(\widetilde{\bf X} = {\bf X}\) and \(\widetilde{\bf Y} = {\bf Y}\) and thus \({\bf P} = {\bf U}\) and \({\bf Q} = {\bf V}\). The latent variables are then expressed with respect to the constraints and \emph{generalized} singular vectors as \({\bf L}_{\bf X} = ({\bf M}_{\bf X}^{\frac{1}{2}}{\bf X}{\bf W}_{\bf X}{\bf P})\) and \({\bf L}_{\bf Y} = ({\bf M}_{\bf Y}^{\frac{1}{2}}{\bf Y}{\bf W}_{\bf Y}{\bf Q})\). These latent variables maximize the weighted covariance (by way of the constraints) subject to orthogonality where \begin{equation} \begin{aligned} {\bf L}_{\bf X}^{T}{\bf L}_{\bf Y} = \\ ({\bf M}_{\bf X}^{\frac{1}{2}}{\bf X}{\bf W}_{\bf X}{\bf P})^{T}({\bf M}_{\bf Y}^{\frac{1}{2}}{\bf Y}{\bf W}_{\bf Y}{\bf Q}) =\\ (\widetilde{\bf X}{\bf U})^{T}(\widetilde{\bf Y}{\bf V}) =\\ {\bf U}^{T}\widetilde{\bf X}^{T}\widetilde{\bf Y}{\bf V} = {\bf \Delta}. \end{aligned} \end{equation} We will see in the following section that the ``weighted covariance'' could be the correlation, which allows us to use the GPLSSVD to perform various types of ``cross-decomposition'' techniques. Like with the GEVD and GSVD, the GPLSSVD produces component or factor scores. The component scores are defined as \begin{equation} {\bf F}_{I} = {\bf W}_{\bf X}{\bf P}{\bf \Delta} \textrm{ and } {\bf F}_{J} = {\bf W}_{\bf Y}{\bf Q}{\bf \Delta}, \end{equation} for the columns of \({\bf X}\) and the columns of \({\bf Y}\), respectively. The optimization in the GPLSSVD can be reframed as the maximization of the component scores where \({\bf F}_{I}^{T}{\bf W}_{\bf X}^{-1}{\bf F}_{I} = {\bf \Lambda} = {\bf F}_{J}^{T}{\bf W}_{\bf Y}^{-1}{\bf F}_{J}\) where \({\bf \Lambda}\) are the eigenvalues, and this maximization is still subject to \({\bf P}^{T}{\bf W}_{\bf X}{\bf P} = {\bf I} = {\bf Q}^{T}{\bf W}_{\bf Y}{\bf Q}\). \hypertarget{decomposition-tuples}{% \subsection{Decomposition tuples}\label{decomposition-tuples}} For simplicity, the GSVD is often referred to as a ``triplet'' or ``the GSVD triplet'' \citep{husson_jan_2016, holmes_multivariate_2008} comprised of (1) the data matrix, (2) the column constraints, and (3) the row constraints. We can use the same concept to also define ``tuples'' for the GEVD and GPLSSVD. To note, the traditional way to present the GSVD triplet is in the above order (data, column constraints, row constraints). However, here I present a different order for the elements in the tuples so that I can (1) better harmonize the tuples across the three decompositions presented here, and (2) simplify the tuples such that the order of the elements within the tuples reflects the matrix multiplication steps. Furthermore, I present two different tuples for each decomposition---a complete and a partial---where the partial is a lower rank solution. The complete decomposition tuples are: \begin{itemize} \item The complete GEVD 2-tuple: \(\mathrm{GEVD(}{\bf X}, {\bf W}_{\bf X}\mathrm{)}\) \item The complete GSVD decomposition 3-tuple: \(\mathrm{GSVD(}{\bf M}_{\bf X}, {\bf X}, {\bf W}_{\bf X}\mathrm{)}\) \item The complete GPLSSVD decomposition 6-tuple: \(\mathrm{GPLSSVD(}{\bf M}_{\bf X}, {\bf X}, {\bf W}_{\bf X}, {\bf M}_{\bf Y}, {\bf Y}, {\bf W}_{\bf Y}\mathrm{)}\). \end{itemize} Additionally, we can take the idea of tuples one step further and allow for the these tuples to also define the desired \emph{returned rank} of the results referred to as ``partial decompositions''. The partial decompositions produce (return) only the first \(C\) components, and are defined as: \begin{itemize} \item The partial GEVD decomposition 3-tuple: \(\mathrm{GEVD(}{\bf X}, {\bf W}_{\bf X}, C\mathrm{)}\) \item The partial GSVD decomposition 4-tuple: \(\mathrm{GSVD(}{\bf M}_{\bf X}, {\bf X}, {\bf W}_{\bf X}, C\mathrm{)}\) \item The partial GPLSSVD decomposition 7-tuple: \(\mathrm{GPLSSVD(}{\bf M}_{\bf X}, {\bf X}, {\bf W}_{\bf X}, {\bf M}_{\bf Y}, {\bf Y}, {\bf W}_{\bf Y}, C\mathrm{)}\). \end{itemize} Overall, these tuples provide short and convenient ways to express the decompositions. And as we will see in later sections, these tuples provide a simpler way to express specific techniques under the same framework (e.g., PLS and CCA via GPLSSVD). \hypertarget{restrictions-and-limitations}{% \subsection{Restrictions and limitations}\label{restrictions-and-limitations}} In general, \pkg{GSVD} was designed around the most common analyses that use the GSVD, GEVD, and GPLSSVD. These techniques are, typically, multidimensional scaling (GEVD), principal components analysis (GEVD or GSVD), correspondence analysis (GSVD), partial least squares (GSVD or GPLSSVD), reduced rank regression (GSVD or GPLSSVD), canonical correlation analysis (GSVD or GPLSSVD), and numerous variants and extensions of all the aforementioned (and more). One of the restrictions of these generalized decompositions is that any constraint matrix must be positive semi-definite. That means, typically, these matrices are square symmetric matrices with non-negative eigenvalues. Often that means constraint matrices are, for examples, covariance matrices or diagonal matrices. \pkg{GSVD} performs checks to ensure adherence to positive semi-definiteness for constraint matrices. Likewise, many techniques performed through the GEVD or EVD also assume positive semi-definiteness. For example, PCA is the EVD of a correlation or covariance matrix. Thus in \pkg{GSVD}, there are checks for positive semi-definite matrices in the GEVD. Furthermore, all decompositions in \pkg{GSVD} check for eigenvalues below a precision level. When found, these very small eigenvalues are effectively zero and not returned by any of the decompositions in \pkg{GSVD}. However, this can be changed with an input parameter to allow for these vectors to be returned (I discuss these parameters in the following section). \hypertarget{other-and-related-decompositions.}{% \subsection{Other and related decompositions.}\label{other-and-related-decompositions.}} To note, the GSVD discussed here is not the same as another technique referred to as the ``generalized singular value decomposition'' \citep{van_loan_generalizing_1976}. The Van Loan generalized SVD has more recently been referred to as the ``quotient SVD (QSVD)'' to help distinguish it from the GSVD defined here \citep{takane_relationships_2003}. Furthermore, there are numerous other variants of the EVD and SVD beyond the GEVD and GSVD presented here. \citet{takane_relationships_2003} provides a detailed explanation of those variants as well as the relationships between the variants. \hypertarget{other-packages-of-note}{% \subsection{Other packages of note}\label{other-packages-of-note}} There are multiple packages that implement methods based on the GSVD or GEVD, where some of do in fact make use of a GSVD or GSVD-like call. Generally, though, the GSVD calls in these packages are meant more for internal (to the package) use instead of a more external facing tool like \pkg{GSVD}. A core (but by no means comprehensive) list of those packages follows. There are at least four packages designed to provide interfaces to specific GSVD techniques, and each of those packages has an internal function meant for GSVD-like decomposition. \begin{itemize} \item \pkg{ExPosition} which includes the function \texttt{genPDQ} \citep{beaton_exposition_2014}. As the author of \pkg{ExPosition}, I designed \texttt{genPDQ} under the most common usages which are generally diagonal matrices (or simply just vectors). I regret that design choice, which is one of the primary motivations why I developed \pkg{GSVD}. \item \pkg{FactoMineR} \citep{le_factominer_2008} includes the function \texttt{svd.triplet}, which also makes use of vectors instead of matrices because the most common (conceptual) uses of the GSVD is that the row and/or column constraints are diagonal matrices. \item \pkg{ade4} \citep{dray_ade4_2007} which includes a function called \texttt{as.dudi} that is the core decomposition step. It, too, makes use of vectors. \item \pkg{amap} \citep{lucas_amap_2019} which includes the function \texttt{acp} which, akin to the previous 3 packages listed, also only makes use of vectors for row and column constraints. \end{itemize} However, there are other packages that also more generally provide methods based on GSVD, GEVD, or GPLSSVD approaches however, they do not necessarily include a GSVD-like decomposition. Instead they make more direct use of the SVD or eigendecompositions, or alternative methods (e.g., alternating least squares). Those packages include but are not limited to: \begin{itemize} \item \pkg{MASS} \citep{venables_modern_2002} includes \texttt{MASS::corresp} which is an implementation of CA, \item \pkg{ca} which includes a number of CA-based methods \citep{nenadic_correspondence_2007}, \item \pkg{CAvariants} is another package that includes standard CA and other variations not seen in other packages \citep{lombardo_variants_2016}, \item \pkg{homals}, and \pkg{anacor}, are packages each that address a number of methods in the CA family \citep{leeuw_gifi_2009, de_leeuw_simple_2009}, \item \pkg{candisc} focuses on canonical discriminant and correlation analyses \citep{friendly_candisc_2020}, \item \pkg{vegan} that includes a very large set of ordination methods with a particular emphasis on cross-decomposition or canonical methods \citep{oksanen_vegan_2019}, and \item \pkg{rgcca} also presents a more unified approach to CCA and PLS under a more unified framework that also includes various approaches to regularization \citep{tenenhaus_rgcca_2017, tenenhaus_regularized_2014}. \item \pkg{ics} is a package for invariant coordinate selection, which can obtain unmixing matrices (i.e., as in independent components analysis) \citep{nordhausen_tools_2008, tyler_invariant_2009}. \end{itemize} \hypertarget{package-description-core-functions-and-features}{% \section{Package description: core functions and features}\label{package-description-core-functions-and-features}} The \pkg{GSVD} package has three primary ``workhorse'' functions: \begin{itemize} \item \code{geigen(X, W, k = 0, tol = sqrt(.Machine\$double.eps), symmetric)}, \item \code{gsvd(X, LW, RW, k = 0, tol = .Machine\$double.eps)}, and \item \code{gplssvd(X, Y, XLW, YLW, XRW, YRW, k = 0, tol = .Machine\$double.eps)} \end{itemize} In \code{geigen()} or \code{gsvd()} each there is one data matrix \code{X}, whereas \code{gplssvd()} has two data matrices \code{X} and \code{Y}. In \code{geigen()} there is a single constraint matrix \code{W}. In \code{gsvd()} there are two constraint matrices, \code{LW} or ``left weights'' for the rows of \code{X} and \code{RW} or ``right weights'' for the columns of \code{X}. The ``left'' and ``right'' references are used because of the association between these weights and the left and right generalized singular vectors. In \code{gplssvd()} there are two constraint matrices per data matrix (so four total constraint matrices): \code{XLW} and \code{XRW} for \code{X}'s ``left'' and ``right'' weights, and \code{YLW} and \code{YRW} for \code{Y}'s ``left'' and ``right'' weights. The \code{geigen()} includes the argument \code{symmetric} to indicate if \code{X} is a symmetric matrix; when missing \code{X} is tested via \code{isSymmetric()}. The \code{symmetric} argument is eventually passed through to, and is the same as, \code{symmetric} in \code{base::eigen()}. All three functions include \code{k} which indicates how many components to return. Finally, all three functions include a tolerance argument \code{tol}, which is passed through to \code{tolerance_svd()} or \code{tolerance_eigen()}. These functions are the same as \code{base::svd()} and \code{base::eigen()}, respectively, with the added tolerance feature. In both cases, the \code{tol} argument is used to check for any eigenvalues or singular values below the tolerance threshold. Any eigen- or singular values below that threshold are discarded, as they are effectively zero. These values occur when data are collinear, which is common in high dimensional cases or in techniques such as Multiple Correspondence Analysis. However, the \code{tol} argument can be effectively turned off with the use of \code{NA}, \code{NULL}, \code{Inf}, \code{-Inf}, \code{NaN}, or any value \(< 0\). In this case, both \code{tolerance_svd()} and \code{tolerance_eigen()} simply call \code{base::svd()} and \code{base::eigen()} with no changes. When using the \code{tol} argument, eigen- and singular values are also checked to ensure that they are real and positive values. If they are not, then \code{geigen()}, \code{gsvd()}, and \code{gplssvd()} stop. The motivation behind this behavior is because the \code{geigen()}, \code{gsvd()}, and \code{gplssvd()} functions are meant to perform routine multivariate analyses---such as MDS, PCA, CA, CCA, or PLS---that require data and/or constraint matrices assumed to be positive semi-definite. Data matrices are the minimally required objects for \code{geigen()}, \code{gsvd()}, and \code{gplssvd()}. All other arguments (input) either have suitable defaults or are allowed to be missing. For example, when any of the constraints (``weights'') are missing, then the constraints are mathematically equivalent to identity matrices (i.e., \({\bf I}\)) which contain \(1\)s on the diagonal with \(0\)s off-diagonal. Table \ref{tab:arguments} shows a mapping between our (more formal) notation above and our more intuitively named arguments for the functions. The rows of Table \ref{tab:arguments} are the three primary functions---\code{geigen()}, \code{gsvd()}, and \code{gplssvd()}---where the columns are the elements used in the formal notation (and also used in the tuple notation). \begin{table} \begin{center} \begin{tabular}{l*{6}{c}r} & ${\bf X}$ & ${\bf Y}$ & ${{\bf M}_{\bf X}}$ & ${{\bf W}_{\bf X}}$ & ${{\bf M}_{\bf Y}}$ & ${{\bf W}_{\bf Y}}$ & $C$ \\ \hline \code{geigen()} & \code{X} & - & - & \code{W} & - & - & \code{k} \\ \code{gsvd()} & \code{X} & - & \code{LW} & \code{RW} & - & - & \code{k} \\ \code{gplssvd()} & \code{X} & \code{Y} & \code{XRW} & \code{XLW} & \code{YRW} & \code{YLW} & \code{k} \\ \end{tabular} \caption{\label{tab:arguments}Mapping between arguments (input) to functions (rows) and notation for the analysis tuples (columns).} \end{center} \end{table} Additionally, there are some ``helper'' and convenience functions used internally to the \code{geigen()}, \code{gsvd()}, and \code{gplssvd()} functions that are made available for use. These include \code{sqrt_psd_matrix()} and \code{invsqrt_psd_matrix()} which compute the square root (\code{sqrt}) and inverse square root (\code{invsqrt}) of positive semi-definite (\code{psd}) matrices (\code{matrix}), respectively. The \pkg{GSVD} package also includes helpful functions for testing matrices: \code{is_diagaonal_matrix()} and \code{is_empty_matrix()}. Both of these tests help minimize the memory and computational footprints for, or check validity of, the constraints matrices. Finally, the three core functions in \pkg{GSVD}---\code{geigen()}, \code{gsvd()}, and \code{gplssvd()}---each have their own class objects but provide overlapping and identical outputs. The class object is hierarchical from a list, to a package, to the specific function: \code{c("geigen","GSVD","list")}, \code{c("gsvd","GSVD","list")}, and \code{c("gplssvd","GSVD","list")} for \code{geigen()}, \code{gsvd()}, and \code{gplssvd()} respectively. Table \ref{tab:values} list the possible outputs across \code{geigen()}, \code{gsvd()}, and \code{gplssvd()}. The first column of Table \ref{tab:values} explains the returned value, where the second column provides a mapping back to the notation used here. The last three columns indicate---with an `X'---which of the returned values are available from the \code{geigen}, \code{gsvd}, or \code{gplssvd} functions. \begin{table} \begin{center} \begin{tabular}{l*{4}{c}r} & What it is & Notation & \code{geigen} & \code{gsvd} & \code{gplssvd} \\ \hline \code{d} & \code{k} singular values & ${\bf \Delta}$ & \checkmark & \checkmark & \checkmark\\ \code{d_full} & all singular values & ${\bf \Delta}$ & \checkmark & \checkmark & \checkmark \\ \code{l} & \code{k} eigenvalues & ${\bf \Lambda}$ & \checkmark & \checkmark & \checkmark \\ \code{l_full} & all eigenvalues & ${\bf \Lambda}$ & \checkmark & \checkmark & \checkmark \\ \code{u} & \code{k} Left singular/eigen vectors & ${\bf U}$ & & \checkmark & \checkmark \\ \code{v} & \code{k} Right singular/eigen vectors & ${\bf V}$ & \checkmark & \checkmark & \checkmark \\ \code{p} & \code{k} Left generalized singular/eigen vectors & ${\bf P}$ & & \checkmark & \checkmark \\ \code{q} & \code{k} Right generalized singular/eigen vectors & ${\bf Q}$ & \checkmark & \checkmark & \checkmark \\ \code{fi} & \code{k} Left component scores & ${\bf F}_{I}$ & & \checkmark & \checkmark \\ \code{fj} & \code{k} Right component scores & ${\bf F}_{J}$ & \checkmark & \checkmark & \checkmark \\ \code{lx} & \code{k} Latent variable scores for \code{X} & ${\bf L}_{\bf X}$ & & & \checkmark \\ \code{ly} & \code{k} Latent variable scores for \code{Y} & ${\bf L}_{\bf Y}$ & & & \checkmark \\ \end{tabular} \caption{\label{tab:values}Mapping of values (output from functions; rows) to their conceptual meanings, notation used here, and which \pkg{GSVD} functions have these values.} \end{center} \end{table} Different fields and traditions use different nomenclature, different descriptions, or different ways of framing optimizations. But conceptually and mathematically, numerous multivariate techniques are much more related than they appear, especially when solved with the EVD and SVD. The \pkg{GSVD} package provides a single framework to unify common multivariate analyses by way of three generalized decompositions. The \code{arguments} (function inputs) and \code{values} (function outputs) help reinforce the mathematical and coneptual equivalence and relationships between techniques, and now via the \pkg{GSVD} package, we see this unification programatically and through analyses. Therefore the common names---across the core functions---for the \code{values} in Table \ref{tab:values} was an intentional design choice. Finally, this package is ``lightweight'' in that it is written in base R, with no dependencies, and has a minimal set of functions to achieve the goals of \pkg{GSVD}. Furthermore, a number of strategies were employed in order to minimize both memory and computational footprints. For example, when constraints matrices are available, they are checked for certain conditions. Specifically, if a matrix is a diagonal matrix is it transformed into a vector, which decreases memory consumption and speeds up some computations (e.g., multiplication). If that same vector is all \(1\)s, then the matrix was an identity matrix, and is then ignored in all computation. \hypertarget{examples-of-multivariate-analyses}{% \section{Examples of multivariate analyses}\label{examples-of-multivariate-analyses}} In this section, I present many of the most commonly used multivariate analyses, and how they can be performed through the \pkg{GSVD} package, as well as how these methods can be framed in various ways. Here, I focus primarily on what are likely the most common: principal components analysis, multidimensional scaling, correspondence analysis (and some of its variations), partial least squares, reduced rank regression \citep[a.k.a. redundnacy analysis or multivariable multivariate regression;][]{van1977redundancy, de2012least}, and canonical correlation analysis. As I introduce these methods, I also introduce how to use various functions (and their parameters) in \pkg{GSVD}. There are other very common multivariate techniques, such as log and power CA methods \citep{greenacre_correspondence_2010, greenacre2009power}, and various discriminant analyses. I forgo the descriptions of these latter cases because they tend to be specific or special cases of techniques I do highlight. For examples: log-linear CA requires additional transformations and then performs CA, and various discriminant analyses reduce to special cases of PLS, RRR, or CCA. The \pkg{GSVD} package contains several toy or illustrative data sets that work as small examples of various techniques. There is also a larger and more realistic data set in the package that I use in the following examples. That data set is a synthetic data set modeled after data from the Ontario Neurodegenerative Disease Research Initiative (ONDRI; \url{https://ondri.ca/}). The synthetic data were generated from real data, but were ``synthesized'' with the \pkg{synthpop} package \citep{synthpop}. This synthetic data set---\code{synthetic_ONDRI}---contains 138 rows (observations) and 17 columns (variables). See \url{https://github.com/ondri-nibs} for more details. The \code{synthetic_ONDRI} data set is particularly useful because it contains a variety of data types, e.g., some quantitative such as cognitive or behavioral scores that are continuous, and brain volumes that are strictly positive integers, as well as categorical and ordinal variables (typically demographic or clinical measures). For each of the following illustrations of techniques, I use particular subsets of the \code{synthetic_ONDRI} data most relevant or appropriate for those techniques (e.g., continuous measures for PCA, distances for MDS, cross-tabulations of categories for CA). \hypertarget{principal-components-analysis}{% \subsection{Principal components analysis}\label{principal-components-analysis}} Generally, there are two ways to approach PCA: with a covariance matrix or with a correlation matrix. First, I show both of these PCA approaches on a subset of continuous measures from the \code{synthetic_ONDRI} dataset. Then I focus on correlation PCA, but with an emphasis on (some of) the variety of ways we can perform correlation PCA with generalized decompositions. PCA is illustrated with a subset of continuous measures from cognitive tasks. We can perform a covariance PCA and a correlation PCA with the generalized eigendecomposition as: \begin{CodeChunk} \begin{CodeInput} R> continuous_data <- synthetic_ONDRI[,c("TMT_A_sec", "TMT_B_sec", + "Stroop_color_sec", "Stroop_word_sec", + "Stroop_inhibit_sec", "Stroop_switch_sec")] R> R> cov_pca_geigen <- geigen( cov(continuous_data) ) R> cor_pca_geigen <- geigen( cor(continuous_data) ) \end{CodeInput} \end{CodeChunk} In these cases, the use here is no different---from a user perspective---of how PCA would be performed with the plain \code{eigen}. For now, the major advantage of the \code{geigen} approach is that the output (values) also include component scores and other measures common to these decompositions, such as singular values, as seen in the output. The following code chunk shows the results of the \code{print} method for \code{geigen} \begin{CodeChunk} \begin{CodeInput} R> cov_pca_geigen \end{CodeInput} \begin{CodeOutput} **GSVD package object of class type 'geigen'.** geigen() was performed on a marix with 6 columns/rows Number of total components = 6. Number of retained components = 6. The 'geigen' object contains: $d_full Full set of singular values $l_full Full set of eigen values $d Retained set of singular values (k) $l Retained set of eigen values (k) $v Eigen/singular vectors $q Generalized eigen/singular vectors $fj Component scores \end{CodeOutput} \begin{CodeInput} R> cor_pca_geigen \end{CodeInput} \begin{CodeOutput} **GSVD package object of class type 'geigen'.** geigen() was performed on a marix with 6 columns/rows Number of total components = 6. Number of retained components = 6. The 'geigen' object contains: $d_full Full set of singular values $l_full Full set of eigen values $d Retained set of singular values (k) $l Retained set of eigen values (k) $v Eigen/singular vectors $q Generalized eigen/singular vectors $fj Component scores \end{CodeOutput} \end{CodeChunk} A more comprehensive approach to PCA is the analysis of a rectangular table---i.e., the observations by the measures---as opposed to the square symmetric matrix of relationships between the variables. The more comprehensive approach to PCA can be performed with \code{gsvd}, as shown below, with its \code{print} method to show the output from \code{gsvd}. For this example of correlation PCA, we introduce a set of constraints for the rows. For standard PCA, the row constraints are just \(\frac{1}{I}\) where \(I\) is the number of rows in the data matrix. \begin{CodeChunk} \begin{CodeInput} R> scaled_data <- scale(continuous_data, center = T, scale = T) R> degrees_of_freedom <- nrow(continuous_data)-1 R> row_constraints <- rep(1/(degrees_of_freedom), nrow(scaled_data)) R> R> cor_pca_gsvd <- gsvd( scaled_data, LW = row_constraints ) R> R> cor_pca_gsvd \end{CodeInput} \begin{CodeOutput} **GSVD package object of class type 'gsvd'.** gsvd() was performed on a matrix with 138 rows and 6 columns Number of components = 6. Number of retained components = 6. The 'gsvd' object contains: $d_full Full set of singular values $l_full Full set of eigen values $d Retained set of singular values (k) $l Retained set of eigen values (k) $u Left singular vectors (for rows of DAT) $v Right singular vectors (for columns of DAT) $p Left generalized singular vectors (for rows of DAT) $q Right generalized singular vectors (for columns of DAT) $fi Left component scores (for rows of DAT) $fj Right component scores (for columns of DAT) \end{CodeOutput} \end{CodeChunk} In the next code chunk, we can see that the results for the common objects---the singular and eigenvalues, the generalized and plain singular vectors, and the component scores---are all identical between \code{geigen} and \code{gsvd}. \begin{CodeChunk} \begin{CodeInput} R> all( + all.equal(cor_pca_geigen$l_full, cor_pca_gsvd$l_full), + all.equal(cor_pca_geigen$d_full, cor_pca_gsvd$d_full), + all.equal(cor_pca_geigen$v, cor_pca_gsvd$v), + all.equal(cor_pca_geigen$q, cor_pca_gsvd$q), + all.equal(cor_pca_geigen$fj, cor_pca_gsvd$fj) + ) \end{CodeInput} \begin{CodeOutput} [1] TRUE \end{CodeOutput} \end{CodeChunk} The use of row constraints helps illustrate the GSVD triplet. The previous correlation PCA example is \(\mathrm{GSVD(}\frac{1}{I-1}{\bf I}, {\bf X}, {\bf I}\mathrm{)}\) where \(\frac{1}{I}{\bf I}\) is a diagaonal matrix of \(\frac{1}{I-1}\) with \(I\) as the number of rows of \({\bf X}\) and \({\bf I}\) is the identity matrix. In this particular case, the row constraints are all equal and thus a simple scaling factor. Alternatively, we could just divide by that scalar as \code{gsvd( scaled_data / sqrt(degrees_of_freedom))} which is equivalent to \code{geigen( cor(continuous_data) )}. The previous PCA triplet---\(\mathrm{GSVD(}\frac{1}{I}{\bf I}, {\bf X}, {\bf I}\mathrm{)}\)---helps us transition to an alternative view of correlation PCA and an expanded use of the GSVD triplet. Correlation PCA can be reframed as covariance PCA with additional constraints imposed on the columns. Let's first compute PCA with row and column constraints via \code{gsvd}, then go into more detail on the GSVD triplet and its computation. \begin{CodeChunk} \begin{CodeInput} R> centered_data <- scale(continuous_data, scale = F) R> degrees_of_freedom <- nrow(centered_data)-1 R> row_constraints <- rep(1/(degrees_of_freedom), nrow(centered_data)) R> col_constraints <- degrees_of_freedom / colSums( (centered_data)^2 ) R> R> cor_pca_gsvd_triplet <- gsvd(centered_data, + LW = row_constraints, + RW = col_constraints) \end{CodeInput} \end{CodeChunk} First, let's test the equality between the previous correlation PCA and this alternative approach. \begin{CodeChunk} \begin{CodeInput} R> mapply(all.equal, cor_pca_gsvd, cor_pca_gsvd_triplet) \end{CodeInput} \begin{CodeOutput} d u "TRUE" "TRUE" v d_full "TRUE" "TRUE" l_full l "TRUE" "TRUE" p fi "TRUE" "TRUE" q fj "Mean relative difference: 23.65976" "Mean relative difference: 0.9420834" \end{CodeOutput} \end{CodeChunk} We can see that almost everything is identical, with two exceptions: \code{q} and \code{fj}. That's because the decomposed matrix is identical across the two, but their constraints are not. The commonalities and the differences stem from how the problem was framed. Let's unpack what we did for the three versions of PCA shown. Let's assume that \({\bf X}\) is a matrix that has been column-wise centered. The covariance matrix for \({\bf X}\) is \({\bf \Sigma} = {\bf X}^{T}(\frac{1}{I-1}{\bf I}){\bf X}\), where \({\bf d} = \mathrm{diag\{}{\bf \Sigma}\mathrm{\}}\), which are the variances per variable, and \({\bf D} = \mathrm{diag\{}{\bf d}\mathrm{\}}\) which is a diagonal matrix of the variances. Each of the three correlation PCA approaches can be framed as the following generalized decompositions: \begin{itemize} \item \(\mathrm{GEVD(}{\bf D}^{-\frac{1}{2}}{\bf S}{\bf D}^{-\frac{1}{2}}, {\bf I}\mathrm{)}\) \item \(\mathrm{GSVD(}\frac{1}{I-1}{\bf I}, {\bf X}{\bf D}^{-\frac{1}{2}}, {\bf I}\mathrm{)}\) \item \(\mathrm{GSVD(}\frac{1}{I-1}{\bf I}, {\bf X}, {\bf D}^{-1} \mathrm{)}\) \end{itemize} The second way of framing PCA---\(\mathrm{GSVD(}\frac{1}{I-1}{\bf I}, {\bf X}{\bf D}^{-\frac{1}{2}}, {\bf I}\mathrm{)}\)---is the canonical ``French way'' of presenting PCA via the GSVD: a centered data matrix (\({\bf X}\)) normalized/scaled by the square root of the variances (\({\bf D}^{-\frac{1}{2}}\)), with equal weights of \({\frac{1}{I}}\) for the rows and an identity matrix for the columns. Though not always practical for simple PCA computation, the use of constraints in a GEVD doublet and GSVD triplet shows that we can frame (or re-frame) various techniques as decompositions of data with respect to constraints. In the case of PCA, the the constraint-based decompositions highlight that we could perform PCA with any (suitable) constraints applied to the rows and columns. So for example, we may have \emph{a priori} knowledge about the observations (rows) and elect to use different weights for each observation (we'll actually see this in the MDS examples). We could also use some known or assumed population variance as column constraints instead of the sample variance, or use column constraints as a set of weights to impose on the variables in the data. \hypertarget{metric-multidimensional-scaling}{% \subsection{(Metric) Multidimensional scaling}\label{metric-multidimensional-scaling}} Metric multidimensional scaling (MDS) is a technique akin to PCA, but specifically for the factorization of distance matrix \citep{torgerson1952multidimensional, borg2005modern}. MDS, like PCA, is also an eigen-technique. Like the PCA examples, I show several ways to perform MDS through the generalized approaches; specifically all through the GEVD. But for this particular example we will (eventually) make use of some known or \emph{a priori} information as the constraints. First we show how to perform MDS as a plain EVD problem and then with constraints as a GEVD problem. In these MDS illustrations, I use a conceptually simpler albeit computationally more expensive approach. For example, I use centering matrices and matrix algebra (in \proglang{R}) to show the steps. Much more efficient methods exist. In these examples we use the same measures as in the PCA example. \begin{CodeChunk} \begin{CodeInput} R> data_for_distances <- synthetic_ONDRI[, + c("TMT_A_sec", "TMT_B_sec", + "Stroop_color_sec", "Stroop_word_sec", + "Stroop_inhibit_sec","Stroop_switch_sec")] R> scaled_data <- scale(data_for_distances, center = T, scale = T) R> R> distance_matrix <- as.matrix( dist( scaled_data ) ) R> R> row_weights <- rep(1/nrow(distance_matrix), nrow(distance_matrix)) R> R> centering_matrix <- diag(nrow(distance_matrix)) - + ( rep(1,nrow(distance_matrix)) R> R> matrix_to_decompose <- centering_matrix + (-(distance_matrix^2)/2) + t(centering_matrix) R> R> mds_geigen <- geigen(matrix_to_decompose) \end{CodeInput} \end{CodeChunk} The results from \code{geigen(matrix_to_decompose)} produce a variety of outputs that align with the concept of eigenvectors, generalized eigenvectors, and component scores. But, more specifically, the results of \code{base::cmdscale(distance_matrix)} are identical to \code{mds_geigen$fj[,1:2]}; that is, MDS scores as viewed through \code{geigen} are component scores. However, the generalized approach allows us to include constraints. In the following example, we can now include a weighting factor per observation as a constraint to impose on that observation. Here we use the inverse of age, so that we effectively downweight older individuals and upweight younger individuals. \begin{CodeChunk} \begin{CodeInput} R> row_weights <- 1/synthetic_ONDRI$AGE R> R> centering_matrix <- + diag(nrow(distance_matrix)) - ( + rep(1,nrow(distance_matrix)) + ) R> R> R> matrix_to_decompose <- + -(centering_matrix R> R> mds_weighted_geigen <- geigen( matrix_to_decompose , W = row_weights, tol = NA ) \end{CodeInput} \end{CodeChunk} In \code{mds_weighted_geigen} we require the use of \code{tol=NA}. This is because \code{matrix_to_decompose} is not positive semi-definite. Recall that one of the key principles of the \pkg{GSVD} package is that we require positive semi-definite matrices for the constraints, and that the design of \code{geigen} also---by default---assume positive semi-definite matrices. This is because most multivariate analyses---from the eigendecomposition perspective---require correlation or covariance matrices which are by definition positive semi-definite. If we were to run \code{geigen(matrix_to_decompose, diag(row_weights))} we would get an error. In fact, we are unable to set an appropriate tolerance parameter in this case because the last few eigenvalues have large \emph{negative} values. But the use of \code{tol=NA} allows for a direct computation of the eigendecomposition, without dropping any of the dimensions (e.g., those below tolerance). For such an analysis, it is typical to discard dimensions with such eigenvalues (which is why it is a default in the package). However, some standard analyses do violate those assumptions. For examples: the weighted MDS here, principal axis factoring, and other approaches to factor analysis where, for example, the diagonal of a correlation matrix is loaded with the communalities. So the \pkg{GSVD} package accommodates these approaches by effectively ignoring the tolerance requirement for eigenvalues. The weighted version of MDS produces the same kind of results as \code{vegan::wcmdscale}, but there exist differences the computation of scores. From the perspective of \pkg{GSVD} the scores are the generalized singular vectors, scaled by the singular values per component, and with respect to a metric (i.e., the constraints). Thus---as we've seen in the previous section---the component scores are \({\bf W}_{\bf X}{\bf Q}{\bf \Delta}\). Many SVD and eigen-based analyses have numerous ways of presenting and visualizing results. In fact, the \pkg{vegan} package presents a slightly different version of the weighted MDS scores compared to \pkg{GSVD}. From the \pkg{GSVD} perspective, \pkg{vegan} produces scores computed from the generalized singular vectors scaled by their respective singular values, or \({\bf Q}{\bf \Delta}\). We can see this in Figure \ref{fig:mds}, which shows the first two components from MDS and the weighted MDS, with component 1 on the horizontal axis and component 2 on the vertical axis. Figures \ref{fig:mds}a, c, and e each show the standard MDS solution where Figures \ref{fig:mds}b, d, and f show the weighted (constrained) MDS solution. Figures \ref{fig:mds}a and b show the eigen/singular vectors scaled by the singular values (\({\bf V}{\bf \Delta}\)), Figures \ref{fig:mds}c and d show the weighted generalized eigen/singular vectors scaled by the singular values (\({\bf W}_{\bf X}{\bf Q}{\bf \Delta}\)), and Figures \ref{fig:mds}c and d show the generalized eigen/singular vectors scaled by the singular values (\({\bf Q}{\bf \Delta}\)). Note that, in general, component scores (e.g., Figures \ref{fig:mds}e and f) should show axes proportion to their eigen or singular values \citep{nguyen_ten_2019}. This is true for virtually all analyses shown in this paper. However, for simplicity and to help with comparisons, here I do not scale the axes. We can see in Figure \ref{fig:mds} that the standard form of MDS produces sets of scores that show no differences in their pattern, but that the weighted version does across the various scores. Furthermore, we can also see that each weighted version set provides a slightly different perspective compared to the standard MDS. \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./mds_vis-1.pdf} } \caption{\label{fig:mds} Metric multidimensional scaling (MDS). Panels A, C, and E show standard MDS where panels B, D, and F show weighted MDS. Panels A and B show the singular vectors, panels C and D show the generalized singular vectors, and panels E and F show the component scores. Note how the configuration of the observations change in the weighted MDS through the different sets of scores (panels B, D, and F).}\label{fig:mds_vis} \end{figure} \end{CodeChunk} \hypertarget{correspondence-analysis}{% \subsection{Correspondence analysis}\label{correspondence-analysis}} Correspondence analysis (CA) is one of---if not \emph{the}---prototypical GSVD triplet method. CA is like PCA, but was originally designed for two-way contingency tables, and was then expanded into multiple correspondence analysis (MCA) for N-way contingency tables \citep{greenacre_theory_1984, lebart_multivariate_1984, escofier-cordier_analyse_1965, greenacre_correspondence_2010, greenacre2006multiple}. MCA is more like PCA in that the rows are typically observations and the columns are measures, where the data are transformed into ``complete disjunctive coding'' (a.k.a. nominal coding, dummy coding, one-hot encoding, and a variety of other names). CA methods can be thought of as a ``\({\chi^2}\) PCA''. Prior to decomposition, the data matrix for CA methods are preprocessed in a way to make the data table analogous to that of a \({\chi^2}\) table, where we decompose the (weighted) deviations under the assumption of independence. The next section goes through three forms of CA. Each form helps highlight the utility and flexibility of the GSVD and make use of various forms of weights. Each illustration is one of the many well-established derivatives of CA. The first example illustrates standard CA (on a two-way contingency table). The second example illustrates MCA on multiple categorical variables, which is then followed by---and compared with---the third example, a \emph{regularized} MCA of the same data. Let's begin with standard CA---which is applied to a two-way contingency table. Here we'll use genotypes from two genes: ApoE and MAPT. Both are risk factors in neurodegenerative disorders. \begin{CodeChunk} \begin{CodeInput} R> observed_matrix <- mapt_by_apoe_table / sum(mapt_by_apoe_table) R> row_probabilities <- rowSums(observed_matrix) R> col_probabilities <- colSums(observed_matrix) R> expected_matrix <- row_probabilities R> deviations_matrix <- observed_matrix - expected_matrix R> R> ca_gsvd <- gsvd( deviations_matrix, + LW = 1/row_probabilities, + RW = 1/col_probabilities ) \end{CodeInput} \end{CodeChunk} In CA, the primarily and almost exclusively visualized items are the component scores: \code{fi} and \code{fj}. Recall that the components scores are the generalized singular vectors, scaled by the singular values, under the metric defined by the constraints, or \({\bf M}_{\bf X}{\bf P}{\bf \Delta}\) and \({\bf W}_{\bf X}{\bf P}{\bf \Delta}\). The reason behind the scores as visuals in CA is that the scores reflect the \({\bf \chi}^2\) distance. The CA component scores are shown in Figure \ref{fig:ca}. \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./ca_vis-1} } \caption{\label{fig:ca} Row and column component scores for Correspondence Analysis (CA). Panel A shows the row component scores, where panel B shows the column component scores.}\label{fig:ca_vis} \end{figure} \end{CodeChunk} \begin{CodeChunk} \begin{table} \caption{\label{tab:mca_data_prep}\label{tab:disj} Illustration of complete disjunctive coding (a.k.a. dummy coding, one-hot encoding) where each level of a categorical variable is represented. A value of '1' indicates the presence of that level for that row ('0' otherwise).} \centering \begin{tabular}[t]{lrrrrr} \toprule & MAPTH1H1 & MAPTH1H2 & MAPTH2H2 & APOEE32 & APOEE33\\ \midrule OND01\_SYN\_0001 & 0 & 0 & 1 & 0 & 0\\ OND01\_SYN\_0011 & 1 & 0 & 0 & 0 & 1\\ OND01\_SYN\_0021 & 0 & 1 & 0 & 0 & 1\\ OND01\_SYN\_0041 & 1 & 0 & 0 & 0 & 1\\ OND01\_SYN\_0083 & 1 & 0 & 0 & 0 & 1\\ \bottomrule \end{tabular} \end{table} \end{CodeChunk} Next I show MCA, which generalizes standard CA to multiple categorical variables. Here we use four variables: MAPT and ApoE like before, and now also include sex, and a clinical measure with a few ordinal levels (but here we treat those as levels of a categorical variable). The computation for MCA is exactly the same as CA, but now the data are disjunctive (see Table \ref{tab:disj}). However, MCA is more akin to PCA with observations on the rows and measures on the columns, as opposed to standard CA which has measures on both the rows and columns. The results of MCA are shown in Figure \ref{fig:mca}, where Figure \ref{fig:mca}a shows the component scores for the rows (observations) and Figure \ref{fig:mca}b shows the columns (measures, some of which are seen in Table \ref{tab:disj}). \begin{CodeChunk} \begin{CodeInput} R> observed_matrix <- disjunctive_data / sum(disjunctive_data) R> row_probabilities <- rowSums(observed_matrix) R> col_probabilities <- colSums(observed_matrix) R> expected_matrix <- row_probabilities R> deviations_matrix <- observed_matrix - expected_matrix R> R> mca_gsvd <- gsvd( deviations_matrix, + LW = 1/row_probabilities, + RW = 1/col_probabilities) \end{CodeInput} \end{CodeChunk} \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./mca_vis-1} } \caption{\label{fig:mca} Row and column component scores from Multiple Correspondence Analysis (MCA). Here, the row component scores (panel A) are observations, where the column component scores (panel B) are levels of categorical variables.}\label{fig:mca_vis} \end{figure} \end{CodeChunk} Up until this point, all constraints matrices have been diagonal matrices. The next example serves two purposes: (1) use a well-established method that has more sophisticated constraints than simple vectors (or diagonal matrices), and (2) highlight a particular ridge regularization approach to show how the GSVD easily incorporates such techniques. Takane \citep{takane_regularized_2006} introduced a variation of MCA called regularized MCA (RMCA) that uses a ridge-like approach. The ridge-like approach requires a cross product matrix for the row (observation) constraints, and a block-diagonal matrix for the column (measures) constraints. The next block of code shows how the procedure works through the GSVD. When the regularization parameter---\(\omega\)---is 0, then this is the same as standard MCA (within a scaling factor). Also note that for each iteration of RMCA, we now make use of the \code{k} parameter where \code{gsvd(..., k = 2)}, so that we only return a subset of possible results. All sets of vectors and scores are just the first two components, with the exception of \code{\$d_full} and \code{\$l_full} which return the full set of singular and eigenvalues, respectively. Note the small changes in the output object that indicate how many full (possible) components exist, and also how many were returned (\code{k=2}). \begin{CodeChunk} \begin{CodeInput} R> omegas <- c(0, 1, 2, 3, 4, 5, 10, 25, 50) R> rmca_results <- vector("list",length(omegas)) R> R> centered_data <- scale(disjunctive_data,scale=F) R> projmat <- t(centered_data) + MASS::ginv( tcrossprod(centered_data) ) + centered_data R> R> for(i in 1:length(omegas)){ + + LW <- diag(nrow(centered_data)) + + (omegas[i] * MASS::ginv(tcrossprod(centered_data))) + RW <- diag(colSums(disjunctive_data)) + (omegas[i] * projmat) + invRW <- t(MASS::ginv(RW)) + + rownames(LW) <- colnames(LW) <- rownames(centered_data) + rownames(invRW) <- rownames(RW) + colnames(invRW) <- colnames(RW) + + rmca_results[[i]] <- gsvd(centered_data, LW = LW, RW = invRW, k = 2) + + } \end{CodeInput} \end{CodeChunk} \begin{CodeChunk} \begin{CodeInput} R> rmca_results[[1]] \end{CodeInput} \begin{CodeOutput} **GSVD package object of class type 'gsvd'.** gsvd() was performed on a matrix with 138 rows and 15 columns Number of components = 11. Number of retained components = 2. The 'gsvd' object contains: $d_full Full set of singular values $l_full Full set of eigen values $d Retained set of singular values (k) $l Retained set of eigen values (k) $u Left singular vectors (for rows of DAT) $v Right singular vectors (for columns of DAT) $p Left generalized singular vectors (for rows of DAT) $q Right generalized singular vectors (for columns of DAT) $fi Left component scores (for rows of DAT) $fj Right component scores (for columns of DAT) \end{CodeOutput} \end{CodeChunk} So what does this regularization do? In Figure \ref{fig:rmca_v_mca} we show a partial view of how this regularization works. In Figures \ref{fig:rmca_v_mca}a and b we see the scree plots, which is the explained variance (eigenvalues) per component. In Figure \ref{fig:rmca_v_mca}c and d we see the component scores for the columns. Figure \ref{fig:rmca_v_mca}a and c show the standard MCA where Figure \ref{fig:rmca_v_mca}b and d show each iteration of the RMCA, where the \(\omega\) parameter is used to color the eigenvalues and component scores on a gradient scale (via \code{ggplot::scale_color_gradient(...,trans="log1p")}). Figure \ref{fig:rmca_v_mca} shows that as the regularization parameter increases, we see that explained variance for the higher dimensions \emph{decreases}, and that in general, the component scores approach zero. Figure \ref{fig:rmca_v_mca}d in particular also shows us that stable measures do not change much under regularization (e.g., \texttt{SEXfemale}) where as rare and relatively unstable measures approach zero and see substantial changes in their component scores (e.g., \texttt{MAPTH2H2} or \texttt{APOEE42}). The CA-based approaches make use of each element in the GSVD. The implementations here show that standard CA and MCA use vectors---algebraically these are diagonal matrices---where RMCA requires more sophisticated matrices than simple diagonal matrices. RMCA illustrates two key concepts: (1) with appropriate sets of constraints, the GSVD---and all of the generalized methods presented here---provides an easy pathway to ridge regularization, and (2) constraints matrices are not always simple diagonal matrices; an important point expanded upon in the next section. \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./unnamed-chunk-1-1} } \caption{\label{fig:rmca_v_mca} MCA (panels A and C) vs. Regularized MCA (panels B and D). Panels A and B show scree plots, which reflect the amount of explained variance per component (and thus decreases as components increase). RMCA shows multiple overlaid screes because there exists one version for each run with $\omega$. Panels C and D show the column component scores (levels of categorical variables) in the plain MCA and as the scores change through the iterations of RMCA. Only the first two components are shown, with the first component on the horizontal axis, and the second component on the vertical axis. Note that the initial RMCA ($\omega = 0$) is equivalent to MCA.}\label{fig:unnamed-chunk-1} \end{figure} \end{CodeChunk} \hypertarget{partial-least-squares-reduced-rank-regression-and-canonical-correlation}{% \subsection{Partial least squares, reduced rank regression, and canonical correlation}\label{partial-least-squares-reduced-rank-regression-and-canonical-correlation}} So far, we have seen various examples of the GEVD and GSVD. The GEVD and GSVD are well-established concepts. Recently, I introduced an extension of the GEVD and GSVD concept to ``two (data) table techniques'', and called it this new extension the ``GPLSSVD'' \citep{beaton_generalization_2019}, which is short for the ``generalized partial least squares-singular value decomposition''. The GPLSSVD provides the same base concepts as the GEVD or GSVD, but specifically for scenarios wherein there are two data matrices. The most common two-table techniques are canonical correlation analysis (CCA), reduced rank regression (RRR; sometimes called redundancy analysis, RDA), and a specific form of partial least squares (PLS) called the PLSSVD, ``partial least squares correlation'' (PLSC), and a variety of other names. This particular form of PLS traces its history to Tucker's interbattery factor analysis. The GPLSSVD requires at minimum two data matrices, and allows for the inclusion of positive semi-definite constraints matrices for the rows and columns of both data matrices. Here, I illustrate four methods that make use of the GPLSSVD. All of these examples emphasize the expression of latent variables, though do recall that at their core, these methods make use of the SVD. Each method makes use of different sets of constraints: (1) PLS-SVD/PLSC which uses no constraints, (2) RRR/RDA which uses one set of constraints, (3) CCA which uses two sets of constraints, and finally (4) partial least squares-correspondence analysis (PLS-CA; \citet{beaton_generalization_2019}; \citet{beaton_partial_2016}) which makes use of all four sets of constraints. The first three methods---PLS, RRR, and CCA---are methods used to analyze numeric, or data that are generally assumed to be continuous values. The final method---PLS-CA---was initially designed for analysis of two data sets that are categorical, but easily extends to a variety of data types \citep{beaton_generalization_2019}. With that in mind, the first three examples will all make use of exactly the same data sets. This helps highlight the similarities and differences between PLS, RRR, and CCA, and shows how the same data can be viewed in multiple ways by imposing various constraints. In the PLS/RRR/CCA examples, we have two data matrices of generally continuous data. One matrix contains age, total gray matter volume (in percentage of total intracranial volume), and total white matter volume (in percentage of total intracranial volume). The other matrix contains the cognitive tasks seen in the PCA and MDS examples. Here, the respective GPLSSVD models for each method are: \begin{itemize} \item PLS: \(\mathrm{GPLSSVD(}{\bf I}_{N}, {\bf X},{\bf I}_{I},{\bf I}_{N},{\bf Y},{\bf I}_{J}\mathrm{)}\) \item RRR: \(\mathrm{GPLSSVD(}{\bf I}_{N}, {\bf X}, ({\bf X}^{T}{\bf X})^{-1},{\bf I}_{N},{\bf Y},{\bf I}_{J}\mathrm{)}\) \item CCA: \(\mathrm{GPLSSVD(}{\bf I}_{N}, {\bf X}, ({\bf X}^{T}{\bf X})^{-1},{\bf I}_{N},{\bf Y},({\bf Y}^{T}{\bf Y})^{-1}\mathrm{)}\) \end{itemize} Before an illustration of these techniques, it is worth noting that PLS, RRR, and CCA are multivariate extensions of univariate concepts. PLS emphasizes the covariance between \({\bf X}\) and \({\bf Y}\), RRR emphasizes the least squares fit (i.e., regression) of \({\bf Y}\) onto space defined by \({\bf X}\), and CCA emphasizes the correlation between \({\bf X}\) and \({\bf Y}\). For the first illustrations of PLS, RRR, and CCA, I use column-wise centered and scaled \({\bf X}\) and \({\bf Y}\) matrices. \begin{CodeChunk} \begin{CodeInput} R> X <- synthetic_ONDRI[, + c("TMT_A_sec", "TMT_B_sec", + "Stroop_color_sec","Stroop_word_sec", + "Stroop_inhibit_sec","Stroop_switch_sec")] R> Y <- synthetic_ONDRI[,c("AGE","NAGM_PERCENT","NAWM_PERCENT")] R> R> scaled_X <- scale(X, center = T, scale = T) R> scaled_Y <- scale(Y, center = T, scale = T) R> R> R> pls_gplssvd <- gplssvd(scaled_X, scaled_Y) R> R> rrr_gplssvd <- gplssvd(scaled_X, scaled_Y, + XRW = MASS::ginv(crossprod(scaled_X))) R> R> cca_gplssvd <- gplssvd(scaled_X, scaled_Y, + XRW = MASS::ginv(crossprod(scaled_X)), + YRW = MASS::ginv(crossprod(scaled_Y))) \end{CodeInput} \end{CodeChunk} All three approaches provide the same types of outputs: singular and eigenvalues, latent variable scores (for the participants), standard and generalized singular vectors, and component scores. The output object is identical for all three approaches because they all make use of \code{gplssvd()}, so let's only look at one of the objects, \texttt{cca\_gplssvd}. Like both the \code{geigen()} and \code{gsvd()} outputs, we have many common objects (e.g., vectors, scores, eigenvalues). However \code{gplssvd()} also provides the latent variable scores---\code{\$lx} and \code{\$ly}---which are row scores---for \({\bf X}\) and \({\bf Y}\), respectively---with respect to the singular vectors. For reference, see also Tables \ref{tab:arguments} and \ref{tab:values}. \begin{CodeChunk} \begin{CodeInput} R> cca_gplssvd \end{CodeInput} \begin{CodeOutput} **GSVD package object of class type 'gplssvd'.** gplssvd() was performed on an X matrix with 138 rows and 6 columns and a Y matrix with 138 rows and 3 columns Number of total components = 3. Number of retained components = 3. The 'gplssvd' object contains: $d_full Full set of singular values $l_full Full set of eigen values $d Retained set of singular values (k) $l Retained set of eigen values (k) $u Left singular vectors (for columns of X) $v Right singular vectors (for columns of Y) $p Left generalized singular vectors (for columns of X) $q Right generalized singular vectors (for columns of Y) $fi Left component scores (for columns of X) $fj Right component scores (for columns of Y) $lx Left (X) latent variable scores (for rows of X) $ly Right (Y) latent variable scores (for rows of Y) \end{CodeOutput} \end{CodeChunk} Before visualizing the results of these three approaches, let's first establish the connections between framing the methods via the GPLSSVD and the more typical ways. Let's do so by pointing out the relationship between \code{base::cancor} and the results in \code{cca_gplssvd}. From \code{cca_gplssvd}, the canonical correlations are the singular values. Recall also that the optimization when framed through the GPLSSVD is that the singular values are also \(\mathrm{diag\{}{\bf L}_{\bf X}^{T}{\bf L}_{\bf Y}\mathrm{\}}\), so \code{diag( t(cca_gplssvd\$lx) \%*\% cca_gplssvd\$ly )} are also the canonical correlations. So from the perspective of GPLSSVD, the relationship between the latent variables also maximize for the canonical correlations. Furthermore, the coefficients (per set of variables) from \code{base::cancor(scaled_X, scaled_Y)} are equal to \({\bf W}_{\bf X}{\bf P}\) and \({\bf W}_{\bf Y}{\bf Q}\) in the GPLSSVD model or alternatively also \({\bf F}_{I}{\bf \Delta}^{-1}\) or \({\bf F}_{J}{\bf \Delta}^{-1}\). The primary difference between the GPLSSVD approach and various implementations of canonical correlation (including \code{base::cancor()}) is that all sets of scores from the GPLSSVD are limited to the rank of the decomposed matrix. That is, if \({\bf X}\) has fewer variables than \({\bf Y}\), then all sets of scores associated with \({\bf Y}\) will be only span the vectors determined by the rank of the relationship between those matrices. Likewise, in reduced rank regression (a.k.a. redundancy analysis) there exist \(\beta\) coefficients which are expressed through the GPLSSVD model above as \({\bf P}{\bf \Delta}\) or \code{rrr_gplssvd\$p \%*\% diag(rrr_gplssvd\$d)}. Finally, it is worth noting that the above approaches to RRR and CCA are not the only ways to frame these techniques through the GPLSSVD. There are multiple and, in some cases, computationally simpler alternatives; however those approaches are conceptually more abstract than how I framed them here. Plus, the \pkg{GSVD} framework highlights how these techniques stem from the same core methods. Let's now visually compare the results from the GPLSSVD for PLS, RRR, and CCA. First let's look at the latent variable scores, which expresses the maximization in GPLSSVD as scores for the rows (observations) of each data matrix. \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./two_tables_vis_ls-1} } \caption{\label{fig:tt_ls}Latent variable scores for PLS, RRR, and CCA where with data matrices are column-wise centered and scaled. The latent variable scores for the rows of each data matrix, with the row items of $\bf X$ on the horizontal axes and with the row items of $\bf Y$ on the vertical axes. Panels A, C, and E show the first latent variable (X vs. Y) and panels B, D, and E show the second latent variable (X vs. Y). Panels A and B show PLS, panels C and D show RRR, and panels E and F show CCA.}\label{fig:two_tables_vis_ls} \end{figure} \end{CodeChunk} Figure \ref{fig:tt_ls} shows the latent variable scores for all three of the analyses. Latent variable scores are the expression of the rows of each data matrix wth respect to the components (a.k.a. latent variables). Each data matrix---i.e., \({\bf X}\) and \({\bf Y}\)---has a set of latent variable scores per component. Typically in PLS, the latent variables for each matrix are plotted against one another, that is, the first column of \({\bf L}_{\bf X}\) vs.~the first column of \({\bf L}_{\bf Y}\). Plotting the latent variables reflects what the GPLSSVD maximizes (i.e., \({\bf L}_{\bf X}^{T}{\bf L}_{\bf Y}\)). So we show these latent variables scores for all of the GPLSSVD approaches (PLS, RRR, and CCA). Figure \ref{fig:tt_ls} shows that the results are highly similar for the latent variable scores across approaches (a point I revisit soon). That is, these techniques show an approximately equivalent perspective of how the rows (observations) express themselves with respect to the latent variables. What about the variables of each data set? Figure \ref{fig:tt_fs} shows the component scores from these same analyses. Figure \ref{fig:tt_fs} shows some clear differences in how variables are related across the techniques---in particular Figure \ref{fig:tt_fs}a from PLS. The similarities and differences in Figure \ref{fig:tt_fs} highlight that while these techniques are very related, they do not necessarily express the relationships between \({\bf X}\) and \({\bf Y}\) in the same way. \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./two_tables_vis_fs-1} } \caption{\label{fig:tt_fs}Component scores for PLS, RRR, and CCA where with data matrices are column-wise centered and scaled. Panels A, C, and E show the variables from the $\bf X$ matrix, where panels B, D, and E show the variables from the $\bf Y$ matrix. The horizontal axes are the first component, where the vertical axes are the second component. Panels A and B show PLS, panels C and D show RRR, and panels E and F show CCA. }\label{fig:two_tables_vis_fs} \end{figure} \end{CodeChunk} Recall that for these analyses, GPLSSVD optimizes for the maximum common information between the two data sets---with respect to their constraints---where \({\bf L}_{\bf X}^{T}{\bf L}_{\bf Y} = {\bf \Delta} = {\bf P}^{T}{\bf W}_{\bf X}[({\bf M}_{\bf X}^{\frac{1}{2}}{\bf X})^{T}({\bf M}_{\bf Y}^{\frac{1}{2}}{\bf Y})]{\bf W}_{\bf Y}{\bf Q}\). Now let's look at these same approaches but where the data matrices are column-wise centered but not scaled. Figures \ref{fig:tt2ls} and \ref{fig:tt2fs} show the latent variable scores and the component scores where the data are only column-wise centered. Now the analyses show much more apparent differences. The differences are because the data transformations in conjunction with the constraints, each play an important role in the results, and thus critical roles for the interpretation of those results. \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./two_table2_vis_ls-1} } \caption{\label{fig:tt2ls}Latent variable scores for PLS, RRR, and CCA, but now with data only column-wise centered. The latent variable scores for the rows of each data matrix, with the row items of $\bf X$ on the horizontal axes and with the row items of $\bf Y$ on the vertical axes. Panels A, C, and E show the first latent variable (X vs. Y) and panels B, D, and E show the second latent variable (X vs. Y). Panels A and B show PLS, panels C and D show RRR, and panels E and F show CCA.}\label{fig:two_table2_vis_ls} \end{figure} \end{CodeChunk} \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./two_table2_vis_fs-1} } \caption{\label{fig:tt2fs}Component scores for PLS, RRR, and CCA, but now with the data only column-wise centered. Panels A, C, and E show the variables from the $\bf X$ matrix, where panels B, D, and E show the variables from the $\bf Y$ matrix. The horizontal axes are the first component, where the vertical axes are the second compnent. Panels A and B show PLS, panels C and D show RRR, and panels E and F show CCA. }\label{fig:two_table2_vis_fs} \end{figure} \end{CodeChunk} Finally, let's look at a GPLSSVD approach that makes use of all data and constraint matrices: PLS-CA. PLS-CA was initially designed as a PLS approach for categorical data \citep{beaton_partial_2016} but can accommodate virtually any data type \citep{beaton_generalization_2019}. Here, let's just focus on the problem of two categorical tables. The data for PLS-CA are processed in the same way they are for MCA, except now there are two tables. For the PLS-CA example, we have genetics in one table and age plus a clinical measure in the other table. All of these measures are categorical or ordinal. In this example and for simplicity, the ordinal data are treated as distinct levels instead of ordered levels. \begin{CodeChunk} \begin{CodeInput} R> plsca_gplssvd <- gplssvd( + X = deviations_matrix_X, + Y = deviations_matrix_Y, + XLW = 1/row_probabilities_X, + YLW = 1/row_probabilities_Y, + XRW = 1/col_probabilities_X, + YRW = 1/col_probabilities_Y + ) \end{CodeInput} \end{CodeChunk} Like the other GPLSSVD techniques, PLS-CA also produces latent variable scores. But we forgo visualizing those in favor of highlighting the singular vectors and scores. That's because PLS-CA is one particular method where we can much more clearly see the differences between the singular vectors, the generalized singular vectors, and the component scores. Figure \ref{fig:plsca} shows all sets of these scores. Figures \ref{fig:plsca}a, c, and e show the scores associated with the columns of one data matrix, where as Figure \ref{fig:plsca}b, d, and f show the scores for the other data matrix. Figures \ref{fig:plsca}a and b show the singular vectors, Figures \ref{fig:plsca}c and d show the \emph{generalized} singular vectors, and Figures \ref{fig:plsca}e and f show the component scores. Figures \ref{fig:plsca} shows component 1 on the horizontal axis with component 2 on the vertical axis for all subfigures. \begin{CodeChunk} \begin{figure} {\centering \includegraphics{./plsca_vis-1} } \caption{\label{fig:plsca}Singular vectors, generalized singular vectors, and component scores from partial least squares-correspondence analysis (PLS-CA). The horizontal axes are the first component, and the vertical axes are the second component. Panels A and B show the singular vectors, panels C and D show the generalized singular vectors, and Panels E and D show the component scores. Panels A, C, and E show the columns from the $\bf X$ matrix, where panels B, D, and F show the columns from the $\bf Y$ matrix. Each visual helps reflect the types and amount of information we can obtain from GPLSSVD methods that make use of various sets of weights.}\label{fig:plsca_vis} \end{figure} \end{CodeChunk} \hypertarget{final-notes}{% \subsection{Final notes}\label{final-notes}} There are several influential factors that drive the core principles that motivated the design and implementation of the \pkg{GSVD} package. These influences span the ``French way'' of data analyses \citep{holmes_discussion_2017, holmes_multivariate_2008}, the relationships between various decompositions \citep{takane_relationships_2003}. Furthermore, many of the most common multivariate techniques perform these non-arbitrary transformations via (positive semi-definitive) metrics used as constraints (see Takane GEVD \& GSVD). The metrics used for constraints include, for examples, \(\chi^2\) distances for Correspondence Analysis, Mahalanobis distances for CCA and related methods. At its core, the \pkg{GSVD} package provides the three primary bases---\code{geigen()}, \code{gsvd()}, and \code{gplssvd()}---of the most common multivariate techniques and numerous variants of those techniques. The package provides a radically simplified way to approach these methods: users generally need only to know what their constraints and (transformed) data should be, instead about how to perform the necessary steps (e.g., matrix multiplication, square roots of matrices). The \code{eigen()} and \code{svd()} functions---and many of their analogs and extensions---provide a very simplified set of outputs. However, with respect to core analysis and interpretation principles, the \pkg{GSVD} package provides comprehensive output common to the numerous eigen-based multivariate techniques. The component scores are examples of important and comprehensive outputs from the \pkg{GSVD} package. The component scores provide a view of the results with respect to (1) the constraints, and (2) the length of the components (singular values). The component scores---and the fact that these are weighted and ``stretched'' versions of the generalized vectors---means that interpretation and visualization requires caution and care. \citet{nguyen_ten_2019} provide a comprehensive view of decompositions, but most importantly make suggestions on how to more appropriately show and interpret visuals. The visuals presented here did not stretch the axis according to the proportions of the singular or eigenvalues. That stretching is important for interpretation, but here many of the visuals are either simple illustrations of output or for simple comparisons. Finally, it is worth noting that many techniques can be performed with any of these decompositions. For examples, PLS, RRR, and CCA could could be performed with the GSVD (or even with two GEVDs one for each set of variables). Likewise, one-table techniques (e.g., CA, or PCA) could be performed as the GSVD or GEVD (as presented earlier here with PCA). However, the direct interfaces of these methods provide a simpler conceptual framework for the techniques than some of these alternative approaches (e.g., CCA as two separate GEVDs). \hypertarget{discussion}{% \section{Discussion}\label{discussion}} The \pkg{GSVD} package provides a core and minimal set of functions designed around a common framework and nomenclature. More importantly, \pkg{GSVD} also conceptually unifies a large array of multivariate techniques. Above, I show only a handful of standard and very common multivariate approaches, and how they conceptually fit into generalized decomposition approaches as well as how to perform them with the \pkg{GSVD} package. The intent and design of \pkg{GSVD} was to provide a unified core for those analysis approaches. Though users could directly perform analyses with the \pkg{GSVD} package, there is likely more benefit to developers and analysts to build analysis methods and packages on top of \pkg{GSVD}. The primary motivations to build upon \pkg{GSVD} (i.e., use \pkg{GSVD} as a dependency) are because (1) \pkg{GSVD} only provides a minimal set of classes and objects for S3 methods, and more importantly (2) numerous multivariate techniques benefit from additional information; for examples, PCA and CA require preprocessing steps that are necessary for external or out of sample data, and thus require such external information (e.g., column centers, or row weights). Though only a relatively small set of multivariate techniques were shown here, the \pkg{GSVD} package was designed for virtually any variation of some of the methods presented here. Such methods include discriminant techniques that generally fall under specific instances of PLS, RRR, or CCA. There are also numerous variations of CA that make use of different---and sometimes asymmetric---sets of weights \citep{gimaret1998non}, use power transformations or various centers \citep{greenacre2009power}, alternate metrics such as the Hellinger distance \citep{rao1995review}, and numerous other alternatives \citep[see, e.g.,][]{beh2012genealogy} Even broader families of techniques that decompose multiple tables also fit within the framework established here. These multi-table techniques are often ways of structuring and transforming sets of tables, and then concatenating them in a way to produce a larger data matrix (with rows and columns). These multi-table techniques also routinely make use of constraints or weights imposed on the columns and rows. Some of those methods include, for examples, constrained PCA \citep{takane2013constrained} and multiple factor analysis \citep{abdi_multiple_2013, becue-bertaut_multiple_2008, escofier_multiple_1994} Finally, the use of returned rank is highly beneficial to statistical learning and machine learning strategies such as sparsification techniques such as the penalized matrix decomposition \citep{witten_penalized_2009} or the constrained SVD \citep{guillemot_constrained_2019}. Lower rank solutions is even more common for iterative algorithms like partial least squares regression (PLSR). The most common implementations of PLSR use a rank 1 SVD solution, followed by deflation of both the \(\bf X\) and \(\bf Y\) matrices. To see such PLSR implementations that depend on \pkg{GSVD}, please see another package I have developed that focuses on two-table/``cross-decomposition'' methods (\citet{beaton_generalization_2019} see also \url{https://github.com/derekbeaton/gpls}). In its current form, the \pkg{GSVD} package is lightweight, has zero dependencies, and is written in base R. The general goals of the \pkg{GSVD} package is both to remain lightweight---with minimal yet core functionality from a user perspective---and as a ``workhorse'' package. The \pkg{GSVD} package should be the core dependency in an analysis package, as opposed to being used as the analysis package. Given that, I plan to provide a substantial overhaul of the \pkg{ExPosition} \citep{beaton_exposition_2014} family of packages in the (near) future, and \pkg{GSVD} is the first step towards that overhaul. Though the \pkg{GSVD} package is intentionally minimal with no dependencies, there are possible benefits to some dependencies in the future. For example, there are numerous packages and tools that could make the \pkg{GSVD} package more memory and computationally efficient. So the future of \pkg{GSVD} could make use the \pkg{Matrix} \citep{bates_matrix_2019} for storage (memory) and other tools for faster decompositions themselves. Specifically, the eigen and singular value decompositions at the core of \code{tolerance_eigen()} and \code{tolerance_svd()} could be sped up and or be more memory efficient with the usage of \pkg{RcppArmadillo} \citep{eddelbuettel_rcpparmadillo_2014}, \pkg{RcppEigen} \citep{bates_fast_2013}, or \pkg{RSpectra} \citep{qiu_rspectra_2019}.

proofpile-arXiv_069-6935

\section{Introduction} \label{sec:introduction} Elastic fields produced by microstructural defects such as dislocations, grain boundaries and precipitates are known to deeply affect the evolution of material properties. Simulations including such elastic effects can be dislocation dynamics~\cite{Amodeo1990,Kubin1992,Ghoniem1999a,Arsenlis2007}, phase field~\cite{Rodney2003,Chen2002} and object kinetic Monte Carlo~\cite{Subramanian2013,Vattre2016}. Whatever the method used, precise values of the elastic field produced by the microstructural defects must be determined. To simulate a large system, it is common to use periodic boundary conditions (PBCs) and the influence of defects outside the simulation box must be taken into account for the calculation of the elastic fields. If elastic fields are calculated by solving the mechanical equilibrium in Fourier space, this contribution is taken into account naturally~\cite{Rodney2003}. Another possibility is to sum the contributions from individual defects located in image boxes, considering that each defect is in an infinite medium. It is known, however, that elastic fields obtained by direct summation over periodic images can contain a spurious component if the lattice sums are not \emph{absolutely} but only \emph{conditionally} convergent~\cite{Cai2003,Kuykendall2013}. A correction scheme has been proposed by W. Cai \emph{et al.} to recover a truly periodic elastic solution~\cite{Cai2003}. Consider, for example, the strain or stress fields produced by dislocation loops and cavities in three-dimensional simulations. For both kinds of defects, elastic fields decay as $1/x^3$ for large values of $x$ due to their non-zero dipolar component, so the sum over periodic images is not absolutely convergent. Similar conditionally convergent sums are present for the calculation of the electric or magnetic fields in materials containing electric or magnetic dipoles, respectively. It is well known that for such systems, conditional convergence is related to a \emph{shape effect}~\cite{Redlack1975,Leeuw1980,Allen1987}, \emph{ie} the value of the sum depends on the shape of the summation domain. If Ewald sum, which makes use of Fourier transform, is used instead, an ``intrinsic'' value, independent of the shape, is obtained~\cite{Redlack1975}. The value given by Ewald sum can be readily deduced from a direct summation, by removing the ``extrinsic'' shape correction which only depends on the dipole moment of the simulation cell~\cite{Redlack1975,Allen1987}. However, it should be remembered that the shape effect is physical and that in general, it should be present if a finite, macroscopic system is simulated. For a material containing electric or magnetic dipoles, the extrinsic correction corresponds to the depolarization and demagnetization fields. For electrostatic problems, Ewald sum corresponds to the very particular case of a system surrounded by a medium of infinite dielectric constant (``tin foil'' boundary condition) \cite{Leeuw1980}. For elastic problems, the situation is more complex. Contrary to electric and magnetic dipoles, the elastic field produced by an elastic dipole is not an intrinsic property of this defect, it depends on the prescribed boundary conditions. If boundaries are sufficiently far from the defect, the elastic solution in an infinite medium is appropriate. In the vicinity of a surface, this solution, however, leads to the appearance of surface tractions which must be canceled if a free surface is considered. An additional mechanical loading can be added if necessary. The question which arises is therefore the following: which correction, if any, should be added to the elastic field computed by direct summation to simulate the elastic field at the center of a macroscopic system of a given shape, with zero surface tractions? We will see that in general, a correction must be added, which depends on the macroscopic shape of the system. This correction also depends on the magnitude of the dipole component in the simulation box. We start by recalling the correction proposed by W. Cai \emph{et al.} to simulate periodic systems. This correction is reformulated in as surface integral over the macroscopic system. This formulation is then used to show that applying the correction amounts to simulating a macroscopic but finite system with uniform loading, related to the elastic dipole density. A correction is proposed to simulate a macroscopic system with zero surface tractions. In the last section, the impact of elastic corrections on the kinetics of dislocation loop growth by absorption of point defects is highlighted. \section{Reformulation of the correction for periodic systems} \label{sec:reformulation-correction-periodic-systems} In this section we investigate the physical meaning of the correction proposed by W. Cai \emph{et al.}~\cite{Cai2003} to remove the component of the strain or stress field linked to non-periodicity of the displacement field, when a direct summation over images is used. We consider a three-dimensional simulation box containing a non-zero elastic dipole component. It means that far from the simulation box, the stress and strain fields produced by all defects contained in the box decay as $1/x^3$, where $x = |\bm{x}|$ and $\bm{x}$ is the position relative to center of the box. This is the case, for example, for a collection of dislocation loops and cavities. In this section we focus on the strain field. The same reasoning can be applied for the stress field. \subsection{Correction for periodic systems} \label{sec:correction-periodic-systems-cai} W. Cai \emph{et al.} have shown that absolutely convergent sums converge to a field which is periodic, so the lack of periodicity is closely linked to the conditional convergence of the direct summation on image boxes. Since absolute convergence is obtained for terms which decay as $1/x^4$, but not $1/x^3$, the first derivative of the strain field is absolutely convergent. Therefore, the strain field $\bm{\varepsilon}$ can be written, by integration of the absolutely convergent field, as \begin{equation} \label{eq-strain-field-decomp-absolute-plus-correction} \varepsilon_{ij}({\bm{x}}) = \varepsilon_{ij}^{\mathrm{PBC}}(\bm{x}) + \varepsilon_{ij}^0, \end{equation} where $\varepsilon_{ij}^{\mathrm{PBC}}$ is the strain field corresponding to the periodic solution of the problem and $\varepsilon_{ij}^0$ is a contribution linked to the non-periodic character of the displacement field $\bm{u}$. By integration, this field reads \begin{equation} \label{eq-displacement-field-cai} u_i(\bm{x}) = u_i^{\mathrm{PBC}}(\bm{x}) + \bm{g}_i\cdot \bm{x} + u_i^{0}, \end{equation} where $u_i^{\mathrm{PBC}}$ is the periodic displacement field. It is related to $\varepsilon_{ij}^{\mathrm{PBC}}(\bm{x})$ by \begin{equation} \label{eq-link-uPBC-eps-PBC} \varepsilon_{ij}^{\mathrm{PBC}}(\bm{x}) = \frac{1}{2}\left(u_{i,j}^{\mathrm{PBC}}(\bm{x}) + u_{j,i}^{\mathrm{PBC}}(\bm{x}) \right), \end{equation} where $u_{i,j} = \partial u_i/ \partial x_j$ and $\bm{g}_i$ is a constant vector such that \begin{equation} \label{eq-link-g-epsilon0} \varepsilon_{ij}^0= \frac{1}{2}(g_{ij}+g_{ji}). \end{equation} In practice, the strain field is calculated by summing over periodic images contained in a given region $\mathcal{V}$, which leads to expression~\eqref{eq-strain-field-decomp-absolute-plus-correction}. The constant field $\bm{\varepsilon}^0$ can be deduced from $\bm{g}_i$, which is computed, for example, by evaluating the displacement field at one corner of the box and at the three adjacent corners. It is important, in this case, to use the same summation domain $\mathcal{V}$. Indeed we will see in next section that $\bm{g}_i$ and thus $\bm{\varepsilon}^0$ depend on the shape of the summation domain. These quantities, however, do not depend on the order of summation (although $u_i^0$, in general, does). To obtain the solution corresponding to a periodic system, it is necessary to subtract $\bm{\varepsilon}^0$ from $\bm{\varepsilon}$. Such corrections are used not only in the framework of dislocation dynamics, but also in atomistic calculations to evaluate formation energies of isolated defects~\cite{Varvenne2013}. In numerical simulations, formation energies contain a spurious component due to the interaction between the defect, which can be modeled as an elastic dipole, and its periodic images. To remove this interaction energy, a direct summation of the strain field on periodic images can be performed and the component $\bm{\varepsilon}^0$ must then be subtracted. We note that in this context, other formulations for the correction of the energy have been recently derived~\cite{Pasianot2016,Dudarev2018}. \subsection{An alternative corrective scheme using surface integrals} \label{sec:alternative-surface-integrals} For an infinite elastic medium, the displacement field generated by an elastic dipole $p_{jk}$, which is the first moment of a localized point-force distribution, reads~\cite{Siems1968,Clouet2018} \begin{equation} \label{eq-displacement-field-elastic-dipole} u_i(\bm{x}) = -p_{jk}G_{ij,k}^{\infty}(\bm{\bar{x}}), \end{equation} where $\bm{\bar{x}} = \bm{x} - \bm{x'}$ is the vector pointing from the dipole location to the point where the field is evaluated and $G_{ij}^{\infty}$ is the elastic Green function in an infinite body. Summation over repeated indices is implied in the following. For an isotropic material, we have \begin{equation} \label{eq-Greeen-function-isotropic-material} G^\infty_{ij}(\bm{\bar{x}}) = \frac{1}{8\pi(1-\nu) 2 \mu} \left( \delta_{ij} \frac{3-4\nu}{\bar{x}}+ \frac{\bar{x}_i \bar{x}_j}{\bar{x}^3}\right). \end{equation} In this equation, $\mu$ is the shear modulus, $\nu$ is the Poisson's ratio and $\delta_{ij}$ is the Kronecker delta. For a system of volume $\mathcal{V}$ containing an array of $N$ identical elastic dipoles $p_{jk}$, the displacement is \begin{equation} \label{eq-displacement-field-N-dipoles} u_i(\bm{x}) = - \sum_{\alpha=1}^N p_{jk} G_{ij,k}^{\infty}(\bm{\bar{x}}^{(\alpha)}). \end{equation} This sum is evaluated inside the material, with $\bm{x} \ne \bm{x}'^{(\alpha)}$ for $\alpha = 1, \dots, N$. For dipoles which are far from $\bm{x}$, the discrete sum can be approximated by an integral. Actually it is possible to perform this integral over the whole volume, since is is absolutely convergent for $x \rightarrow 0$ (it behaves as $1/x^2$~\cite{Leathem1912}). The result is not guaranteed to be the same as the discrete sum, but what is important is that we capture the contribution from faraway sources. The displacement is \begin{equation} \label{eq-integral-displacement-N-dipoles} u_i(\bm{x}) = -\int_{\mathcal{V}} P_{jk} G_{ij,k}^{\infty}(\bm{\bar{x}}) \mathrm{d}V', \end{equation} where $P_{jk} = p_{jk}/V$ is an elastic dipole density and $V = \mathcal{V}/N$ is the volume corresponding to a single dipole. Using Ostrogradsky's theorem and the fact that $G_{ij,k}^{\infty}(\bm{\bar{x}}) = -G_{ij,k'}^{\infty}(\bm{\bar{x}})$, we obtain \begin{equation} \label{eq-integral-displacement-dipoles-with-gauss} u_i(\bm{x}) = \int_{\mathcal{S}} P_{jk} n_k G_{ij}^{\infty}(\bm{\bar{x}}) \mathrm{d}S', \end{equation} with $\bm{n}$ the outward-pointing normal to the surface $\mathcal{S}$ which delimits $\mathcal{V}$. The elastic strain can be readily deduced: \begin{equation} \label{eq-elastic-strain-from-dipole-density} \varepsilon_{ij}(\bm{x}) = \frac{1}{2}\int_{\mathcal{S}} P_{lk}n_k \left[ G_{il,j}^{\infty}(\bm{\bar{x}}) + G_{jl,i}^{\infty}(\bm{\bar{x}}) \right] \mathrm{d}S'. \end{equation} This expression corresponds to the strain produced by surface forces $\bm{f} = \bm{P} \mathrm{d}\bm{S'}$~\cite{Leibfried1978}. We see that the direct sum of the strain field created by dipoles in $\mathcal{V}$ has a contribution which is due to surface forces on the boundary of the summation domain. Since a periodic system has no surfaces, the contribution of these surface forces to the strain field must be subtracted from the direct sum to recover a periodic system. It thus appears that Eq.~\eqref{eq-elastic-strain-from-dipole-density} corresponds to the spurious field $\bm{\varepsilon}^0$. The same volume must in principle be used for the discrete sum and the contribution of surface forces. Actually, since the function to integrate over $\mathcal{S}$ varies as $1/x^2$, the integral does not depend on the volume itself, but only on the \emph{shape} of the volume. Within the framework of anisotropic elasticity, efficient numerical evaluations of the derivative of elastic Green function can be used to compute the integral~\cite{Barnett1972}. In isotropic elasticity, the integral can be written under the following form, owing to Eq.~\eqref{eq-Greeen-function-isotropic-material}: \begin{multline} \label{eq-elastic-strain-from-dipole-density-isotropic} \varepsilon_{ij}(\bm{x}) = \frac{1}{32\pi (1-\nu) \mu} \int_{\mathcal{S}} P_{lk}n_k \left(-\delta_{il} (3-4\nu) \frac{\bar{x}_j}{\bar{x}^3} + \delta_{lj} \frac{\bar{x}_i}{\bar{x}^3} + \delta_{ij} \frac{\bar{x}_l}{\bar{x}^3} - 3 \frac{\bar{x}_i\bar{x}_j\bar{x}_l}{\bar{x}^5}\right.\\ \left. - \delta_{jl}(3-4\nu) \frac{\bar{x}_i}{\bar{x}^3} + \delta_{li} \frac{\bar{x}_j}{\bar{x}^3} + \delta_{ij} \frac{\bar{x}_l}{\bar{x}^3} - 3 \frac{\bar{x}_i\bar{x}_j\bar{x}_l}{\bar{x}^5} \right) \mathrm{d}S'. \end{multline} For a cuboid shaped box, this integral can be calculated analytically (\ref{sec:surface-integral-strain-correction}). It takes a particularly simple form when the field is estimated at $(0,0,0)$ (Eqs.~\eqref{eq-corr-eps-11-center} and~\eqref{eq-corr-eps-12-center}). This result can be used to correct a discrete sum over a cuboid-shaped domain with the simulation box at the center of the domain. To validate the explicit form of the correction given in Eq.~\eqref{eq-elastic-strain-from-dipole-density}, we consider a cubic simulation box of edge length $l = 10$~nm ($V = l^3$), containing an interstitial prismatic dislocation loop of radius $r=2$~nm along $x_3$ axis. A cuboid-shaped domain is used for the discrete sum: each image box is identified by a tuple $(n_1,n_2,n_3)$ and summation indices run from $-n_{\mathrm{neighbours}}$ to $n_{\mathrm{neighbours}}$ in the three directions. The triplet $(0,0,0)$ corresponds to the simulation box. Isotropic elasticity is used, so that results can also be compared to the analytical solution (Eq.~\eqref{eq-corr-eps-11-center}). The elastic dipole tensor of a dislocation loop is~\cite{Clouet2018} \begin{equation} \label{eq-elastic-dipole-loop} p_{ij} = -C_{ijkl} S_k b_l = -\mu (S_ib_j + S_jb_i) - \frac{2\nu \mu}{1-2\nu} \delta_{ij} S_k b_k, \end{equation} where $C_{ijkl}$ are the elastic constants, $\bm{b}$ is the Burgers vector ($\bm{b} = -b \bm{e}_{3}$) and $\bm{S}$ is the surface vector defining the area of the loop~\cite{Rovelli2018}. Here $\bm{S} = S \bm{e}_3$ with $ S = \pi r^2$. Note that $\bm{S}\cdot \bm{b} = S_k b_k = -bS$ due to the interstitial character of the loop. Parameters corresponding to aluminum are used: Burgers vector of magnitude $b = 0.2338$~nm, Poisson's ratio $\nu = 0.35$, shear modulus $\mu = 26$~GPa (not necessary for the evaluation of the strain correction). The elastic dipole density tensor is $P_{11} = P_{22} = 1.1064$~eV/nm$^3$, $P_{33} = 2.0548$~eV/nm$^3$ and $P_{ij} = 0$ for $i \neq j$, so owing to Eq.~\eqref{eq-corr-eps-12-center} components $\varepsilon_{ij}$ are also zero for $i \neq j$. In addition, $\varepsilon_{11}^0 = \varepsilon_{22}^0$, so only $\varepsilon_{11}^0$ and $\varepsilon_{33}^0$ are shown in Fig.~\ref{fig-comp-volume-surface}. \begin{figure}[htbp] \centering \includegraphics[width=\linewidth]{comp-volume-surface.pdf} \caption{Comparison of the two approaches (discrete sum and surface integral) to evaluate the correction corresponding to a periodic elastic solution. The simulation box of edge length $l = 10$~nm contains an interstitial prismatic dislocation loop of radius $r=2$~nm along $x_3$ axis (see text for details). Two components are represented: (a) $\varepsilon_{11}^0$ and (b) $\varepsilon_{33}^0$. The reference solution (Eq.~\eqref{eq-corr-eps-11-center}) is shown in gray. } \label{fig-comp-volume-surface} \end{figure} For the approach based on direct summation of displacement fields (Eq.~\eqref{eq-link-g-epsilon0}), the displacement field is evaluated at four locations in the simulation box (the origin is at the center of the box) : $(-l/2,-l/2,-l/2)$, $(l/2,-l/2,-l/2)$, $(-l/2,l/2,-l/2)$ and $(-l/2,-l/2,l/2)$, so the number of Green function evaluations to determine the spurious strain field is $ n_{\textrm{Green}} = 4 (2n_{\mathrm{neighbours}}+1)^3$. For the surface approach (Eq.~\eqref{eq-elastic-strain-from-dipole-density}), a Gaussian quadrature is used to calculate the integral. For $n_{\textrm{Gauss}}$ integration points in one direction, the number of Green function evaluations is $n_{\textrm{Green}} = 6 n_{\mathrm{Gauss}}^2$. To compare the two methods, the strain field is represented as a function of $n_{\textrm{Green}}$. It is clear that the two approaches converge to the same result given by \eqref{eq-corr-eps-11-center} and~\eqref{eq-corr-eps-12-center}. The surface approach appears to converge faster than the direct sum approach, although in both cases values are reasonably well converged for a few hundreds of Green function evaluations, corresponding to $n_{\mathrm{neighbours}} = 1$ and $n_{\mathrm{Gauss}} = 5$. Therefore it appears that the surface method is preferable if the computation of the strain correction is required to be fast and precise. The surface method can be readily generalized to a collection of defects with non-zero elastic dipole components, once the elastic dipole density tensor is calculated. \section{Simulation of macroscopic systems with prescribed tractions} \label{sec:simulation-macroscopic-systems} We consider the case of a simulation box embedded into a macroscopic, finite system obtained by replication of the simulation box around it (Fig.~\ref{fig-schematic-compensation} (a)). Surface tractions $\bm{T} = \bm{\sigma} \bm{n}$ are imposed, where $\bm{n}$ is the normal to the surface of the system. A particular case is $\bm{T} = 0$, which corresponds to a system with free surfaces. In this section we derive the field that must be added to the discrete sum over image boxes which constitute the macroscopic system, in order to be representative of a finite system with zero surface tractions. The more general case of a given stress state can be readily obtained by adding the corresponding stress field. To obtain the solution corresponding to zero surface tractions, a common method is to add a field which cancels surface tractions $\bm{T}$ produced by the solution for an infinite medium~\cite{Giessen1995}. This can be done, for example, by finite element (FE) solving of the elastic problem with prescribed tractions $-\bm{T}$. The traction field can be quite complicated, with steep variations on the scale of the simulation box, due to the distribution of defects in the box. However, if the simulation box is in the middle of the macroscopic system, far from surfaces, the effect of surface tractions can be accurately modelled by taking into account only their average value over a surface $S_k = l_i l_j$ defined by the box dimensions $(l_1,l_2, l_3)$. The average value over $S_k$ of the field created by a discrete set of defects of periodicity $l_i$ and $l_j$ is well approximated by integrals over a continuous distribution of dipole density, except near the edges of the system where the discrete nature of sources can be more significant. However, such regions represent a small part of the surface and their contribution to the field in the middle of the macroscopic system is small. Higher order multipole contributions can be safely neglected if the simulation box is far from the surfaces. It is therefore envisageable to determine the field to add to the simulation box by performing a FE solving of the elastic problem with surface tractions determined by surface integrals similar to Eq.~\eqref{eq-elastic-strain-from-dipole-density} for the stress field, evaluated outside the system, close to the surface. Actually it is possible to avoid the numerical solving phase and to derive a simple expression for this field. \begin{figure}[htbp] \includegraphics[width=\textwidth]{schematic_all-crop.pdf} \caption{Schematic representation of the correction to remove the non-periodic part of the elastic solution. (a) System with periodic images: the stress field computed by direct sum over a finite set of images contains a spurious component which corresponds to a non-periodic displacement field. (b) This component can be rewritten as a contribution from surface forces $\bm{P} \mathrm{d}\bm{S}$, where $\bm{P}$ is the elastic dipole density. (c) By removing these surface forces, a periodic solution is obtained. Surface tractions on the boundary of the macroscopic system (red dashed lines) are $\bm{\sigma} \bm{n} = -\bm{P}\bm{n}$, where $\bm{n}$ is the outward-pointing normal to the surface.} \label{fig-schematic-compensation} \end{figure} As noted in the previous section, the contribution of faraway defects to the elastic field in the simulation box can be accurately described by surface forces $\bm{f}^0 = \bm{P}\mathrm{d}\bm{S'}$ (Figs.~\ref{fig-schematic-compensation} (a) and (b)). To remove the spurious field and simulate a periodic system, we have seen that we have to add to the field in the simulation box (and in the macroscopic system) the contribution from an opposite distribution of surface forces $-\bm{f}^{0} = -\bm{P}\mathrm{d}\bm{S'}$ (Fig.~\ref{fig-schematic-compensation}-(c)). The stress field, in the interspace between the two distributions of forces $\bm{f}^0$ and $-\bm{f}^0$, is such that \begin{equation} \label{eq-stress-state-interspace} \bm{\sigma} \bm{n} = -\bm{P}\bm{n}. \end{equation} This expression can be obtained either by performing the integral in Eq.~\eqref{eq-elastic-strain-from-dipole-density} (in isotropic elasticity), or more simply by applying equilibrium equation of elasticity on a small volume straddling one of the two distributions of forces. The same method is used, for example, to determine the electric field due to an infinite plane of charges. The two distributions of surface forces correspond to a capacitor, where the electric field is constant if the two planes are close enough to each other. Eq.~\eqref{eq-stress-state-interspace} means that by adding the field due to $-\bm{f}^{0} = -\bm{P}\mathrm{d}\bm{S'}$, which cancels the shape effect and leads to a periodic solution, we impose a loading of the material equal to $\bm{\sigma} = -\bm{P}$. Therefore, to cancel surface tractions, this field should be removed. This result can also be obtained directly by noting that applying the correction from Ref.~\cite{Cai2003} amounts to considering a periodic system with no imposed deformation ; it has been shown that in this case, the average stress on the simulation box, or on a group of simulation boxes, is $-\bm{P}$~\cite{Clouet2008}. Finally, the field inside the simulation box, which corresponds to zero surface tractions on the macroscopic system, can be written as follows: \begin{equation} \label{eq-field-final-with-correction-and-zero-traction} \bm{\sigma} = \bm{\sigma}^{\mathrm{sum}} - \bm{\sigma}^0 + \bm{P}, \end{equation} where $\bm{\sigma}^{\mathrm{sum}} = \bm{\sigma}^{\mathrm{PBC}} + \bm{\sigma}^0$ comes from the sum over the defects in the macroscopic system, $\bm{\sigma}^0$ is the field created by surface forces $\bm{f}^0 = \bm{P}\mathrm{d}\bm{S'}$ distributed over the surface of the macroscopic system and $\bm{P}$ is the dipole density inside the simulation box. As mentioned in the previous section, $\bm{\sigma}^0$ is known analytically in isotropic elasticity and can be evaluated numerically in anisotropic elasticity by either of the two methods described in the previous section. It is important to notice that the solution with zero tractions does not depend on the shape of the sample (it is simply $\bm{\sigma} = \bm{\sigma}^{\mathrm{PBC}} + \bm{P}$). This is markedly different from the local electric and magnetic fields in systems containing electric and magnetic dipoles, which depend on the shape of the sample. To validate this expression, we consider the same system as in the previous section, \emph{ie} a prismatic loop of radius $r = 2$~nm in a cubic simulation box of edge length $l = 10$~nm. This box is duplicated 21 times along each direction to create the macroscopic system. Surface tractions produced by the solution in an infinite medium, resulting from the discrete sum over the loops, is shown in Fig.~\ref{fig-stresses-001}-(a,d) for $\sigma_{33}$ and $\sigma_{13}$ on the upper surface of normal $[001]$. They exhibit steep variations, correlated with the loop positions. However, tractions averaged over the simulation box dimensions have a much smoother profile (Fig.~\ref{fig-stresses-001}-(b,e)). This profile is mostly due to the shape effect, which can be removed by adding the field $-\bm{\sigma}^0$. By adding further the elastic dipole density $\bm{P}$, average surface tractions become essentially zero (Fig.~\ref{fig-stresses-001}-(c,f)). \begin{figure}[htbp] \includegraphics[width=\textwidth]{tractions_all.png} \caption{Stresses $\sigma_{33}$ (a-c) and $\sigma_{13}$ (d-f) on a surface of normal $[001]$ of a macroscopic system containing $21\times 21\times 21$ simulation boxes: (a,d) stress $\bm{\sigma} = \bm{\sigma}^{\mathrm{sum}}$ due to the contribution of defects inside the system (b,e) average stress $\bm{\sigma} = \langle \bm{\sigma}^{\mathrm{sum}}\rangle$ over the simulation box dimensions (c,f) average stress corrected by the non-periodic part $\bm{\sigma}^0$ and the dipole density: $\bm{\sigma} = \langle\bm{\sigma}^{\mathrm{sum}}\rangle - \bm{\sigma}^0 + \bm{P}$. The simulation box contains one prismatic loop (see text for details).} \label{fig-stresses-001} \end{figure} The accuracy of expression~\eqref{eq-field-final-with-correction-and-zero-traction} is assessed by performing reference FE calculations (see for example~\cite{Giessen1995}). Surface tractions $\bm{T}$ are obtained by summing the contributions of all the loops, as in Fig.~\ref{fig-stresses-001}-(a,d). A typical elastic solution with prescribed tractions $\bm{-T}$ is shown in Fig.~\ref{fem_config}. As noted before, in the simulation box located in the middle of the macroscopic system, the details of the surface tractions do not impact the solution, only the average value, linked to the elastic dipole density, is important. The FE solution in the middle of the macroscopic system is compared to the analytical solution, $\bm{\sigma} = -\bm{\sigma}^0 + \bm{P}$, for different aspect ratios $l_1/l_3$ (Fig.~\ref{fig-comp-fem-analytic}). The agreement is very good, which proves that a continuous description of the traction fields, including only the dipole component, is precise enough. Corrections on $\sigma_{13}$, $\sigma_{23}$ and $\sigma_{33}$ slowly converge to zero as $l_1/l_3$ approaches infinity, since $\bm{\sigma}^0 \bm{e}_3$ approaches $\bm{P}\bm{e}_3$ in the interspace between two infinite distributions of surface forces $\pm \bm{P}\mathrm{d}S'\bm{e}_3$. \begin{figure}[htbp] \centering \includegraphics[width=0.7\textwidth]{fem_21x21x21_modif_axis2.png} \caption{Stress component $\sigma_{33}$ calculated by FE modelling of a macroscopic system. Surface tractions are set to the opposite of the tractions generated by the collection of loops inside the system (see for example Fig.~\ref{fig-stresses-001}-(a,d) for one of the surfaces). The macroscopic system contains $21\times 21 \times 21$ simulation boxes. Each box has a single prismatic loop in the middle (see text for details). The system is cut half-way along $\bm{e}_{2}$ for the purpose of visualisation. } \label{fem_config} \end{figure} \begin{figure}[htbp] \centering \includegraphics[width=0.7\textwidth]{comp_FEM_analytic.pdf} \caption{Correction for the stress field in the middle of a macroscopic system, corresponding to a traction-free system, for different aspect ratios $l_1/l_3$. To obtain these ratios, the macroscopic system is made of $11\times 11\times 41$, $21\times 21\times 21$ and $41\times 41\times 11$ simulation boxes. Each box has a single prismatic loop in the middle (see text for details). Reference FE simulations are compared to the analytical result derived in the present work, $\bm{\sigma} = -\bm{\sigma}^0 + \bm{P}$.} \label{fig-comp-fem-analytic} \end{figure} \section{Application: loop evolution under irradiation} \label{sec:application} In the previous section we have seen that a correction must be added to the field calculated by direct summation over near images, in order to obtain a solution corresponding to zero surface tractions on the macroscopic system. Its magnitude is proportional to the elastic dipole density, as the correction proposed in Ref.~\cite{Cai2003} which corresponds to a solution for a fully periodic system. For boxes with large elastic dipole densities, dislocation and point defect behaviours may be affected by the correction. In this section we investigate the effect of the two corrections on the loop growth under irradiation, using an object kinetic Monte Carlo (OKMC) approach~\cite{Vattre2016,Carpentier2017}. As before, we use typical parameters for aluminum (see section~\ref{sec:alternative-surface-integrals}). Two interstitial Frank loops of different radii (2 and 3 nm) are introduced in a cubic box of edge length $l = 10$~nm, at $(l/2,l/2,l/4)$ and $(l/2,l/2,3l/4)$. The normal to their habit plane is $\bm{e}_3$. Six vacancies and self-interstitials are introduced in the box per second, which corresponds to damage rate of $10^{-4}$~dpa/s (displacements per atom). They diffuse in the simulation box with periodic boundary conditions until they are absorbed by one of the loops. Simulations are performed at $T = 300$~K. The emission of point defects by loops can be neglected at this temperature. The migration of point defects occurs by successive hops between stable positions in the lattice. The jump frequency is given by \begin{equation} \label{eq-jump-frequency} \nu = \nu_0 \exp{\left(-\frac{E^{\mathrm{m}}}{k_{\mathrm{B}}T}\right)}, \end{equation} where $\nu_0= 10^{13}$~Hz is an attempt frequency and $E^{\mathrm{m}}$ is the migration energy for the considered jump. It reads~\cite{Siems1968} \begin{equation} \label{eq-migration-energy} E^{\mathrm{m}} = E^{\mathrm{m}}_0 - p_{ij}^{\mathrm{sad}}\varepsilon_{ij} + p_{ij}^{\mathrm{sta}}\varepsilon_{ij}, \end{equation} with $E^{\mathrm{m}}_0$ the migration energy without any strain, $p_{ij}^{\mathrm{sta}}$ and $p_{ij}^{\mathrm{sad}}$ the elastic dipoles at the stable and saddle positions and $\varepsilon_{ij}$ the strain field, which is assumed to be the same at both positions. We also suppose here that the elastic dipoles do not depend on the strain, \emph{ie} polarizability effects are not considered~\cite{Schober1984}. Elastic dipoles of vacancies and self-interstitials in aluminum at stable and saddle configurations can be found in Ref.~\cite{Carpentier2017}. If the local strain field is corrected by $\bm{\varepsilon}^{\mathrm{corr}}$, the migration barrier becomes \begin{equation} \label{eq-migration-energy-corr} E^{\mathrm{m}} = E^{\mathrm{m}}_0 - (p_{ij}^{\mathrm{sad}} - p_{ij}^{\mathrm{sta}})\varepsilon_{ij} - (p_{ij}^{\mathrm{sad}} - p_{ij}^{\mathrm{sta}})\varepsilon_{ij}^{\mathrm{corr}}. \end{equation} Since in general elastic dipoles are not equal at stable and saddle positions, the elastic correction can alter the point defect diffusion. In particular, we can expect an effect of the correction in the simulation of phenomena such as void swelling or irradiation creep, for which the influence of the elastic field created by dislocations and cavities on the diffusion of point defects is important~\cite{Heald1975}. If the magnitude of $\varepsilon_{ij}^{\mathrm{corr}}$ is the same as $\varepsilon_{ij}$, results could change appreciably. The evolution of the two loops is given in Fig.~\ref{fig-loop-evolution} with different elastic corrections, averaged over 1000 simulations for each condition. Whatever the correction, one sees that the larger loop grows, while the smaller loop shrinks. This result is in agreement with bias calculations on single loops, which show that the bias increases with loop size~\cite{Bullough1981,Jourdan2015,Rouchette2015}. Differences in loop evolution are clearly observed for the various corrections envisaged, although they remain small from an experimental point of view. Loops exhibit the fastest evolution if the correction for zero surface tractions is used. The slowest evolution is obtained with the correction for a fully periodic system. Simulations were also performed with simplified elastic dipoles. They were taken purely hydrostatic, with the same value at stable and saddle positions, deduced from the trace of the DFT dipole tensors at stable position. No difference in loop evolution is seen in this case, in agreement with Eq.~\eqref{eq-migration-energy-corr}. \begin{figure}[htbp] \centering \includegraphics[width=0.7\textwidth]{loop_evolution.pdf} \caption{Evolution of the radius of two Frank loops of initial radii (a) 3 nm and (b) 2 nm in a cubic box of edge length $l = 10$~nm. Vacancies and self-interstitials are introduced simultaneously, simulating an electron irradiation with a damage rate equal to $10^{-4}$~dpa/s. Corrections $-\bm{\sigma}^0$ and $-\bm{\sigma}^0 + \bm{P}$ are added to the stress field calculated as a sum of contributions from nearby image boxes ($\bm{\sigma}^{\mathrm{sum}}$), to account for different boundary conditions. ``Anisotropic'' case corresponds to elastic dipoles obtained from DFT calculations. For the ``isotropic'' case, dipoles are assumed to be the same at stable and saddle points and are purely hydrostatic. Their trace is given by DFT results at stable position.} \label{fig-loop-evolution} \end{figure} Since vacancies and interstitials are produced at the same rate, the number of interstitials in loops stays constant, providing no defects remain in the matrix. This means that the elastic correction is also roughly the same at any time. The stress correction for a fully periodic simulation is $\sigma_{11} = \sigma_{22} = 0.290$~GPa and $\sigma_{33} = 0.489$~GPa , while for a macroscopic system with zero tractions it is $\sigma_{11} = \sigma_{22} = -0.159$~GPa and $\sigma_{33} = -0.368$~GPa. These stress levels are quite substantial and could also affect processes such as dislocation glide. It therefore appears crucial to apply the stress correction corresponding to the desired boundary conditions. \section{Conclusions} \label{sec:conclusions} From a simulation box with a non-zero elastic dipole component, the aim of this work is to determine the effect, in the box, of prescribed tractions at the boundary of a macroscopic system built by replicating the simulation box around it. The starting point is a reformulation of the correction proposed by Cai \emph{et al.}~\cite{Cai2003} to obtain a fully periodic elastic solution. Using this formulation, based on surface integrals, we show that the correction only depends on the shape of the macroscopic system, and that applying the correction is equivalent to simulating a macroscopic but finite system with surface tractions $-\bm{P}\bm{n}$, where $\bm{P}$ is the elastic dipole density and $\bm{n}$ an outward-pointing normal unit vector. By removing these tractions, a system containing a homogeneous distribution of defects, with zero surface tractions, is simulated. The elastic solution thus obtained does not depend anymore on the shape of the macroscopic system. Elastic corrections are applied in OKMC simulation boxes to simulate the evolution of dislocation loops under irradiation, due to the absorption of point defects. It is shown that the dislocation loop evolution depends on the correction if point defects have different properties at stable and saddle points. It can be expected that these elastic corrections not only have an influence on point defect diffusion, but also on dislocation movement. Therefore it appears important to be aware of the type of system that is simulated when elastic corrections are applied, and to apply the desired correction. \section{Acknowledgments} \label{sec:acknowledgments} The author is grateful to E. Clouet for useful discussions and for his comments on the manuscript. Part of this work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under Grant Agreements No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

proofpile-arXiv_069-7235

\section{Approximating the Congruency} \label{sec:approximation} Consider two time series $S,T$, an arbitrary orthogonal matrix $M\in\ensuremath{\mathcal{MO}}(k)$, and a vector $v\in\ensuremath{\mathbb{R}}^k$. Using the triangle inequality, we obtain \begin{align*} \label{eq:delta} &\ensuremath{\textsf{\upshape d}}(s_i, M\cdot s_j+v) \leq \\ &\, \ensuremath{\textsf{\upshape d}}(s_i, M\cdot t_i+v)+\ensuremath{\textsf{\upshape d}}(t_i, t_j)+\ensuremath{\textsf{\upshape d}}(M\cdot t_j+v, s_j) \\ \Rightarrow &\left| \ensuremath{\textsf{\upshape d}}(s_i, s_j)-\ensuremath{\textsf{\upshape d}}(t_i, t_j)\right| \leq \\ &\,\ensuremath{\textsf{\upshape d}}(s_i, M\cdot t_i+v)+\ensuremath{\textsf{\upshape d}}(s_j, M\cdot t_j+v) \addtocounter{equation}{1}\tag{\theequation} \end{align*} i.\,e. we can estimate the congruence distance $\sum_{i=0}^{n-1} \ensuremath{\textsf{\upshape d}}(s_i, M\cdot t_i+v)$ without actually solving the optimization problem. We unroll this idea to propose two approximating algorithms in Section~\ref{sec:approxdelta}~and~\ref{sec:approxgreedy}. Considering the well-known \emph{self-similarity matrix} of a time series, the left hand side of Equation~(\ref{eq:delta}) matches the entry of the difference between two self-similarity matrices. Usually, the self-similarity matrix is used to analyze a time series for patterns (e.\,g. using Recurrence Plots \cite{recurrenceplots}). The important property that makes the self-similarity matrix useful for approximating the congruence distance, is its invariance under transformations considered for the congruence distance, i.\,e. rotation, translation, and mirroring. The \emph{self-similarity matrix} of an arbitrary time series $T=(t_0,\dots,t_{n-1})$ is defined as follows: \begin{displaymath} \Delta T \ \ \coloneqq \ \ \big(\; \ensuremath{\textsf{\upshape d}}\left(t_i,t_j\right)\; \big)_{0\leq i,j < n} \end{displaymath} Note, that $\Delta T_{i,j} = \Delta T_{j,i}$ and $\Delta T_{i,i}=0$. In fact, the self-similarity matrix $\Delta T$ completely describes the sequence $T$ up to congruence, i.e., up to rotation, translation, and mirroring of the whole sequence in $\ensuremath{\mathbb{R}}^k$ \cite{Congruence}: Two time series $S$ and $T$ are congruent iff they have the same self-similarity matrix, i.\,e. \begin{align} \label{eq:congiffdelta} &\exists M\in\ensuremath{\mathcal{MO}}(k),v\in\ensuremath{\mathbb{R}}^k: \nonumber \\ &\quad S=M\cdot T+v \ \iff \ \Delta S=\Delta T. \end{align} \subsection{Metric Approximation} \label{sec:approxdelta} Equation~(\ref{eq:delta})~and~(\ref{eq:congiffdelta}) yield the approach for approximating the congruency: We measure the congruency of two time series $S$ and $T$ via a metric on their self-similarity matrices. \begin{definition}[Delta Distance] \label{def:deltadistance} Let $S,T$ be two time series of length $n$. The \emph{delta distance} $d^\Delta(S,T)$ is defined as follows: \begin{align*} &d^\Delta(S,T) \coloneqq \\ &\quad \frac 1 2 \max_{0<\delta<n} \sum_{i=0}^{n-1} \left| \ensuremath{\textsf{\upshape d}}\left(s_i,s_{(i+\delta)\% n}\right) - \ensuremath{\textsf{\upshape d}}\left( t_i,t_{(i+\delta)\% n} \right) \right| \end{align*} \end{definition} \begin{proposition} The delta distance satisfies the triangle inequality. \end{proposition} \begin{proof} Consider three time series $R,S,$ and $T$ and fixiate a $\delta^*$ which maximizes $d^\Delta(R,T)$ in Definition~\ref{def:deltadistance}. Then \begin{align*} &d^\delta(R,T) = \\ &\quad\sum_{i=0}^{n-1} \left| \ensuremath{\textsf{\upshape d}}\left( r_i,r_{(i+\delta^*)\% n} \right)-\ensuremath{\textsf{\upshape d}}\left( t_i,t_{(i+\delta^*)\% n} \right) \right| \\ &\leq\sum_{i=0}^{n-1} \left| \ensuremath{\textsf{\upshape d}}\left( r_i,r_{(i+\delta^*)\% n} \right)+\ensuremath{\textsf{\upshape d}}\left( s_i,s_{(i+\delta^*)\% n} \right) \right| + \\ &\quad\left| \ensuremath{\textsf{\upshape d}}\left( s_i,s_{(i+\delta^*)\% n} \right)-\ensuremath{\textsf{\upshape d}}\left( t_i,t_{(i+\delta^*)\% n} \right) \right| \\ &\leq d^\Delta(R,S)+d^\Delta(S,T) \end{align*} prooves the triangle inequality. \end{proof} Since $\ensuremath{\textsf{\upshape d}}$ is symmetric, $d^\Delta$ inherits its symmetry. Hence, $d^\Delta$ is a pseudo metric on the set of time series of length $n$ where all time series of an equivalence class are congruent to each other. We omit providing pseudo code since the computation matches the formula in Definition~\ref{def:deltadistance}. The complexity of computing the delta distance $d^\Delta$ grows quadratically with the length of the time series. Our next aim is to show that the the delta distance $d^\Delta$ provides a lower bound on the congruence distance $d^C$, as formulated in the following theorem. \begin{theorem} \label{thm:quadmaxest} For all time series $S$ and $T$, the following holds: \begin{align*} d^\Delta(S,T) \ \ \leq \ \ d^C(S,T). \end{align*} \end{theorem} \begin{proof} Fixiate a $\delta^*$ which maximizes $d^\Delta(S,T)$ in Definition~\ref{def:deltadistance}. Using the triangle inequality as in Equation~(\ref{eq:delta}) yields \begin{align*} &d^\Delta(S,T) = \\ &\quad\frac 1 2\sum_{i=0}^{n-1} \left| \ensuremath{\textsf{\upshape d}}\left( s_i,s_{(i+\delta^*)\% n} \right)-\ensuremath{\textsf{\upshape d}}\left( t_i,t_{(i+\delta^*)\% n} \right)\right| \\ &= \frac 1 2\sum_{i=0}^{n-1} \left| \ensuremath{\textsf{\upshape d}}\left( s_i,s_{(i+\delta^*)\% n} \right)- \right. \\ &\quad\quad \left. \ensuremath{\textsf{\upshape d}}\left( M\cdot t_i+v,M\cdot t_{(i+\delta^*)\% n}+v \right)\right| \\ &\leq \frac 1 2\sum_{i=0}^{n-1} \left( \ensuremath{\textsf{\upshape d}}\left(s_i,M\cdot t_i+v\right)+ \right. \\ &\quad\quad \left. \ensuremath{\textsf{\upshape d}}\left(s_{(i+\delta^*)\% n}, M\cdot t_{(i+\delta^*)\% n}+v \right) \right) \\ &= \sum_{i=0}^{n-1} \ensuremath{\textsf{\upshape d}}\left( s_i,M\cdot t_i+v \right) \end{align*} for arbitrary $M\in\ensuremath{\mathcal{MO}}(k)$ and $v\in\ensuremath{\mathbb{R}}^k$. Hence, $d^\Delta(S,T)\leq d^C(S,T)$. \end{proof} In this section, we provided the delta distance, which is a metric lower bound to the congruence distance. \subsection{Greedy Approximation} \label{sec:approxgreedy} The approach of the delta distance is simple: For time series $S$ and $T$, it only sums up values along a (wrapped) diagonal in $|\Delta S-\Delta T|$ and chooses the largest value. However, another combination of elements within $|\Delta S-\Delta T|$ as addends might provide a better approximation of the congruence distance. Since it is a computational expensive task, to try all combinations, we try to find a good combination using a greedy algorithm for selecting the entries of $|\Delta S-\Delta T|$. The greedy algorithm first sorts the elements $d_{i,j} = \left|\ensuremath{\textsf{\upshape d}}(s_i,s_j)-\ensuremath{\textsf{\upshape d}}(t_i,t_j)\right|$ in descending order and stores them in a sequence $Q=\left( d_{i_1,j_1}, d_{i_2,j_2},\cdots \right)$. While iterating over the sequence $Q$, it adds $d_{i_r,j_r}$ to a global sum and masks the indices $i_r$ and $j_r$ as already seen. Elements in the queue which access already seen indices are skipped, thus each index is used at most once. Basically, this is the reason, why the greedy delta distance (denoted as $d^G(S,T)$) is a lower bound to the congruence distance. Theorem~\ref{thm:greedylb} proves the last statement and Algorithm~\ref{alg:approxgreedy} provides the pseudo code for the computation. \begin{algorithm} \caption{Greedy Delta Distance} \begin{lstlisting} Algorithm: greedydelta Input: time series $S, T$ of length $n$ Output: distance $d$ let $Q=()$ // empty sequence for $i=0,\ldots,n-2$ for $j=i+1,\ldots,n-1$ append $d_{i,j}\coloneqq\left|\ensuremath{\textsf{\upshape d}}\left( s_i,s_j \right)-\ensuremath{\textsf{\upshape d}}\left( t_i,t_j \right)\right|$ to $Q$ sort $Q$ // (descending) let $S=\emptyset$ let $d = 0$ for each $d_{i_a,j_a}$ in $Q$ if $i_a\in S$ or $j_a\in S$ continue let $d=d+d_{i_a,j_a}$ let $S=S\cup \left\{ i_a,j_a \right\}$ return $d$ \end{lstlisting} \label{alg:approxgreedy} \end{algorithm} The complexity is dominated by sorting $n^2$ elements, which takes $n^2\cdot\log(n^2)$ steps. \begin{theorem} \label{thm:greedylb} For all time series $S$ and $T$, the following holds: \begin{align*} d^G(S,T) \ \ \leq \ \ d^C(S,T). \end{align*} \end{theorem} \begin{proof} Let $Q^*=\left( d_{i_1,j_1},\cdots,d_{i_r,j_r} \right)$ be the list of elements from the queue in Algorithm~\ref{alg:approxgreedy} which have not been skipped. Since each index is appears at most once in this list, the following inequality holds for arbitrary orthogonal matrices $M$ and vectors $v\ensuremath{\mathbb{R}}^k$: \begin{align*} &d^G(S,T) = \sum_{a=1}^r d_{i_a,j_a} \\ &\leq \sum_{a=1}^r \left| \ensuremath{\textsf{\upshape d}}\left( s_{i_a},s_{j_a} \right)-\ensuremath{\textsf{\upshape d}}\left( M\cdot t_{i_a}+v,M\cdot t_{j_a}+v \right) \right| \\ &\leq \sum_{a=1}^r \ensuremath{\textsf{\upshape d}}\left( s_{i_a},t_{i_a} \right)+\ensuremath{\textsf{\upshape d}}\left( M\cdot t_{i_a}+v,M\cdot t_{j_a}+v \right) \\ &\leq \sum_{i=0}^{n-1} \ensuremath{\textsf{\upshape d}}\left( s_i,M\cdot t_i+v \right) \end{align*} Hence, $d^G(S,T)\leq d^C(S,T)$. \end{proof} \subsection{Runtime improvement} \label{sec:runtimeimprovement} The complexity of the delta distance and greedy delta distance is linear regarding the dimensionality but quadratic in length. In this section, we motivate an optimization for both algorithms. Time series usually do not contain random points, but they come from continuous processes in the real world, i.\,e. the distance between two successive elements is rather small. Hence, the distances $\ensuremath{\textsf{\upshape d}}\left( t_i,t_j \right)$ and $\ensuremath{\textsf{\upshape d}}\left( t_i,t_{j+1} \right)$ are probably close to each other if $i \ll j$, i.\,e. if $j$ is much larger than $i$. This insight leads to the idea, to only consider elements $\ensuremath{\textsf{\upshape d}}\left( t_i,t_j \right)$ where $|i-j|$ is a power of two, i.\,e. we consider less elements for larger temporal distances. \paragraph*{The Fast Delta Distance:} Adapting the idea to the delta distance $d^\Delta$ yields the following definition. \begin{definition}[Fast Delta Distance] \label{def:fastdeltadistance} Let $S,T$ be two time series of length $n$. The \emph{fast delta distance} $\tilde d^\Delta(S,T)$ is defined as follows: \begin{align*} &\tilde d^\Delta(S,T) \coloneqq \\ &\frac 1 2 \max_{0<\delta<n, \delta\in 2^\ensuremath{\mathbb{N}}} \sum_{i=0}^{n-1} \left| \ensuremath{\textsf{\upshape d}}\left(s_i,s_{(i+\delta)\% n}\right) - \ensuremath{\textsf{\upshape d}}\left( t_i,t_{(i+\delta)\% n} \right) \right| \end{align*} \end{definition} Since we omit some values $\delta$ in Definition~\ref{def:deltadistance}, the fast version $\tilde d^\Delta$ is a lower bound to $d^\Delta$, i.\,e. the following theorem holds: \begin{theorem} \label{thm:fastdeltalb} For all time series $S$ and $T$, the following holds: \begin{align*} \tilde d^\Delta(S,T) \ \ \leq \ \ d^\Delta(S,T) \end{align*} \end{theorem} Especially, the fast delta distance is also a lower bound to the congruence distance. For time series of length $n$ the complexity of the fast delta distance $\tilde d^\Delta$ improves to $n\log n$. On the other hand, equivalence classes regarding the fast delta distance might include time series which are not congruent. \paragraph*{The Fast Greedy Delta Distance:} Incorporating the idea for improving the runtime into the greedy delta distance simply changes Line~$7$ of Algorithm~\ref{alg:approxgreedy}: We only consider values for the variable $j$, which add a power of $2$ to the variable $i$. Algorithm~\ref{alg:fastgreedy} provides the line to change in Algorithm~\ref{alg:approxgreedy} in order to achieve the fast greedy delta distance. \begin{algorithm} \caption{Distinction between Greedy Delta Distance (cf. Algorithm~\ref{alg:approxgreedy}) and Fast Greedy Delta Distance} \begin{lstlisting}[firstnumber=7] for $j\in 2^{\ensuremath{\mathbb{N}}}$ with $j\leq n-1$ \end{lstlisting} \label{alg:fastgreedy} \end{algorithm} The fast greedy delta distance is again dominated by the sorting of elements. This time, $n\log n$ elements have to be sorted, thus its complexity is $n\log(n) \log(n\log n)=n \log(n)^2$. Hence, the fast versions both have quasi linear runtime regarding length and linear runtime regarding dimensionality. An inequality such as in Theorem~\ref{thm:fastdeltalb} does not exist for the fast greedy delta distance. Also, there is no correlation between the (fast) delta distance and the (fast) greedy distance. Though, the evaluation shows that the greedy delta distance provides a much better approximation in most cases. Call for Section~\ref{sec:experiments} for an evaluation of their tightness to the congruence distance. \section{Conclusion and Future Work} \label{sec:conclusion} In this paper, we analyzed the problem of measuring the congruence between two time series. We provided four measures for approximating the congruence distance which are at least $2$ orders of magnitude faster than the congruence distance. The first (namely, the delta distance) provides the additional ability to be used in metrix indexing structures. The second (greedy delta distance) loses this benefit, but seems to achieve a better approximation. Both approximations have linear complexity regarding the dimensionality but at least quadratic complexity regarding the length of the time series. The other two approximations address this problem at a cost of approximation quality. They have quasi-linear runtime regarding the length. In practical applications, time series distance functions need to be robust against time warping. The approximations provided in this work are based on comparing self-similarity matrices of time series. Based on this idea, our next step is to develop a time warping distance function measuring the congruency. \section{Evaluation} \label{sec:experiments} Since the exact computation of the congruence distance is a computational hard problem (an thus not feasable in practical applications), we are mainly interested in the evaluation of the approximations. Unfortunately, there is no direct algorithm for the computation of the congruence distance and we have to consider the computation of the congruence distance as a nonlinear optimization problem. For two time series $S$ and $T$, we will denote the distance value computed by an optimizer with $d^O(S,T)$. Since an optimizer might not find the global optimum, all values for the congruence distance (computed by an optimizer) in this section, are in fact upper bounds to the correct but unknown value of the congruence distance, i.\,e. $d^C(S,T)\leq d^O(S,T)$. This given circumstance complicates the evaluation of our approximations to the congruence distance. To estimate the tightness of the approximations, we first evaluate our optimizer on problems for which we know the correct results (cf.~Section~\ref{sec:evaloptimizer}). In those cases, where the error of the optimizer is small, the estimation of the tightness of our approximations is accurate. On the other hand, when the error of the optimizer increases, our estimation of the tightness of our approximations are loose and the approximation might be tighter than the experiments claim. For a detailed explanation, consider a lower bound $\ell(S,T)$ for the congruence distance (e.\,g. $\ell$ might be one of $d^G$, $\tilde d^G$, $d^\Delta$, or $\tilde d^\Delta$) and suppose $d^C(S,T)=d^O(S,T)-\varepsilon$, i.\,e. $\varepsilon\geq 0$ is the error of the optimizer. Then, we have the following correlation between the estimated tightness and the real tightness: \begin{align*} \frac{\ell(S,T)}{d^C(S,T)} &= \frac{\ell(S,T)}{d^O(S,T)-\varepsilon} \geq \frac{\ell(S,T)}{d^O(S,T)} \end{align*} Hence, for small errors $\varepsilon$, the estimated tightness is accurate and for large errors $\varepsilon$ we underestimate the tightness. In Section~\ref{sec:tightness}~and~\ref{sec:speedup}, we evaluate the tightness and the speedup of approximations to the (optimizer based) congruence distance, respectively. \subsection{Congruence Distance: An Optimization Problem} \label{sec:evaloptimizer} Consider fixed time series $S$ and $T$ in $\ensuremath{\mathbb{R}}^k$ with length $n$. The congruence distance is a nonlinear optimization problem with equality based constraints. The function to minimize is \begin{align*} f\left( M,v \right) &= \sum_{i=0}^{n-1} \ensuremath{\textsf{\upshape d}}\left( s_i,M\cdot t_i+v \right) \end{align*} while the $k^2$ equality based constraints correspond the the constraints for orthogonal matrices: \begin{align*} M\cdot M^T &= I. \end{align*} As a initial ``solution'' for the optimizer, we simply choose $M=I$ and $v=0$. \begin{figure} \centering \includegraphics[width=.49\linewidth]{images/RAM_exactmatch_boxplot_error} \includegraphics[width=.49\linewidth]{images/RAM_exactmatch_boxplot_runtime} \caption{Boxplot: distance values (left) and runtimes (right) from our Optimizer on congruent time series.} \label{fig:zerodist} \end{figure} We manually transformed time series $T$ with a random orthogonal matrix $M^*$ and a small random vector $v^*$ and solved the optimization problem $d^C(T,M^*\cdot T+v^*)$ to examine whether our optimizer is working properly. Clearly, we expect the optimizer to find a solution with value $0$. Whenever the optimizer claimed large distance values, we concluded that the optimizer is not working. We tried different optimizer strategies and chose an augmented lagrangian algorithm \cite{AUGLAGRANGE} with the BOBYQA algorithm \cite{BOBYCA} as local optimizer for further experiments because it promised the best performance with these experiments. We used the implementations provided by the NLopt library \cite{NLOPT}. We used the \ensuremath{\texttt{RAM}}{}{} dataset generator \cite{RAMgenerator} to evaluate the optimizer on time series with varying dimensionality. We solved $400$ optimization problems for varying dimensionality and removed all of those runs where the optimizer did not find a reasonable solution (i.\,e. runs where the optimizer yielded solutions larger than $100$). Figure~\ref{fig:zerodist} shows the distance values proposed by the optimizer (and therefore the error it makes) per dimensionality up to dimensionality $4$. For higher dimensionalities, the optimizer completely failed to find any reasonable value near $0$ although we gave it enough resources of any kind (e.\,g. number of iterations, computation time, etc.). Figure~\ref{fig:zerodist} also shows that the computation times rapidly increase with increasing dimensionality. Because of the raising error and runtime with increasing dimensionality, an evaluation of the congruence distance on higher dimensionality is not feasable. Hence, we can only consider up to $4$-dimensional time series in all further experiments. \subsection{Tightness of Approximations} \label{sec:tightness} \begin{figure} \centering \includegraphics[width=.49\linewidth]{images/RAM_tightness_maxsum} \includegraphics[width=.49\linewidth]{images/RAM_tightness_fastmaxsum} \includegraphics[width=.49\linewidth]{images/RAM_tightness_greedy} \includegraphics[width=.49\linewidth]{images/RAM_tightness_fastgreedy} \caption{Average tightness of the delta distance (top left), the fast delta distance (top right), the greedy delta distance (bottom left), and the fast greedy delta distance (bottom right) to the congruence distance, respectively.} \label{fig:ramtightness} \end{figure} In order to evaluate the tightness of the (fast) delta distance and (fast) greedy delta distance as lower bounds to the congruence distance, we used the \ensuremath{\texttt{RAM}}{}{} dataset generator \cite{RAMgenerator} as well as a real world dataset with 2-dimensional time series (character trajectories \cite{UCIDatasets}, contains over 2800 time series). Other real world datasets with higher dimensionality have not been suitable because the optimizer failed to compute the congruence distance. Since making the (greedy) delta distance time warping aware is future work, we have to deal with time warping another way. We simply preprocess our datasets, such that each time series, seen as a trajectory, moves with constant speed, i.\,e. for each \emph{dewarped} time series, the following holds: \begin{align*} \ensuremath{\textsf{\upshape d}}\left( t_i,t_{i+1} \right) \approx \ensuremath{\textsf{\upshape d}}\left( t_{i+1},t_{i+2} \right). \end{align*} We achieve this property by simply reinterpolating the time series regarding the arc length. Figure~\ref{fig:ramtightness} shows the tightness of the approximations on \ensuremath{\texttt{RAM}}{}{} datasets. As we expected, the greedy delta distance provides the tightest approximation to the congruence distance (provided by our optimizer). As we observed in Section~\ref{sec:evaloptimizer}, the error of our optimizer increases with increasing dimensionality. Hence, the tightness of the optimizer to the real congruence distance is decreasing. Since we can observe a similar behaviour here (the tightness of the approximation is decreasing with increasing dimensionality), the reason might be the inaccuracy of the optimizer. Either way, we can see that the tightness is above $50\%$ in most cases. Especially when using the greedy delta distance, the tightness is above $75\%$ in most cases. On the character trajectories dataset, the delta distance and the greedy delta distance achieved a tightness of $63\%$ and $83\%$, respectively. \subsection{Speedup of Approximations} \label{sec:speedup} \begin{figure} \centering \includegraphics[width=.49\linewidth]{images/RAM_speedup} \caption{Average speedup of the approximations to our optimizer} \label{fig:ramspeedup} \end{figure} Figure~\ref{fig:ramspeedup} shows the speedup of the approximations to the optimizer. As expected, the speedup increases exponentially with increasing dimensionality. While the fast delta distance is the fastest algorithm, it also provides the worst approximation (compare with Figure~\ref{fig:ramtightness}). On the other hand, the greedy delta distance provides the best approximation while being the slowest algorithm. Still, the greedy delta distance is multiple orders of magnitudes faster than our optimizer. The following speedups have been achieved on the character trajectory dataset: $1642$ with the delta distance; $8040$ with the fast delta distance; $321$ with the greedy delta distance; $2287$ with the fast greedy delta distance. The results are similar to those on the \ensuremath{\texttt{RAM}}{}{} generated datasets. \section{Introduction}\label{sec:introduction} Multimedia retrieval is a common application which requires finding similar objects to a query object. We consider examples such as gesture recognition with modern virtual reality motion controllers and classification of handwritten letters where the objects are multi-dimensional time series. In many cases, similarity search is performed using a distance function on the time series, where small distances imply similar time series. A \emph{nearest neigbor query} to the query time series can be a $k$-nearest neighbor ($k$-NN) query or an $\epsilon$-nearest neighbor ($\epsilon$-NN) query: A $k$-NN query retrieves the $k$ most similar time series; an $\epsilon$-NN query retrieves all time series with a distance of at most $\epsilon$. In our examples, the time series of the same classes (e.\,g., same written characters or same gestures) differ by temporal as well as spatial displacements. \emph{Time warping distance functions} such as dynamic time warping (\ensuremath{\texttt{DTW}}{})~\cite{DTWSakoe} and the Dog-Keeper distance (\ensuremath{\texttt{DK}}{})~\cite{WDK17,computingfrechet} are robust against temporal displacements. They map pairs of time series representing the same trajectory to small distances. Still, they fail when the time series are rotated or translated in space. The distance functions defined and analyzed in this paper measure the (approximate) congruence of two time series. Thereby, the distance between two time series $S$ and $T$ shall be $0$ iff $S$ can be transformed into $T$ by rotation, translation, and mirroring; in this case, $S$ and $T$ are said to be \emph{congruent}. A value greater than $0$ shall correlate to the amount of transformation needed to turn the time series into congruent ones. The classical \textsc{Congruence} problem basically determines whether two point sets $A,B\subseteq\mathbb R^k$ are congruent considering isometric transformations (i.\,e., rotation, translation, and mirroring) \cite{Heffernan:1992:ADA:142675.142697,Alt:1988:CSS:44611.44614}. For $2$- and $3$-dimensional spaces, there are results providing algorithms with runtime $\mathcal O(n\cdot\log n)$ when $n$ is the size of the sets \cite{Alt:1988:CSS:44611.44614}. For larger dimensionalities, they provide an algorithm with runtime $\mathcal O(n^{k-2}\cdot\log n)$. For various reasons (e.\,g. bounded floating point precision, physical measurement errors), the \emph{approximated} \textsc{Congruence} problem is of much more interest in practical applications. Different variations of the approximated \textsc{Congruence} problem have been studied (e.\,g. what types of transformations are used, is the assignment of points from $A$ to $B$ known, what metric is used) \cite{Heffernan:1992:ADA:142675.142697,Alt:1988:CSS:44611.44614,Indyk:2003:ACN:636968.636974,Alt96discretegeometric}. The \textsc{Congruence} problem is related to our work, since the problem is concerned with the existence of isometric functions such that a point set maps to another point set. The main difference is, that we consider ordered lists of points (i.\,e. time series) rather than pure sets. It turned out, that solving the approximated \textsc{Congruence} problem is NP-hard regarding length and dimensionality \cite{Congruence}. With this work, we contribute by evaluating the congruence distance with an implementation based on a nonlinear optimizer. We propose two approximations to the congruence distance which have linear runtime regarding the dimensionality and (quasi-) quadratic runtime regarding the length of the time series. We improve the complexity of both approximations at cost of approximation quality, such that their complexity is (quasi-) linear regarding the length of the time series. We evaluate the approximations experimentally. \subsection{Basic Notation} \label{sec:notation} We denote the natural numbers including zero with $\ensuremath{\mathbb{N}}$ and the real numbers with $\ensuremath{\mathbb{R}}$. For $a,b\in\ensuremath{\mathbb{N}}$ we denote the modulo operator by $a\%b$. The set of all powers of two is denoted via $2^\ensuremath{\mathbb{N}}\coloneqq\left\{ 1,2,4,8,\cdots \right\}$. Elements of a $k$-dimensional vector $v\in\ensuremath{\mathbb{R}}^k$ are accessed using subindices, i.\,e. $v_3$ is the third element of the vector. Sequences (here also called time series) are usually written using capital letters, e.\,g. $S=(s_0,\cdots,s_{n-1})$ is a sequence of length $n$. Suppose $s_i\in\ensuremath{\mathbb{R}}^k$, then $s_{i,j}$ denotes the $j$-th element of the $i$-th vector in the sequence $S$. The projection to the $j$-th dimension is denoted via $S^j$, i.\,e. $S^j=(s_{0,j},\cdots,s_{n-1,j})$. The Euclidean norm of a vector $v$ is denoted via $\|v\|_2$, thus $\ensuremath{\textsf{\upshape d}}(v,w)\coloneqq\|v-w\|_2$ denotes the Euclidean distance between $v$ and $w\in\ensuremath{\mathbb{R}}^k$. We denote the set of $k$-dimensional orthogonal matrices with $\ensuremath{\mathcal{MO}}(k)$, the identity matrix with $I$ and the transposed of a matrix $M$ with $M^T$. For a matrix $M$ in $\ensuremath{\mathbb{R}}^k$, we denote the matrix holding the absoloute values with $|M|\coloneqq \left( |M_{i,j}| \right)_{0\leq i,j < k}$. \subsection{Congruence Distance} While \ensuremath{\texttt{DTW}}{}{} compares two time series $S$ and $T$, it is (nearly) invariant under time warpings. In detail, consider $\sigma(S)$ and $\tau(T)$ as warpings by duplicating elements (e.\,g. $\sigma(S)=\left( s_0,s_1,s_1,s_1,s_2,s_3,s_3,s_4,\cdots \right)$), then $\ensuremath{\texttt{DTW}}{}$ minimizes the L1 distance under all time warps: \begin{align*} \ensuremath{\texttt{DTW}}{}(S,T) \coloneqq \min_{\sigma,\tau} \sum_{i=0}^{|\sigma(S)|-1} \ensuremath{\textsf{\upshape d}}\left( \sigma(S)_i, \tau(T)_i\right) \end{align*} with $|\sigma(S)|=|\tau(T)|$. On the other hand, the congruence distance is invariant under all isometric transformations. The difference to \ensuremath{\texttt{DTW}}{}{} is, that it minimizes the L1 distance by multiplying an orthogonal matrix and adding a vector: \begin{align} \label{eq:congruence} d^C(S,T) &\coloneqq \min_{M,v} f(M,v) \nonumber \\ &\coloneqq \min_{M,v} \sum_{i=0}^{n-1} \ensuremath{\textsf{\upshape d}}\left( s_i, M\cdot t_i+v\right) \end{align} The computation of $d^C(S,T)$ is an optimization problem where $f$ (cf. Equation~(\ref{eq:congruence})) corresponds to the objective function. For time series in $\ensuremath{\mathbb{R}}^k$, the orthogonality of $M$ yields a set of $k^2$ equality based constraints.

proofpile-arXiv_069-7356

\section{Introduction} The field of cavity optomechanics has developed rapidly in the past few decades. As a solid-state quantum platform, the optomechanical system has exhibited both great potential for testing fundamental problems in quantum mechanics and wide applications in quantum information processing and quantum precise measurement (see reviews \cite{OM_rev0,OM_rev,OM_rev2}). Because of the radiation-pressure-induced nonlinearity between optics and mechanics, cavity optomechanical systems allow the study of a variety of nonlinear phenomena, such as bistability \cite{Samuel,Bist}, multi-stability \cite{LC_OM1,MultiS1,Fan}, self-sustained oscillations \cite{LC_OM1,Zhang2014,LC_OM5}, and chaotic motion \cite{Chaos1,Chaos2}. However, noise is unavoidable in open optomechanical systems, and compared to all-optical systems the mechanical freedom in an optomechanical system is more sensitive to thermal noise because of its relatively low resonance frequency. Thus, the effect of noise on optomechanical nonlinearity has become an important topic \cite{Aldana,Suchoi}. In nonlinear systems, noise and random fluctuations do not always act destructively. Some physical mechanisms allow noise to act beneficially, such as stochastic resonance (SR) \cite{Benzi81,Benzi82} and coherence resonance (CR) \cite{CR1,CR0,CR2}. As a mechanism for enhancing the response of a nonlinear system to a weak external signal, SR has been observed and investigated in various fields (see review \cite{Gammaitoni}) and used widely to detect \cite{Sun2015,Wang2017,Shao2018}, amplify \cite{amp1,amp2}, and reconstruct \cite{Cao14,Han17} weak signals. Recently, SR has been introduced in the field of cavity optomechanics, examples being chaos-mediated optomechanical SR \cite{Monifi} and SR in a tristable optomechanical system \cite{Fan}. However, although optomechanical SR is now being investigated, there are more aspects to be explored. CR is the phenomenon of purely noise-induced temporal regularity, which has been observed in excitable \cite{CR0,Lee2010} and nonexcitable \cite{Liu2001,CR2} systems but is yet to be reported in the context of an optomechanical system. In the present work, we study the constructive effects of thermal noise on the nonlinear behavior of a basic optomechanical system with self-sustained oscillations. Our investigation comprises two parts, the first being a study of purely noise-induced novel dynamics in the form of (i) noise-sustained quasi-coherent oscillations near the Hopf bifurcation and (ii) noise-induced random switching between self-sustained oscillations and equilibrium in the overlapping regime of self-sustained oscillations and bistability. In particular, (i) is often referred to as CR, and the present investigation may be the first of optomechanical CR. In the second part, we study how a subthreshold periodic signal and thermal noise affect system performance. We find that large-amplitude periodic switching between self-sustained oscillations and equilibrium can be observed numerically, indicating the occurrence of SR and an amplification of the external signal by noise. More interestingly, there is a double-peak structure in the power spectrum, which differs from typical SR behavior. Based on experimentally feasible conditions, our results will help to realize optomechanical SR and CR experimentally in a single setup. The rest of this paper is structured as follows. In the Sec. 2, we introduce the model and give linear stability analysis of our system. In Sec. 3, we study noise induced system dynamics in the absence of an external signal, including the CR phenomenon and noise induced state switching. In Sec. 4, we study noise induced system dynamics in the presence of a weak periodic force, in particular, the stochastic resonance phenomenon with self-sustained oscillations. Finally, we conclude our paper in Sec. 5. \begin{figure*} \centering \includegraphics[width=5in]{fig1.eps} \caption{(a) Optomechanical system driven by a coherent optical field $E_d$ and a periodic force signal $F_s$. (b) System stability diagram. The parameter plane is divided into four regions: (i) one stable fixed point (grey); (ii) three fixed points (red); (iii) parametric instability (blue); (iv) overlap region (purple). The white dashed line indicates the detuning ($\Delta=-1.38\omega_m$ ) used in the following numerical calculations. The system parameters are $(g,\kappa,\gamma_m) = (0.21,1,0.25) \times \omega_m.$} \label{fig1} \end{figure*} \section{Model and theory} We consider a basic optomechanical system, namely a driven optical cavity with a freely oscillating end-mirror; see Fig.~\ref{fig1}(a). The optical mode inside the cavity is coupled with the mechanical oscillation mode of the mirror via radiation pressure. The cavity is driven coherently by a field with amplitude $E_d$ and frequency $\omega_d$. A weak force $F_s$ with frequency $\omega_f$ acts on the oscillating mirror as a probe signal. In the weak driving regime, i.e., the radiation pressure induced mirror displacement is small compared to the cavity length, nonlinear modulation on the cavity frequency by the mirror displacement can be ignored \cite{law95,Zaitsev2012,OM_rev}. Thus, the system Hamiltonian after rotating at the driving frequency $\omega_d$ can be approximately given by (with $\hbar=1$) \begin{eqnarray} \label{H}\nonumber \hat{H} &=&\frac{1}{2}\omega _{m}\left( \hat{x}^{2}+\hat{p}^{2}\right) -\Delta\hat{a}^\dag\hat{a}-g\hat{a}^{\dagger }\hat{a}\hat{x}+iE_{d}(\hat{a}-\hat{a}^\dag)\\ &+&F_s \hat{x}\cos(\omega_f t), \end{eqnarray}% where $\hat{a}^{\dagger }\left( \hat{a}\right) $ is the creation (annihilation) operator of the optical mode; $\hat{x}$ and $\hat{p}$ are the dimensionless position and momentum operators, respectively, of the mechanical mode, which satisfy $[\hat{x},\hat{p}]=i$; $\omega_m$ is the mechanical resonance frequency; $g$ is the optomechanical coupling coefficient; and $\Delta=\omega_c-\omega_d$ is the detuning between the driving field and the cavity resonance frequency $\omega_c$. The equations of motion for the system operators are then \begin{eqnarray} \frac{d\hat{a}}{dt} &=&-( -i\Delta +\kappa/2 -ig\hat{x})\hat{a}-E_{d}, \\ \frac{d\hat{a}^\dag}{dt} &=&-( i\Delta +\kappa/2 +ig\hat{x})\hat{a}^\dag-E_{d}, \\ \frac{d\hat{p}}{dt} &=&-\gamma_m\hat{p}-\omega _{m}\hat{x}+g\hat{a}^\dag\hat{a}-F_s\cos(\omega_f t)+\xi,\\ \frac{d\hat{x}}{dt} &=&\omega _{m}\hat{p}. \end{eqnarray} Here, the optical decay rate $\kappa$ and the mechanical decay rate $\gamma_m$ are introduced phenomenologically. $\xi$ is the thermal white noise in the mechanical mode and satisfies $\left\langle {\hat \xi (t){{\hat \xi }^ \dag }(t')} \right\rangle=2D_m \delta (t - t')$ and $\left\langle {\xi (t)} \right\rangle=0$. Note that any effect of thermal noise on the optical mode is neglected because $k_B T\ll \hbar \omega_c$ for room temperature and below. We begin by studying the stability properties of the system in the absence of the weak signal $F_s$. By setting the time derivatives in the equations of motion to zero, we easily obtain the steady-state equation for the mechanical freedom, namely \begin{eqnarray} \omega_m(\kappa^2/4+(\Delta+gx_s)^2)x_s-gE^2_d=0. \end{eqnarray} This is a cubic equation of the steady-state mechanical position $x_s$ and has three real roots at most. The stability of these solutions derived above can be investigated by evaluating the eigenvalues of the associated Jacobian matrix, i.e., the solution is stable if the real part of each eigenvalue is negative. Therefore, we linearize the system operators around their steady-state values, namely $\hat{a} \rightarrow \alpha_s+\delta\hat{a}$, $\hat{a}^\dag\rightarrow \alpha^*_s +\delta\hat{a}^\dag$, $ \hat{x}\rightarrow x_s +\delta\hat{x}$, and $\hat{p}\rightarrow p_s +\delta\hat{p}$. By ignoring higher-order terms, we have the linearized equation of motion as \begin{eqnarray} \dot{y} = J\hat{y}+\xi_0, \end{eqnarray} where $\hat{y}=[\delta\hat{a},\delta\hat{a}^\dag,\delta\hat{p},\delta\hat{x}]^T$, $\xi_0=[0, 0,\xi,0]^T$, and the Jacobian matrix $J$ is given by \begin{equation} J= \left (\begin{matrix} i(\Delta+gx_s)-\frac{\kappa}{2}{\rm{ }}&0&0& {\rm{ }}ig\alpha_s\\ 0&{\rm{ }}-i{\rm{(}}\Delta{\rm{ + }}gx_s{\rm{)}}-\frac{\kappa}{2}&0& -ig{\alpha^*_s}\\ g{\alpha^*_s}&g\alpha_s&-\gamma_m &-\omega _m\\ 0&0&\omega_m &0 \end{matrix}\right). \end{equation} The eigenvalues of $J$ are the roots of the characteristic polynomial $\det(\lambda I_4-J)={\lambda ^4} + {c_1}{\lambda ^3} + {c_2}{\lambda ^2} + {c_3}\lambda + {c_4}$, where \begin{eqnarray} {c_1} &=&\gamma_m + \kappa, \\ {c_2} &=& {(gx_s + \Delta )^2} + \frac{1}{4}{\kappa ^2} + \gamma_m \kappa + {\omega ^2_m}, \\ {c_3} &=& \gamma_m {(gx_s + \Delta )^2} + \frac{1}{4}\gamma_m {\kappa ^2} + \kappa {\omega ^2_m}, \\ {c_4} &=& ({(gx_s + \Delta )^2} + 2gx_s(\Delta + x_sg) + \frac{1}{4}{\kappa ^2}){\omega ^2_m}. \end{eqnarray} It is usually difficult to compute the eigenvalues of $J$ directly. An easier method is to determine the stability properties by evaluating the coefficients $c_1$ to $c_4$ using the Routh--Hurwitz stability criterion \cite{RHC}. Thus, we write the Hurwitz determinants as \begin{eqnarray} D_1&=& c_1,\\ D _2&=& \left|\begin{matrix} c_1 & c_3\\ 1 & c_2 \end{matrix}\right|,\\ D _3&=& \left|\begin{matrix} c_1 & c_3 & 0\\ 1 & c_2 & c_4\\ 0 & c_1 & c_3 \end{matrix}\right|,\\ D _4&=& \left|\begin{matrix} c_1 & c_3 & 0 & 0\\ 1 & c_2 & c_4 &0\\ 0 & c_1 & c_3 &0\\ 0 & 1 & c_2 & c_4 \end{matrix}\right|=c_4 D_3. \end{eqnarray} According to the Routh--Hurwitz stability criterion, our system is stable if and only if all Hurwitz determinants are positive, i.e., $D_{j} (j=1,2,3,4)>0$. Furthermore, determine the existence of self-sustained oscillations, we employ Liu's criterion \cite{Liu} for a simple Hopf bifurcation (i.e., a transition from a fixed point to a limit cycle), whereby two roots of the characteristic polynomial cross the imaginary axis and all other roots have negative real parts. Applying Liu's Hopf-bifurcation criterion to our model, a Hopf bifurcation occurs when the following conditions are satisfied: \begin{eqnarray} D_3 =0, D_1>0, D_2>0, c_4>0, \rm{and} \: \frac{d D_3}{d\lambda}>0. \end{eqnarray} Based on these criteria, we obtain the stability diagram of our system in the $E_d$--$\Delta$ plane as shown in Fig.~\ref{fig1}(b). The plane is divided into several regions with different stability properties. The grey region contains one stable fixed point. The red region inside the black curve and outside the orange curve is bistable, containing two stable fixed points and one unstable fixed point. The blue region inside the orange curve is a region of parametric instability in which the system exhibits limit-cycle, period-doubling, and chaotic behaviors as the driving strength is increased. The most complicated region is the overlap between bistability and parametric instability, in which one stable fixed point (the lowest one), one unstable fixed point (the middle one), and one limit cycle or chaotic solution (the highest one) coexist. This overlapping region and its vicinity comprise the parameter regime of interest herein. \section{System response to pure noise} In this section, we study purely noise-induced novel dynamics in our system near and inside the overlapping regime of bistability and parametric instability. We begin by considering the situation in which the driving strength $E_d$ is below the Hopf bifurcation, meaning that there is no self-sustained oscillation. According to the bifurcation diagram in Fig.~1(b), when the detuning is set at $\Delta=-1.38\omega_m$ [labelled by the white dashed line in Fig.~1(b)] the bifurcation occurs at about $E_d=3\omega_m$, below which the system has one or three fixed points and above which the system experiences self-sustained oscillation. First, we take the driving strength to be $E_d=2.85\omega_m$. Under these conditions and in the absence of noise, a system variable (specifically the mechanical position $x$) already at its equilibrium value will remain so [see Fig.~\ref{CR}(a)]. When relatively low noise ($D_m=0.005$) is introduced in the system, the mechanical position undergoes a quasi-coherent small-amplitude oscillation with certain variations in the oscillation amplitude. As the noise amplitude $D_m$ is increased to an intermediate value (i.e., $D_m=0.08$), the amplitude of the noise-sustained oscillation grows and the regularity decreases slightly. If we continue to increase the noise strength, the oscillation amplitude increases further and the regularity worsens further. This type of noise-induced quasi-coherent oscillation is usually referred to as CR and has been observed in a wide range of systems including excitable and non-excitable ones. However, to the best of our knowledge it has not been reported in an optomechanical system until now, making the present study the first example of CR in an optomechanical system. To better understand the CR in our model, we evaluate the peak width ($\Delta_\omega$), the peak height ($H_\omega$), and the coherence factor $\beta=\omega H_\omega/\Delta_\omega$ \cite{CR2,PhysRevE.92.042909} from the spectra of the mechanical position while varying the noise strength. As shown in Fig.~\ref{CR}, the spectral peak widens monotonically as the noise increases, while there is a clear mono-peak structure in the curves of height $H_\omega$ and coherence factor $\beta$, indicating the existence of resonance behavior that depends on the noise strength. The optimal noise corresponding to the best coherence factor is $D_m \approx 0.08$, which is consistent with the result in the time domain [Fig.~\ref{CR}(a)]. \begin{figure*} \centering \includegraphics[width=5in]{CR_180918.eps} \caption{Noise-sustained quasi-coherent oscillations. (a--c) Single trajectories of system dynamics (the mechanical position $x$) with different noise strengths. (d--f) Peak height $H_\omega$, peak width $\Delta_\omega$, and coherence factor $\beta$ as functions of noise strength $D_m$. The system parameters are $(g,\kappa,\gamma,E_d,\Delta) = (0.21, 1,0.25, 2.85,-1.38) \times \omega_m$.}\label{CR} \end{figure*} \begin{figure*} \centering \includegraphics[width=5in]{rand_switch180917.eps} \caption{ Time series of system variable ($x$) and their power spectra with (a,b) no thermal noise ($D_m=0$) and (c,d) thermal noise ($D_m=0.55$). The system parameters are the same as in Fig.~\ref{CR} except for $E_d=3.11\omega_m$.}\label{switch} \end{figure*} We turn next to how noise affects the system dynamics inside the overlapping regime described above. We assume that initially the mechanical position is on the upper branch of the bistable structure. Here, ``bistability'' does not mean that there are two branches with stable fixed points. Instead, the upper branch here corresponds to limit-cycle dynamics or chaotic motion. Without noise, the mechanical position $x$ oscillates coherently [Fig.~\ref{switch}(a)]. In its spectrum [Fig.~\ref{switch}(b)], apart from the main peak associated with the limit cycle, there are peaks at double and higher frequencies that arise from period doubling. After adding a certain level of mechanical noise ($D_m=0.5$) to the system, random switching between the upper and lower branches is observed numerically, that is similar to other noise-induced bistable transitions. However, upon zooming in, we find that the dynamics of the high-amplitude state have a certain periodicity, which is verified by the single-peak power spectrum in Fig.~4(b). Interestingly, the inclusion of noise suppresses the double- and higher-frequency components as evidenced by only a single peak remaining in the spectrum, corresponding to the main peak in the noise-free case. These phenomena are evidence for a stochastic transition between the self-sustained oscillations and the fixed-point dynamics. Similar noise-induced intermittency between limit-cycle and fixed-point dynamics has been observed experimentally in optomechanics \cite{Suchoi}. This transition provides a basis for the occurrence of SR between self-sustained oscillations and fixed-point dynamics as discussed in the next section. \section{System response to noise and an external signal} \begin{figure*} \centering \includegraphics[width=5in]{SNRs.eps} \caption{Stochastic resonance (SR) with self-sustained oscillations. (a) Signal-to-noise ratio (SNR) versus noise strength $D_m$. The red squares are the original data for the SNR calculated from trajectories and the blue curve is a fit to the SNR data. Points A, B, and C are representative of the low-noise, optimal, and high-noise regimes, respectively. (b) Spectra at noise levels A (yellow), B (blue), and C (red). (c) Time series of mechanical position $x$ with no noise. (d--f) Typical trajectories of mechanical position $x$ at noise levels A--C, respectively. The parameters are $(g,\kappa,\gamma,E_d,\Delta,F_s,f_s) = (0.21, 1,0.25, 3.11,-1.38,1.5,0.05) \times \omega_m$.} \label{SR} \end{figure*} In this section, we study the system dynamics subject to an external periodic force and thermal noise and show how the input signal can be amplified by thermal noise through the mechanism of SR. For typical SR to occur, one needs three ingredients (nonlinearity or a threshold, a certain amount of noise, and a subthreshold periodic signal) and a frequency-scale or time-scale matching condition to be satisfied. The results in the preceding sections show that the optomechanical nonlinearity provides a bistable structure for state switching. Thermal noise exists naturally in the system and its amplitude can be tuned by controlling the operating temperature. The only remaining ingredient is a subthreshold signal, which is inadequate for activating bistable state switching. For instance, as seen in Fig.~\ref{SR}(c), the mechanical force with amplitude $F_s=1.5\omega_m$ and frequency $f_s=0.05\omega_m$ causes only a small-amplitude modulation of the self-sustained oscillations in the upper branch without driving it to the lower branch. The SR matching condition is usually that the average period of the noise-induced switching between states should be comparable with half the period of the external signal \cite{Gammaitoni}. The average period of noise-induced random switching is determined by the noise amplitude and the system parameters. Therefore, to satisfy this condition, one can sweep either the noise strength or the system parameters, or sweep the signal. Here, we choose to fix the signal frequency and sweep the thermal noise over a large range and search for an optimal noise strength for matching. In Fig.~\ref{SR}(a), we plot the signal-to-noise ratio (SNR) [dB] as a function of the noise strength $D_m$. Here, the SNR is defined as the ratio of the spectrum peak to the spectrum background at the central frequency of the peak. We see that the SNR curve peaks at a nonvanishing finite noise strength, which is a characteristic feature of SR. We choose three points (A, B, and C) and study the spectral and temporal properties at different noise levels. The spectra for points A--C are shown in Fig.~\ref{SR}(b), where there are two peaks (a narrow peak at low frequency and a wide peak at high frequency) in every spectrum, unlike the single-peak SNR curve for typical SR. The narrow peak is at a frequency of around $0.008\omega_m$, consistent with the signal frequency $f_s/(2\pi)$. Therefore, this peak is the SR peak due to synchronization to the external signal $F_s$. The wide peak is centered at around $0.15\omega_m$, almost the same as the peak position in Fig.~\ref{switch}, and corresponds to the frequency of the intrinsic self-sustained oscillations in the high-amplitude state. We turn next to the temporal performance of the system with different noise levels (points A--C). For reference, we plot in Fig.~\ref{SR}(c) the system dynamics (i.e., the mechanical position $x$) in the absence of noise. The signal clearly involves a slow modulation of the self-sustained oscillations on the upper branch without switching to the lower stable branch, which indicates that the signal is below threshold. With the lowest noise level (point A), the switching between the oscillating and stable states is not particularly regular. When the noise is increased to the intermediate level (point B), we observe excellent switching regularity that is synchronized well with the signal. Increasing the noise further to the highest level (point C), the switching becomes much noisier because of the strong noise background, resulting in the reduction in the SR peak. \section{Conclusion} We studied noise-induced CR and SR in an optomechanical system with self-sustained oscillation dynamics. We analyzed the stability properties theoretically and showed the existence of an overlapping region of bistability and self-sustained oscillations. In the vicinity of this region, we found the occurrence of CR that could sustain long-term quasi-coherent oscillations of the system, a result that is the first report of CR in cavity optomechanics. Inside the overlapping region and with the imposition of a subthreshold signal with a matching condition, we observed by numerical means thermal noise-induced periodic switching between self-sustained oscillations and equilibrium, resulting in the external signal being amplified. Unlike the single-peak spectrum of typical SR, there were two resonance peaks in the power spectrum, with the additional peak corresponding to the self-sustained oscillations. Our results shed further light on the interplay between noise and nonlinearity in an optomechanical system and provide a new means for understanding SR. \section*{Acknowledgment} The authors thank Dr Zhenglu Duan for helpful discussions. \section*{Funding} National Natural Science Foundation of China (No.11504145 and No.11664014); Natural Science Foundation of Jiangxi Province (No.20161BAB211013 and No.20161BAB201023). \bibliographystyle{unsrt}

proofpile-arXiv_069-7397

\section{Introduction} Semiconductor lasers, which constitute a major part of semiconductor photonics, are efficient devices and enabling technology that have opened many novel prospects in the modern information society from miniature semiconductor lasers which drive tech gadgets to the lasers that are employed in the modern communication systems. Their complex dynamics and nonlinear features, particularly in spatially extended systems such as Vertical-Cavity Surface Emitting Lasers (VCSELs), have attracted strong scientific interest as well as emerging technological significance \cite{VCSEL}. Most important of them have been the Cavity Solitons (CSs) and Light Bullets (LBs) studied in variety of configurations from injected semiconductor lasers \cite{Barbay11,Nature2002,IEEE06,McIntyre10,Prati10,Eslami462014,Eslami612014,Taghavi212018,Anbardan212019,Anbardan4742020} to the ones with delayed feedback \cite{Tlidi09,Panajotov10,Tlidi12,Garbin17,Pimenov18,Scroggie09} to those with intracavity saturable absorber \cite{CSL05,CSL07,Columbo10,Elsass10,Turconi15,Eslami892014,Eslami692015,Eslami192016,Eslami982018}. The semiconductor laser with an intracavity saturable absorber is the most interesting for applications due to its compactness and the possibility of integration in an all-optical circuit. However, fabrication of these devices poses the challenge of positioning the gain (pumped) and absorption (not pumped) elements as close as possible to allow for maximum interaction through the intracavity field. In a common design, the saturable absorber consisting of a quantum-well is placed either in the upper mirror stack or inside a second cavity coupled to the one containing the gain medium, see for example \cite{VCSELsa1,VCSELsa2}. An alternative approach has also been introduced where optical pumping is used which allows the coherent properties of the pump source to be used in the design of a compact and monolithic cavity for self-pulsing or bistable lasers \cite{Elsass10}. A schematic figure of the VCSEL with a saturable absorber used in our study is shown in Fig.~\ref{scheme}.\\ \begin{figure} \includegraphics[width=1\columnwidth]{fig1.jpg} \caption{Schematic representation of a VCSEL with an intracavity saturable absorber. DBR, QW, and SA respectively stand for Distributed Bragg Reflector, Quantum Well, and Saturable Absorber.} \label{scheme} \end{figure} \begin{figure*} \includegraphics[width=0.6\columnwidth]{fig2a.jpg}\quad \includegraphics[width=0.6\columnwidth]{fig2b.jpg} \quad \includegraphics[width=0.6\columnwidth]{fig2c.jpg} \caption{Turbulent branches for $\delta_1=0$ and $\delta_1=0.22$ in pump current $\mu$ and $r$ scans. $r$ is fixed at 2.4 for (a); $\mu=5$ (below laser threshold) for (b) and $\mu=5.35$ (above laser threshold) for (c). For the turbulent branches in (a) we have depicted the time average of maximum intensities.} \label{branches} \end{figure*} Since more than a decade, investigation of rare and extreme pulses in optical systems has been a hot topic among researchers of nonlinear photonics \cite{Solli07,Akhmediev10,roadmap}. These extreme events in optical wave domain, referred to as rogue waves (RWs), are generally characterized by their probability of occurrence deviating from standard Gaussian statistical models and the peak height exceeding a certain system-dependent threshold which all roughly imply a value several times higher than the long-time average. Later on, spatiotemporal complexity associated with broad-area optical systems (both active and passive) inspired researchers to investigate 2D RWs \cite{Oppo13,Gibson16,Eslami17,Rimoldi17,Eslami20}. Statistics and dynamics of transverse RWs in broad-area semiconductor lasers with a saturable absorber were numerically studied recently in \cite{Rimoldi17,Eslami20}, for the underlying mechanisms and controlling their occurrences, and experimentally in \cite{Selmi16}.\\ Since broad-area semiconductor lasers have a broad stripe width of the active region ($\sim 200 \mu m$), the effects related to transverse carrier diffusion are important in the study of such a structure. The effect is minimum for the case where localized structures are considered but it is expected to play a significant role when extended turbulent structures are investigated. Here we focus on the effect of carrier transverse diffusion on the properties of the turbulent solutions and on the statistical and dynamical behavior of RWs in a broad-area semiconductor laser with an intracavity saturable absorber by slightly changing the model used in \cite{CSL07,Rimoldi17,Eslami20}. We extend the results in \cite{Rimoldi17,Eslami20} and show that inclusion of lateral carrier diffusion has a significant effect on the number of emitted RWs and their temporal dynamics. We particularly illustrate that transverse drift of carriers in a turbulent state eases its chaoticity to some extent and causes a reduction in the peak value of intensities and suppression of the RW formation for high values of carrier lifetimes ratio. However, RW emission is enhanced in nonzero carrier diffusion coefficients when the ratio of the carrier lifetimes in the two materials is decreased. Broader temporal width for RWs is also evidenced by the presence of nonzero transverse carrier diffusion coefficient.\\ To further approach the realistic view, we also studied the effects of a disk-shaped pump profile on RWs since in experimental conditions the pump shape controls the current-density distribution which is nonuniform across the transverse section. We show that when the carrier lifetimes ratio is lower, the number of emitted RWs increases in the presence of the finite circular pump. This is reversed for increased values of the carrier lifetimes ratio where the emission of RWs is suppressed by the effect of finite circular pump. Temporal dynamics of RWs is also discussed and shown that finite pump reduces their temporal width.\\ In section 2 model equations are detailed and the inclusion of transverse carrier diffusion coefficient is discussed. Section 3 and 4 are respectively devoted to the turbulent solutions and RWs under the effect of transverse carrier diffusion. Section 5 discusses the finite pump effects and conclusions are drawn in section 6. \begin{figure} \includegraphics[width=1\columnwidth]{fig3.jpg} \caption{Power spectrum of oscillations of the maximum intensity point in a turbulent state obtained for $\mu=5.2$ and $r=2.4$. The dominant frequency of oscillations increases from 1.53 GHz to 1.80 GHz accompanied by a considerable drop in its amplitude when diffusion coefficient is increased from zero to 0.22.} \label{fft} \end{figure} \begin{figure*} \includegraphics[width=0.6\columnwidth]{fig4a.jpg}\quad \includegraphics[width=0.6\columnwidth]{fig4b.jpg}\quad \includegraphics[width=0.6\columnwidth]{fig4c.jpg} \caption{Dominant frequency of intensity oscillations versus $\mu$ compared for $\delta_1=0$ and $\delta_1=0.22$ when $r$ is fixed at 2.4 (a). The same versus $r$ for $\mu=5.0$ (b) and $\mu=5.35$ (c).} \label{freqmur} \end{figure*} \section{Model equations: inclusion of lateral carrier diffusion} The effective model appropriate for RW studies in a broad-area semiconductor laser with an intracavity saturable absorber is basically the one used in \cite{CSL07} for investigation of CSs but with an additional term representing diffusion coefficient for the electric field $\delta_F$ that sets a filter to the number of spatial frequencies involved in the dynamics \cite{Rimoldi17,Eslami20}. While the diffusion term in the electric field equation makes the model completely suitable for studying spatiotemporal turbulence in terms of structural stability, we shall include lateral carrier diffusion in the equations for population dynamics in the active and passive media. The diffusion length $l$ is, by definition, the average distance that the excess carriers can cover before they recombine and depends on the lifetime and mobility of the carriers scaling as $\sqrt{C\tau}$ where $C$ is the diffusion constant and $\tau$ is the carrier lifetime before recombination. In our case of a semiconductor laser with a saturable absorber, carrier lifetimes are different in the two amplifying and absorbing medium which lead to a different diffusion coefficient for carriers. Then, we will have the following calculations for the diffusion lengths in the active and passive materials respectively: \begin{equation} l_1=\sqrt{C\tau_c}, \quad l_2=\sqrt{C\tau_c/r}, \end{equation} where $\tau_c$ is the carrier lifetime in the active material and $r$ is the ratio of the carrier lifetime in the amplifier to that in the absorber. In the model equations, the transverse diffusion coefficients for carriers in the amplifier and absorber are then defined respectively as \begin{align} \delta_1&=l_1^2/\Bar{a}=C\tau_c/\Bar{a},\\ \nonumber \delta_2&=l_2^2/\Bar{a}=C\tau_c/r\Bar{a}=\delta_1/r, \end{align} where $\Bar{a}$ is the diffraction coefficient. We scan $\delta_1$ from zero (no lateral diffusion at all) to 0.22 which translates to diffusion lengths in the order of zero to a few microns. With these arrangements, the full set of equations used in our simulations is written as \begin{eqnarray} \nonumber \partial_t F&=&[(1-i\alpha)D+(1-i\beta)d-1+(\delta_F+i)\nabla_{\bot}^2]F\,,\\ \nonumber \partial_t D&=&b[\mu-D(1+|F|^2)-BD^2+\delta_1 \nabla_{\bot}^2 D]\,,\\ \partial_t d&=&rb[-\gamma-d(1+s|F|^2)-Bd^2+\dfrac{\delta_1}{r} \nabla_{\bot}^2 d]\,, \label{model} \end{eqnarray} where $F$ is the slowly varying amplitude of the electric field, and $D$, $d$ are the population variables defined as \begin{equation} D=\eta_1(N_1/N_{1,0}-1),\; d=\eta_2(N_2/N_{2,0}-1). \label{popeqs} \end{equation} In Eq.~\ref{popeqs}, $N_1$ and $N_2$ are the carrier densities in the active and passive materials, respectively; $N_{1,0}$ and $N_{2,0}$ are their transparency values, and $\eta_1$, $\eta_2$ are dimensionless coefficients related to gain and absorption, respectively. $\delta_F$ is the diffusion coefficient for the electric field that protects structural stability of the equations. The parameters $\alpha$ ($\beta$) and $b$ are the linewidth enhancement factor in the active (passive) material and the ratio of the photon lifetime to the carrier lifetime in the active material. $\mu$ is the pump parameter of the active material, $\gamma$ is the absorption parameter of the passive material, $s$ is the saturation parameter, and $B$ is the coefficient of radiative recombination. Time is scaled to the photon lifetime, and space to the diffraction length. Typically a time unit is a few ps and a space unit $\sim 4\, \mu$m. The integration of the dynamical equations is performed by a split-step method separating the time and space derivatives on a $256\times256$ (and occasionally $512\times512$) spatial grids with space step 0.25 implying the physical distance of 1 $\mu$m between two consecutive grid points. This integration method consists in separating the linear part (the Laplacian), integrated by a FFT algorithm, and the nonlinear part, integrated via a Runge-Kutta method. In all simulations, we use periodic boundary conditions unless stated otherwise. We have used the following parameter values throughout the paper: $\delta_F=0.01, s=1, B=0.1, \alpha=2, \beta=1, b=0.01$ and $\gamma=2$. $r$ and $\mu$ are used as the control parameters.\\ \begin{figure} \includegraphics[width=0.49\columnwidth]{fig5a.jpg} \includegraphics[width=0.49\columnwidth]{fig5b.jpg}\\ \includegraphics[width=0.49\columnwidth]{fig5c.jpg} \includegraphics[width=0.49\columnwidth]{fig5d.jpg} \caption{RW ratio (left panel) and kurtosis (right panel) compared for $\delta_1=0$ and $\delta_1=0.22$ in $r$ scan; below laser threshold $\mu=5.0$ (a,b) and above laser threshold $\mu=5.35$ (c,d).} \label{RWkbeab} \end{figure} \section{Turbulent solutions affected by transverse carrier diffusion} In this section we present the effects of lateral carrier diffusion on the dynamics of turbulent solutions where RWs are likely to be emitted. In Fig.~\ref{branches} we only focus on the turbulent solutions along with the homogeneous steady state curve although, depending on parameters, there could be other branches of solutions such as stationary CSs, self-pulsing CSs and chaotic CSs which are detailed in \cite{Rimoldi17,Eslami20}. Fig.~\ref{branches} shows the solutions space where the maximum intensity of turbulent solutions averaged in time is compared in the two cases of $\delta_1=0$ and $\delta_1=0.22$. It is observed from Fig.~\ref{branches}(a) that the inclusion of transverse carrier diffusion has larger effects in lower pump current $\mu$ values which is consistent with Fig.~\ref{branches}(b) and (c) where turbulent branches are sketched in $r$ scan respectively for below $\mu=5.0$ and above $\mu=5.35$ laser threshold. We note that for $r=1$, where equal carrier lifetimes are considered for both amplifier and absorber materials, the time-averaged maximum intensity of turbulent states grows almost linearly with $\mu$, see \cite{Eslami20}.\\ \begin{figure} \includegraphics[width=0.49\columnwidth]{fig6a.jpg} \includegraphics[width=0.49\columnwidth]{fig6g.jpg}\\ \includegraphics[width=0.49\columnwidth]{fig6b.jpg} \includegraphics[width=0.49\columnwidth]{fig6h.jpg}\\ \includegraphics[width=0.49\columnwidth]{fig6c.jpg} \includegraphics[width=0.49\columnwidth]{fig6i.jpg}\\ \includegraphics[width=0.49\columnwidth]{fig6d.jpg} \includegraphics[width=0.49\columnwidth]{fig6j.jpg}\\ \includegraphics[width=0.49\columnwidth]{fig6e.jpg} \includegraphics[width=0.49\columnwidth]{fig6k.jpg}\\ \includegraphics[width=0.49\columnwidth]{fig6f.jpg} \includegraphics[width=0.49\columnwidth]{fig6l.jpg}\\ \caption{RW ratio and kurtosis difference from $\delta_1=0$ in percentage versus $\delta_1$. $r=2.0$ (a,b) and (g,h), $r=2.2$ (c,d) and (i,j), and $r=2.4$ (e,f) and (k,l). The left panel shows the pump regime of below laser threshold ($\mu=5.0$) and the right panel shows that of above laser threshold ($\mu=5.35$). We remind that for $r=2.4$ larger diffusion coefficient than $\delta_1=0.22$ switches the laser to the off state.} \label{RWkdelta} \end{figure} Transverse diffusion of carriers can also have consequences in terms of the dominant frequencies of intensity oscillations in a turbulent state as it allows drift of carriers within a finite area before recombination to emit a photon. We have compared the situation with $\delta_1=0$ to that with $\delta_1=0.22$ for a typical case of $\mu=5.2$ and $r=2.4$ in Fig.~\ref{fft} which clearly shows an increase in the dominant frequency and a reduction in its amplitude. The trends of variations in the dominant frequency of intensity oscillations in $\mu$ and $r$ scans are shown in Fig.~\ref{freqmur} for the two values of $\delta_1=0$ and $\delta_1=0.22$. It is observed from Fig.~\ref{freqmur}(a) that increase in the dominant frequencies persists for the entire range of $\mu$ values for nonzero lateral diffusion coefficient in spite of the fact that for higher carrier densities provided by large pump currents the difference from $\delta_1=0$ is small. The trend for dominant frequency of intensity oscillations is decreasing in $r$ scan while still maintaining its larger value compared to the case with zero diffusion at the same $r$, as depicted in Fig.~\ref{freqmur}(b) for pump current of below laser threshold and (c) for that above laser threshold.\\ \begin{figure} \includegraphics[width=0.49\columnwidth]{fig7a.jpg} \includegraphics[width=0.49\columnwidth]{fig7b.jpg} \caption{PDFs calculated over spatiotemporal intensity maxima and the corresponding RW thresholds. Weibull fit is used to distinguish those spatiotemporal events which deviate from the normal. Number of RWs increases from 76 for $\delta_1=0$ (a) to 130 for $\delta_1=0.22$ (b) in a simulation window of 50 ns. Other values are $\mu=5.0$ and $r=2.0$.} \label{PDF2.0} \end{figure} \begin{figure} \includegraphics[width=0.49\columnwidth]{fig8a.jpg} \includegraphics[width=0.49\columnwidth]{fig8b.jpg}\quad \caption{Trajectories of spatiotemporal maxima in sub-space made up of optical gain (D+d-1) and intensity for $\delta_1=0$ (a) and $\delta_1=0.22$ (b). RW ratio increases in (b) where the area relevant to extreme intensities is more touched by the trajectories. Other values are $\mu=5.0$ and $r=2.0$.} \label{gainInt2.0} \end{figure} \section{Rogue waves affected by transverse carrier diffusion} RWs in this system of two dimensions are generated in the peaks of a turbulent state and satisfy the widely-accepted threshold defined for characterizing RWs, i.e. $\langle I \rangle + 8 \sigma$, where $\sigma$ is the standard deviation. However, we take a rather stringent threshold condition for RWs which uses the probability density function (PDF) of spatiotemporal maxima instead of total intensity and distinguishes the RWs among many spatiotemporal maxima according to $I_{RW}=\langle I_{max} \rangle + 8 \sigma$ \cite{Rimoldi17}. In this case, the Weibull distribution \begin{equation} \dfrac{a}{b}\left(\dfrac{I_{max}}{b}\right)^{a-1}\exp{\left[-\left(\dfrac{I_{max}}{b}\right)^a\right]}, \end{equation} is a better choice to evidence the deviations leading to characterizing a wave as rogue wave \cite{Rimoldi17,Eslami20}. We also make use of two other indicators to measure the degree of rogueness, RW ratio and kurtosis respectively corresponding to the number of spatiotemporal events with intensities exceeding the intensity threshold $I_{RW}$ to the total number of spatiotemporal events during simulation and the ratio of the fourth moment about the mean to the square of the variance given by \begin{equation} \mathcal{K}=\dfrac{\dfrac{1}{n}\sum_{i=1}^n(I_i-\langle I \rangle )^4}{\left[\dfrac{1}{n}\sum_{i=1}^n (I_i-\langle I \rangle)^2 \right]^2}. \end{equation} \begin{figure} \includegraphics[width=0.49\columnwidth]{fig9a.jpg} \includegraphics[width=0.49\columnwidth]{fig9b.jpg} \caption{PDFs calculated over spatiotemporal intensity maxima and the corresponding RW thresholds. Weibull fit is used to distinguish those spatiotemporal events which deviate from the normal. Number of RWs decreases from 750 for $\delta_1=0$ (a) to 354 for $\delta_1=0.18$ (b) in the simulation window of 50 ns. Other values are $\mu=5.0$ and $r=2.2$.} \label{PDF2.2} \end{figure} \begin{figure} \includegraphics[width=0.49\columnwidth]{fig10a.jpg} \includegraphics[width=0.49\columnwidth]{fig10b.jpg} \caption{Trajectories of spatiotemporal maxima in sub-space made up of optical gain (D+d-1) and intensity for $\delta_1=0$ (a) and $\delta_1=0.18$ (b). Reduction of RW ratio is apparent in (b) by the smaller occupied area. Other values are $\mu=5.0$ and $r=2.2$.} \label{gainInt2.2} \end{figure} A comparison is made between zero and a nonzero diffusion coefficient ($\delta_1=0.22$) in Fig.~\ref{RWkbeab} in terms of RW ratio and kurtosis for the cases of below ($\mu=5.0$) and above ($\mu=5.35$) laser threshold when $r$ is scanned. The figures suggest that RW ratio is essentially unaffected by carrier transverse diffusion for small $r$ values and decreases for larger $r$ values. We note that RW ratio values in Fig.~\ref{RWkbeab}(a) is almost twice those in Fig.~\ref{RWkbeab}(c) which suggests that smaller pump currents (below laser threshold) favor the formation of RWs. The same can be argued for kurtosis in Fig.~\ref{RWkbeab} (b) and (d). We should also note that turbulent state relaxes to laser off solution for larger diffusion coefficients than 0.24 at $r=2.4$ and beyond.\\ \begin{figure} \includegraphics[width=0.49\columnwidth]{fig11a.jpg} \includegraphics[width=0.49\columnwidth]{fig11b.jpg} \caption{Temporal width of the RWs in terms of FWHM (a) and that of the corresponding optical gain $\tau_{gain}$ (b) versus normalized RW intensities compared for the cases of zero and nonzero carrier diffusion coefficient. Note that RW intensity FWHM and $\tau_{gain}$ are both normalized to the intensity of the related RW. Other values are $\mu=5.0$ and $r=2.2$.} \label{FWHMP} \end{figure} \begin{figure*} \includegraphics[width=0.6\columnwidth]{fig12a.jpg}\quad \includegraphics[width=0.6\columnwidth]{fig12b.jpg}\quad \includegraphics[width=0.6\columnwidth]{fig12c.jpg}\\ \caption{Typical transverse profiles for output intensity (a), carrier density in the amplifier (b) and that in the absorber (c) under finite current injection represented by Eq.~\ref{circeq}. $w_0$ is fixed at 120 in all the simulations discussed in this section unless stated otherwise.} \label{circprofs} \end{figure*} In Fig.~\ref{RWkdelta}, deviations of RW ratio and kurtosis from their values at zero diffusion coefficient are depicted for several values of $\delta_1$ in percentage. In these results, the pump current is kept below laser threshold at $\mu=5.0$ (left panel) and above laser threshold at $\mu=5.35$ (right panel) and several $r$ values are considered. Apart from the fact that larger transverse diffusion coefficients have more notable consequences in altering RW ratio and kurtosis values, positive and negative deviations of their values from those of zero diffusion are identified from Fig.~\ref{RWkdelta}. Positive deviations in RW ratio and kurtosis values occur for small $r$ which increases their values for nonzero diffusion coefficients, as it is evident from Fig.~\ref{RWkdelta}(a,b) and (g,h), that is, both below and above the laser threshold. The same goes for Fig.~\ref{RWkdelta}(i,j) in spite of minor irregularities. In contrast, when the value of $r$ is larger, RW ratio and kurtosis reduce and we observe negative differences from zero diffusion coefficient, as seen from figures \ref{RWkdelta}(c-f) and (k,l).\\ As an example, PDFs calculated over spatiotemporal intensity maxima along with the respective RW threshold and Weibull fits are shown in Fig.~\ref{PDF2.0} for $\mu=5.0$, $r=2.0$ and two different values of diffusion coefficients: zero and $\delta_1$ at which RW ratio increases to its maximum value according to Fig.~\ref{RWkdelta}(a,b). These specific cases are also shown in terms of their (optical gain-intensity) sub-space plots in Fig.~\ref{gainInt2.0}. From the two figures it is evident that RW emission is enhanced in terms of both numbers and peak intensities.\\ In Fig.~\ref{RWkdelta}(c,d), we show a situation where $r$ value is larger and $\mu$ is below laser threshold meaning that carrier density is low and transverse diffusion can have a different consequence. In Fig.~\ref{PDF2.2}, we have shown PDFs calculated over spatiotemporal intensity maxima along with the respective RW thresholds and Weibull fits for $\mu=5.0$, $r=2.2$ and two different values of $\delta_1$: no diffusion and diffusion coefficient for which RW ratio reaches its minimum. The corresponding trajectories in (optical gain-intensity) sub-space plots are also shown in Fig.~\ref{gainInt2.2}. These figures confirm that nonzero diffusion coefficient reduces the number of RWs and their peak intensities when larger ratio of lifetimes is considered for carriers in active and passive media. This is in contrast with the regimes having smaller ratio of carrier lifetimes.\\ Temporal dynamics of RWs is also expected to be affected by the transverse carrier diffusion. This can be seen from the temporal width of RWs when nonzero carrier diffusion coefficient is considered. In Fig.~\ref{FWHMP}(a) normalized temporal full-width half-maximum (FWHM) values of RW intensities are depicted with respect to normalized RW intensities for $\delta_1=0$ and $\delta_1=0.22$. It is clearly seen that for nonzero carrier diffusion coefficients, RWs with the same intensity have broader temporal FWHM as a consequence of the degree of freedom corresponding to carrier drift. This is confirmed in Fig.~\ref{FWHMP}(b) where normalized temporal width of the optical gain $\tau_{gain}$ is plotted for the RWs with respect to normalized RW intensities. \section{Finite pump effects} In experimental conditions, the current-density distribution is controlled by the pump shape and is not uniform all across the transverse section. However, in numerical studies it is customary to use periodic boundary conditions which simulate an infinite pumped area. A natural and widely used pump shape in experiments is that of a circular disc which can be simulated in our model by replacing $\mu$ with \begin{equation} \label{circeq} \mu(w)=\begin{cases} \mu_0, & \text{$w\leq w_0$}\\ \mu_0 \exp{\left(-\dfrac{w-w_0}{\delta_1}\right)}, & \text{$w>w_0$}. \end{cases} \end{equation} where $\delta_1$ and $w_0$ are respectively the effective diffusion length of the injection carrier and the radius of the circular disc. $\mu_0$ is the current density within the pump area (in the top flat part) \cite{YuIEEE}. The transverse profile of the output intensity, carriers in the amplifier and absorber materials affected by the circular pump shape are shown in Fig.~\ref{circprofs}.\\ We want to check if the statistics and dynamics of RWs are altered by the inclusion of such a pump shape since the finite injection area removes the unrealistic effects of the periodic boundary condition. First, we compare infinite and finite pump shapes for $\delta_1=0$ by the deviation percentage of RW density (number of RWs per unit area) in circular pump from that in flat pump considering different circular pump radii. Fig.~\ref{RWkcp} shows that in both regimes of below and above laser threshold, RW density increases in presence of finite pump for smaller $r$ values and positive deviations are observed. However, we see negative deviations from infinite pump for larger $r$ and the values for RW density are reduced. The same comparison is made for zero and nonzero transverse carrier diffusion coefficients and is shown in Fig.~\ref{FWHMPC} for temporal dynamics of the RWs in both conditions. It is seen from Fig.~\ref{FWHMPC}(a) that infinite flat pump unrealistically gives the RWs broader temporal width at the same intensity. This fact is supported by Fig.~\ref{FWHMPC}(b) where the temporal widths of the optical gain for RWs are narrower for the same intensity when finite circular profile replaces infinite flat profile. The difference persists but reduces when a nonzero carrier diffusion coefficient is considered, see for example Fig.~\ref{FWHMPC}(c) and (d) for $\delta_1=0.22$.\\ \begin{figure} \includegraphics[width=0.49\columnwidth]{fig13a.jpg} \includegraphics[width=0.49\columnwidth]{fig13c.jpg}\\ \includegraphics[width=0.49\columnwidth]{fig13b.jpg} \includegraphics[width=0.49\columnwidth]{fig13d.jpg}\\ \caption{RW density difference for circular pump shape from that of infinite flat pump versus pump radius for $\delta_1=0$. The left panel shows the pump regime of below laser threshold $\mu=5.0$ and the right panel shows that of above laser threshold $\mu=5.35$. $r=2.2$ (a,c), $r=2.4$ (b) and $r=2.55$ (d).} \label{RWkcp} \end{figure} \begin{figure} \includegraphics[width=0.49\columnwidth]{fig14a.jpg} \includegraphics[width=0.49\columnwidth]{fig14b.jpg}\\ \includegraphics[width=0.49\columnwidth]{fig14c.jpg} \includegraphics[width=0.49\columnwidth]{fig14d.jpg} \caption{Temporal width of the RWs in terms of FWHM (a) and that of the corresponding optical gain $\tau_{gain}$ (b) versus normalized RW intensities compared for the cases of infinite flat pump and finite circular pump at zero carrier diffusion coefficient. (c) and (d) are the same for $\delta_1=0.22$. The ratio of the RW intensity FWHM values in finite circular pump to those in infinite flat pump on average are 0.70 and 0.58 for $\delta_1=0.22$ and $\delta_1=0$, respectively. Note that RW intensity FWHM and $\tau_{gain}$ are both normalized to the intensity of the related RW. Other values are $\mu=5.0$ and $r=2.2$.} \label{FWHMPC} \end{figure} \section{Conclusion} We extended the results of \cite{Rimoldi17,Eslami20} by inclusion of a transverse carrier diffusion coefficient in the equations describing a broad-area semiconductor laser with a saturable absorber and performed numerical simulations to show that dynamical properties of the turbulent solutions and thus the emitted rogue waves at the peaks of spatiotemporal maxima are altered as a result of carrier transverse drift. Unlike localized structures for which the small carrier diffusion coefficient has no significant role, extended spatiotemporal structures respond to carrier transverse diffusion depending on the ratio of the carrier lifetimes in the active and passive media given by $r$. It is particularly shown that more rogue waves are emitted in presence of carrier transverse diffusion when carriers in the active material are given shorter lifetime by small $r$ value assuming that carrier lifetime in the passive material is fixed. The opposite happens for longer carrier lifetimes in the active material, larger $r$ value, and the formation of rogue waves are increasingly suppressed for larger lateral carrier diffusion coefficients. It is also illustrated that longer duration of rogue waves is followed by the nonzero transverse carrier diffusion. The importance of pump shape in experimental situations is also discussed and the statistical and dynamical behavior of rogue waves are studied under a disk-shaped pump. It is shown that less number of rogue waves per unit area for larger ratio of carrier lifetimes $r$ and more number of rogue waves per unit area for smaller $r$ values is emitted compared to the case with infinite flat pump. Moreover, it turned out that the unrealistic situation associated to infinite flat pump is responsible for longer temporal width of rogue waves since they are emitted in shorter duration under a finite pump.

proofpile-arXiv_069-7638

\section*{Introduction} \textit{Introduction.}---Quantum metrology utilizes quantum entanglement and coherence to enhance the measurement precision over classical metrology \cite{Caves1981,Giovannetti2004a,Pezze2018}. In the absence of noise, the precision of classical metrology scales as $n^{-1/2}$, which is limited by the central-limit theorem with $n$ being the number of resources implemented in the measurements. In contrast, quantum metrology can reach the Heisenberg limit, scaling as $n^{-1}$. However, in practice, any quantum system is inevitably subject to decoherence which results from the interaction with the environment \cite{Breuer2002}. Due to decoherence, there will be no improvement in the measurement precision over classical metrology, although the measurement time may be effectively shortened \cite{Huelga1997}. {In order to overcome this shortcoming, various methods have been put forward, e.g., squeezing \cite{Ma2011b}, purification \cite{Yamamoto2022}, and one-way quantum-computing-based teleportation \cite{Matsuzaki2018}.} It was shown that instead of maximally entangled states, partially entangled initial states may reduce the error to some extent \cite{Huelga1997}. When the system interacts with a specific bath with a band structure, ideal precision may be retrieved as a result of the existence of the bound state \cite{Bai2019a,Ai2010}. {When there is a correlation between the baths for individual qubits, an auxiliary qubit can be introduced to reach the Heisenberg limit \cite{Monz2011,He2021,Kukita2021}.} Interestingly, the nonlinear interaction between the system and the physical quantity to be measured is used to surpass the Heisenberg limit \cite{Boixo2007,Nie2015,Choi2008,Roy2008,Napolitano2011}. However, the nonlinear interaction may not be easily realized in practical measurements. Then, a question naturally comes to our minds: Can we figure out some practical method to improve the measurement precision in the presence of noise? It is shown that there are generally three stages in open quantum dynamics, including the well-known exponential decay in the intermediate stage \cite{Nakazato1996}. {However, in the first stage, the open quantum system decays in a Gaussian way, where the quantum Zeno effect (QZE) \cite{Ai2010a,Ai2013,Harrington2017,Virzi2022,Kiilerich2015,Do2019,Burgarth2014} happens.} {It has been pointed out that due to the QZE, metrology using maximally entangled states is superior to the one using product states by a factor of $n^{1/4}$ \cite{Chin2012,Matsuzaki2011}.} On the other hand, recently we have theoretically proposed and experimentally demonstrated in a nuclear magnetic resonance (NMR) platform, to simulate the quantum dynamics for various Hamiltonians and spectral densities \cite{Wang2018b,Zhang2021,Chen2022}. In these works, the bath-engineering technique \cite{Soare2014a,Soare2014,Khaneja2005b,Li2017} enables a theoretically exact, fully controllable, and practically efficient quantum simulation \cite{Buluta2009,Georgescu2014} approach. {Using the approach, we show that the non-Markovianity of the open quantum dynamics should be essentially characterized from both aspects of global and local points of view, e.g., quantum mutual information v.s. quantum Fisher information flows \cite{Chen2022,Luo2012,Lu2010,Altherr2021}.} It is this theoretically exact and fully controllable characteristic that enables us to experimentally verify the quantum metrology scheme proposed in Refs.~\cite{Chin2012,Matsuzaki2011}, which requires homogeneity of the qubits. \textit{Protocol.}---{The bath-engineering technique offers a way to engineer arbitrary environments by modulating the control field \cite{Soare2014a,Soare2014}. By applying a time-dependent magnetic field to the total system, we can artificially add a spectral characteristic noise, so as to simulate an arbitrary noisy environment.} The Hamiltonian of the total system can be written as $H(t)=H_\textrm{S}+H_\textrm{SB}(t)$, {where $H_\textrm{S}=\sum_{m}\varepsilon_m\vert m\rangle\langle m\vert+\sum_{m\neq n}J_{mn}\vert m\rangle\langle n\vert$ is the system Hamiltonian, and $H_\textrm{SB}(t)$ is the noise Hamiltonian to simulate the system-bath couplings.} When simulating the pure-dephasing noise, cf. Fig.~\ref{fig:scheme}(a), the noise Hamiltonian is $H_\textrm{SB}(t)=\sum_m\beta_m(t)\vert m\rangle\langle m\vert$. $\beta_{m}(t)$'s are stochastic errors generated by performing amplitude and phase modulations on a carrier \cite{Wang2018b,Zhang2021} \begin{equation} \beta_{m}(t)=\alpha_{z}\sum_{j=1}^{J}\omega_{j}F(\omega_{j})\cos\left(\omega_{j}t+\psi_{j}^{(m)}\right), \label{eq:beta} \end{equation} where $\alpha_{z}$ is the amplitude of the dephasing noise, $\omega_\textrm{b}$ ($J\omega_\textrm{b}$) is the base (cutoff) frequency with $\omega_{j}=j\omega_\textrm{b}$, $F(\omega_{j})$ is the modulation function of a specific spectral characteristic noise, and $\psi_{j}^{(m)}$'s are a set of random numbers. For a noise of Gaussian type, the average over the ensemble is equivalent to the average over the time \cite{Goodman2015}. { The ensemble-averaged decoherence factor reads $\Gamma(t) =\alpha_{z}^{2}\sum_{j=1}^{J}F(\omega_{j})^{2}\sin^{2}\frac{\omega_{j}t}{2}$, as determined by the power spectral density of the noise, which is the Fourier transform of the second-order correlation function $\langle \beta_{m}(t+\tau)\beta_{m}(t)\rangle$ \cite{Soare2014a,Soare2014}. } \begin{figure} \includegraphics[scale=0.75]{fig1.pdf}\caption{(a) Single-qubit Ramsey experiment for quantum metrology under non-Markovian noise. The free precession is governed by non-Markovian dynamics, which can be simulated using the ensemble-averaging technique. This averaging can be visualized in the Bloch sphere: A large number of identical initial probes undergo phase diffusion owing to their different modulated Hamiltonians, leading to an attenuation in amplitude for the overall spin signal. This attenuation gives rise to the error in quantum-metrology experiments. (b) Quantum circuits for multi-qubit Ramsey experiment under non-Markovian noise with unentangled and entangled probes, respectively. The evolution $U$ is the simulation of non-Markovian dynamics. For the case of entangled probes, the readout is typically some collective measurement \cite{Ma2011b,Huelga1997}.} \label{fig:scheme} \end{figure} We consider a system Hamiltonian $H_\textrm{S}=\sum_{k=1}^{n}\omega_{0}\sigma_{k}^{z}/2$, which is subject to a pure-dephasing noise \cite{Chin2012}. We present an illustration for the case with a single qubit in Fig.~\ref{fig:scheme}(a). Initially, the qubit is prepared at $(\vert0\rangle+\vert1\rangle)/\sqrt{2}$. Afterwards, it unitarily evolves at the equator of the Bloch sphere with rate $\omega_{0}+\beta_1(t)$. In order to mimic the pure dephasing, we generate a large number of realizations with the same initial state and obtain the final result by averaging over the random ensemble with different rate $\beta_1(t)$ after a second $\pi/2$ pulse. We assume a Drude-Lorentz spectral density, i.e., $J_{k}(\omega)=J(\omega)=2\lambda\gamma\omega/(\omega^{2}+\gamma^{2})$, where $\lambda$ is the reorganization energy and $\gamma$ is the relaxation rate. A generic spectral density can be decomposed into a summation of Lorentzian forms. The modulation function then becomes $F(\omega_{j})=[\lambda\gamma\omega_{0}\coth(\beta\omega_{j})/(\omega_{j}^{3}+\gamma^{2}\omega_{j})]^{1/2}$. {The quantum dynamics of the system is exactly described by the time-local master equation \cite{Breuer2002,Tao2020} \begin{equation} \frac{\partial}{\partial t}\rho(t)=-i[H_\textrm{S},\rho]+\sum_{k}\gamma_{k}(t)(\sigma^{z}_{k}\rho\sigma^{z}_{k}-\rho). \label{eq:ME} \end{equation} Note that $\gamma_{k}(t)$ is explicitly dependent on time. According to the bath-engineering technique, the ensemble-averaged decoherence rate is $\gamma_{k}(t)=\frac{\alpha_z^{2}}{2}\sum_{j=1}^{J}\omega_{j}F(\omega_{j})^{2}\sin(\omega_{j}t)$. } In Fig.~\ref{fig:scheme}(b), we present the quantum circuit for the quantum metrology scheme for the product and entangled states, respectively. For the former case, the probes are prepared at $[(\vert0\rangle+\vert1\rangle)/\sqrt{2}]^{\otimes n}$ and then evolve under the unitary evolution $U=\exp(-i\int_0^t d\tau[H_\textrm{S}+\sum_k\beta_k(\tau)\sigma_k^z])$, which will not induce entanglement between different qubits but mimic the noise. Then, we perform the individual measurements on each qubit, respectively, since all qubits are disentangled. However, for the latter case, since the probes are initialized at the maximally entangled state $(\vert0\rangle^{\otimes n}+\vert1\rangle^{\otimes n})/\sqrt{2}$, we only perform a collective measurement after the same $U$. { For open quantum dynamics governed by the master equation~(\ref{eq:ME}), the variances of the measured frequency for the unentangled and entangled probes are respectively $\delta\omega^2_0\vert_u=\exp[2\Gamma(t_u)]/(nTt_u)$ and $\delta\omega^2_0\vert_e=\exp[2n\Gamma(t_e)]/(n^2Tt_e)$, where $\Gamma(t)=\int_0^t\gamma_k(t^\prime)dt^\prime$ is the decoherence factor for a single qubit, and the subscript $u$ ($e$) refers to the unentangled (entangled) state. The optimal times to perform the measurement are determined by $2t_u\gamma_k(t_u)=1$ and $2nt_e\gamma_k(t_e)=1$, respectively. When the decoherence rate $\gamma_{k}(t)=c$ is time-independent, the variances $\delta\omega^2_0=2ce/(nT)$ for both initial states are the same, although the optimal times are different, i.e., $t_u=1/(2c)$ v.s. $t_e=1/(2nc)$. However, when the QZE occurs, i.e., $\gamma_{k}(t)=2ct$, the variance for the unentangled probes is inferior to that for the entangled ones, as $\delta\omega^2_0\vert_u=2\sqrt{ce}/(nT)>\delta\omega^2_0\vert_e=2\sqrt{nce}/(n^2T)$ with $t_u=(4c)^{-1/2}>t_e=(4nc)^{-1/2}$.} Hereafter, we shall give an introduction to the experimental details. \begin{figure*} \includegraphics[scale=1]{fig2}\caption{(a) Molecular structure and relevant parameters of the 7-qubit NMR processor. Chemical shifts (diagonal, Hz), scalar coupling strengths (off-diagonal, Hz), and relaxation times ($T_1$ and $T_2$) are all listed in the table. (b) Pulse sequence for the 7-qubit Ramsey experiment using the entangled probe under non-Markovian noise. To simulate the non-Markovian dynamics, the rotating angles $\theta_m^j$ are carefully designed by ensemble-averaging modulation. A step-by-step description of the sequence can be found in SM \cite{sup}.} \label{fig:circuit} \end{figure*} \emph{Experiment.}---The experiments are performed on a Bruker 600 MHz spectrometer at room temperature. {The sample is $^{13}$C-labeled trans-crotonic acid dissolved in $d_6$-acetone, which consists of three $^1$H and four $^{13}$C nuclear spins, forming a 7-qubit quantum processor.} The molecular structure of the sample is shown in Fig.~\ref{fig:circuit}(a). The internal Hamiltonian of this system can be described as \begin{equation} \mathcal{H}_{\rm{NMR}}=-\sum_{i=1}^7 \frac{\omega_i}{2} \sigma^z_{i}+\sum_{i<j,=1}^7\frac{\pi}{2} J_{ij}\sigma^z_{i}\sigma^z_{j}, \end{equation} where $\omega_i/2\pi$ is the Larmor frequency of the $i$-th spin, and $J_{ij}$ is the scalar coupling strength between spins $i$ and $j$. The corresponding parameters are listed in Fig.~\ref{fig:circuit}(a), as well as the relaxation times $T_1$ and $T_2$. Overall, the experiment aims to observe the enhancement of quantum metrology with entangled probes under the non-Markovian circumstance. This is demonstrated using the Ramsey magnetometry, which typically contains three major stages: (i) preparation of initial probe states, (ii) accumulation of an energy-splitting dependent phase, and (iii) measurement of that phase. In the following, we will describe each stage in detail. (i) Prepare the two types of probe states: the unentangled probe $\ket{\psi_0}_u=[(\vert0\rangle+\vert1\rangle)/\sqrt{2}]^{\otimes n}$, and the maximally entangled probe $\ket{\psi_0}_e = (\ket{0}^{\otimes n}+\ket{1}^{\otimes n})/\sqrt{2}$. For the unentangled probe, the processor is initialized to the pseudo-pure state $\ket{0}^{\otimes n}$ (up to an identity matrix) from the thermal equilibrium state $\rho_{0}$ using the cat-state method \cite{Knill2000a}. Followed by a 2-ms-shaped pulse to perform a collective single-qubit rotation $R_y(\pi/2)$ on all qubits, the system is prepared into $\ket{\psi_0}_u$ with over $99\%$ fidelity. The pulse sequence can be found in Supplemental Material (SM) \cite{sup}. {For the entangled probe, we directly start from the thermal equilibrium $\rho_{0}$, and eventually prepare the state $\rho_e=\ket{0}^{\otimes n}\bra{1}+\ket{1}^{\otimes n}\bra{0}$ with the aid of nearest-neighbour scalar couplings and gradient-echo techniques \cite{sup}, as shown in Fig.~\ref{fig:circuit}(b). Despite not the exact form of $\ket{\psi_0}_e$, the dynamics of $\rho_e$ for quantum metrology purpose is the same since the frequency information is just encoded in the phase accumulation of coherent terms \cite{sup}.} In addition, the creation of $\rho_e$ avoids regular initialization of the pseudo-pure state, which reduces the sequence complexity remarkably. (ii) {For the target system $H_\text{S}= \frac{\omega_0 }{2}\sum_{i=1}^n\sigma^z_{i}$ with $\omega_0$ being the energy splitting to be measured, let the probes precess unperturbedly for a duration $t$ under the non-Markovian noise. This non-Markovian circumstance is simulated by the aforementioned bath-engineering technique. Explicitly, we introduce an additional time-dependent Hamiltonian term in the form of $H_\text{SB}^m(t)=\sum_{i=1}^n\beta^m_{i}(t)\sigma^z_{i}$, where $\beta_{i}^m(t)$'s are stochastic errors subject to Drude-Lorentz spectral density as shown in Eq.~(\ref{eq:beta}), and $m$ is the sampling number so as to approximate the non-Markovian noise via the ensemble-averaging approach \cite{sup}. So the total Hamiltonian in the $m$-th experiment can be written as $H_\text{S}+H_\text{SB}^m(t)$. Note that the evolution of this Hamiltonian corresponds to a collective single-qubit rotation about the $z$-axis, which can be easily implemented in most physical systems. In addition, this time-dependent Hamiltonian can be discretized into time-independent slices for experimental realization (here we use $10^3$ slices \cite{sup}). Regarding the experimental parameters, we set $\omega_0=10$~kHz and choose $m=20$ samples for ensemble average, which can already approximate the non-Markovian environment with a high precision \cite{sup}.} \begin{figure*} \includegraphics[scale=0.8]{fig3} \caption{(a) {Dynamics of population $P(t)$ for the unentangled (top panel) and entangled (bottom panel) probes.} Blue curves are obtained by numerically fitting the experimental data (yellow triangles) using Eq.~(\ref{population}), and red lines are envelopes with the form $(1-e^{-\Gamma(t)})/2$, which depicts the time-dependent decoherence factor $\Gamma(t)$. The star symbol represents the optimal measurement point determined by the experiment. (b) Standard deviation $\delta\omega_0$ of the measured frequency with respect to the variation of $t$. {Dashed (solid) curves are fitting results corresponding to the unentangled (entangled) probes for different number of qubits. Triangles and dots are the optimal measurement points. The scaling behavior of these optimal times when changing $n$ is shown in the inset. }} \label{fig:7qubit} \end{figure*} (iii) {Read out the accumulated phase. For the unentangled probes, the final phase is encoded in each individual qubit, so the readout only involves single-qubit measurements, which is similar to the standard Ramsey interferometry experiment. For the entangled probes, the final phase is retained in the relative phase between $|00...0\rangle$ and $|11...1\rangle$, and we apply the multi-coherence measurement (MCM) technique \cite{Laflamme2002a} to extract that phase. As shown in Fig.~\ref{fig:circuit}(b), this technique involves a series of elementary quantum gates, which can be extended to other systems as well. Subsequently, the transition probabilities are computed by averaging over the ensemble with $m=20$ samples to simulate the non-Markovian decay.} \emph{Results.}---{We have conducted Ramsey experiments for $n=2\sim7$ qubits. Taking the 7-qubit case as an example, the transition probability $P(t)$ is shown in Fig.~\ref{fig:7qubit}(a) for both the unentangled (top panel) and entangled (bottom panel) probes.} Compared to the unentangled case, $P(t)$ in the entangled case manifests a much faster oscillation owing to the rapid accumulation ($\sim 7$ times) of phase. We use the theoretical form \begin{equation} P(t)=\frac{1}{2}[1-\cos(n\omega_0 t)e^{-n\Gamma(t)}] \label{population} \end{equation} to fit the experimental data which gives the energy-splitting frequency $\omega_0$ and decoherence factor $\Gamma(t)$. For the unentangled probes, the measured frequency is $\omega_0|_u=9.871$~kHz and the decoherence factor is $\Gamma_u = (0.370\pm0.029)t^2$. For the entangled probes, the measured frequency is $\omega_0|_e=10.142$~kHz and the decoherence factor is $\Gamma_e=(2.701\pm0.138)t^2$. The minimum standard deviations of the frequencies are $\delta\omega_0|_u= 0.169\pm0.003$~kHz and $\delta\omega_0|_e= 0.105\pm0.001$~kHz at the optimal measurement times $t_u=0.797\pm0.032$~ms and $t_e=0.288\pm0.003$~ms, respectively. Hence, the ratio of the sensitivity enhancement is $r=\delta\omega_0|_u/\delta\omega_0|_e = 1.608\pm0.035\approx 7^{\frac{1}{4}}$, indicating that the entangled probes are indeed superior to the unentangled ones under the non-Markovian environment. {Note that the readout of the NMR experiment is from the ensemble average result, which means the statistical error by repeating experiments is almost negligible. So, in this work, the error bars represent experimental uncertainties by error propagation from the fitting uncertainties in Fig.~\ref{fig:7qubit}(a); see SM for details \cite{sup}.} \begin{figure} \includegraphics[scale=1]{fig4}\caption{Experimental results of the ratio $r = \delta\omega_0|_u/\delta\omega_0|_e$ (dots) and the minimum standard deviation $\delta\omega_0|_e$ (triangles) for different number of qubits $n$. Under the non-Markovian noise, the sensitivity is enhanced by $r = n^{1/4}$ (solid red) with the entangled probes, reaching the QZE limit; $\delta\omega_0|_e$ scales as $n ^{-3/4}$ (solid blue). In the absence of noise, the sensitivity enhancement is governed by the Heisenberg limit $r = n^{1/2}$ (dashed red), and $\delta\omega_0|_e$ scales as $n ^{-1}$ (dashed blue).} \label{fig:r} \end{figure} For the other numbers of qubits, the experimentally measured frequencies $\omega_u$ and $\omega_e$, in association with their respective decoherence factors $\Gamma_u$ and $\Gamma_e$, can be found in SM \cite{sup}. The standard deviation of the estimated frequency is obtained by processing the experimental data, as shown in Fig.~\ref{fig:7qubit}(b), {where the dashed and solid lines are for the unentangled and entangled probes,} respectively. For the unentangled probes, the optimal measurement time $t_u$ is almost the same for different $n$'s and the minimum standard deviation $\delta\omega_0|_u$ is proportional to $n^{-1/2}$. For the entangled probes, $t_e$ and $\delta\omega_0|_e$ decrease with the growth of $n$. By varying the number of qubits, we compute the ratio $r = \delta\omega_0|_u/\delta\omega_0|_e$ to estimate the improvement of sensitivity with the entangled probes; see the solid red curve in Fig.~\ref{fig:r}. As predicted, the experimentally-observed $r$ scales as $r \sim n^{1/4}$, which is the QZE limit. For comparison, we have performed another group of experiments in the absence of noise \cite{sup}. In this noiseless scenario, the ratio of sensitivity $r$ reaches the Heisenberg limit, scaling as $r \sim n^{1/2}$ by the dashed red curve in Fig.~\ref{fig:r}. In addition, we depict distinct tendencies of the minimum standard deviation $\delta\omega_0|_e$ of the entangled probes for non-Markovian and noiseless environments, respectively. As shown in Fig.~\ref{fig:r}, with the growth in number of qubits, experimental results show that $\delta\omega_0|_e \propto n ^{-3/4}$ for the non-Markovian noise by solid blue curve, while $\delta\omega_0|_e \propto n ^{-1}$ for the noiseless case by dashed blue curve. Definitely, our high-precision experiment is sufficient to verify that the measurement accuracy of the entangled probes can reach the QZE limit under the non-Markovian noise. \textit{Conclusion.}---We experimentally demonstrate that the entangled probes can enhance the sensitivity of quantum metrology by the QZE. When the coherence decays quadratically with time, the entangled probes optimize the time for performing the measurement. On the contrary, when the decoherence rate is time-independent, the entangled probes cannot effectively improve the precision but shortens the measurement time. We remark that since our quantum simulation approach can subtly engineer the parameters of both the system and bath, it may provide an effective platform for experimentally verifying various quantum metrology schemes, e.g., achieving ideal precision by coupling to a bath with a band structure~\cite{Bai2019a}. And it can also be implemented in other physical systems \cite{Soare2014a,Potocnik2018}. \textit{Acknowledgement.}---This work is supported by the National Key Research and Development Program of China (2019YFA0308100), Beijing Natural Science Foundation (1202017), Beijing Normal University (2022129), National Natural Science Foundation of China (11674033, 12075110, 11975117, 11905099, 11875159, 11905111 and U1801661), Guangdong Basic and Applied Basic Research Foundation (2019A1515011383), Guangdong International Collaboration Program (2020A0505100001), Science, Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20190902092905285, KQTD20190929173815000, JCYJ20200109140803865, and JCYJ20180302174036418), Pengcheng Scholars, Guangdong Innovative and Entrepreneurial Research Team Program (2019ZT08C044), and Guangdong Provincial Key Laboratory (2019B121203002).

proofpile-arXiv_069-7967

\section{Introduction} As data-driven technologies continue to be adopted and deployed in high-stakes decision-making environments, the need for fast, interpretable algorithms has never been more important. As one such candidate, it has become increasingly common to use decision trees, constructed by adaptive recursive partitioning, for inferential tasks on a predictive or causal model. These applications are spurred by the appealing connection between decision trees and rule-based decision-making, particularly in clinical, legal, or business contexts, as the determination of the output mimics the way a human user may think and reason \citep{berk2020statistical}. Decision trees are ubiquitous in empirical work not only because they offer an interpretable decision-making methodology \citep{murdoch2019definitions, rudin2019stop}, but also because their construction relies on data-adaptive implementations that take into account the specific features of the underlying data generating process. See \cite{Hastie-Tibshirani-Friedman2009_book} for a textbook introduction. While data-adaptive, rule-based tree learning is powerful, it is not without its pitfalls. In this paper, we provide theoretical evidence of these shortcomings in commonly encountered data situations. Focusing on the simplest possible data generating process (i.e., a homoskedastic constant regression/treatment effect model), we show that decision trees will exhibit, at best, poly-logarithmic rates of convergence, as a function of the sample size $n$, uniformly over the entire support of the covariates. Furthermore, when adding sample-splitting to the tree construction, which is often regarded as an improvement over canonical tree fitting \citep{athey2016recursive}, we show that the resulting decision trees can be inconsistent, uniformly over the covariate support, as soon as the depth of the tree is at least a constant multiple of $\log\log(n)$ (e.g., $\log\log(n)\approx 3$ for $n=1 \text{ billion}$ observations). Our results paint a rather bleak picture of decision trees, if the goal is to use them for statistical learning \emph{pointwise} (or \textit{uniformly}) over the entire support of the covariates; they can produce unreliable estimates even in large samples for the simplest possible statistical model underlying the data generation. Thankfully, in such settings, we are able to show that random forests are provably superior and exhibit optimal performance when the constituent trees do not. This improvement comes at the cost of losing interpretability and introducing two additional tuning parameters (subsample size and number of candidate variables to consider at each node). To formalize our results, we consider the canonical regression model where the observed data $\{(y_i,\mathbf{x}_i^T) : i = 1, 2, \dots n\}$ is a random sample satisfying \begin{equation} \label{eq:model} y_i = \mu(\mathbf{x}_i)+ \varepsilon_i, \qquad \mathbb{E}[\varepsilon_i \mid \mathbf{x}_i]=0, \qquad \mathbb{E}\big[\varepsilon_i^2 \mid \mathbf{x}_i\big]=\sigma^2(\mathbf{x}_i), \end{equation} with $\mathbf{x}_i = (x_{i1}, x_{i2}, \dots, x_{ip})^T$ a vector of $p$ covariates taking values on some support set $\mathcal{X}$. The parameter of interest is the conditional mean response function $\mu(\mathbf{x}_i) = \mathbb{E}[y_i \mid \mathbf{x}_i]$, which may be assumed to belong to some smooth, or otherwise appropriately restricted, set of functions. The goal is to use the observed data together with an algorithmic procedure to learn $\mu(\mathbf{x})$ for all values of $\mathbf{x}\in\mathcal{X}$. While there are many ways to grow a decision tree (i.e., a partition of $\mathcal{X}$), our focus throughout this paper will be on the CART algorithm \citep{breiman1984}, by far the most popular in practice. A decision tree is a hierarchically organized data structure constructed in a top down, greedy manner through recursive binary splitting. According to conventional CART methodology, a parent node $ \mathrm{t} $ (i.e., a region in $\mathcal{X}$) in the tree is divided into two child nodes, $ \mathrm{t}_L $ and $ \mathrm{t}_R $, by minimizing the sum-of-squares error (SSE) \begin{equation} \label{eq:se} \sum_{\mathbf{x}_i \in \mathrm{t}}(y_{i} - \beta_1 \mathbf{1}(x_{ij} \leq \tau) - \beta_2 \mathbf{1}(x_{ij} > \tau))^2, \end{equation} with respect to the child node outputs, split point, and split direction, $(\beta_1, \beta_2, \tau, j)$, with $\mathbf{1}(\cdot)$ denoting the indicator function. Because the splits occur along values of a single covariate, the induced partition of the input space $ \mathcal{X} $ is a collection of hyper-rectangles. The solution of \prettyref{eq:se} yields estimates $(\hat\beta_1, \hat\beta_2, \hat \tau, \hat\jmath)$, and the resulting refinement of $ \mathrm{t} $ produces child nodes $ \mathrm{t}_L = \{\mathbf{x} \in \mathrm{t} : x_{\hat\jmath} \leq \hat \tau\} $ and $ \mathrm{t}_R = \{\mathbf{x} \in \mathrm{t} : x_{\hat\jmath} > \hat \tau\} $. The normal equations imply that $\hat\beta_1 = \overline y_{\mathrm{t}_L} = \frac{1}{\#\{\mathbf{x}_i \in \mathrm{t}_L\}}\sum_{\mathbf{x}_i \in \mathrm{t}_L} y_i $ and $\hat\beta_2 = \overline y_{\mathrm{t}_R} = \frac{1}{\#\{\mathbf{x}_i \in \mathrm{t}_R\}}\sum_{\mathbf{x}_i \in \mathrm{t}_R} y_i $, the respective sample means after splitting the parent node at $ x_{\hat\jmath} = \hat \tau $, where $\#A$ denotes the cardinality of the set $A$. These child nodes become new parent nodes at the next level of the tree and can be further refined in the same manner, and so on and so forth, until a desired depth is reached. To obtain a maximal decision tree $T_K$ of depth $K$, the procedure is iterated $ K $ times until (i) the node contains a single data point $(y_i, \mathbf{x}^T_i)$ or (ii) all input values $ \mathbf{x}_i $ and/or all response values $y_i$ within the node are the same. In a conventional regression problem, where the goal is to estimate the conditional mean response $ \mu(\mathbf{x}) $, the tree output for $ \mathbf{x} \in \mathrm{t} $ is the within-node sample mean $ \overline y_{\mathrm{t}} $, i.e., if $ T $ is a decision tree, then $ \hat \mu(T)(\mathbf{x}) = \overline y_{\mathrm{t}} = \frac{1}{\#\{\mathbf{x}_i \in \mathrm{t}\}}\sum_{\mathbf{x}_i \in \mathrm{t}}y_i $. However, one can aggregate the data in the node in a number of ways, depending on the target estimand. For example, CART methodology is also commonly used for classification tasks (e.g., propensity score estimation in causal inference settings), in particular, where the outcome variable $y_i\in\{0,1\}$ takes on binary values. In this case, the classification tree output is the majority vote of the class instances in the node. Because the canonical splitting criterion for binary classification, the \emph{Gini index}, is equivalent to \prettyref{eq:se}, the results presented in this paper are directly applicable. In addition, decision tree methodology can also be employed for conditional quantile regression and its various downstream tasks, such as estimating quantiles, constructing confidence intervals, or performing outlier detection \cite[and references therein]{meinshausen2006quantile}. These methods also require high pointwise accuracy of decision trees, and thus our results will have methodological implications in those settings as well. Finally, in multi-step semiparametric settings, it is often the case that preliminary unknown functions (e.g., propensity scores in causal inference settings) are estimated using modern machine learning methods such as CART \cite[see, for example,][and references therein]{chernozhukov2022locally}. Our results reveal that reliance on fast uniform convergence rates for decision tree methodology may not be guaranteed, as we show below that decision trees will have a convergence rate slower than any polynomial-in-$n$, over the entire support $\mathcal{X}$. This finding implies that other machine learning procedures such as neural networks \citep[and references therein]{farrell2021deep} may be preferable in those multi-step semiparametric settings, if such methods could be shown to be uniformly consistent with sufficiently fast rates of convergence. From a big picture perspective, our main methodological message is to warn against mechanical application of flexible, adaptive machine learning methodologies for tasks that require good quality estimates at specific covariate values of interest. Machine learning procedures that are currently deployed in practice (for canonical regression problems) are trained to approximately minimize the empirical mean squared error. As such, they enjoy good out-of-sample accuracy for an average-case value of the covariates, i.e., if accuracy is measured via the integrated mean squared error (IMSE). However, if the task requires a more stringent form of convergence, such as uniform convergence, it is unknown if those procedures meet such additional demands. Our results are the first to formally show that this is not the case for decision trees, despite them having small IMSE. \section{Causal Inference and Policy Decisions}\label{sec: Causal Inference and Policy Decisions} As mentioned earlier, recursive partitioning is now a common tool of choice in the analysis of heterogeneous causal treatment effects and the design of heterogeneous policy interventions \citep[and references therein]{athey2019machine,yao2021survey}. Here the observed data is a random sample $ \{(y_i, \mathbf{x}^T_i, d_i):i=1,2,\dots,n\}$, where $y_i$ is the outcome of interest, $\mathbf{x}_i$ is a set of pre-treatment covariates, and $d_i$ is a binary treatment indicator variable. Employing standard potential outcomes notation, \begin{equation*}\label{eq:rubin} y_i = y_i{(1)} \cdot d_i + y_i{(0)} \cdot (1-d_i), \end{equation*} where $y_i{(1)}$ is the potential outcome under treatment ($d_i=1$) and $y_i{(0)}$ is the potential outcome under control ($d_i=0$). This paradigm is fundamental to most applied sciences; for example, it can be used to model the effectiveness of a drug therapy, behavioral intervention, marketing campaign, or government program. In cases where the individual treatment effect $ y_i{(1)} - y_i{(0)} $ varies across different subgroups, a natural goal is to estimate the \emph{heterogeneous} average treatment effect (ATE) for each covariate value $ \mathbf{x} \in \mathcal{X}$, namely, $ \theta(\mathbf{x}) = \mathbb{E}[y_i{(1)}-y_i{(0)} \mid \mathbf{x}_i=\mathbf{x}] $. In recent years, there has been an explosion of machine learning technologies adapted for heterogeneous causal effect estimation, owing to the abundance of data produced from large-scale experiments and observational studies. Among these machine learning algorithms, recursive partitioning estimators (specifically, \emph{causal decision trees}) stand out as natural contenders, as they are well-suited for grouping data according to the treatment effect size, conditional on observable characteristics \citep[e.g.,][]{su2009subgroup, athey2016recursive}. We now discuss CART methodology in the context of heterogeneous causal effect estimation, one popular application of decision trees where accurate pointwise estimates over the entire support $\mathcal{X}$ are essential. In experimental settings, where $ (y_i{(0)},y_i{(1)},\mathbf{x}_i^T) \Perp d_i $, the \textit{conditional} ATE is identifiable because \begin{align*} \theta(\mathbf{x}_i) &= \mathbb{E}[y_i \mid \mathbf{x}_i, \;d_i=1] - \mathbb{E}[y_i \mid \mathbf{x}_i,\;d_i=0]\\ &= \mathbb{E}\Bigg[y_i \frac{d_i-\xi}{\xi(1-\xi)} \mid \mathbf{x}_i \Bigg], \end{align*} where the probability of treatment assignment $\xi=\mathbb{P}(d_i=1)$ is known by virtue of the known randomization mechanism. It follows that $\theta(\mathbf{x})$, $\mathbf{x}\in\mathcal{X}$, can be estimated using decision tree methodology in at least two ways, namely, for a decision tree $T$, \begin{equation} \label{eq:imp} \hat\theta_\text{reg}(T)(\mathbf{x}) = \frac{1}{\#\{\mathbf{x}_i \in \mathrm{t}: d_i = 1\}}\sum_{\mathbf{x}_i \in \mathrm{t}: d_i = 1}y_i - \frac{1}{\#\{\mathbf{x}_i \in \mathrm{t}: d_i = 0\}}\sum_{\mathbf{x}_i \in \mathrm{t}: d_i = 0}y_i \end{equation} or \begin{equation} \label{eq:ipw} \hat\theta_\text{ipw}(T)(\mathbf{x}) = \frac{1}{\#\{\mathbf{x}_i \in \mathrm{t}\}}\sum_{\mathbf{x}_i \in \mathrm{t}} y_i \frac{d_i-\xi}{\xi(1-\xi)}, \end{equation} where recall $\mathrm{t}$ denotes the unique (terminal) node containing $\mathbf{x}\in\mathcal{X}$. In this spirit, we consider a tree-based approach for analyzing treatment effect heterogeneity in randomized control trials, which may also be used to design personalized treatment assignments based on pre-intervention observable characteristics. While our forthcoming results are stated for the regression problem \prettyref{eq:model}, they are also directly applicable to the causal decision tree estimators above that involve minimizing a SSE criterion. This is precisely because $ \hat\theta_\text{reg}(T)(\mathbf{x}) $ and $\hat\theta_\text{ipw}(T)(\mathbf{x})$ can be implemented using conventional CART methodology. That is, we implement $ \hat\theta_\text{reg}(T)(\mathbf{x}) $ following a plug-in approach that estimates $ \mathbb{E}[y_i \mid \mathbf{x}_i, \;d_i=1] $ and $ \mathbb{E}[y_i \mid \mathbf{x}_i, \;d_i=0] $ separately with regression trees and conventional CART methodology. Alternatively, we fit a regression tree with CART methodology to the transformed outcome $ y_i(d_i-\xi)/(\xi(1-\xi)) $ to implement $\hat\theta_\text{ipw}(T)(\mathbf{x})$. Yet another (more principled) approach \citep{athey2016recursive} implements $\hat\theta_\text{reg}(T)(\mathbf{x}) $ by growing a decision tree using a slightly modified version of the SSE criterion \prettyref{eq:se} (referred to as \emph{adjusted expected MSE}) that directly targets the conditional ATE, together with sample-splitting, where different samples are used for constructing the partition and estimating the effects of each subpopulation (a procedure referred to as \emph{honest}). Our theory implies that, for a constant treatment effect model, the aforementioned causal decision tree estimators will exhibit, at best, poly-logarithmic rates of convergence. Furthermore, in more interesting cases, shallow (honest) causal decision tree estimators will be shown to be inconsistent, as a function of the sample size $n$, for some $\mathbf{x}\in\mathcal{X}$, even in settings with very large sample sizes. Finally, we will also show that random forest methodology, while hurting the interpretability and introducing additional tuning parameters, can overcome the limitations of decision trees by restoring nearly optimal pointwise (for all $\mathbf{x}\in\mathcal{X}$) convergence rates. \section{Homoskedastic Constant Regression Model} To formalize the pitfalls of pointwise regression estimation using decision trees, we consider the simplest possible data generating process. \begin{assumption}[Location Regression Model]\label{ass:DGP} The observed data $\{(y_i,\mathbf{x}_i^T) : i = 1, 2, \dots, n\}$ is a random sample satisfying \prettyref{eq:model} and the following: \begin{enumerate} \item $ \mu(\mathbf{x}) \equiv \mu $ constant for all $\mathbf{x}\in\mathcal{X}\subseteq\mathbb{R}^p$. \item $\mathbf{x}_i$ has a continuous distribution. \item $ \mathbf{x}_i \Perp \varepsilon_i$ for all $i=1,2,\dots,n$. \item $ \mathbb{E}\big[|\varepsilon_i|^{2+\nu}\big]<\infty$ for some $\nu>0$. \end{enumerate} \end{assumption} Because trees are invariant with respect to monotone transformations of the coordinates of $ \mathbf{x} $, without loss of generality, we assume henceforth that the marginal distributions of the covariates are uniformly distributed on $\mathcal{X} = [0, 1]^p$, i.e., $ x_j \thicksim U([0, 1])$ for $ j = 1, 2, \dots, p$. Under \prettyref{ass:DGP}, the regression model \prettyref{eq:model} becomes the standard location (or \emph{intercept-only regression}) model with homoskedastic errors: \begin{equation} \label{eq:pure} y_i = \mu + \varepsilon_i, \qquad \mathbb{E}\big[\varepsilon_i^2\big]=\sigma^2. \end{equation} This statistical model is perhaps the most canonical member of any interesting set of data generating processes. In particular, the regression function belongs to all classical smoothness function classes, as well as to the set of functions with bounded total variation. See, for example, \cite{gyorfi2002distribution} for review and further references. As a consequence, our results will also shed light in settings where uniformity over any of the aforementioned classes of functions is of interest, since our lower bounds can be applied directly in those cases. To be more precise, if $ \hat\mu(T)(\mathbf{x}) $ is the output from a decision tree $ T $, then for any class of data generating processes $\mathcal{P}$ containing the model defined by \prettyref{ass:DGP}, $\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{P}( \sup_{\mathbf{x} \in \mathcal{X}}|\hat\mu(T)(\mathbf{x}) - \mu(\mathbf{x})| > \epsilon) \geq \mathbb{P}(\sup_{\mathbf{x} \in \mathcal{X}}|\hat\mu(T)(\mathbf{x}) - \mu| > \epsilon)$, for any $ \epsilon > 0 $. Because $\mathcal{P}$ will include the model defined by \prettyref{ass:DGP} in all relevant (both theoretically and practically) cases, our results also highlight fundamental limitations of CART regression methods from a uniform (over $\mathcal{P}$) perspective, whenever interest lies on estimation of $\mu(\mathbf{x})$ for all $\mathbf{x}\in\mathcal{X}$. Since the main purpose of this paper is to explore the limits of decision tree methodology, we do not aim for generality, but rather consider the simplest possible data generating process (\prettyref{ass:DGP}). In the context of causal inference and treatment effects (e.g., \prettyref{sec: Causal Inference and Policy Decisions}), the assumptions correspond to a constant treatment effect model, the most basic case of practical interest. Importantly, \prettyref{ass:DGP} removes issues related to smoothing (or misspecification) bias because the regression function $\mu(\mathbf{x})$ is constant for all $\mathbf{x}\in\mathcal{X}$, which shows that our results will not be driven by standard (boundary or other smoothing) bias in nonparametrics \citep{fan1996local}. Indeed, if the distribution of $ \varepsilon_i $ is symmetric, then we have $ \mathbb{E}[\hat\mu(T)(\mathbf{x})-\mu] = - \mathbb{E}[\hat\mu(T)(\mathbf{x})-\mu] \implies \mathbb{E}[\hat\mu(T)(\mathbf{x})] = \mu, $ owing to the fact that the split points $ \hat \tau $ are symmetric functions of the $\varepsilon_i$. Our results will be driven instead by the fact that decision tree methodology can generate small cells containing only a handful of observations, thereby making the estimator imprecise in certain regions of $\mathcal{X}$. In other words, inconsistency is due to a \emph{large variance} problem, not a \emph{large bias} problem. The location (or constant treatment effect) model is the simplest instantiation of a regression model of practical interest because the regression function is supersmooth and the curse of dimensionality is absent. We should expect any competitive nonparametric estimator to separate a constant signal from noise or, in the language of causal inference, to estimate accurately (constant) treatment effects when they happen to be homogeneous. \prettyref{ass:DGP} also approximately captures another common modeling situation in machine learning and data science, in which the marginal distribution $ y_i \mid x_{ij} $ is noisy (i.e., the marginal projections $ \mathbb{E}[y_i \mid x_{ij}] $ are constant and contain no signal). Because splits in trees are determined using only marginal information, here the split at the root node would be essentially fitting the location model. \section{Decision Stumps} \label{sec:decision_stumps} For each variable $ j = 1, 2, \dots, p $, the data $ \{ x_{ij}: \mathbf{x}_i \in \mathrm{t} \} $ is relabeled so that $ x_{ij} $ is increasing in the index $ i = 1, 2, \dots, n(\mathrm{t}) $, where $ n(\mathrm{t}) = \#\{ \mathbf{x}_i \in \mathrm{t} \} $. Then, minimization of the objective \prettyref{eq:se} can be equivalently recast as maximizing the so-called \emph{impurity gain}: \begin{equation} \label{eq:Delta} \begin{aligned} & \sum_{\mathbf{x}_l \in \mathrm{t}}(y_l-\mu)^2 - \sum_{\mathbf{x}_l \in \mathrm{t}}\big(y_l - \overline y_{\mathrm{t}_L} \mathbf{1}(x_{lj} \leq \tau) - \overline y_{\mathrm{t}_R} \mathbf{1}(x_{lj} > \tau)\big)^2 \\ & \qquad\qquad\qquad = \Bigg(\frac{1}{\sqrt{i}}\sum_{l=1}^{i}(y_l-\mu)\Bigg)^2 + \Bigg(\frac{1}{\sqrt{n(\mathrm{t})-i}}\sum_{l=i+1}^{n(\mathrm{t})}(y_l-\mu) \Bigg)^2, \end{aligned} \end{equation} with respect to the index $ i $ and variable $ j $. The maximizers are denoted by $ (\hat\imath, \hat\jmath) $, and the optimal split point $ \hat \tau $ that minimizes \prettyref{eq:se} can be expressed as $ x_{\hat\imath\hat\jmath} $. We start by considering the case when the tree is depth one ($K=1$), i.e., a decision stump. The tree output can then be written as \begin{equation} \label{eq:stump} \hat\mu(T_1)(\mathbf{x}) = \hat\beta_1\mathbf{1}(x_{\hat\jmath} \leq \hat \tau) + \hat\beta_2\mathbf{1}(x_{\hat\jmath} > \hat \tau)= \begin{cases} \frac{1}{\#\{\mathbf{x}_i: x_{i \hat\jmath} \leq x_{\hat\imath\hat\jmath}\}} \sum_{\mathbf{x}_i:x_{i \hat\jmath} \leq x_{\hat\imath\hat\jmath}} y_i, \quad x_{\hat\jmath} \leq x_{\hat\imath\hat\jmath} \\ \frac{1}{\#\{\mathbf{x}_i: x_{i \hat\jmath} > x_{\hat\imath\hat\jmath}\}} \sum_{\mathbf{x}_i:x_{i \hat\jmath} > x_{\hat\imath\hat\jmath}} y_i, \quad x_{\hat\jmath} > x_{\hat\imath\hat\jmath} \end{cases}\hspace{-3mm}, \end{equation} where $x_{\hat\jmath}$ denotes the value of the $\hat\jmath$-th component of $\mathbf{x}$. The following theorem characterizes the location of the CART split index $ \hat\imath $ at the root node. \begin{theorem}\label{thm:master} Suppose \prettyref{ass:DGP} holds and $ p = 1 $, and let $ \hat\imath $ be the CART split index at the root node. For each $ \gamma \in (0, 1/2] $ with $ \log_2(1/\gamma) \in \mathbb{N} $ and $ \delta \in (0, 1) $, there exist positive constants $ A $ and $ B $, and a positive integer $ N = N(\gamma, \delta) $ such that, for all $n \geq N$, \begin{equation} \label{eq:split_range} \mathbb{P}\Big(A n^{\gamma} \leq \hat\imath \leq B n^{2\gamma} \;\; \text{or} \;\; n-B n^{2\gamma} \leq \hat\imath \leq n-A n^{\gamma} \Big) \geq 1-\delta. \end{equation} Furthermore, there exist positive constants $ a $, $ b$, $ A $, and $ B $, and a positive integer $N = N(a, b, A, B)$ such that, for all $ n \geq N $, \begin{equation} \label{eq:split_log} \mathbb{P}\big(\hat\imath \leq A\log^a(n)\;\;\text{or}\;\; \hat\imath \geq n- A\log^a(n) \big) \geq B(\log(n))^{-b}. \end{equation} \end{theorem} This theorem formally characterizes the regions of the support $\mathcal{X}$ where the first CART split index $ \hat\imath $, at the root node, is most likely to realize. As a consequence, the theorem also characterizes the effective sample size of the resulting cells (recall the data is ordered so that $ \hat \tau = x_{\hat\imath \hat\jmath} $ and hence $ \hat\imath = \#\{\mathbf{x}_i : x_{ij} \leq \hat \tau \} $). First, \prettyref{thm:master} shows that with arbitrary high probability, $ \hat\imath $ will concentrate near its extremes, from the beginning of any tree construction. The arbitrarily slow polynomial rates do not contradict—but are rather precluded by—existing polynomial convergence guarantees \citep[e.g.,][]{wager2018estimation}, which a priori require that each split generates two child nodes that contain a constant fraction of the number of observations in the parent node, i.e., $ n(\mathrm{t}_L) \gtrsim n(\mathrm{t}) $ and $ n(\mathrm{t}_R) \gtrsim n(\mathrm{t}) $. By implication, \prettyref{thm:master} shows that such assumptions requiring \emph{balanced} cells almost surely, which are typically imposed in the literature, are in general incompatible with standard decision tree constructions employing conventional CART methodology \citep[e.g.,][and references therein]{behr2022provable}. The slow convergence rates for the decision stump occur because the optimal split point concentrates near the boundary of the support \citep{ishwaran2015effect}, i.e., $ \hat \tau \approx 0 $ or $ \hat \tau \approx 1 $, causing the two nodes in the stump to be imbalanced, with one containing a much smaller number of samples, and therefore rendering a situation where local averaging is inaccurate. To be more precise, after the first split when $ n(\mathrm{t}) = n $, CART will generate two unbalanced cells with high ($1-\delta$) probability; either $ n^{\gamma} \lesssim n(\mathrm{t}_L) \lesssim n^{2\gamma} $ or $ n^{\gamma} \lesssim n(\mathrm{t}_R) \lesssim n^{2\gamma} $ for large $ n $. Second, and more importantly for our purposes, \prettyref{thm:master} shows that just the first split of a decision tree construction can generate a cell containing, at most, $\log^a(n)$ observations, with probability at least $(\log(n))^{-b}$, up to a constant factor. It will follow from this result that, on the event considered in \prettyref{eq:split_log}, too few observations will be available on one of the cells after the first split for CART to deliver a polynomial-in-$n$ consistent estimator of $\mu$, thereby making the decision tree procedure exhibit slow poly-logarithmic rates, for some $\mathbf{x}\in\mathcal{X}$. \subsection{Convergence Rates} \prettyref{thm:master} appears to be new in the literature. It arises from a careful study of the maximum of \prettyref{eq:Delta} over different ranges of the split index. We employ a generalization \citep[Equation (3)]{berkes2006almost} of the classic Darling-Erd\"{o}s limit law for the maximum of normalized sums of i.i.d. mean zero random variables \citep{darling1956limit}, the proof of which relies on the scaling property of Brownian motion. That is, under \prettyref{ass:DGP}, for any non-decreasing function $1 \leq h(m) \leq m$ for which $\lim_{m\to\infty} h(m)=\infty$ and any $ w \in \mathbb{R}$, \begin{equation} \begin{aligned}\label{eq:darling-erdos-main} \mathbb{P}\Bigg(\max_{m/h(m) \leq i \leq m} \Bigg|\frac{1}{\sqrt{i}}\sum_{l=1}^{i}(y_l-\mu)\Bigg| < \lambda(h(m),w) \Bigg) \to \exp(-\exp(-w)), \end{aligned} \end{equation} as $m\to\infty$, where $$\lambda(h(m),w) = {\textstyle \sqrt{2\sigma^2 \log\log h(m)}} + \frac{\sigma\log\log\log h(m)}{2\sqrt{2 \log \log h(m)}}+\frac{\sigma(w-\log(\sqrt{\pi}))}{\sqrt{2\log \log h(m)}}.$$ Once the location of the first CART split point is well-understood, we can study the resulting CART estimator $\hat\mu(T_1)(\mathbf{x})$ of the unknown regression function. The following statements hold for the pointwise prediction error of the decision stump. \begin{theorem}\label{thm:rates} Suppose \prettyref{ass:DGP} holds and $ p = 1 $, and let $\hat\mu(T_1)(x)$ be the CART estimator of the regression function at the root node. For each $\gamma \in (0, 1/2] $ with $ \log_2(1/\gamma) \in \mathbb{N} $ and $ \delta \in (0, 1) $, there exist positive constants $ C = C(\gamma, \delta) $ and $ D= D(\gamma, \delta)$, and a positive integer $N = N(\gamma, \delta)$ such that, for all $ n \geq N $, \begin{equation}\label{eq:rate_constant2} \mathbb{P}\Bigg(\sup_{x\in\mathcal{X}}|\hat\mu(T_1)(x) - \mu| \geq C \sigma n^{-\gamma}\sqrt{\log\log(n)}\Bigg) \geq 1-\delta, \end{equation} and \begin{equation} \label{eq:rate_constant} \mathbb{P}\Big(|\hat\mu(T_1)(x) - \mu| \geq C \sigma n^{-\gamma}\sqrt{\log\log(n)}\Big) \geq 1/2-\delta, \end{equation} for all $ x \in [0, D n^{\gamma-1}) \cup (1- D n^{\gamma-1}, 1]$. Furthermore, there exist positive constants $ c $, $ d $, $ C $, and $ D $, and a positive integer $N = N(c, d, C, D)$ such that, for all $ n \geq N $, \begin{equation} \label{eq:rate_log} \mathbb{P}\Big(|\hat\mu(T_1)(x) - \mu| \geq C \sigma(\log(n))^{-c}\Big) \geq D(\log(n))^{-d}, \end{equation} for all $x\in\{0,1\}$. \end{theorem} The theorem above shows that decision stumps can have, at most, $n^{\gamma}$ convergence for evaluation points that are within $n^{\gamma-1}$ distance from the boundary of $\mathcal{X}$ (see \prettyref{eq:rate_constant}), for arbitrarily small $\gamma\in(0,1/2]$, and, at most, poly-logarithmic convergence at the boundaries of the covariate space (see \prettyref{eq:rate_log}). This happens because the two nodes in the stump are highly imbalanced with non-trivial probability under \prettyref{ass:DGP}, with one containing a much smaller number of samples—thereby making local estimation difficult. An immediate implication of \prettyref{thm:rates} in the context of heterogeneous (in $\mathbf{x}\in\mathcal{X}$) causal effect estimation is that the CART estimators discussed in \prettyref{sec: Causal Inference and Policy Decisions} can have poor performance in some regions of the covariate support, particularly near the boundaries of $\mathcal{X}$. \subsection{Past Work} \prettyref{thm:rates} contributes to the literature in several ways. Our results indicate that when the goal is to approximate the unknown conditional expectation pointwise for all $\mathbf{x}\in\mathcal{X}$, as it is the case in the analysis of heterogeneity in causal inference settings, decision trees will exhibit extremely slow convergence rates in some regions of the support, making those methods suboptimal from an approximation perspective. The phenomenon revealed in Theorems \ref{thm:master} and \ref{thm:rates} has been observed in various forms since the inception of CART \cite[Section 11]{breiman1984}. Historically, the phenomenon characterized in \prettyref{thm:master} has been called the \emph{end-cut preference}, where splits along noisy directions tend to concentrate along the end points of the parent node. More specifically, \citet[Theorem 11.1]{breiman1984} and \citet[Theorem 4]{ishwaran2015effect} showed that for each $ \epsilon \in (0, 1) $, $ \mathbb{P}( \hat\imath \leq \epsilon n \; \text{or} \;\hat\imath \geq (1-\epsilon)n) \rightarrow 1 $ as $n\rightarrow \infty $, but, unlike our \prettyref{thm:master}, this result is too weak to imply rates of estimation slower than the optimal $\sqrt{n}$ rate. In accordance with \prettyref{thm:rates}, simulation results from \citet[Supplement, Section B]{wager2018estimation}, and many others, also suggested that adaptive causal trees can have slow convergence at the boundaries of the support $\mathcal{X}$, but no formal theory supporting that numerical evidence was available in the literature until now. \citet{tang2018when} give sufficient theoretical conditions under which non-adaptive random forests (i.e., where the decision nodes are independent of the data) will be inconsistent, but those conditions do not apply to commonly used forest implementations nor are they shown to be realized by the data generating mechanism. \cite{buhlmann2002analyzing} and \cite{banerjee2007confidence} show that the minimizers $(\hat\beta_1, \hat\beta_2, \hat \tau)$ of \prettyref{eq:se} at the root node converge to the population minimizers $ (\beta^*_1, \beta^*_2, \tau^*) $ at a cube-root $ n^{1/3} $ rate when the regression model \prettyref{eq:model} satisfies specific regularity assumptions. Because the decision stump \prettyref{eq:stump} can be expressed as $ \hat\mu(T_1)(x) = \hat\beta_1\mathbf{1}(x \leq \hat \tau) + \hat\beta_2\mathbf{1}(x > \hat \tau) $, their results can be used to study the asymptotic properties of $\hat\mu(T_1)(x)$. Among other things, they posit that the population minimizers $ (\beta^*_1, \beta^*_2, \tau^*) $ are unique and that the regression function $ \mu(x) $ is continuously differentiable and has nonzero derivative at $ \tau^* $. \prettyref{thm:rates} shows that the results in \cite{buhlmann2002analyzing} and \cite{banerjee2007confidence} are not uniformly valid in the sense that excluding the constant regression function from the allowed class of data generating processes is necessary for their results to hold for $x\in\mathcal{X}$. \subsection{Uniform Minimax Rates} Letting $\mathcal{P}$ be any set of data generating processes of interest that includes the location model in \prettyref{ass:DGP}, Markov's inequality applied to \prettyref{eq:rate_log} from \prettyref{thm:rates} yields $$\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{E}\Bigg[ \sup_{x\in\mathcal{X}}(\hat\mu(T)(x) - \mu(x))^2 \Bigg] \gtrsim \sigma^2 (\log n)^{-(2c+d)},$$ where $ T $ is any tree constructed using conventional CART methodology with at least one split. Therefore, decision trees grown with CART methodology exhibit, at best, poly-logarithmic uniform convergence rates, when uniformity over the full support of the data $\mathcal{X}$, and over possible data generating processes, is of interest. \subsection{Higher Dimensions} The high probability bound \prettyref{eq:rate_constant2} in \prettyref{thm:rates} continues to hold in higher dimensions ($p>1$). For example, suppose that $ x_{i1}, x_{i2}, \dots, x_{ip} $ are independent, which, by symmetry, implies the optimal splitting direction $\hat\jmath $ satisfies $ \mathbb{P}( \hat\jmath =j ) = 1/p $ for all $ j $. Then, granting \prettyref{ass:DGP}, for each $ \gamma \in (0, 1/2] $ and $ \delta \in (0, 1) $, there exists a constant $ C = C(\gamma, \delta) $, and positive integer $N = N(\gamma, \delta) $ such that, for all $ n \geq N $, \begin{equation} \label{eq:rate_p} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\mu(T_1)(\mathbf{x}) - \mu| \geq C \sigma n^{-\gamma}\sqrt{\log\log(n)}\Bigg) \geq 1-\delta. \end{equation} Thus, the decision stump in multiple dimensions \emph{cannot} converge at a polynomial rate, i.e., its rate is slower than any polynomial-in-$n$. \subsection{Honest Trees} While \prettyref{thm:rates} deals with depth $K=1$ adaptive trees (i.e., the same data is used for determining the split points and terminal node output), analogous results hold for honest trees. The honest tree output is \begin{equation} \label{eq:honest_tree} \tilde\mu(T)(\mathbf{x}) = \frac{1}{\#\{\tilde \mathbf{x}_i \in \mathrm{t}\}}\sum_{\tilde \mathbf{x}_i \in \mathrm{t}} \tilde y_i, \quad \mathbf{x} \in \mathrm{t}, \end{equation} where $( \tilde y_i, \tilde \mathbf{x}_i^T) $, $i=1,2,\dots,n$, are independent samples from those which were used to construct the decision nodes (i.e., the partition of $ \mathcal{X} $), and $ n(\mathrm{t}) = \#\{\tilde \mathbf{x}_i \in \mathrm{t}\} > 0 $. To simplify calculations, we define $ \tilde\mu(T)(\mathbf{x}) = \mu(\mathbf{x}) $ if $ n(\mathrm{t}) = 0 $, an event that occurs with vanishingly small probability. Conditional on the data used to construct the partition, the honest decision stump $ \tilde \mu(T_1)(x) $ at $ x = 0 $ is an average of (approximately) $ \hat\imath $ response values, and so we expect its variance (equal to mean squared error) to be approximately $ \sigma^2/\hat\imath $. The problem is that, according to \prettyref{thm:master}, the split index $ \hat\imath $ is much smaller than $ n $, with high probability. More rigorously, using a conditioning argument and \prettyref{eq:split_log}, it follows that $ \tilde\mu(T_1)(x) $ converges uniformly no faster than \begin{equation} \label{eq:honest_lower} \mathbb{E}\Bigg[\sup_{x\in\mathcal{X}}(\tilde\mu(T_1)(x)-\mu)^2\Bigg] \geq \sigma^2\mathbb{E}\Bigg[\frac{(1-2^{-\hat\imath})^2}{\hat\imath}\Bigg] \gtrsim \sigma^2(\log(n))^{-(a+b)}. \end{equation} \subsection{Simulation Evidence} \label{sec:numerical_stump} We illustrate the implications of Theorems \ref{thm:master} and \ref{thm:rates} numerically with $p=1$. In \prettyref{fig:boundary}, we plot the pointwise root mean squared error (RMSE) $ \sqrt{\mathbb{E}[(\hat\mu(T_1)(x)-\mu)^2]} $, approximated by $500$ replications, when $ \mu = 0 $, $ \varepsilon_i \thicksim N(0, 1) $, and $ n = 1000 $. In \prettyref{fig:boundary_causal}, we consider the context of the causal model discussed in \prettyref{sec: Causal Inference and Policy Decisions}, with a constant treatment effect $\theta(x) = 1 $ and $ \mathbb{E}[y_i(0)]= 0 $, $ d_i \thicksim \text{Bern}(0.5) $, and $ \varepsilon_i \thicksim N(0, 1) $. We plot the pointwise RMSE for an honest causal decision stump with output based on the regression estimator $\hat\theta_\text{reg}(T_1)(x)$ constructed using the adjusted expected MSE splitting criterion proposed by \cite{athey2016recursive}. The transformed outcome tree, $\hat\theta_\text{ipw}(T_1)(x) $, exhibits similar empirical behavior. Both plots corroborate with \prettyref{thm:rates}: the decision stump has smallest pointwise RMSE near the center of the covariate space, but the performance degrades as the evaluation points move closer to the boundary. \begin{figure} \centering \begin{subfigure}[t]{0.4\textwidth} \centering \includegraphics[width=1\textwidth]{boundary} \caption{Pointwise RMSE of decision stump.} \label{fig:boundary} \end{subfigure} \hspace{1.5cm} \begin{subfigure}[t]{0.4\textwidth} \centering \includegraphics[width=1\textwidth]{boundary_causal} \caption{Pointwise RMSE of causal decision stump.} \label{fig:boundary_causal} \end{subfigure} \caption{Pointwise RMSE of decision stumps for location model.} \label{fig:stump} \end{figure} The following section investigates further the role of sample-splitting (i.e., honesty) in the construction of deeper trees, and shows an even stronger result: honest trees will be inconsistent on some (at least countably many) regions of $\mathcal{X}$ whenever the trees are grown up to depth $K\approx \log\log(n)$. In other words, shallow (honest) regression trees can be uniformly inconsistent, a result that is intuitively anticipated from Theorems \ref{thm:master} and \ref{thm:rates} because even after one single split there is non-trivial probability of having small cells with only a few observations, and repeating this process further down the tree can only exacerbate the issue. The main results in this section were derived in the simplest possible case (constant regression model, $p=1$, $K=1$, etc.), but the main conclusions are applicable more generally. The key phenomenon captured by Theorems \ref{thm:master} and \ref{thm:rates} are only exacerbated in multi-dimensional settings ($p>1$) or for multi-level decision trees ($K>1$). We already demonstrated this fact for multi-dimensional covariates in \prettyref{eq:rate_p}, and we will formalize the shortcomings associated with deeper honest trees in the next section. \section{Inconsistency with Deeper Trees} The previous section provides a pessimistic view on depth one ($K=1$) decision trees: decision stumps can have slow (at best, poly-logarithmic) convergence for the simplest regression models in some regions of $\mathcal{X}$. We now discuss formally situations where decision trees can be \emph{inconsistent }(i.e., fail to converge) altogether, if grown only to depth $K\approx \log\log(n)$. As it is customary in the literature, we will focus on trees constructed using sample-splitting (honesty), which are believed to offer better empirical performance \citep{athey2016recursive}. \begin{definition}[CART with sample-splitting (CART+)] At each level of the tree, generate new data $ \{(\tilde y_i, \tilde \mathbf{x}_i^T) : i = 1, 2, \dots, n\} $. Each node $ \mathrm{t} $ from the parent level is further refined by selecting a split direction and split point that minimizes the CART squared error criterion \prettyref{eq:se} with data $ \{ (\tilde y_i, \tilde\mathbf{x}_i^T) : \tilde \mathbf{x}_i \in \mathrm{t} \}. $ The output of the tree $ T $ at a point $ \mathbf{x} $ belonging to a terminal node $ \mathrm{t} $ is $ \tilde\mu(T)(\mathbf{x}) = \frac{1}{\#\{\tilde \mathbf{x}_i \in \mathrm{t} \}}\sum_{\tilde \mathbf{x}_i \in \mathrm{t}} \tilde y_i $ if $ n(\mathrm{t}) = \#\{\tilde \mathbf{x}_i \in \mathrm{t} \} > 0 $ and $ \tilde\mu(T)(\mathbf{x}) = \mu(\mathbf{x}) $ if $ n(\mathrm{t}) =0 $. \end{definition} The only difference between conventional CART and CART with sample-splitting (CART+) is that the split points at each level are chosen using a fresh (statistically independent) random sample. In fact, all the results in this section remain valid if sample-splitting is not done at the output level, but only for symmetry do we consider CART+ herein. The adaptive properties of the tree are retained either way, as the nodes are still refined by minimizing the empirical squared error \prettyref{eq:se}. One can construct a depth $ K $ tree with this methodology by splitting the original dataset $ \{(y_i, \mathbf{x}_i^T): i = 1,2 , \dots, n\} $ into $K$ disjoint subsets of size $ \lfloor n/K \rfloor $. For our purposes, problems will arise as soon as the depth $ K $ is approximately $ \log\log(n) $ and so there is little practical difference between CART+ and the original CART algorithm when the sample size is large. CART+ serves as a phenomenological model of conventional CART and allows us to analyze its pointwise (and uniform in $\mathbf{x}\in\mathcal{X}$) behavior. Importantly, the formulation of CART+ ensures that the split points have a desirable Markovian property: a split point $ \tilde \tau \in [a, b] $ conditioned on its immediate ancestor split points $ \tilde a = a $ and $ \tilde b = b $ is independent of all ancestor split points, including $ \tilde a $ and $ \tilde b $. \begin{theorem} \label{thm:main} Suppose \prettyref{ass:DGP} holds and $ p = 1 $. Consider a maximal depth $ K_n \gtrsim \log\log(n)$ tree $T_{K_n}$ constructed with CART+ methodology. Then, there exists a positive constant $ Q $, and a positive integer $N$ such that, for all $n \geq N$, $$ \mathbb{P}\Bigg(\sup_{x\in\mathcal{X}}|\tilde\mu(T_{K_n})(x)-\mu| > Q \Bigg) > Q^2. $$ \end{theorem} This theorem shows that very shallow trees grown with the conventional squared error criterion can be pointwise (and hence uniform in $\mathbf{x}\in\mathcal{X}$) inconsistent. To put the iterated logarithm scaling of the depth $K$ into perspective, if $ n = 1\; \text{billion} $, then $ \log\log(n) \approx 3 $, a typical depth used in practice. The pointwise error in \prettyref{thm:main} should be contrasted with the IMSE. Under \prettyref{ass:DGP}, \begin{equation} \label{eq:imse_rate} \mathbb{E}\Bigg[\int_{\mathcal{X}}(\tilde \mu(T_K)(x)-\mu)^2 \mathbb{P}_x(dx)\Bigg] \leq \frac{2^{K+1}\sigma^2}{n+1}. \end{equation} Therefore, the IMSE of the pointwise inconsistent depth $ K \asymp \log\log(n) $ decision tree decays at the optimal $\sqrt{n}$ rate, up to poly-logarithmic factors. This shows that the performance of the tree varies widely depending on whether the input $ x $ is average or worst case. The intuition for \prettyref{thm:main} is based similarly on \prettyref{thm:master}, but for depth $K$ trees constructed with CART+ methodology. That is, honest trees of depth only $K \approx \log\log(n)$ will generate cells near the boundaries of the support $\mathcal{X}$ containing a finite number of observations with probability bounded away from zero. Choosing $ \gamma = 1/4 $, the inequality \prettyref{eq:split_range} implies that, with probability bounded away from zero, the number of observations in a child node $ \mathrm{t}' $ of a parent node $ \mathrm{t} $ (near the boundary of $ \mathcal{X} $) satisfies $ n(\mathrm{t}') \lesssim \sqrt{n(\mathrm{t})} $. It turns out that the number of times this occurs after $ K $ splits is stochastically dominated by a negative binomial random variable, providing a lower bound on the probability that a maximal depth $ K $ tree will have, at most, $ n(\mathrm{t}) \lesssim n^{2^{-K}} $ observations in terminal nodes near the boundary of $ \mathcal{X} $. The bound $ n^{2^{-K}} $ is a constant whenever $K$ exceeds a constant multiple of $ \log\log(n)$. It is important to note that the aforementioned inconsistency of honest regression trees need not occur at the boundary of the support $\mathcal{X}$. By a symmetry argument, if $ \tilde \tau $ is any split point that occurs at a fixed depth in the tree, then $ \tilde\mu(T_K)(\tilde \tau) $ will also fail to converge to $ \mu $ if the tree has depth $ K \gtrsim \log\log(n) $. In other words, after the first (finite) $ J \geq 1 $ splits, inconsistency will occur at any of the (approximately) $2^{J}+1$ endpoints associated with the $2^J$ cells, whenever $ J+K \gtrsim \log\log(n)$ and $ n\rightarrow \infty$. Therefore, \prettyref{thm:main} establishes that there are at least countably many locations on $\mathcal{X}$ where a maximal depth $K$ decision tree will be inconsistent, whenever $ K \gtrsim \log\log(n)$ and $ n \rightarrow \infty$ (since $J$ is finite, it can be absorbed in the constant underlying the condition $ K \gtrsim \log\log(n)$). \section{Pruning} \label{sec:pruning} Pruning is a well-established strategy for mitigating some of the ill consequences of working with trees, such as overfitting. In some cases, however, pruning will not help. Indeed, as the previous section has revealed, depth one trees can have extremely slow convergence near the boundary of the covariate space. While this phenomenon holds for location models, it can also manifest with models that have a strong dependence on the covariates. For example, if the first split at the root of the tree is along a variable $ x_j $ such that the marginal projection $ \mathbb{E}[y \mid x_j ] $ is constant---resembling the location model in \prettyref{ass:DGP} marginally---then, according to the previous discussion, the tree will almost always have one node with very few observations, but no amount of pruning at lower depths will help. The \emph{checkerboard} model \citep{bengio2010} in $ p = 2$ dimensions is an example where $y$ is marginally independent of both covariates. That is, if $ y_i = \text{sgn}( x_{i1}-0.5)\text{sgn}(x_{i2}-0.5) + \varepsilon_i $, where $ \mathbf{x}_i \thicksim U([0,1]^2) $ and $ \varepsilon_i \thicksim N(0, 1) $ are independent, then $ y_i $ given $ x_{ij} = x_j $ is distributed as a symmetric two-component Gaussian mixture, free from $ x_j $. To illustrate the point above numerically on a model with a smooth regression function, suppose $ y_i = (x_{i1}-0.5)(x_{i2}-0.5) + \varepsilon_i $, where $ \mathbf{x}_i \thicksim U([0,1]^2) $ and $ \varepsilon_i \thicksim N(0, 1) $ are independent. As $ \mathbb{E}[y_i \mid x_{ij} = x_j] = 0 $ for $ j = 1, 2$, the response variable has no marginal dependence on either covariate. \prettyref{fig:boundary_prune} displays the results of a computer experiment with $ n = 1000 $ and $500$ replications. The plot shows the pointwise RMSE of a pruned tree $ T$ with output $ \hat\mu(T)(\mathbf{x}) $ at $ \mathbf{x} = (0, x_2)$ as $ x_2 $ ranges from $0$ to $1$. Similarly, \prettyref{fig:boundary_prune_causal} shows the result of fitting a pruned causal tree $T$ with output $\hat\theta_\text{reg}(T)(\mathbf{x})$, constructed using honesty and the adjusted expected MSE splitting criterion proposed by \cite{athey2016recursive}. The experiment consists of $500$ replications from the model $ y_i = d_i(x_{i1}-0.5)(x_{i2}-0.5) + \varepsilon_i $, where $ d_i \thicksim \text{Bin}(0.5) $, $ \mathbf{x}_i \thicksim U([0,1]^2) $, and $ \varepsilon_i \thicksim N(0, 1) $ are independent, and $n = 1000 $. We do not include the transformed outcome tree $\hat\theta_\text{ipw}(T)(\mathbf{x}) $ as it also produces a similar plot. In both cases, the numerical evidence indicates that pruning does not mitigate the lack of uniform consistency over $\mathcal{X}$ and the poor performance near the boundary persists. \begin{figure} \centering \begin{subfigure}[t]{0.4\textwidth} \centering \includegraphics[width=1\textwidth]{boundary_prune} \caption{Pointwise RMSE for pruned tree at $ \mathbf{x} = (0, x_2)^T$.} \label{fig:boundary_prune} \end{subfigure} \hspace{1.5cm} \begin{subfigure}[t]{0.4\textwidth} \centering \includegraphics[width=1\textwidth]{boundary_prune_causal} \caption{Pointwise RMSE for pruned causal tree at $ \mathbf{x} = (0, x_2)^T$.} \label{fig:boundary_prune_causal} \end{subfigure} \caption{Pointwise RMSE of pruned trees for models where $ \mathbf{x} $ and $ y $ are dependent.} \end{figure} \section{Random Forests} At this point, the curious reader may wonder whether ensemble learning can address some of the convergence issues with decision trees. Here we consider \emph{honest random forests}, developed by \cite{wager2018estimation}. Specifically, for each tree in the ensemble, we randomly sample a subset $ S \subset \{ 1, 2, \dots, n\} $ of size $ s $ and, among the data $ \{(\mathbf{x}_i, y_i)\}_{ i \in S} $, use half for determining the splits and the other half for estimating the conditional mean in the terminal nodes (the division of $ S $ into two equally sized subsets occurs randomly). More specifically, for each $ S \subset \{1, 2, \dots, n \} $ with $ |S| = s $, let $ S_0 $ denote the portion used for determining the splits and $ S_1 $ be the portion used for estimating the conditional mean in the terminal nodes. The set of all such subsamples is denoted by $ \mathcal{S} = \{S= S_1\cup S_0 \subset \{1, 2, \dots, n\}: S_0\cap S_1 = \emptyset, \; |S_0|=|S_1|=s/2\} $. In addition, at each node, a particular variable is split if it yields the smallest SSE \prettyref{eq:se} among a random selection $ M \subset \{1, 2, \dots, p \} $ of $ m = mtry $ candidate directions. The set of all candidate variable selections is denoted by $ \mathcal{M} = \{M \subset \{1, 2, \dots p\} : |M| = m\}$. This idea can be applied to regression trees to obtain a regression forest, or causal decision tree estimators \prettyref{eq:imp} or \prettyref{eq:ipw} to obtain a causal forest, though, for simplicity, here we only consider the regression setting. To get a sense of the improvement that forests offer over trees, we specialize to the case where the constituent trees in the forest are honest decision stumps (i.e., honest trees \prettyref{eq:honest_tree} with depth $K=1$). The decision stump output $\hat\mu(T_1)(\mathbf{x}) $ constructed in this way is denoted by $ \hat\mu(T(M, S))(\mathbf{x}) $ and the (regression) random forest output is $ \hat\mu_B(\mathbf{x}) = B^{-1}\sum_{b=1}^B \hat\mu(T(M_b, S_b))(\mathbf{x}) $, where $ (M_1, S_1), (M_2, S_2), \dots, (M_B, S_B) $ are independent copies of $ (M, S) $. When the number of trees $ B$ is large, the honest random forest can be approximated by $$ \hat \mu(\mathbf{x}) = \frac{1}{{n \choose s}{s \choose s/2}{p \choose m}}\sum_{S \in\mathcal{S}}\sum_{M \in\mathcal{M}}\hat\mu(T(M, S))(\mathbf{x}). $$ The next theorem provides an upper bound on its pointwise error. \begin{theorem}\label{thm:forests} Suppose \prettyref{ass:DGP} holds, and, additionally, that $ x_{i1}, x_{i2}, \dots, x_{ip} $ are independent. If $ s = o(n^{1/3})$ and $ m = o(p/s) $, then for all $ \mathbf{x} \in \mathcal{X} $, $$\mathbb{E}\big[(\hat \mu(\mathbf{x}) - \mu)^2\big] \leq (\sigma^2/n)(1+(s/2)(m/p)+o(1)), \quad n \rightarrow \infty, \; p \rightarrow \infty. $$ \end{theorem} This theorem showcases explicitly the effect of both subsampling and the random variable selection mechanism---each is important for reducing variance. According to past work that utilizes the Hoeffding-Serfling variance inequality for U-statistics \citep{wager2018estimation, buhlmann2002analyzing}, subsampling allows us to achieve a pointwise error $$ \mathbb{E}\big[(\hat\mu(\mathbf{x}) - \mu)^2\big] \lesssim \sigma^2s/n, $$ which is significantly better than the slow poly-logarithmic or polynomial rates for individual trees (see \prettyref{thm:rates}), but still suboptimal since $ s $ is typically chosen to grow with the sample size to reduce bias when it exists. The result becomes more interesting when we account for the random variable selection mechanism, because it further reduces the error by decorrelating the constituent trees. Therefore, if the dimensionality $ p $ is large relative to $ s $ and $ m = o(p/s) $, then it is possible to achieve the \emph{exact} optimal $\sqrt{n}$ rate---a vast improvement over the $ n^{\gamma} $ rate for individual trees. (This specification corroborates with recent empirical work by \cite{mentch2020randomization}, which suggests that $ m $ should be small when the signal-to-noise ratio is low.) The price paid for such improvement is the inclusion of two additional tuning parameters for implementation ($s$ and $m$), and the loss of interpretability for the resulting estimates. \begin{figure} \centering \begin{subfigure}[t]{0.4\textwidth} \centering \includegraphics[width=1\textwidth]{boundary_forest} \caption{Pointwise RMSE of random forest with $ s = 100 $ and $ m = 1 $.} \label{fig:boundary_forest} \end{subfigure} \hspace{1.5cm} \begin{subfigure}[t]{0.4\textwidth} \centering \includegraphics[width=1\textwidth]{boundary_forest_causal} \caption{Pointwise RMSE of causal forest with $ s = 100 $ and $ m = 1 $.} \label{fig:boundary_forest_causal} \end{subfigure} \caption{Pointwise RMSE of random forests for location model.} \label{fig:forest} \end{figure} In \prettyref{fig:forest}, we plot the pointwise RMSE of a regression forest and causal forest for the models in \prettyref{sec:numerical_stump}. Compared to \prettyref{fig:boundary} and \prettyref{fig:boundary_causal}, we see that random forests have considerably better performance than a single tree near the boundary. When $ p = 1 $, \cite{banerjee2007confidence} and \cite{buhlmann2002analyzing} investigated formally the properties of decision trees under assumptions that rule out the location model in \prettyref{ass:DGP}. They also showed that subsampling can reduce variance, similar to our result in \prettyref{thm:forests}. However, because the decision stump exhibits large bias in their setting, one cannot deduce from their results how random forests would improve the pointwise mean square error, which accounts for both bias and variance. Additionally, unlike \prettyref{thm:forests}, the random variable selection mechanism was not explored by \cite{banerjee2007confidence} and \cite{buhlmann2002analyzing} because their results are limited to the one-dimensional setting $ p = 1 $. As a consequence, \prettyref{thm:forests} complements prior literature by studying the pointwise mean squared error performance of random forest under the the location model with $p\geq 1$, and thus formalizes a beneficial aspect of random feature selection for decision tree ensembles. Finally, while \prettyref{thm:forests} concerns a depth one ($K=1$) random forest construction, it is possible to explore multi-level honest tree ensembles. \prettyref{thm:main} showed that shallow honest trees constructed with the CART+ procedure can produce pointwise inconsistent estimates of the regression function $\mu$. In contrast, using the Hoeffding-Serfling variance inequality for U-statistics, it can be shown that an ensemble of depth $ K \asymp \log\log(n) $ trees constructed with CART+ methodology on subsampled data will have pointwise error $ \sqrt{\mathbb{E}[(\hat\mu(\mathbf{x}) - \mu)^2]} = O(\sigma\sqrt{s/n})$, for all $\mathbf{x}\in\mathcal{X}$. This result provides a concrete example where an ensemble of shallow inconsistent decision trees can be consistent with nearly optimal convergence rates, and is, to the best of our knowledge, the first time that such a result has been shown in the literature for practical trees based on CART methodology. \section{Conclusion} This article studied the delicate pointwise properties of axis-aligned recursive partitioning, focusing on heterogeneous causal effect estimation, where accurate pointwise estimates over the entire support of the covariates are essential for valid statistical learning (e.g., point estimation, testing hypotheses, confidence interval construction). Specifically, we called into question the use of causal decision trees for such purposes by demonstrating that, for a standard location model, depth one decision trees (e.g., decision stumps) constructed using CART methodology exhibit, at best, poly-logarithmic uniform convergence rates, with pointwise convergence rates slower than any polynomial in boundary regions of the support of the covariates. Even more dramatic, when using sample-splitting (honesty), shallow decision trees were shown to be inconsistent even in large samples. Pruning was unable to overcome these limitations, but ensemble learning with both subsampling and random feature selection was successful at restoring near-optimal convergence rates for pointwise estimation for the specific simple class of data generating processes that we considered. While our emphasis was on direct use of decision trees for causal effect estimation, the methodological implications are similar for multi-step semi-parametric settings, where preliminary unknown functions (e.g., propensity scores) are estimated with machine learning tools, as well as conditional quantile regression, both of which require estimators with high pointwise accuracy. In conclusion, our results have important implications for heterogeneous prediction and causal inference learning tasks employing decision trees. Whenever the goal is to produce accurate pointwise regression estimates over the entire support of the conditioning variables, even shallow decision trees trained with a large number of samples can exhibit poor performance. Consequently, adaptive recursive partitioning should be used with caution for heterogeneous prediction or causal inference purposes, especially in high-stakes environments where high pointwise accuracy is crucial. \newpage

proofpile-arXiv_069-8110

\section{Introduction} A lower bound $\mathrm{scal}\geq c>0$ on the scalar curvature of a Riemannian $n$-manifold does not constrain either its diameter or volume, provided \(n>2\). However, there are other types of upper bounds on the ``size'' of such manifolds, in terms of ``dilations of topologically significant maps'', as explained by Gromov~\cite[Sec.~4]{gromov-dozen}. Since shrinking areas of surfaces on manifolds with $\mathrm{scal}>0$ always increases scalar curvature, it is natural to ask in which cases this is the \emph{only} way of doing so. Motivated by this question, following Gromov~\cite[Sec.~4]{gromov-dozen}, we call a metric $\mathrm g_0$ on a closed manifold $M$ \emph{area-extremal} (for scalar curvature) if, for all metrics $\mathrm g_1$ on~$M$, \begin{equation}\label{eq:g1-competitor-global} \wedge^2\mathrm g_1\succeq\wedge^2\mathrm g_0 \quad\text{ and }\quad \mathrm{scal}(\mathrm g_1)\geq\mathrm{scal}(\mathrm g_0) \end{equation} imply that $\mathrm{scal}(\mathrm g_1) = \mathrm{scal}(\mathrm g_0)$; and \emph{area-rigid} if \eqref{eq:g1-competitor-global} implies $\mathrm g_1=\mathrm g_0$. The first condition in \eqref{eq:g1-competitor-global} means that areas measured with \(\mathrm g_1\) are no smaller than those measured with \(\mathrm g_0\). For instance, this is the case whenever the stronger condition \(\mathrm g_1\succeq \mathrm g_0\) holds, i.e., whenever distances are no smaller under \(\mathrm g_1\) than under \(\mathrm g_0\). Thus, area-extremality can be summarized as the impossibility of increasing scalar curvature without shrinking the area of some surface (or infinitesimal 2-plane). Round metrics on spheres are area-rigid, by a result of Llarull \cite{llarull1}. Expanding on work of Min-Oo \cite{min-oo-scal}, Goette and Semmelmann \cite{goette-semmelmann2} proved that any metric $\mathrm g$ with positive-semidefinite curvature operator ($R\succeq0$) on a manifold with nonzero Euler characteristic is area-extremal, and area-rigid if $\tfrac{\mathrm{scal}}{2}\, \mathrm g\succ\operatorname{Ric} \succ0$. Furthermore, Goette and Semmelmann \cite{goette-semmelmann1} proved that K\"ahler metrics with \(\operatorname{Ric}\succeq 0\) are area-extremal, and area-rigid if \(\operatorname{Ric}\succ0\). However, since these previous results only apply to either spheres or manifolds with special holonomy, finding broader criteria for area-extremality/area-rigidity and describing particular classes of these metrics remain important problems \cite[Prob.~C]{gromov-dozen}. In this paper, we prove new area-extremality and area-rigidity criteria in dimension $4$, relying on a unique characterization of sectional curvature bounds. Namely, by the so-called \emph{Finsler--Thorpe trick}, an orientable Riemannian $4$-manifold has $\sec\geq0$ if and only if there exists a function $\tau$ such that $R+\tau \, *\succeq0$, where \(R\) denotes the curvature operator, and \(*\) the Hodge star operator, each acting on 2-vectors, see \Cref{prop:FTtrick} for details. Other geometric applications of the Finsler--Thorpe trick have recently appeared in \cite{bm-iumj,bkm-siaga,bkm-geography,bk-rf}. \subsection{Main results} Our first result is the following extremality/rigidity criterion: \begin{mainthm}\label{mainthm1} Let $(M^4,\mathrm g)$ be a closed simply-connected Riemannian manifold with $\sec\geq0$. If $\tau\colon M\to\mathds{R}$ such that $R+\tau \,*\succeq0$ can be chosen nonpositive or~nonnegative, then $\mathrm g$ is area-extremal. If, in addition, $\tfrac{\mathrm{scal}}{2}\, \mathrm g\succ\operatorname{Ric} \succ0$, then $\mathrm g$ is area-rigid. \end{mainthm} As a consequence of \cite[Thm.~D]{bm-iumj}, the $4$-manifolds $(M^4,\mathrm g)$ to which \Cref{mainthm1} applies either have \emph{definite} intersection form, or are isometric to $\S^2\times \S^2$ endowed with a product metric. By the classical work of Donaldson and Freedman, the former are homeomorphic to \(\S^4\) or to a connected sum $\mathds{C} P^2\#\dots\#\mathds{C} P^2$ of finitely many copies of $\mathds{C} P^2$. Conjecturally, such manifolds only admit metrics with $\sec>0$ if at most one summand is used (i.e., if $M^4$ is either $\S^4$ or $\mathds{C} P^2$), and with $\sec\geq0$ if at most two summands are used (i.e., if $M^4$ is either $\S^4$, $\mathds{C} P^2$, or $\mathds{C} P^2\#\mathds{C} P^2$). Previously known area-extremal metrics on simply-connected $4$-manifolds $M$~are: \begin{enumerate}[$\;\bullet$] \item metrics with $R\succ0$, hence $M$ is diffeomorphic to $\S^4$, see e.g.~\cite[Thm.~1.10]{wilking-survey}; \item metrics with $R\succeq0$, hence $M$ is isometric to either a product metric on $\S^2\times \S^2$ or a K\"ahler metric on $\mathds{C} P^2$, see e.g.~\cite[Thm.~1.13]{wilking-survey}; \item K\"ahler metrics on $\mathds{C} P^2\# \overline{\mathds{C} P^2}$ with \(\operatorname{Ric}\succ 0\), which exist by Yau's solution~\cite{yau-calabiconj} to the Calabi conjecture, and are area-rigid by \cite{goette-semmelmann1}. \end{enumerate} Thus, using \Cref{mainthm1}, we obtain new examples of area-rigid metrics: \begin{maincor}\label{maincor:CP2CP2} The following hold: \begin{enumerate}[\rm (i)] \item On $\mathds{C} P^2$, metrics in a neighborhood of the Fubini--Study metric are area-rigid; \item On $\mathds{C} P^2\# \mathds{C} P^2$, Cheeger metrics with vanishing neck length are area-rigid. \end{enumerate} \end{maincor} To our knowledge, \Cref{maincor:CP2CP2} (i) gives the first example of an \emph{open set} of area-rigid metrics on a closed manifold other than the sphere. Moreover, note that both (i) and (ii) yield the existence of area-rigid metrics with \emph{generic holonomy}. Recall that \emph{Cheeger metrics} on the connected sum of two compact rank one symmetric spaces are metrics with $\sec\geq0$ obtained gluing complements of disks along a ``neck'' region isometric to a round cylinder $\S^n\times [0,\ell]$ of arbitrary length $\ell\geq0$, see \cite{cheeger}. In particular, the neck region does not have $\operatorname{Ric}\succ0$. For this reason, $\ell=0$ is necessary in \Cref{maincor:CP2CP2} (ii), since then $\operatorname{Ric}\succ0$ on the (dense) complement of the neck hypersurface. Meanwhile, Cheeger metrics on $\mathds{C} P^2\# \mathds{C} P^2$ with $\ell>0$ are area-extremal but not area-rigid, see \Cref{thm:cheeger-metrics} and \Cref{rem:noneck}. We note that $\mathds{C} P^2\# \mathds{C} P^2$ admits no metrics with $R\succeq0$, nor complex structures, and thus no previous methods can be used to identify area-extremal metrics on this manifold. As a consequence of \Cref{maincor:CP2CP2} and previous examples, there are area-extremal metrics with $\sec\geq0$ on all closed simply-connected $4$-manifolds currently known to admit metrics with $\sec\geq0$, namely, $\S^4$, $\mathds{C} P^2$, $\S^2\times \S^2$, and $\mathds{C} P^2\# \pm\mathds{C} P^2$, which are conjectured to be all of them. Furthermore, metrics in a neighborhood of the round metric on \(\S^4\) have $R\succ0$ and are thus area-rigid by \cite{goette-semmelmann2}, so we conclude there is an open set of area-rigid metrics with $\sec>0$ on each of the two simply-connected closed $4$-manifolds known (and conjectured to be all) to admit metrics with \(\sec>0\). \smallskip There is a natural extension of the above notions of area-extremality and area-rigidity to Riemannian manifolds \emph{with boundary}. Given such a manifold $(M,\mathrm g)$, we denote by $\mathrm I\!\mathrm I_{\partial M}$ the second fundamental form of $\partial M$ with respect to the inward unit normal, and by $H(\mathrm g)=\operatorname{tr}\mathrm I\!\mathrm I_{\partial M}$ the mean curvature of $\partial M$. A metric $\mathrm g_0$ on $M$ is \emph{area-extremal} (for scalar curvature) if all metrics $\mathrm g_1$ satisfying \eqref{eq:g1-competitor-global} on $M$ with \begin{equation}\label{eq:g1-competitor-local} \mathrm g_1|_{\partial M}=\mathrm g_0|_{\partial M} \quad\text{ and }\quad H(\mathrm g_1)\geq H(\mathrm g_0) \end{equation} must have $\mathrm{scal}(\mathrm g_1) = \mathrm{scal}(\mathrm g_0)$ as well as $H(\mathrm g_1) = H(\mathrm g_0)$; and \emph{area-rigid} if \eqref{eq:g1-competitor-global} and \eqref{eq:g1-competitor-local} imply $\mathrm g_1=\mathrm g_0$. It is striking that area-extremality, in the above sense for $4$-manifolds with boundary, holds locally around any point with $\sec>0$, as follows: \begin{mainthm}\label{mainthm-local1} If \((X^4,\mathrm g)\) is a Riemannian \(4\)-manifold, then $\mathrm g$ is area-extremal on sufficiently small convex neighborhoods $M$ of any point in $X$ at which $\mathrm g$ has $\sec>0$. \end{mainthm} In particular, \Cref{mainthm-local1} implies that any metric deformation supported in a sufficiently small convex subset $M$ of a $4$-manifold with $\sec>0$ either preserves both $\mathrm{scal}$ and $H$, or else decreases the area of some surface in $M$, or $\mathrm{scal}$ somewhere on $M$, or $H$ somewhere on $\partial M$. On the other hand, recall that for generic $\mathrm g$, \emph{any} sufficiently small deformation of the function $\mathrm{scal}|_M$ is realized as the (restriction to $M$ of the) scalar curvature of a metric near $\mathrm g$, by a result of Corvino~\cite{corvino}. \Cref{mainthm-local1} follows from an extension of \Cref{mainthm1} to $4$-manifolds $(M^4,\mathrm g)$ with convex boundary, i.e., with $\mathrm I\!\mathrm I_{\partial M}\succeq0$. (Note that a geodesic ball of sufficiently small radius in any Riemannian manifold has convex boundary.) \begin{mainthm}\label{mainthm-local2} Let $(M^4,\mathrm g)$ be an orientable compact Riemannian $4$-manifold with $\sec\geq0$ and $\mathrm I\!\mathrm I_{\partial M}\succeq0$. If $\tau\colon M\to \mathds{R}$ such that $R+\tau\,*\succeq0$ can be chosen nonpositive or nonnegative, then $\mathrm g$ is area-extremal. If, in addition, $\frac{\mathrm{scal}}{2}\,\mathrm g\succ\operatorname{Ric}\succ0$, then $\mathrm g$ is area-rigid. \end{mainthm} Similar criteria for manifolds with boundary have been recently proven in all even dimensions under the more restrictive assumption $R\succeq0$. Lott \cite{lott-spin} showed that metrics with $R\succeq0$ and $\mathrm I\!\mathrm I_{\partial M}\succeq0$ on compact manifolds with boundary and nonzero Euler characteristic are area-extremal (in a more general sense; namely, relaxing the first condition in \eqref{eq:scal-comp-local} to \(\mathrm g_1|_{\partial M}\succeq \mathrm g_0|_{\partial M}\)). Cecchini and Zeidler \cite{CZ} obtained area-extremality results for certain warped product metrics on \(M\times [-1,1],\) where \(M\) has $R\succeq0$ and nonvanishing Euler characteristic. It is noteworthy that \Cref{mainthm-local2} does not require \(M\) to be simply-connected, nor to have nonzero Euler characteristic. Indeed, by the Soul Theorem, any $(M^4,\mathrm g)$ as in \Cref{mainthm-local2} is a disk bundle over a totally geodesic closed submanifold, whose topology is constrained by the fact it has $\sec\geq0$ and dimension $\leq3$. In turn, this has topological implications on $M$ which are sufficient to apply our methods and prove area-extremality. \Cref{mainthm-local2} yields new examples of area-extremal and area-rigid metrics; e.g., the metrics on the complement of a ball in $\mathds{C} P^2$ used in the construction of Cheeger metrics on \(\mathds{C} P^2\# \mathds{C} P^2\), described above. No other previous criteria apply to this manifold, since it is diffeomorphic to the normal disk bundle of \(\mathds{C} P^1\subset\mathds{C} P^2\), which is known not to admit metrics with $R\succeq0$ and $\mathrm I\!\mathrm I_{\partial M}\succeq0$, see~\cite{Noronha}. Examples with vanishing Euler characteristic are provided by standard metrics with $R\succeq0$ on products of spheres and disks, see \Cref{ex:examples-local}. With different boundary conditions, such examples are also addressed in the related result~\cite[Thm.~1.3]{lott-spin}. Finally, note that the round hemisphere $\S^4_+$ is area-rigid as a consequence of \Cref{mainthm-local2}, or \cite[Cor.~1.2]{lott-spin}, so the counterexamples to the Min-Oo conjecture constructed in \cite{bmn} must shrink areas somewhere on $\S^4_+$, a fact that was shown in~\cite{miao-tam}. \smallskip Both Theorems \ref{mainthm1} and \ref{mainthm-local2} follow from more general results (\Cref{thm,local}), in which we prove area-extremality/area-rigidity in a broader sense, also discussed in \cite{llarull1, goette-semmelmann1,gromov-dozen,lott-spin}. Namely, metrics can be compared similarly to \eqref{eq:g1-competitor-global} and \eqref{eq:g1-competitor-local} but using maps other than the identity, including maps between different manifolds, which Gromov describes as allowing for competitors with ``topological modifications''. Indeed, Theorems \ref{mainthm1} and \ref{mainthm-local2} are simplified versions of such statements on comparisons with self-maps of nonzero degree, see \Cref{cor:self-maps,cor:boundary-self}. \subsection{Outline of proofs} For the reader's convenience, we briefly describe a general framework to prove area-extremality/area-rigidity based on spin geometry, which is used in the above results. Let $(M,\mathrm g_0)$ and $(N,\mathrm g_1)$ be oriented Riemannian manifolds and \(f\colon N\to M\) be a spin map; e.g., one may take $N=M$ and $f=\operatorname{id}$; see \Cref{4closed} for details. Consider the Dirac operator \(D(\mathrm g_0,\mathrm g_1)\) on spinors over \(N\) twisted with the pullback bundle via $f$ of the spinor bundle over \(M.\) The Bochner--Lichnerowicz--Weitzenb\"ock formula for \(D(\mathrm g_0,\mathrm g_1)\) is given by \[D(\mathrm g_1,\mathrm g_0)^2=\nabla^*\nabla+\tfrac14\mathrm{scal}(\mathrm g_1)+\mathcal{R}(R,\mathrm d f),\] where \(\mathcal R(R,\mathrm d f)\) is a bundle endomorphism that depends only on the curvature operator \(R\) of \((M,\mathrm g_0)\) and \(\mathrm d f\colon TN\to TM\). Algebraic considerations show that \begin{equation*} \mathcal{R}(R,\mathrm d f)=\mathcal{T}(R,\mathrm d f)-\tfrac14 \operatorname{tr}(R)\circ f-\tfrac14\operatorname{tr}\!\big( F^*\circ R\circ F\big)\circ f, \end{equation*} where $F\colon\wedge^2 TN\to\wedge^2 TM$ is the map $F(v\wedge w)=\mathrm d f(v)\wedge \mathrm d f(w)$, and $\mathcal{T}(R,\mathrm d f)\succeq0$ whenever $R\succeq0$, see \Cref{rt,nneg}. For instance, if \(f=\operatorname{id}\) and \(\mathrm g_1=\mathrm g_0\), then \(\mathcal{T}(R,\mathrm d f)\) is the curvature term in the Weitzenb\"ock formula for the Hodge Laplacian on forms. Further algebraic considerations (see \Cref{fscal}) show that if \(\wedge^2\mathrm g_1\succeq f^*\wedge^2\mathrm g_0\) and $R$ has \(\sec\geq0\), then \( \operatorname{tr}\!\big(F^*\circ R\circ F\big)\leq\operatorname{tr}(R)=\tfrac12\mathrm{scal}(\mathrm g_0)\). We thus search for conditions on \(R\) such that \(\sec\geq0\) and \(\mathcal T(R,\mathrm d f)\succeq0\) whenever \(\wedge^2\mathrm g_1\succeq f^*\wedge^2\mathrm g_0\), since then \[D(\mathrm g_0,\mathrm g_1)^2\succeq \nabla^*\nabla+\tfrac14\big(\mathrm{scal}(\mathrm g_1)-\mathrm{scal}(\mathrm g_0)\circ f\big)\] and so \(\mathrm{scal}(\mathrm g_1)\geq\mathrm{scal}(\mathrm g_0)\circ f\) forces \(\mathrm{scal}(\mathrm g_1)=\mathrm{scal}(\mathrm g_0)\circ f\) if the underlying topologies imply, by way of the Atiyah--Singer Index Theorem, that \(\ker D(\mathrm g_1,\mathrm g_0)\neq\{0\}\). In \Cref{pi}, we show that the curvature assumption in \Cref{mainthm1} ensures that \(\mathcal T(R,\mathrm d f)\succeq0\), up to restricting this endomorphism to an appropriate subbundle. In \Cref{indextheory}, we give topological conditions on $M$ and $N$ sufficient to have a nontrivial section \(\xi\) of that subbundle with \(D(\mathrm g_0,\mathrm g_1)\xi=0\). We then use \(\xi\) to prove our most general area-extremality and area-rigidity result for closed manifolds, \Cref{thm}, which in turn implies \Cref{mainthm1}. In \Cref{examples}, we describe metrics satisfying the required curvature assumption, proving \Cref{maincor:CP2CP2}. Following \cite{lott-spin}, the above method is extended to manifolds with boundary in \Cref{4boundary}, proving \Cref{local} along with two corollaries, which imply Theorems \ref{mainthm-local1} and \ref{mainthm-local2}. The convexity assumption $\mathrm I\!\mathrm I_{\partial M}\succeq0$ is used to ensure that \(\nabla^*\nabla\succeq0\), while the existence of a nontrivial section $\xi$ of the appropriate subbundle involves an application of the Atiyah--Patodi--Singer Index Theorem for manifolds with boundary. \subsection*{Acknowledgements} It is our great pleasure to thank Anusha Krishnan, Claude LeBrun, and John Lott for valuable conversations regarding Grove--Ziller metrics, K\"ahler metrics, and Index Theory on manifolds with boundary, respectively. \section{Preliminaries}\label{prelim} In this section, we fix conventions, definitions, and notations, and recall basic facts from linear algebra and spin geometry, closely following \cite{lm-book}. Throughout, $V$ and $W$ denote (finite-dimensional) oriented real inner product spaces. \subsection{Linear algebra}\label{subsec:linalgebra} Given a linear map $l\colon W\to V$, its \emph{adjoint} $l^*\colon V\to W$ is the linear map such that $\langle l(w),v\rangle_V = \langle w,l^*(v)\rangle_W$ for all $v\in V$ and $w\in W$. The space of linear maps $l\colon W\to V$ is denoted $\operatorname{Hom}(W,V)$, and, if $V=W$, we write $\operatorname{End}(V)=\operatorname{Hom}(V,V)$. We identify $\operatorname{Hom}(W,V)=W\otimes V$ by means of $(w\otimes v)(\cdot)=\langle w,\cdot\rangle v$. The subspaces of \emph{symmetric} and \emph{skewsymmetric} endomorphisms of $V$, i.e., $l\in\operatorname{End}(V)$ such that $l^*=l$ and $l^*=-l$, are denoted $\operatorname{Sym}^2(V)\subset \operatorname{End}(V)$ and $\wedge^2 V\subset \operatorname{End}(V)$, respectively, and $\operatorname{End}(V)=\operatorname{Sym}^2(V)\oplus\wedge^2 V$. The special orthogonal group of $V$, i.e., the group of linear isometries of $V$ is denoted $\mathsf{SO}(V)\subset \operatorname{Sym}^2(V)$. All of $\operatorname{End}(V)$, $\operatorname{Sym}^2(V)$, $\wedge^2 V$, $\operatorname{Hom}(W,V)$, $\operatorname{Hom}(\wedge^2 W,\wedge^2 V)$, etc., are endowed with the compatible inner products determined by those in $V$ and $W$. For example, if a linear map $L\colon \wedge^2 W\to\wedge^2 V$ is of the form $L=\wedge^2 l$ for some $l\colon W\to V$,~i.e., \begin{equation}\label{eq:L} \phantom{, \quad\text{ for all } w_1,w_2\in W} L(w_1\wedge w_2)=l(w_1)\wedge l(w_2), \quad\text{ for all } w_1,w_2\in W, \end{equation} then its adjoint \(L^*\colon \wedge^2V\to \wedge^2W\) is given by $L^*=\wedge^2 (l^*)$. We will make repeated use of the following elementary fact from Linear Algebra: \begin{lemma}[Singular Value Decomposition]\label{lemma:diag} Given any linear map $l\colon W\to V$ between real inner product spaces of the same dimension, there exist orthonormal bases $\{w_i\}$ of $W$ and $\{v_i\}$ of $V$, and real numbers $\lambda_i\geq0$, such that $l(w_i)=\lambda_i\,v_i$. \end{lemma} \begin{proof} Let $A\in\operatorname{Sym}^2(W)$ be the linear map such that $\langle A x,y\rangle_W=\langle l(x),l(y)\rangle_V$ for all $x,y\in W$. Clearly, $A$ is symmetric and positive-semidefinite, so it can be diagonalized by an orthonormal basis $\{w_i\}$ of $W$, on which $A=\mathrm{diag}(\lambda_i^2)$ for some $\lambda_i\geq 0$. By the above, $\{l(w_i)\}$ are pairwise orthogonal vectors in $V$ that span the image $l(W)$. The corresponding unit-length vectors form a set that can be extended to an orthonormal basis $\{v_i\}$ of $V$, and, by construction, $l(w_i)=\lambda_i\,v_i$. \end{proof} We say that a linear map $l\colon W\to V$ is \emph{nonincreasing} if $\|l(w)\|\leq \|w\|$ for all $w\in W$, or, equivalently, if all $\lambda_i\geq0$ arising from \Cref{lemma:diag} satisfy $\lambda_i\leq 1$. Note that if $l\colon W\to V$ is nonincreasing, then so is $L=\wedge^2l\colon \wedge^2 W\to\wedge^2 V$ given in \eqref{eq:L}. \subsection{Algebraic curvature operators} For convenience, we treat endomorphisms $R\in\operatorname{Sym}^2(\wedge^2 V)$ both as a symmetric linear maps $R\colon \wedge^2 V\to \wedge^2 V$ and as linear maps $V\times V\ni (x,y)\mapsto R_{x,y}\in \wedge^2 V\subset \operatorname{End}(V)$ such that $R_{x,y}=-R_{y,x}$, where \begin{equation}\label{eq:curvop-sign} \phantom{\quad\text{for all}x,y,z,w\in V.} \langle R_{x,y}(z),w\rangle=\langle R(x\wedge y),w\wedge z\rangle, \quad\text{ for all }x,y,z,w\in V. \end{equation} The linear subspace of $\operatorname{Sym}^2(\wedge^2 V)$ formed by those $R$ that satisfy the \emph{first Bianchi identity} $R_{x,y}(z)+R_{y,z}(x)+R_{z,x}(y)=0$ is denoted $\operatorname{Sym}^2_b(\wedge^2 V)$, and its elements are called \emph{algebraic curvature operators}. These are pointwise models at each $V=T_pM$ for the curvature tensor/operator $R$ of a Riemannian manifold $(M,\mathrm g)$. Accordingly, the \emph{sectional curvature} of a $2$-plane $\sigma=x\wedge y\in\wedge^2 V$ with respect to $R$ is \begin{equation*} \sec_R(\sigma)=\langle R_{x,y}(y),x\rangle=\langle R(\sigma),\sigma\rangle , \end{equation*} while $\operatorname{Ric}_{R}(x,y)$ is the trace of the endomorphism $z\mapsto R_{z, x}(y)$, and $\mathrm{scal}_{R}=2\operatorname{tr} R$. As usual, by $\sec_R\geq0$ and $\operatorname{Ric}_R\succeq0$ we mean $\sec_R(\sigma)\geq0$ for all $2$-planes $\sigma\subset V$ and $\operatorname{Ric}_R(x,x)\geq0$ for all $x\in V$, respectively; similarly for $\sec_R>0$ and $\operatorname{Ric}_R\succ0$. The orthogonal complement of $\operatorname{Sym}^2_b(\wedge^2 V)$ in $\operatorname{Sym}^2(\wedge^2 V)$ can be identified with $\wedge^4 V$, where $\omega\in\wedge^4 V\subset\operatorname{Sym}^2(\wedge^2 V)$ is given by $\langle \omega(\alpha),\beta \rangle=\langle \omega,\alpha\wedge\beta\rangle$. In particular, if $\dim V=4$, this is a one-dimensional space spanned by the Hodge star operator $*\colon\wedge^2 V\to\wedge^2 V$. Moreover, $\sigma\in\wedge^2V$ satisfies $\sigma\wedge\sigma=0$ if and only if $\langle *\sigma,\sigma\rangle=0$, i.e., the quadric defined by $*$ in $\wedge^2 V$ is precisely the Pl\"ucker embedding of the oriented Grassmannian of $2$-planes $\operatorname{Gr}_2^+(V)\subset\wedge^2V$. As shown by Finsler~\cite{finsler}, a quadratic form $\langle R(\sigma),\sigma\rangle$ is nonnegative when restricted to the quadric $\langle *\sigma,\sigma\rangle=0$ if and only if some linear combination of $R$ and $*$ is positive-semidefinite, yielding: \begin{proposition}[Finsler--Thorpe trick]\label{prop:FTtrick} Let $R\in \operatorname{Sym}^2_{b}(\wedge^2 V)$ be an algebraic curvature operator on $V$, with $\dim V=4$. Then $\sec_R\geq0$, respectively $\sec_R>0$, if and only if there exists $\tau\in\mathds{R}$ such that $R+\tau\, *\succeq0$, respectively $R+\tau\, *\succ0$. \end{proposition} \begin{remark} The above has been referred to as \emph{Thorpe's trick}, as it was rediscovered by Thorpe~\cite{Thorpe72}, see~\cite{bkm-siaga} for details. In the mathematical optimization and control literature, this fact is known as \emph{$S$-lemma}, or \emph{$S$-procedure}, see \cite{slemma-survey}. \end{remark} It is an easy consequence of convexity that the set of $\tau\in\mathds{R}$ such that $R+\tau\,*\succeq0$ for a fixed $R\in \operatorname{Sym}^2_{b}(\wedge^2 V)$ with $\sec_R\geq0$, as in \Cref{prop:FTtrick}, is a closed interval $[\tau_{\mathrm{min}},\tau_{\mathrm{max}}]$, which degenerates to a single point (i.e., $\tau_{\mathrm{min}}=\tau_{\mathrm{max}}$) if and only if $R$ has $\sec_R\geq0$ but does not have $\sec_R>0$, see \cite[Prop.~3.1]{bkm-siaga}. \subsection{Clifford Algebra and Spinors}\label{subsec:CAandS} If $\dim_\mathds{R} V=2n$, then the complex Clifford algebra \(\mathds{C}\ell(V)\) associated to \(V\) has a unique irreducible complex representation \(S(V)\), which is a complex vector space of dimension $2^{n}$ of so-called \emph{(Dirac) spinors}. We endow $S(V)$ with a Hermitian inner product for which the action of unit vectors in \(\mathds{C}\ell(V)\) is isometric. The complex volume element \(\omega_\mathds{C}\in\mathds{C}\ell(V)\), which is given by \(\omega_{\mathds{C}}=(\sqrt{-1})^n e_1e_2\dots e_{2n}\) for any orthonormal basis \(\{e_1,\dots,e_{2n}\}\) of \(V\), satisfies \(\omega_\mathds{C}^2=1\). Thus, it induces splittings of $\mathds{C}\ell(V)$ and $S(V)$ as orthogonal direct sums of eigenspaces of \(\omega_\mathds{C}\) with eigenvalue \(\pm1\), denoted \begin{equation}\label{eq:clpm-spm} \mathds{C}\ell(V)=\mathds{C}\ell^+(V)\oplus \mathds{C}\ell^-(V) \quad\text{ and }\quad S(V)=S^+(V)\oplus S^-(V), \end{equation} respectively. We write $S:=S(V)$ and $S^\pm:=S^\pm(V)$ to simplify notation, when the inner product space $V$ in question is clear from the context. The homomorphism \(\mathds{C}\ell(V)\to \operatorname{End}(S)\) defining the representation $S$ is an isomorphism; and, just as in the real case discussed above, the Hermitian inner product $\langle,\rangle$ on \(S\) allows us to identify \(\operatorname{End}(S) =S\otimes S\) via \((\phi\otimes \psi)(\cdot)=\langle \phi,\cdot \rangle \, \psi\), for all \(\phi,\psi\in S\). The composition of these isomorphisms is an isomorphism \(\mathds{C}\ell(V)\cong S\otimes S\) which is \(\mathds{C}\ell(V)\)-equivariant with respect to left multiplication on \(\mathds{C}\ell(V)\) and multiplication on the second factor of \(S\otimes S\). Thus, in light of \eqref{eq:clpm-spm}, it restricts to isomorphisms \begin{equation}\label{eq:cl+cl-} \mathds{C}\ell^+(V)\cong S\otimes S^+\quad \text{ and }\quad\mathds{C}\ell^-(V)\cong S\otimes S^-. \end{equation} As a vector space, the Clifford algebra \(\mathds{C}\ell(V)\) is isomorphic to the complexified exterior algebra \(\wedge^*_\mathds{C} V=\bigoplus_p \wedge^p V\otimes_\mathds{R} \mathds{C}\), via the linear map given on orthonormal basis elements by \(e_{i_1}\dots e_{i_p}\mapsto e_{i_1}\wedge\dots\wedge e_{i_p}\). This is a \emph{$\mathds{Z}_2$-graded} isomorphism: the natural splitting \(\mathds{C}\ell(V)=\mathds{C}\ell^0(V)\oplus \mathds{C}\ell^1(V)\) arising from the $\mathds{Z}_2$-grading of $\mathds{C}\ell(V)$ is mapped to the splitting $\wedge^*_\mathds{C} V=\wedge^{\mathrm{even}}_\mathds{C} V\oplus \wedge^{\mathrm{odd}}_\mathds{C} V$ into exterior powers of even and odd degrees. The action of \(\mathds{C}\ell^0(V)\cong \wedge^{\mathrm{even}}_\mathds{C} V\) on $S$ preserves \(S^\pm\) while that of \(\mathds{C}\ell^1(V)\cong \wedge^{\mathrm{odd}}_\mathds{C} V\) interchanges these subspaces. Since the $\mathds{Z}_2$-graded isomorphism \(\wedge^*_\mathds{C} V\to \mathds{C}\ell(V)\) conjugates the duality isomorphism \(\wedge^pV\to \wedge^{2n-p} V\) given by $(\sqrt{-1})^{p(p-1)+n}\,*$, where \(*\) is the Hodge star operator, and left multiplication by \(\omega_\mathds{C}\) in \(\mathds{C}\ell(V)\), it follows that \begin{align*} \wedge_\mathds{C}^{\mathrm{even}} V&=\wedge_\mathds{C}^{+,{\mathrm{even}}} V \oplus \wedge_\mathds{C}^{-,{\mathrm{even}}} V, && \wedge_\mathds{C}^{\mathrm{odd}} V=\wedge_\mathds{C}^{+,{\mathrm{odd}}} V\oplus \wedge_\mathds{C}^{-,{\mathrm{odd}}} V, \\ \mathds{C}\ell^0(V)&=\mathds{C}\ell^{+,0}(V) \oplus \mathds{C}\ell^{-,0}(V), && \mathds{C}\ell^1(V)=\mathds{C}\ell^{+,1}(V)\oplus \mathds{C}\ell^{-,1}(V), \end{align*} where vertically aligned spaces are isomorphic, i.e., \begin{equation}\label{decompositions} \begin{aligned} \wedge_\mathds{C}^{\pm,{\mathrm{even}}}V &\cong \mathds{C}\ell^{\pm,0}(V)\cong S^\pm \otimes S^\pm,\\ \wedge_\mathds{C}^{\pm,{\mathrm{odd}}}V &\cong \mathds{C}\ell^{\pm ,1}(V)\cong S^\mp \otimes S^\pm , \end{aligned} \end{equation} and this notation is compatible with \eqref{eq:cl+cl-}, i.e., $\mathds{C}\ell^\pm(V)=\mathds{C}\ell^{\pm,0}(V)\oplus\mathds{C}\ell^{\pm,1}(V)$. \subsection{Spin group} The double cover of the special orthogonal group $\mathsf{SO}(V)$ is the \emph{spin group} $\mathsf{Spin}(V)$, and it can be realized as $\mathsf{Spin}(V)\subset \mathds{C}\ell^0(V)$, see \cite[Chap.~I]{lm-book} for details. Thus, its Lie algebra is isomorphic to the Lie algebra $\mathfrak{so}(V)$ of $\mathsf{SO}(V)$, which is identified with $\wedge^2 V$ as usual, i.e., $(x\wedge y)(\cdot)=\langle x,\cdot \rangle y-\langle y,\cdot\rangle x$ corresponds to an infinitesimal rotation in the $2$-plane of $V$ spanned by $x$ and $y$. The inverse of the Lie algebra isomorphism $\Xi_0$ induced by the double cover \(\mathds{C}\ell^0(V)\supset\mathsf{Spin}(V)\to \mathsf{SO}(V)\) is the map $\Xi^{-1}_0 \colon \wedge^2V\cong \mathfrak{so}(V)\to\mathfrak{spin}(V)\subset \mathds{C}\ell^0(V)$ given on $x\wedge y$, where $x,y\in V$ are orthogonal vectors, by \begin{equation}\label{coverderiv} \Xi^{-1}_0(x\wedge y)=\tfrac12 xy, \end{equation} cf.~\cite[Prop.~I.6.2]{lm-book}. Note that \(\Xi_0^{-1}\) differs by a factor of \(\tfrac12\) from the restriction to $\wedge^2 V$ of the isomorphism \(\wedge^*_\mathds{C} V\to \mathds{C}\ell(V)\) mentioned above, for which $x\wedge y\mapsto xy$. \subsection{Spinor bundles and Dirac operators} Let $(M^{2n},\mathrm g)$ be an oriented Riemannian manifold of dimension $2n$, and denote by \(\nabla^\text{LC}\) its Levi-Civita connection on \(TM\). Applying the above constructions pointwise, i.e., to each tangent space $V=T_pM$, $p\in M$, we obtain the \emph{spinor bundle} $S(TM)$ over $M$ and analogous isomorphisms and splittings compatible with the natural connections induced by \(\nabla^\text{LC}\) on each of these bundles. In particular, we note that \(\omega_\mathds{C}\) is parallel. Once again, to simplify notation, we write $S:=S(TM)$ and $S^\pm:=S^\pm(TM)$ if the Riemannian manifold $(M,\mathrm g)$ is clear from the context, as well as $S_\mathrm g$ and~$S^\pm_\mathrm g$, respectively, to indicate the Riemannian metric $\mathrm g$ being used, when necessary. The connection on \(S_\mathrm g\) induced by \(\nabla^\text{LC}\) is obtained applying the map \eqref{coverderiv} to the connection forms. Namely, given a local orthonormal frame \(\{e_1,\dots,e_{2n}\}\) of $TM$ we can choose a local frame for \(S\) such that the connection \(\nabla^S\) on \(S\) is given by \begin{equation*} \nabla^S_{v}=\mathrm d+\sum_{i<j}\mathrm g\big(\nabla^\text{LC}_{v}e_i,e_j\big)\frac{e_ie_j}{2}. \end{equation*} The curvature tensor \(R^S\colon TM\times TM\to\wedge^2 S\subset \operatorname{End}(S)\) of \(\nabla^S\) is given by \begin{equation}\label{spinorcurv} R^S_{x,y}=\Xi_0^{-1}\circ R_{x,y}=\sum_{i<j} \mathrm g\big( R_{x,y}(e_i),e_j \big)\frac{e_i e_j}{2}, \end{equation} where \(R\colon TM\times TM\to\wedge^2TM\subset \operatorname{End}(TM)\) is the curvature tensor of \(\nabla^\text{LC}\). The \emph{Dirac operator} is the first-order differential operator on sections of $S$ given~by \begin{equation*} D(\phi)=\sum_{i=1}^{2n}e_i\nabla^S_{e_i}\phi. \end{equation*} More generally, if \(E\) is a complex vector bundle over \(M\), with a connection \(\nabla^E\), we can consider the \emph{spinor bundle twisted by} $E$, which is the bundle \(S\otimes E\) endowed with the tensor product connection \(\nabla\) and Clifford multiplication on the \(S\) factor, which makes it a Clifford bundle with a natural \emph{twisted Dirac operator} \(D_E\). Similarly to the above, \(D_E\) acts on a decomposable local section \(\phi\otimes \varepsilon\) of \(S\otimes E\) by \begin{equation}\label{eq:twisted-Dirac} D_E(\phi\otimes \varepsilon)=\sum_{i=1}^{2n}(e_i\nabla^S_{e_i}\phi)\otimes \varepsilon+(e_i \phi)\otimes \nabla^E_{e_i}\varepsilon. \end{equation} The \emph{Bochner--Lichnerowicz--Weitzenb\"ock formula}, cf.~\cite[Thm.~II.8.17]{lm-book}, relates $D_E^2$ and the connection Laplacian $\nabla^*\nabla$ acting on sections of $S\otimes E$ as follows \begin{equation}\label{twistlich} D^2_E=\nabla^*\nabla+\tfrac{1}{4}\mathrm{scal}(\mathrm g)+\sum_{i<j}e_ie_j\otimes R^E_{e_i,e_j}, \end{equation} where \(R^E\colon TM \times TM\to \wedge^2 E\subset \operatorname{End}(E)\) is the curvature tensor of \(\nabla^E\), and $\mathrm{scal}(\mathrm g)$ is the scalar curvature of $(M,\mathrm g)$. For example, if $E$ is the trivial bundle, one recovers the well-known formula $D^2=\nabla^*\nabla+\tfrac14\mathrm{scal}(\mathrm g)$ for sections of $S$; while if $E=S$, the twisted Dirac operator $D_S$ on $S\otimes S$ is conjugate to \(\mathrm d+\mathrm d^*\) acting on \(\wedge_\mathds{C}^* TM^*\cong \mathds{C}\ell(TM) \cong S\otimes S\), via the isomorphisms above. \section{Pointwise Inequalities}\label{pi} In this section, we analyze algebraic properties and provide estimates for two types of curvature terms: the last term $\mathcal R$ in the Bochner--Lichnerowicz--Weitzenb\"ock formula \eqref{twistlich} if $E=f^*(S(TM))$ is the pullback by $f\colon N\to M$ of the spinor bundle of $M$, and (a modification of) the curvature term $\mathcal T$ in the Weitzenb\"ock formula for the Hodge Laplacian on differential forms on $M$. This is done pointwise, so we work with oriented real inner product spaces \(V\) and \(W\) of the same dimension, a linear map $l\colon W\to V$ which encodes $\mathrm d f$, and the induced map $L=\wedge^2 l$ as in \eqref{eq:L}. \begin{definition}\label{def:RT} For every \(R\in\operatorname{Sym}^2(\wedge^2 V)\) and $L\in\operatorname{Hom}(\wedge^2 W,\wedge^2 V)$, we define two elements in the space of endomorphisms $\operatorname{End}\!\big(S(W)\otimes S(V)\big)$ by means of \begin{align*} \mathcal{R}(R,L)&:=-2\sum_{i}\beta_i\otimes R(L(\beta_i)),\\ \mathcal{T}(R,L)&:=-\sum_i\big(L^*(\alpha_i)\otimes1+1\otimes \alpha_i\big)\circ\big(L^*(R(\alpha_i))\otimes 1+1\otimes R(\alpha_i)\big), \end{align*} where \(\{\alpha_i\}\) and \(\{\beta_i\}\) are orthonormal bases of \(\wedge^2 V\) and \(\wedge^2 W\), respectively, and \begin{equation}\label{eq:inclusions} \wedge^2 V \subset \operatorname{End}\!\big(S(V)\big) \quad \text{ and } \quad \wedge^2 W \subset \operatorname{End}\!\big(S(W)\big) \end{equation} via the respective actions of $\wedge^2 V$ and $\wedge^2 W$ on $S(V)$ and $S(W)$, determined by the map \(\Xi_0^{-1}\) in \eqref{coverderiv}. Moreover, we canonically identify $\operatorname{End}\!\big(S(W)\big) \otimes \operatorname{End}\!\big(S(V)\big)$ and $\operatorname{End}\!\big(S(W)\otimes S(V)\big)$, and $\circ$ in the definition of $\mathcal T(R,L)$ is composition in the latter. \end{definition} The endomorphisms $\mathcal{R}(R,L)$ and $\mathcal{T}(R,L)$ of $S(W)\otimes S(V)$ do not depend on the choices of \(\{\alpha_i\}\) and \(\{\beta_i\}\). Indeed, identifying $\wedge^2 W\otimes \wedge^2 V = \operatorname{Hom}(\wedge^2 W,\wedge^2 V)$, and considering $R\in\operatorname{Sym}^2(\wedge^2 V)\subset\operatorname{End}(\wedge^2 V)=\wedge^2 V\otimes\wedge^2 V$, we have that \begin{equation*} \mathcal R(R,L) = -2\, R\circ L\quad \text{ and }\quad \mathcal T(R,L)=-c((T_L\otimes T_L)(R)), \end{equation*} where $T_L\colon \wedge^2 V\to\operatorname{End}(S(W)) \otimes \operatorname{End}(S(V))$ is given by $T_L(\alpha)=(L^*(\alpha)\otimes 1+1\otimes \alpha)$, and $c$ is the composition $c(A\otimes B)=A\circ B$ for all $A,B\in \operatorname{End}\!\big(S(W)\otimes S(V)\big)$. Clearly, the maps $R\mapsto \mathcal{R}(R,L)$, $L\mapsto \mathcal R(R,L)$, and $R\mapsto \mathcal{T}(R,L)$ are linear. As elements of $\mathds{C}\ell(W)\otimes\mathds{C}\ell(V)\cong \operatorname{End}(S(W))\otimes \operatorname{End}(S(V))$, both $\mathcal R(R,L)$ and $\mathcal T(R,L)$ have even degree, i.e., belong to $\mathds{C}\ell^0(W)\otimes\mathds{C}\ell^0(V)$, and hence, as endomorphisms, they restrict to endomorphisms of $S^+(W)\otimes S^+(V)$ and of $S^-(W)\otimes S^-(V)$. \begin{lemma}\label{rt} For all algebraic curvature operators \(R\in\operatorname{Sym}_b^2(\wedge^2V)\), we have \begin{equation*} \mathcal{R}(R,L)=\mathcal{T}(R,L)-\tfrac14\operatorname{tr}(L^*\circ R\circ L)-\tfrac18\mathrm{scal}_{R}. \end{equation*} \end{lemma} \begin{proof} By \Cref{lemma:diag}, we may choose orthonormal bases \(\{\alpha_i\}\) of $\wedge^2V$ and \(\{\beta_i\}\) of $\wedge^2 W$ such that \(L(\beta_i)=\lambda_i\alpha_i\) for some \(\lambda_i\geq0\). Note that $L^*(\alpha_i)=\lambda_i\beta_i$. Since \(R\) is symmetric, we may write $R(\alpha_i)=\sum_j R_{ij}\alpha_j$ with $R_{ij}=R_{ji}$, and hence \begin{align*} \mathcal{R}(R,L)&=-\sum_i \beta_i\otimes R(\lambda_i\alpha_i)-\sum_i \beta_i\otimes \lambda_i \sum_j R_{ij}\alpha_j \\ &=-\sum_i \lambda_i\beta_i\otimes R(\alpha_i)-\sum_j \sum_i R_{ji} \lambda_i \beta_i\otimes \alpha_j \\ &=-\sum_i L^*(\alpha_i)\otimes R(\alpha_i)-\sum_j L^*(R(\alpha_j))\otimes \alpha_j\\ &=\mathcal{T}(R,L)+\sum_i L^*(\alpha_i)\circ L^*(R(\alpha_i))\otimes 1+\sum_i 1\otimes \alpha_i\circ R(\alpha_i). \end{align*} A routine argument using the symmetries of Clifford multiplication and of the curvature operator $R$, including the Bianchi identity, see \cite[Thm.~II.8.8]{lm-book}, implies \begin{equation*} \sum_i \alpha_i\circ R(\alpha_i)=-\tfrac14\operatorname{tr}(R)=-\tfrac18\mathrm{scal}_{R}. \end{equation*} Since \(L^*\circ R\circ L\in\operatorname{Sym}^2(\wedge^2 W)\) has the same symmetries as \(R\), the above also implies \begin{equation*} \sum_i L^*(\alpha_i)\circ L^*(R(\alpha_i))=\sum_i\beta_i\circ(L^*\circ R\circ L(\beta_i))=-\tfrac14\operatorname{tr}(L^*\circ R\circ L).\qedhere \end{equation*} \end{proof} We now estimate the terms in the right-hand side of the identity in \Cref{rt}. The following two lemmas were observed in~\cite[Sec~1.1]{goette-semmelmann2}, but, for completeness, we supply their proofs below using our notations. \begin{lemma}\label{fscal} If \(R\in\operatorname{Sym}^2_b(\wedge^2V)\) is an algebraic curvature operator with $\sec_R\geq0$ and $l\colon W\to V$ is a linear map such that $L=\wedge^2 l$ is nonincreasing, then \begin{equation*} \operatorname{tr}(L^*\circ R\circ L)\leq\tfrac12\mathrm{scal}_{R}. \end{equation*} If, in addition, \(\tfrac12\mathrm{scal}_{R} \succ \operatorname{Ric}_{R}\succ 0\), then equality above implies \(l\) is an isometry. \end{lemma} \begin{proof} By \Cref{lemma:diag}, we may choose orthonormal bases \(\{v_i\}\) of \(V\) and \(\{w_i\}\) of \(W\) such that \(l(w_i)=\lambda_i v_i\) for some \(\lambda_i\geq0\). Then \(\{w_i\wedge w_i\}_{i<j}\) is an orthonormal basis of $\wedge^2 W$ and $(R\circ L)(w_i\wedge w_j)=\lambda_i\lambda_j \, R(w_i\wedge w_j)$ for all $i<j$, so \begin{equation*} \operatorname{tr}(L^*\circ R\circ L)=\sum_{i<j}\lambda_i^2\lambda_j^2\, \sec_{R}(v_i\wedge v_j). \end{equation*} Since \(L\) is nonincreasing, \(\lambda_i\lambda_j\leq1\) for all \(i < j\), which proves the desired inequality. If, in addition, \(\tfrac12\mathrm{scal}_{R} \succ \operatorname{Ric}_{R}\succ 0\), and $\operatorname{tr}(L^*\circ R\circ L)=\tfrac12\mathrm{scal}_{R}$, then \begin{equation}\label{scaleq} \sum_{i<j}(1-\lambda_i^2\lambda_j^2)\,\sec_R(v_i\wedge v_j)=0. \end{equation} Since \(\operatorname{Ric}_{R}\prec \tfrac12\mathrm{scal}_{R}\), for each fixed \(a\), \(\sum_{b}\sec_R(v_a\wedge v_b)<\sum_{i<j}\sec_R(v_i\wedge v_j)\), or equivalently, \begin{equation*} 0<\sum_{\substack{i<j\\ i,j\neq a}}\sec_R(v_i\wedge v_j). \end{equation*} Since $\sec_R\geq0$, this implies that there exists \(i,j\neq a\) such that \(\sec_R(v_i\wedge v_j)>0\). Together with \eqref{scaleq}, this implies \(\lambda_i\lambda_j=1\). But \(\lambda_i\lambda_a\leq1\) and \(\lambda_j\lambda_a\leq1\), so we conclude that \(\lambda_a\leq1\) for all \(a\), i.e., \(l\) is nonincreasing. Since \(\operatorname{Ric}_{R}\succ 0,\) for each \(a\) there exists \(b\neq a\) such that \(\sec_R(v_a\wedge v_b)>0.\) Again it follows that \(\lambda_a\lambda_b=1\), and we conclude that \(\lambda_a=\lambda_b=1,\) so \(l\) is an isometry. \end{proof} \begin{lemma}\label{nneg} If $R\in\operatorname{Sym}^2(\wedge^2 V)$ is such that \(R\succeq0\), then \(\mathcal{T}(R,L)\succeq0\). \end{lemma} \begin{proof} Choose an orthonormal basis \(\alpha_i\) of $\wedge^2 V$ that diagonalizes $R$, i.e., such that \(R(\alpha_i)=\rho_i\,\alpha_i\). Since $R\succeq0$, we have that \(\rho_i\geq0\). Then, by \Cref{def:RT}, \begin{equation*} \mathcal{T}(R,L)=-\sum_i\rho_i\left(L^*(\alpha_i)\otimes 1+1\otimes \alpha_i\right)^2. \end{equation*} Since $\alpha_i\in\mathfrak{spin}(V)$ and $L^*(\alpha_i)\in\mathfrak{spin}(W)$, i.e., these are elements of $\mathds{C}\ell^0(V)$ and $\mathds{C}\ell^0(W)$ in the images of the corresponding isomorphisms \eqref{coverderiv}, the endomorphisms $L^*(\alpha_i)\otimes 1+1\otimes \alpha_i$ of $S(W)\otimes S(V)$ are skewsymmetric, so the conclusion follows. \end{proof} Let us now assume that both $V$ and $W$ are $4$-dimensional. We denote the Hodge star operators of $V$ and $W$ by \(*^V\in\operatorname{Sym}^2(\wedge^2V)\) and \(*^W\in\operatorname{Sym}^2(\wedge^2W)\), and similarly for \(\omega_\mathds{C}^V\in\mathds{C}\ell(V)\) and \(\omega_\mathds{C}^W\in\mathds{C}\ell(W)\). Recall from \Cref{subsec:CAandS} and \eqref{coverderiv} that \(\Xi_0^{-1}(*(x\wedge y))=\tfrac12\,\omega_\mathds{C} \,xy\) if $x$ and $y$ are orthogonal. \begin{lemma}\label{star} If the linear map \(l\colon W\to V\) is such that $L=\wedge^2l$ is nonincreasing and \(\dim_\mathds{R} V=\dim_\mathds{R} W=4\), then the restriction of \(\mathcal{T}(*^V,L)\) to \(S^+(W)\otimes S^+(V)\) is positive-semidefinite. \end{lemma} \begin{proof} By \Cref{lemma:diag}, we may choose orthonormal bases \(\{v_i\}\) of \(V\) and \(\{w_i\}\) of \(W\) such that \(l(w_i)=\lambda_i v_i\) for some \(\lambda_i\geq0\). The assumption on \(l\) ensures that \(\lambda_i\lambda_j\leq1\) for all \(i\neq j\). We first note that, with these choices, \begin{equation*} L^*\big(*^V(v_i\wedge v_j)\big)=\mu_{ij}*^W (w_i\wedge w_j), \end{equation*} where \(|\mu_{ij}|=\big\|L^* (*^V (v_i\wedge v_j))\big\|\leq1\), since \(*^V\) is an isometry of $\wedge^2 V$. Symmetries of Clifford multiplication in \(\mathds{C}\ell^{0}(W)\otimes \mathds{C}\ell^{0}(V)\cong\operatorname{End}\!\big(S^+(W)\otimes S^+(V)\big)\) and the fact that $\Xi_0^{-1}(*(x\wedge y))=\tfrac12\,\omega_\mathds{C} \,xy$ if $x$ and $y$ are orthonormal imply that: \begin{align*} L^*\big(*^V (v_i\wedge v_j)\big)\otimes1+1\otimes*^V (v_i\wedge v_j) &=\tfrac12\left(\mu_{ij}\,\omega^W_\mathds{C} w_iw_j\otimes 1+1\otimes\omega^V_\mathds{C} v_i v_j\right)\\ &=\tfrac12\left(\mu_{ij}\,w_iw_j\omega^W_\mathds{C}\otimes 1+1\otimes v_i v_j\omega^V_\mathds{C}\right)\\ &=\tfrac12\left(\mu_{ij}\,w_iw_j\otimes 1+1\otimes v_iv_j\right), \end{align*} where the last equality holds because the above endomorphisms are restricted to the tensor product $S^+(W)\otimes S^+(V)$ of $+1$-eigenspaces of $\omega_\mathds{C}^W$ and $\omega_\mathds{C}^V$. Therefore, \begin{align*} \mathcal{T}(*^V,L) &=-\tfrac14\sum_{i<j}\left(\lambda_i\lambda_j\, w_iw_j\otimes 1+1\otimes v_iv_j\right)\circ\left(\mu_{ij}\, w_iw_j\otimes 1+1\otimes v_iv_j\right)\\ &=\tfrac14\sum_{i<j}\left(-\lambda_i\lambda_j+w_iw_j\otimes v_iv_j\right)\circ\left(-\mu_{ij}+w_iw_j\otimes v_iv_j\right), \end{align*} and, since \(\left(w_iw_j\otimes v_iv_j\right)^2=1\), and \(|\lambda_i\lambda_j|\leq1\) as well as \(|\mu_{ij}|\leq1\), we conclude that each of the above summands is positive-semidefinite. \end{proof} \begin{remark} Under the hypotheses of \Cref{star}, it also follows that the restriction of $\mathcal T(*^V,L)$ to $S^-(W)\otimes S^-(V)$ is \emph{negative}-semidefinite. \end{remark} \section{Extremality and rigidity on closed 4-manifolds}\label{4closed} In this section, we prove \Cref{mainthm1} and Corollary \ref{maincor:CP2CP2} in the Introduction, by showing that a certain twisted Dirac operator has nontrivial kernel and using the results of \Cref{pi} to analyze the curvature term in the corresponding Bochner--Lichnerowicz--Weitzenb\"ock formula. We begin by recalling and generalizing the notions of area-extremality and area-rigidity for scalar curvature discussed in the Introduction to also account for ``topologically modified'' competitors, cf.~Gromov~\cite[Sec.~4]{gromov-dozen} and \cite[Sec.~$5\tfrac49$]{gromov-96}. \begin{definition}\label{def:extremality-global} A closed oriented Riemannian manifold $(M,\mathrm g_M)$ is \emph{area-extremal with respect to a class $\mathcal C=\{ f\colon (N,\mathrm g_N)\to (M,\mathrm g_M) \}$} of competitors, consisting of closed oriented Riemannian manifolds $(N,\mathrm g_N)$ with $\dim M=\dim N$ and smooth spin maps $f\colon N\to M$ of nonzero degree, if the inequalities \begin{equation}\label{eq:scal-comp} \wedge^2\mathrm g_N\succeq f^*\wedge^2 \mathrm g_M \qquad \text{and}\qquad \mathrm{scal}(\mathrm g_N)\geq \mathrm{scal}(\mathrm g_M)\circ f \end{equation} imply $\mathrm{scal}(\mathrm g_N)=\mathrm{scal}(\mathrm g_M)\circ f$. If, in addition, there exists $q\in M$ such that \eqref{eq:scal-comp} implies $\mathrm d f(p)\colon T_pN\to T_{q}M$ is a linear isometry for all competitors $f\colon N\to M$ in $\mathcal C$ and all $p\in f^{-1}(q)$, then $(M,\mathrm g_M)$ is called \emph{area-rigid at $q\in M$ with respect to $\mathcal C$}. If $(M,\mathrm g_M)$ is area-rigid at all of its points with respect to $\mathcal C$, then it is simply called \emph{area-rigid with respect to $\mathcal C$.} \end{definition} Recall that a smooth map \(f\colon N\to M\) is \emph{spin} if it is compatible with second Stiefel--Whitney classes, i.e., $f^*w_2(M)=w_2(N)$, and $\wedge^2\mathrm g_N\succeq f^* \wedge^2 \mathrm g_M$ means that $f\colon (N,\mathrm g_N)\to (M,\mathrm g_M)$ is area-nonincreasing, i.e., $\wedge^2 \mathrm d f$ is nonincreasing, namely \begin{equation*} \big\|x\wedge y\big\|_{\mathrm g_N} \geq \big\|\mathrm d f(p) x\wedge \mathrm d f(p) y\big\|_{\mathrm g_M} \end{equation*} for all $x,y\in T_{p}N$ and all $p\in N$. For example, this holds if $f\colon (N,\mathrm g_N)\to (M,\mathrm g_M)$ is distance nonincreasing, see \Cref{subsec:linalgebra}. \begin{remark}\label{rem:f=id} The notions of area-extremality and area-rigidity for closed manifolds in the Introduction correspond to using the class $\mathcal C^{\operatorname{id}}_0:=\{\operatorname{id}\colon (M,\mathrm g_1)\to (M,\mathrm g_0)\}$ in \Cref{def:extremality-global}, cf.~\eqref{eq:g1-competitor-global}. Note that competitors given by any diffeomorphisms $f\colon (M,\mathrm g_1)\to (M,\mathrm g_0)$ reduce to the above case, pulling back $\mathrm g_1$ by $f^{-1}$. \end{remark} \subsection{Index theory}\label{indextheory} Let $(M,\mathrm g_M)$ and $(N,\mathrm g_N)$ be closed oriented Riemannian $4$-manifolds, and denote by \(S(TM)\) and \(S(TN)\) their (locally defined) spinor bundles. Given a spin map $f\colon N\to M$, the twisted spinor bundle \(S(TN)\otimes f^*S(TM)\) is globally defined; namely, it is the spinor bundle of the spin bundle \(TN\oplus f^*TM\). Let \(E=f^*S^+(TM)\), and consider the twisted Dirac operator \begin{equation}\label{eq:DE-global} D_E\colon \Gamma(S(TN)\otimes E)\longrightarrow\Gamma(S(TN)\otimes E). \end{equation} Recall from \eqref{eq:twisted-Dirac} that, if we denote by $\nabla^{S_N}$ and $\nabla^{S_M}$ the connections on $S(TN)$ and $S(TM)$ respectively, and by \(\{e_i\}\) a local $\mathrm g_N$-orthonormal frame for \(TN\), then on a decomposable local section \(\phi\otimes f^*\psi\) of \(S(TN)\otimes E\), the operator \(D_E\) acts as \[D_E(\phi\otimes \psi)=\sum_{i=1}^4 \left(e_i\nabla^{S_N}_{e_i}\phi\right)\otimes f^*\psi+(e_i\phi)\otimes f^*\left(\nabla^{S_M}_{\mathrm d f(e_i)}\psi\right) .\] We respectively denote by \(\chi(X)\) and \(\sigma(X)\) the Euler characteristic and signature of an oriented $4$-manifold \(X\), and by $\deg(f)$ the degree of $f\colon N\to M$. \begin{lemma}\label{kernel2} If $f\colon N\to M$ has $\deg(f)\neq0$ and \begin{equation}\label{eq:nontrivialKerDE} 2\chi(M)+3\sigma(M)>\frac{\sigma(N)}{\deg(f)}, \end{equation} then the restriction of \(D_E\) to \(\Gamma(S^+(TN)\otimes E)\) has nontrivial kernel. \end{lemma} \begin{proof} If \(TM\otimes\mathds{C}\cong \lambda_1\oplus\overline\lambda_1\oplus\lambda_2\oplus\overline\lambda_2\) is a formal splitting into complex line bundles, then \(S^+(TM)\cong \lambda_1^{\frac12}\otimes \lambda_2^{\frac12}\oplus {\overline \lambda}_1^{\frac12}\otimes{\overline \lambda}_2^{\frac12}\), cf.~\cite[p.~238]{lm-book}. Thus, the Chern character of $S^+(TM)$ is given by \(\mathrm{ch}(S^+(TM))=2+\tfrac14 p_1(TM)+\tfrac12 e(TM)\), where \(p_1\) is the first Pontryagin class and \(e\) is the Euler class. By the Atiyah--Singer Index Theorem, \begin{align*} \text{ind}\big(D_E|_{S^+(TN)\otimes E}\big)&=\left<\hat{A}(TN)\cdot f^*\mathrm{ch}(S^+(TM)),[N]\right>\\&=\left<-\tfrac{1}{12}p_1(TN)+\tfrac14 f^*p_1(TM) +\tfrac12 f^*e(TM),[N]\right>\\&=-\tfrac14\sigma(N)+\text{deg}(f)\left(\tfrac34\sigma(M)+\tfrac12\chi(M)\right)>0, \end{align*} and hence \(\ker\big(D_E|_{S^+(TN)\otimes E}\big)\) is nontrivial. \end{proof} Note that all hypotheses of \Cref{kernel2} are satisfied if $\deg(f)=1$ and $M=N$ has vanishing first Betti number $b_1(M)=0$, e.g., if $M$ is simply-connected, since then $\chi(M)=2+b_+(M)+b_-(M)$ and $\sigma(M)=b_+(M)-b_-(M)$, where $b_\pm(M)$ are the self-dual/anti-self-dual second Betti numbers, so \eqref{eq:nontrivialKerDE} simplifies to $4+4b_+(M)>0$. \subsection{Extremality and Rigidity} We now combine the pointwise inequalities from \Cref{pi} with the Bochner--Lichnerowicz--Weitzenb\"ock formula \eqref{twistlich} for the twisted Dirac operator \eqref{eq:DE-global} and \Cref{kernel2} to prove our main result on area-extremality and area-rigidity of closed $4$-manifolds, in the sense of \Cref{def:extremality-global}. \begin{theorem}\label{thm} A closed oriented Riemannian $4$-manifold \((M,\mathrm g_M)\) with $\sec\geq0$ such that $R+\tau\,*\succeq0$ for a nonpositive $\tau\colon M\to \mathds{R}$ is area-extremal with respect to \begin{equation*} \mathcal C_0:=\left\{f\colon (N,\mathrm g_N)\to (M,\mathrm g_M) \; : \; 2\chi(M)+3\sigma(M)>\frac{\sigma(N)}{\deg(f)}\right\}. \end{equation*} If, in addition, $\frac{\mathrm{scal}(\mathrm g_M)}{2}\mathrm g_M\succ\operatorname{Ric}(\mathrm g_M)\succ0$ at $q\in M$, then $(M,\mathrm g_M)$ is area-rigid at $q\in M$ with respect to $\mathcal C_0$. \end{theorem} \begin{proof} Let $f\colon (N,\mathrm g_N)\to (M,\mathrm g_M)$ be a competitor in $\mathcal C_0$, and let $E=f^*S^+(TM)$. The Bochner--Lichnerowicz--Weitzenb\"ock formula for the square \(D_E^2\) of the twisted Dirac operator \eqref{eq:DE-global} is given by \eqref{twistlich}, setting \( \mathrm g=\mathrm g_N\). From \eqref{eq:curvop-sign} and \eqref{spinorcurv}, \[R^{{S_M}}_{x,y}=-\Xi_0^{-1} \big( R(x\wedge y)\big),\] where $R\colon \wedge^2 TM\to\wedge^2 TM$ is the curvature operator of $(M,\mathrm g_M)$. Working pointwise with $W=T_pN$, $V=T_{f(p)}M$, $l\colon W\to V$ given by $\mathrm d f(p)$, and $L=\wedge^2 l$, we see that \eqref{twistlich} can be written using the endomorphism $\mathcal R(R,L)$ from \Cref{def:RT} as \begin{equation}\label{d2} D_E^2=\nabla^*\nabla+\tfrac{1}{4}\mathrm{scal}(\mathrm g_N)+\mathcal{R}(R, L). \end{equation} Let \(\xi\in\Gamma(S^+(TN)\otimes E)\) be a nonzero section in the kernel of \(D_E\), which exists by \Cref{kernel2}. Applying \eqref{d2} and integrating, we obtain, using \Cref{rt}, \begin{equation}\label{eq:pfeq1} \begin{aligned} 0&=\int_{N} \langle \nabla^*\nabla\xi, \xi\rangle+\tfrac14\int_N \mathrm{scal}(\mathrm g_N)\|\xi\|^2+\int_N\left<\mathcal{R}(R,L)\xi,\xi\right>\\ &=\int_{N}\|\nabla \xi\|^2+\int_N\left<\mathcal{T}(R,L)\xi,\xi\right> \\ &\quad +\tfrac14\int_N\left( \mathrm{scal}(\mathrm g_N)-\tfrac12\mathrm{scal}(\mathrm g_M)\circ f-\tfrac12\operatorname{tr}(L^*\circ R \circ L)\circ f\right)\|\xi\|^2. \end{aligned} \end{equation} Linearity of $R\mapsto \mathcal{T}(R,L)$ and \Cref{nneg,star} imply that, on \(S^+(TN)\otimes E\), \begin{equation}\label{eq:pfeq2} \mathcal{T}(R,L)=\mathcal{T}(R+\tau\, *,L) -\tau \;\mathcal{T}(*,L)\succeq0, \end{equation} since $\tau\leq 0$. Thus, combining \eqref{eq:scal-comp}, \eqref{eq:pfeq1}, and \eqref{eq:pfeq2} with \Cref{fscal}, which may be applied because $\sec_R\geq0$ and $L$ is nonincreasing by \eqref{eq:scal-comp}, we conclude that \begin{align*} 0&\geq\int_{N}\|\nabla \xi\|^2+\tfrac14\int_N\left(\mathrm{scal}(\mathrm g_N)-\tfrac12\mathrm{scal}(\mathrm g_M)\circ f-\operatorname{tr}(L^*\circ R \circ L)\circ f\right)\|\xi\|^2\\ &\geq\int_{N}\|\nabla \xi\|^2+\tfrac14\int_N\left(\mathrm{scal}(\mathrm g_N)-\mathrm{scal}(\mathrm g_M)\circ f\right)\|\xi\|^2\geq0. \end{align*} Therefore, \(\nabla\xi\) vanishes identically and hence \(\|\xi\|\neq0\) is constant, so it follows that $\mathrm{scal}(\mathrm g_N)=\mathrm{scal}(\mathrm g_M)\circ f$ everywhere. Furthermore, \(\operatorname{tr}(L^*\circ R \circ L)=\tfrac12\mathrm{scal}(\mathrm g_M)\), so, if $\frac{\mathrm{scal}(\mathrm g_M)}{2}\mathrm g_M\succ\operatorname{Ric}(\mathrm g_M)\succ0$ at $q=f(p)$, then \(l\) is an isometry by \Cref{fscal}. \end{proof} Let us briefly discuss some situations in which the hypotheses of \Cref{thm} are satisfied. If \((M,\mathrm g_M)\) is simply-connected and its curvature operator $R$ satisfies $R+\tau\,*\succeq0$ with $\tau\leq0$, then, by the proof of \cite[Thm.~D]{bm-iumj}, either \((M,\mathrm g_M)\) is isometric to \(\S^2\times \S^2\) and $\tau\equiv0$, hence it is area-extremal with respect to any class of competitors by \cite{goette-semmelmann2}, or else $M$ has negative-definite intersection form, i.e., $b_+(M)=0$, and hence \(\sigma(M)=-b_-(M)=-b_2(M)\). In this latter case, restricting ourselves to \emph{self-map competitors}, i.e., setting \(N=M\), the class $\mathcal C_0$ simplifies to \begin{equation*} \mathcal C_0^{\mathrm{self}}:=\left\{f\colon (M,\mathrm g_1)\to (M,\mathrm g_0) \; : \; 4+\left(\frac{1}{\deg(f)}-1\right)b_2(M)>0 \right\}. \end{equation*} Clearly, $\mathcal C^{\operatorname{id}}_0\subset\mathcal C_0^{\mathrm{self}}$, and this proves \Cref{mainthm1} in the Introduction, since we may choose the orientation of $M$ for which $\tau\leq 0$ in order to apply \Cref{thm}, see \Cref{rem:f=id}. Moreover, if \(b_2(M)\leq4\), then $\mathcal C_0^{\mathrm{self}}$ contains all (smooth spin) self-maps $f\colon (M,\mathrm g_1)\to (M,\mathrm g_0)$ of nonzero degree. Thus, more generally, we have: \begin{corollary}\label{cor:self-maps} A closed simply-connected Riemannian $4$-manifold $(M,\mathrm g_M)$ whose curvature operator $R$ satisfies $R+\tau\,*\succeq0$ with $\tau\leq0$ is area-extremal with respect to any (smooth spin) self-maps of degree $1$, and (smooth spin) self-maps of arbitrary nonzero degree if \(b_2(M)\leq 4\). If, in addition, $\frac{\mathrm{scal}(\mathrm g_M)}{2}\mathrm g_M\succ\operatorname{Ric}(\mathrm g_M)\succ0$ at $q\in M$, then $(M,\mathrm g_M)$ is area-rigid at $q\in M$ with respect to the same classes of self-maps. \end{corollary} \subsection{Examples}\label{examples} The only closed simply-connected $4$-manifolds currently known to admit metrics with $\sec\geq0$ are $\S^4$, $\mathds{C} P^2$, $\S^2\times \S^2$, $\mathds{C} P^2\#\mathds{C} P^2$, and $\mathds{C} P^2\#\overline{\mathds{C} P^2}$, and this list is conjectured to be exhaustive. We now analyze the existence of families of area-extremal and area-rigid metrics on these manifolds, starting with those known to admit $\sec>0$. \begin{theorem}\label{thm:S4RP4CP2} The spaces of Riemannian metrics on $\S^4$ and $\mathds{C} P^2$ have nonempty open subsets consisting of area-rigid metrics with respect to self-maps of any nonzero degree, that contain the round metric and the Fubini--Study metric, respectively. \end{theorem} \begin{proof} Any metric on \(\S^4\) with $R\succ0$ is area-rigid with respect to self-maps of any nonzero degree by \Cref{cor:self-maps} (taking $\tau\equiv0$), or by \cite{goette-semmelmann2}. In particular, there is a neighborhood of the round metric consisting of such metrics. Let \(\mathrm g_{\mathds{C} P^2}\) be the Fubini--Study metric on \(\mathds{C} P^2\) and denote by $R_{\mathds{C} P^2}$ its curvature operator, with $1\leq \sec\leq 4$. In a self-dual/anti-self-dual basis of $\wedge^2 T_p\mathds{C} P^2$, i.e., a basis that diagonalizes the Hodge star operator as $*_{\mathds{C} P^2}=\mathrm{diag}(1,1,1,-1,-1,-1)$, we have $R_{\mathds{C} P^2}=\mathrm{diag}(0,0,6,2,2,2)$, so $R_{\mathds{C} P^2}+\tau\, *_{\mathds{C} P^2}\succeq0$ if and only if $0\leq \tau\leq 2$. Thus, to apply \Cref{thm}, we must set $M$ to be $\overline{\mathds{C} P^2}$, i.e., $\mathds{C} P^2$ with the opposite orientation, which is isometric to $\mathds{C} P^2$ and hence has the same curvature operator $R_{\overline{\mathds{C} P^2}}=R_{\mathds{C} P^2}$, but $*_{\overline{\mathds{C} P^2}}=-*_{\mathds{C} P^2}$, so $R_{\overline{\mathds{C} P^2}}+\tau\,*_{\overline{\mathds{C} P^2}}\succ0$ if and only if $-2<\tau<0$. We may then take, e.g., $\tau\equiv-1$, and, by continuity, there exists a neighborhood \(\mathcal U\) of \(\mathrm g_{\mathds{C} P^2}\) in the $C^2$-topology formed by metrics $\mathrm g$ that also satisfy $R_\mathrm g+\tau\,*_\mathrm g\succ0$ for the constant function $\tau\equiv-1$. Furthermore, since $\mathrm g_{\mathds{C} P^2}$ is an Einstein metric, \[\frac{\mathrm{scal}(\mathrm g_{\mathds{C} P^2})}{2}\mathrm g_{\mathds{C} P^2}\succ\frac{\mathrm{scal}(\mathrm g_{\mathds{C} P^2})}{4}\mathrm g_{\mathds{C} P^2}=\operatorname{Ric}(\mathrm g_{\mathds{C} P^2})\succ0,\] so, up to shrinking $\mathcal U$, we may assume that all \(\mathrm g\in\mathcal U\) satisfy $\frac{\mathrm{scal}(\mathrm g)}{2}\mathrm g\succ\operatorname{Ric}(\mathrm g)\succ0$. Therefore, since $b_2(\mathds{C} P^2)=1$, it follows from \Cref{cor:self-maps} that \((\mathds{C} P^2,\mathrm g)\) is area-rigid with respect to self-maps of any nonzero degree, for all $\mathrm g\in\mathcal U$. \end{proof} Let us consider the remaining simply-connected $4$-manifolds known to admit $\sec\geq0$. On $\S^2\times \S^2$, any product of metrics on $\S^2$ with $\sec>0$ has $R\succeq0$ and $\frac{\mathrm{scal}(\mathrm g)}{2}\mathrm g \succ\operatorname{Ric}(\mathrm g)\succ0$, thus it is area-rigid with respect to self-maps of any nonzero degree by \Cref{cor:self-maps}; or, in a more general sense, by \cite{goette-semmelmann2}. On \({\mathds{C}}P^2\#\overline{{\mathds{C}}P}^2\) and \({\mathds{C}}P^2\#{\mathds{C}}P^2\), metrics with $\sec\geq0$ were constructed by Cheeger \cite{cheeger}, using a gluing method that endows the connected sum $M_1\# M_2$ of any two (oriented) compact $n$-dimensional rank one symmetric spaces with $\sec\geq0$. These \emph{Cheeger metrics} contain a \emph{neck} region isometric to a round cylinder $\S^{n-1}\times [0,\ell]$, to which the complement of a ball in each $M_i$ is glued. The length $\ell\geq0$ of the neck can be chosen arbitrarily, including $\ell=0$. There do not exist metrics on \({\mathds{C}}P^2\#\overline{{\mathds{C}}P}^2\) that satisfy the hypotheses of \Cref{thm}, as a consequence of \cite[Thm.~D]{bm-iumj}. However, \({\mathds{C}}P^2\#\overline{{\mathds{C}}P}^2\) admits K\"ahler metrics of positive Ricci curvature \cite{yau-calabiconj}, which are area-rigid by \cite{goette-semmelmann1}. Meanwhile, existence of area-extremal or area-rigid metrics on \({\mathds{C}}P^2\#{\mathds{C}}P^2\) cannot be established with any previous results, since they either require metrics that are K\"ahler or have positive-semidefinite curvature operator, and \({\mathds{C}}P^2\#{\mathds{C}}P^2\) admits neither. We overcome these restrictions as follows: \begin{theorem}\label{thm:cheeger-metrics} The Cheeger metrics on \({\mathds{C}}P^2\#{\mathds{C}}P^2\) are area-extremal with respect to self-maps of any nonzero degree, and area-rigid with respect to these maps on all points in the complement ${\mathds{C}}P^2\#{\mathds{C}}P^2 \setminus (\S^3\times [0,\ell])$ of the neck region. \end{theorem} \begin{proof} The manifold \({\mathds{C}}P^2\#{\mathds{C}}P^2\) is obtained gluing together two equally oriented copies of the normal disk bundle $\nu(\mathds{C} P^1)$ of $\mathds{C} P^1\subset\mathds{C} P^2$ along their boundary $\S^3$, with an orientation-reversing diffeomorphism $\alpha$. Since bi-invariant metrics on $\mathsf{SU}(2)\cong\S^3$ are round (in particular, they support orientation-reversing isometries), Cheeger metrics on each copy of $\nu(\mathds{C} P^1)$ are isometric to the metrics with $\sec\geq0$ constructed by Grove and Ziller~\cite{grove-ziller-annals} on $\nu(\mathds{C} P^1)\cong \mathsf G\times_{\mathsf K} D^2$ that are invariant under the action of $\mathsf G=\mathsf{SU}(2)$ with orbit space $\nu(\mathds{C} P^1)/\mathsf G=\left[0,r_{\mathrm{max}}\right]$, where $0$ corresponds to the singular orbit with isotropy $\mathsf K=\S^1$ and all other (principal) orbits have trivial isotropy $\mathsf H=\{1\}$. A detailed treatment of Grove--Ziller metrics on a related disk bundle $\mathsf G'\times_{\mathsf K'} D^2$ over $\mathds{C} P^1=\mathsf G/\mathsf K=\mathsf G'/\mathsf K'$ is given in \cite{bk-rf}, where $\mathsf G'=\mathsf{SO}(3)$, $\mathsf K'=\mathsf{SO}(2)_{1,2}$, and $\mathsf H'=\mathds{Z}_2$, including the explicit computation of their curvature operator, see \cite[Prop.~3.5]{bk-rf}. This computation applies mutatis mutandis to $\mathsf G\times_{\mathsf K} D^2$, as the Lie groups $(\mathsf G,\mathsf K,\mathsf H)$ and $(\mathsf G',\mathsf K',\mathsf H')$ have isomorphic Lie algebras. Namely, one must only replace the constant $\rho(b)=b/2$ with $\rho(b)=2b$ when passing from $\mathsf G'\times_{\mathsf K'} D^2$ to $\mathsf G\times_{\mathsf K} D^2$. Then \cite[Prop.~3.5]{bk-rf} implies that, in a basis induced by an orthonormal frame, the curvature operator $R$ of a Grove--Ziller metric $\mathrm g$ on $\nu(\mathds{C} P^1)\cong \mathsf G\times_{\mathsf K} D^2$ is block diagonal, i.e., $R=\mathrm{diag}(R_1,R_2,R_3)$, where \begin{equation}\label{eq:curvopGZ} R_1 = \begin{bmatrix} \frac{4b^2 - 3\varphi^2}{4b^4} & -\frac{\varphi'}{b^2} \\ -\frac{\varphi'}{b^2} & -\frac{\varphi''}{\varphi} \end{bmatrix}, \quad R_2 = R_3 = \begin{bmatrix} \frac{\varphi^2}{4b^4} & \frac{\varphi'}{2b^2} \\ \frac{\varphi'}{2b^2} & 0 \end{bmatrix}, \end{equation} $b>0$ is an arbitrary constant, $\varphi\colon \left[0,r_{\mathrm{max}}\right]\to\mathds{R}$ is a nonnegative smooth function satisfying $\varphi(0)=0$, $\varphi'(0)=1/2$, $\varphi^{(2k)}(0)=0$ for all $k\in\mathds N$, $\varphi'(r) \geq 0$ for all $r\in \left[0,r_{\mathrm{max}}\right]$, $\varphi''(r)\leq 0$ for all $r\in\left[0,r_{\mathrm{max}}\right]$, and $\varphi(r)\equiv b$ for all $r\in \left[r_0,r_{\mathrm{max}}\right]$, with $0<r_0\leq r_{\mathrm{max}}$ and $r_0$ chosen sufficiently large. Clearly, all data on $(\nu(\mathds{C} P^1),\mathrm g)$ which is invariant under isometries, such as $R$, is determined by its value along a unit speed horizontal geodesic parametrized on $\left[0,r_{\mathrm{max}}\right]$, and $(\nu(\mathds{C} P^1),\mathrm g)$ is isometric to the round cylinder $\S^3(2b)\times [r_0,r_{\mathrm{max}}]$ near its boundary, where $\S^3(2b)$ is a round $3$-sphere of radius~$2b$. Thus, gluing two copies of $(\nu(\mathds{C} P^1),\mathrm g)$ along $\S^3(2b)$ using an orientation-reversing isometry $\alpha$ produces a smooth metric on \({\mathds{C}}P^2\#{\mathds{C}}P^2=\nu(\mathds{C} P^1)\cup_{\alpha} \nu(\mathds{C} P^1)\), which we also denote by $\mathrm g$. The preimage of $\left[r_0,r_{\mathrm{max}}\right]$ under the projection map $\nu(\mathds{C} P^1)\to \left[0,r_{\mathrm{max}}\right]$ is half of the neck region in \({\mathds{C}}P^2\#{\mathds{C}}P^2\); in particular, the neck region is isometric to $\S^3(2b)\times [0,\ell]$, where $\ell=2(r_{\mathrm{max}}-r_0)\geq0$. Note that while $r_0>0$ must be chosen sufficiently large in order for $\varphi$ as above to exist, the value of $r_{\mathrm{max}}\geq r_0$ is arbitrary. The matrix of the Hodge star operator $*$ is also block diagonal in the basis used in \eqref{eq:curvopGZ}, namely $* = \operatorname{diag}(H, H, H)$, where \begin{equation*} H= \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}. \end{equation*} Therefore, the unique function $\tau\colon \nu(\mathds{C} P^1)\to\mathds{R}$ such that $R+\tau\,*\succeq0$ is $\tau=-\frac{\varphi'}{2b^2}$, and hence the unique function $\tau \colon \mathds{C} P^2\# \mathds{C} P^2\to\mathds{R}$ such that $R+\tau\,*\succeq0$ satisfies $\tau\leq 0$ and $\tau=0$ along the neck region. Routine computations with \eqref{eq:curvopGZ} yield \begin{equation*} \begin{aligned} \operatorname{Ric}(\mathrm g)&=\operatorname{diag}\left(\tfrac{\varphi^2}{2b^4}-\tfrac{\varphi''}{\varphi},\; \tfrac{1}{b^2}-\tfrac{\varphi^2}{2b^4},\; \tfrac{1}{b^2}-\tfrac{\varphi^2}{2b^4}, \;-\tfrac{\varphi''}{\varphi} \right),\\ \mathrm{scal}(\mathrm g)&=\tfrac{2}{b^2}-\tfrac{\varphi^2}{2b^4}-2\tfrac{\varphi''}{\varphi},\\ \tfrac{\mathrm{scal}(\mathrm g)}{2}\mathrm g - \operatorname{Ric}(\mathrm g)&=\operatorname{diag}\left( \tfrac{4b^2-3\varphi^2}{4b^4},\; \tfrac{\varphi^2}{4b^4}-\tfrac{\varphi''}{\varphi},\;\tfrac{\varphi^2}{4b^4}-\tfrac{\varphi''}{\varphi},\;\tfrac{4b^2-\varphi^2}{4b^4}\right), \end{aligned} \end{equation*} for all $r\in [0,r_{\mathrm{max}}]$. In particular, $\tfrac{\mathrm{scal}(\mathrm g)}{2}\mathrm g \succ \operatorname{Ric}(\mathrm g) \succeq0$ on all of $\mathds{C} P^2\# \mathds{C} P^2$, and $\operatorname{Ric}(\mathrm g)\succ0$ outside the neck region. Thus, as $b_2(\mathds{C} P^2\# \mathds{C} P^2)=2$, \Cref{cor:self-maps} implies that $\mathrm g$ is area-extremal with respect to self-maps of any nonzero degree, and area-rigid with respect to these maps on all points outside the neck region. \end{proof} \begin{remark}\label{rem:noneck} The statement concerning area-rigidity in \Cref{thm:cheeger-metrics} is sharp. In fact, given a Cheeger metric $\mathrm g_0$ on $\mathds{C} P^2\# \mathds{C} P^2$ with neck length $\ell_0\geq0$, one can easily produce nonisometric competitor Cheeger metrics $\mathrm g_1$ that satisfy \eqref{eq:g1-competitor-global} by elongating the neck length to any $\ell_1>\ell_0$. However, Cheeger metrics with \emph{vanishing} neck length are area-rigid (at all points) with respect to \emph{diffeomorphisms}, since any isometry of the complement $\mathds{C} P^2\# \mathds{C} P^2\setminus(\S^3\times\{0\})$ of the neck hypersurface clearly extends to a global isometry of $\mathds{C} P^2\# \mathds{C} P^2$. \end{remark} \Cref{thm:S4RP4CP2,thm:cheeger-metrics} and \Cref{rem:noneck} imply \Cref{maincor:CP2CP2} in the Introduction. \begin{remark} Although both halves of \(\nu(\mathds{C} P^1)\cup_{\alpha} \nu(\mathds{C} P^1)={\mathds{C}}P^2\#{\mathds{C}}P^2\) have cohomogeneity one $\mathsf{SU}(2)$-actions, these actions do not extend to \({\mathds{C}}P^2\#{\mathds{C}}P^2\). Indeed, the gluing isometry $\alpha\colon \S^3(2b)\to\S^3(2b)$ is orientation-reversing and hence cannot be $\mathsf{SU}(2)$-equivariant. Using an orientation-\emph{preserving} isometry $\beta$ instead, the resulting manifold is $\nu(\mathds{C} P^1)\cup_{\beta} \nu(\mathds{C} P^1)=\mathds{C} P^2\# \overline{\mathds{C} P^2}$, which carries a cohomogeneity one $\mathsf{SU}(2)$-action and invariant Cheeger metrics with $\sec\geq0$. Nevertheless, the corresponding function $\tau$ assumes \emph{opposite} signs on each half due to the flip in orientation, hence \Cref{thm} and \Cref{cor:self-maps} do not apply, cf.~\cite[Thm.~D]{bm-iumj}. \end{remark} \section{Extremality and rigidity on 4-manifolds with Boundary}\label{4boundary} In this section, we prove Theorems \ref{mainthm-local1} and \ref{mainthm-local2} in the Introduction, adapting the approach from the previous section to the case of compact manifolds with boundary. We begin by generalizing the notions of area-extremality and area-rigidity for closed manifolds in \Cref{def:extremality-global} to manifolds with boundary, along the lines of \cite{lott-spin}. Given a compact Riemannian manifold $(M,\mathrm g)$ with boundary $\partial M$, we denote by $\mathrm I\!\mathrm I_{\partial M}$ the second fundamental form of $\partial M$ with respect to the inward unit normal, and by $H(\mathrm g)=\operatorname{tr} \mathrm I\!\mathrm I_{\partial M}$ the mean curvature of $\partial M$ computed accordingly. \begin{definition}\label{def:extremality-local} A compact oriented Riemannian manifold $(M,\mathrm g_M)$ with boundary is \emph{area-extremal with respect to a class $\mathcal C=\{ f\colon (N,\mathrm g_N)\to (M,\mathrm g_M) \}$} of competitors, consisting of compact oriented Riemannian manifolds $(N,\mathrm g_N)$ with boundary such that $\dim M=\dim N$, and smooth boundary-preserving spin maps $f\colon N\to M$ of nonzero degree, if the inequalities \begin{equation}\label{eq:scal-comp-local} \begin{aligned} &\wedge^2\mathrm g_N\succeq f^*\wedge^2 \mathrm g_M, &\quad \mathrm{scal}(\mathrm g_N)&\geq \mathrm{scal}(\mathrm g_M)\circ f,\\ &\mathrm g_N|_{\partial N}\succeq f^* \mathrm g_M|_{\partial M}, &\quad H(\mathrm g_N)&\geq H(\mathrm g_M)\circ f \end{aligned} \end{equation} imply $\mathrm{scal}(\mathrm g_N)=\mathrm{scal}(\mathrm g_M)\circ f$ and \(H(\mathrm g_N)= H(\mathrm g_M)\circ f\). If, in addition, there exists $q\in M$ such that \eqref{eq:scal-comp-local} implies $\mathrm d f(p)\colon T_pN\to T_{q}M$ is a linear isometry for all competitors $f\colon N\to M$ in $\mathcal C$ and all $p\in f^{-1}(q)$, then $(M,\mathrm g_M)$ is called \emph{area-rigid at $q\in M$ with respect to $\mathcal C$}. If $(M,\mathrm g_M)$ is area-rigid at all of its points with respect to $\mathcal C$, then it is simply called \emph{area-rigid with respect to $\mathcal C$.} \end{definition} The notions of area-extremality and area-rigidity for manifolds with boundary discussed in the Introduction correspond to using the class \[\mathcal C^{\text{id}}_\partial =\big\{ \operatorname{id}\colon (M,\mathrm g_1)\to (M,\mathrm g_0) : \mathrm g_1|_{\partial M}=\mathrm g_0|_{\partial M}\big\}.\] Note that in the case $\partial M=\emptyset$, the class $\mathcal C^{\text{id}}_\partial$ is simply $\mathcal C^{\text{id}}_0$ from \Cref{rem:f=id}. \subsection*{Index Theory} Let \((M,\mathrm g_M)\) and \((N,\mathrm g_N)\) be oriented Riemannian $4$-manifolds as in \Cref{4closed}, except here we allow these manifolds to have nonempty boundary and assume \(f|_{\partial N}\colon \partial N\to \partial M\) is an oriented isometry. The twisted Dirac operator \(D_E\) is defined on sections of \(S(TN)\otimes f^*S^+(TM)\) as in \Cref{4closed}, see \eqref{eq:DE-global}. Let \(r\colon N\to\mathds{R}{}\) be the distance from \(\partial N\) and choose \(\varepsilon>0\) smaller than the focal radius of $\partial N$, so that the normal exponential map of \(\partial N\) is a diffeomorphism from \(\partial N\times [0,\varepsilon]\) to \(U=\{x\in N : r(x)\leq \varepsilon\}\). Let \(\nu=\partial _r,\) so \(\nu|_{\partial N}\) is the inward unit normal. There is an isomorphism \[\mathds{C}\ell(T\partial N)\longrightarrow \mathds{C}\ell^0(TN)|_{\partial N}\] generated by \( T\partial N\ni v \mapsto -\nu v\in\mathds{C}\ell(TN).\) Using this isomorphism, we identify \(S(T\partial N)\) and \(S^+(TN)|_{\partial N}\). Let \(\nabla^{\partial N}\) be the connection on \(S(T\partial N)=S^+(TN)|_{\partial N}\) induced by the restriction of $\mathrm g_N$ to \(\partial N\). Similarly, let \(\nabla^{\partial M}\) be the connection induced by $\mathrm g_M|_{\partial M}$ on \(S(T\partial M)=S^+(TM)|_{\partial M}.\) The connections \(\nabla^{\partial N}\) and \(f^*\nabla^{\partial M}\) induce a connection \(\nabla^\partial \) on \(S^+(TN)\otimes E|_{\partial N}\), which is different from the connection \(\nabla\) induced by the connections $\nabla^{S_N}$ and $f^*\nabla^{S_M}$ used to define~\(D_E\) in \eqref{eq:DE-global}. Using parallel translation along normal geodesics, we can identify \begin{equation}\label{collar} \Gamma\left(S(TN)\otimes E|_U\right)\cong C^\infty\big([0,\varepsilon],\,\Gamma\left(S(TN)\otimes E|_{\partial N}\right)\!\big), \end{equation} that is, sections of $S(TN)\otimes E$ on the collar neighborhood $U$ are $1$-parameter families of sections of $S(TN)\otimes E$ on the boundary $\partial N$. Using this identification, we have \[D_E|_{S^+(TN)\otimes E}=\nu\left(\partial_r+B+ A\right),\] where, in a local orthonormal frame \(\{e_i\}\) for \(TN\) such that \(e_1=-\nu,\) \begin{equation}\label{eq:BtildeA} B=\sum_{i=2}^4e_1e_i\nabla^\partial_{e_i}, \qquad A=\sum_{i=2}^4e_1e_i\left(\nabla_{e_i}-\nabla^\partial_{e_i}\right). \end{equation} Recall that Clifford multiplication is only on the first factor of \(S(TN)\otimes E\), so \(B\) is a first order self-adjoint elliptic differential operator on \(S(TN)\otimes E|_{\partial N}\) and \(A\) is a linear endomorphism of the same bundle. Let \(P_{>0}\) and \(P_{\leq0}\) be the projections onto the subspaces of \(\Gamma(S^+(TN)\otimes E|_{\partial N})\) spanned by eigenvectors of \(B\) with positive and nonpositive eigenvalues, respectively. We impose boundary conditions on \(D_E\) by setting $D_E^+:=D_E|_{\Gamma^+}$, where \begin{equation}\label{eq:gamma+def} \Gamma^+:=\{\xi\in\Gamma(S^+(TN)\otimes E) : P_{>0}\left(\xi|_{\partial N}\right)=0\}. \end{equation} The adjoint $D_E^-$ of \(D_E^+\) is the restriction of \(D_E\) to \[\Gamma^-:=\{\xi\in\Gamma(S^-(TN)\otimes E) : P_{\leq0}\left(\nu\xi|_{\partial N}\right)=0\}.\] We use the convention that \(\sigma(M)\) is the signature of the bilinear form induced by the cup product on the image of \(H^2(M,\partial M)\) in \(H^2(M)\), and similarly for $\sigma(N)$. \begin{lemma}\label{lem:section} If \(2\chi(M)+3\sigma(M)+2b_0(\partial M)+2b_2(\partial M)>\sigma(N)\), then $D_E^+$ has nontrivial kernel. \end{lemma} \begin{proof} With the boundary conditions given by $\Gamma^+$, the operator \(D_E^+\) is Fredholm, see e.g.~\cite{Grubb}. We claim that its index is \begin{equation}\label{eq:ind-bdy} \text{ind}\big(D_E^+\big)=\tfrac14\big(-\sigma(N)+2\chi(M)+3\sigma(M)+2b_0(\partial M)+2b_2(\partial M)\big), \end{equation} which implies the desired result, since \(\text{ind}(D_E^+)=\dim\ker(D_E^+)-\dim\ker(D_E^-).\) In order to prove \eqref{eq:ind-bdy}, note that a continuous deformation of the map $f\colon N\to M$ and of the metrics $\mathrm g_M$ and $\mathrm g_N$ that keeps the boundary conditions fixed yields a continuous deformation of Fredholm operators on the same space, which does not change the index. In particular, deforming \(\mathrm g_M\) and \(\mathrm g_N\) as needed, we may assume that \(\mathrm g_N|_U=\mathrm g_N|_{\partial N}+\mathrm d r^2\) with respect to the identification \(U\cong \partial N\times [0,\varepsilon].\) Since \(B\) depends only on \(f|_{\partial N}\) and its tangential derivatives, we may modify \(f|_{U}\) so that it is a diffeomorphism onto its image and modify \(\mathrm g_M\) so that \(f|_{U}\) is an isometry. Under the identification \eqref{collar}, we now see that \(A=0\), hence \(D_E^+=\nu(\partial_r+B)\). Thus, the adjoint operator is given near the boundary by \[D_E^-=(-\partial_r+B)(-\nu)=\nu(\partial_r+\nu B\nu).\] Let \(P'_{\geq0}\) and \(P'_{<0}\) be the projections onto the subspaces of \(\Gamma(S^-(TN)\otimes E|_{\partial N})\) spanned by eigenvectors of \(\nu B\nu\) with nonnegative and negative eigenvalues, respectively. Since \(\nu\) is skewsymmetric, we have that \begin{equation*} \begin{aligned} \Gamma^-&=\{\xi\in\Gamma(S^-(TN)\otimes E) : P'_{\geq0}(\xi|_{\partial N})=0\},\\ \Gamma^+&=\{\xi\in\Gamma(S^+(TN)\otimes E) : P'_{<0}\left(\nu\xi|_{\partial N}\right)=0\}. \end{aligned} \end{equation*} These are exactly the Atiyah--Patodi--Singer boundary conditions for the operator \(D_E^-\) with boundary operator \(\nu B\nu\) and adjoint \(D_E^+\). (Note that the conditions for \(D_E^+\) and \(D_E^-\) differ by the role of the strict and nonstrict inequalities.) Thus, applying the Atiyah--Patodi--Singer Index Theorem~\cite[Thm.~4.2]{APS} to \(D_E^-\), we compute \begin{equation}\label{eq:APS-1} \begin{aligned} \text{ind}\big(D_E^+\big)=-\text{ind}\big(D_E^-\big)&=-\int_{N} a(D_E^-)+\frac{h(\nu B\nu)+\eta(\nu B\nu)}{2}\\ &=\int_{N} a(D_E^+)+\frac{h(B)-\eta(B)}{2}, \end{aligned} \end{equation} where \(a\) is the integrand in the Atiyah--Singer Index Theorem (in the closed case), which has opposite signs for adjoints, \(h(\cdot)=\dim\ker(\cdot)\) is the dimension of the kernel, and \(\eta\) is the \(\eta\)-spectral invariant, which commutes with sign changes. (Note that \(\nu B \nu=-(\nu^{-1}B\nu)\) has opposite spectrum to \(B\).) By the discussion in the closed case (see \Cref{kernel2}), we have that \[a(D_E^+)=-\tfrac{1}{12} p_1(\mathrm g_N) +\tfrac{1}{4} f^*p_1(\mathrm g_M)+\tfrac{1}{2} f^*e(\mathrm g_M),\] where \(p_1\) and \(e\) denote the Pontryagin and Euler forms from Chern--Weil theory. Since \(f|_{\partial N}\) is an oriented isometry, there is an isomorphism of \(S^+(TN)|_{\partial N} \cong f^*S^+(TM)|_{\partial M}\) identifying \(\nabla^{\partial N}\) with \(f^*\nabla^{\partial M}\). Under the isomorphisms \[S^+(TN)\otimes E|_{\partial N}\cong S^+(TN)\otimes S^+(TN)|_{\partial N}\cong \wedge_\mathds{C}^{+,\text{even}}TN^*|_{\partial N}\cong\wedge_\mathds{C}^\text{even}T\partial N^*,\] where the final isomorphism is given by restriction, and its inverse on \(\wedge_\mathds{C}^{2p}T\partial N^*\) is given by \(\alpha\mapsto \alpha-({-1})^{p}\,* \alpha\), the operator \(B\) corresponds to \((-1)^p(*_{\partial N}\mathrm d_{\partial N}-\mathrm d_{\partial N}*_{\partial N})\) on \(\wedge_\mathds{C}^{2p}T\partial N^*.\) The latter is the boundary operator in the Atiyah--Patodi--Singer Signature Theorem~\cite[Thm.~4.14]{APS}, from which we obtain that \[\int_N\tfrac13 p_1(\mathrm g_N) =\sigma(N)+\eta(B), \qquad\text{and}\qquad \int_M\tfrac13 p_1(\mathrm g_M) =\sigma(M)+\eta(B),\] where we are using that \(f\) is an isometry on the boundary, hence both boundary operators have the same spectrum. Moreover, since \(f\) is an oriented diffeomorphism near the boundary, it follows that \begin{equation}\label{eq:intaDE} \begin{aligned} \int_N a(D_E^+)&=-\int_N\tfrac{1}{12} p_1(\mathrm g_N)+\int_M\tfrac14 p_1(\mathrm g_M) + \tfrac12 e(\mathrm g_M)\\ &=-\tfrac14 {\sigma(N)}+\tfrac34 \sigma(M)+\tfrac12 \eta(B)+\tfrac12 \chi(M). \end{aligned} \end{equation} Thus, \eqref{eq:ind-bdy} follows from \eqref{eq:APS-1} and \eqref{eq:intaDE}, as the kernel of \(\pm(*_{\partial N}\mathrm d_{\partial N}-\mathrm d_{\partial N}*_{\partial N})\) is the space of even harmonic forms on $\partial N\cong\partial M$, and hence \(h(B)=b_0(\partial M)+b_2(\partial M)\) by Hodge theory. \end{proof} \subsection{Extremality and Rigidity} We now combine the Bochner--Lichnerowicz--Weitzenb\"ock formula with \Cref{lem:section} to prove our main result on area-extremality and area-rigidity of compact $4$-manifolds with boundary (see \Cref{def:extremality-local}), closely following the proof of \Cref{thm}, but also handling boundary terms. \begin{theorem}\label{local} A compact oriented Riemannian $4$-manifold \((M,\mathrm g_M)\) with $\sec\geq0$ such that $R+\tau\,*\succeq0$ for a nonpositive $\tau\colon M\to \mathds{R}$ and convex boundary $\partial M$, i.e., \(\mathrm I\!\mathrm I_{\partial M}\succeq0\), is area-extremal with respect to \begin{equation*} \mathcal C_{\partial}=\left\{f\colon (N,\mathrm g_N)\to (M,\mathrm g_M) :\!\! \begin{array}{l} f|_{\partial N}\text{ is an oriented isometry onto \(\partial M\), and}\\ 2\chi(M)+3\sigma(M)+2b_0(\partial M)+2b_2(\partial M)>\sigma(N) \end{array}\!\!\!\right\}. \end{equation*} If, in addition, $\frac{\mathrm{scal}(\mathrm g_M)}{2}\mathrm g_M\succ\operatorname{Ric}(\mathrm g_M)\succ0$ at $q\in M$, then $(M,\mathrm g_M)$ is area-rigid at $q\in M$ with respect to $\mathcal C_{\partial}$. \end{theorem} \begin{proof} Let $f\colon (N,\mathrm g_N)\to (M,\mathrm g_M)$ be a competitor in $\mathcal C_{\partial}$, and let $E=f^*S^+(TM)$. As in \Cref{thm}, working pointwise with $W=T_pN$, $V=T_{f(p)}M$, $l\colon W\to V$ given by $\mathrm d f(p)$, and $L=\wedge^2 l$, we have from \eqref{d2} that \[ D_E^2=\nabla^*\nabla+\tfrac{1}{4}\mathrm{scal}(\mathrm g_N)+\mathcal{R}(R, L).\] Let \(\xi\in\Gamma(S^+(TN)\otimes E)\) be a nonzero section in the kernel of \(D_E\), which exists by \Cref{lem:section}. Using \Cref{rt,fscal,nneg,star} as in the proof of \Cref{thm}, we have: \begin{equation}\label{lich} \begin{aligned} 0&=\int_N\langle\nabla^*\nabla\xi,\xi\rangle+\tfrac14\int_N \mathrm{scal}(\mathrm g_N)\|\xi\|^2 +\int_N\left<\mathcal{R}(R,L)\xi,\xi\right>\\ &\geq \int_N\langle\nabla^*\nabla\xi,\xi\rangle+\tfrac14\int_N\big(\mathrm{scal}(\mathrm g_N)-\mathrm{scal}(\mathrm g_M)\circ f\big)\|\xi\|^2. \end{aligned} \end{equation} A standard computation using the divergence theorem implies that \begin{equation}\label{adjoint} \int_N\langle\nabla^*\nabla\xi,\xi\rangle=\int_N\|\nabla\xi\|^2+\int_{\partial N}\langle \nabla_\nu\xi,\xi\rangle, \end{equation} where, as above, $\nu$ is the inward unit normal along $\partial N$. In a local \(\mathrm g_N\)-orthonormal frame \(\{e_i\}\) of $N$ with \(e_1=-\nu\), since \(D_E\xi=0,\) we may use \eqref{eq:BtildeA} to compute \begin{equation}\label{eq:nablanuxi} \nabla_\nu\xi=\textstyle\sum\limits_{i=2}^4\nu e_i\nabla_{e_i}\xi=-B\xi- A\xi. \end{equation} As \(\xi\in \Gamma^+,\) we have \(\langle B\xi,\xi\rangle\leq 0\) from \eqref{eq:gamma+def}. To analyze the endomorphism \(A\) at the boundary, let \(\widetilde\nu\) be an the inward unit normal field of \(\partial M\). For all \(v\in T\partial N\) \begin{align*} \nabla_{v}^{S_N}-\nabla_v^{\partial N}&=\tfrac12 \mathrm I\!\mathrm I_{\partial N}(v)\cdot \nu\in \mathds{C}\ell(TN)|_{\partial N},\\ \nabla_{l(v)}^{S_M}-\nabla_{l(v)}^{\partial M}&=\tfrac12 \mathrm I\!\mathrm I_{\partial M}(l(v))\cdot \widetilde\nu\in \mathds{C}\ell(TM)|_{\partial M}, \end{align*} where we use the same symbol \(\mathrm I\!\mathrm I_{\partial N}\) to denote the symmetric endomorphism of \(T\partial N\) induced by $\mathrm I\!\mathrm I_{\partial N}$, namely so that \(\mathrm I\!\mathrm I_{\partial N}(x,y)=\mathrm g_N(\nabla^{N}_x y,\nu )=\mathrm g_N (\mathrm I\!\mathrm I_{\partial N}(x),y)\) for all \(x,y\in T\partial N,\) and similarly for \(\mathrm I\!\mathrm I_{\partial M}\). Using the above and \eqref{eq:BtildeA}, we compute \begin{align*} A=&-\nu\sum_{i=2}^{4}e_i\left(\nabla_{e_i}^{S_N}\otimes 1+1\otimes \nabla^{S_M}_{l(e_i)}-\nabla^{\partial N}_{e_i}\otimes 1-1\otimes \nabla^{\partial M}_{l(e_i)}\right)\\ =&\tfrac12 \sum_{i=2}^{4} e_i\, \mathrm I\!\mathrm I_{\partial N}(e_i)\otimes 1+\tfrac12 \sum_{i=2}^{4}\nu\, e_i\otimes{\widetilde\nu}\, \mathrm I\!\mathrm I_{\partial M}(l(e_i)). \end{align*} By the symmetries of \(\mathrm I\!\mathrm I_{\partial N}\) and of Clifford multiplication, the first term above is \(-\tfrac12 H(\mathrm g_N).\) The second term can be written using the pointwise formalism from \Cref{pi}, as follows. Given $p\in \partial N$, since \(f|_{\partial N}\colon\partial N\to\partial M\) is an isometry and $\{e_2,e_3,e_4\}$ is an orthonormal frame of $T_p\partial N\subset W$, its image \(\{l(e_2),l(e_3),l(e_4)\}\) is an orthonormal frame of \(T_{f(p)}\partial M\subset V\). Let \(Q\colon \wedge^2 V\to \wedge^2 V\) be the symmetric endomorphism such that \(Q(\widetilde\nu\wedge l(e_i))=\widetilde\nu\wedge \mathrm I\!\mathrm I_{\partial M}(l(e_i))\) and \(Q(l(e_i)\wedge l(e_j))=0\), for \(2\leq i,j\leq 4,\) and note that $Q\in \operatorname{Sym}^2_b(\wedge^2 V)$, i.e., $Q$ satisfies the first Bianchi identity. Moreover, let \(l'\colon W \to V\) be the linear map such that \(l'(e_i)=l(e_i)\) for \(2\leq i\leq 4\) and \(l'(\nu)=\widetilde \nu\), and set $L'=\wedge^2 l'$. From \Cref{def:RT}, we have that \[-\mathcal{R}(Q,L')=\sum_{i<j}e_ie_j\otimes Q(l(e_i)\wedge l(e_j))=\tfrac12\sum_{j\geq2}\nu \, e_j\otimes \widetilde\nu \,\mathrm I\!\mathrm I_{\partial M}(l(e_j)),\] so we conclude that \(A=-\tfrac12 H(\mathrm g_N)-\mathcal{R}(Q,L').\) Applying \Cref{rt,fscal,nneg}, since \(\mathrm I\!\mathrm I_{\partial M}\succeq0\) and \(l'\) is an isometry, it follows that \(Q\succeq0\) and \[-\langle A\xi,\xi\rangle\geq\tfrac12 \big(H(\mathrm g_N) - \operatorname{tr}(Q) \big)\|\xi\|^2=\tfrac12\big( H(\mathrm g_N)-H(\mathrm g_M) \big)\|\xi\|^2.\] From \eqref{adjoint} and the above, the left-hand side of \eqref{eq:nablanuxi} can be bounded from below: \begin{align*} \int_N\langle \nabla^*\nabla\xi,\xi\rangle&=\int_N \|\nabla\xi\|^2-\int_{\partial N}\langle B\xi,\xi\rangle-\int_{\partial N}\langle A\xi,\xi\rangle\\ &\geq\int_N \|\nabla\xi\|^2+\tfrac12\int_{\partial N} \big(H({\mathrm g_N})-H({\mathrm g_M})\circ f\big)\|\xi\|^2. \end{align*} Combined with \eqref{lich}, this yields: \begin{align*} 0&\geq\int_N\|\nabla\xi\|^2+\tfrac14\int_N\big(\mathrm{scal}(\mathrm g_N)-\mathrm{scal}(\mathrm g_M)\circ f\big)\|\xi\|^2\\ &\quad+\tfrac12\int_{\partial N}\big(H({\mathrm g_N})-H({\mathrm g_M})\circ f\big)\|\xi\|^2. \end{align*} The stated conclusions now follow exactly as in the proof of \Cref{thm}. \end{proof} The following result implies \Cref{mainthm-local2} in the Introduction, since \(\mathcal C^\mathrm{id}_\partial\subset \mathcal C_\partial^{\mathrm{self}}.\) \begin{corollary}\label{cor:boundary-self} A compact oriented Riemannian $4$-manifold \((M,\mathrm g_0)\) with $\sec\geq0$ such that $R+\tau\,*\succeq0$ for a nonpositive or nonnegative $\tau\colon M\to \mathds{R}$ and convex boundary $\partial M$, i.e., \(\mathrm I\!\mathrm I_{\partial M}\succeq0\), is area-extremal with respect to \begin{equation}\label{eq:classC-local-self} \mathcal C_\partial^{\mathrm{self}}=\big\{f\colon (M,\mathrm g_1)\to (M,\mathrm g_0) \; : \; f|_{\partial M}\text{ is an oriented isometry onto }\partial M \big\}. \end{equation} \end{corollary} \begin{proof} Choose the orientation of \(M\) for which \(\tau\leq0\), and let \(N\) be either \(M\) or \(\overline M\), i.e., $M$ with an orientation that will be fixed later. Let \(f\colon N\to M\) be a boundary-preserving spin map and \(\mathrm g_1\) be a metric on \(N\) such that \[\mathrm{scal}(\mathrm g_1)\geq \mathrm{scal}(\mathrm g_0)\circ f, \quad \wedge^2\mathrm g_1\succeq f^*\wedge^2\mathrm g_0, \quad H({\mathrm g_1})\geq H({\mathrm g_0})\circ f,\] and \(f|_{\partial N}\) is an isometry between \(\mathrm g_1|_{\partial N}\) and \(\mathrm g_0|_{\partial N}.\) Orient \(N\) such that \(f|_{\partial N}\) \emph{preserves} orientation, i.e., is an oriented isometry. The conclusion will follow from \Cref{local} by showing that the topological condition in $\mathcal C_{\partial}$ is satisfied by $f\colon N\to M$. By the Soul Theorem, see e.g.~\cite[Sec.~9-10]{eschenburg}, since $(M,\mathrm g_0)$ has \(\sec\geq0\) and \(\mathrm I\!\mathrm I_{\partial M}\succeq0 \), it is diffeomorphic to the total space of a disk bundle over a closed totally geodesic submanifold \(\Sigma\subset M\). We proceed case-by-case in terms of \(0\leq \dim\Sigma\leq 3\). If \(\dim\Sigma\leq 1,\) then \(\chi(M)=\chi(\Sigma)\geq0\), and \(H^2(M)=H^2(\Sigma)=0\), which implies \(\sigma(M)=\sigma(N)=0.\) Thus, the topological condition in $\mathcal C_{\partial}$ is satisfied, as \(b_0(\partial M)>0.\) If \(\dim\Sigma=3,\) then \(M\to\Sigma\) is an interval bundle which restricts to a covering map \(\partial M\to \Sigma.\) Such a map is homotopic to the inclusion \(\partial M\to M\), and thus \(H^2(M)\to H^2(\partial M)\) is injective and \(H^2(M,\partial M)\to H^2(M)\) is trivial. Therefore, \(\sigma(M)=\sigma(N)=0\), so the conclusion follows as in the previous case, since $\chi(\Sigma)=0$. Finally, assume \(\dim\Sigma=2,\) in which case the Gauss--Bonnet Theorem implies that \(\chi(\Sigma)\geq0\). If \(\Sigma\) is nonorientable, then \(H^2(M)=H^2(\Sigma)=0\) and \(\sigma(M)=0\), so the topological condition in $\mathcal C_{\partial}$ is satisfied as in the previous cases. Else, \(\Sigma\) is orientable, so \(b_2(M)=b_2(\Sigma)=1\) and \(|\sigma(M)|\leq 1.\) If \(\Sigma\) is diffeomorphic to \(\S^2\), then \(2\chi(M)+3\sigma(M)-\sigma(N)=4+(3\pm1)\sigma(M)\geq0\) and hence the topological condition in $\mathcal C_{\partial}$ is again satisfied. If \(\Sigma\) is diffeomorphic to \(T^2\), then \(\partial M\) is diffeomorphic to an oriented \(\S^1\)-bundle over \(T^2\) so \(b_2(\partial M)\geq 2.\) Once again, the topological condition in $\mathcal C_{\partial}$ is satisfied, as \(2b_0(\partial M)+2b_2(\partial M)\geq 6\geq(-3\pm1)\sigma(M)\). \end{proof} Our final main result implies \Cref{mainthm-local1} in the Introduction because \(\mathcal C^\mathrm{id}_\partial\subset \mathcal C_\mathrm{loc}\) if $|\sigma(M)|<4$, and is captured by the vague but clear statement that, near a point where $\sec>0$, a Riemannian metric on a $4$-manifold cannot be modified to have greater scalar curvature without decreasing the area of some tangent $2$-plane. \begin{corollary} If \((X,\mathrm g)\) is a Riemannian \(4\)-manifold and \(p\in X\) is a point at which $\sec>0$, then there is a neighborhood \(M\cong D^4\) of \(p\) such that \(\mathrm g|_M\) is area-extremal with respect to \begin{equation*} \mathcal C_\mathrm{loc}=\left\{f\colon (N,\mathrm g_N)\to (M,\mathrm g|_M) : \!\! \begin{array}{l} f|_{\partial N}\text{ is an oriented isometry onto \(\partial M\),} \\ \text{and }|\sigma(N)|<4 \end{array} \right\}. \end{equation*} \end{corollary} \begin{proof} At each stage of the proof, we shrink \(M\) if needed, in order to make a series of assumptions. First, we may choose a neighborhood \(M\) of $p\in X$ such that $\sec>0$ at all points of $M$. Then, by the Finsler--Thorpe trick (see \Cref{prop:FTtrick}), there exists a function \(\tau\colon M\to \mathds{R}\) such that the curvature operator \(R\) of \((M,\mathrm g|_M)\) satisfies \(R+\tau\,* \succ0.\) Moreover, there is an open interval $(\tau_\mathrm{min}(x),\tau_\mathrm{max}(x))$ of possible values for $\tau$ at each point $x\in M$ that depends continuously on $x$, so we may choose $\tau$ to be continuous and \(\tau(p)\neq0.\) Shrinking \(M\) if necessary, we may assume \(\tau\neq 0\) throughout \(M,\) and, changing the orientation of \(M\) if necessary, we may assume \(\tau<0\) on all $M$. Finally, we shrink \(M\) to be the closure of a convex ball in $X$ containing $p\in X$. Then \((M,\mathrm g|_M)\) satisfies all the hypotheses of \Cref{local} and, since \(3\sigma(M)+2\chi(M)+2b_0(\partial M)+2b_2(\partial M)=4,\) the corollary follows. \end{proof} \begin{example}\label{ex:examples-local} Let \(\S^n_+\) denote the unit \(n\)-dimensional hemisphere. Standard metrics on \(\S^4_+,\) \(\S^1\times\S^3_+,\) \(\S^2\times \S^2_+\), and \(\S^3\times [-1,1]\) have positive-semidefinite curvature operator and totally geodesic boundary, and are hence area-extremal with respect to $\mathcal C_\partial$ by \Cref{local}. In particular, they are area-extremal with respect to \(\mathcal{C}_\partial^\mathrm{self}\subset\mathcal C_\partial\) defined in \eqref{eq:classC-local-self}. The first and third spaces are also area-rigid, as they satisfy the additional hypothesis $\frac{\mathrm{scal}}{2}\mathrm g\succ\operatorname{Ric}\succ0$. Note that the second and fourth spaces have vanishing Euler characteristic. Similar results are implied by \cite[Thm.~1.3]{lott-spin}. The $\mathsf{SU}(2)$-invariant metrics with totally geodesic boundary on the normal disk bundle \(\nu(\mathds{C} P^1)\) of $\mathds{C} P^1\subset\mathds{C} P^2$, described in the proof of \Cref{thm:cheeger-metrics}, also satisfy the hypotheses of \Cref{local}. Thus, these metrics are area-extremal with respect to \(\mathcal{C}_\partial\) and \(\mathcal{C}_\partial^\mathrm{self}\). Moreover, if such a metric has vanishing neck length, then it is area-rigid with respect to diffeomorphisms. Indeed, the neck in that case consists solely of the boundary $\partial \nu(\mathds{C} P^1)$ and $\frac{\mathrm{scal}}{2}\mathrm g\succ\operatorname{Ric}\succ0$ everywhere else, so the same continuity argument in \Cref{rem:noneck} applies. This example is notable because \(\nu(\mathds{C} P^1)\) does not admit metrics with both \(R\succeq 0\) and \(\mathrm I\!\mathrm I_{\partial\nu(\mathds{C} P^1)}\succeq0,\) to which previous methods could be applied to prove area-extremality: a simply-connected manifold admitting such a metric is diffeomorphic to a \emph{trivial} disk bundle over its soul, see e.g.~\cite{Noronha}. \end{example} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2}

proofpile-arXiv_069-8176

\section{Introduction} Biproducts were introduced by Mac Lane in his book \emph{Homology} to study split extensions in the context of abelian categories. Semidirect products are appropriate to study group split extensions. Although these concepts have been thoroughly developed over the last decades in the context of protomodular and semi-abelian categories \cite{BB,Bourn,BournJanelidze,semiabelian}, the notion of \emph{relative biproduct} introduced by Mac Lane to study relative split extensions seems to have been forgotten (see \cite{Homology}, p.~263). On the other hand, much work has been done in extending the tools and techniques from groups \cite{MacLane2} to monoids \cite{DB.NMF.AM.MS.13, NMF14,DB.NMF.AM.MS.16,Faul,Fleischer,Ganci,Leech,NMF et all,Wells} and even more general settings \cite{GranJanelidzeSobral}. However, as it has been observed several times, it is not a straightforward task to take a well-known result in the category of groups (or any other semi-abelian category) and materialize it in the category of monoids not to mention more general situations. We will argue that a convenient reformulation of relative biproduct (called \emph{semibiproduct}) can be used to study group and monoid extensions as a unified frame of work. Even though semidirect products are suitable to describe all group split extensions they fail to capture those group extensions that do not split. The key observation to semibiproducts in reinterpreting relative biproducts (see \cite{Homology}, diagram (5.2), p.~263 and compare with diagram $(\ref{diag: biproduct in a Mag-category})$ in Definition \ref{def: semibiproduct}) is that although an extension may fail to split as a monoid extension or as a group extension, it necessarily splits as an extension of pointed sets. The main result (Theorem \ref{thm: equivalence}) establishes an equivalence of categories between pointed semibiproducts of monoids (Definition \ref{def: pointed semibiproduct of monoids}) and pointed monoid action systems (Definition \ref{def: pseudo-action}). The 14 classes of non-isomorphic pointed semibiproducts of 2-element monoids are listed in Section \ref{sec: eg}. We start with some motivation in Section \ref{sec: motivation}, introduce $\mathbf{Mag}$-categories and semibiproducts in Section \ref{Sec: Mag-categories}, restrict to the pointed case in Section~\ref{sec: stability} while studying some stability properties and pointing out some differences and similarities between groups, monoids and unitary magmas. From Section \ref{Sec: Sbp} on we work towards the main result and restrict our attention to monoids. \section{Motivation}\label{sec: motivation} It is well known that a split extension of groups \[\xymatrix{X\ar[r]^k & A \ar[r]^{p} & B,} \] with a specified section $s\colon{B\to A}$, can be completed into a diagram of the form \[\xymatrix{X\ar@<-.5ex>[r]_{k} & A\ar@<-.5ex>@{..>}[l]_{q}\ar@<.5ex>[r]^{p} & B \ar@{->}@<.5ex>[l]^{s},}\] in which $q\colon{A\to X}$ is the map uniquely determined by the formula $kq(a)=a-sp(a)$, $a\in A$. Furthermore, the information needed to reconstruct the split extension as a semidirect product is encoded in the map $\varphi\colon{B\times X\to X}$, uniquely determined as $\varphi(b,x)=q(s(b)+k(x))$. If writing the element $\varphi(b,x)\in X$ as $b\cdot x$ we see that $k(b\cdot x)$ is equal to $s(b)+k(x)-s(b)$ and that the group $A$ is recovered as the semidirect product $X\rtimes_{\varphi}B$. In the event that the section $s$, while being a zero-preserving map, is not necessarily a group homomorphism then the classical treatment of group extensions prescribes a different procedure (see e.g.\ \cite{Northcott}, p.~238). However, the results obtained here suggest that non-split extensions may be treated similarly to split extensions, and moreover the same approach is carried straightforwardly into the context of monoids. Indeed, when $s$ is not a homomorphism, in addition to the map $\varphi$, we get a map $\gamma\colon{B\times B\to X}$, determined as $\gamma(b,b')=q(s(b)+s(b'))$ and the group $A$ is recovered as the set $X\times B$ with group operation \[(x,b)+(x',b')=(x+\varphi(b,x')+\gamma(b,b'),b+b')\] defined for every $x,x'\in X$ and $b,b'\in B$. Note that $X$ needs not be commutative. However, instead of simply saying that $\varphi$ is an action and that $\gamma$ is a factor system, we have to consider two maps $\varphi$ and $\gamma$ which in conjunction turn the set $X\times B$ into a group with a prescribed operation (Section \ref{Sec: Act}). This is precisely what we call a semibiproduct of groups. Observe that when $s$ is a homomorphism it reduces to the usual notion of semidirect product. Almost every step in the treatment of groups is carried over into the context of monoids. However, while in groups all extensions are obtained as semibiproducts, in monoids we have to restrict our attention to those for which there exists a section $s$ and a retraction $q$ satisfying the condition $a=kq(a)+sp(a)$ for all $a\in A$ (Section \ref{Sec: Sbp}). Consequently, in addition to the maps $\varphi$ and $\gamma$ obtained as in groups, a new map $\rho\colon{X\times B\to X}$, determined as $\rho(x,b)=q(k(x)+s(b))$ needs to be taken into consideration. Hence, the monoid $A$ is recovered as the set $\{(x,b)\in X\times B\mid\rho(x,b)=x\}$ with operation \begin{equation}\label{eq: operation} (x,b)+(x',b')=(\rho(x+\varphi(b,x')+\gamma(b,b'),b+b'),b+b') \end{equation} which is defined for every $x,x'\in X$ and $b,b'\in B$. \section{$\mathbf{Mag}$-categories and semibiproducts}\label{Sec: Mag-categories} Let $\mathbf{Mag}$ denote the category of magmas and magma homomorphisms and let $U\colon{\mathbf{Mag}\to \mathbf{Set}}$ be the forgetful functor into the category of sets and maps. By a $\mathbf{Mag}$-category we mean a category $\mathbf{C}$ together with a bifunctor $\mathrm{map}\colon{\mathbf{C}^{\text{op}}\times \mathbf{C}\to \mathbf{Mag}}$ and a natural inclusion $\varepsilon\colon{\mathrm{hom}_{\mathbf{C}}\to U\circ \mathrm{map}}$. If $\mathbf{C}$ is a concrete category over sets in which a meaningful map addition is available then a bifunctor $\mathrm{map}$ is obtained as follows. For every pair of objects $(A,B)$ in $\mathbf{C}$, $\mathrm{map}(A,B)$ is the magma of underlying maps from object $A$ to object $B$ equipped with component-wise addition. In particular $\mathrm{map}(A,B)$ contains $\mathrm{hom}_{\mathbf{C}}(A,B)$ as a subset since $$\varepsilon_{A,B}\colon{\mathrm{hom}_{\mathbf{C}}(A,B)\to U(\mathrm{map}(A,B))}$$ is required to be a natural inclusion. As expected the category $\mathbf{Mag}$ is a $\mathbf{Mag}$-category with $\mathrm{map}(A,B)$ the magma of all maps from $U(A)$ to $U(B)$. If $f$ is a magma homomorphism from $A$ to $B$ then $\varepsilon_{A,B}(f)$ is nothing but $f$ considered as a map between the underlying sets of $A$ and $B$. In the same way the categories of groups, abelian groups, monoids and commutative monoids are $\mathbf{Mag}$-categories. However, there is a significant distinction between the Ab-category of abelian groups, the linear category of commutative monoids and the Mag-categories of groups or monoids. If $A$ is an object in an Ab-category then $\mathrm{hom}(A,A)$ is a ring. If $A$ is an object in a linear category then $\mathrm{hom}(A,A)$ is a semiring. In contrast, if $A$ is a group (or a monoid) then $\mathrm{hom}(A,A)$ is a subset of the near-ring $\mathrm{map}(A,A)$. \begin{definition}\label{def: semibiproduct} Let $(\mathbf{C},\mathrm{map},\varepsilon)$ be a $\mathbf{Mag}$-category. A \emph{semibiproduct} is a tuple $(X,A,B,p,k,q,s)$ represented as a diagram of the shape \begin{equation} \label{diag: biproduct in a Mag-category} \xymatrix{X\ar@<-.5ex>[r]_{k} & A\ar@<-.5ex>@{..>}[l]_{q}\ar@<.5ex>[r]^{p} & B \ar@{..>}@<.5ex>[l]^{s}} \end{equation} in which $p\colon{A\to B}$ and $k\colon{X\to A}$ are morphisms in $\mathbf{C}$, whereas $q\in \mathrm{map}(A,X)$ and $s\in \mathrm{map}(B,A)$. Furthermore, the following conditions are satisfied: \begin{eqnarray} ps={1_B}\label{eq: biproduct1}\\ qk={1_{X}},\label{eq: biproduct2}\\ kq+sp={1_A}.\label{eq: biproduct3} \end{eqnarray} \end{definition} There is an obvious abuse of notation in the previous conditions. This is justified because we will be mostly concerned with the case in which $\mathbf{C}$ is the category of monoids and $\mathrm{map}(A,B)$ is the set of zero-preserving maps. In rigorous terms, condition $ps=1_B$ should have been written as $\mathrm{map}(1_B,p)(s)=\varepsilon_{B,B}(1_B)$ whereas condition $qk=1_X$ should have been written as $\mathrm{map}(k,1_X)(q)=\varepsilon_{X,X}(1_X)$. In the same way the condition $\mathrm{map}(1_A,k)(q)+\mathrm{map}(p,1_A)(s)=\varepsilon_{A,A}(1_A)$ should have been written in the place of $kq+sp=1_A$. For the moment we will not develop this concept further but rather concentrate our attention on the concrete cases of groups and monoids. \section{Stability properties of pointed semibiproducts}\label{sec: stability} From now on the category $\mathbf{C}$ is assumed to be either the category of groups or the category of monoids (occasionally we will refer to the category of unitary magmas) and $\mathrm{map}(A,B)$ is the magma of zero-preserving maps with component-wise addition. In each case the category $\mathbf{C}$ is pointed and the composition of maps is well defined. It will be convenient to consider \emph{pointed semibiproducts} by requiring two further conditions, namely \begin{eqnarray} pk=0_{X,B},\quad qs=0_{B,X}. \end{eqnarray} However, as it is well known, in the case of groups this distinction is irrelevant. \begin{proposition} Every semibiproduct of groups is pointed. \end{proposition} \begin{proof} We have $pk=p1_Ak=p(kq+sp)k=pkqk+pspk=pk+pk$. And $s=1_As=(kq+sp)s=kqs+sps=kqs+s$. Hence we may conclude $pk=0$ and $kqs=0$. Since $k$ is a monomorphism $qs=0$. \end{proof} The previous proof also shows that a semibiproduct of monoids is pointed as soon as the monoid $A$ admits right cancellation. Clearly, this is not a general fact. \begin{proposition} Let $A$ be a monoid. The tuple $(A,A,A,1_A,1_A,1_A,1_A)$ is a semibiproduct of monoids if and only if $A$ is an idempotent monoid. \end{proposition} \begin{proof} Condition $(\ref{eq: biproduct3})$ in this case becomes $a=a+a$ for all $a\in A$. \end{proof} Every pointed semibiproduct of monoids has an underlying exact sequence. \begin{proposition}\label{thm:kernel of p}\label{thm:cokernel of k} Let $(X,A,B,p,k,q,s)$ be a pointed semibiproduct of monoids. The sequence \[\xymatrix{X\ar[r]^k & A \ar[r]^{p} & B} \] is an exact sequence. \end{proposition} \begin{proof} Let $f\colon{Z\to A}$ be a morphism such that $pf=0$. Then the map $\bar{f}=qf$ is a homomorphism \begin{eqnarray*} qf(z+z')&=&q(fz+fz')=q(kqf(z)+spf(z)+kqf(z')+spf(z'))\\ &=& q(kqf(z)+0+kqf(z')+0)\\ &=& qk(qf(z)+qf(z'))=qf(z)+qf(z') \end{eqnarray*} and it is unique with the property $k\bar{f}=f$. Indeed, if $k\bar{f}=f$ then $qk\bar{f}=qf$ and hence $\bar{f}=qf$. This means that $k$ is the kernel of $p$. Let $g\colon{A\to Y}$ be a morphism and suppose that $gk=0$. It follows that $g=gsp$, \begin{equation*} g=g1_A=g(kq+sp)=gkq+gsp=0+gsp=gsp, \end{equation*} and consequently the map $\bar{g}=gs$ is a homomorphism, indeed \begin{eqnarray*} gs(b)+gs(b')=g(sb+sb')=gsp(sb+sb')=gs(b+b'). \end{eqnarray*} The fact that $\bar{g}=gs$ is the unique morphism with the property $\bar{g}p=g$ follows from $\bar{g}ps=gs$ which is the same as $\bar{g}=gs$. Hence $p$ is the cokernel of $k$ and the sequence is exact. \end{proof} The following results show that pointed semibiproducts are stable under pullback and in particular split semibiproducts of monoids are stable under composition. \begin{proposition}\label{thm:stable under pullback} Pointed semibiproducts of monoids are stable under pullback. \end{proposition} \begin{proof} Let $(X,A,B,p,k,q,s)$ be a pointed semibiproduct of monoids displayed as the bottom row in the following diagram which is obtained by taking the pullback of $p$ along an arbitrary morphism $h\colon{C\to B}$, with induced morphism $\langle k,0 \rangle$ and map $\langle sh,1 \rangle$, \begin{eqnarray} \vcenter{\xymatrix{X\ar@<-.5ex>[r]_(.35){\langle k,0 \rangle}\ar@{=}[d]_{} & A\times_B C\ar@{->}@<0ex>[d]^{\pi_1}\ar@<-.5ex>@{..>}[l]_(.6){q\pi_1}\ar@<.5ex>[r]^(.6){\pi_2} & C\ar@{->}[d]^{h} \ar@{->}@<.5ex>@{..>}[l]^(.35){\langle sh,1\rangle}\\ X\ar@<-.5ex>[r]_{k} & A \ar@<-.5ex>@{..>}[l]_{q}\ar@<.5ex>[r]^{p} & B \ar@{->}@<.5ex>@{..>}[l]^{s}.}} \end{eqnarray} We have to show that the top row is a pointed semibiproduct of monoids. By construction we have $\pi_2\langle sh,1\rangle=1_C$, $\pi_2\langle k,0\rangle=0$, $q\pi_1\langle sh,1\rangle=qsh=0$, $q\pi_1\langle k,0\rangle=qk=1_X$. It remains to prove the identity \[(a,c)=(kq(a),0)+(sh(c),c)=(kq(a)+sh(c),c)\] for every $a\in A$ and $c\in C$ with $p(a)=h(c)$, which follows from $a=kq(a)+sp(a)=kq(a)+sh(c)$. \end{proof} The previous results are stated at the level of monoids but are easily extended to unitary magmas. The particular case of semidirect products has been considered in \cite{GranJanelidzeSobral} and the notion of composable pair of pointed semibiproducts is borrowed from there. We say that a pointed semibiproduct $(X,A,B,p,k,q,s)$ can be composed with a pointed semibiproduct $(C,B,D,p',k',q',s')$ if the tuple $$(A\times_B C,A,D,p'p,\pi_1,q'',ss'),$$ in which $q''$ is such that $\pi_1q''=kq+sk'q'p$ and $\pi_2q''=q'p$, is a pointed semibiproduct. Note that in the case of groups the map $q$ is uniquely determined as $q(a)=a-sp(a)$ for all $a\in A$. However this is not the case for monoids nor for unitary magmas. \begin{proposition} A pointed semibiproduct of monoids $(X,A,B,p,k,q,s)$ can be composed with $(C,B,D,p',k',q',s')$, another pointed semibiproduct of monoids, if and only if the map $s$ is equal to the map $sk'q'+ss'p'$. \end{proposition} \begin{proof} Let us observe that the tuple $(A\times_B C,A,D,p'p,\pi_1,q'',ss')$ is a pointed semibiproduct if and only if $\pi_1q''+ss'p'p=1_A$. Indeed, the kernel of the composite $p'p$ is obtained by taking the pullback of $p$ along $k'$, the kernel of $p'$, as illustrated \begin{eqnarray} \vcenter{\xymatrix{ & A\times_B C\ar@{->}@<0ex>[d]_{\pi_1}\ar@<.5ex>[r]^(.6){\pi_2} & C\ar@{->}[d]_{k'} \ar@{->}@<.5ex>@{..>}[l]^(.35){\langle sk',1\rangle}\\ X\ar@<-.5ex>[r]_{k} & A \ar@<-.5ex>@{..>}[l]_{q}\ar@<.5ex>[r]^{p} \ar[d]_{p'p} & B \ar@{->}@<.5ex>@{..>}[l]^{s} \ar[d]_{p'}\\ & D\ar@{=}[r] & D.}} \end{eqnarray} In order to obtain a pointed semibiproduct we complete de diagram with a map $q''$ such that $\pi_1q''=kq+sk'q'p$ and $\pi_2q''=q'p$ as illustrated \begin{eqnarray} \vcenter{\xymatrix{ & A\times_B C\ar@{->}@<-0.5ex>[d]_{\pi_1}\ar@<.5ex>[r]^(.6){\pi_2} & C\ar@{->}[d]_{k'} \ar@{->}@<.5ex>@{..>}[l]^(.35){\langle sk',1\rangle}\\ X\ar@<-.5ex>[r]_{k} & A \ar@<-.5ex>@{..>}[l]_{q} \ar@<-.5ex>@{..>}[u]_{q''} \ar@<.5ex>[r]^{p} \ar@<-.5ex>[d]_{p'p} & B \ar@{->}@<.5ex>@{..>}[l]^{s} \ar[d]_{p'}\\ & D \ar@<-.5ex>@{..>}[u]_{ss'} \ar@{=}[r] & D.}} \end{eqnarray} The map $q''$ is well defined, $p(kq+sk'q'p)=pkq+psk'q'p=k'q'p$. Moreover, $p'pss'=1_D$, $p'p\pi_1=p'k'\pi_2=0$, $q''ss'=0$ and we observe \begin{eqnarray*} q''\pi_1 &=&\langle kq+sk'q'p,q'p \rangle \pi_1\\ &=&\langle kq\pi_1+sk'q'p\pi_1,q'p\pi_1 \rangle \\ &=&\langle kq\pi_1+sk'q'k'\pi_2,q'k'\pi_2 \rangle \\ &=&\langle kq\pi_1+sp\pi_1,\pi_2 \rangle \\ &=&\langle \pi_1,\pi_2 \rangle =1_{A\times_B C}. \end{eqnarray*} It remains to analyse the condition $\pi_1q''+ss'p'p=1_A$. If $s=sk'q'+ss'p'$ then we have $\pi_1q''+ss'p'p=kq+sk'q'p+ss'p'p$ and hence $kq+sp=1_A$. Conversely, having $\pi_1q''+ss'p'p=1_A$ we get $kq+sk'q'p+ss'p'p=1_A$ and $kqs+sk'q'ps+ss'p'ps=s$ so $sk'q'+ss'p'=s$. \end{proof} Note that associativity is used to convert $(kq+sk'q'p)+ss'p'p$ into $kq+(sk'q'p+ss'p'p)$. Moreover, if the map $s$ is a homomorphism then condition $s=sk'q'+ss'p'$ is trivial. A pointed semibiproduct $(X,A,B,p,k,q,s)$ in which the map $s$ is a homomorphism is called a pointed split semibiproduct. This means that pointed split semibiproducts of monoids are stable under composition. \section{The category of pointed semibiproducts of monoids}\label{Sec: Sbp} The purpose of this section is to introduce the category of pointed semibiproducts of monoids, denoted $\mathbf{Psb}$. \begin{definition}\label{def: pointed semibiproduct of monoids} A \emph{pointed semibiproduct of monoids} consists of a tuple $(X,A,B,p,k,q,s)$ that can also be represented as a diagram of the shape \begin{equation} \label{diag: biproduct} \xymatrix{X\ar@<-.5ex>[r]_{k} & A\ar@<-.5ex>@{..>}[l]_{q}\ar@<.5ex>[r]^{p} & B \ar@{..>}@<.5ex>[l]^{s}} \end{equation} in which $X$, $A$ and $B$ are monoids (not necessarily commutative), $p$, $k$, are monoid homomorphisms, while $q$ and $s$ are zero-preserving maps. Moreover, the following conditions are satisfied: \begin{eqnarray} ps&=&1_B\\ qk&=&1_X\\ kq+sp&=&1_A\\ pk&=&0_{X,B}\\ qs&=&0_{B,X}. \end{eqnarray} \end{definition} A morphism in $\mathbf{Psb}$, from the object $(X,A,B,p,k,q,s)$ to the object $(X',A',B',p',k',q',s')$, is a triple $(f_1,f_2,f_3)$, displayed as \begin{equation}\label{diag:morphism of semi-biproduct} \vcenter{\xymatrix{X\ar@<-.5ex>[r]_{k}\ar@{->}[d]_{f_1} & A\ar@{->}@<0ex>[d]^{f_2}\ar@<-.5ex>@{..>}[l]_{q}\ar@<.5ex>[r]^{p} & B\ar@{->}[d]^{f_3} \ar@{->}@<.5ex>@{..>}[l]^{s}\\ X'\ar@<-.5ex>[r]_{k'} & A' \ar@<-.5ex>@{..>}[l]_{q'}\ar@<.5ex>[r]^{p'} & B' \ar@{->}@<.5ex>@{..>}[l]^{s'}}} \end{equation} in which $f_1$, $f_2$ and $f_3$ are monoid homomorphisms and moreover the following conditions are satisfied: $f_2k=k'f_1$, $p'f_2=f_3p$, $f_2s=s'f_3$, $q'f_2=f_1q$. \begin{theorem}\label{thm: a+a' and associativity} Let $(X,A,B,p,k,q,s)$ be a pointed semibiproduct of monoids. For every $a,a'\in A$ the element $a+a'\in A$ can be written in terms of $q(a)$, $q(a')$, $p(a)$ and $p(a')$ as \begin{equation}\label{eq: a+a'} k(q(a)+q(sp(a)+ kq(a'))+q(sp(a)+ sp(a')))+s(p(a)+p(a')). \end{equation} \end{theorem} \begin{proof} We observe: \begin{eqnarray*} a+a'&=& kqa+(spa+kqa')+spa' \qquad(kq+sp=1)\\ &=& kqa+kq(spa+kqa')+sp(spa+kqa')+spa' \\ &=& kqa+kq(spa+kqa')+spa+spa' \qquad( ps=1,pk=0)\\ &=& kqa+kq(spa+kqa')+kq(spa+spa')+sp(spa+spa')\\ &=& kqa+kq(spa+kqa')+kq(spa+spa')+s(pa+pa')\\ &=& k(qa+q(spa+kqa')+q(spa+spa'))+sp(a+a'). \end{eqnarray*} \end{proof} The previous result suggests a transport of structure from the monoid $A$ into the set $X\times B$ as motivated with formula $(\ref{eq: operation})$ in Section \ref{sec: motivation}. However, as we will see, in order to keep an isomorphism with $A$ we need to restrict the set $X\times B$ to those pairs $(x,b)$ for which there exists $a\in A$ such that $x=q(a)$ and $b=p(a)$. \section{The category of pointed monoid action systems}\label{Sec: Act} The purpose of this section is to introduce the category of pointed monoid action systems, which will be denoted as $\mathbf{Act}$. This category is obtained by requiring the existence of a categorical equivalence between $\mathbf{Act}$ and $\mathbf{Psb}$ (see Theorem \ref{thm: equivalence}). \begin{definition}\label{def: pseudo-action} A \emph{pointed monoid action system} is a five-tuple $$(X,B,\rho,\varphi,\gamma)$$ in which $X$ and $B$ are monoids, $\rho\colon{X\times B\to X}$, $\varphi\colon{B\times X\to X}$, $\gamma\colon{B\times B\to X}$ are maps such that the following conditions are satisfied for every $x\in X$ and $b,b'\in B$: \begin{eqnarray} \rho(x,0)=x,\quad \rho(0,b)=0\label{eq:act01}\\ \varphi(0,x)=x,\quad \varphi(b,0)=0\label{eq:act02}\\ \gamma(b,0)=0=\gamma(0,b)\label{eq:act03}\\ \rho(x,b)=\rho(\rho(x,b),b)\label{eq:act04}\\ \varphi(b,x)=\rho(\varphi(b,x),b)\label{eq:act05}\\ \gamma(b,b')=\rho(\gamma(b,b'),b+b')\label{eq:act06} \end{eqnarray} and moreover the following condition holds for every $x,x',x''\in X$ and $b,b',b''\in B$, \begin{eqnarray}\label{eq:act07} \rho(\rho(x+\varphi(b,x')+\gamma(b,b'),b+b')+\varphi(b+b',x'')+\gamma(b+b',b''),b''')=\nonumber\\ =\rho(x+\varphi(b,\rho(x'+\varphi(b', x'') + \gamma(b', b''),{b'+b''}))+\gamma(b, b'+b''),{b'''})\quad \end{eqnarray} where $b'''=b+b'+b''$. \end{definition} A morphism of pointed monoid action systems, say from $(X,B,\rho,\varphi,\gamma)$ to $(X',B',\rho',\varphi',\gamma')$ is a pair $(f,g)$ of monoid homomorphisms, with $f\colon{X\to X'}$ and $g\colon{B\to B'}$ such that for every $x\in X$ and $b,b'\in B$ \begin{eqnarray} f(\rho(x,b))&=&\rho'(f(x),{g(b)}),\label{eq:act08}\\ f(\varphi(b, x))&=&\varphi'(g(b), {f(x)}),\label{eq:act09}\\ f(\gamma(b, b'))&=&\gamma'(g(b), g(b')).\label{eq:act10} \end{eqnarray} \begin{theorem}\label{thm: functor R syntehic construction} There exists a functor $R\colon{\mathbf{Act}\to \mathbf{Mon}}$ such that for every morphism in $\mathbf{Act}$, say $(f,g)\colon{(X,B,\rho,\varphi,\gamma)\to (X',B',\rho',\varphi',\gamma')}$, the diagram \begin{equation}\label{diag:semi-biproduct with R} \xymatrix{X\ar@<-.5ex>[r]_(.3){\langle 1,0\rangle}\ar@{->}[d]_{f} & R(X,B,\rho,\varphi,\gamma)\ar@{->}@<.0ex>[d]^{R(f,g)}\ar@<-.5ex>@{..>}[l]_(.7){\pi_X}\ar@<.5ex>[r]^(.7){\pi_B} & B\ar@{->}[d]^{g} \ar@{->}@<.5ex>@{..>}[l]^(.3){\langle 0,1\rangle}\\ X'\ar@<-.5ex>[r]_(.3){\langle 1,0 \rangle} & R{(X',B',{\rho',\varphi',\gamma'})} \ar@<-.5ex>@{..>}[l]_(.7){\pi_{X}}\ar@<.5ex>[r]^(.7){\pi_B} & B \ar@{->}@<.5ex>@{..>}[l]^(.3){{\langle 0 ,1 \rangle}}} \end{equation} is a morphism in $\mathbf{Psb}$. \end{theorem} The functor $R$ realizes a pointed monoid action system $(X,B,\rho,\varphi,\gamma)$ as a synthetic semibiproduct diagram \begin{equation}\label{diag:synthetic semi-biproduct} \xymatrix{X\ar@<-.5ex>[r]_(.5){\langle 1,0\rangle} & R\ar@<-.5ex>@{..>}[l]_(.5){\pi_X}\ar@<.5ex>[r]^(.5){\pi_B} & B \ar@{..>}@<.5ex>[l]^(.5){\langle 0,1\rangle}} \end{equation} in which $R=R(X,B,{\rho,\varphi,\gamma})=\{(x,b)\in X\times B\mid \rho(x,b)=x\}$ is equipped with the binary synthetic operation \begin{equation}\label{eq: semibiproduct sunthetic operation} (x,b)+(x',b')=(\rho(x+\varphi(b,x')+\gamma(b,b'),b+b'),b+b') \end{equation} which is well defined for every $x,x'\in X$ and $b,b'\in B$ due to condition $(\ref{eq:act04})$ and is associative due to condition $(\ref{eq:act07})$. It is clear that $\pi_B$ is a monoid homomorphism and due to conditions $(\ref{eq:act01})$--$(\ref{eq:act03})$ we see that the maps $\langle 1,0 \rangle $ and $\langle 0 ,1\rangle $ are well defined and moreover $\langle 1,0\rangle $ is a monoid homomorphism. Finally, we observe that a pair $(x,b)\in X\times B$ is in $R$ if and only if $(x,b)=(x,0)+(0,b)$. Further details can be found in the preprint \cite{NMF.20a-of}. \section{The equivalence} In order to establish a categorical equivalence between $\mathbf{Act}$ and $\mathbf{Psb}$ we need a procedure to associate a pointed monoid action system to every pointed semibiproduct of monoids in a functorial manner. \begin{theorem}\label{thm: pseudo-actions} Let $(X,A,B,p,k,q,s)$ be an object in $\mathbf{Psb}$. The system $(X,B,\rho,\varphi,\gamma)$ with \begin{eqnarray} \rho(x,b)=q(k(x)+s(b))\\ \varphi(b,x)=q(s(b)+k(x))\\ \gamma(b,b')=q(s(b)+s(b')) \end{eqnarray} is an object in $\mathbf{Act}$. Moreover, if $(f_1,f_2,f_3)$ is a morphism in $\mathbf{Psb}$ then $(f_1,f_3)$ is a morphism in $\mathbf{Act}$. \end{theorem} \begin{proof} To see that the system $(X,B,\rho,\varphi,\gamma)$ is a well defined object in $\mathbf{Act}$ we recall that $q$ and $s$ are zero-preserving maps and hence conditions $(\ref{eq:act01})$--$(\ref{eq:act03})$ are satisfied. Conditions $(\ref{eq:act04})$--$(\ref{eq:act06})$ are obtained by applying the map $q$ to both sides of equations \begin{eqnarray*} k(x)+s(b)=kq(k(x)+s(b))+s(b)\\ s(b)+k(x)=kq(s(b)+k(x))+s(b)\\ s(b)+s(b')=kq(s(b)+s(b'))+s(b+b') \end{eqnarray*} which hold because $(X,A,B,p,k,q,s)$ is a pointed semibiproduct of monoids. Condition $(\ref{eq:act07})$ follows from Theorem \ref{thm: a+a' and associativity} with $a=k(x)+s(b)$, $a'=k(x')+s(b')+k(x'')+s(b'')$ on the one hand whereas on the other hand $a=k(x)+s(b)+k(x')+s(b')$, $a'=k(x'')+s(b'')$. Moreover, the pair $(f_1,f_3)$ is a morphism of actions as soon as the triple $(f_1,f_2,f_3)$ is a morphism of semibiproducts, indeed we have \begin{eqnarray*} f_1(\rho(x,b))&=&f_1q(k(x)+s(b))=q'f_2(k(x)+s(b))\\ &=&q'(k'f_1(x)+s'f_3(b)))=\rho'(f_1(x),f_3(b)) \end{eqnarray*} and similarly for $\varphi$ and $\gamma$ thus proving conditions $(\ref{eq:act08})$--$(\ref{eq:act10})$. \end{proof} The previous result describes a functor from the category of pointed semibipro\-ducts of monoids into the category of pointed monoid action systems, let us denote it by $P\colon{\mathbf{Psb}\to\mathbf{Act}}$. The synthetic construction of Theorem \ref{thm: functor R syntehic construction} produces a functor in the other direction, let us denote it $Q\colon{\mathbf{Act}\to\mathbf{Psb}}$. We will see that $PQ=1$ whereas $QP\cong 1$. \begin{theorem}\label{thm: equivalence} The categories $\mathbf{Psb}$ and $\mathbf{Act}$ are equivalent. \end{theorem} \begin{proof} Theorem \ref{thm: pseudo-actions} tells us that $P(X,A,B,p,k,q,s)=(X,B,\rho,\varphi,\gamma)$ and $P(f_1,f_2,f_3)=(f_1,f_3)$ is a functor from $\mathbf{Psb}$ to $\mathbf{Act}$ whereas Theorem \ref{thm: functor R syntehic construction} gives a functor $Q$ in the other direction. It is clear that $Q(X,B,\rho,\varphi,\gamma)=(X,R,B,\pi_B,\langle 1,0\rangle,\pi_X,\langle 0,1\rangle)$ is the synthetic realization displayed in $(\ref{diag:synthetic semi-biproduct})$ and hence it is a pointed semibiproduct. Moreover $Q(f,g)=(f,R(f,g),g)$ with $R(f,g)$ illustrated as in $(\ref{diag:semi-biproduct with R})$ and defined as $R(f,g)(x,b)=(f(x),g(b))$ is clearly a morphism of semibiproducts. We observe that $PQ(X,B,\rho,\varphi,\gamma)=(X,B,\rho,\varphi,\gamma)$ due to conditions $(\ref{eq:act05})$ and $(\ref{eq:act06})$. This proves $PQ=1$, in order to prove $QP\cong 1$ we need to specify natural isomorphisms $\alpha$ and $\beta$ as illustrated \begin{equation} \vcenter{\xymatrix{A\ar@<.5ex>[r]^(.3){\alpha_A}\ar[d]_{f_2} & RP(X,A,B,p,k,q,s)\ar@<.5ex>[l]^(.7){\beta_A}\ar[d]^{R(f_1,f_3)}\\ A' \ar@<.5ex>[r]^(.3){\alpha_{A'}} & RP(X',A',B',p',k',q',s')\ar@<.5ex>[l]^(.7){\beta_{A'}} }} \end{equation} and show that they are compatible with diagrams $(\ref{diag:morphism of semi-biproduct})$ and $(\ref{diag:semi-biproduct with R})$. Indeed it is a routine calculation to check that $\alpha(a)=(q(a),p(a))$ and $\beta(x,b)=k(x)+s(b)$ are well defined natural isomorphisms compatible with semibiproducts. Further details can be found in the preprint \cite{NMF.20a-of}. \end{proof} \section{Examples}\label{sec: eg} Several examples can be found in the preprint \cite{NMF.20a-of}. Here we list all the possible pointed semibiproducts of monoids $(X,A,B,p,k,q,s)$ in which $X$ and $B$ are monoids with two elements. This particular case is interesting because it gives a simple list with all the possible components of an action system $(X,B,\rho,\varphi,\gamma)$. The equivalence of Theorem \ref{thm: equivalence} then gives us an easy way of checking all the possibilities. Let us denote by $M$ and $G$ the two monoids with two elements, $M$ being the idempotent monoid while $G$ being the group, both expressed in terms of multiplication tables as \[M=\begin{pmatrix} 1 & 2 \\ 2 & 2 \end{pmatrix}, \quad B=\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}.\] Note that we are using multiplicative notation so that $2\cdot 2=2$ in $M$, whereas in $G$ we have $2\cdot 2=1$. Due to restrictions $(\ref{eq:act01})$--$(\ref{eq:act03})$ we have the following two possibilities for each component $\rho$, $\varphi$ and $\gamma$: \[\rho_0=\begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix} , \quad \rho_1=\begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix},\] \[\varphi_0=\begin{pmatrix} 1 & 2 \\ 1 & 2 \end{pmatrix}, \quad \varphi_1=\begin{pmatrix} 1 & 2 \\ 1 & 1 \end{pmatrix},\] \[\gamma_0=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \quad \gamma_1=\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}.\] The following list shows all the possible 14 cases of pointed semibiproducts of monoids $(X,A,B,p,q,k,q,s)$ in which $X$ and $B$ are either $M$ or $G$ via the equivalence of Theorem \ref{thm: equivalence}. \begin{multicols}{2} \begin{enumerate} \item $(G,G,\rho_0,\varphi_0,\gamma_0)$ \item $(G,G,\rho_0,\varphi_0,\gamma_1)$ \item $(G,M,\rho_0,\varphi_0,\gamma_0)$ \item $(G,M,\rho_0,\varphi_0,\gamma_1)$ \item $(G,M,\rho_0,\varphi_1,\gamma_0)$ \item $(G,M,\rho_1,\varphi_1,\gamma_0)$ \item $(M,G,\rho_0,\varphi_0,\gamma_0)$ \item $(M,G,\rho_0,\varphi_0,\gamma_1)$ \item $(M,G,\rho_1,\varphi_1,\gamma_1)$ \item $(M,M,\rho_0,\varphi_0,\gamma_0)$ \item $(M,M,\rho_0,\varphi_0,\gamma_1)$ \item $(M,M,\rho_0,\varphi_1,\gamma_0)$ \item $(M,M,\rho_0,\varphi_1,\gamma_1)$ \item $(M,M,\rho_1,\varphi_1,\gamma_0)$ \end{enumerate} \end{multicols} Note that the cases with $\gamma_0$ correspond to split extensions while the cases with $\rho_0$ correspond to Schreier extensions. The cases with $\rho_1$ correspond to $R=\{(1,1),(1,2),(2,1)\}$ since $(2,2)$ fails to be in $R$ because $\rho_1(2,2)=1\neq 2$. If interpreting $\varphi$ as an action then the map $\varphi_0$ is the trivial action whereas $\varphi_1$ is a non-trivial action. \section{Conclusion} A new tool has been introduced for the study of monoid extensions from which a new notion of action has emerged in order to establish the categorical equivalence of Theorem \ref{thm: equivalence}. A clear drawback to this approach is the necessity of handling morphisms and maps at the same level. We have solved the problem by extending the hom-functor through an appropriate profunctor (Definition \ref{def: semibiproduct}). Other possible solutions would consider maps as an extra structure in higher dimensions \cite{Brown,NMF.15,NMF.20a-in} or as imaginary morphisms \cite{BZ,MontoliRodeloLinden}. Developing further a categorical framework in which to study semibiproducts seems desirable due to several important cases occurring in different settings. For example, semibiproduct extensions can be studied in the context of preordered monoids \cite{NMF.20b} and preordered groups \cite{Preord}, where the maps $q$ and $s$ are required to be monotone maps rather than zero-preserving maps. The context of topological monoids \cite{Ganci} should also be worthwhile studying with $q$ and $s$ required to be continuous maps. \section*{Acknowledgements} This work was supported by Fundação para a Ciência e a Tecnologia (FCT UID-Multi-04044-2019), Centro2020 (PAMI -- ROTEIRO\-/0328\-/2013 -- 022158) and by the Polytechnic of Leiria through the projects CENTRO\--01\--0247: FEDER\--069665, FEDER\--069603, FEDER\--039958, FEDER\--039969, FEDER\--039863, FEDER\--024533.

proofpile-arXiv_069-8208

\section{Introduction} A $k$-{\it simplex\/} is a $k$-dimensional convex polytope with $k+1$ vertices. For $k=0,1,2,3$ respectively, a $k$-simplex is usually called a {\it vertex\/}, {\it edge\/}, {\it triangle\/}, {\it tetrahedron\/}. When $k$ is not important, a $k$-simplex is just called a {\it simplex\/}. A {\it simplicial complex\/} is a finite collection $\cal C$ of simplices, all in the ambient Euclidean space, such that \begin{itemize} \item If $S \in \cal C$ and $S'$ is a sub-simplex of $S$ then $S' \in \cal C$. \item If $S,T \in \cal C$ then $S \cap T$ is either empty or in $\cal C$. \end{itemize} Informally, the simplices in a simplicial complex fit together cleanly, without crashing through each other. The {\it support\/} $|{\cal C\/}|$ of $\cal C$ is the union of all the simplices in $\cal C$. Often we blur the distinction between $\cal C$ and $|{\cal C\/}|$ and think of a simplicial complex as a union of simplices. A simplicial complex may be described with no mention of the ambient space containing it, but there is always the understanding that in principle one can find an isomorphic complex in some Euclidean space. For example, let $\R\P^2_6$ be the quotient of the regular icosahedron by the antipodal map. This simplicial complex has $6$ vertices, $15$ edges, and $10$ faces. One can reconstruct $\R\P^2_6$ in $\R^5$ by fixing some $5$-simplex $\Sigma \subset \R^5$, the convex hull of vertices $v_1,...,v_6$, then mapping vertex $k$ of $\R\P^2_6$ to $v_k$ and extending linearly. The complex $\R\P^2_6$ is called a $6$-{\it vertex triangulation\/} of the real projective plane $\R\P^2$ because its support is homeomorphic to $\R\P^2$. This triangulation has the fewest number of vertices amongst triangulations of $\R\P^2$, so it is called a {\it minimal triangulation\/} of $\R\P^2$. (Smaller examples like the quotient of the regular octahedron by the antipodal map fail to be simplicial complexes.) Here are some other examples related to minimal triangulations. \begin{itemize} \item The boundary of a tetrahedron is the unique $4$-vertex minimal triangulation of the $2$-sphere. More generally, the boundary of a $(k+1)$ simplex is the unique minimal triangulation of the $k$-sphere. \item If you identify the opposite sides of the big hexagon in Figure 2 below, you get the unique minimal triangulation $T_7^2$ of the $2$-torus. $T_7^2$ has $14$ triangles, $21$ edges, and $7$ vertices. \item In $1980$, W. K\"{u}hnel discovered $\C\P^2_9$, the unique $9$-vertex minimal triangulation of the complex projective plane $\C\P^2$. This triangulation has $36$ $4$-simplices and a symmetry group of order $54$. \item In $1992$, U. Brehm and W. K\"{u}hnel [{\bf BK\/}] defined $\H\P_{15}^2$ (and two variants), a $15$-vertex simplicial complex with $490$ $8$-simplices. In $2019$, D. Gorodkov [{\bf G\/}] proved that $\H\P_{15}$ and the variants are PL homeomorphic to the quaternionic projective plane $\H\P^2$. \item In 2021, K. Adiprasito, S. Avvakumov, R. Karasev [{\bf AAK\/}] proved that real projective space can be triangulated using a sub-exponential number of simplices. \end{itemize} The survey article by B. Datta [{\bf D\/}] has a wealth of information about minimal triangulations up to the year $2007$, and a large number of references. The subject of this paper is $\C\P_2^9$, the complex discovered by Wolfgang K'\"{u}hnel. In [{\bf KB\/}], K\"{u}hnel and T. Banchoff, establish many interesting properties of $\C\P_9^2$ and give a rather intricate proof that $\C\P_9^2$ really is homeomorphic to $\C\P^2$. Since [{\bf KB\/}], there has been a lot of work done trying to understand $\C\P_9^2$ from various points of view. In particular, there are a number of proofs that $\C\P_9^2 \cong \C\P^2$, and also a number of proofs that $\C\P_9^2$ is the only minimal triangulation of $\C\P^2$. See the article by B. Morin and M. Yoshida [{\bf MY\/}] for a survey of these proofs. See also the paper by B. Bagchi and B. Datta [{\bf BB\/}]. I learned about $\C\P_9^2$ from Tom Banchoff, my colleague at Brown, some years ago. Last semester, while teaching algebraic topology, I mentioned this example to my class and this got me thinking about it. I subsequently found a really nice proof that $\C\P_9^2 \cong \C\P^2$. I will give the proof in this paper. The idea is to recall the trisection of $\C\P^2$ into $3$ bi-disks, and then to see this trisection inside a symmetry-breaking subdivision of $\C\P_9^2$. The picture developed here is related to the $10$-vertex triangulation $\C\P^2_{10}$ of $\C\P^2$ that Banchoff and K\"{u}hnel [{\bf BK\/}] construct by building outward from $T_7^2$. Indeed, Denis Gorodkov (in a private communication) very recently refined the subdivision idea here to find a ``path'' from $\C\P^2_9$ to $\C\P^2_{10}$ using something akin to bi-stellar flips. My proof also has a close kinship with the ``red-white-blue discussion'' in \S 1.3 of the M.P.I.M. preprint by Morin and Yoshida that is the precursor to [{\bf MY\/}] (and has the same title). This discussion is, in turn, related to Figure 8 in [{\bf KB\/}]. Morin and Yoshida describe the red-white-blue discussion as a ``topological insight'' but they don't really push it forward into a proof. I think that my picture is very similar, but clarified by the special subdivision. The approach here possibly could shed light on Gorodkov's result that $\H\P^2_{15} \cong \H\P^2$. The same subdivision and trisection idea goes through for $\H\P^2_{15}$ almost {\it verbatim\/}, and I can see computationally that each of the $3$ sub-complexes is shellable and therefore PL homeomorphic to an $8$-ball. However, the high dimensional topology involved in analyzing $\H\P_{15}^2$ makes a direct topological analysis of the whole complex formidable. For instance, the sub-complex that plays the role of $T^2_7$ has $288$ $6$-simplices. This monster ought to be homeomorphic to $(S^3 \times S^3 \times S^3)/S^3$ in a way that respects the $3$-fold symmetry. In \S 2 I will recall the trisection of $\C\P^2$ and discuss a few key properties of the {\it central torus\/} in this decomposition. In \S 3 I will describe $\C\P_9^2$ and then explain my symmetry-breaking subdivision. In \S 4 I will find the trisection inside the subdivision and construct a homeomorphism from $\C\P_9^2$ to $\C\P^2$ which respects the trisections. I thank Tom Banchoff, Kenny Blakey, Joe Hlavinka, Wolfgang K\"{u}hnel, Tyler Lane, and Dennis Sullivan for helpful discussions. \section{The Smooth Trisection} The complex projective plane $\C\P^2$ is the space of scale equivalence classes of nonzero vectors in $\C^3$. We denote the equivalence class of $(z_1,z_4,z_7) \in \C^3$ by $[z_1:z_4:z_7] \in \C\P^2$. We chose the variable names so that they line up with the notation for $\C\P_9^2$. We have the trisection $\C\P^2=\beta_1 \cup \beta_4 \cup \beta_7$, where $\beta_j$ is the set where $\max(|z_1|,|z_4|,|z_7|)=|z_j|$. Points in $\beta_1$ may be written uniquely as $[1:w:z]$ with $|w|, |z| \leq 1$. Thus $\beta_1$ is the product of $2$ unit disks. So are $\beta_4$ and $\beta_7$. These bi-disks have disjoint interiors and are permuted by the map $$\Sigma: (z_1,z_4,z_7) \to (z_4,z_7,z_1)$$ The boundary $\partial \beta_1$ is a $3$-sphere, and it decomposes into the solid tori $\beta_{14}$ and $\beta_{17}$. Here $\beta_{ij}=\beta_i \cap \beta_j$. To see that $\beta_{14}$ is a solid torus, note that $\beta_{14}$ consists of points of the form $[1:u:z]$ with $|z| \leq |u|=1$ and is therefore the product of the unit disk and the unit circle. The {\it central torus\/} $\beta_{147}=\beta_{14} \cap \beta_{17}=\beta_1 \cap \beta_4 \cap \beta_7$ consists of points where $|z_1|=|z_4|=|z_7|$. We discuss $\beta_{147}$ in more detail, with a view towards recognizing it inside $\C\P_9^2$. \newline \newline {\bf Hexagonal Structure:\/} Let $\R^3_0 \subset \R^3$ denote the plane of points whose coordinates sum to $0$. Let $H \subset \R^3_0$ be the regular hexagon whose vertices are all permutations of $(1,-1,0)$. Let $\overline H$ be the flat torus obtained by identifying the opposite sides of $H$ by translations. The translation vectors are the cyclic permutations of $\pm (1,1,-2)$. The map $$(x_1,x_4,x_7) \to [\underline x_1: \underline x_4: \underline x_7], \hskip 30 pt \underline x=e^{\frac{2 \pi i x}{3}}$$ induces a homeomorphism $\overline H \to \beta_{147}$. The main point behind this fact is that $[\underline 1:\underline 1: \underline{-2}]=[\underline 0:\underline 0:\underline 0]$, etc. We equip $\beta_{147}$ with the metric which makes $\overline H \to \beta_{147}$ an isometry. The $3$ fixed points of $\Sigma$ lie in $\beta_{147}$ and correspond to the points on $\overline H$ represented by the center and vertices of $H$. \newline \newline {\bf A Contractible Loop:\/} The line in $\R_0^3$ where $x_1=x_4$ bisects $H$ and contains the midpoints of a pair of opposite sides. This line gives rise to a geodesic loop in $\overline H$. See the loop $a_{14}$ in Figure 2 below. The corresponding loop $\alpha_{14} \subset \beta_{147}$ is given by $[1:1:u]$, where $u$ ranges through all unit complex numbers. The loop $\alpha_{14}$ is contractible in $\beta_{14}$: It bounds the disk in $\beta_{14}$ consisting of points $[1:1:z]$ with $|z| \leq 1$. \section{The Complex and its Subdivision} The vertices of $\C\P_9^2$ are labeled $1,...,9$. Here are $16$ of the $36$ $4$-simplices of $\C\P_9^2$ listed on p. 15 of [{\bf KB\/}]. \begin{itemize} \item $15289$\ $12389$\ $13689$\ $45289$\ $42389$\ $43689$ \item $14256$\ $14356$\ $14259$\ $14368$ \item $14726$\ $14768$ ($14783$\ $14735$\ $14759$\ $14792$) \end{itemize} Comparing our list to [{\bf KB\/}], we have sometimes permuted the vertices so as to highlight the indices $1,4,7$. The other $24$ $4$-simplices are orbits of the first $12$ under the action of the {\it fundamental permutation\/}: $$S=(147)(258)(369).$$ For instance, $14726$ has orbit $14726 \to 14759 \to 14783$. The four simplices in parentheses are listed for the sake of making our tetrahedron list below more transparent. In [{\bf KB\/}] the authors exhibit a symmetry group of order $54$ acting on $\C\P_9^2$. Our proof only uses $S$. Figures 1 and 2 below exhibit the additional symmetry $(23)(56)(89)$. \newline Let $[ij]$ be the midpoint of the edge $i \leftrightarrow j$. Let $[ijk]$ be the center of the triangle $ijk$. We only care about the points $[14], [17], [47], [147]$, though $[258]$ and $[369]$ also arise naturally in Figure 2 below. Let the {\it rank\/} of a simplex be the number of vertices which belong to the set $\{1,4,7\}$. Our list above goes by rank. We divide each rank $k$ simplex into $k!$ smaller simplices, as follows: The rank $1$ simplices are untouched. The rank $2$ simplex $14abc$ divides into $$1[14]abc \hskip 15 pt 4[14]abc,$$ and likewise with the indices $1,4,7$ permuted. The rank $3$ simplex $147ab$ divides into {\footnotesize $$ 1[14][147]ab\hskip 10 pt 1[17][147]ab\hskip 10 pt 4[14][147]ab\hskip 10 pt 4[47][147]ab\hskip 10 pt 7[17][147]ab\hskip 10 pt 7[47][147]ab. $$ \/} We replace our original $36$ simplices with the subdivided simplices. Since there are respectively $18,12,6$ simplices of rank $1,2,3$ we get a total of $$(1,2,6) \cdot( 18,12,6)=78=3 \times 26$$ {\it new\/} simplices. (The rank $1$ simplices count as ``new''.) Each new simplex has exactly one vertex from the set $\{1,4,7\}$. \section{The Combinatorial Trisection} We have $\C\P_9^2=B_1 \cup B_4 \cup B_7,$ where $B_j$ is the union of the $26$ new simplices having $j \in \{1,4,7\}$ as a vertex. Each $B_j$ is the cone to vertex $j$ of $\partial B_j$. Hence $B_i$ and $B_j$ have disjoint interiors for $i \not = j$. We get $13$ tetrahedra contained in $B_{14}=B_1 \cap B_4$ by subdividing the simplices on our list above and omitting $1$ or $4$: \begin{itemize} \item $5 2 8 9$\hskip 10 pt $2 3 8 9 $\hskip 10 pt $3 6 8 9 $ \item $[14] 2 5 6 $\hskip 10 pt $[14] 3 5 6 $\hskip 10 pt $[14] 2 5 9$\hskip 10 pt $[14] 3 6 8 $ \item {\footnotesize $[14] [147] 2 6$\hskip 10 pt $[14] [147] 6 8 $\hskip 10 pt $[14] [147] 8 3$\hskip 10 pt $[14] [147] 3 5$\hskip 10 pt $[14] [147] 5 9$\hskip 10 pt $[14] [147] 9 2 $\/} \end{itemize} The images of these $13$ tetrahedra under $S^{-1}$ lie in $B_{17}$ and are totally distinct from the ones above. This accounts for all $26$ tetrahedra in $\partial B_1$. Hence, the $13$ above are the complete list of tetrahedra comprising $B_{14}$, and moreover $B_{14}$ and $B_{17}$ have disjoint interiors. To see that $B_{14}$ is a solid torus, write $B_{14}=B_{14}' \cup B_{14}''$, where $B_{14}'$ is the union of the first $3$ tetrahedra above and $B_{14}''$ is the union of the last ten. $B_{14}'$ is a $3$-ball because it is the join of the path $5236$ with the segment $89$, and $B_{14}''$ is a $3$-ball because it is the cone to vertex $[14]$ of $\partial B_{14}''$, a $10$-triangle triangulation of the $2$-sphere. Figure 1 shows $\partial B_{14}'$ and $\partial B_{14}''$. Each one is drawn as the union of $2$ combinatorial hexagons glued along their boundaries according to the labels. $B_{14}' \cap B_{14}''$ is the union of the $2$ disjoint grey triangles $259$ and $368$. Topologically, we get $B_{14}$ by gluing two $3$-balls together along a pair of disjoint disks in their boundaries. The orientations of the gluings are such that the result is a solid torus. \begin{center} \resizebox{!}{2.2in}{\includegraphics{links1.eps}} \newline Figure 1: $\partial B_{14}'$ and $\partial B_{14}''$. Glue the hex boundaries together. \end{center} \begin{center} \resizebox{!}{3.7in}{\includegraphics{links2.eps}} \newline Figure 2: The universal covering of the triangulation of $B_{147}$. \end{center} We get the triangulation of $B_{147}=\partial B_{14}$ by gluing the two triangulations from Figure 1 along the grey triangles. Figure 2 shows the universal cover of the triangulation. We get back to $B_{147}$ by gluing the opposite sides of the big hexagon by translations. This triangulation of $B_{147}$ is exactly $T_7^2$. We also note that $S$ acts on $B_{147}$ fixing $[147]$, $[258]$, $[369]$, points which respectively correspond to the center and vertices of the hexagon. From all this structure we see that (after suitably scaling) there is an isometry $h_{147}: B_{147} \to \beta_{147}$ which conjugates $S$ to $\Sigma$ and which maps the blue loop $a_{14}$ to $\alpha_{14}$. Note that $a_{14}$ is contractible in $B_{14}$ because $a_{14} \subset B_{14}''$, and recall that $\alpha_{14}$ is contractible in $\beta_{14}$. Hence $h_{147}$ extends to a homeomorphism $h_{14}: B_{14} \to \beta_{14}$. Define $h_{17}=\Sigma^{-1} \circ h_{14} \circ S$ and $h_{47}=\Sigma \circ h_{14} \circ S^{-1}$. This gives us homeomorphisms $h_{17}: B_{17} \to \beta_{17}$ and $h_{47}: B_{47} \to \beta_{47}$. The maps $h_{ij}$ all agree on $B_{147}$ because $h_{147}$ conjugates $S$ to $\Sigma$. The union $h=h_{14} \cup h_{17} \cup h_{47}$ is a homeomorphism from $\partial B_1 \cup \partial B_4 \cup \partial B_7$ to $\partial \beta_1 \cup \partial \beta_4 \cup \partial \beta_7$ which respects the individual pieces and their intersections. Since $B_j$ and $\beta_j$ are cones over $\partial B_j$ and $\partial \beta_j$ we can extend $h$, by coning, to a homeomorphism from $\C\P_9^2=B_1 \cup B_4 \cup B_7$ to $\C\P^2=\beta_1 \cup \beta_4 \cup \beta_7$. \section{References} \noindent [{\bf AAK\/}], K. Adiprasito, S. Avvakumov, R. Karasev, {\it A subexponential size triangulation of $\R\P^n$,\/} arXiv: math:2009.02703v2 (2021) \newline \newline \noindent [{\bf BB\/}] B. Bagchi and B. Datta, {\it A short proof of the uniqueness of K\"{u}hnel's $9$-vertex complex projective plane\/}, Advances in Geometry (2001) pp 157-163. \newline \newline [{\bf BK\/}] T. Banchoff and W K\"{u}hnel, {\it Equilibrium triangulations of the complex projective plane\/}, Geometriae Dedicata {\bf 44\/} (1992) pp 311-333 \newline \newline [{\bf BrK\/}] U. Brehm and W K\"{u}hnel, {\it $15$-vertex triangulations of an $8$-manifold\/}, Math Annalen {\bf 294\/} (1992) pp 167--193 \newline \newline [{\bf D\/}] B. Datta, {\it Minimal Triangulations of Manifolds\/}, arXiv:math/0701735v1 (2007) \newline \newline [{\bf G\/}] D. Gorodkov, {\it A $15$-Vertex Triangulation of the Quaternionic Projective Plane\/}, Discrete Computational Geometry, (Sept 2019) pp 348-373 (2019) \newline \newline [{\bf KB\/}]: W. K\"{u}hnel and T. F. Banchoff, {\it The $9$-Vertex Complex Projective Plane\/}, Math. Intelligencer. Vol 5, No 3 (1983) pp 11-22 \newline \newline [{\bf MY\/}] A. Morin and M. Yoshida, {\it The K\"{u}hnel trangulation of the complex projective plane from the point of view of complex crystallography\/}, Mem, Fac. Sci. Kyushu Univ. Ser. A {\bf 45\/} (1991) pp 55-142 \noindent \end{document} \end{document}

proofpile-arXiv_069-8227

\section{Introduction} Recently, Kandi\'{c} and \v{S}ivic \cite{KS} have studied order analogs of Gerstenhaber's theorem stating that the dimension of the unital algebra generated by two commuting $n \times n$ matrices is at most $n$. They showed that the dimension of the unital algebra generated by two positive $n \times n$ matrices $A$ and $B$ is at most $n (n+1)/2$ provided its commutator $[A, B] = AB-BA$ is also positive (see Theorem \ref{one}). Here positivity of a matrix means that it has nonnegative entries. We prove an extension of this result in the case when the matrix $A$ has distinct eigenvalues (see Theorem \ref{finitely}). Under the same assumption on $A$ we then consider the unital algebra generated by the super left-commutant of $A$, that is the collection of all positive matrices $B$ such that $[A,B] \ge 0$. We prove that the dimension of this algebra is at most $n (n+1)/2$ (see Corollary \ref{super left-commutant}). If $A$ is a positive diagonal matrix with distinct diagonal entries, then this upper bound is attained (see Theorem \ref{distinct_diagonal}). It has been also shown in \cite{KS} that $9$ is the largest dimension of the unital algebra generated by two positive idempotent matrices $E$ and $F$ satisfying $E F \ge F E$ (see Corollary \ref{main_matrix}). Moreover, the paper \cite{KS} provides a nontrivial example showing that this upper bound can be attained. In our paper this result is proved in more transparent way that also gives some insight in constructing the just-mentioned example. Extensions to the vector lattice setting are also considered. In \cite{GLS}, it is shown that a unital algebra generated by two $n \times n$ matrices with quadratic minimal polynomials is at most $2n$-dimensional if $n$ is even, and at most $(2n - 1)$-dimensional if $n$ is odd. Examples in \cite{GLS} show that the bounds on dimensions are sharp even in the case of idempotents. We give such examples in which the two idempotents are also positive matrices. \vspace{3mm} \section{Preliminaries} Since some of our results hold in general setting of vector lattices, we recall some basic definitions and properties of vector lattices and operators on them. For the terminology and details not explained here we refer the reader to \cite{AA} or \cite{AB}. Let $L$ be a vector lattice with the positive cone $L^+$. The band $$ S^d :=\{x\in L:\; |x|\wedge |y|=0 \textrm{ for all }y \in S\}$$ is called the {\it disjoint complement} of a set $S$ of $L$. A band $B$ of $L$ is said to be a {\it projection band} if $L=B\oplus B^d$. If every band of $L$ is a projection band, we say that the vector lattice $L$ has {\it the projection property}. Let $A$ be a positive (linear) operator on a vector lattice $L$. The {\it null ideal} $\cN(A)$ is the ideal in $L$ defined by $$ \cN(A) = \{ x \in L : A |x| = 0 \} . $$ When $\cN(A) = \{0\}$, we say that the operator $A$ is {\it strictly positive}. The {\it range ideal} $\cR(A)$ of $A$ is the ideal generated by the range of $A$, that is, $$ \cR(A) = \{ y \in L : \, \exists x \in L^+ \textrm{\ \ such that \ } |y| \le A x \} . $$ An operator $A$ on $L$ is called {\it order continuous} if every net $\{x_\alpha\}$ order converging to zero is mapped to the net $\{A x_\alpha\}$ order converging to zero as well. It is easy to verify that the null ideal of an order continuous positive operator is always a band of $L$. On the other hand, the range ideal of an order continuous positive operator is not necessarily a band, even if the operator is idempotent. \begin{example} Let $E : l^2 \to l^2$ be the order continuous positive operator defined by $E x = \langle x, u \rangle u$, where $u = (2^{-1/2}, 2^{-2/2}, 2^{-3/2}, 2^{-4/2}, 2^{-5/2}, \ldots) \in l^2$. Since $\|u\|_2 = 1$, we have $E^2 = E$. Clearly, $\cR(E)$ is the ideal generated by $u$, and it is not equal to $l^2$, as $(2^{-1/2}, 2 \cdot 2^{-2/2}, 3 \cdot 2^{-3/2}, 4 \cdot 2^{-4/2}, 5 \cdot 2^{-5/2}, \ldots) \not\in \cR(E)$. On the other hand, we have $\cR(E)^d = \{0\}$. \end{example} We will make use of the following simple lemma. \begin{lemma} \label{zero} Let $L$ be an Archimedean vector lattice. Let $A$ be a positive operator on $L$ such that $\cR(A)^d = \{0\}$, and let $B$ be an order continuous positive operator on $L$ such that $B A = 0$. Then $B= 0$. \end{lemma} \begin{proof} Assume first that $0 \le y \in \cR(A)$. Then there is a positive vector $x \in L^+$ such that $y \le A x$. It follows that $0 \le B y \le B A x = 0$, and so $B y = 0$. Assume now that $y \in L$. Since $\cR(A)^{d d} = L$, there exists a net $\{y_\alpha\} \subset \cR(A)$ order converging to $y$. As $By_\alpha = 0$ and $B$ is order continuous, we obtain that $By = 0$. \end{proof} A family $\cF$ of operators on a $n$-dimensional vector space $X$ is {\it reducible} if there exists a nontrivial subspace of $X$ that is invariant under every operator from $\mathcal F$. Otherwise, $\cF$ is {\it irreducible}. A family $\cF$ is said to be {\it triangularizable} if there is a basis of $X$ such that all operators in $\cF$ have upper triangular representation with respect to that basis. Clearly, triangularizability is equivalent to the existence of a chain of invariant subspaces $$ \{0\} = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_n = X $$ with the dimension of $M_j$ equal to $j$ for each $j=0, 1, 2, \ldots, n$. Any such chain is called a {\it triangularizing chain} of $\cF$. Order analogs of these concepts are defined as follows. A family $\mathcal F$ of operators on an $n$-dimensional vector lattice $L$ is said to be {\it ideal-reducible} if there exists a nontrivial ideal of $L$ that is invariant under every operator from $\mathcal F$. Otherwise, we say that $\mathcal F$ is {\it ideal-irreducible}. A family $\mathcal F$ is said to be {\it ideal-triangularizable} if it is triangularizable and at least one of (possibly many) triangularizing chains of $\mathcal F$ consists of ideals of $L$. More information on triangularizability can be found in \cite{RR}. For a complex $n \times n$ matrix $A$, the {\it commutant} $\{A\}^\prime$ is the algebra of all matrices $B$ such that $A B = BA$. For a family $\mathcal F$ of complex $n \times n$ matrices, let $\lin (\cF)$ and $\alg (\cF)$ denote the subspace and the algebra generated by the family $\mathcal F$, respectively. By $J_n$ we denote the nilpotent $n \times n$ Jordan block. For $i, j \in \{1,2, \ldots, n \}$, let $E_{i j}$ denote the $n \times n$ matrix whose entries are all $0$ except in the $(i,j)$ cell, where it is $1$. Let $A$ be a positive $n \times n$ matrix. The {\it super left-commutant} $\langle A ]$ is the collection of all positive matrices $B$ such that $[A,B] \ge 0$. Similarly, the {\it super right-commutant} $[ A \rangle$ is the collection of all positive matrices $B$ such that $[A,B] \le 0$. Since $B \in \langle A ]$ if and only if $B^T \in [ A^T \rangle$, we will consider super left-commutants only. It is easy to verify that $\langle A ]$ is an additive and multiplicative semigroup of positive matrices. It follows easily that $\lin (\langle A ]) = \alg (\langle A ])$. More about super commutants can be found in \cite{AA}. \vspace{3mm} \section{Positive matrices} Searching for order analogs of Gerstenhaber's theorem, Kandi\'{c} and \v{S}ivic \cite{KS} have recently proved the following theorem \cite[Theorem 3.2]{KS}. In fact, this theorem also follows from \cite[Proposition 4.3]{KS_Pos}. \begin{theorem} \label{one} Let $A$ and $B$ be positive $n \times n$ matrices such that $[A, B] \ge 0$. Then the unital algebra $\cA$ generated by $A$ and $B$ is triangularizable, and so its dimension is at most $n (n+1)/2$. \end{theorem} This result raises a question under which conditions the whole super left-commutant $\langle A ]$ is triangularizable. A possible answer is given by the following theorem and its corollary. \begin{theorem} \label{finitely} Let $A$ be a positive $n \times n$ matrix with distinct eigenvalues. Let $B_1$, $B_2$, $\ldots$, $B_k$ be positive matrices such that $[A, B_i] \ge 0$ for all $i=1, \ldots, k$. Then the unital algebra $\cA$ generated by $A$, $B_1$, $B_2$, $\ldots$, $B_k$ is triangularizable, and so its dimension is at most $n (n+1)/2$. \end{theorem} \begin{proof} Let $S= A + B_1+ \ldots + B_k$. Then, up to similarity with a permutation matrix, we may assume that $$ S=\left( \begin{matrix} S_{11} & S_{12} & S_{13} & \ldots & S_{1m} \\ 0 & S_{22} & S_{23} & \ldots & S_{2m} \\ 0 & 0 & S_{33} & \ldots & S_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & S_{mm} \end{matrix} \right) \qquad \text{and} \qquad A=\left( \begin{matrix} A_{11} & A_{12} & A_{13} & \ldots & A_{1m} \\ 0 & A_{22} & A_{23} & \ldots & A_{2m} \\ 0 & 0 & A_{33} & \ldots & A_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & A_{mm} \end{matrix} \right) , $$ where $S_{11}$, $S_{22}$, $\ldots$, $S_{mm}$ are ideal-irreducible matrices. Since $0 \le A \le S$ and $0 \le B_i \le S$ for each $i$, every product of length $l$ formed from the matrices $A$, $B_1$, $\ldots$, $B_k$ is dominated by $S^l$. It follows that every member of $\cA$ has the same block form as $S$, except that the diagonal blocks are not necessarily ideal-irreducible. Since $[A, S] =\sum_{i=1}^k [A, B_i] \ge 0$, we have $[A_{j j}, S_{j j}] \ge 0$ for all $j$. Fix $j \in \{1, 2, \ldots, m\}$. By \cite[Theorem 2.1]{BDFRZ}, we obtain that $[A_{j j}, S_{j j}] = 0$, as $S_{j j}$ is ideal-irreducible. Let $B^{(i)}_{jj}$ denotes the $(j,j)$-block of the matrix $B_i$. Since $0 = [A_{jj}, S_{jj}] = \sum_{i=1}^k [A_{jj}, B^{(i)}_{jj}]$ and $[A_{jj}, B^{(i)}_{jj}] \ge 0$, we conclude that $[A_{jj}, B^{(i)}_{jj}] = 0$ for all $i=1, \ldots, k$. Since $A$ has distinct eigenvalues, the same holds for $A_{jj}$, and so its commutant $\{A_{j j}\}^\prime$ is diagonalizable. If $\cA_{j j}$ denotes the algebra of all compressions to the $(j,j)$-block of the members of $\cA$, then $\cA_{j j} \subseteq \{A_{j j}\}^\prime$, and so the algebra $\cA_{j j}$ is diagonalizable as well. It follows that the whole algebra $\cA$ is triangularizable. \end{proof} \begin{corollary} \label{super left-commutant} Let $A$ be a positive $n \times n$ matrix with distinct eigenvalues. Then the algebra $\alg (\langle A ]) = \lin (\langle A ])$ is triangularizable, and so its dimension is at most $n (n+1)/2$. \end{corollary} \begin{proof} Because of finite-dimensionality there exist positive matrices $B_1$, $B_2$, $\ldots$, $B_k$ in $\langle A ]$ such that $\alg (\langle A ]) = \lin \{B_i: i=1, \ldots, k\}$. We now apply Theorem \ref{finitely}. \end{proof} If the matrix $A$ in Corollary \ref{super left-commutant} is diagonal, more can be said. Consider first the special case. \begin{proposition} \label{left-commutant} Let $A$ be a positive diagonal $n \times n$ matrix with strictly decreasing diagonal entries. Then the algebra generated by $\langle A ]$ is equal to the algebra generated by $A$ and $J_n$, and it coincides with the algebra of all upper triangular matrices. \end{proposition} \begin{proof} Let $\cA$ be the algebra generated by $A$ and $J_n$, let $\cS$ be the algebra generated by the super left-commutant $\langle A ]$, and let $\cUT$ be the algebra of all upper triangular matrices. Since $[A, J_n] \ge 0$, we have $J_n \in \langle A]$, and so $\cA \subseteq \cS$. As $A$ has distinct diagonal entries, there exists a polynomial $p_i$ such that $p_i(A) = E_{i i}$, so that $E_{i i} \in \cA$ for all $i=1,2, \ldots, n$. For any $1 \le i < j \le n$ we have $E_{i i} J_n^{j-i} = E_{i j}$, and so $\cUT \subseteq \cA \subseteq \cS$. By Corollary \ref{super left-commutant}, the dimension of the algebra $\cS$ is at most $n (n+1)/2$, so that we finally conclude that $\cS = \cUT = \cA$. \end{proof} It is well-known that the commutant of a complex diagonal $n \times n$ matrix with distinct diagonal entries is equal to the algebra of all diagonal matrices, and so it has dimension $n$. The following theorem says that the super left-commutant of a positive diagonal matrix with distinct diagonal entries spans maximal triangularizable algebra. \begin{theorem} \label{distinct_diagonal} Let $A$ be a positive diagonal $n \times n$ matrix with distinct diagonal entries. Then the algebra generated by $\langle A ]$ is permutation similar to the algebra of all upper triangular matrices, and so its dimension is $n (n+1)/2$. \end{theorem} \begin{proof} There exists a permutation matrix $P$ such that the positive diagonal matrix $P^T A P$ has strictly decreasing diagonal entries. Now, we apply Proposition \ref{left-commutant}. \end{proof} Examples show that in Corollary \ref{super left-commutant} we cannot omit the assumption on the eigenvalues of $A$. As a trivial example, we can take $A$ to be the identity matrix. More interesting examples can be obtained if we want, in addition, that the matrix $A$ is ideal-irreducible. \begin{example} Let $A = e e^T$ be an ideal-irreducible $n \times n$ matrix, where $e = (1,1, \ldots, 1)^T$. Then $\langle A ]$ consists of all positive multiples of doubly stochastic matrices. Recall that a positive $n \times n$ matrix $S$ is doubly stochastic if $S e = e$ and $S^T e = e$, that is, each of its rows and columns sums to $1$. Clearly, the super left-commutant $\langle A ]$ is not triangularizable provided $n\ge 3$. The dimension of the algebra generated by $\langle A ]$ is $(n-1)^2 + 1 = n^2 - 2 n +2$ that is greater than $n (n+1)/2$ when $n \ge 5$. \end{example} \vspace{3mm} \section{Positive idempotents with positive commutators} Let us begin with a supplement of \cite[Theorem 6.3]{KS}. \begin{theorem} \label{one_idem} Let $E$ a positive idempotent operator on a Archimedean vector lattice $L$, and let $A$ be an operator on $L$ such that either $A E \ge E A$ or $A E \le E A$. (a) If $\cN(E) = \{0\}$, then $A E = E A E$ and $(AE- EA)^2 = 0$. (b) If $\cR(E) = L$, then $E A= E A E$ and $(AE- EA)^2 = 0$. (c) If $\cN(E) = \{0\}$ and $\cR(E) = L$, then $A E = E A$. \noindent Suppose, in addition, that the operators $A$ and $E$ are order continuous. (d) If $\cR(E)^d = \{0\}$, then $E A= E A E$ and $(AE- EA)^2 = 0$. (e) If $\cN(E) = \{0\}$ and $\cR(E)^d = \{0\}$, then $A E = E A$. \end{theorem} \begin{proof} We consider only the case that $A E \ge E A$, as the other case can be treated similarly. (a) It follows from $E (A E - E A E) = 0$ that $A E- E A E = 0$, since $A E - E A E = (A E - E A) E \ge 0$ and $\cN(E) = \{0\}$. Now, we have $$ E (AE- EA)^2 = E A E AE - EA EA - EA^2 E + EA EA = E A (E AE- A E) = 0 . $$ Since $(AE- EA)^2 \ge 0$ and $\cN(E) = \{0\}$, we conclude that $(AE- EA)^2 = 0$. (b) Since $E A E - E A= E (A E - E A) \ge 0$ and $\cR(E) = L$, the equality $ (E A - E A E) E = 0$ implies that $E A - E A E = 0$. We now compute $$ (AE- EA)^2 E = A E A E - A E A E - E A^2 E + E A E A E = (E A- E A E) A E = 0 . $$ Since $(AE- EA)^2 \ge 0$ and $\cR(E) = L$, we conclude that $(AE- EA)^2 = 0$. (c) This is a direct consequence of (a) and (b). (d) Using Lemma \ref{zero} the proof goes similar lines as the proof of (b). (e) This follows from (a) and (d). \end{proof} The following result is a slight improvement of \cite[Theorem 6.4]{KS}. \begin{theorem} \label{two_idem} Let $L$ be a vector lattice, and let $E$ and $F$ be positive idempotent operators on a vector lattice $L$ such that either $E F\ge F E$ or $E F \le F E$. Let $\cA$ be the unital algebra generated by $E$ and $F$. (i) If either $\cN(E) = \{0\}$ or $\cR(E) = L$, then $$ \cA = \lin \{I, E, F, E F, F E, F E F\} . $$ (ii) If $\cN(E) = \{0\}$ and $\cR(E) = L$, then $$ \cA = \lin \{I, E, F, E F\} . $$ \noindent If the operators $E$ and $F$ are order continuous, the assumption $\cR(E) = L$ can be replaced by a (weaker) condition $\cR(E)^d = \{0\}$. \end{theorem} \begin{proof} If $\cN(E) = \{0\}$, then Theorem \ref{one_idem}(a) gives that $E F E = F E$. If $\cR(E) = L$, then Theorem \ref{one_idem}(b) implies that $E F E = E F$. In both cases, we conclude that $\cA = \lin \{I, E, F, E F, F E, F E F\}$, proving the assertion (i). The assertion (ii) is a direct consequence of Theorem \ref{one_idem}(c). The last assertion holds because of Theorem \ref{one_idem}(d) and (e). \end{proof} The proof of the main result of this section (Theorem \ref{main}) is based on the following key result. \begin{theorem} \label{key} Let $L$ be a vector lattice with the projection property. Let $E$ be an order continuous positive idempotent operator on $L$. Let $A$ be an order continuous positive operator on $L$ such that either $A E \ge E A$ or $A E \le E A$. Then $$ (AE- EA)^2 E = E (AE- EA)^2 = 0 , $$ or equivalently $$ (E A)^2 E = E A^2 E . $$ \end{theorem} \begin{proof} Since $E$ is order continuous, its null ideal $\cN(E)$ is a band of $L$. Let us define the bands $L_1$, $L_2$, $L_3$ and $L_4$ by $L_1 = \cN(E) \cap \cR(E)^d$, $L_2 = \cN(E) \cap \cR(E)^{d d}$, $L_3 = \cN(E)^d \cap \cR(E)^{d d}$ and $L_4 = \cN(E)^d \cap \cR(E)^d$. With respect to the band decomposition $L = L_1 \oplus L_2 \oplus L_3 \oplus L_4$, the idempotent $E$ has the form $$ E=\left( \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & X & Z \\ 0 & 0 & G & Y \\ 0 & 0 & 0 & 0 \end{matrix} \right) , $$ where $G$, $X$, $Y$ and $Z$ are positive operators on the appropriate bands. It follows from $E^2 = E$ that $G^2 = G$, $X G = X$, $G Y = Y$ and $Z = X Y$. Therefore, we have $$ E=\left( \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & X G & X G Y \\ 0 & 0 & G & G Y \\ 0 & 0 & 0 & 0 \end{matrix} \right) = \left( \begin{matrix} 0 \\ X \\ I \\ 0 \end{matrix} \right) G \left( \begin{matrix} 0 & 0 & I & Y \end{matrix} \right) , $$ $\cN(G) = \{0\}$ and $\cR(G)^d = \{0\}$. Writing $A = [A_{ij}]_{i,j=1}^4$ with respect to the same decomposition, we compute $$ A E=\left( \begin{matrix} 0 & 0 & (A_ {12} X + A_{13}) G & (A_ {12} X + A_{13}) G Y \\ 0 & 0 & (A_ {22} X + A_{23}) G & (A_ {22} X + A_{23}) G Y \\ 0 & 0 & (A_ {32} X + A_{33}) G & (A_ {32} X + A_{33}) G Y \\ 0 & 0 & (A_ {42} X + A_{43}) G & (A_ {42} X + A_{43}) G Y \end{matrix} \right) $$ and $$ E A=\left( \begin{matrix} 0 & 0 & 0 & 0 \\ X G (A_ {31} + Y A_{41}) & X G (A_ {32} + Y A_{42}) & X G (A_ {33} + Y A_{43}) & X G (A_ {34} + Y A_{44}) \\ G (A_ {31} + Y A_{41}) & G (A_ {32} + Y A_{42}) & G (A_ {33} + Y A_{43}) & G (A_ {34} + Y A_{44}) \\ 0 & 0 & 0 & 0 \end{matrix} \right) \ . $$ We now consider two cases. {\it Case 1}: $A E \ge E A$. Comparing the $(3,1)$-block we conclude that $G (A_ {31} + Y A_{41}) = 0$. Since $A_ {31} + Y A_{41} \ge 0$ and $\cN(G) = \{0\}$, we have $A_ {31} + Y A_{41} = 0$, and so $A_ {31}= 0$ and $Y A_{41}=0$. As $\cN(Y) = \{0\}$ we obtain that $A_{41}=0$. Similarly, the equality $G (A_ {32} + Y A_{42}) = 0$ implies that $A_ {32}=0$ and $A_{42}=0$. Comparing the $(3,3)$-block we have $A_{3 3} G \ge G (A_ {33} + Y A_{43})$. If we apply the idempotent $G$ on both sides, we obtain that $G Y A_{43} G = 0$. Since $\cN(G) = \{0\}$, $\cR(G)^d = \{0\}$ and $\cN(Y) = \{0\}$, we use Lemma \ref{zero} to conclude that $A_{43}=0$. Consequently, we have the inequality $A_{3 3} G \ge G A_ {33}$, and so $A_{3 3} G = G A_{3 3}$, by Theorem \ref{one_idem} (c). This shows that the commutator of $A$ and $E$ has the form $$ A E - E A =\left( \begin{matrix} 0 & 0 & + & + \\ 0 & 0 & + & + \\ 0 & 0 & 0 & + \\ 0 & 0 & 0 & 0 \end{matrix} \right) , $$ where each $+$ denotes a positive block (that can be also $0$). It follows that $$ (A E - E A)^2 =\left( \begin{matrix} 0 & 0 & 0 & + \\ 0 & 0 & 0 & + \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right) , $$ and so $$ E (A E - E A)^2 =0 = (A E - E A)^2 E . $$ {\it Case 2}: $A E \le E A$. Comparing the $(1,3)$-block we obtain that that $(A_ {12} X + A_{13}) G = 0$. Since $ A_ {12} X + A_{13}$ is a positive order continuous operator and $\cR(G)^d = \{0\}$, we have $A_ {12} X + A_{13} = 0$ by Lemma \ref{zero}, and so $A_ {12} X = 0$ and $A_{13}=0$. Because of $\cR(X)^d = \{0\}$ we finally obtain that $A_{12}=0$. Similarly, the equality $ (A_ {42} X + A_{43}) G = 0$ implies that $A_ {42}=0$ and $A_{43}=0$. Comparing the $(3,3)$-block we have $(A_ {32} X + A_{33}) G \le G A_ {33}$. Applying the idempotent $G$ on both sides, we obtain that $G A_{32} X G = 0$. Since $\cN(G) = \{0\}$, $\cR(G)^d = \{0\}$ and $\cR(X)^d = \{0\}$, we use Lemma \ref{zero} to conclude that $A_{32}=0$. Consequently, we have the inequality $A_{33} G \le G A_ {33}$, and so $A_{3 3} G = G A_{3 3}$, by Theorem \ref{one_idem} (c). This shows that the commutator of $A$ and $E$ has the form $$ E A - A E =\left( \begin{matrix} 0 & 0 & 0 & 0 \\ + & 0 & + & + \\ + & 0 & 0 & + \\ 0 & 0 & 0 & 0 \end{matrix} \right) , $$ It follows that $$ (A E - E A)^2 =\left( \begin{matrix} 0 & 0 & 0 & 0 \\ + & 0 & 0 & + \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right) , $$ and so $$ E (A E - E A)^2 =0 = (A E - E A)^2 E . $$ \end{proof} The main result of this section now easily follows. With a different proof it was already shown in \cite[Theorem 6.6]{KS} under slightly weaker assumption, as the order continuity of the idempotent $E$ was not needed. \begin{theorem} \label{main} Let $L$ be a vector lattice with the projection property. Let $E$ and $F$ be order continuous positive idempotent operators on $L$ such that $E F \ge F E$. Then $(E F)^2 E = E F E$ and $(F E)^2 F = F E F$, and so the unital algebra $\cA$ generated by $E$ and $F$ is equal to the linear span of the set $$ \{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\} . $$ In particular, the dimension of $\cA$ is at most $9$. \end{theorem} \begin{proof} Applying Theorem \ref{key} twice we obtain that $(E F)^2 E = E F E$ and $(F E)^2 F = F E F$. The remaining conclusions of the theorem are then clear. \end{proof} Theorem \ref{main} can be slightly generalized using an extension theorem for positive order continuous operators \cite[Theorem 1.65]{AB}. \begin{theorem} \label{Veksler} Let $L$ be an Archimedean vector lattice. Let $E$ and $F$ be order continuous positive idempotent operators on $L$ such that $E F \ge F E$. Then the unital algebra generated by $E$ and $F$ is equal to the linear span of the set $$ \{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\} . $$ \end{theorem} \begin{proof} Let $L^\delta$ be the Dedekind completion of $L$. Since $L$ is order dense in $L^\delta$, the operator $E : L \to L^\delta$ is order continuous. By \cite[Theorem 1.65]{AB}, there is a unique order continuous linear idempotent extension $E_0: L^\delta \to L^\delta$. If $F_0$ is an order continuous linear idempotent extension of $F$, then $E_0 F_0 \ge F_0 E_0$, and so we can apply Theorem \ref{main} to complete the proof. \end{proof} Theorem \ref{main} can be also slightly extended in the case when the order dual $L^{\sim}$ separates points of a vector lattice $L$. This condition is satisfied for normed lattices. \begin{theorem} \label{without_oc} Let $L$ be a vector lattice whose order dual $L^{\sim}$ separates points of $L$. Let $E$ and $F$ be positive idempotent operators on $L$ such that $E F \ge F E$. Then the unital algebra $\cA$ generated by $E$ and $F$ is equal to the linear span of the set $$ \{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\} . $$ \end{theorem} \begin{proof} By \cite[Theorem 1.73]{AB}, the order adjoint $T^\sim$ of a positive operator $T$ on $L$ is necessarily order continuous. Therefore, $E^\sim$ and $F^\sim$ are order continuous positive idempotent operators on $L^\sim$ such that $E^\sim F^\sim = (F E)^\sim \le (E F)^\sim = F^\sim E^\sim$. By Theorem \ref{main}, the unital algebra $\cA^\sim = \{ A^\sim : A \in \cA \}$ that is generated by $E^\sim$ and $F^\sim$ is equal to the linear span of the set $$ \{I, E^\sim, F^\sim, E^\sim F^\sim, F^\sim E^\sim, E^\sim F^\sim E^\sim, F^\sim E^\sim F^\sim, (E^\sim F^\sim)^2, (F^\sim E^\sim)^2\} . $$ Since the order dual $L^{\sim}$ separates points of $L$, the conslusion of the theorem follows. \end{proof} In the special case of matrices, Theorem \ref{main} gives the following result. \begin{corollary} \label{main_matrix} Let $E$ and $F$ be positive idempotent $n \times n$ matrices such that $E F \ge F E$. Then the unital algebra generated by $E$ and $F$ is equal to the linear span of the set $\{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\}$, and so its dimension is at most $9$. \end{corollary} The upper bound of Theorem \ref{main} (and Corollary \ref{main_matrix}) can be attained, as \cite[Example 6.8]{KS} shows. Let us rewrite this example in such a way that the idempotent $F$ has the block form appeared in the proof of Theorem \ref{main} and the unital algebra generated by $E$ and $F$ is ideal-triangularizable. \begin{example} \label{example_KS} Define the ideal-triangularizable idempotent positive matrices $E$ and $F$ by $$ E = \left( \begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right) \ \ \ \textrm{and} \ \ \ F = \left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 & 1 & 1 \cr 0 & 0 & 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right) . $$ Direct verifications prove that $E F \ge F E$ and that the elements of the set $$ \{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\} $$ are linearly independent matrices. \end{example} Several papers have been published about semigroups of idempotents, called {\it bands} in the abstract semigroup theory (see e.g. \cite{FMRR}). So, the following corollary of Theorem \ref{main} is perhaps interesting. \begin{corollary} Let $L$ be a vector lattice with the projection property. Let $E$ and $F$ be order continuous positive idempotent operators on $L$ such that $E F \ge F E$. Suppose that the semigroup $\cS$ generated by $E$ and $F$ consists of positive idempotents. Then $$ \cS = \{E, F, E F, F E, E F E, F E F\} , $$ and so the unital algebra generated by $E$ and $F$ is at most $7$-dimensional. \end{corollary} The bound $7$ in the last theorem cannot be improved. An example showing this can be obtained from Example \ref{example_KS} by deleting the last row and the last column of the matrices $E$ and $F$. \vspace{3mm} \section{General positive idempotents} If in Corollary \ref{main_matrix} positivity of the commutator $[E, F]$ is removed, the dimension of the unital algebra generated by $E$ and $F$ is at most $2 n$, as we have the following theorem (proved in \cite{GLS}). \begin{theorem} \label{quadratic} A unital algebra generated by two $n \times n$ matrices with quadratic minimal polynomials is at most $2n$-dimensional if $n$ is even, and at most $(2n - 1)$-dimensional if $n$ is odd. \end{theorem} Examples in \cite{GLS} show that the bounds on dimensions are sharp even in the case of idempotents. Now, we give such examples in which the two idempotents are also positive matrices. \begin{example} \label{even} Let $n \in \bN$ be an even integer, so that $n = 2 k$ for some $k \in \bN$. If $k=1$, then define positive idempotents $E$ and $F$ by $$ E = \left( \begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix} \right) \ \ \ \textrm{and} \ \ \ F = \left( \begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix} \right) . $$ Then the algebra $\cA$ generated by $E$ and $F$ is equal to the algebra of all $2 \times 2$ matrices. In the proof of this conclusion one can use the fact that $2 E_{1 1} = E F \in \cA$. So, the dimension of $\cA$ is $4$ when $n=2$. Assume now that $k\ge 2$. Define positive idempotents $E$ and $F$ by $$ E = \left( \begin{matrix} I & 2I \\ 0 & 0 \end{matrix} \right) \ \ \ \textrm{and} \ \ \ F = \left( \begin{matrix} I & 0 \\ P & 0 \end{matrix} \right) , $$ where $I$ is the $k \times k$ identity matrix and $P$ is the $k \times k$ permutation matrix corresponding to the largest cycle: $$ P = \left( \begin{matrix} 0 & 0 & 0 & \ldots & 0 & 1\cr 1 & 0 & 0 & \ldots & 0 & 0\cr 0 & 1 & 0 & \ldots & 0 & 0\cr \vdots & \vdots & \vdots & \ddots & \vdots &\vdots \cr 0 & 0 & 0 & \ldots & 0 & 0 \cr 0 & 0 & 0 & \ldots & 1 & 0 \end{matrix} \right) . $$ Define $$ C = E - F = \left( \begin{matrix} 0 & 2 I \\ - P & 0 \end{matrix} \right) . $$ Then we claim that the following matrices from the unital algebra $\cA$ generated by $E$ and $F$ are linearly independent: $$ C^{2 j} = \left( \begin{matrix} (-2 P)^j & 0 \\ 0 & (-2 P)^j \end{matrix} \right) \ \ \ \textrm{and} \ \ \ C^{2 j+1} = \left( \begin{matrix} 0 & 2 (- 2 P)^j \\ - P (-2 P)^{j} & 0 \end{matrix} \right) , \ \ \ \textrm{and} $$ $$ C^{2 j} E = \left( \begin{matrix} (-2 P)^j & 2 (-2 P)^j \\ 0 & 0 \end{matrix} \right) \ \ \textrm{and} \ \ C^{2 j+1} E = \left( \begin{matrix} 0 & 0 \\ - P (-2 P)^{j} & (-2 P)^{j+1} \end{matrix} \right) , j = 1, 2, \ldots, k . $$ To verify this claim, assume that $$ \sum_{j=1}^k (a_j C^{2 j} + b_j C^{2 j+1} + c_j C^{2 j} E + d_j C^{2 j+1} E) = 0 $$ for some numbers $a_j$, $b_j$, $c_j$, $d_j$, $j = 1, 2, \ldots, k$. It follows that $$ \sum_{j=1}^k (a_j+c_j) (-2 P)^{j} = 0 , $$ $$ \sum_{j=1}^k (b_j+c_j) 2 (-2 P)^{j} = 0 , $$ $$ \sum_{j=1}^k (b_j+d_j) P (-2 P)^{j} = 0 , $$ $$ \sum_{j=1}^k (a_j (-2 P)^{j} +d_j (-2 P)^{j+1})= 0 .$$ From the first three equations we obtain that $a_j = b_j = - c_j = - d_j$. Inserting this in the last equation, we get that $$ a_1 (-2P) + \sum_{j=2}^k (a_j - a_{j-1}) (-2 P)^{j} - a_k (-2)^{k} (-2 P)= 0 .$$ This yields $a_1=a_2=\ldots = a_k$ and $a_1 = (-2 )^k a_k$, and so $a_j = 0$ for all $j$. This complete the proof of the claim. Since the dimension of $\cA$ is at most $2 n= 4 k$ by Theorem \ref{quadratic}, this dimension must be equal to $2 n$. \end{example} \begin{example} Let $n = 2 k +1$ for some $k \in \bN$. Take $E$, $F$ and $C = E- F$ as in Example \ref{even}, and let $$ \tilde{E} = \left( \begin{matrix} 1 & 0 \\ 0 & E \end{matrix} \right) \ \ \ \textrm{and} \ \ \ \tilde{F} = \left( \begin{matrix} 0 & 0 \\ 0 & F \end{matrix} \right) . $$ Then $\tilde{E}$ and $\tilde{F}$ are positive idempotent matrices, and the algebra $\cA$ generated by them has dimension $4 k+1 = 2 n - 1$. In the proof of the last claim (for $k \ge 2$) one can use the fact that $$ \left( \begin{matrix} 1 & 0 \\ 0 & (-2)^k I \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & C^{2 k} \end{matrix} \right) = (\tilde{E}-\tilde{F})^{2k} \in \cA \ \ \ \textrm{and} \ \ \ \left( \begin{matrix} 1 & 0 \\ 0 & 4^k I \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & (-2)^k I \end{matrix} \right)^2 \in \cA $$ imply that $$ \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right) \in \cA . $$ \end{example} {\it Acknowledgments.} The author acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0222). He is also thankful to Klemen \v{S}ivic for useful observations, and to Marko Kandi\'{c} for stating Theorem \ref{Veksler} and for providing comments on the manuscript.

proofpile-arXiv_069-8608

\section{Take-away messages} \label{sec:conc} In this paper, we argued that the generation and use of synthetic data is a critical ingredient in facilitating collaborations involving sensitive data. The cost of establishing formal data sharing agreements limits the impact of these ad hoc collaborations in government, social sciences, health, or other areas where data is heavily encumbered by privacy rules. A good fake dataset has two properties: it is representative of the original data, and it provides strong guarantees against privacy violations. We discussed several use cases for fake data generation, and presented DataSynthesizer, a privacy-preserving synthetic data generator for tabular data. Given a dataset, \ds~ can derive a structurally and statistically similar dataset, at a configurable level of statistical fidelity, while ensuring strong privacy guarantees. \ds~ is designed with usability in mind. The system supports three intuitive modes of operation, and requires minimal input from the user. We see fake data generators like \ds~ used in a variety of application contexts, both as stand-alone libraries and as components of more comprehensive data sharing platforms. As part of ongoing work, we are studying how best to deliver these features to data owners, and determining how additional requirements can be met. \ds~ is open source, and is available for download at \url{https://github.com/DataResponsibly}. The work on \ds~ is part of the Data, Responsibly project, and is a component of the Fides framework~\cite{DBLP:conf/ssdbm/StoyanovichHAMS17}, which operationalizes responsibility in data sharing, integration, analysis and use. \section{Introduction} \label{sec:intro} Collaborative projects in the social and health sciences increasingly require sharing sensitive, privacy-encumbered data. Social scientists, government agencies, health workers, and non-profits are eager to collaborate with data scientists, but many projects fail before they begin due to delays incurred by data sharing agreements --- our colleagues report that 18 months is a typical timeframe to establish such agreements! Data scientists typically require access to data before they can understand the problem or even determine whether they can help. But data owners cannot share data without significant legal protections in place. Beyond legal concerns, there is a general reluctance to share sensitive data with non-experts before they have ``proven themselves,'' since they typically are not familiar with the context in which the data was collected and may be distracted by spurious results. To bootstrap these collaborations prior to data sharing agreements being established, we advocate generating datasets that are \emph{structurally and statistically similar} to the real data but that are 1) obviously synthetic to put the data owners at ease, and 2) offer strong privacy guarantees to prevent adversaries from extracting any sensitive information. These two requirements are not redundant: strong privacy guarantees are not always sufficient to convince data owners to release data, and even seemingly random datasets may not prevent subtle privacy attacks. With this approach, data scientists can begin to develop models and methods with synthetic data, but maintain some degree of confidence that their work will remain relevant when applied to the real data once proper data sharing agreements are in place. As an initial exploration of these ideas, we have developed a tool named \ds~ that generates structurally and statistically similar synthetic datasets based on real, private datasets. Given a sensitive dataset, \ds~ infers the domain of each attribute and derives a probabilistic model of the attribute values, possibly adding noise to ensure differential privacy. The description of this derived model is saved in a \emph{dataset description file}. The tool then generates synthetic datasets of arbitrary size by sampling from the stored distribution. \ds~ can operate in one of three modes, which differ in how the probabilistic model is derived: In \emph{correlated attribute mode}, we learn a differentially private Bayesian network capturing the correlation structure between attributes, then draw samples from this model to construct the result dataset. In cases where the correlated attribute mode is too computationally expensive or when there is insufficient data to derive a reasonable model, one can use \emph{independent attribute mode}. In this mode, a histogram is derived for each attribute, noise is added to the histogram to achieve differential privacy, and then samples are drawn for each attribute. Finally, for cases of extremely sensitive data, one can use \emph{random} mode that simply generates type-consistent random values for each attribute. We give a brief overview of the implementation of the tool, and of the kinds of user interaction it supports, in Section~\ref{sec:specto}. We envision various extensions to this basic approach. For example, joining multiple sensitive datasets requires care: the join of two synthetic datasets does not necessarily have the same properties as a synthetic dataset derived from the join of two rel datasets. But in many cases, joins between the real data are expressly forbidden: linking education and housing datasets is important to understand the effects of homelessness on graduation rates, but FERPA laws preclude this kind of linking. Beyond linking, we see value in mixing real data and fake data in order to adjust statistical properties or ensure anonymity requirements, adversarially generating fake datasets to assess bias and accuracy of external models, and even generating complete``cities'' of fake data based on the real data exhaust from city operations. In the latter case, we see the resulting interconnected datasets as a research instrument that can attract researchers who may otherwise be turned off by the administrative hurdles in getting access to data. We describe all of these extensions in more detail in Section \ref{sec:vision}. We briefly survey related work in Section~\ref{sec:related} and conclude in Section~\ref{sec:conc}. \section{Related Work} \label{sec:related} In our work on \ds~ we leverage recent advances in practical differential privacy~\cite{DBLP:conf/sigmod/HayMMCZ16} and privacy-preserving generation of synthetic datasets~\cite{DBLP:conf/icde/LuMG14,DBLP:conf/sigmod/ZhangCPSX14}. In particular, we make use of the privacy-preserving learning of the structure and conditional probabilities of a Bayesian network in PrivBayes~\cite{DBLP:conf/sigmod/ZhangCPSX14}, and are inspired in our implementation by the work on DPBench~\cite{DBLP:conf/sigmod/HayMMCZ16}. Other recent approaches in privacy-preserving data generation include the work on plausible deniability in data synthesis~\cite{DBLP:journals/pvldb/BindschaedlerSG17}, on perturbed Gibbs sampling for private data release~\cite{DBLP:conf/ichi/ParkGS13,DBLP:journals/tdp/ParkG14}, and on sampling from differentially private copula functions~\cite{DBLP:journals/pvldb/LiXZJ14}. Data sharing systems, including SQLShare \cite{jain:16} and DataHub \cite{bhardwaj:15}, aim to facilitate collaborative data analysis, but do not incorporate privacy preserving features or purport to manage sensitive data. We see these systems efforts as a potential delivery vector for \ds~ capabilities. \section{Demonstration Scenario} \label{sec:scenarios} \julia{I will restructure or remove this section once the use cases are in place. Likely the first 2 paragraphs will go into the System section, and the last 2 paragraphs will be dropped once we have usecases.} \bill{I'd sat remove, or redistribute the important points that aren't made elsewhere.} We now briefly describe the main components of the \ds~ system. The high-level architecture of the system is presented in Figure~\ref{fig:arch}. The system has three modules --- DataDescriber, \dg~ and ModelInspector. Each component of the system will be explained in detail in Section~\ref{sec:sys}. A data owner interacts with \ds~ through a Web-based UI that serves as a wrapper for 1) invoking the \dd~ library to compute a data summary, 2) generating a synthetic dataset from the summary using DataGenerator, and 3) inspecting and comparing datasets and data summaries with ModelInspector. \dd~ can be invoked on any CSV file. In particular, we can invoke it on both the input file and the output file, and compare the resulting summaries. Based on this comparison, we will convey an important point to the demonstration attendees: While the input and output datasets themselves are clearly very different, the statistical descriptions of the datasets are very similar. What sort of a comparison is drawn between the input and the output depends on the mode of operation of DataSynthesizer. When the tool in invoked in correlated attribute mode, a comparison of the learned Bayesian networks and of the pair-wise attribute correlations is shown to the user. In independent attribute mode, per-attribute histograms are presented, along with the ranges of attribute values. During the demonstration, we will showcase the functionality of \ds~ on a variety of datasets. We will prepare several interesting datasets for the interactive session, including an urban homelessness dataset from the University of Washington Data Science Institute (where the relevant tasks include making targeted service recommendations), the criminal sentencing dataset from a ProPublica investigation~\cite{propublica} (where the relevant tasks include analyzing the bias of recidivism models), and several datasets from the UCI Machine Learning Repository~\cite{Lichman:2013}. During the demonstration we will encourage users to download additional datasets from portals such as \url{data.seattle.gov} and \url{data.ny.gov} and to run the tool. Attendees will play the role of a data owner who is preparing a synthetic dataset for release to a set of new collaborators. After reviewing the raw dataset, they will generate a summary model and inspect the results, verifying that the results are sensible. Then, the attendees will generate a synthetic dataset from this model and inspect individual records, observing that the records are visibly ``scrambled''. Finally, they will generate a new model from the synthetic dataset and observe that the statistical properties are intact. At this point, we will explain the privacy guarantees that are enforced, and will illustrate them using the before-and-after visualizations. \section{DataSynthesizer: Safe Tabular Data} \label{sec:specto} We instantiate our vision in an open-source tool called DataSynthesizer, which takes a private dataset as input and generates synthetic datasets that can be shared safely. \ds~ generates fake data using state-of-the-art differential privacy mechanisms~\cite{DBLP:conf/tcc/DworkMNS06}. Differential privacy is a family of techniques that guarantee that the output of an algorithm is statistically indistinguishable on a pair of \emph{neighboring} databases --- a pair of databases that differ by only one row. That is, the presence or absence of a single individual in the input to the algorithm will be undetectable when one looks at the output. An important property of differential privacy is that its effectiveness degrades with repeated queries~\cite{DBLP:conf/uss/HaeberlenPN11}. To prevent leaking private information through adversaries repeatedly sending data generation requests, the system administrator can assign a unique random seed for each person who requires a synthetic dataset. \subsection{Overview of the implementation} \label{sec:unconditioned_distributions} We now briefly describe the implementation of DataSynthesizer; see Ping et al.~\cite{DBLP:conf/ssdbm/PingSH17} for additional details. \ds~ is implemented in Python 3. The data owner can interact with the tool through Jupyter notebooks or through a Web-based UI. \ds~ assumes that the private dataset is presented in CSV format. The tool is designed to work with minimal input from the user. For example, it is not necessary to specify data types of the attributes, the tool determines these automatically. The input dataset is first processed by the \dd~ module, which infers attribute data types and estimates their distributions. For each attribute identified as categorical (one with a small number of distinct values, such as gender, ethnicity and education level), \dd~ computes the frequency distribution of each distinct value. For non-categorical numerical and datetime attributes, \dd~ derives an equi-width histogram to represent the distribution. For non-categorical string attributes, their minimum and maximum lengths are recorded. This information is recorded in a dataset description, which is then used to generate fake datasets. Additional processing steps may be required depending on the specific mode of operation chosen by the data owner. There are three such modes, which we describe next. \subsubsection*{Random mode} \label{sec:build_random} When invoked in random mode, \dg~ generates type-consistent random values for each attribute. If an attribute is of type string, then a random string is generated with length that falls within the observed range of lengths in the dataset. \subsubsection*{Independent attribute mode} \label{sec:unconditioned_distributions} \begin{figure*} \centering \includegraphics[width=5.3in]{figs/datacomp.png} \caption{Data comparison} \label{fig:datacomp} \end{figure*} \begin{figure*} \centering \includegraphics[width=5.3in]{figs/histcomp.png} \caption{Histogram comparison} \label{fig:histcomp} \end{figure*} \begin{figure*} \centering \includegraphics[width=5in]{figs/heatmap_INDEP.png} \caption{Pair-wise correlations: independent mode.} \label{fig:heatmap_indep} \end{figure*} \begin{figure*} \centering \includegraphics[width=5in]{figs/heatmap_CORR.png} \caption{Pair-wise correlations: correlated attribute mode.} \label{fig:heatmap_corr} \end{figure*} When invoked in independent attribute mode, \dd~ implements a differentially private mechanism by adding controlled noise into the learned per-attribute distributions (histograms). The noise is from a Laplace distribution with location 0 and scale $\frac{1}{n \epsilon}$, where $n$ is the size of the input, denoted $Lap(\frac{1}{n \epsilon})$, setting $\epsilon=0.1$ by default. When Laplace noise is added to histogram frequencies, the value may become negative. In that case the value is reset to 0~\cite{DBLP:conf/sigmod/ZhangCPSX14}. To generate a synthetic privacy-preserving dataset, the \dg~ module is invoked and generates a synthetic dataset by sampling. Each row is sampled independently. The value of each attribute in each row is sampled independently from the corresponding noisy histogram using uniform sampling. \subsubsection*{Correlated attribute mode} \label{sec:build_Bayesian_networks} Attribute values are often correlated, e.g., \textit{age} and \textit{income} of a person. When invoked in correlated attribute mode, \dd~ uses the GreedyBayes algorithm to construct Bayesian networks (BN) to model correlated attributes~\cite{DBLP:conf/sigmod/ZhangCPSX14}. The Bayesian network gives the sampling order for generating attribute values, see Figures~\ref{fig:bn_before} and~\ref{fig:bn_after} for examples. The distribution from which a dependent attribute is sampled is called a conditioned distribution. When constructing a noisy conditioned distribution, $Lap(\frac{4(d-k)}{n \cdot \epsilon})$ is injected to preserve privacy. Here, $d$ is the number of attributes, $k$ is the maximum number of parents of a BN node, and $n$ is the number of tuples in the input dataset. We construct conditional distributions according to Algorithm 1 of~\cite{DBLP:conf/sigmod/ZhangCPSX14}. The parents of a dependent attribute can be categorical or numerical, whose distributions are modeled by bar charts and histograms, respectively. The conditions for this dependent attribute are the legal values of the categorical parents and the intervals of the numerical parents. Here, the intervals are formed in the same way as the unconditioned distributions of the parent attributes. For example, the \textit{age} attribute has intervals \{[10, 20),[20, 30),[30, 40)\} in its unconditioned distribution. Assume \textit{education} only depends on \textit{age}. Its conditioned distributions will be under the same intervals, i.e., age $ \in [10,20)$, age $ \in [20,30)$ and age $ \in [30,40)$ respectively. \subsection{Interacting with DataSynthesizer} \ds~ provides several built-in functions to inspect the similarity between the private input dataset and the output synthetic dataset. With the {\em Data Comparison} view (Figure~\ref{fig:datacomp}), the data owner can quickly test whether the tuples in the synthetic dataset are detectable by inspecting and comparing the raw data. With the {\em Comparison by histogram} view (Figure~\ref{fig:histcomp}), the user can compare the estimates of the per-attribute probability distributions in the input dataset to those in the synthetic dataset, with the expectation that these histograms will be similar in independent attribute and correlated attribute modes (as shown in Figure~\ref{fig:histcomp}), but that they would be dis-similar in random attribute mode. With the {\em Comparison by heatmap} view (Figures~\ref{fig:heatmap_indep} and~\ref{fig:heatmap_corr}), the user can inspect pair-wise attribute correlations in the original dataset, and compare these to the correlations in the synthetic dataset. We quantify correlations using mutual information (MI), a common measure of mutual dependence between two random variables. Consider, for example, the comparison by heatmap in independent attribute mode presented in Figure~\ref{fig:heatmap_indep}. Blue grid cells correspond to little or no correlation (MI close to 0), yellow cells correspond to moderate correlation (MI around 0.5), while red cells correspond to strong correlation (MI close to 1). Observe that marital status and relationship exhibit moderate correlation in the original dataset but that they do not correlate in the synthetic dataset (the corresponding cells are yellow on the left side of Figure~\ref{fig:heatmap_indep} and blue on the right side of this figure). This effect is expected, since independent attribute mode removes any correlations between attributes. \begin{figure*}[t!] \centering \begin{minipage}{3in} \centering \includegraphics[width=2.8in]{figs/bn_reduced_before.pdf} \caption{Bayesian network: Adult Income~\cite{Lichman:2013}.} \label{fig:bn_before} \end{minipage} \begin{minipage}{3in} \centering \includegraphics[width=2.8in]{figs/bn_reduced_after.pdf} \caption{Bayesian network: synthetic.} \label{fig:bn_after} \end{minipage} \end{figure*} Next, consider Figure~\ref{fig:heatmap_corr}, which presents heatmaps for correlated attribute mode. Observe that marital status and relationship exhibit a similar level of correlation in the synthetic dataset as they did in the original. \section{System Overview} \label{sec:sys} \julia{We now have a working Web-based UI in addition to Jupyter notebooks. I'll update this section to include some interesting screenshots.} \begin{figure}[t!] \centering \includegraphics[width=\columnwidth]{figs/architecture.pdf} \caption{The \ds~ system architecture.} \label{fig:arch} \end{figure} \ds~ is an end-to-end system that takes a private dataset as input and generates synthetic datasets. The system is implemented in Python 3. It assumes that the private dataset is presented in CSV format. The input dataset is first processed by the \dd~ module. The domains and the estimates of distributions of the attributes are inferred and saved in a dataset description file. For each attribute identified as categorical, \dd~ computes the frequency distribution of values, represented as a bar chart. \dg~ then samples from this distribution when deriving the synthetic dataset. For non-categorical numerical and datetime attributes, \dd~ derives an equi-width histogram to represent the distribution. \dg~ draws samples from this histogram during data generation using uniform sampling. For non-categorical string attributes, their minimum and maximum lengths are recorded. \dd~ generates random strings within the length range during data generation stage. We now describe each of the modules of \ds~ in turn. \subsection{DataDescriber} \subsubsection{Inferring data types and domains} The domain of an attribute is the set of its legal values. The data type is an important ingredient of the attribute domain. \ds~ supports four data types. The system allows users to explicitly specify attribute data types. If an attribute data type is not specified by the user, it is inferred by the DataDescriber. For each attribute, \dd~ first detects whether it is numerical, and if so --- whether it is an \textit{integer} or a \textit{float}. If the attribute is non-numerical, \dd~ attempts to parse it as \textit{datetime}. Any attribute that is neither numerical nor \textit{datetime} is considered a \textit{string}. \begin{table} \centering \caption{Data types supported by DataSynthesizer} \label{tab:table_data_types} \begin{tabular}{ll} \toprule Data Type & Example\\ \midrule \textit{integer} & id, age, ... \\ \textit{float} & score, rating, ... \\ \textit{string} & name, gender, ... \\ \textit{datetime} & birthday, event time, ... \\ \bottomrule \end{tabular} \end{table} The data type of an attribute restricts its domain, which may be limited further if the attribute is categorical, where only specific values are legal. For example, in a dataset of students, the attribute \textit{degree} may only take on values \textit{BS}, \textit{BA}, \textit{MS}, or \textit{PhD}. The domain of \textit{degree} is \{\textit{BS}, \textit{BA}, \textit{MS}, \textit{PhD}\}. Note that \textit{integer}, \textit{float} and \textit{datetime} attributes can also be categorical. In general, an attribute is considered to be categorical if a limited number of distinct values for that attribute is observed in the input dataset. \dd~ has a parameter \textit{categorical threshold} that defaults to 10 and represents the maximum number of distinct values for a categorical attribute. The {\em categorical threshold}, like any threshold, may be challenging, or even impossible, to set in a way that reflects user preferences for all attributes in the input. Some attributes may appear categorical, but the user may prefer to treat them as numerical instead. For example, age of elementary school children may take on only a handful of distinct values but the user may nonetheless wish to generate data for this attribute from a continuous range. The opposite situation may also arise --- an attribute may take on 200 or so distinct values, as is the case with country names, and so would not be considered categorical under any reasonable threshold. Still, the user may prefer to treat this attribute as categorical, so that a valid country name, rather than a random string, is generated for this attribute in the synthesized dataset. For this reason, \ds~ allows users to specify a data type, and state whether an attribute is categorical, over-riding defaults on a per-attribute basis. \begin{figure*}[t!] \centering \begin{minipage}{3in} \centering \includegraphics[width=3in]{figs/AdultIncome_full_age_before.pdf} \caption{Histogram on \emph{age}: Adult Income~\cite{Lichman:2013}.} \label{fig:age_before} \end{minipage} \begin{minipage}{3.1in} \centering \includegraphics[width=3.1in]{figs/AdultIncome_full_age_after.pdf} \caption{Histogram on \emph{age}: synthetic.} \label{fig:age_after} \end{minipage} \end{figure*} Note that the actual datatype of a categorical attribute is immaterial in terms of the statistical properties and the privacy guarantees in synthetic data generation. For example, an attribute such as \textit{sex} may be encoded as {\sf M}/{\sf F}, as {\sf 0}/{\sf 1} or using a Boolean flag (e.g., {\sf True} for male and {\sf False} for female). In all cases, the tool will compute the frequencies of each attribute value in the input, and will subsequently sample values from the resulting data summary. There might be missing values in the input dataset, and these are important to represent in the summary. \dd~ calculates the missing rate for each attribute --- the number of observed missing values divided by $n$, the size of the dataset. \subsubsection{Differential privacy} \label{sec:diffrential_privacy_implentation} Differential privacy is a family of techniques that guarantee that the output of an algorithm is statistically indistinguishable on a pair of \emph{neighboring} databases; that is, a pair of databases that differ by only one tuple. This concept is formalized with the following definition. ($\epsilon$-Differential Privacy~\cite{DBLP:conf/tcc/DworkMNS06}) Let $\epsilon$ be a positive number. Let $\mathcal{A}$ be a randomized algorithm taking a dataset as input. Let $D$ be a dataset. For any $D'$ that differs from $D$ on at most one tuple, for any legal output $O$ of $\mathcal{A}$, $Pr(\mathcal{A}(D)=O) \leq e^\epsilon \times Pr(\mathcal{A}(D')=O)$. When $\epsilon$ approaches 0, $Pr(\mathcal{A}(D)=O) = Pr(\mathcal{A}(D')=O)$. That is, the presence or absence of a single individual in the input to the algorithm will be undetectable when one looks at the output. \subsubsection{Independent attribute mode} \label{sec:unconditioned_distributions} \begin{figure*}[t!] \centering \begin{minipage}{3in} \centering \includegraphics[width=3in]{figs/AdultIncome_full_before.pdf} \caption{Pair-wise correlations: Adult Income~\cite{Lichman:2013}.} \label{fig:matrix_before} \end{minipage} \begin{minipage}{3in} \centering \includegraphics[width=3in]{figs/AdultIncome_full_after.pdf} \caption{Pair-wise correlations: synthetic.} \label{fig:matrix_after} \end{minipage} \end{figure*} \begin{figure*}[t!] \centering \begin{minipage}{3in} \centering \includegraphics[width=2.8in]{figs/bn_reduced_before.pdf} \caption{Bayesian network: Adult Income~\cite{Lichman:2013}.} \label{fig:bn_before} \end{minipage} \begin{minipage}{3in} \centering \includegraphics[width=2.8in]{figs/bn_reduced_after.pdf} \caption{Bayesian network: synthetic.} \label{fig:bn_after} \end{minipage} \end{figure*} When invoked in independent attribute mode, \dd~ performs frequency-based estimation of the unconditioned probability distributions of the attributes. The distribution is captured by bar charts for categorical attributes, and by histograms for numerical attributes. Precision of the histograms can be refined by a parameter named \textit{histogram size}, which represents the number of bins and is set to 20 by default. \dd~ implements a differentially private mechanism, adding controlled noise into the learned distributions. The noise is from a Laplace distribution with location 0 and scale $\frac{1}{n \epsilon}$, where $n$ is the size of the input, denoted $Lap(\frac{1}{n \epsilon})$, setting $\epsilon=0.1$ by default. \dd~ also adds Laplace noise to the missing rate. When Laplace noise is added to histogram frequencies, the value may become negative. In that case the value is reset to 0~\cite{DBLP:conf/sigmod/ZhangCPSX14}. \subsubsection{Correlated attribute mode} \label{sec:build_Bayesian_networks} Attribute values are often correlated, e.g., \textit{age} and \textit{income} of a person. When invoked in correlated attribute mode, \dd~ uses the GreedyBayes algorithm to construct Bayesian networks (BN) to model correlated attributes~\cite{DBLP:conf/sigmod/ZhangCPSX14}. \begin{algorithm}[t!] \caption{GreedyBayes($D$, $A$, $k$)} \begin{algorithmic}[1] \REQUIRE Dataset $D$, set of attributes $A$, maximum number of parents $k$\\ \STATE Initialize $\mathcal{N}=\emptyset$ and $V=\emptyset$. \STATE Randomly select an attribute $X_1$ from $A$. \STATE Add $(X_1, \emptyset)$ to $\mathcal{N}$; add $X_1$ to $V$. \FOR {$i=2, .., |A|$} \STATE Initialize $\Omega = \emptyset$\\ \STATE $p = min(k, |V|)$\\ \FOR {each $X \in A \backslash V$ and each $\Pi \in {V \choose p}$} \STATE Add $(X, \Pi)$ to $\Omega$ \ENDFOR \STATE Compute mutual information based on $D$ for all pairs in $\Omega$. \STATE Select $(X_i, \Pi_i)$ from $\Omega$ with maximal mutual information. \STATE Add $(X_i, \Pi_i)$ to $\mathcal{N}$. \ENDFOR \RETURN $\mathcal{N}$ \end{algorithmic} \label{alg:greedy_bayes} \end{algorithm} Algorithm \ref{alg:greedy_bayes} constructs a BN $\mathcal{N}$ from input dataset $D$, attributes $A$, and the maximum number of BN node parents $k$, which defaults to 4. In this algorithm, $V$ is the set of visited attributes, and $\Pi$ is a subset of $V$ that will become parents of node $X$ if added to $\mathcal{N}$. Which attributes $\Pi$ are selected as parents of $X$ is determined greedily, by maximizing mutual information $(X, \Pi)$. Algorithm \ref{alg:greedy_bayes} learns the structure of the BN with privacy guarantees when the mutual information computation is differentially private ~\cite{DBLP:conf/sigmod/ZhangCPSX14}. The Bayesian networks constructed in Algorithm \ref{alg:greedy_bayes} gives the sampling order for generating attribute values. The distribution from which a dependent attribute is sampled is called a conditioned distribution. When constructing noisy conditioned distributions, $Lap(\frac{4(d-k)}{n \cdot \epsilon})$ is injected to preserve privacy. Here, $d$ is the number of attributes, $k$ is the maximum number of parents of a BN node, and $n$ is the number of tuples in the input dataset. We construct conditional distributions according to Algorithm 1 of~\cite{DBLP:conf/sigmod/ZhangCPSX14}. The parents of a dependent attribute can be categorical or numerical, whose distributions are modeled by bar charts and histograms, respectively. The conditions for this dependent attribute are the legal values of the categorical parents and the intervals of the numerical parents. Here, the intervals are formed in the same way as the unconditioned distributions of the parent attributes. For example, the \textit{age} attribute has intervals \{[10, 20),[20, 30),[30, 40)\} in its unconditioned distribution. Assume \textit{education} only depends on \textit{age}. Its conditioned distributions will be under the same intervals, i.e., age $ \in [10,20)$, age $ \in [20,30)$ and age $ \in [30,40)$ respectively. \subsection{DataGenerator} Given a dataset description file generated by DataDescriber, \dg~ samples synthetic data from the distributions in this file. The size of the output dataset is specified by the user, and defaults to $n$, the size of the input dataset. \begin{algorithm}[t!] \caption{DataGenerator($n, \mathcal{M}, \mathcal{S}, A_U, s$)} \begin{algorithmic}[1] \REQUIRE number of tuples $n$ to generate, mode $\mathcal{M}$, dataset description $\mathcal{S}$, uniform attributes $A_U$, seed $s$\\ \STATE Set seed = $s$ for pseudo-random number generator. \IF {$\mathcal{M}$ is independent attribute mode} \STATE Read all attributes $A$ from $\mathcal{S}$. \FOR {$X \in A$} \IF {$X \in A_U$} \STATE Read the domain of $X$ from $\mathcal{S}$. \STATE Sample $n$ values uniformly from its domain. \ELSE \STATE Read the distribution of $X$ from $\mathcal{S}$. \STATE Sample $n$ values from its distribution. \ENDIF \ENDFOR \ELSIF {$\mathcal{M}$ is correlated attribute mode} \STATE Read Bayesian network $\mathcal{N}$ from $\mathcal{S}$. \STATE Sample root attribute from an unconditional distribution. \STATE Sample remaining attributes from conditional distributions. \ENDIF \RETURN Sampled dataset \end{algorithmic} \label{alg:data_generator} \end{algorithm} Algorithm~\ref{alg:data_generator} describes the data generation process. When invoked in random mode, \dg~ generates type-consistent random values for each attribute. When invoked in independent attribute mode, \dg~ draws samples from bar charts or histograms using uniform sampling. Finally, when invoked in correlated attribute mode, \dd~ samples attribute values in appropriate order from the Bayesian network. An important property of differential privacy is that its performance degrades with repeated queries. To prevent leaking private information through adversaries repeatedly sending data generation requests, the system administrator can assign a unique random seed for each person who requires a synthetic dataset. To support this, \dg~ provides per-user seed functionality. \subsection{Model Inspector} \label{sec:mi} \mi~ provides several built-in functions to inspect the similarity between the private input dataset and the output synthetic dataset. The data owner can quickly test whether the tuples in the synthetic dataset are detectable by inspecting and comparing the first 5 and last 5 tuples in both datasets. The synthetic dataset should have similar distributions of attribute values as the input dataset. In independent attribute mode, \mi~ allows users to visually compare attribute distributions in the form of bar charts or histograms. Figures~\ref{fig:age_before} and~\ref{fig:age_after} present such summaries for the attribute \emph{age} from the Adult Income dataset~\cite{Lichman:2013}. The system also computes the correlation coefficient of an appropriate kind, depending on attribute type, and the KL-divergence value, which measure how much the ``after'' probability distribution diverges from the ``before'' probability distribution for a given attribute. In correlated attribute mode, \mi~ presents ``before'' and ``after'' pairwise mutual information matrices (Figures~\ref{fig:matrix_before} and~\ref{fig:matrix_after}) and describes Bayesian networks (such as those shown graphically in Figures~\ref{fig:bn_before} and~\ref{fig:bn_after}), enabling an at-a-glance comparison of the statistical properties of the datasets. \section{Extensions and Discussion} \label{sec:vision} We see synthetic data as a fundamental component of ``people analytics,'' where sensitive, private data must be used to make high-risk decisions. Beyond the capabilities of the current \ds~ tool, we envision a number of usage scenarios and corresponding extensions; we describe these extensions in this section. \subsection{Enabling collaboration} As described in the introduction, our primary motivating use case is to reduce friction in the early stages of collaboration between data providers and outside data scientists. Our hypothesis is if data scientists are allowed to ``get their hands dirty'' with synthetic data, they are more likely to internalize the problem being solved and develop effective solutions more efficiently. In our own experience, we find that ``whiteboard'' solutions designed prior to seeing the data often become irrelevant once the data is available --- attributes are different than we expected, data sizes are too small to train advanced models, biases in the data prevent certain kinds of analyses from taking place, inconsistent values complicate debugging (e.g., the string ``N/A'' in a column of integers). Exposure to these challenges early helps shape the conversation and reduce effort as data sharing agreements are being prepared. \subsection{Fake Linked Data} The value of the municipal datasets that motivate our approach enjoys a network effect: each pair of datasets enables new insights. For example, in the City of Seattle, a study is underway to determine the effect of housing instability on high school graduation rates: Do children who endure periods of homelessness graduate on time? Although the question is simple, it involves linking two extremely sensitive datasets: student data (protected by FERPA and requiring consent of parents to use) and homelessness data (protected by HIPAA and local privacy norms). In fact, the typical agreements governing the use of each of these datasets explicitly forbid linking them with any other datasets, and these typical agreements must be revised on a case by case basis to enable such studies. To bootstrap collaborations over linked data, we might like to use the same approach we have described: generate fake education data, and then generate fake homelessness data. But this na\"{i}ve will not work: synthetic records will not necessarily link with other synthetic records in a statistically similar way as in the real data. An apparent solution is to simply join the two datasets, then generate a synthetic dataset from the result. But this approach entails one data provider sharing their data with another provider, which is explicitly what we need to avoid. To solve this problem, we need to estimate the statistical properties of the joined dataset, and use that information to guide the data synthesis process independently for the education and homelessness data. We observe that there are three classes of joined tuples: housing records $H$ that have no corresponding education records, education records $E$ that have no corresponding housing record, and linked pairs of housing and education records $HE$. We want to estimate the number in each category, $|H|$, $|E|$, $|HE|$. To produce these estimates without sharing information, we can use locality sensitive hashing techniques~\cite{LSH} to independently map education tuples and housing tuples into a common space. Locality sensitive hashing algorithms have the property that similar inputs are mapped to similar outputs, without coordination. For example, integers can be mapped to their most significant $n$ bits. For structured data, one simple approach is to concatenate the values in the tuples, split this long string into n-grams, sort the n-grams lexicographically, then truncate the sorted list to retain the first $k$ n-grams. This way, similar tuples will map to similar sequences of n-grams. Using this approach (or more sophisticated approaches that make use of domain knowledge), we can independently map tuples from different providers into a shared space to determine how likely they are to match, and therefore estimate the counts $|H|$, $|E|$, $|HE|$. Recall that $|H|$ is the number of housing tuples for which no nearby education tuples exist, and $|E|$ is the number of education tuples for which no nearby housing tuples exist. We can assume the remaining tuples join to produce linked pairs. Armed with these estimates, we can generate ids for education and housing tuples to ensure that an appropriate number of joined tuples are produced. Further, we can generate attribute values guided by the same LSH techniques to ensure that joined tuples share similar values. \subsection{Mixing Real and Synthetic Data} In many data sharing situations, data must be aggregated as an attempt at anonymization. Although aggregation approaches typically offer limited formal protection in practical cases~\cite{demontjoye:13,DBLP:conf/pods/DinurN03}, they are often written into data sharing policies that must be obeyed. For example, energy providers are strongly incentivized to deliver upgrades designed to improve efficiency. But assessing the efficacy of these upgrades using consumption data, normalized by weather and project specifications, is difficult. Once again, the problem is to share sensitive data: the energy consumption of customers is a signal-rich resource that could be used for ``cybercasing,'' for example to predict when people leave their homes. To mitigate such risks, the rules that govern data sharing are designed to prevent disambiguation of aggregate measures. For instance, a geographic estimate of energy usage must aggregate no fewer than 100 consumers, and no one consumer can represent more than 10\% of the total usage. The final calculation must require a certain number of days of recorded usage data. We see a novel use case for synthetic data to ``fill out'' the aggregates to meet anonymity requirements, as a kind of tuple-level imputation process. Synthetic data may also be mixed with real data to repair global statistical anomalies, a kind of tuple-level imputation. For example, as reported in a recent New York Times article on urban homelessness~\cite{nytimeshomeless}: ``Last year, the total number of sheltered and unsheltered homeless people in the city was 75,323, which included 1,706 people between ages 18 and 24. The actual number of young people is significantly higher, according to the service providers, who said the census mostly captured young people who received social services.'' This representativeness gap can be filled with synthetic data to help data scientists triage their methods, or obscure the fact that the gap exists, in case data collection activities are sensitive. \subsection{Adversarial Fake Data Generation} Data providers are reluctant to share data for more than just privacy reasons. As decisions are shifted from humans to algorithms, the opportunity for, and impact of, discrimination becomes more acute. To earn the trust of data providers and demonstrate that proposed methods are robust to biased data, we envision generating intentionally biased datasets to explore ``corner cases." Consider for example a hiring scenario. A cluster of job applicants who are similar in terms of experience, skills, and current position should tend to all receive offers to interview. If an African-American candidate in the cluster does not receive an offer when Caucasian candidates do, there is evidence that individual fairness is violated~\cite{DBLP:conf/innovations/DworkHPRZ12}, and, specifically in this case, that there is {\em disparate treatment} --- an illegal practice of treating an entity differently based on a protected characteristic such as race or gender. This situation is illustrated in Figure \ref{fig:racist}. Beyond disparate treatment, adversarial synthetic datasets can be generated to test more general cases of violation of individual and group fairness, under different interpretations of these measures~\cite{DBLP:conf/innovations/DworkHPRZ12,DBLP:journals/corr/Zliobaite15a}. \begin{figure}[t!] \centering \includegraphics[width=2in]{figs/individualfairness.pdf} \caption An illustration of a pathological dataset to evaluate disparate treatment. The symbol \textsf{w} represents a white candidate and the symbol \textsf{b} represents a black candidate. Red indicates that a hypothetical model rejected the candidate and green indicates that a hypothetical model accepted the candidate. At lower left, candidates are both low-skill and low-experience, and all are rejected. At upper right, a cluster of predominately white candidates and one black candidate receives inconsistent outcomes: if the one black candidate is rejected while similar candidates are accepted, there is evidence of disparate treatment. Generating specific situations to test a model's response is a role that synthetic data generation can play.} \label{fig:racist} \end{figure} Testing fairness and bias properties of algorithmic decision-making systems is particularly important in cases where black-box third party tools are used, and where intervening on the inputs and analyzing the impact of the interventions on the outputs is one of only a handful of methods to infer system behavior. In enacting such interventions, it is particularly important to generate inputs that are realistic, and that systematically explore cases that may not have been present in the actual historical data that was used to train the model. Existing approaches for this problem rely on random sampling of input data to measure the response of black box models \cite{DBLP:conf/sp/DattaSZ16,DBLP:journals/corr/TramerAGHHHJL15}, but random sampling cannot necessarily generate the pathological datasets that may occur in practice. Beyond statistical bias, the ability to generate pathological datasets in terms of scale, anomalous values, and unexpected correlations can aid in debugging and stress-testing of external models, in the same way that benchmark datasets can help expose problems in, say, database systems \cite{arasu:11}. To generate these pathological datasets, we can make a three extensions to the existing \ds~ tool: First, we can allow users to edit the distribution derived from the real data to produce extreme values. Second, to allow even more precise control, we can design preconfigured pathological distributions to simulate, for example, individual fairness situations. That is, using the annotations on the original data to distinguish protected attributes from non-protected attributes, we can generate clusters of similar tuples intentionally. Third, to assess systematic robustness (as opposed to statistical robustness) we can intentionally inject pathological values into attributes --- missing values, inconsistent types, and extreme values (say, an age of -2 years). \subsection{Synthetic Cities: Comprehensive Interconnected Administrative Datasets} We envision combining all these techniques to generate, and incrementally improve, an administrative projection of entire virtual city to support research without data sharing encumbrances. Unlike population synthesis approaches in urban planning \cite{farooq:13} and economics \cite{axtell:16} which use agent-based models to study the emergent dynamics of an entire city from the ground up, our approach focuses on modeling \emph{only the administrative data that would result from the dynamics of the city}, which provides a more realistic way of evaluating solutions. Since in practice researchers will typically only have access to the administrative data, models developed based on untestable assumptions about human behavior are difficult to evaluate, and interventions based on these models are difficult to trust. But the approaches are ultimately complementary, since artificial administrative data can be used to evaluate agent-based models, and agent-based models can be used as a source of artificial administrative data when the true datasets are not available.

proofpile-arXiv_069-8727

\section{Introduction} Reproducibility is a cornerstone of the scientific method, and a strongly desirable principle for any exact-science discipline -- computer science included. By enabling third parties to check experimental results, it indeed ensures the validity of published contributions and stimulates new ones, ultimately providing a better, community-shared understanding of the state of the art. Unfortunately, reproducibility of research in networking exhibits huge room for improvement. Despite recent enablers of reproducibility like network simulators~\cite{Weingartner:2009}, virtualization tools~\cite{mininet-conext12} and testbeds~\cite{planetlab-03}, articles still often come with a rough description of the performed experiments, and limited or no support (code, dataset, etc.) to reproduce the presented evaluation. As a consequence, reviewers have to believe the claimed results. Also, researchers willing to compare new proposals with the state of the art must re-implement technical contributions (to the best of their understanding) and re-create experiments from scratch: As confirmed by the few initiatives on reproducing research results like the Stanford CS244 course~\cite{stanford-cs244-ccr}, this is a challenging, time-consuming task, that further provides no guarantees of achieving a fair comparison against the original proposal. We believe that the networking community should find more effective ways to ease support for reproducibility, at least for a significant part of the experiments in every publication. Of course, achieving reproducibility is not straightforward. In this work, we focus on traffic engineering (TE). For TE, a primary challenge consists in having access to both experimental settings, evaluation datasets, and evaluated algorithms, which unfortunately is costly if not impossible. The original experimental setting is generally hard to re-create: for example, testing infrastructures (e.g., servers, network equipment and software) are often not shared among researchers, may not be affordable for all research institutes, and tend to quickly become unsupported over time (e.g., because of new software releases). Also, evaluations on real, production-network data involve privacy concerns, as they contain sensitive information that can rarely be disclosed. Finally, releasing code (e.g., for algorithms, system prototypes, etc.) implies a significant effort that often comes with little benefits for authors. We propose an approach based on \textit{repeatability}, a looser form of reproducibility focused on reproducing a given set of experiments in a local setting (e.g., on a different server with respect to the one used in the original papers). By itself, repeatability avoids the main roadblocks deriving from rebuilding the original experimental setting. To tackle the other challenges, we argue for more effective practices and tools that facilitate repeatability. As a concrete step in this direction, we publicly release\footnote{see {\tt https://github.com/svissicchio/Repetita}\xspace} REPETITA\xspace, a software platform that we have also used to develop our own research contributions~\cite{srls-infocom17}. REPETITA\xspace eases repeatable evaluation of TE algorithms: It automates most of the technical operations involved (topology pre-processing, evaluation setup, analyses, etc.) and enables users to run experiments on more than 250 topologies (with 5 synthetic traffic matrices each) with a single command-line instruction. Admittedly, REPETITA\xspace is not a silver bullet for any possible evaluation on TE. Its dataset can be improved by adding more (e.g., different types and more updated) real topologies of networks, as well as real traffic data. The implemented algorithms mainly focus on optimization with two basic primitives (IGP and Segment Routing) -- even if the configuration of explicit paths, like MPLS tunnels and OpenFlow-installed paths, is also supported. The evaluation metrics are far from covering the full spectrum of possible evaluations: they mainly focus on bandwidth optimization (e.g., in contrast with recent TE systems like~\cite{B4:2013,swan-sigcomm13}). Even the type of possible experiments is limited, as it restricts to the static optimization of forwarding paths, with no knowledge of transient states and system dynamics. Nevertheless, we believe that no repeatability-oriented approach can be successful if it does not involve a community effort and commitment. We designed and implemented REPETITA\xspace to be extendable, so that it can benefit from the community push that we argue for. For example, its plugin-oriented architecture enable to easily enrich the amount of supported algorithms and primitives (e.g., underlying protocols and router capabilities). Our framework is also easy to use. We hope that usability can (i) provide researchers with immediate incentives and technical means to balance the practical value of experiments on private datasets with the scientific importance of reproducible ones, in future publications on TE; (ii) spur the creation of benchmarks for TE algorithms, ideally merging contributions from the wide research and operator communities; (iii) reviewers and external readers to check evaluation results; and (iv) stimulate approaches that support repeatable evaluations of research contributions in other areas. In the following, we describe how REPETITA\xspace fills a gap in the state of the art (\S\ref{sec:related-work}), how it is designed (\S\ref{sec:framework}) and implemented (\S\ref{sec:implementation}), and how it can be used for both repeating experiments and facilitating new insightful analyses (\S\ref{sec:eval}). \section{What do we Lack?}\label{sec:related-work} \myitem{Many networking contributions are hard to reproduce.} Position papers~\cite{reproduciblecs-peng11}, methodologies~\cite{recomp-manifesto} and repositories of data~\cite{beasley1990or,xcsp3} have been published to facilitate experiment reproducibility in several areas of computer science. In our experience, however, most networking contributions are hard to reproduce and fairly compare against. For instance, the majority of papers on wide-area traffic engineering (from earlier on IGP weight optimization~\cite{Fortz_Internet:2000} to more recent approaches based on MPLS~\cite{elwalid2001mate,Kandula:2005-texcp}, SDN~\cite{B4:2013,swan-sigcomm13} and SR~\cite{belllabs-infocom15}) present experiments made with proprietary code, often on a restricted dataset of private networks. \myitem{The networking community is pushing towards open-source code.} Software Defined Networking (SDN) originated from the definition of OpenFlow~\cite{mckeown2008openflow}, an open-source interface to network devices. Consistently, many SDN proposals, from projects about SDN programming languages (e.g.,~\cite{frenetic-11}) to those on network controllers (e.g.,~\cite{nox-08}), have publicly released the produced code -- an encouraging step towards reproducibility. A few recent works also consider the problem of fairly comparing open-source proposals. Prominently, they focus on tools to benchmark SDN controllers (e.g.,~\cite{sdload-14,hvbench-16}) and OpenFlow switches (e.g.,~\cite{oflops-12}), under specific network conditions or workloads. While close in spirit with those contributions, this paper has a different scope (traffic engineering), that also comes with its own challenges like working around confidentiality of data (network topologies, traffic matrices, etc.) used in experiments. \myitem{We still miss approaches and tools that make reproducibility easy.} The literature describes a pletora of tools that potentially enable repeatability, including simulators (e.g.,~\cite{Weingartner:2009}), emulation platforms (e.g.,~\cite{mininet-conext12}) and testbeds (e.g.,~\cite{planetlab-03}). They provide the low-level means to perform network experiments at scale, but everything else (network setup, approaches to be evaluated, experimental configuration, etc.) must be added on top of them. As such, those tools make repeatability just possible. We however believe that the networking community still lacks proven methodologies, tools and practices that make reproducibility not only possible, but also easy and affordable for researchers (and reviewers). For example, a past proposal to enable TE experiments in non-SDN networks is TOTEM~\cite{totem-comcom}. TOTEM defines a set of TE algorithms, but no standard experiment or analysis on them. Further, the code is not maintained since years. Hence, newer algorithms and support for cutting-edge technologies (from OpenFlow to Segment Routing) are not implemented. Even worse, it is unclear how to support such algorithms and technologies in TOTEM, since the tool is mainly a collection of heterogeneous scripts only sharing input and output format. By releasing an extendable platform that includes recent TE approaches and automates all the main operations to run repeatable experiments, we hope to overcome the difficulties of tools like TOTEM and provide immediate incentives for reproducibility in the TE field. \section{Repeatability Made Simpler}\label{sec:framework} In this section, we overview REPETITA\xspace. We describe its design (\S\ref{subsec:framework-arch}), workflow (\S\ref{subsec:framework-basics}), and key benefits (\S\ref{subsec:framework-benefits}). \subsection{Designing REPETITA\xspace}\label{subsec:framework-arch} \begin{figure}[] \centering \includegraphics[width=\columnwidth]{architecture-crop.pdf} \caption{REPETITA\xspace architecture.} \label{fig:architecture} \end{figure} As shown in Figure~\ref{fig:architecture}, REPETITA\xspace design is based on modeling and separating three main concepts: settings, solvers, and scenarios. Altogether, they define the \textit{experiments and analyses} that are performed in a run of REPETITA\xspace. \myitem{Settings.} A setting represents the problem instance considered in an experiment. It therefore groups all the input data needed to run an experiment. In the TE case, settings include topologies, traffic matrices, and routing configurations. A \textit{topology} represents a network and contains all its characteristics that are relevant for traffic engineering. We identify nodes, links, link capacities and delays as a basic set of such characteristics. A \textit{traffic matrix} encompasses data (sources, destinations and volumes) about traffic flows assumed to traverse the network in a given experiment. A \textit{routing configuration} contains additional information to compute forwarding paths on the given topology and traffic matrix. Primarily, it includes link weights for shortest-path computations done by current intra-domain routing protocols (called IGPs). Also, it encompasses other protocol-specific configurations, like Segment Routing settings as well as explicit paths reflecting configured MPLS tunnels or OpenFlow rules. \myitem{Solvers.} A solver is an object under test. In our case, a solver is a TE algorithm that given a setting, updates the routing configuration inside the setting for the corresponding topology and traffic matrix. This computation typically aims at optimizing network performance, e.g., maximizing bandwidth, avoiding congestion or reducing delay. Solvers are pieces of code that are either standalone or integrated in REPETITA\xspace. In the latter case, REPETITA\xspace offers \textit{TE libraries} implementing building blocks like computation of IGP shortest paths, and the calculation of link utilizations induced by a given routing configuration. \myitem{Scenarios.} A scenario models the dimensions along which the output of different solvers is analyzed. In other words, scenarios define analyses of interest for REPETITA\xspace users, and provide automated support to perform such analyses\footnote{\small Note that scenario has a different meaning in TOTEM where it only defines events occurring during an experiment.}. For example, a scenario can define how good a TE algorithm (solver) is at optimizing bandwidth on a given network and traffic matrix (setting). In this sense, scenarios implement the abstractions (i.e., the evaluation metrics) that ensure comparability between different solvers. \vspace*{0.07in} We believe that REPETITA\xspace architecture can fit a broader class of networking problems, as for example congestion control (where solvers are congestion control schemes, settings encompass traffic rates and network conditions, and scenarios evaluate performance metrics as throughput) and security (where solvers are defense techniques, settings models networks, targets and attacks, and scenarios quantify success rate of solvers as well as consumed resources and time). \subsection{Running REPETITA\xspace}\label{subsec:framework-basics} REPETITA\xspace provides the software infrastructure that links together the building blocks discussed in \S\ref{subsec:framework-arch}. We now list the steps sequentially performed by REPETITA\xspace to carry out a (repeatable) experiment. Note that the framework also provides direct access to the functions used in those steps, hence it can be used as a software library (e.g., to solve multi-commodity flow problems~\cite{shahrokhi1990maximum} and calculate lower bounds for any TE algorithm). The first step consists in processing the input. REPETITA\xspace parses topology and demands files using custom parsers: Those parsers create setting objects from files of tabular format, where every line describes an attribute (e.g., IGP weight, traffic volume, etc.) of a network element (node, link or demand). Once such files are parsed, REPETITA\xspace instantiates solvers and scenarios required in the experiments to be performed. Scenarios are then asked to run. For the simplest possible scenario, this method call translates into passing the configured setting to the solver stored in the scenario, and reporting the differences (e.g., in terms of maximum link utilization) between the routing configuration before and after the solver's optimization. More advanced scenarios can also modify the original settings or create new ones. For example, consider a scenario that requires to evaluate what happens in the case of any single link failure (like the robustness analysis described in \S\ref{subsec:impl-scenarios}). Such a scenario instantiates many new settings, each corresponding to a single link failure on the initial topology -- and runs the configured solver on all the new settings. Scenarios also implement two key functions. First, they enforce that every solver's execution terminates after a (potentially infinite) time bound, e.g., stopping it after the given time bound and extracting the best routing configuration computed by the solver so far. Second, scenarios process the routing configurations returned by the solvers. Namely, they translate each of those configurations into paths, map the input traffic matrix to the computed paths, and evaluate metrics like link utilization or routing overhead. Such a translation leverages functions implemented in the REPETITA\xspace TE libraries, for independence from solver implementations. Results of scenarios' post-processing are finally reported on the screen (by default) or on a pre-configured output file. \remove{ \myitem{First, REPETITA\xspace creates settings.} It processes files provided in input, typically a topology and a demands file, using a custom parser. The input file format is tabular, with every line describing an attribute (e.g., IGP weight, traffic volume, etc.) of a network component (node, link or demand). Settings are then shared with Scenario and Solver objects: the former can modified REPETITA\xspace relies on the configured scenarios to identify the actual settings to be instantiated. For example, if a scenario requires to evaluate what happens in the case of any single link failure (like the robustness analysis described in \S\ref{subsec:impl-scenarios}), REPETITA\xspace instantiates many settings per topology, each corresponding to a single link failure on that topology. \myitem{Second, REPETITA\xspace feeds solvers with settings.} It extracts information from the created settings; translates the extracted data into the formats expected by solvers; possibly computes non-optimized paths (e.g., shortest IGP paths for SR algorithms); and sequentially runs the solvers. We use REPETITA\xspace TE libraries for the computation of non-optimized paths for SR, factoring out a pre-processing procedure common to all SR solvers. REPETITA\xspace also enforces that a solver's execution terminates after a (potentially-indefinite) time bound, e.g., stopping it after the given time bound and extracting the solution computed that far. \myitem{Finally, REPETITA\xspace evaluates the configured scenarios.} After a solver terminates, the framework collects its output, i.e., the generated traffic-engineering configuration (e.g., new IGP weights or Segment Routing detours). It then translates that configuration into a mapping of traffic flows to paths, and evaluates metrics (e.g., link utilization or routing overhead) specified by every scenario. Such a translation is performed by a function in the REPETITA\xspace TE libraries, for independence from solver implementations. } \subsection{Benefiting from REPETITA\xspace Design}\label{subsec:framework-benefits} REPETITA\xspace is designed to achieve usability and extensibility. \noindent Technical documentation, instructions to run experiments and examples to add new solvers are provided at {\tt https://github.com/svissicchio/Repetita}\xspace. \myitem{REPETITA\xspace is easy to use.} Users can run the implemented solvers on supported settings and scenarios by executing the framework from the command line. The following working example instructs REPETITA\xspace to run a solver called \texttt{defoCP}, once, for 1 second, on the Abilene network. \begingroup \small \begin{center} \begin{Verbatim}[frame=single] repetita -graph Abilene.graph -demands Abilene.demands -solver defoCP -t 1 -scenario SingleSolverRun \end{Verbatim} \end{center} \endgroup \noindent A few more CLI options enable users to adjust parameters like the output file name (REPETITA\xspace results are printed to the CLI standard output by default). \myitem{REPETITA\xspace is easy to run on already implemented solvers.} For ease of adoption, we do not require solvers to be implemented within our framework. In contrast, our framework can run external solvers (written in any programming language) as long as they are packaged as executables that take as input a network and a traffic matrix. To add such solvers, it is sufficient to specify information about them in a textual file (i.e., external\_solvers/solvers-specs.txt) inside the REPETITA\xspace directory. Figure~\ref{fig:externalconf_example} reports a snippet of such file, displaying the description of an example solver (getRandomPaths.py) written in Python. As shown in the figure, limited information must be specified: general data like the name (to refer the solver within REPETITA\xspace) and the optimization objective; specification of the CLI commands to run the executable and extract the time taken by its last run; a description on how to interpret the output of the executable. More details on the file format are provided in the file itself. \begin{figure} \begingroup \small \begin{center} \begin{Verbatim}[frame=single] // general information about the solver name = randomTunnels optimization objective = 'undefined' // how to run the external solver run command = python external_solvers/getRandomPaths.py $TOPOFILE $DEMANDFILE $OUTFILE // how to interpret the output of the last solver's run optimization effect = setExplicitPaths field separator = '; ' key field = 0 value field = 2 // how to get the time taken by the last solver's run gettime command = cat $OUTFILE | grep 'execution time' | awk -v FS=': ' '{print $2}' \end{Verbatim} \end{center} \endgroup \vspace{-0.4 cm} \caption{Configuration of an external solver (in Python), added to REPETITA\xspace.} \label{fig:externalconf_example} \vspace{-0.3cm} \end{figure} \myitem{REPETITA\xspace is easy to extend.} Its architecture supports separate evolution (e.g., modification or replacement) of the dataset, the solvers and the evaluation analyses. Topologies and traffic data are controlled with a command-line parameter, that specifies the directory containing them. Additional analyses can be implemented by defining new scenario objects in the REPETITA\xspace code. Beyond using external solvers, new algorithms can also be integrated within the framework. Two options are viable, as detailed on the REPETITA\xspace Web page. The first option is to re-implement the original solver using the REPETITA\xspace TE libraries, as we did for our 156-line long implementation of the IGP weight optimization heuristic based on~\cite{Fortz_Internet:2000}. A more lightweight alternative consists in adding the new solvers, unmodified, along with solver-specific wrappers, i.e., software objects that convert input and output of method calls between the framework and the original solver. We used the latter option to support the original DEFO solver~\cite{defo-sig15} (in Scala) in our current REPETITA\xspace implementation. \section{Repeat, for Real!}\label{sec:implementation} We now describe settings (\S\ref{subsec:impl-settings}), solvers (\S\ref{subsec:impl-solvers}) and scenarios (\S\ref{subsec:impl-scenarios}) supported by our Java-based implementation. \subsection{Topologies and Traffic Matrices}\label{subsec:impl-settings} We first detail how settings are built, and why. \myitem{We collected realistic topologies} used in the evaluation of previous TE papers. We relied on 3 public data sources: the Rocketfuel project~\cite{rocketfuel-ton04}, the synthetic topologies used in DEFO~\cite{defo-sig15}, and the Internet Topology Zoo \cite{knight2011internet}. The former two sources provide a limited number of topologies, either inferred from Internet measurements (Rocketfuel) or artificially generated to stress-test algorithm scalability (DEFO). Internet Topology Zoo gathers real topologies (often WANs) as reported by official Web sites of commercial companies. \myitem{We post-processed the available topologies to have a complete dataset.} Original topologies in the Internet Topology Zoo dataset were incomplete in several cases. We completed them as follows. First, we kept only the largest connected component of any disconnected topology. Second, for topologies where no link capacity is specified, we set the same value on all links. Third, when capacity information are specified only for a subset of network links, we set the capacity of any link with missing capacity as equal to the average capacity across all links. Further, we normalized link capacities to avoid too large variations (e.g., Kbps access links versus Gbps core links) leading to unavoidable traffic bottlenecks on which all TE algorithms would perform equally bad. In particular, we impose that the value of any link capacity is not less than $1/20$-th of the largest link capacity in the topology. \myitem{We considered $3$ heuristics to assign IGP weights.} Intra-domain routing protocols (i.e., IGPs) are based on shortest-path computation. We therefore need IGP weights of all the links in every topology to compute pre- and post-TE paths. Unfortunately, the Internet Topology Zoo dataset does not contain this information. Our framework implementation then uses $3$ standard heuristic assignments of IGP weights. The first heuristic assigns unary weight to all links: It is motivated by simplicity, and by the intuition that unitary weights may provide the most flexibility for protocols built on top of IGP (like Segment Routing~\cite{defo-sig15}). The second heuristic sets IGP weights as inversely proportional to link capacities, i.e., reflecting a default that has been classically adopted by vendors\footnote{\small e.g., see http://www.cisco.com/c/en/us/support/docs/ip/open-shortest-path-first-ospf/7039-1.html\#t6}. The third option is to consider weights optimized for a given traffic matrix, i.e., according to our IGP-WO solver (see \S\ref{subsec:impl-solvers}). \myitem{We synthesized traffic matrices.} We adopted a randomized gravity model that has been shown to generate realistic traffic matrices~\cite{syntheticTMs-ccr05} and is often used to evaluate TE algorithms (\cite{Kandula:2005-texcp},\cite{defo-sig15},...). We generated $5$ traffic matrices per topology to have some diversity while keeping the dataset manageable\footnote{\small For a couple of topologies, we could have also added old traffic matrices (e.g.,~\cite{geantTM-ccr06}): We decided not to do so, in order to have the same number and type of matrices for all topologies.}. We scaled the traffic matrices so that the maximally utilized link is loaded at 90\% of its capacity in the optimal solution of the corresponding multi-commodity flow problem (using the REPETITA\xspace LP described in \S\ref{subsec:impl-solvers}) The maximal link utilization is thus larger or equal than 90\% in any real TE solution. We are well aware that operators (especially in WANs) do not typically run production networks at a so high utilization rate. However, we applied this scaling factor to build cases where very effective bandwidth-optimizing TE algorithms are absolutely needed -- cases for which recent TE approaches in private networks even argue~\cite{B4:2013,swan-sigcomm13}. To match more common settings, traffic-matrix values can be divided by a constant factor (e.g., $3$ for an optimal maximum link utilization of $30\%$). \subsection{TE Algorithms}\label{subsec:impl-solvers} REPETITA\xspace includes a linear program that computes a \textbf{baseline} for bandwidth-optimization algorithms. \myitem{Linear Program (LP), solving the multi-commodity flow problem.} The LP provides a theoretical lower bound for the classic TE goal of minimizing the maximum link utilization. It is indeed allowed to arbitrarily split traffic among any possible path, and ignores IGP weights. We used the LP for scaling the traffic matrices in our datasets (as just described), and as a baseline in our experiments (see \S\ref{sec:eval}). We report the LP formal definition in the Appendix. \vspace*{0.07in} In addition, REPETITA\xspace currently includes three \textbf{solvers}. To show the possibility to implement algorithms based on qualitatively-different primitives, we encoded a weight optimization algorithm as well as recent proposals based on the newer Segment Routing (SR) protocol. To ensure practicality of the implemented algorithms and apples-to-apples comparison, all the solvers assume that traffic is equally split among all the paths used from any traffic source to any destination -- a feature called even load balancing. \myitem{IGP Weight Optimization (IGP-WO), based on~\cite{Fortz_Internet:2000}.} Given a traffic matrix and a topology, this algorithm runs a local search to find IGP weights that minimize the load on the maximally-utilized link. We implemented an approach based on~\cite{Fortz_Internet:2000} in 156 lines of code by leveraging REPETITA\xspace TE libraries (e.g., for shortest-path computation). \myitem{Mixed Integer Linear Programming (MILP) optimization, inspired by~\cite{belllabs-infocom15}.} We use standard optimization tools (i.e., the Gurobi optimizer\footnote{\small see www.gurobi.com}) to solve mixed integer linear programs inspired by the first work on SR-based TE, from Bhatia et al.~\cite{belllabs-infocom15}. As in~\cite{belllabs-infocom15}, our formulation admits that 2 IGP shortest paths are stitched together, sharing a common node or \textit{detour}, thanks to SR. In contrast to~\cite{belllabs-infocom15}, though, we removed the assumption that traffic flows could be split with arbitrary ratios at the network ingress -- as none of the other algorithms relies on the same assumption. We also reduced the number of variables, using a modified node-link formulation~\cite{pioro2004routing}, lowering the consumed memory. We detail our MILP model in the Appendix. \myitem{Constraint Programming heuristics (DEFO), as proposed in~\cite{defo-sig15}.} DEFO~\cite{defo-sig15} is a Constraint Programming heuristic designed to trade optimality for time efficiency and scalability. This algorithm is heuristic in two ways. First, it does not completely explore the solution space but iteratively samples it using a randomization technique called Large Neighborhood Search. Second, at every iteration, it greedily selects demands (from the biggest to the smallest) and an unconstrained number of detours per demand. We plugged DEFO in REPETITA\xspace by implementing a software wrapper for the original DEFO code, as described in \S\ref{subsec:framework-benefits}. \subsection{Scenarios}\label{subsec:impl-scenarios} REPETITA\xspace readily supports the following scenarios. \myitem{Max-congestion Analysis.} This scenario computes the maximum link utilization of paths returned by solvers. It therefore evaluates the quality of solutions found by input solvers with respect to a classic traffic-engineering goal (see, e.g.,~\cite{Fortz_Internet:2000}), i.e., minimization of maximal link utilization. \myitem{Overhead Analysis.} Primitives like segment routing enables more flexibility in the choice of forwarding paths at the cost of additional overhead (e.g., information added to packets and router configuration to implement detours). This scenario quantifies such overhead by keeping track of changed IGP weights, traffic demands re-routed with SR, and modified explicit paths. \myitem{Robustness Analysis.} REPETITA\xspace finally allows users to evaluate the robustness of a TE configuration with respect to failures. For any given topology and routing configuration (returned by a solver), the robustness analysis scenario evaluates how the maximum link utilization changes after the failure of every link that does not disconnect the network. To this end, it recomputes the paths associated to the given routing configuration (e.g., IGP weights or SR paths) on all the topologies resulting from removing any single link from the original network. \begin{figure*}[] \centering \begin{subfigure}[t]{.65\columnwidth} \includegraphics[width=\textwidth]{repeta-boxplot-smalltopos.pdf} \caption{Small topologies (<30 nodes).} \label{fig:load-small} \end{subfigure} \begin{subfigure}[t]{.65\columnwidth} \includegraphics[width=\textwidth]{repeta-boxplot-mediumtopos.pdf} \caption{Medium topologies.} \label{fig:load-medium} \end{subfigure} \begin{subfigure}[t]{.65\columnwidth} \includegraphics[width=\textwidth]{repeta-boxplot-largetopos.pdf} \caption{Large topologies (>100 nodes).} \label{fig:load-large} \end{subfigure} \caption{Max-congestion analysis results obtained with REPETITA\xspace. Top and bottom of the boxes represents $25$-th and $75$-th percentiles, the thick line in the boxes the median, and top and bottom whiskers minimum and maximum values, respectively. The dashed horizontal line is the theoretical lower bound, computed by REPETITA\xspace.} \label{fig:boxplot-load} \vspace{-1em} \end{figure*} \section{Repetita Iuvant\protect\footnote{\small L\MakeLowercase{atin for ``repeating does good''}}} \label{sec:eval} We ran our REPETITA\xspace implementation using a 40-core Intel(R) Xeon(R) 3.10GHz machine with 128GB memory. We limited multithreading of any given solver to 4 cores. The used JVM is OpenJDK version 1.8.0\_91. Experimental results are summarized in the following. \myitem{REPETITA\xspace eases reproduction of published evaluations.} We reproduced experiments described in~\cite{defo-sig15}, running each of them $20$ times. Our results confirmed what reported in the original publication, with minimal differences (at most $3\%$) in the maximal link utilization that should be ascribed to the randomized nature of the DEFO algorithm. \myitem{REPETITA\xspace enables new, large-scale comparisons, not provided in the original papers.} We compared MILP and DEFO performance on all the Internet Topology Zoo networks with link weights assigned as inverse of their capacity. We ran both solvers with two time limits (30 and 300 seconds), for a total of 5,200 experiments (1,300 settings per solver with a given time limit). This is an original head-to-head comparison between the two algorithms. Those experiments are meant to be a \textit{demonstration} of how our framework can be used to evaluate different TE algorithms according to several metrics. The evaluated solvers are indeed based on different principles: MILP looks for optimal solutions, DEFO trades optimality guarantees for speed and scalability. The following results quantify the tradeoffs achieved by those two algorithms. First, we evaluated the quality of solutions returned by MILP and DEFO, as shown in Figure~\ref{fig:boxplot-load}. In few cases (about 10\%), both the solvers are quite far from the theoretical lower bound, likely because of intrinsic limitations of segment routing (e.g., SR cannot force traffic on any possible path) and the assumed flow unsplittability (for practicality). The remaining experiments highlight fundamental differences between the considered algorithms, mostly depending on the size of input topologies. For small and medium topologies (Figures~\ref{fig:load-small} and~\ref{fig:load-medium}), both solvers are within $10\%$ of the theoretical lower bound, most of the time. Nevertheless, MILP is closer to the theoretical optimum than DEFO, as expected since the latter implements a heuristic approach. For bigger topologies (Figure~\ref{fig:load-large}), however, DEFO definitely outperforms MILP, as the latter often needs much more time to optimize SR paths. Also, the larger variation of results suggests that the allowed execution time is more critical for MILP than for DEFO, as the latter tends to converge faster to good TE solutions. We also ran the overhead analysis for the two solvers. Table~\ref{tab:overhead} stresses the huge difference between the high overhead needed by MILP (to reach optimal traffic spread), and the little one induced by DEFO (which moves far less flows). \begin{table}[h] \centering \footnotesize \begin{tabular}{@{}lclclclcl@{}} \hline \emph{Experiments} && \emph{25\%} && \emph{Median} && \emph{75\%} && \emph{95\%} \\ \hline MILP (30s) && 42.7\% && 56.6\% && 65.9\% && 75.2\% \\ MILP (5m) && 42.7\% && 56.9\% && 66.1\% && 75.2\% \\ DEFO (30s) && 1.1\% && 2.7\% && 6.1\% && 16.2\% \\ DEFO (5m) && 1.6\% && 4.2\% && 8.8\% && 20\% \\ \hline \end{tabular} \caption{Overhead comparison performed with REPETITA\xspace. The table reports the percentage of demands re-routed by MILP and DEFO, in the same experiments as in Figure~\ref{fig:boxplot-load}.} \label{tab:overhead} \vspace{-14pt} \end{table} \myitem{REPETITA\xspace facilitates new analyses.} We extended the comparison between MILP and DEFO with metrics not considered in previous papers. We ran the robustness analysis scenario to assert whether solutions computed with a congestion minimization focus are robust to failures. We provided MILP and DEFO with a 300s time limit, and collected their output for all topologies and one demand file per topology. The robustness analysis scenario then simulated single link failures and compared the maximum link utilization for each solution with their respective theoretical lower bound. On a total of $17,144$ possible link failures (across all networks), congestion appeared in $9,489$ cases for DEFO paths, and in $11,174$ MILP cases. In contrast, REPETITA\xspace lower bound algorithm could not avoid congestion in only $3,289$ cases. This observation opens an interesting question on whether and to what extent SR can be used to optimize both link utilization during normal operation while simultaneously providing some guarantees in the case of failures. \myitem{REPETITA\xspace motivates new research questions}, by showing the impact of TE algorithms in specific settings. In particular, our experiments highlight two aspects. First, they confirm the potential relevance of TE algorithms whenever links are run at high utilization. For example, Figure~\ref{fig:load-large} shows huge differences in the ability to quickly remove congestion (as for online TE) between two algorithms, despite relying on the same primitive (SR). This observation complements the more system-oriented perspective taken by recent works~\cite{B4:2013,swan-sigcomm13}, focused on how to robustly extract input data and try to enforce TE decisions. Second, our experiments motivate questions about TE primitives and technologies. As an illustration, we compared results obtained by IGP-WO with those returned by our SR algorithms (keeping the best result between MILP and DEFO, for each experiment). Figure~\ref{fig:pie} summarizes such comparison, when all the solvers are run for 5 minutes per experiment, on all topologies with inverse-capacity link weight assignment. While SR enables a substantially better solution in $25\%$ of the cases, solutions obtained by re-weighting links and using SR are quite similar from the congestion-avoidance perspective in most cases (within $1\%$ in $57\%$ of our experiments), with IGP-WO being even more effective than SR algorithms for about $7\%$ of the settings. This raises questions about the real expressive power of segment routing, and its usefulness in (which) network setups. \begin{figure}[t] \vspace{1em} \centering \includegraphics[width=0.9\columnwidth]{repeta-pie-crop} \caption{Systematic comparison through REPETITA\xspace between maximum link utilization induced state-of-the-art algorithms relying on weight optimization or SR.} \label{fig:pie} \end{figure} \section{Conclusions}\label{sec:conclusions} Repeatability of experiments is a key ingredient of science, and a largely-unfulfilled need in the networking community. This paper shows that we can overcome major obstacles to repeatability. Prominently, we can simplify the release of code in a way that allows repeatable evaluation and comparison with the state of the art. We presented REPETITA\xspace, a software framework that automates most of the experimental setup and evaluation process for traffic-engineering algorithms. To this end, REPETITA\xspace (i) includes a large and diversified input dataset, (ii) implements procedures (e.g., computation of shortest paths and traffic-to-path mapping), baselines (e.g., multi-commodity flow solutions) and cutting-edge algorithms for traffic engineering, and (iii) includes a pre-defined set of analyses tailored to traffic-engineering. Our experiments confirm the practicality of REPETITA\xspace for both repeating previous evaluations and performing new ones, that both deepen pros and cons of specific algorithms and open new research questions. We invite other researchers to join our effort, integrating our released code and proposing similar tools for other areas. Our work also aims at raising a serious discussion on how to work around real-data confidentiality (e.g., for network topologies and traffic data). Our proposal is to lower the need for proprietary data rather than keep looking for a method to share them. We bootstrapped a repository that we envision to collect real or realistic network topologies for repeatable TE evaluations. While current topologies mainly refer to specific networks (Internet transit provider ones, at per-router level), we plan to further enrich our repository, possibly integrating feedback from operators and vendors. \section*{Acknowledgements} We would like to thank Olivier Bonaventure and Renaud Hartert for initial discussion on repeatability in networking and traffic engineering. We are also grateful to Nikola Gvodziev for comments on a previous version of this paper. This work has been partially supported by the ARC grant 13/18-054 from Communaut\'{e} fran\c{c}aise de Belgique.

proofpile-arXiv_069-8797

\section{Introduction} In recent years, with the development of deep learning, researchers have found that using deeper and larger convolution neural network as the backbone of object detection, the accuracy of object detection is improved more. with the improvement of detection accuracy of object detection, computer vision moves from non-critical areas to key areas (such as unmanned driving and medical fields). However, in order to ensure the detection accuracy, a larger convolution neural network has to be used as backbone of object detection, resulting in a decrease in the detection speed and an increase in the cost of computing equipment. Therefore, many researchers propose many methods to improve the detection speed on the premise of ensuring the detection accuracy, such as the method of reducing the number of floating-point operations of convolution neural network by depth-wise separable convolution~\cite{MobileNets,MobileNetV2}point wise group convolution and channel shuffle~\cite{shufflenetv1,shufflenetv2}. Although considerable speed-up results have been achieved, these methods require carefully designed and tuned backbone of network. Many researchers believe that although the deeper backbone of network has a larger network capacity, so in the image classification, object detection and other tasks have a better performance. However, some specific tasks do not need such a large capacity, so in the case of ensuring the accuracy of convolution neural network, compression, quantization, channel subtraction on convolutional neural networks~\cite{5,6,7,8,9}. On the other hand, there are some studies on knowledge distillation that show~\cite{10, 11, 12, 13}, use a deeper model and a lighter model as teacher net and student net, then training student net with true label and soft label that teacher net output or intermediate result of teacher net output. It can greatly improve the performance of student net on specific tasks. But most of these methods require very complex cost functions and training methods, and these methods are mostly used for image classification, and two-stage object detection, etc., and are rarely used for one-stage object detection. Therefore, we need a knowledge distillation method that is simpler and more effective and can be applied to one-stage object detection. This paper proposes a simple and effective knowledge distillation neural network architecture, and it can obviously improve the performance of student net in one-stage object detection. Different from the conventional knowledge distillation method, We separated the backbone of deeper object detection neural network and lighter object detection neural network as teacher net and student net respectively, then we use the feature map generated by the teacher net as true samples, and the feature map generated by the student net as fake samples,referring to architecture of generative adversarial Nets~\cite{14}. Finally, we design a neural network as a discriminator and use true samples and fake samples to do the generation adversarial training. \noindent There are two main contributions: 1. Propose a clean and effective architecture of knowledge distillation that does not require the design of complex cost functions, and can be applied to one-stage object detection. 2. Using the architecture of generation adversarial nets to avoid complex knowledge migration design, let the student net automatically obtain dark knowledge from the teacher net. \section{Related Works} \noindent {\bf CNN For Detection}: The deep learning architecture of object detection is mainly divided into two types:1) one is the one-stage object detection, such as the SSD proposed by Liu W et al. ~\cite{15}, which directly returns the position and category of the object by the convolutional neural network; 2)two is the two-stage object detection, such as fast rcnn ~\cite{16} proposed by Girshick et al., and later Faster-RCNN ~\cite{17} and R-FCN ~\cite{18}, etc., it first regress the proposal boxes by the convolutional neural network; then identify each proposal box again; Finally return to the correct location and category. \noindent {\bf Network Compression}: many researchers believe that deep neural networks are over-parameterized, and it has too many redundant neurons and connections. He Y et al. [8] thought each layer of neurons of convolutional neural networks are sparse, and they use lasso regression to find the most representative neuron per layer of convolutional neural networks reconstructs the output of this layer. Zhuang Z et al. ~\cite{9} believe that layer-by-layer channel pruning affects the discriminating ability of Convolutional neural networks, so the auxiliary ability of convolutional neural networks is preserved by adding auxiliary loss in the fine-tune and pruning stages. \noindent {\bf Network quantification}: Wu J et al. ~\cite{20} use the k-means clustering algorithm to accelerate and compress the convolutional layer and the fully connected layer of the model to obtain better quantization results by reducing the estimation error of the output response of each layer, and proposed an effective training scheme to suppress the multi-layer cumulative error after quantization. Jacob B ~\cite{21} et al. proposed a method that quantify weights and inputs as uint8, and bias to unit32, as the same time, the forward uses quantization, and the backward correction error is not quantized to ensure that the convolutional neural networks performance and speed of inference during training. \noindent {\bf Knowledge Distillation}: A method for transfer knowledge of large network to small network. Hinton et al. ~\cite{6} used the result of teacher net output as the soft label of student net, and advocated the use of temperature cross entropy instead of L2 loss.Romero et al. ~\cite{19} believe that student net needs more unlabeled data to get as close as possible and imitate teacher net.When Chen G et al. ~\cite{12} distilled the two-stage object detection, they respectively extracted the middle feature map of teacher net and the dark knowledge of rpn/rcnn to let student net go to mimic.There are also some researchers who give the attention information of teacher net to the student network. For example. For example, Zagoruyko S ~\cite{22} et al. proposed spatial-attention, which is a method of transmitting the thermal information of teacher net to student net. Yim J et al. ~\cite{23} used the relationship between layer and layer of teach net as the goal of student network learning. However their knowledge distillation requires the design of very complex loss functions, and the extraction of complex dark knowledge,and these methods are mostly in the two-stage object detection, rarely used in one-stage object detection. In order to have a simple and effective way of distilling knowledge, we refer to the architecture of the generated adversarial nets ~\cite{14} to take the feature map generated by the teacher network and the student network as true samples and fake samples, respectively. Finally we design a neural network as a discriminator and use true samples and fake samples to do the generation adversarial training to improve the performance of the student network in one-stage object detection. \begin{figure*} \begin{center} \includegraphics[width=16cm, height =8cm]{pictures/GAN_dist.png} \end{center} \caption{Teacher Net and SSD-Head are the backbone network and head network of the larger and fully trained SSD model, respectively. Student Net is a smaller network. D Net is a module consisting of multiple 6 small discriminant networks.} \label{fig:short} \end{figure*} In this paper, we use the one-stage object detection SSD ~\cite{15} as our object detection. The architecture of SSD is mainly divided into two parts, 1) backbone of network, as feature extractor. 2) SSD-Head, use the features extracted by the backbone of network to detect the category and location of the object. In order to obtain a better knowledge distillation effect, it is important to make rational use of these two parts. \section{Method} \subsection{Overall Structure} Fig 1 is the overall structure of our model. We first use a SSD model with a larger capacity and full training, and split the SSD model into backbone of network and SSD-Head. We use the backbone of network as the teacher net,and pick a smaller network as student net. We use multiple feature maps generated by teacher net as true samples, and multiple feature maps generated by student net as fake samples, then send the true sample and the fake sample to each of the corresponding discriminative networks (fig 2) in D Net,at the same time, input the fake sample into the SSD-Head. \begin{figure}[H] \begin{center} \includegraphics[width=6cm, height =4cm]{pictures/GAN_dist2.png} \end{center} \caption{One of the discriminative networks in the D-Net module, consisting of multiple downsampled convolutional layers.} \label{fig:short} \end{figure} \subsection{Training Process} Our training process has two stages.The first stage is mainly the use of the generation adversarial training (GAN). First,we train each discriminating network in the D Net module to identify whether the input sample is true sample or fake sample, and freeze the weights of teacher net and student net. Then train the student net to generate a fake sample to trick each of the discriminating networks in the D Net module, and normal SSD training is performed on student net and SSD-Head.During the training process,weight of the teacher net and weights of each discriminative network in the D Net module are frozen,as Equation 1. \begin{equation}\label{eq:12} \begin{split} & L\mathop{{}}\nolimits_{{Dt}}=D{ \left( {Teacher{ \left( {x, \theta \mathop{{}}\nolimits_{{t}}} \right) }, \theta \mathop{{}}\nolimits_{{d}}} \right) } \\ & L\mathop{{}}\nolimits_{{Ds}}=D{ \left( {Student{ \left( {x, \theta \mathop{{}}\nolimits_{{s}}} \right) }, \theta \mathop{{}}\nolimits_{{d}}} \right) } \\ & L\mathop{{}}\nolimits_{{conf}}=L\mathop{{}}\nolimits_{{conf}}{ \left( {Student{ \left( {x, \theta \mathop{{}}\nolimits_{{s}}} \right) }}, \theta \mathop{{}}\nolimits_{{head}} \right) } \\ & L\mathop{{}}\nolimits_{{loc}}=L\mathop{{}}\nolimits_{{loc}}{ \left( {Student{ \left( {x, \theta \mathop{{}}\nolimits_{{s}}} \right) }},\theta \mathop{{}}\nolimits_{{head}} \right) } \\ & L\mathop{{}}\nolimits_{{D}}=\frac{{1}}{{N}}{ \left( {L\mathop{{}}\nolimits_{{Dt}}-L\mathop{{}}\nolimits_{{Ds}}} \right) } \\ & L\mathop{{}}\nolimits_{{G}}=\frac{{1}}{{N}}{ \left( {D{ \left( {Student{ \left( {x, \theta \mathop{{}}\nolimits_{{s}}} \right) }, \theta \mathop{{}}\nolimits_{{d}}} \right) }+L\mathop{{}}\nolimits_{{conf}}+L\mathop{{}}\nolimits_{{loc}}} \right) } \\ \end{split} \end{equation} N in Equation 1 represents the size of batchsizes; D represents the discriminant network, Teacher and Student represent teacher net and student net, respectively; θt, θs, and θd represent the weight of the teacher net, the weight of the student net, and the weights of each discriminant network in the D Net module, respectively. Lconf represents the loss function of the classification in the SSD, and Lloc represents the loss function of the bounding box in the SSD. The second stage is that we normal SSD training on student net and SSD-Head alone,after many epochs of generation adversarial training . \section{Experiment} In this section, we will experiment with the PASCAL VOC and cocoto validate our approach, including 20 categories. Our hardware device is two NVIDIA GTX 1080Ti GPUs. The software framework is gluoncv. \subsection{Training Process} All models used adam with 0.0005 learning rate in the first phase and SGD with 0.0005 weight decay and 0.9 momentum in the second phase. The first stage trains epochs is 180 and the second stage epochs is 90. The stduent net is MobilenetV1, MobilenetV2 and resnet18, teacher net VGG16, ResNet50 and ResNet101. These models have pretrained under ImageNet. Due to insufficient time, we only use MobilenetV1 as Student Net and ResNet50 as Teacher Net do experiment on coco. \subsection{Results} We compare the native SSD with the SSD of GAN knowledge distillation under different Teacher nets,up to the student net 2.8mAP on PASCAL VOC. When the teacher net is ResNet101 and student net is ResNet18, the improvement is not as good as teacher net is ResNet50.Because precision of SSD of ResNet101 is not as good as precision of SSD of ResNet50 on PASCAL VOC,and SSD of ResNet101 is only one mAP higher than SSD of ResNet50 on coco.So in order to complete the experiment efficiently,We only use ResNet50 as teacher net on coco,and we will use different teacher net in the near future. From the table.1, we found that the lower the mAP on the data set, the more obvious the effect of the student net after GAN knowledge distillation.So we use moblienetV0.75 as the student net on coco,and We have increased 5 mAP for SSD of moblienetV0.75 on coco. \begin{table} \begin{center} \begin{tabular}{|l|l|c|} \hline Student net & Teacher net & voc 2007 test\\ \hline\hline & - & 75.4 \\ MobilenetV1 & VGG16 & 77.3(+1.9)\\ & ResNet50 & 77.6(+2.2)\\ & ResNet101 & 77.6(+2.2)\\ \hline\hline & - & 75.9 \\ MobilenetV2 & VGG16 & 77.2(+1.4)\\ & ResNet50 & 77.7(+1.7)\\ & ResNet101 & 77.5(+1.5)\\ \hline\hline & - & 74.8 \\ ResNet18 & VGG16 & 77.2(+2.6)\\ & ResNet50 & 77.6(+2.8)\\ & ResNet101 & 77.3(+2.7)\\ \hline \end{tabular} \end{center} \caption{Different student nets are not used GAN-knowledge distillation and the use of a GAN-knowledge } \end{table} \begin{table} \begin{center} \begin{tabular}{|l|l|c|} \hline Student net & Teacher net & coco 2017 val\\ \hline\hline & - & 21.7 \\ MobilenetV1 & VGG16 & -\\ & ResNet50 & 25.4(+3.7)\\ & ResNet101 & - \\ \hline\hline & - & 18.0 \\ MobilenetV1\underline{\hspace{0.5em}}0.75 & VGG16 & -\\ & ResNet50 & 23.0(+5.0)\\ & ResNet101 & - \\ \hline\hline & - & 22 \\ MobilenetV2 & VGG16 & -\\ & ResNet50 & 24.6(+2.6)\\ & ResNet101 & -\\ \hline \end{tabular} \end{center} \caption{ moblienetv1 use GAN-knowledge distillation in coco.} \end{table} We also use our method to improve the two stage object detection,such as faster rcnn.We found that faster rcnn of roialign is 4.7 mAP higher than faster rcnn of roipooling in Pascal VOC 2007 test,as shown in Table 3.Our method gets 6.8 absolute gain in mAP for faster rcnn of ResNet50. \begin{table} \begin{center} \begin{tabular}{|l|l|c|} \hline Student net & Teacher net & voc 2007 test\\ \hline\hline & - & 67.0 \\ ResNet50(roipooling+GAN-KD) & ResNet101 & 73.8(+6.8)\\ \hline\hline & - & 71.7 \\ ResNet50(roialign+GAN-KD) & ResNet101 & 74(+2.3)\\ \hline\hline & - & 69.0 \\ ResNet50(Imitation) & ResNet101 & 72.0(+3.0)\\ \hline \end{tabular} \end{center} \caption{ Teacher net is the faster rcnn of the ResNet101, and uses Pascal Voc 2007 trainval as the training set. The mAP is 74.8+ on the Pascal Voc 2007 test set. The first row and the second row is our method, and the third row is Distilling Object Detectors with Fine-grained Feature Imitation\cite{wang2019distilling}} \end{table} \section{Conclusion} At present, most of the methods of knowledge distillation are aimed at the two-stage object detection. We propose a clean and effective knowledge distillation method for the one-stage object detection. The feature map generated by the teacher net is taken as true samples, and the feature map generated by the student net is taken as fake samples. Then We make the distribution of student net to fit the distribution of teacher net by using true samples and fake samples to do the generation adversarial training. Through unsupervised learning then generation adversarial training, we avoid manual design and extract feature from teacher net to let the student net fit. In this way, we make the whole training process simple and efficient, and greatly improve the detection accuracy of the student net. Our approach provides new ideas for the further development of knowledge distillation. {\small

proofpile-arXiv_069-9023

\section{Introduction} The Carleson embedding theorem (CET) is a classical theorem in harmonic analysis with many applications to PDE. It states that a Carleson measure gives an $L^2$ embedding for a function. The Carleson embedding theorem first appeared in L. Carleson's solution of the free interpolation problem \cite{Ca1958} and later was used in his celebrated proof of the Corona Theorem \cite{Ca1962}. In this paper we are primarily using the language of dyadic cubes. See for example \cite{NT1997} for the formulation therein and the illustration of its proof via Bellman functions. One may consider any underlying measure here, so the weighted version is just the same as the unweighted version with a very similar proof. A weighted bilinear embedding theorem (BET) was an important tool in the early days of sharp weighted theory. Indeed, it was a crucial ingredient in the first sharp weighted estimates for classical singular operators \cite{P2007}. It states that a Carleson measure gives rise to a bilinear estimate, featuring two different functions. In this note we are concerned with the failure of its matrix weighted analogs. \ An unweighted CET with matrix Carleson measure holds trivially, derived from the scalar case. Other than in the scalar case, the extension to the weighted setting is not trivial. First versions imposed the so--called $A_2$ property of the matrix weight introduced by Treil--Volberg in \cite{TV1997}, a condition that had been absent in the scalar case. Recently Culiuc--Treil \cite{CT} obtained the matrix weighted version of this theorem without any restriction on the weight, other than it being a matrix weight. \ The so--called matrix $A_2$ conjecture asks for the exact growth of the matrix weighted norm estimate of the Hilbert transform acting on vector functions. The conjectured growth estimate is that of a linear dependence on the matrix $A_2$ characteristic of the weight. Motivated by this problem, we consider the question of a bilinear version of the matrix weighted Carleson lemma that might be useful for this task. Recently one of the authors with Pott and Reguera proved an apparently weak version, featuring a matrix weight, but a scalar Carleson sequence instead of a matrix sequence and an estimate involving inner products instead of norms \cite{RPP}. We show that this formulation is optimal in that any improvement of the statement fails. Indeed, the failure stems from the maximal excentricity of the matrix weight and not from any increase in the $A_2$ characteristic. We show that even assuming the $A_2$ condition, one can get an infinite bilinear embedding. We show that a bilinear estimate can be obtained in terms of the square root of the conditioning number and this condition is necessary. \ The failure of BET is natural and requires only a very simple example. It is however an important notable difference to the scalar case. It is also in a contrast to a positive result on a matrix two--weighted $T1$ theorem by Bickel--Culiuc--Treil--Wick \cite{BCTW}. There is an array of difficulties encountered in the task to find various sharp weighted estimates in the matrix weighted case. Further, it is also very difficult to get definite negative answers. As of today, most optimal estimates elude us. We mention \cite{IKP2017} for some interesting positive results in the matrix weighted setting. The first quantitative estimate for the Hilbert transform in the matrix weighted setting was given in \cite{BPW2016} as well as upper and lower square function estimates. Some of these estimates in \cite{BPW2016} were close to their scalar weighted analogs, but none of them matched the sharp scalar estimate. We now know that the estimate for the square function with matrix weight does not change its dependence on the $A_2$ characteristic as compared to the scalar case. The result is due to Hyt\"onen--Petermichl--Volberg \cite{HPV2019} and Treil \cite{T} where both use a stopping time argument known as sparse domination. The classical sharp scalar weighted result by Hukovi\'c--Treil--Volberg \cite{HTV} was proved via Bellman functions and only required the simple Carleson Lemma, not the bilinear version. The estimate for the Hilbert transform is still open, with best to date estimate by Nazarov--Petermichl--Treil--Volberg \cite{NPTV2017}, missing the sharp conjecture by a half power of the $A_2$ characteristic. For most known operator norms, the question of sharpness is unsettled, but there usually is just a raised power on the dependence of the $A_2$ charateristic, not complete failure of the estimate, such as what we see in the case of BET. \ The first proof of a version of the scalar bilinear Carleson lemma is found in \cite{PW2002} and \cite{P2007} and was rather complicated, using tools and a construction implicit in the seminal article by Nazarov--Treil--Volberg \cite{NTV1999}. The argument features a rather cleverly gaged Bellman function and three conditions on the measure sequence rather than one. It was understood for some time by the experts that two of the arising conditions were redundant. We show that this redundancy is still true in the presence of a matrix weight and a matrix sequence. A previous result only allowed for scalar Carleson sequences. Since BET fails with matrix Carleson sequences, this redundancy result does not have this particular application, but it is an interesting estimate in its own right, useful for other matrix weighted tasks such as certain maximal function estimates. It can also be used in combination with CET to alter the testing condition, i.e. changing the Carleson sequence. This step was important in some scalar proofs in the non--homogenous setting, see \cite{TTV} and \cite{DP2019}. \section{Notation and detailed history} Let us say $Q_0 = [0, 1]$ is endowed with a dyadic filtration and let $\mathcal{D}$ be the dyadic grid. We call a $d \times d$ matrix--valued function $W$ a weight if $W (x)$ is positive semidefinite almost everywhere and if $W$ and $W^{- 1}$ are locally integrable. One defines $L_{\mathbbm{C}^d}^2 (W)$ to be the set of vector functions with \begin{eqnarray*} \| f \|^2_{L_{\mathbbm{C}^d}^2 (W)} = \int_{Q_0} \| W^{1 / 2} (x) f (x) \|_{\mathbbm{C}^d}^2 \mathrm{d} x = \int_{Q_0} \langle W (x) f (x), f (x) \rangle_{\mathbbm{C}^d} \mathrm{d} x < \infty . \end{eqnarray*} When $d = 1$ this becomes the classical weighted space $L_{\mathbbm{C}}^2 (w)$. Let us denote by $\langle \cdot \rangle_Q$ the average of a scalar, vector or matrix function over the cube $Q$. By $\| \cdot \|_{\tmop{op}}$ we mean the operator norm of the matrix. \ The dyadic formulation of the Carleson embedding theorem reads as follows: let $(\alpha_Q)$ be a sequence of non--negative scalars and let $w$ be a weight. Then for $f$ supported on $Q_0$ \begin{eqnarray*}\lefteqn{ \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \alpha_Q \langle w \rangle^2_Q \lesssim \langle w \rangle_K, \forall K \in \mathcal{D} (Q_0)}\\ & \Leftrightarrow & \sum_{Q \in \mathcal{D} (Q_0)} \alpha_Q \langle f w^{1/2} \rangle^2_Q \lesssim \| f \|^2_{L^2_{\mathbbm{C}}} . \end{eqnarray*} There is no difference if the weight is identical to $1$, with the proof in \cite{NT1997} being exactly the same when switching to weighted averages and renormalizing appropriately. One can rewrite the assumption and conclusion renaming $\beta_Q = \alpha_Q \langle w \rangle^2_Q$ and $g=fw^{-1/2}$ with \begin{eqnarray*} \frac{1}{w (K)} \sum_{Q \in \mathcal{D} (K)} \beta_Q \lesssim 1, \forall K \in \mathcal{D} (Q_0) \Leftrightarrow \sum_{Q \in \mathcal{D} (Q_0)} \beta_Q \langle g \rangle^2_{Q, w} \lesssim \| g \|^2_{L^2_{\mathbbm{C}} (w)}, \end{eqnarray*} with $ \langle g \rangle_{Q, w} = \frac{1}{w (Q)} \int_Q g w. $ There is as an immediate consequence a CET for matrix Carleson sequences $(A_Q)$ and the matrix weight $W =\tmmathbf{1}$: \begin{eqnarray*} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} A_Q \lesssim \tmmathbf{1}, \forall K \in \mathcal{D} (Q_0) \Leftrightarrow \sum_{Q \in \mathcal{D} (Q_0)} \langle A_Q \langle f \rangle_Q, \langle f \rangle_Q \rangle_{\mathbbm{C}^d} \lesssim \| f \|^2_{L^2_{\mathbbm{C}^d}} . \end{eqnarray*} Here, the left hand side inequality is understood in the sense of operators and its necessity `$\Leftarrow$' is seen by testing the right hand inequality on functions $f = e\tmmathbf{1}_K$ for any vector $e$ and any $K \in \mathcal{D} (Q_0)$. The deduction of the implication `$\Rightarrow$' from the scalar case is easily observed by taking trace and using equivalence of norms at the cost of a dimensional constant. See \cite{NPTV2002} for details. A weighted version does not follow from the scalar case, though. The matrix weighted CET was proved only in 2015: \begin{theorem}[Culiuc, Treil] \label{WCET}Let $W$ be a matrix weight of size $d \times d$ and $(A_Q)$ a sequence of positive semidefinite matrices of size $d \times d$. Then for $f$ supported in $Q_0$ \begin{eqnarray*} \lefteqn{ \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \langle W \rangle_Q A_Q \langle W \rangle_Q \lesssim \langle W \rangle_K, \forall K \in \mathcal{D} (Q_0)}\\ &\Leftrightarrow& \sum_{Q \in \mathcal{D} (Q_0)} \| A_Q^{1 / 2} \langle W^{1 / 2} f \rangle_Q \|_{\mathbbm{C}^d}^2 \lesssim \| f \|^2_{L^2_{\mathbbm{C}^d}} \end{eqnarray*} \end{theorem} The successful argument added a twist to the `Bellman function with a parameter' invented in \cite{NPTV2002}, that managed to make certain non--commutative obstacles disappear, which arise from differentiating functions with matrix variables. Theorem \ref{WCET} implies via a simple linearization argument the norm estimate below on a Doob type maximal function with matrix measure. The obtained norm estimate does not assume the $A_2$ condition. See for example \cite{RPP} for an exposition of the well known argument as well as a motivation of the definition of this maximal function. \begin{theorem} \label{MAX}Let $W$ be a matrix weight of size $d \times d$ then $M_W : L^2_{\mathbbm{C}^d} \rightarrow L^2_{\mathbbm{C}}$ defined by \begin{eqnarray*} M_W f (x) = \sup_{Q \in \mathcal{D} (Q_0), x \in Q} \| W^{1 / 2} (x) \langle W \rangle^{- 1}_Q \langle W^{1 / 2} f \rangle_Q \|_{\mathbbm{C}^d}, \end{eqnarray*} is bounded. \end{theorem} We now give some background on the occurrance and motivation for the weighted bilinear Carleson lemma in the scalar case. In 1999, Nazarov--Treil--Volberg published their paper \cite{NTV1999} on the necessary and sufficient conditions on the scalar weights $w$ and $v$ so that the operators $T_{\sigma} : L^2 (w) \rightarrow L^2 (v), h_{_{} Q} \mapsto \sigma_Q h_Q$ are uniformly bounded. Here \begin{eqnarray*} h_Q = - | Q |^{- 1 / 2} \tmmathbf{1}_{Q_-} + | Q |^{- 1 / 2} \tmmathbf{1}_{Q_+} \end{eqnarray*} are the Haar functions and $\sigma_Q = \pm 1$. Wittwer then proved that $T_{\sigma} : L^2 (w) \rightarrow L^2 (w)$ has operator norm uniformly bounded by a linear function of the scalar $A_2$ characteristic of the weight \begin{eqnarray} [w]_{A_2} = \sup_Q \langle w \rangle_Q \langle w^{- 1} \rangle_Q . \label{sA2} \end{eqnarray} This estimate is sharp, as it was shown to be true for classical operators such as the Hilbert transform \cite{P2007}. In these early proofs the idea consisted of `disbalancing' the Haar basis by finding coefficients $a^w_Q$ and $b^w_Q$ \begin{eqnarray} h_Q = a^w_Q \tmmathbf{1}_Q + b^w_Q h^w_Q \label{whaar} \end{eqnarray} so that the system $h^w_Q$ becomes orthonormal in $L^2 (w)$ and similarly for $w^{- 1}$. In its dualized form, the operators $T_{\sigma}$ are uniformly dominated by the sum \begin{eqnarray*} \sum_{Q \in \mathcal{D} (Q_0)} | \langle w^{1/2}f, h_Q \rangle_{L^2} \langle w^{-1/2}g, h_Q \rangle_{L^2} | \lesssim [w]_{A_2} \| f \|_{L_{\mathbbm{C}}^2 } \| g \|_{L_{\mathbbm{C}}^2 } . \end{eqnarray*} When replacing the Haar functions by the equation (\ref{whaar}) above, we obtain four sums. The sum featuring the two non--cancellative terms arising from the indicator functions being the most difficult. It is for this sum that the bilinear Carleson lemma came to life. The first version appeared in \cite{NTV1999} in the two weight case with a rather complicated necessary condition on the two weights $w$ and $v$. A simplified version with easy to test conditions on the single weight $w$ appeared in \cite{PW2002} \cite{P2007}: \begin{eqnarray*}\lefteqn{ \frac{1}{| K |} \sum_{Q \in K} \alpha_Q \lesssim 1, \forall K \in \mathcal{D} (Q_0),}\\ &&\quad \frac{1}{| K |} \sum_{Q \in K} \frac{\alpha_Q}{\langle w \rangle_Q} \lesssim \langle w^{- 1} \rangle_K, \forall K \in \mathcal{D} (Q_0), \\ &&\quad\frac{1}{| K |} \sum_{Q \in K} \frac{\alpha_Q}{\langle w^{- 1} \rangle_Q} \lesssim \langle w \rangle_K, \forall K \in \mathcal{D} (Q_0)\\ &\Rightarrow& \sum_{Q \in \mathcal{D} (Q_0)} \frac{\alpha_Q}{\langle w \rangle_Q \langle w^{- 1} \rangle_Q} \langle w^{1 / 2} | f | \rangle_Q \langle w^{- 1 / 2} | g | \rangle_Q \lesssim \| f \|_{L^2_{\mathbbm{C}^d}} \| g \|_{L^2_{\mathbbm{C}^d}} \text{} . \end{eqnarray*} The conditions are not necessary. Indeed it has been known that two conditions are redundant: \begin{eqnarray}\lefteqn{ \frac{1}{| K |} \sum_{Q \in K} \alpha_Q \lesssim 1, \forall K \in \mathcal{D}(Q_0)}\\ \nonumber&\Rightarrow& \frac{1}{| K |} \sum_{Q \in K} \frac{\alpha_Q}{\langle w \rangle_Q} \lesssim \langle w^{- 1} \rangle_K \tmop{and} \frac{1}{| K |} \sum_{Q \in K} \frac{\alpha_Q}{\langle w^{- 1} \rangle_Q} \lesssim \langle w \rangle_K, \forall K \in \mathcal{D}(Q_0). \label{red} \end{eqnarray} But not even the strongest condition is necessary for the conclusion of the bilinear embedding. The interest lied in the simpler testing condition and its successful application to sharp weighted estimates. The first proofs of this bilinear lemma were quite complicated. Indeed, in 2008 Nazarov--Treil--Volberg \cite{NTV2008} characterized two--weight estimates for individual multipliers $T_{\sigma}$ again in the flavor of a $T 1$ theorem. There is a positive extension to the matrix case by Bickel--Culiuc--Treil--Wick \cite{BCTW}. \ Here is the best to date matrix weighted version of BET featuring a scalar sequence and an inner product: \begin{theorem}[Petermichl, Pott, Reguera] \label{siBET}Let $(\alpha_Q)$ be a sequence of non--negative scalars. Then for $f, g$ supported in $Q_0$ \begin{eqnarray*}\lefteqn{ \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \alpha_Q \lesssim 1, \forall K \in \mathcal{D} (Q_0)}\\ &\Rightarrow & \sum_{Q \in \mathcal{D} (Q_0)} \alpha_Q | \langle \langle W \rangle^{- 1}_Q \langle W^{1 / 2} f \rangle_Q, \langle W^{- 1} \rangle^{- 1}_Q \langle W^{- 1 / 2} g \rangle_Q \rangle_{\mathbbm{C}^d} | \lesssim \| f \|_{L^2_{\mathbbm{C}^d}} \| g \|_{L^2_{\mathbbm{C}^d}}. \end{eqnarray*} \end{theorem} We also study matrix analogs for the redundancy whose scalar version is the implication in (\ref{red}). Here is the redundancy with scalar sequence and matrix weight, which was proved recently: \begin{theorem}[Petermichl, Pott, Reguera] \label{sRED}Let $(\alpha_Q)$ be a non--negative sequence and $W$ a matrix weight, then \begin{eqnarray*}\lefteqn{ \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \alpha_Q \leqslant 1, \forall K \in \mathcal{D} (Q_0)}\\ & \Rightarrow & \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \alpha_Q \langle W^{- 1} \rangle_Q^{- 1} \leqslant 4 \langle W \rangle_K, \forall K \in \mathcal{D} (Q_0) . \end{eqnarray*} \end{theorem} \section{Main Results} In this paper we prove the sharpness of the bilinear Embedding Theorem \ref{siBET}, i.e. the failure of any improvement: \begin{theorem} \label{BETfailure}Let $(A_Q)$ be a sequence of positive semidefinite matrices. The Carleson condition \begin{eqnarray}\label{conditionBETfailure} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} A_Q \lesssim \tmmathbf{1}, \forall K \in \mathcal{D} (Q_0) \end{eqnarray} does not imply the existence of a constant so that \begin{eqnarray}\label{miBETfailure}\lefteqn{ \sum_{Q \in \mathcal{D} (Q_0)} | \langle A_Q \langle W \rangle^{- 1}_Q \langle W^{1/2} f \rangle_Q, \langle W^{- 1} \rangle^{- 1}_Q \langle W^{- 1/2} g \rangle_Q \rangle_{\mathbbm{C}^d} |}\\ \nonumber& \lesssim &\| f \|_{L_{\mathbbm{C}^d}^2 } \| g \|_{L_{\mathbbm{C}^d}^2} \end{eqnarray} for all matrix weights $W$ and functions $f, g$ supported in $Q_0$. The Carleson condition also does not imply \begin{eqnarray}\label{mnBETfailure}\lefteqn{ \sum_{Q \in \mathcal{D} (Q_0)} \| A^{1 / 2}_Q \langle W \rangle^{- 1}_Q \langle W^{1/2} f \rangle_Q \|_{\mathbbm{C}^d} \| A^{1 / 2}_Q \langle W^{- 1} \rangle^{- 1}_Q \langle W^{- 1/2} g \rangle_Q \|_{\mathbbm{C}^d}}\\ \nonumber & \lesssim & \| f \|_{L_{\mathbbm{C}^d}^2} \| g \|_{L_{\mathbbm{C}^d}^2 } \end{eqnarray} for all matrix weights $W$ and functions $f, g$ supported in $Q_0$, not even if we only allow scalar sequences $A_Q = m_Q$. \end{theorem} We find that a natural bilinear Carleson embedding theorem only holds under a very strong condition on the matrix weight: \begin{definition} Let $W$ be a matrix weight. We define the conditioning number of a matrix weight as \begin{eqnarray*} [W]_{C_2} = \sup_x \kappa (W (x)) = \sup_x \| W (x) \|_{\tmop{op}} \| W^{- 1} (x) \|_{\tmop{op}} = \frac{\lambda_{\max} (W (x))}{\lambda_{\min} (W (x))}, \end{eqnarray*} where $\lambda_{\max}$ and $\lambda_{\min}$ return the maximal respectively minimal eigenvalue. \end{definition} This is in a sharp contrast to the matrix $A_2$ condition in \cite{TV1997}, the analog of the scalar version in equation (\ref{sA2}): \begin{definition} Let $W$ be a matrix weight. Then the matrix $A_2$ condition is \begin{eqnarray*} [W]_{A_2} = \sup_Q \| \langle W \rangle^{1 / 2}_Q \langle W^{- 1} \rangle^{1 / 2}_Q \|_{\tmop{op}}^2 . \end{eqnarray*} \end{definition} See \cite{TV1997} for a list of elementary properties of this characteristic, such as $[W]_{A_2} \geqslant 1$. We prove: \begin{theorem} \label{C2BET}Let $(A_Q)$ be a sequence of positive semidefinite matrices and $W$ a matrix weight. Then \begin{eqnarray*}\lefteqn{ \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} A_Q \lesssim \tmmathbf{1}, \forall K \in \mathcal{D} (Q_0)}\\ & \Rightarrow & \sum_{Q \in \mathcal{D} (Q_0)} \| A^{1 / 2}_Q \langle W \rangle^{- 1}_Q \langle W^{1 / 2} f \rangle_Q \|_{\mathbbm{C}^d} \| A^{1 / 2}_Q \langle W^{- 1} \rangle^{- 1}_Q \langle W^{- 1 / 2} g \rangle_Q \|_{\mathbbm{C}^d} \\ && \quad \lesssim [W]^{1 / 2}_{C_2} \| f \|_{L_{\mathbbm{C}^d}^2} \| g \|_{L_{\mathbbm{C}^d}^2 .} \end{eqnarray*} The condition $[W]_{C_2}$ is necessary and the power $1 / 2$ is optimal. \end{theorem} Further, we deduce from Theorem \ref{sRED} in full generality: \begin{theorem} \label{RED}Let $(B_Q)$ be a sequence of positive semidefinite matrices such that \begin{eqnarray*} \frac{1}{| {L} |} \sum_{Q \in \mathcal{D} ({ L})} B_Q \lesssim \tmmathbf{1}, \forall { L} \in \mathcal{D} (Q_0) . \end{eqnarray*} Then \begin{eqnarray*}\lefteqn{ \frac{1}{| { K} |} \sum_{Q \in \mathcal{D} ({ K})} \langle B_Q \langle W \rangle_{{ K}}^{- 1 / 2} \langle W^{- 1} \rangle^{- 1 / 2}_Q { e}, \langle W \rangle^{- 1 / 2}_{{ K}} \langle W^{- 1} \rangle^{- 1 / 2}_Q { e} \rangle_{\mathbbm{C}^d}}\\ &&\quad \lesssim \| { e} \|_{\mathbbm{C}^d}^2, \forall {K} \in \mathcal{D} (Q_0) . \end{eqnarray*} and \begin{eqnarray*}\lefteqn{ \frac{1}{| { K} |} \sum_{Q \in \mathcal{D} ({ K})} \langle B_Q \langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_{{ K}} { e}, \langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_{{K}} {e} \rangle_{\mathbbm{C}^d} }\\ &&\quad \lesssim \| { e} \|_{\mathbbm{C}^d}^2, \forall {\color{black} K} \in \mathcal{D} (Q_0) . \end{eqnarray*} for all vectors ${e }$ and all matrix weights $W$. \end{theorem} Notice that in Theorem \ref{RED} putting ${ f} = \langle W \rangle^{- 1 / 2}_{{ K}} { e}$ the last inequality becomes \begin{eqnarray*}\lefteqn{ \frac{1}{| {\color{black} K} |} \sum_{Q \in \mathcal{D} ({K})} \langle B_Q \langle W^{- 1} \rangle^{- 1 / 2}_Q { f}, \langle W^{- 1} \rangle^{- 1 / 2}_Q {\color{black} f} \rangle_{\mathbbm{C}^d}}\\ &&\quad \lesssim \| \langle W \rangle^{1 / 2}_{{ K}} { { f}} \|_{\mathbbm{C}^d}^2, \forall { K} \in \mathcal{D} (Q_0), \end{eqnarray*} that rewrites as the operator inequality resembling (\ref{red}): \begin{eqnarray*} \frac{1}{| { K} |} \sum_{Q \in \mathcal{D} ( K)} \langle W^{- 1} \rangle^{- 1 / 2}_Q B_Q \langle W^{- 1} \rangle^{- 1 / 2}_Q \lesssim \langle W \rangle_K,\forall {K} \in \mathcal{D} (Q_0) . \end{eqnarray*} \section{Bilinear embedding theorem} We prove Theorems \ref{BETfailure} and \ref{C2BET} in this section. \begin{proof}[Theorem \ref{C2BET}] First, we can assume that the sequence consists of scalars, by switching to the Carleson sequence of maximal eigenvalues $\alpha_Q = \| A_Q \|_{\tmop{op}}$ at the loss of a dimensional constant. Then we have to show the inequality \begin{eqnarray}\lefteqn{ \sum_{Q \in \mathcal{D} (Q_0)} \alpha_Q \| \langle W \rangle^{- 1}_Q \langle W^{1 / 2} f \rangle_Q \|_{\mathbbm{C}^d} \| \langle W^{- 1} \rangle^{- 1}_Q \langle W^{- 1 / 2} g \rangle_Q \|_{\mathbbm{C}^d}}\\ \nonumber&&\quad \lesssim [W]^{1 / 2}_{C_2} \| f \|_{L_{\mathbbm{C}^d}^2} \| g \|_{L_{\mathbbm{C}^d}^2} . \label{C2ineq} \end{eqnarray} In order to prove this inequality, let us define \begin{eqnarray*} {\mu} (\mathcal{K}) = \sum_{Q \in \mathcal{K}} \alpha_Q \end{eqnarray*} for any collection $\mathcal{K}$ of dyadic cubes. Let $F$ be any non--negative function defined on the dyadic cubes. Then let denote $\{ F (Q) > \lambda \}$ the collection of cubes so that $F (Q) > \lambda$. It follows that \begin{eqnarray*} \int^{\infty}_0 \left( \sum_{Q \in \{ F (Q) > \lambda \}} \alpha_Q \right) \mathrm{d} \lambda = \int^{\infty}_0 {\mu} (\{ F (Q) > \lambda \}) \mathrm{d} \lambda = \sum_{Q \in \mathcal{D} (Q_0)} F (Q) \alpha_Q, \end{eqnarray*} which is the classical fact on Choquet integrals. Let us pose \begin{eqnarray*} F (Q) = \| \langle W \rangle^{- 1}_Q \langle W^{1 / 2} f \rangle_Q \|_{\mathbbm{C}^d} \| \langle W^{- 1} \rangle^{- 1}_Q \langle W^{- 1 / 2} g \rangle_Q \|_{\mathbbm{C}^d} \end{eqnarray*} and let for any $\lambda > 0$ the set $\mathcal{J}_{\lambda}$ denote the collection of maximal dyadic intervals for which $F (Q) > \lambda$. So the integrand above becomes \begin{eqnarray*} \sum_{Q \in \{ F (Q) > \lambda \}} \alpha_Q \leqslant \sum_{Q \in \mathcal{J}_{\lambda}} \sum_{Q' \in \mathcal{D} (Q)} \alpha_Q \leqslant \sum_{Q \in \mathcal{J}_{\lambda}} | Q | . \end{eqnarray*} Now let \begin{eqnarray*} \Phi (x) = M_W f (x) M_{W^{- 1}} g (x) . \end{eqnarray*} Observe that with \begin{eqnarray*} A^W_Q f (x) := W^{1 / 2} (x) \langle W \rangle^{- 1}_Q \langle W^{1 / 2} f \rangle_Q \end{eqnarray*} and \begin{eqnarray*} A^{W^{- 1}}_Q g (x) = W^{- 1 / 2} (x) \langle W^{- 1} \rangle^{- 1}_Q \langle W^{- 1 / 2} g \rangle_Q \end{eqnarray*} by Cauchy Schwarz for all $x$ and all $Q$ there holds \begin{eqnarray*}\lefteqn{ F (Q) }\\ & = & \| W^{- 1 / 2} (x) W^{1 / 2} (x) \langle W \rangle^{- 1}_Q \langle W^{1 / 2} f \rangle_Q \|_{\mathbbm{C}^d} \\ &&\quad \| W^{1 / 2} (x) W^{- 1 / 2} (x) \langle W^{- 1} \rangle^{- 1}_Q \langle W^{- 1 / 2} g \rangle_Q \|_{\mathbbm{C}^d}\\ & \leqslant & \| W^{- 1 / 2} (x) \|_{\tmop{op}} \| W^{1 / 2} (x) \|_{\tmop{op}} \| A^W_Q f (x) \|_{\mathbbm{C}^d} \| A^{W^{- 1}}_Q g (x) \|_{\mathbbm{C}^d} . \end{eqnarray*} Observe that \begin{eqnarray*} \sup_y \| W^{- 1 / 2} (y) \|_{\tmop{op}} \| W^{1 / 2} (y) \|_{\tmop{op}} = [W]^{1 / 2}_{C_2} \end{eqnarray*} and for $x \in Q$ we get \begin{eqnarray*} F (Q) \lesssim [W]^{1 / 2}_{C_2} \Phi (x) . \end{eqnarray*} So if $x \in Q$ with $F (Q) > \lambda$ then also $[W]^{1 / 2}_{C_2} \Phi (x) > \lambda$. So \begin{eqnarray*} \sum_{Q \in \mathcal{J}_{\lambda}} | Q | \leqslant | \{ x \in Q_0 : [W]^{1 / 2}_{C_2} \Phi (x) > \lambda \} | . \end{eqnarray*} Integrating with respect to $\mathrm{d} \lambda$ gives \begin{eqnarray*}\lefteqn{ \sum_{Q \in \mathcal{D} (Q_0)} \alpha_Q F (Q)}\\ & \leqslant & \int^{\infty}_0 | \{ x \in Q_0 : [W]^{1 / 2}_{C_2} \Phi (x) > \lambda \} | \mathrm{d} \lambda \\ &=& [W]^{1 / 2}_{C_2} \int_{Q_0} M_W f (x) M_{W^{- 1}} g (x) \mathrm{d} x. \end{eqnarray*} Using the above estimate of the maximal function, Theorem \ref{MAX} and an application of Cauchy Schwarz finishes the proof of estimate (\ref{C2ineq}). \end{proof} Now, we prove Theorem \ref{BETfailure}, the failure of any improvement of Theorem \ref{siBET}. \begin{proof}[Theorem \ref{BETfailure}] Let us take the case $d = 2$ and let $a$ and $b$ be orthogonal unit vectors. Let $W = a a^{\ast} + \varepsilon^2 b b^{\ast}$ and thus $W^{- 1} = a a^{\ast} + \varepsilon^{- 2} b b^{\ast}$. Letting $f = W^{1 / 2} b\tmmathbf{1}_{Q_0}$ and $g = W^{- 1 / 2} a\tmmathbf{1}_{Q_0}$ we get \begin{eqnarray*} \| f \|_{L_{\mathbbm{C}^d}^2} = \langle W b, b \rangle_{\mathbbm{C}^d}^{1 / 2} = \varepsilon \; \tmop{ and } \; \| g \|_{L_{\mathbbm{C}^d}^2} = \langle W^{- 1} a, a \rangle_{\mathbbm{C}^d}^{1 / 2} = 1. \end{eqnarray*} This gives us the order $\varepsilon$ for the right hand sides of inequalities (\ref{miBETfailure}) and (\ref{mnBETfailure}). Further, we choose the Carleson sequence $A_{Q_0} =\tmmathbf{1}$ and $A_Q=\tmmathbf{0}$ for all other cubes. The Carleson intensity in inequality (\ref{conditionBETfailure}) is 1. We get for the sum in inequality (\ref{mnBETfailure}) only one term: \begin{eqnarray} \| A^{1 / 2}_{Q_0} \langle W \rangle^{- 1}_{Q_0} \langle W^{1 / 2} f \rangle_{Q_0} \|_{\mathbbm{C}^d} \| A^{1 / 2}_{Q_0} \langle W^{- 1} \rangle^{- 1}_{Q_0} \langle W^{- 1 / 2} g \rangle_{Q_0} \|_{\mathbbm{C}^d} = 1. \label{BETtest} \end{eqnarray} Letting $\varepsilon \to 0$, we have shown the failure of conclusion (\ref{mnBETfailure}). The conditioning number of the weight $W$ is $[W]_{C_2} = \varepsilon^2$. Again letting $\varepsilon \rightarrow 0$ shows the necessity of $[W]^{1 / 2}_{C_2}$ occurring in Theorem \ref{C2BET}. To see that the inequality (\ref{miBETfailure}) also fails, choose $A_{Q_0} = 2^{- 1} (a+b) (a+b)^{\ast} $ and $A_Q= \tmmathbf{0}$ for all other cubes. We see that \begin{eqnarray*} \langle A^{1 / 2}_{Q_0} \langle W \rangle^{- 1}_{Q_0} \langle W^{1 / 2} f \rangle_{Q_0}, A^{1 / 2}_{Q_0} \langle W^{- 1} \rangle^{- 1}_{Q_0} \langle W^{- 1 / 2} g \rangle_{Q_0} \rangle_{\mathbbm{C}^d} \gtrsim 1, \end{eqnarray*} showing the failure of conclusion (\ref{miBETfailure}). It is easy to replace the matrix Carleson sequence by the scalar sequence $\alpha_{Q_0} = 1$ and $\alpha_Q=0$ for all other cubes as the expression (\ref{conditionBETfailure}) does not change. \end{proof} As we see, the bilinear Carleson Lemma fails violently - using constant weights with a high discrepancy in their eigenvalues is sufficient. It is natural that this quantity determines the growth of the estimate and not features measured by the matrix $A_2$ characteristic. Notice that $[W]_{A_2} = 1$ for our example. \ \section{Redundant Carleson condition} In this section we prove Theorem \ref{RED}. We sketch the proof of Theorem \ref{sRED}. Consider the matrix valued Bellman function of matrix variables $U, V$ and scalar variable $m$ \begin{eqnarray*} B (U, V, m) = U - (m + 1)^{- 1} V^{- 1} . \end{eqnarray*} This function has domain $\tmmathbf{1} \leqslant V^{1 / 2} U V^{1 / 2}$ and $0 \leqslant m \leqslant 1.$ There holds the size estimate \begin{eqnarray*} 0 \leqslant B (U, V, m) \leqslant U. \end{eqnarray*} Indeed, $0 \leqslant U - V^{- 1} \leqslant U - (m + 1)^{- 1} V^{- 1} \leqslant U.$ The function $B$ is also concave: Dropping the linear dependence on $U$, its Hessian acting on the matrix difference $\Delta V$ and scalar $\Delta m$ is a positive multiple of \begin{eqnarray*}\lefteqn{ - 2 V^{- 1} \Delta V V^{- 1} \Delta V V^{- 1} }\\ &&\quad - 2 V^{- 1} \Delta V V^{- 1} (m + 1)^{- 1} \Delta m - 2 (m + 1)^{- 2} V^{- 1} (\Delta m)^2 . \end{eqnarray*} Observe that \begin{eqnarray*} V^{- 1} \Delta V V^{- 1} \Delta V V^{- 1} \geqslant 0, \quad V^{- 1} (\Delta m)^2 \geqslant 0. \end{eqnarray*} If we add positive multiples of these non--negative terms to the Hessian, we can write it as a negative perfect square and therefore concavity follows. We also have \begin{eqnarray*} \partial B / \partial m = (m + 1)^{- 2} V^{- 1} \geqslant 4^{- 1} V^{- 1} . \end{eqnarray*} One can check that the variables $$U_K = \langle W \rangle_K, \; V_K = \langle W^{- 1} \rangle_K, \; m_K = | K |^{- 1} \sum_{Q \in \mathcal{D} (K)} \alpha_Q$$ lie in the domain of $B$. We see that $m_K - | K |^{- 1} \alpha_K = 2^{- 1} (m_{K_-} + m_{K_+})$. The usual Bellman dynamics argument gives the estimate on the operator sum: \begin{eqnarray*}\lefteqn{ | K | \langle W \rangle_K = | K | U_K \geqslant | K | B (U_K, V_K, m_K)}\\ && \quad = | K | (B (U_K, V_K, m_K) \\ && \quad \quad - B (U_K, V_K, m_K - | K |^{- 1} \alpha_K) + B (U_K, V_K, m_K - | K |^{- 1} \alpha_K)\\ && \quad \geqslant 4^{- 1} V_K^{- 1} \alpha_K + | K | B (U_K, V_K, m_K - | K |^{- 1} \alpha_K)\\ && \quad \geqslant 4^{- 1} V_K^{- 1} \alpha_K + | K_- | B (U_{K_-}, V_{K_-}, m_{K_-}) + | K_+ | B (U_{K_+}, V_{K_+}, m_{K_+}) \end{eqnarray*} Iterating this argument gives the desired estimate for scalar sequences $(\alpha_Q)$. We note that if the Carleson measure is not scalar, then the function to consider may be \begin{eqnarray*} U - V^{- 1 / 2} (M +\tmmathbf{1})^{- 1} V^{- 1 / 2} . \end{eqnarray*} The concavity of this function is unclear to us. Now we argue that we can conclude anyways. \begin{proof}[Theorem \ref{RED}] The required observation is the following: \begin{eqnarray*}\lefteqn{ \frac{1}{| L|} \sum_{Q \in \mathcal{D} (L)} B_Q \lesssim \tmmathbf{1}, \forall L\in \mathcal{D} (Q_0) }\\ & \Leftrightarrow & \frac{1}{| L |} \sum_{Q \in \mathcal{D} ( L)} \tmop{tr} (B_Q) \lesssim 1, \forall { L} \in \mathcal{D} (Q_0)\\ & \Leftrightarrow & \frac{1}{| { L} |} \sum_{Q \in \mathcal{D} ({ L})} \| B_Q \|_{\tmop{op}} \lesssim 1, \forall { L} \in \mathcal{D} (Q_0) . \end{eqnarray*} The implied constants may depend upon $d$. Applying Theorem \ref{sRED} to the Carleson sequence $b_Q = \| B_Q \|_{\tmop{op}}$ gives us \begin{eqnarray*} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \langle W^{- 1} \rangle^{- 1 / 2}_Q b_Q \langle W^{- 1} \rangle_Q^{- 1 / 2} \lesssim \langle W \rangle_K, \forall K \in \mathcal{D} (Q_0) \end{eqnarray*} and thus since $b_Q \tmmathbf{1} \geqslant B_Q$ we obtain \begin{eqnarray*} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \langle W^{- 1} \rangle^{- 1 / 2}_Q B_Q \langle W^{- 1} \rangle_Q^{- 1 / 2} \lesssim \langle W \rangle_K, \forall K \in \mathcal{D} (Q_0) \end{eqnarray*} from which it follows that \begin{eqnarray*}\lefteqn{ \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \langle W \rangle^{- 1 / 2}_K \langle W^{- 1} \rangle^{- 1 / 2}_Q B_Q \langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_K}\\ &&\quad \lesssim \tmmathbf{1}, \forall K \in \mathcal{D} (Q_0) . \end{eqnarray*} Also \begin{eqnarray*} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \langle W^{- 1} \rangle^{- 1 / 2}_Q b_Q \langle W^{- 1} \rangle_Q^{- 1 / 2} \lesssim \langle W \rangle_K, \forall K \in \mathcal{D} (Q_0) \end{eqnarray*} implies in particular \begin{eqnarray*} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \tmop{tr} (\langle W \rangle^{- 1 / 2}_K \langle W^{- 1} \rangle^{- 1 / 2}_Q b_Q \langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_K) \lesssim 1, \forall K \in \mathcal{D} (Q_0) \end{eqnarray*} and with \begin{eqnarray*}\lefteqn{ \tmop{tr} (\langle W \rangle^{- 1 / 2}_K \langle W^{- 1} \rangle^{- 1 / 2}_Q b_Q \langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_K)}\\ &&\quad = \tmop{tr} (^{} \langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_K b_Q \langle W \rangle^{- 1 / 2}_K \langle W^{- 1} \rangle^{- 1 / 2}_Q) \end{eqnarray*} thus \begin{eqnarray*} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \tmop{tr} (\langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_K b_Q \langle W \rangle^{- 1 / 2}_K \langle W^{- 1} \rangle^{- 1 / 2}_Q) \lesssim 1, \forall K \in \mathcal{D} (Q_0) \end{eqnarray*} and so \begin{eqnarray*} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_K b_Q \langle W \rangle^{- 1 / 2}_K \langle W^{- 1} \rangle^{- 1 / 2}_Q \lesssim \tmmathbf{1}, \forall K \in \mathcal{D} (Q_0) . \end{eqnarray*} So we also obtain with $b_Q \tmmathbf{1} \geqslant B_Q$ that \begin{eqnarray*} \frac{1}{| K |} \sum_{Q \in \mathcal{D} (K)} \langle W^{- 1} \rangle^{- 1 / 2}_Q \langle W \rangle^{- 1 / 2}_K B_Q \langle W \rangle^{- 1 / 2}_K \langle W^{- 1} \rangle^{- 1 / 2}_Q \lesssim \tmmathbf{1}, \forall K \in \mathcal{D} (Q_0) . \end{eqnarray*} \end{proof} \ \ \ \

proofpile-arXiv_069-9045

\section{Introduction} \label{sect:introduction} Gravitational interactions in dense stellar systems such as globular clusters can lead to the formation of a dense core with a number of tight black-hole (BH) binaries (e.g., \citealt{1987degc.book.....S,2000ApJ...528L..17P,2008gady.book.....B,2010MNRAS.402..371B,2013MNRAS.429.2997L,2014MNRAS.444...29L,2015PhRvL.115e1101R,2017MNRAS.464L..36A,2017MNRAS.469.4665P}). The evolution of these BH binaries is of great importance for the understanding of BH mergers in dense stellar systems, which have received much interest in the past years with the direct detection of gravitational waves (e.g., \citealt{2016PhRvL.116x1103A,2016PhRvL.116f1102A,2017PhRvL.118v1101A,2017ApJ...851L..35A,2017PhRvL.119n1101A,2017ApJ...848L..12A}). Traditionally, so-called strong encounters, which are associated with energy exchange between the binary and a passing object (a star, or another compact object) are believed to dominate the binary BH evolution. More distant encounters, with periapsis distance $Q\gtrsim 2a$, where $a$ is the binary semimajor axis, are not believed to play an important role, since energy changes are exponentially suppressed (e.g., \citealt{1975MNRAS.173..729H,2003CeMDA..87..411R}), and the eccentricity changes are typically small \citep{1995ApJ...445L.133R,1996MNRAS.282.1064H,2016ApJ...818...21L,2018MNRAS.476.4139H}. However, in a recent paper \citep{2019arXiv190607189S}, we showed that these more distant encounters, i.e., weak encounters of the `secular' kind, are in fact important for some properties of merging BH binaries, such as the ratio of the number of mergers occurring inside the cluster to the number of mergers of BH binaries that are ejected from the cluster. Therefore, it is of importance to understand in more detail the effects of secular encounters on binaries in dense stellar clusters. A seminal work addressing this topic is \citet{1996MNRAS.282.1064H}, who derived expressions for the scalar eccentricity change to the quadrupole and octupole expansion orders in the ratio $r/R$, where $r$ is the binary separation, and $R$ is the perturber separation. In the secular limit, the binary's mean motion is much faster than the perturber's motion, such that it is justified to average the equations of motion (or, more fundamentally, the Hamiltonian) over the binary, resulting in the so-called `single-averaging' (SA) equations of motion. \citet{1996MNRAS.282.1064H} subsequently computed the scalar eccentricity change $\Delta e$ by integrating the equations of motion over the perturber, assuming either a parabolic or hyperbolic orbit. In their approach, \citet{1996MNRAS.282.1064H} assumed that the binary's eccentricity and angular-momentum vectors are constant during the perturber's passage. This corresponds to a `first-order' (FO) approach in the parameter $\epsilon_\mathrm{SA}$, which measures the strength of the perturbation (see equation~\ref{eq:epssa} below). However, as we showed in \citeauthor{2019arXiv190409624H} (\citeyear{2019arXiv190409624H}; hereafter Paper I) and in \citet{2019arXiv190607189S}, even if the binary is considered to be `hard' in the orbital energy sense, i.e., its orbital energy is much larger than the typical kinetic energy of stars in the cluster \citep{1975MNRAS.173..729H,1993ApJ...403..256H,1996ApJ...467..359H}, it can be `soft' in the angular-momentum sense if its eccentricity is very high. In the latter case, the binary can still be susceptible to perturbations near its apoapsis (note that the binary orbital speed at apoapsis is proportional to $[(1-e)/(1+e)]^{1/2}$). For such binaries that are hard in the orbital energy sense but soft in the angular-momentum sense, the FO expressions can break down, i.e., the eccentricity change according to the FO theory can be such that the new eccentricity is $e' = e+\Delta e>1$. In reality, $\Delta e$ is such that $e'<1$ in the secular limit, and the behaviour can be well described by modified expressions that are second order (SO) in $\epsilon_\mathrm{SA}$, and which were derived in Paper I. This was achieved by developing a Fourier expansion of the equations of motion in order to obtain a description of the instantaneous response of the eccentricity and angular-momentum vectors of the binary to the perturber, and substituting the resulting expressions back into the equations of motion. This approach was used in an earlier work in the context of hierarchical triples (i.e., with a bound third object) by \citet{2016MNRAS.458.3060L}. In hierarchical triples, the instantaneous response of the binary to the perturber's phase can lead to long-term modulations of the secular evolution, including orbital flips (e.g., \citealt{1936MNRAS..97...56B,2004AJ....128.2518C,2005MNRAS.358.1361I,2012arXiv1211.4584K,2012ApJ...757...27A,2013PhRvL.111f1106S,2014MNRAS.439.1079A,2014MNRAS.438..573B,2018MNRAS.481.4602L,2018MNRAS.481.4907G}). The expressions in Paper I~ were valid to SO in $\epsilon_\mathrm{SA}$, and, for simplicity, we assumed that the octupole expansion-order terms were zero, which is the case for binaries with equal component masses ($m_1=m_2$), and/or when the perturber is very distant. Here, we extend this previous work, and derive expressions to third order (TO) in $\epsilon_\mathrm{SA}$, and including all associated octupole-order terms. The TO terms give corrections to the lower-order terms, which can be important if $\epsilon_\mathrm{SA}$ is large (see \S~\ref{sect:TO:test_ex} below for examples). Due to the excessive number of terms in the analytic expressions for hyperbolic perturbers (see Table~\ref{table:N_terms}), we here restrict to parabolic encounter orbits, in which case the number of terms up to including TO in $\epsilon_\mathrm{SA}$ is more manageable. Our results are incorporated into a freely-available \textsc{Python} script (see the link in \S~\ref{sect:TO:results}). In addition, in this paper we study in more detail the effects of post-Newtonian terms combined with secular encounters (\S~\ref{sect:PN}). Also, in \S~\ref{sect:steady_state}, we study analytically the steady-state binary eccentricity distribution of cumulative secular encounters according to our analytic expressions. This provides insight into the importance of weak encounters for the eccentricity evolution of binaries. Also, we show that the FO terms give a significantly different steady-state distribution compared to when both the FO and SO terms are included. \section{Third-order perturbation theory including octupole expansion order terms} \label{sect:TO} \subsection{Overview} \label{sect:TO:overview} As in Paper I, we consider a binary (component masses $m_1$ and $m_2$, with $m=m_1+m_2$) with semimajor axis $a$ and eccentricity $e$, perturbed by a third passing body with mass $M$ and periapsis distance to the binary center of mass $Q>a$. We describe the binary's orientation using its eccentricity $\ve{e}$ and dimensionless angular-momentum vector $\ve{\j}$, where $\j=\sqrt{1-e^2}$, or, alternatively, using orbital elements (see Paper I~for the definitions). \begin{table} \begin{tabular}{lcc} \toprule Terms of order & \multicolumn{2}{c}{Number of terms in $\Delta e$} \\ & $E=1$ & $E>1$ \\ \midrule $\epsilon_\mathrm{SA}$ (all) & 16 & 60 \\ $\quad \epsilon_\mathrm{SA}$ & 2 & 8 \\ $\quad \epsilon_\mathrm{SA} \epsilon_\mathrm{oct}$ & 14 & 52 \\ \midrule $\epsilon_\mathrm{SA}^2$ (all) & 193 & 55,895\\ $\quad \epsilon_\mathrm{SA}^2$ & 17 & 1,871 \\ $\quad \epsilon_\mathrm{SA}^2 \epsilon_\mathrm{oct}$ & 60 & 16,035 \\ $\quad \epsilon_\mathrm{SA}^2 \epsilon_\mathrm{oct}^2$ & 116 & 37,989 \\ \midrule $\epsilon_\mathrm{SA}^3$ (all) & 1,146 & 2,931,541 \\ $\quad \epsilon_\mathrm{SA}^3$ & 54 & 38,366 \\ $\quad \epsilon_\mathrm{SA}^3 \epsilon_\mathrm{oct}$ & 175 & 289,496 \\ $\quad \epsilon_\mathrm{SA}^3 \epsilon_\mathrm{oct}^2$ & 311 & 856,072 \\ $\quad \epsilon_\mathrm{SA}^3 \epsilon_\mathrm{oct}^3$ & 606 & 1,747,607 \\ \bottomrule \end{tabular} \caption{Number of terms appearing in the analytic expressions for $\Delta e$ formulated using the orbital vectors $\ve{e}$ and $\ve{\j}$, obtained by counting the number of terms after applying the \textsc{Expand} function in \textsc{Mathematica}. For the FO, SO, and TO terms in $\epsilon_\mathrm{SA}$, we give the total number of terms (indicated with `all') including all octupole-order terms. We also separately state the number of terms associated with different orders of the octupole parameter $\epsilon_\mathrm{oct}$. The second and third columns correspond to parabolic ($E=1$) and hyperbolic ($E>1$) perturber orbits, respectively. } \label{table:N_terms} \end{table} The perturber can be on a parabolic orbit (eccentricity $E=1$), or a hyperbolic orbit ($E>1$). Although we derived all results for both parabolic and hyperbolic orbits, we do not present results for hyperbolic orbits since the number of terms in this case to high orders of $\epsilon_\mathrm{SA}$ and/or including octupole-order expansion terms is excessively large. This is illustrated in Table~\ref{table:N_terms}, in which the number of terms appearing in our expressions for the scalar eccentricity change $\Delta e$ is given for the first, second, and third perturbation orders $\epsilon_\mathrm{SA}$, and for different orders of the expansion octupole parameter $\epsilon_\mathrm{oct}$ (see below for the definitions of $\epsilon_\mathrm{SA}$ and $\epsilon_\mathrm{oct}$). The number of terms in the case of hyperbolic orbits is close to 3 million for the $\epsilon_\mathrm{SA}^3$ term (including all octupole-order terms). In the case of parabolic orbits, the number of terms at order $\epsilon_\mathrm{SA}^3$ is three orders of magnitudes lower, and still manageable (at least, using computer algebra software), with 1,146 terms. We consider secular encounters, i.e., encounters for which the perturber angular speed at all times is much lower than the binary mean motion (typically, this implies that the binary is `hard' in the energy sense). The strength of the perturber is characterized by the quantity $\epsilon_\mathrm{SA}$, which is defined as (cf. Paper I) \begin{align} \label{eq:epssa} \epsilon_\mathrm{SA} \equiv \left [ \frac{M^2}{m(m+M)} \left ( \frac{a}{Q} \right )^3 \left(1+E \right )^{-3} \right ]^{1/2}. \end{align} For equal-mass systems ($m_1=m_2=M$) and parabolic orbits, this reduces to $\epsilon_\mathrm{SA} = (a/Q)^{3/2}/(4\sqrt{3})$. As in Paper I, we expand the Hamiltonian in the ratio of the binary separation, $r$, to the perturber's separation $R$ to the binary's center of mass, and average over the binary's orbital phase. Up to and including third order in $(r/R)$ (i.e., octupole order), the resulting single-averaged (SA) equations of motion read \begin{subequations} \label{eq:EOM_SA_gen} \begin{align} \nonumber \frac{\mathrm{d}\ve{e}}{\mathrm{d} \theta} &= \epsilon_\mathrm{SA} (1+E \cos \theta) \Biggl \{ -3 \left(\ve{\j}\times \ve{e}\right) - \frac{3}{2} \left (\ve{\j} \cdot \unit{R} \right) \left(\ve{e} \times \unit{R} \right )+ \frac{15}{2} \left( \ve{e}\cdot \unit{R} \right ) \left ( \ve{\j} \times \unit{R} \right )\\ & + \epsilon_\mathrm{oct} (1+E \cos \theta) \frac{15}{16} { \Biggl [ 16 \left(\ve{e} \cdot \unit{R} \right ) \left(\ve{\j}\times \ve{e} \right ) - \left(1-8e^2 \right) \left(\ve{\j} \times \unit{R} \right ) + 10 \left ( \ve{e} \cdot \unit{R} \right ) \left ( \ve{\j} \cdot \unit{R} \right ) \left ( \ve{e} \times \unit{R} \right ) + 5 \left(\ve{\j}\cdot \unit{R} \right )^2 \left ( \ve{\j}\times \unit{R} \right ) - 35 \left ( \ve{e} \cdot \unit{R} \right )^2 \left ( \ve{\j} \times \unit{R} \right ) \Biggl ] \Biggl \} }; \\ \nonumber \frac{\mathrm{d}\ve{\j}}{\mathrm{d} \theta} &= \epsilon_\mathrm{SA} (1+E \cos \theta) \Biggl \{ -\frac{3}{2} \left ( \ve{\j} \cdot \unit{R} \right ) \left( \ve{\j} \times \unit{R} \right ) + \frac{15}{2} \left ( \ve{e} \cdot \unit{R} \right ) \left ( \ve{e} \times \unit{R} \right ) \\ & + \epsilon_\mathrm{oct} (1+E \cos \theta) \frac{15}{16} \Biggl [ - \left(1-8e^2 \right) \left(\ve{e} \times \unit{R} \right ) + 10 \left ( \ve{e} \cdot \unit{R} \right ) \left ( \ve{\j} \cdot \unit{R} \right ) \left ( \ve{\j} \times \unit{R} \right ) + 5 \left(\ve{\j}\cdot \unit{R} \right )^2 \left ( \ve{e}\times \unit{R} \right ) - 35 \left ( \ve{e} \cdot \unit{R} \right )^2 \left ( \ve{e} \times \unit{R} \right ) \Biggl ] \Biggl \}. \end{align} \end{subequations} Here, $\theta$ is the true anomaly of the perturber's orbit; generally, $-L<\theta<L$, where \begin{align} \label{eq:theta0_def} L\equiv \arccos\left ( -\frac{1}{E} \right ). \end{align} Evidently, $L=\pi$ for parabolic orbits. The octupole-order terms in equation~(\ref{eq:EOM_SA_gen}) are associated with the `octupole parameter' $\epsilon_\mathrm{oct}$ (analogous to the octupole parameter in hierarchical triples, e.g., \citealt{2011ApJ...742...94L,2011PhRvL.107r1101K,2013ApJ...779..166T,2014ApJ...791...86L}, or more generally in hierarchical systems, \citealt{2016MNRAS.459.2827H}), which is defined according to \begin{align} \label{eq:epsoct} \epsilon_\mathrm{oct} \equiv \frac{|m_1-m_2|}{m_1+m_2} \frac{a}{Q} \frac{1}{1+E}. \end{align} Note that the quadrupole-order terms are $\propto \epsilon_\mathrm{SA} (1+E \cos \theta)$, whereas the octupole-order terms are $\propto \epsilon_\mathrm{SA} \epsilon_\mathrm{oct} (1+E \cos \theta)^2$. In Paper I, we focused on equal-mass binaries ($m_1=m_2$), such that $\epsilon_\mathrm{oct}=0$, independent of $a/Q$. Here, we relax this assumption, and derive the expressions including the octupole-order terms. \subsection{Analytic results} \label{sect:TO:results} We follow the same procedure as in Paper I~(which was based on \citealt{2016MNRAS.458.3060L}) to derive the changes of the eccentricity and angular-momentum vectors. In summary, we develop the Fourier expansions of the equations of motion, equations~(\ref{eq:EOM_SA_gen}), in terms of $\theta$. Assuming that $\ve{e}$ and $\ve{\j}$ can be written as Fourier expansions of $\theta$, we subsequently derive relations between the Fourier coefficients of $\ve{e}$ and $\ve{\j}$, and the known Fourier coefficients of the equations of motion. We then integrate the equations of motion taking into account the Fourier expansions of $\ve{e}$ and $\ve{\j}$. Physically, this amounts to taking into account the instantaneous response of the binary to the perturber when calculating the net changes to the binary's orbital vectors. For more details on the procedure, we refer to Paper I. We denote the results for the changes of the eccentricity and angular momentum vectors in the following compact form, \begin{subequations} \label{eq:Delta_e_Delta_j} \begin{align} \Delta \ve{e} &= \epsilon_\mathrm{SA} \left [ \ve{f}_{\scriptscriptstyle e}^{(0)} + \epsilon_\mathrm{oct} \ve{f}_{\scriptscriptstyle e}^{(1)} \right ] + \epsilon_\mathrm{SA}^2 \left [ \ve{g}_{\scriptscriptstyle e}^{(0)} + \epsilon_\mathrm{oct} \ve{g}_{\scriptscriptstyle e}^{(1)} + \epsilon_\mathrm{oct}^2 \ve{g}_{\scriptscriptstyle e}^{(2)} \right ] + \epsilon_\mathrm{SA}^3 \left [ \ve{h}_{\scriptscriptstyle e}^{(0)} + \epsilon_\mathrm{oct} \ve{h}_{\scriptscriptstyle e}^{(1)} + \epsilon_\mathrm{oct}^2 \ve{h}_{\scriptscriptstyle e}^{(2)} + \epsilon_\mathrm{oct}^3 \ve{h}_{\scriptscriptstyle e}^{(3)} \right ]; \\ \Delta \ve{\j} &= \epsilon_\mathrm{SA} \left [ \ve{f}_{\scriptscriptstyle \j}^{(0)} + \epsilon_\mathrm{oct} \ve{f}_{\scriptscriptstyle \j}^{(1)} \right ] + \epsilon_\mathrm{SA}^2 \left [ \ve{g}_{\scriptscriptstyle \j}^{(0)} + \epsilon_\mathrm{oct} \ve{g}_{\scriptscriptstyle \j}^{(1)} + \epsilon_\mathrm{oct}^2 \ve{g}_{\scriptscriptstyle \j}^{(2)} \right ] + \epsilon_\mathrm{SA}^3 \left [ \ve{h}_{\scriptscriptstyle \j}^{(0)} + \epsilon_\mathrm{oct} \ve{h}_{\scriptscriptstyle \j}^{(1)} + \epsilon_\mathrm{oct}^2 \ve{h}_{\scriptscriptstyle \j}^{(2)} + \epsilon_\mathrm{oct}^3 \ve{h}_{\scriptscriptstyle \j}^{(3)} \right ]. \end{align} \end{subequations} Here, $\ve{f}^{(i)}$, $\ve{g}^{(i)}$, and $\ve{h}^{(i)}$ (with subscripts indicating association with $\Delta \ve{e}$ or $\Delta \ve{\j}$) are vector functions of the initial binary eccentricity and angular-momentum vectors (or, equivalently, the initial binary orbital elements); $i$ in $\ve{f}^{(i)}$ denotes the order of $\epsilon_\mathrm{oct}$ with which $\ve{f}^{(i)}$ is associated (similarly for $\ve{g}^{(i)}$ and $\ve{h}^{(i)}$). For small eccentricity changes, the scalar eccentricity change is given by $\Delta e = \unit{e} \cdot \Delta \ve{e}$. Below, we give the explicit expressions for the relevant terms $\unit{e} \cdot \ve{f}_{\scriptscriptstyle e}^{(0)}$, etc., up to and including TO in $\epsilon_\mathrm{SA}$, and valid for parabolic orbits (expressed in terms of the orbital vectors). We include the complete octupole expressions for terms FO and SO in $\epsilon_\mathrm{SA}$, whereas we do not explicitly show the octupole-order terms associated with $\epsilon_\mathrm{SA}^3$, since the latter expressions are excessively long. The complete expressions for the vector functions appearing in equations~(\ref{eq:Delta_e_Delta_j}) are given explicitly in Appendix~\ref{app:fgh} for the case of parabolic orbits, where we exclude the octupole-order terms at TO in $\epsilon_\mathrm{SA}$. The complete TO expressions in $\epsilon_\mathrm{SA}$, i.e., including all octupole-order terms, are implemented in a freely-available \textsc{Python} script\footnote{\href{https://github.com/hamers/flybys}{https://github.com/hamers/flybys}}. \begin{align} \nonumber \Delta e &= \frac{15}{2e} \pi e_z \epsilon_\mathrm{SA} (e_y \j_x-e_x \j_y) - \frac{15}{32e} \pi \epsilon_\mathrm{SA} \epsilon_\mathrm{oct} \Biggl [e_x^2 (3 e_y \j_z-73 e_z \j_y)+10 e_x \j_x (7 e_y e_z+\j_y \j_z)+e_y \j_z \left(3 e_y^2-32 e_z^2-15 \j_x^2-5 \j_y^2+4\right) \\ \nonumber &\quad +e_z \j_y\left(-3 e_y^2+32 e_z^2+5 \j_x^2+5 \j_y^2-4\right)\Biggl ] \\ \nonumber &+ \frac{3}{8e} \pi \epsilon_\mathrm{SA}^2 \Biggl [ e_z^2 \left(-75 \pi e_x^2+18 \pi \j_x^2-25 \j_x \j_y+18 \pi \j_y^2\right)+3 \pi e_x^2 \left(6 \j_y^2-\j_z^2\right)+e_y \left(e_x \left(-36 \pi \j_x \j_y-25 \j_y^2+25 \j_z^2\right)+2 e_z \j_z (25 \j_x+6 \pi \j_y)\right) \\ \nonumber &\quad +2 e_x e_z \j_z (6 \pi \j_x-25 \j_y)+e_y^2 \left(-3 \pi \left(25 e_z^2+\j_z^2\right)+18 \pi \j_x^2+25 \j_x \j_y\right)\Biggl] \\ \nonumber &-\frac{15}{512e} \pi \epsilon_\mathrm{SA}^2 \epsilon_\mathrm{oct} \Biggl[ e_y \biggl \{ 7 \j_x^2 \left(121 e_x^2+682 e_z^2-\j_y^2-91 \j_z^2-20\right)-6 \j_x \left(662 \pi e_x^2 \j_y-1281 e_x e_z \j_z+2 \pi \j_y \left(-16 e_z^2-15 \j_y^2+10 \j_z^2+12\right)\right) \\ \nonumber &\quad +e_x^2 \left(3549 \j_z^2-3066 \j_y^2\right)+2712 \pi e_x e_z \j_y \j_z+42 e_z^2 \left(73 \j_y^2-80 \j_z^2\right)+49 \j_x^4+180 \pi \j_x^3 \j_y+21 \left(20-43 \j_y^2\right) \j_z^2\biggl \}+3 e_x^2 e_z \j_z (1100 \pi \j_x-3759 \j_y) \\ \nonumber &\quad +e_y^2 \left(e_x \left(-12 \pi \left(730 e_z^2+\j_y^2+15 \j_z^2\right)+1992 \pi \j_x^2+3773 \j_x \j_y\right)+21 e_z \j_z (28 \pi \j_x-243 \j_y)\right)+e_x \biggl \{ 12 \pi \j_x^2 \left(276 e_z^2+5 \left(\j_z^2-3 \j_y^2\right)\right)\\ \nonumber &\quad +7 \j_x \j_y \left(-676 e_z^2+\j_y^2-62 \j_z^2+20\right)+12 \pi \left(320 e_z^4+4 e_z^2 \left(65 \j_y^2+8 \j_z^2-10\right)-15 \j_y^4+3 \j_y^2 \left(5 \j_z^2+4\right)-4 \j_z^2\right)-49 \j_x^3 \j_y\biggl \} \\ \nonumber &\quad +e_y^3 \left(-707 \j_x^2+12 \pi \j_x \j_y+2037 \j_z^2\right)+3 e_z \j_z \left(7 \j_y \left(160 e_z^2+51 \j_x^2-20\right)-4 \pi \j_x \left(96 e_z^2+35 \j_x^2-12\right)-140 \pi \j_x \j_y^2+301 \j_y^3\right) \\ \nonumber &\quad + e_x^3 \left(-\left(60 \pi \left(146 e_z^2-33 \j_y^2+3 \j_z^2\right)+847 \j_x \j_y\right)\right) \Biggl] \\ \nonumber &-\frac{225}{8192e} \pi \epsilon_\mathrm{SA}^2 \epsilon_\mathrm{oct}^2 \Biggl [e_x^4 \left(4 \pi \left(9 e_y^2+5329 e_z^2+54 \left(\j_z^2-19 \j_y^2\right)\right)+7343 \j_x \j_y\right)+e_x^3 \biggl \{ 10 e_z \j_z (2191 \j_y-1216 \pi \j_x)-7 e_y \left(1049 \j_x^2-1280 \pi \j_x \j_y \right. \\ \nonumber &\quad \left. -2045 \j_y^2+612 \j_z^2\right)\biggl \}+e_x^2 \biggl \{ -8 \pi \j_x^2 \left(602 e_y^2+1342 e_z^2-85 \j_y^2-105 \j_z^2\right)-\j_x \left(2 \j_y \left(6048 e_y^2-4207 e_z^2-1901 \j_z^2+336\right)+14350 e_y e_z \j_z \right. \\ \nonumber &\quad \left. +2237 \j_y^3\right)+8 \pi \left(9 e_y^4+e_y^2 \left(2573 e_z^2+64 \j_y^2+18 \j_z^2+12\right)-608 e_y e_z \j_y \j_z-2336 e_z^4-2 e_z^2 \left(871 \j_y^2+48 \j_z^2-146\right)+135 \j_y^4-65 \j_y^2 \j_z^2-108 \j_y^2 \right. \\ \nonumber &\quad \left. +12 \j_z^2\right)-281 \j_x^3 \j_y\biggl \}+e_x \biggl \{ e_y^3 \left(-\left(2219 \j_x^2+192 \pi \j_x \j_y-7679 \j_y^2+2772 \j_z^2\right)\right)+2 e_y^2 e_z \j_z (9359 \j_y-2656 \pi \j_x)+e_y \left(\j_x^2 \left(-11690 e_z^2+4032 \j_y^2 \right. \right. \\ \nonumber &\quad \left. \left. +722 \j_z^2+672\right)-32 \pi \j_x \j_y \left(-164 e_z^2+40 \j_y^2+15 \j_z^2-22\right)-2 \j_y^2 \left(2219 e_z^2-3839 \j_z^2+504\right)+2352 \left(8 e_z^2-1\right) \j_z^2+281 \j_x^4-480 \pi \j_x^3 \j_y \right. \\ \nonumber &\quad \left. +55 \j_y^4\right)+2 e_z \j_z \left(\j_y \left(-30016 e_z^2-8817 \j_x^2+3752\right)+16 \pi \j_x \left(304 e_z^2+105 \j_x^2-38\right)+2800 \pi \j_x \j_y^2-7029 \j_y^3\right)\biggl \}+36 \pi e_y^6-e_y^4 \biggl \{ 4 \pi \left(183 e_z^2 \right. \\ \nonumber &\quad \left. -10 \j_y^2+18 \j_z^2-24\right)+240 \pi \j_x^2+7679 \j_x \j_y\biggl \}+2 e_y^3 e_z \j_z (992 \pi \j_y-4067 \j_x)+e_y^2 \biggl \{ -40 \pi \j_x^2 \left(2 e_z^2-30 \j_y^2-39 \j_z^2+8\right)-\j_x \j_y \left(3374 e_z^2+55 \j_y^2 \right. \\ \nonumber &\quad \left. +3802 \j_z^2-1008\right)+4 \pi \left(832 e_z^4+4 e_z^2 \left(123 \j_y^2+48 \j_z^2-58\right)+125 \j_y^4-10 \j_y^2 \left(7 \j_z^2+12\right)-24 \j_z^2+16\right)+300 \pi \j_x^4-1795 \j_x^3 \j_y\biggl \} \\ \nonumber &\quad +2 e_y e_z \j_z \biggl \{\j_x \left(19264 e_z^2+3945 \j_y^2-2408\right)+16 \pi \j_y \left(272 e_z^2+55 \j_y^2-34\right)+5517 \j_x^3-240 \pi \j_x^2 \j_y\biggl \}+4 e_z^2 \biggl \{ 10 \pi \j_x^2 \left(64 e_z^2-17 \j_y^2-8\right)\\ \nonumber &\quad +3 \j_x \j_y \left(224 e_z^2+191 \j_y^2-28\right)+\pi \left(1024 e_z^4-128 e_z^2 \left(11 \j_y^2+2\right)-245 \j_y^4+176 \j_y^2+16\right)+75 \pi \j_x^4+519 \j_x^3 \j_y\biggl \}\Biggl ] \\ \nonumber &+\frac{3}{512e} \pi \epsilon_\mathrm{SA}^3 \Biggl [-1200 \pi e_x^4 \j_z+15 e_x^3 \left(466 e_y \j_z+80 \pi e_z \j_x+\left(369+384 \pi ^2\right) e_z \j_y\right)-3 e_x^2 \biggl \{ 1000 \pi e_y^2 \j_z+15 e_y e_z \left(\left(128 \pi ^2-159\right) \j_x+160 \pi \j_y\right) \\ \nonumber &\quad +2 \j_z \left(20 \pi \left(160 e_z^2+\j_y^2\right)+120 \pi \j_x^2+\left(581+192 \pi ^2\right) \j_x \j_y\right)\biggl \}+e_x \biggl \{ 9390 e_y^3 \j_z+15 e_y^2 e_z \left(40 \pi \j_x+\left(811+384 \pi ^2\right) \j_y\right)+4 e_y \j_z \left(840 e_z^2 \right. \\ \nonumber &\quad \left. +8 \left(36 \pi ^2-7\right) \j_x^2+180 \pi \j_x \j_y-3 \left(217+96 \pi ^2\right) \j_y^2\right)+e_z \left(-120 e_z^2 \left(80 \pi \j_x+\left(133-48 \pi ^2\right) \j_y\right)+720 \pi \j_x^3-313 \j_x^2 \j_y+120 \pi \j_x \left(9 \j_y^2+32 \j_z^2\right)\right.\\ \nonumber &\quad \left. -927 \j_y^3-616 \j_y \j_z^2+384 \pi ^2 \j_y \j_z^2\right)\biggl \}-1800 \pi e_y^4 \j_z-15 e_y^3 e_z \left(\left(384 \pi ^2-387\right) \j_x+520 \pi \j_y\right)-6 e_y^2 \j_z \biggl \{ 20 \pi \left(80 e_z^2-7 \j_y^2\right)+80 \pi \j_x^2+\left(371 \right. \\ \nonumber &\quad \left. -192 \pi ^2\right) \j_x \j_y\biggl \}+e_y e_z \biggl \{ -360 \left(21+16 \pi ^2\right) e_z^2 \j_x+87 \j_x^3+720 \pi \j_x^2 \j_y+285 \j_x \j_y^2-8 \left(7+48 \pi ^2\right) \j_x \j_z^2+120 \pi \j_y \left(9 \j_y^2+16 \j_z^2\right)\biggl \}\\ \nonumber &\quad +96 e_z^2 \j_x \j_z (20 \pi \j_x+49 \j_y)\Biggl ] \\ & + \epsilon_\mathrm{SA}^3 \mathcal{O}(\epsilon_\mathrm{oct}) + \mathcal{O} \left (\epsilon_\mathrm{SA}^4 \right ). \end{align} \subsection{Tests and examples} \label{sect:TO:test_ex} \begin{figure} \center \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig1a} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig1b} \caption { An example of the scalar eccentricity change as a function of $Q/a$ according to numerically solving the SA equations of motion, equations~(\ref{eq:EOM_SA_gen}), and using analytic expressions: first order (FO; dashed lines), second order (SO; solid lines), and third order (TO; dotted lines) in $\epsilon_\mathrm{SA}$ (refer to the legend). Blue (red) lines and symbols correspond to positive (negative) $\Delta e$. The FO, SO, and TO lines include octupole-order terms up to and including FO, SO, and TO, respectively, although in this case $m_1=m_2$ such that $\epsilon_\mathrm{oct}=0$. The right-hand panel is a zoomed-in version of the left-hand panel. The vertical dotted green line shows the value of $Q/a$ for which the `adiabatic ratio' of the angular speed of the perturber at periapsis to the binary mean motion (see equation 1 of Paper I) is 0.5. To the left of this line, we expect the secular approximation to break down. } \label{fig:test_TO} \end{figure} \begin{figure} \center \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig2a} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig2b} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig2c} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig2d} \caption { Similar to Fig.~\ref{fig:test_TO}, here showing two examples with nonzero octupole-order terms -- the binary component masses are unequal; we set $m_1=M=1$ with $m_2=2/3$ and $m_2=1/10$ in the left and right-hand panels, respectively. The bottom panels are zoomed-in versions of the top panels. The (non-horizontal) black dotted lines show results from the analytic SO terms {\it without} the octupole-order terms derived in this paper. } \label{fig:test} \end{figure} In Fig.~\ref{fig:test_TO}, we show an example of the scalar eccentricity change $\Delta e$ as a function of $Q/a$ according to numerically solving the SA equations of motion, equations~(\ref{eq:EOM_SA_gen}), and using the analytic expressions: FO, SO, and TO in $\epsilon_\mathrm{SA}$. The right-hand panel is a zoomed-in version of the left-hand panel. Here, we choose the same initial conditions as in the bottom-right-hand panel of Fig. 5 in Paper I. In the latter panel in Paper I, some deviation was found between the SA and SO results. Here, we also include the TO terms, and it can be seen that the addition of the TO terms solves the (minor) discrepancy. Next, we focus on the octupole-order terms. In Fig.~\ref{fig:test}, we adopt similar initial conditions as in Section 5.1 of Paper I, i.e., with $m_1\neq m_2$, such that $\epsilon_\mathrm{oct} \neq 0$. In Paper I, the octupole-order corrections were only included to FO in $\epsilon_\mathrm{SA}$ (and, therefore, to FO in $\epsilon_\mathrm{oct}$). Consequently, there was significant deviation of the analytic prediction from the 3-body and SA integrations. Here, we include all octupole-order terms to the corresponding orders of $\epsilon_\mathrm{SA}$, i.e., in Fig.~\ref{fig:test}, the FO, SO, and TO lines include octupole-order terms up to and including FO, SO, and TO in $\epsilon_\mathrm{oct}$, respectively. In the left (right)-hand panels, we set $m_1/m_2=1.5$ ($m_1/m_2=10$). The (non-horizontal) black dotted lines in Fig.~\ref{fig:test} show results from the analytic SO terms {\it without} the octupole-order terms derived in this paper, and illustrate that the octupole-order contribution can be important. As shown in Fig.~\ref{fig:test}, the additional octupole-order terms derived here give good agreement with the SA integrations. In particular, the value of $Q/a$ for which the sign change of $\Delta e$ occurs is accurately predicted. The bottom panels are zoomed-in versions of the top panels. In the bottom panels, it becomes clear that the TO terms give slightly better agreement with the SA integrations. The additional number of terms introduced by the TO terms is significant, however, as shown in Table~\ref{table:N_terms}. \subsubsection{Further exploration of parameter space} \label{sect:TO:test_ex:ex} \begin{figure} \center \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig3a} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig3b} \caption { Scalar eccentricity change as a function of $i$ (left-hand panel), and of $\omega$ (right-hand panel). Blue (red) colors correspond to positive (negative) $\Delta e$. The dependence on $\Omega$ is very weak, and is not shown. The fixed parameters assumed in each panel are indicated in the top of each panel. } \label{fig:ex1} \end{figure} \begin{figure} \center \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig4a} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig4b} \caption{ Similar to Fig.~\ref{fig:ex1}, here assuming a low initial eccentricity of $e=0.01$.} \label{fig:ex2} \end{figure} \begin{figure} \center \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig5a} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig5b} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig5c} \caption{ Similar to Fig.~\ref{fig:ex1}, here assuming different fixed values of $\omega$ and $\Omega$. } \label{fig:ex3} \end{figure} Here, we explore in more detail the dependence of $\Delta e$ on the assumed parameters. In Fig.~\ref{fig:ex1}, we plot $\Delta e$ as a function of the inclination $i$ (left-hand panel) and the argument of periapsis $\omega$ (right-hand panel), for fixed other parameters (the dependence on the longitude of the ascending node $\Omega$ for the fixed other parameters is very weak, and not shown). For the chosen fixed parameters, $\Delta e$ is positive for $i$ around $90^\circ$, whereas $\Delta e<0$ if $i$ is close to 0 or $180^\circ$. This sign change is not captured by the FO expression, which predicts positive $\Delta e$ throughout (specifically, $\Delta e \propto \sin^2 i \geq 0$). Considered as a function of $\omega$, $\Delta e$ changes sign multiple times. The FO expression predicts a simple dependence, $\Delta e \propto \sin 2 \omega$, which approximately captures the dependence on $\omega$. However, the numerical SA integrations show a more elaborate behaviour which is well captured by the SO and TO expressions. In Fig.~\ref{fig:ex2}, the initial eccentricity is assumed to be $e=0.01$, in contrast to $e=0.999$ in Fig.~\ref{fig:ex1}. The inclination dependence (left-hand panel) is now asymmetric, and this behaviour is captured by the FO, SO, and TO expressions. The detailed features as a function of $\omega$ in Fig.~\ref{fig:ex1} are not apparent in the lower-eccentricity example in Fig.~\ref{fig:ex2}. In Fig.~\ref{fig:ex3}, we set $e=0.99$, and choose different values for the fixed $\omega$ and $\Omega$. The scalar eccentricity change is positive for most inclinations except for small inclinations, which is captured by all analytic methods. Considered as a function of $\omega$, $\Delta e$ shows cyclical behaviour, which is not entirely captured by the FO expression, but accurately with the higher-order expressions. For these parameters, the dependence of $\Delta e$ on $\Omega$ is no longer very weak, and is shown in the third panel of Fig.~\ref{fig:ex3}. The dependence is sinusoidal, and is well captured by all analytic expressions. \section{Post-Newtonian terms} \label{sect:PN} Here, we consider in more detail compared to Paper I~the effects of post-Newtonian (PN) terms. We restrict to the lowest-order non-dissipative PN terms, i.e., those associated with $c^2$, where $c$ is the speed of light. As is well known, averaged over the mean motion of the binary, the 1PN terms give rise to precession of $\ve{e}$ around $\ve{\j}$, described by \citep{1972gcpa.book.....W} \begin{align} \label{eq:1PN_t} \frac{\mathrm{d} \ve{e}}{\mathrm{d} t} = 3 n_ {\mathrm{bin}} \frac{ r_{\mathrm{g}}}{a} \frac{1}{1-e^2} e \, \unit{q}, \end{align} where $ r_{\mathrm{g}} \equiv Gm/c^2$ is the binary's gravitational radius, $n_ {\mathrm{bin}}\equiv \sqrt{Gm/a^3}$ is the binary's mean motion, and $\unit{q} \equiv \unit{\j} \times \unit{e}$ is a unit vector perpendicular to both $\unit{e}$ and $\unit{\j}$. Formulated in terms of the perturber's true anomaly $\theta$ (and, again, averaged over the binary's mean motion), 1PN precession give rise to an additional term to the equations of motion, equations~(\ref{eq:EOM_SA_gen}), which is given by \begin{align} \label{eq:1PN} \left. \frac{\mathrm{d} \ve{e}}{\mathrm{d} \theta}\right |_{\mathrm{1PN}} = \epsilon_\mathrm{1PN} \frac{e \, \unit{q}}{1-e^2} \frac{1}{(1+E \cos \theta)^2}, \end{align} where \begin{align} \epsilon_\mathrm{1PN} \equiv 3 \frac{ r_{\mathrm{g}}}{a} \left [ \frac{m}{m+M} \left ( \frac{Q}{a} \right )^3 (1+E)^3 \right ]^{1/2}. \end{align} From equation~(\ref{eq:1PN}), it is immediately clear that the 1PN term $\mathrm{d} \ve{e}/\mathrm{d}\theta $ diverges when integrating it from $\theta=-L$ to $\theta=L$. Physically, this means that the amount of 1PN-induced precession is infinite over a time span corresponding to $t\rightarrow -\infty$ to $t\rightarrow \infty$. Unfortunately, this implies that the procedure used in Paper I~to compute Fourier coefficients based on the equations of motion directly cannot straightforwardly be extended to include the 1PN terms directly in the equations of motion. Another important implication is that the infinite precession of the binary corresponding to $-L<\theta<L$ implies that the orientation of the binary when the perturber passes it is ill-defined. Consequently, the 1PN terms give rise to an uncertainty in the problem -- the initial argument of periapsis of the binary can no longer be considered to be a true initial parameter. Therefore, we will consider the {\it statistical} properties of the eccentricity changes over an ensemble of the initial arguments of periapsis, $\omega_{\mathrm{i}}$, where we take the initial distribution of $\omega_{\mathrm{i}}$ to be uniform between 0 and $2\pi$. The Newtonian predictions for the eccentricity change in this case can be straightforwardly computed from the analytic expressions derived in Paper I~and in \S~\ref{sect:TO}. Specifically, by expressing the scalar eccentricity change in terms of orbital elements and averaging over $\omega$, we find that the mean and root-mean-squared (rms) scalar eccentricity change to TO in $\epsilon_\mathrm{SA}$ are given by (setting $\epsilon_\mathrm{oct}=0$ and $E=1$ for simplicity) \begin{subequations} \label{eq:stat_N} \begin{align} \nonumber \langle \Delta e \rangle_\omega &\equiv \int_{0}^{2\pi} \mathrm{d} \omega \, \Delta e = \frac{9}{512} \pi ^2 \epsilon_\mathrm{SA}^2 e \left[4 \left(81 e^2-56\right) \cos (2 i)+\left(39 e^2+36\right) \cos (4 i)-299 e^2+124\right] \\ \nonumber &\quad -\frac{9}{4096} \pi \epsilon_\mathrm{SA}^3 e \sqrt{1-e^2} \cos (i) \Biggl [176 e^2 \sin (2 i-4 \Omega )-44 e^2 \sin (4 i-4 \Omega )+854 e^2 \sin (2 i-2 \Omega )-1001 e^2 \sin (4 i-2 \Omega ) \\ \nonumber &\qquad -854 e^2 \sin (2 (i+\Omega ))+44 e^2 \sin (4 (i+\Omega ))+1001 e^2 \sin (4 i+2 \Omega )-176 e^2 \sin (2 i+4 \Omega )-880 \pi e^2 \cos (2 i-2 \Omega ) \\ \nonumber &\qquad +660 \pi e^2 \cos (4 i-2 \Omega )-880 \pi e^2 \cos (2 (i+\Omega ))+660 \pi e^2 \cos (4 i+2 \Omega )+40 \pi \left(49 e^2-4\right) \cos (2 i)-120 \pi \left(11 e^2+4\right) \cos (4 i) \\ \nonumber &\qquad -294 e^2 \sin (2 \Omega )+264 e^2 \sin (4 \Omega )+440 \pi e^2 \cos (2 \Omega )+3360 \pi e^2+64 \sin (2 i-4 \Omega )-16 \sin (4 i-4 \Omega )+616 \sin (2 i-2 \Omega ) \\ \nonumber &\qquad -364 \sin (4 i-2 \Omega )-616 \sin (2 (i+\Omega ))+16 \sin (4 (i+\Omega ))+364 \sin (4 i+2 \Omega )-64 \sin (2 i+4 \Omega )-320 \pi \cos (2 i-2 \Omega ) \\ &\qquad +240 \pi \cos (4 i-2 \Omega )-320 \pi \cos (2 (i+\Omega )) + 240 \pi \cos (4 i+2 \Omega )+504 \sin (2 \Omega )+96 \sin (4 \Omega )+160 \pi \cos (2 \Omega )+640 \pi \Biggl ]; \\ \nonumber \left \langle (\Delta e)^2 \right \rangle^{1/2}_\omega &\equiv \left [ \int_{0}^{2\pi} \mathrm{d} \omega \, (\Delta e)^2 \right ]^{1/2} = \frac{15}{4 \sqrt{2}} \pi \epsilon_\mathrm{SA} e \sqrt{1-e^2} \sin ^2(i) \\ \nonumber &\quad + \frac{75}{128 \sqrt{2}} \pi \epsilon_\mathrm{SA}^2 e \left(1-e^2\right) \cos (i) (3 \cos (2 i-2 \Omega )+3 \cos (2 (i+\Omega ))-6 \cos (2 i)+2 \cos (2 \Omega )+6) \\ \nonumber &\quad -\frac{3}{1310720 \sqrt{2}}\pi \epsilon_\mathrm{SA}^3 \frac{e}{\sqrt{1-e^2}} \csc ^2(i) \Biggl [ -348456 \pi ^2 \cos (6 i) e^4-51100 \cos (6 i) e^4-69867 \pi ^2 \cos (8 i) e^4-63800 \cos (2 i-4 \Omega ) e^4 \\ \nonumber &\qquad +232560 \cos (4 i-4 \Omega ) e^4-25160 \cos (6 i-4 \Omega ) e^4-607460 \cos (2 i-2 \Omega ) e^4+19880 \cos (4 i-2 \Omega ) e^4-232540 \cos (6 i-2 \Omega ) e^4 \\ \nonumber &\qquad +1640240 \cos (2 \Omega ) e^4+352800 \cos (4 \Omega ) e^4-607460 \cos (2 (i+\Omega )) e^4+232560 \cos (4 (i+\Omega )) e^4+19880 \cos (4 i+2 \Omega ) e^4\\ \nonumber &\qquad -232540 \cos (6 i+2 \Omega ) e^4-63800 \cos (2 i+4 \Omega ) e^4-25160 \cos (6 i+4 \Omega ) e^4-82800 \pi \sin (2 i-2 \Omega ) e^4+309600 \pi \sin (4 i-2 \Omega ) e^4\\ \nonumber &\qquad -13200 \pi \sin (6 i-2 \Omega ) e^4+427200 \pi \sin (2 \Omega ) e^4+82800 \pi \sin (2 (i+\Omega )) e^4-309600 \pi \sin (4 i+2 \Omega ) e^4+13200 \pi \sin (6 i+2 \Omega ) e^4\\ \nonumber &\qquad -8084769 \pi ^2 e^4+241400 e^4-228288 \pi ^2 \cos (6 i) e^2+7700 \cos (6 i) e^2-92016 \pi ^2 \cos (8 i) e^2+53800 \cos (2 i-4 \Omega ) e^2 \\ \nonumber &\qquad -375120 \cos (4 i-4 \Omega ) e^2+18520 \cos (6 i-4 \Omega ) e^2+503020 \cos (2 i-2 \Omega ) e^2+73640 \cos (4 i-2 \Omega ) e^2+202580 \cos (6 i-2 \Omega ) e^2\\ \nonumber &\qquad-1558480 \cos (2 \Omega ) e^2-674400 \cos (4 \Omega ) e^2+503020 \cos (2 (i+\Omega )) e^2-375120 \cos (4 (i+\Omega )) e^2+73640 \cos (4 i+2 \Omega ) e^2\\ \nonumber &\qquad +202580 \cos (6 i+2 \Omega ) e^2+53800 \cos (2 i+4 \Omega ) e^2+18520 \cos (6 i+4 \Omega ) e^2+27600 \pi \sin (2 i-2 \Omega ) e^2-247200 \pi \sin (4 i-2 \Omega ) e^2\\ \nonumber &\qquad+68400 \pi \sin (6 i-2 \Omega ) e^2-302400 \pi \sin (2 \Omega ) e^2-27600 \pi \sin (2 (i+\Omega )) e^2+247200 \pi \sin (4 i+2 \Omega ) e^2-68400 \pi \sin (6 i+2 \Omega ) e^2\\ \nonumber &\qquad +7626288 \pi ^2 e^2+210200 e^2+12 \left(\left(-69475+871678 \pi ^2\right) e^4+\left(48825-948656 \pi ^2\right) e^2+185728 \pi ^2+20650\right) \cos (2 i)\\ \nonumber &\qquad -100 \left\{\left(-34+21045 \pi ^2\right) e^4-2 \left(2381+21864 \pi ^2\right) e^2+13008 \pi ^2+4796\right\} \cos (4 i)+351744 \pi ^2 \cos (6 i)+43400 \cos (6 i)\\ \nonumber &\qquad -34992 \pi ^2 \cos (8 i)+10000 \cos (2 i-4 \Omega )+142560 \cos (4 i-4 \Omega )+6640 \cos (6 i-4 \Omega )+104440 \cos (2 i-2 \Omega )\\ \nonumber &\qquad -93520 \cos (4 i-2 \Omega )+29960 \cos (6 i-2 \Omega )-81760 \cos (2 \Omega )+321600 \cos (4 \Omega )+104440 \cos (2 (i+\Omega ))+142560 \cos (4 (i+\Omega ))\\ \nonumber &\qquad -93520 \cos (4 i+2 \Omega )+29960 \cos (6 i+2 \Omega )+10000 \cos (2 i+4 \Omega )+6640 \cos (6 i+4 \Omega )+55200 \pi \sin (2 i-2 \Omega )\\ \nonumber &\qquad -62400 \pi \sin (4 i-2 \Omega )-55200 \pi \sin (6 i-2 \Omega )-124800 \pi \sin (2 \Omega )-55200 \pi \sin (2 (i+\Omega ))+62400 \pi \sin (4 i+2 \Omega ) \\ &\qquad +55200 \pi \sin (6 i+2 \Omega )-1392144 \pi ^2-451600 \Biggl ]. \end{align} \end{subequations} Note that, when only the FO terms are included, $\langle \Delta e \rangle_{\omega} = 0$. \begin{figure} \center \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig6a} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig6b} \caption{ Mean and rms scalar eccentricity changes as a function of $Q/a$ (keeping $a$ fixed while varying $Q$) for a system with $m_1=m_2=M=40\,\mathrm{M}_\odot$, $a=1\,\,\textsc{au}$, $e=0.999$, $i=90^\circ$, and $\Omega=-\pi/4$. Shown are points obtained by numerically integrating the SA equations of motion, equations~(\ref{eq:EOM_SA_gen}) (open circles: mean; stars: rms), and the lines correspond to the Newtonian prediction, equations~(\ref{eq:stat_N}) (solid: TO mean; dotted: TO rms). Blue (red) lines correspond to positive (negative) values. The 1PN terms were excluded (included) in the left (right)-hand panels. The vertical yellow dashed lines show equation~(\ref{eq:Qdiva_1PN}), and the vertical black dotted lines show equation~(\ref{eq:Qdiva_1PNb}).} \label{fig:PN} \end{figure} In Fig.~\ref{fig:PN}, we plot the mean and rms scalar eccentricity changes as a function of $Q/a$ (keeping $a$ fixed while varying $Q$) for a system with $m_1=m_2=M=40\,\mathrm{M}_\odot$, $a=1\,\,\textsc{au}$, $e=0.999$, $i=90^\circ$, and $\Omega=-\pi/4$. Shown are points each based on 20 integrations with different $\omega_{\mathrm{i}}$, obtained by numerically integrating the SA equations of motion, equations~(\ref{eq:EOM_SA_gen}), and the lines correspond to the Newtonian prediction, equations~(\ref{eq:stat_N}). The 1PN terms were excluded (included) in the left (right)-hand panels. In the numerical integrations, a finite range of $\theta$ should be specified; we integrate from $\theta=-0.95L$ to $\theta=0.95L$. We remark that the numerical results are converged with respect to the integration range. In the Newtonian case, the numerically-obtained SA results agree well with the analytic results. Some deviation for the rms values in particular can be seen at very small $Q/a$, for which higher-order terms in $\epsilon_\mathrm{SA}$ become important. When the 1PN terms are included, the scalar eccentricity changes are consistent with the Newtonian results at small $Q/a$, in which case the rate of Newtonian perturbation is relatively strong compared to the rate of 1PN precession, i.e., 1PN terms are not important. As $Q/a$ increases, the rms eccentricity change in particular shows a drop with respect to the Newtonian case. Also, the mean starts to fluctuate, and change sign. This can be attributed to quenching of the Newtonian torques due to rapid 1PN precession. The value of $Q/a$ for which 1PN terms start to become important can be estimated by equating the time-scale associated with the Newtonian terms with the 1PN precession time-scale. The former can be estimated by \begin{align} t_{\mathrm{TB}} \sim n_ {\mathrm{bin}} \frac{M}{m} \left ( \frac{a}{Q} \right )^3. \end{align} Equating the latter to the 1PN time-scale implied by equation~(\ref{eq:1PN_t}), we find for the critical $Q/a$ for 1PN terms to become important \begin{align} \label{eq:Qdiva_1PN} \left (\frac{Q}{a} \right )_{\mathrm{crit}} \sim \left [ \frac{1}{3} \left (1-e^2 \right ) \frac{a}{ r_{\mathrm{g}}}\frac{M}{m} \right ]^{1/3} \simeq 7.5, \end{align} where the numerical value assumes the parameters in Fig.~\ref{fig:PN}, and which is shown in the latter figure with the vertical yellow dashed lines. It can be seen that the value of $Q/a$ for which 1PN terms start to decrease the typical eccentricity change can be reasonably estimated with this equation. Alternatively, the regime in which the 1PN terms dominate can be estimated by equating the time-scale of the passage of the perturber to the 1PN precession time-scale. The former can be estimated as $t_{\mathrm{fly}} \sim \sqrt{Q^3/[G(m+M)]}$. This gives \begin{align} \label{eq:Qdiva_1PNb} \left (\frac{Q}{a} \right )_{\mathrm{crit,2}} \sim \left [ \frac{1}{3} \left (1-e^2 \right ) \frac{a}{ r_{\mathrm{g}}}\frac{m+M}{m} \right ]^{2/3} \simeq 117, \end{align} where the numerical value again assumes the parameters in Fig.~\ref{fig:PN}, and which is shown with the vertical black dotted lines. Approximately, $(Q/a)_{\mathrm{crit,2}} \sim (Q/a)_{\mathrm{crit}}^2$. Equation~(\ref{eq:Qdiva_1PNb}) gives a less conservative estimate for the value of $Q/a$ for which the 1PN terms dominate; the rms $\Delta e$ is already significantly below the Newtonian value at $Q/a = (Q/a)_{\mathrm{crit,2}}$. \section{Steady-state distribution} \label{sect:steady_state} In Paper I~and in the previous sections, we considered in detail the effects of a single encounter on a binary. However, in dense stellar systems such as globular clusters, a typical binary will encounter a large number of perturbers. Although not all encounters will be of the secular type, it is still of theoretical interest to determine the steady-state distribution of encounters in the secular regime. More specifically, encounters are typically thought to give rise to equipartition, i.e., to a uniform distribution in phase space in terms of angular momentum. In other words, in this case, the distribution function $f(\mathcal{E},\mathcal{L})$ is independent of $\mathcal{L}$, where $\mathcal{E}$ is the orbital energy $\mathcal{E}$ and $\mathcal{L}$ is the angular momentum (note that $\mathcal{L} \propto \j$). The probability density function (PDF) in energy and angular space, $N(\mathcal{E},\mathcal{L})$, is related to the distribution function according to (e.g., \citealt{2013degn.book.....M}) \begin{align} N(\mathcal{E},\mathcal{L}) = 8 \pi^2 \mathcal{L} f(\mathcal{E},\mathcal{L}) P(\mathcal{E},\mathcal{L}), \end{align} where $P(\mathcal{E},\mathcal{L})$ is the orbital period. Therefore, if $f(\mathcal{E},\mathcal{L})$ is independent of $\mathcal{L}$ (and if $P$ is independent of $\mathcal{L}$, which is the case for a binary), then the number distribution in angular momentum is $N(\mathcal{L}) \propto \mathcal{L}$, which implies a (normalized) PDF in eccentricity \begin{align} \label{eq:thermal} N(e) = 2e. \end{align} The well-known distribution in equation~(\ref{eq:thermal}) is also referred to as a thermal distribution \citep{1919MNRAS..79..408J,ambartsumian1937,1975MNRAS.173..729H}. However, it is not a priori clear if a uniform distribution in phase space is reached, and whether strong encounters or more distant encounters contribute to the steady state. Therefore, it is relevant to consider the steady-state distribution arising from secular encounters, which we address here using our analytic expressions for the scalar eccentricity change. Since the calculations are already lengthy when the FO and SO terms are included for parabolic encounters, we will here focus on these simplified cases. We do not include octupole-order terms, nor PN terms, or allow for hyperbolic encounters. We carry out the calculations with only the FO terms included, and with the FO and SO terms both included, to illustrate the importance of adding the SO terms in the (purely Newtonian) steady-state calculations. To determine the angular-momentum steady-state distribution, we consider the Fokker-Planck equation in angular-momentum space, which is given by (e.g., \citealt[Section 5.1.1]{2013degn.book.....M}) \begin{align} \frac{\partial N(\mathcal{R},t)}{\partial t} = - \frac{\partial}{\partial \mathcal{R}} [ N(\mathcal{R},t) \left \langle \Delta \Rt \right \rangle ] + \frac{1}{2} \frac{\partial^2}{\partial \mathcal{R}^2} \left [ N(\mathcal{R},t) \left \langle (\Delta \Rt)^2 \right \rangle \right ]. \end{align} Here, $\mathcal{R} \equiv \j^2 = 1-e^2$ is the squared normalized angular momentum\footnote{$\mathcal{R}$ here is not to be confused with the ratio of the angular speed of the perturber to the binary's mean motion, as defined in equation~(1) of Paper I.}, and the brackets denote diffusion coefficients. The steady state is defined by $\partial N(\mathcal{R},t)/ \partial t = 0$, implying \begin{align} - N(\mathcal{R},t) \left \langle \Delta \Rt \right \rangle + \frac{1}{2} \frac{\partial}{\partial \mathcal{R}} \left [ N(\mathcal{R},t) \left \langle (\Delta \Rt)^2 \right \rangle \right ] = C, \end{align} where $C$ is a constant. Here, we consider `zero-flux' solutions, i.e., $C=0$. In this case, it is straightforward to show that the steady-state solution is (e.g., \citealt[Appendix D]{2014MNRAS.443..355H}) \begin{align} \label{eq:gen_steady_state} N(\mathcal{R}) \propto \exp \left [ \int \mathrm{d} \mathcal{R} \frac{2 \left \langle \Delta \Rt \right \rangle - \left \langle (\Delta \Rt)^2 \right \rangle'}{ \left \langle (\Delta \Rt)^2 \right \rangle} \right ], \end{align} where the prime denotes the derivative with respect to $\mathcal{R}$. We compute the diffusion coefficients in $\mathcal{R}$ from our analytic expressions for $\Delta e$. Generally, a diffusion coefficient $\langle x \rangle$ in some quantity $x$ measures the change of $x$ per unit of time. In our case, the unit of time corresponds to a single passage of a perturber; also, note that the time unit drops out in the integrand in equation~(\ref{eq:gen_steady_state}). From the definition $\mathcal{R} \equiv 1-e^2$, $\Delta \mathcal{R} = -2e \Delta e - (\Delta e)^2$, where we retain the second-order term in $\Delta e$. Subsequently, we compute the diffusion coefficients by averaging over all angles of the binary orbit relative to the perturber assuming isotropic orientations, i.e., \begin{subequations} \begin{align} \left \langle \Delta \Rt \right \rangle &= \frac{1}{2\pi} \frac{1}{2\pi} \frac{1}{2} \int_{0}^{2\pi} \mathrm{d} \omega \int_{0}^{2\pi} \mathrm{d} \Omega \int_{0}^{\pi} \mathrm{d} i \, \sin i \left [ -2e \Delta e - (\Delta e)^2 \right ]; \\ \left \langle (\Delta \Rt)^2 \right \rangle &= \frac{1}{2\pi} \frac{1}{2\pi} \frac{1}{2} \int_{0}^{2\pi} \mathrm{d} \omega \int_{0}^{2\pi} \mathrm{d} \Omega \int_{0}^{\pi} \mathrm{d} i \, \sin i \left [ -2e \Delta e - (\Delta e)^2 \right ]^2. \end{align} \end{subequations} In principle, when computing the diffusion coefficients, one should also average over all perturber properties, such as $M$, $E$, and impact parameters $b$. Here, we make the simplifying assumption of a single-mass perturber population, and parabolic orbits ($E=1$). Also, we assume for simplicity that the octupole-order terms are negligible ($\epsilon_\mathrm{oct}=0$). With regards to the impact parameter $b$, we take two approaches. \begin{itemize}[leftmargin=5mm] \item In the first approach (1), we consider $b$ to be fixed and therefore do not average over it, meaning that the results will be valid for a given impact parameter. \item In the second approach (2), we also integrate the diffusion coefficients over impact parameters assuming a distribution $N(b) \propto b$. Since we consider fixed $M$ and $E=1$ (such that $b=Q$, where $Q$ is the perturber's periapsis distance), this implies a (normalized) distribution of perturber strengths \begin{align} N(\epsilon_\mathrm{SA}) = - \frac{4}{3} \frac{\epsilon_\mathrm{SA}^{-7/3}}{\epsilon_\mathrm{SA,2}^{-4/3} - \epsilon_\mathrm{SA,1}^{-4/3}}, \end{align} where $\epsilon_\mathrm{SA,1}$ and $\epsilon_\mathrm{SA,2}$ are the values of $\epsilon_\mathrm{SA}$ corresponding to the strongest (weakest) perturber (i.e., smallest and largest $Q$, respectively). \end{itemize} Below, we state the results separately based on the expressions with the FO terms taken into account only, and also with both the FO and SO terms included. Due to their complexity, we do not include the TO terms here. Moreover, we find numerically that the addition of the TO terms does not significantly affect the steady-state distribution (see \S~\ref{sect:steady_state:MC} below). \begin{figure} \center \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig7a} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig7b} \caption{ Steady-state distributions for fixed $e$ as a function of $\epsilon_\mathrm{SA}$ (in the case of approach 1), or as a function of $\epsilon_\mathrm{SA,1}$ (in the case of approach 2). Left (right)-hand panels correspond to the distributions with the FO (FO+SO) terms. In each panel, black solid lines correspond to approach (1), whereas blue dashed lines correspond to approach (2). Different values for the eccentricity are assumed, indicated with different line widths ($e$ increases with increasing line width; refer to the legends). } \label{fig:steady_state_eps} \end{figure} \subsection{Steady-state distribution based on FO terms} \label{sect:steady_state:FO} In approach (1) (considering $\epsilon_\mathrm{SA}$ to be fixed) and including the FO terms in $\epsilon_\mathrm{SA}$ only, the diffusion coefficients are given by (assuming parabolic orbits, and $\epsilon_\mathrm{oct}=0$) \begin{subequations} \label{eq:DFC_FO} \begin{align} \left \langle \Delta \Rt \right \rangle &= -\frac{15}{4} \pi^2 \epsilon_\mathrm{SA}^2 \mathcal{R} (1-\mathcal{R}); \\ \left \langle (\Delta \Rt)^2 \right \rangle &= \frac{15}{112} \pi^2 \epsilon_\mathrm{SA}^2 (1-\mathcal{R})^2 \left(112 + 225\pi^2 \mathcal{R} \epsilon_\mathrm{SA}^2 \right). \end{align} \end{subequations} Applying equation~(\ref{eq:gen_steady_state}), the implied zero-flux steady-state distribution in terms of the eccentricity is given by \begin{align} \label{eq:steady_state_FO_1} N(e) \propto \frac{1}{1-e^2} e^{\frac{112}{225 \pi ^2 \epsilon_\mathrm{SA}^2+112}-3} \left(112+225 \pi ^2 \left(1-e^2\right) \epsilon_\mathrm{SA}^2\right)^{-\frac{56}{225 \pi ^2 \epsilon_\mathrm{SA}^2+112}-1} \end{align} The distribution in equation~(\ref{eq:steady_state_FO_1}) is nearly independent of $\epsilon_\mathrm{SA}$, unless $\epsilon_\mathrm{SA} \gtrsim 10^{-2}$. To illustrate, we show with black solid lines in the left-hand panel of Fig.~\ref{fig:steady_state_eps} the distributions (modulo a normalization factor) as a function of $\epsilon_\mathrm{SA}$. In the limit of $\epsilon_\mathrm{SA}\rightarrow0$, equation~(\ref{eq:steady_state_FO_1}) reduces to the simple expression \begin{align} \label{eq:steady_state_FO_1_zero} N(e) \propto \frac{1}{e^2(1-e^2)}. \end{align} In approach (2), when making the assumption $\epsilon_\mathrm{SA,2} \ll \epsilon_\mathrm{SA,1}$, the implied eccentricity distribution is \begin{align} \label{eq:steady_state_FO_2} N(e) \propto \frac{1}{1-e^2} e^{\frac{448}{225 \pi ^2 \epsilon_\mathrm{SA,1}^2+448}-3} \left(448+225 \pi ^2 \left(1-e^2\right) \epsilon_\mathrm{SA,1}^2\right)^{-\frac{224}{225 \pi ^2 \epsilon_\mathrm{SA,1}^2+448}-1}. \end{align} This distribution is again weakly dependent on $\epsilon_\mathrm{SA,1}$, which is illustrated with the blue dashed lines in the left-hand panel of Fig.~\ref{fig:steady_state_eps}. In the limit of $\epsilon_\mathrm{SA,1}\rightarrow0$, equation~(\ref{eq:steady_state_FO_2}) reduces to the same expression as in approach (1), i.e., equation~(\ref{eq:steady_state_FO_1_zero}). \subsection{Steady-state distribution based on FO and SO terms} \label{sect:steady_state:SO} When the SO terms are included, the diffusion coefficients in approach (1) are given by \begin{subequations} \label{eq:DFC_SO} \begin{align} & \left \langle \Delta \Rt \right \rangle = -\frac{3}{1120} \pi ^2 (1-\mathcal{R}) \epsilon_\mathrm{SA}^2 \left(3 \left(625+8158 \pi ^2\right) \mathcal{R}^2 \epsilon_\mathrm{SA}^2+\mathcal{R} \left(6776-24900 \pi ^2 \epsilon_\mathrm{SA}^2\right)+100 \left(75 \pi ^2 \epsilon_\mathrm{SA}^2-28\right)\right); \\ \nonumber & \left \langle (\Delta \Rt)^2 \right \rangle = \frac{3}{174254080} \pi ^2 (1-\mathcal{R})^2 \epsilon_\mathrm{SA}^2 \Biggl [ 918 \pi ^2 \left(3125000+113795625 \pi ^2+434516724 \pi ^4\right) \mathcal{R}^4 \epsilon_\mathrm{SA}^6 \\ \nonumber &\quad -3240 \pi ^2 \mathcal{R}^3 \left(251074980 \pi ^4 \epsilon_\mathrm{SA}^2+17 \pi ^2 \left(1771875 \epsilon_\mathrm{SA}^2-5501464\right)-9116250\right) \epsilon_\mathrm{SA}^4+3 \mathcal{R}^2 \left(225664380000 \pi ^6 \epsilon_\mathrm{SA}^4 \right. \\ \nonumber &\quad \left. +11475 \pi ^4 \left(828125 \epsilon_\mathrm{SA}^2-11198592\right) \epsilon_\mathrm{SA}^2-3536 \pi ^2 \left(1078125 \epsilon_\mathrm{SA}^2-4188008\right)+388960000\right) \epsilon_\mathrm{SA}^2 \\ &\quad -1600 \mathcal{R} \left(167214375 \pi ^6 \epsilon_\mathrm{SA}^6-108438750 \pi ^4 \epsilon_\mathrm{SA}^4+18436704 \pi ^2 \epsilon_\mathrm{SA}^2-544544\right)+120000 \pi ^2 \left(354375 \pi ^4 \epsilon_\mathrm{SA}^4-229500 \pi ^2 \epsilon_\mathrm{SA}^2+38896\right) \epsilon_\mathrm{SA}^2\Biggl ]. \end{align} \end{subequations} The general steady-state distribution implied by equations~(\ref{eq:DFC_SO}) is excessively complicated. Here, we simplify the expression by expanding $N(e)$ in $\epsilon_\mathrm{SA}$, assuming $\epsilon_\mathrm{SA}\ll 1$. To order $\epsilon_\mathrm{SA}^2$, the result is \begin{align} \label{eq:steady_state_SO_1} N(e) \propto e^{-\frac{4}{25} -\frac{3 \left(57500+1372461 \pi ^2\right) \epsilon_\mathrm{SA}^2}{70000}} \left(1-e^2\right)^{\frac{543 \pi ^2 \epsilon_\mathrm{SA}^2}{14}} \exp \left[-\frac{3 \epsilon_\mathrm{SA}^2 \left(182500+7730161 \pi ^2\right) \left(1-e^2\right)}{140000}\right]. \end{align} Similarly to the FO result, this distribution is insensitive to $\epsilon_\mathrm{SA}$, unless $\epsilon_\mathrm{SA} \gtrsim 10^{-2}$ (see the right-hand panel of Fig.~\ref{fig:steady_state_eps}). Taking the limit $\epsilon_\mathrm{SA}\rightarrow 0$, the (normalized) distribution simplifies to \begin{align} \label{eq:steady_state_SO_1_zero} N(e) = \frac{21}{25} e^{-4/25}. \end{align} In approach (2) and assuming $\epsilon_\mathrm{SA,2}\ll \epsilon_\mathrm{SA,1}$, the steady-state distribution is \begin{align} N(e) \propto e^{-\frac{4}{25} -\frac{3 \left(57500+1372461 \pi ^2\right) \epsilon_\mathrm{SA,1}^2}{280000}} \left(1-e^2\right)^{\frac{543 \pi ^2 \epsilon_\mathrm{SA,1}^2}{56}} \exp \left [- \frac{3 \epsilon_\mathrm{SA,1}^2\left(182500+7730161 \pi ^2\right) \left(1-e^2\right)}{560000} \right ] \end{align} Similarly to the FO case, this reduces to equation~(\ref{eq:steady_state_SO_1_zero}) in the limit $\epsilon_\mathrm{SA,1}\rightarrow0$. \subsection{Monte Carlo results} \label{sect:steady_state:MC} To verify the distributions derived above, we perform numerical Monte Carlo (MC) experiments in which perturbers are continuously sampled assuming an isotropic orientation relative to the binary, and the effect on the binary eccentricity is computed for each encounter using our analytic FO and SO expressions (we also implemented the TO terms, but their inclusion does not significantly affect the steady-state distribution). Similarly to the analytic expressions, we assume a single perturber mass and parabolic encounter orbits. The masses are set to $m_1=m_2=M$, such that $\epsilon_\mathrm{oct}=0$. The inner and outer boundaries in terms of $Q$ in the MC experiments are set to be $2\,a$ and $80\,a$, respectively; note that the resulting steady-state distribution is insensitive to these values. The results are shown in Fig.~\ref{fig:steady_state}. In the left (right)-hand panels, we show results from the MC experiments in which the FO terms (FO and SO terms) were included. We set the initial eccentricity distribution of the binary to be thermal, i.e., $N(e)=2e$, but we note that the final distributions are independent of the initial distributions (e.g., an initially uniform or delta function distribution). The FO and SO analytic predictions (assuming $\epsilon_\mathrm{SA}\rightarrow 0$ and $\epsilon_\mathrm{SA,1}\rightarrow 0$) are shown in each panel with red dashed and dotted lines, respectively. The analytic distributions are in agreement with the numerical results. In particular, with the FO and SO terms, the steady-state distribution is fairly flat, and well described by $N(e)\propto e^{-4/25}$ as derived above. Note that, in the case of the FO terms only, the analytic distribution, equation~(\ref{eq:steady_state_FO_1_zero}), diverges when integrating over eccentricities from 0 to 1; therefore, we normalized between two boundary eccentricities, $e_1$ and $e_2$, and the normalization depends on the precise choices of these values (in Fig.~\ref{fig:steady_state}, $e_1=0.01$, and $e_2=0.9999$). \begin{figure} \center \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig8a} \includegraphics[scale = 0.46, trim = 8mm 0mm 8mm 0mm]{figs/fig8b} \caption{ Steady-state eccentricity distributions from the MC experiments in which the FO terms (left-hand panel) and FO and SO terms (right-hand panel) were included. The initial eccentricity distribution of the binary (black dotted lines) is assumed to be thermal; note that the final distributions are independent of the initial distributions. The final eccentricity distributions are shown with the solid red lines. The FO and SO analytic predictions are shown in each panel with red dashed and dotted lines, respectively. } \label{fig:steady_state} \end{figure} \section{Discussion} \label{sect:discussion} \subsection{High-order terms} \label{sect:discussion:higher} Having presented here higher-order terms compared to Paper I, a natural question is whether even higher-order terms can be important. In principle, fourth-order terms in $\epsilon_\mathrm{SA}$ can be derived using the same procedures. However, as hinted at in Table~\ref{table:N_terms}, the associated number of terms is so large that analytic expressions, although explicit, can no longer be considered to be useful. Moreover, for the fourth-order terms to be important, $\epsilon_\mathrm{SA}$ must be close to unity, and the secular approximation in this case becomes questionable. Higher-order terms could also be derived in the expansion of $r/R$ (the next-order terms in the Hamiltonian are $\propto [r/R]^4$, and are known as the hexadecupole-order terms). This has been done for bound triples (e.g., \citealt{2010A&A...522A..60L,2015MNRAS.452.3610A,2016CeMDA.124...73C,2018MNRAS.481.4602L}), and higher-order multiplicity systems \citep{2015MNRAS.449.4221H,2016MNRAS.459.2827H}. However, in most situations, these higher-order terms do not significantly affect the dynamics. Moreover, similarly to the above, higher-order expansion orders are only important for relatively large $r/R$, and in this case the secular approximation may no longer be adequate. \subsection{Steady-state distribution} \label{sect:discussion:steady} In \S~\ref{sect:steady_state}, we determined the steady-state eccentricity distribution based on our analytic expressions, and verified that our analytic results based on the Fokker-Planck equation are consistent with numerical Monte Carlo experiments. We found that the distributions based on the FO and the FO+SO terms are different from a thermal distribution. The natural question to ask is why this is the case. As discussed in \S~\ref{sect:steady_state}, a thermal distribution results from equipartition in angular momentum. The latter argument does not make a distinction between strong and weak encounters, whereas in our analysis we considered weak (i.e., secular) encounters only. Moreover, in our treatment of the scalar eccentricity change, the perturber is not allowed to exchange angular momentum (nor energy) with the binary -- the binary is perturbed by the third body, but the third body is not affected by the binary. This `test particle' approach may be another reason for a steady state that is different from thermal. The above results indicate that the more distant encounters give rise to a different steady-state distribution compared to `all' encounters, which are also allowed to exchange angular momentum (and energy) with the binary. Of course, the latter situation applies to real clusters. Nevertheless, it is instructive to understand the individual contributions from `weak' and `strong' encounters, as the `weak' type of encounters are typically ignored in studies of binaries in clusters (e.g., \citealt{2018PhRvD..97j3014S,2018MNRAS.481.5445S,2019PhRvD..99f3006S}). \section{Conclusions} \label{sect:conclusions} In this paper, we extended previous work in which we analytically calculated the secular effects of a perturber moving on a hyperbolic or parabolic orbit on a binary system. In this approximation, the binary mean motion is much faster than the perturber's motion, and it is justified to average the equations of motion over the binary. The main results are given below. \medskip \noindent 1. Specifying to parabolic encounters, we derived expressions for the changes of the eccentricity and angular-momentum vectors of the binary to third order (TO) in the perturbation parameter $\epsilon_\mathrm{SA}$ (cf. equation~\ref{eq:epssa}), and including terms up to and including octupole order in the expansion of the ratio $r/R$ of the binary separation $r$ to the perturber separation $R$. The latter appear in particular for relatively close encounters, and unequal binary component masses ($m_1\neq m_2$). Our methodology is based on Fourier expansions of the equations of motion and the eccentricity and angular-momentum vectors. Starting from the second-order (SO) in $\epsilon_\mathrm{SA}$, this technique allows to incorporate effects associated with the changing eccentricity and angular-momentum vectors as the perturber passes the binary. The results are summarized in \S~\ref{sect:TO:results}, where a link is given to a publicly-available \textsc{Python} code which can be used to quickly evaluate the expressions. In \S~\ref{sect:TO:test_ex}, we demonstrated the correctness of the TO and octupole expressions, and illustrated the behaviour of secular encounters for parts of the parameter space. \medskip \noindent 2. We considered the effects of the lowest-order post-Newtonian (PN) terms, which (after averaging over the binary's orbital phase) give rise to precession of the binary's argument of periapsis. We were unable to find an analytic expression for the scalar eccentricity changes with the addition of the 1PN terms using the methodology of Fourier expansions, due to a divergence occurring in the integrated equations of motion associated with the 1PN terms. Physically, the latter corresponds to the infinite precession of the binary when considering an infinite time span, irrespective of the binary's properties (i.e., for any $a$, $e$, and $m_1$ and $m_2$). Nevertheless, we showed a numerical example of the quenching of the Newtonian perturbation on the binary when the 1PN are included, and the strength of the perturbation is weak (such that the 1PN terms dominate). We found that the transition between the two regimes (1PN terms not being important, and the 1PN terms quenching the perturbation) can be estimated using equations~(\ref{eq:Qdiva_1PN}) or (\ref{eq:Qdiva_1PNb}). \medskip \noindent 3. We determined the steady-state binary eccentricity distribution in response to the cumulative effect of secular encounters by computing the associated angular-momentum diffusion coefficients, and applying the Fokker-Planck equation. For simplicity, we restricted to the steady-state distribution according to both our first-order (FO) and second-order (SO) scalar eccentricity change expressions for parabolic encounters, and setting the octupole-order terms to zero ($m_1=m_2$). We found that the steady-state distributions in both the FO and SO cases are weakly dependent on the perturber impact parameter, unless $\epsilon_\mathrm{SA} \gtrsim 10^{-2}$. In the limit $\epsilon_\mathrm{SA} \rightarrow 0$, we found that the steady-state eccentricity distribution is $N(e) \propto [e^2(1-e^2)]^{-1}$, whereas for the SO expression, we found $N(e) \propto e^{-4/25}$. The FO expression is strongly biased to circular and highly eccentric orbits. The SO distribution, on the other hand, is much flatter, with some preference for circular orbits, and no high-eccentricity tail. We verified these analytic expressions numerically by comparing them to results from Monte Carlo experiments. \section*{Acknowledgements} We thank Scott Tremaine for stimulating discussions and comments on the manuscript, and the referee, Nathan Leigh, for a very helpful report. A.S.H. gratefully acknowledges support from the Institute for Advanced Study, and the Martin A. and Helen Chooljian Membership. J.S. acknowledges support from the Lyman Spitzer Fellowship. \bibliographystyle{mnras}

proofpile-arXiv_069-9122

\section{Introduction} The formation of dendritic structures during solidification processes is a well-known phenomenon. Thermal supercooling in the case of pure metals and constitutional supercooling in the case of alloys have traditionally been used to explain the instability that gives rise to such structures, and a detailed review of the theory of dendritic growth during solidification may be found in the reviews~\cite{Kurz1984,langer1980,langer1987,kessler1988,pelce2012,kurz1990, trivedi1994,trivedi1994solidification}. Hence, while the study of dendritic growth during solidification has been studied quite extensively leading to an almost complete understanding of the parameters that lead to the selection of a unique dendritic tip radius and velocity, the same is not true for the case of solid-state dendritic growth, wherein the number of experimental observations of such growth is itself limited. Yoo et al.~\cite{Henry1995} observed that during the precipitation of $\gamma'$-phase in Ni-base superalloys, after giving a specified heat treatment to the alloy, i.e., when it is cooled to the ageing temperatures very close to the solvus temperature, the coherent precipitates grow into dendritic structures, where the arms of the dendrite grow along the $\langle111\rangle$ directions. Doherty~\cite{Doherty1983} discussed various factors that appear to explain the shape instabilities in some solid-state reactions, such as the solid-state dendrites of $\gamma$-brass precipitates in the $\beta$-brass matrix in the Cu-Zn alloy. Husain et al.~\cite{Husain1999} reported that the precipitation of the $\gamma_2$-phase in Cu-Al $\beta$-phase alloys yield dendritic morphologies. It is also observed that the rapid bulk diffusion and fast interfacial reaction kinetics would promote the formation of such morphologies. They have predicted the occurrence of dendritic morphologies during solid-state precipitation in many binary systems such as Cu-Zn, Cu-Al, Ag-Al, Ag-Zn, Cu-Ga, Au-Zn, Ni-Zn, Cu-An, Ag-Cd, and Cu-Sn. The authors attribute crystallographic similarities between the parent phase and the precipitates to be the dominating factor that gives rise to the formation of the dendritic structures. In a later study, Yoo et al.~\cite{Yoo2005} have observed spherical precipitates changing into flower-like structures as the supersaturation in the matrix is increased. They have shown that the results are concurrent with the Mullins-Sekerka theory, where they comment that the morphological instabilities should be evident if the point effect of the diffusion is significant in the vicinity of the precipitate-matrix interface regardless of the nature of the matrix phase and the ageing stage. Khan et al.~\cite{AKhan2004} have also noticed the formation of solid-state dendritic structures during the elevated temperature treatment of maraging steel. There, the oxy-nitride phase acquires dendritic morphology in the martensitic matrix. At a fundamental level, the presence of a supersaturation in the matrix makes the interface unstable to diffusive instabilities of the Mullins-Sekerka type, and therefore the observation of such dendritic structures during solid-state reactions is not surprising. However, there is an important difference from the case of solidification, wherein the solid-solid interface in precipitation reactions is typically coherent in the initial stages of precipitation, and thereby the precipitate and the matrix are also elastically stressed. Since the elastic-effects scale with the size of the precipitate, the presence of coherency will therefore naturally influence the selection of length scales during dendritic evolution. Leo and Sekerka~\cite{Leo1989} have extended the linear stability analysis performed by Mullins and Sekerka~\cite{Mullins1963} by including the influence of elastic coherency stresses and have investigated the morphological stability of the precipitate grown from the solid solution. The analysis shows that the elastic-effects manifest through the boundary conditions based on local equilibrium that determines the compositions at a coherent solid-solid interface. The elastic-effects increase in importance relative to capillarity as the scale of the system increases, and so will become very important at small initial supersaturation, as the supersaturation is inversely proportional to the critical nucleus radius. Elastic fields can also influence the selection of the fastest-growing harmonic, with stabilizing elastic fields favoring long-wavelength harmonics while destabilizing elastic fields favoring short-wavelength harmonics. The stability analysis reveals the dependence of the different elastic parameters in either amplification or decay of the perturbations at the solid-solid interface. While such a linear stability analysis gives an idea about the characteristics of the diffusive instability, they cannot explain the selection of a unique dendritic tip radius and velocity, as is known from solidification experiments. The selection requires the presence of anisotropy in either the interfacial energy or the attachment kinetics, and the selected growth shape of the dendrite may be predicted using the well established microsolvability theory~\cite{kessler1986,kessler1986steady,barbieri1987,barbieri1989}. Formulation of such a theory is presently lacking for solid-state dendritic growth, and a possible strategy here is the use of dynamical phase-field simulations that naturally lead to the selection of a unique tip radius and velocity. Previously, Greenwood and coworkers~\cite{greenwood2009} have proposed a phase-field model for microstructural evolution which includes the effects of the elastic strain energy. The authors show that the solid-state dendritic structures undergo a transition from surface energy anisotropy dominated $\langle 10 \rangle$ growth directions to elastically driven $\langle 11 \rangle$ growth directions induced by changes in the elastic anisotropy, the surface anisotropy, and the supersaturation in the matrix. Our work, in this paper, will be an extension of this investigation where we will study dendritic tip selection using a phase-field model as a function of the different elastic and interfacial parameters as well as the supersaturation. Similar to dendrites formed during the solidification experiments, our studies will aim at identifying the variation of the classical selection constant ($\sigma^{*}$) as a function of the material and process parameters, and in particular, the strength of anisotropy in the stiffness matrix and the interfacial energies. Thus, our motivation in this paper is to determine the effect of several physical properties on the formation of the solid-state dendritic structures and characterize the influence on the selection constant ($\sigma^{*}$). Here, we will utilize a diffuse interface formulation where we solve the Allen-Cahn equation along with the evolution of the diffusion potential of the relevant species that ensure mass conservation. While solving the evolution equations of the order parameter and the diffusion potential, we ensure that the mechanical equilibrium is satisfied in all parts of the system. The simulations show the evolution of a precipitate morphology in the supersaturated matrix with imposed anisotropies in the elastic energy as well as the interfacial energy. We will systematically study the effect of change in different properties of the phases on the selection of radius of the dendrite tip and velocity. \section{Model formulation} We formulate a phase-field model based on a grand-potential functional which includes the interfacial properties as well as the thermodynamics of the bulk phases in the system, \begin{align} \Psi(\mu,\mathbf{u},\phi) &= \int_V \Bigg[\gamma W a^{2}\left(\mathbf{n}\right)|\nabla \phi|^{2} + \dfrac{1}{W} w\left(\phi\right) + \psi(\mu,\phi) \Bigg] dV \nonumber\\ &+ \int_V f_{el}\left(\mathbf{u},\phi\right)dV, \end{align} where $V$ is the total volume of the system undergoing solid-state phase transformation. The phase-field order parameter $\phi$ that describes the presence and absence of precipitate ($\alpha$ phase) and matrix ($\beta$ phase), acquires value $\phi=1$ in the precipitate phase whereas $\phi=0$ in the matrix phase. The double obstacle potential which writes as, \begin{align} w\left(\phi\right) &= \dfrac{16}{\pi^2} \gamma \phi\left(1-\phi\right) \qquad \phi \in [0,1],\nonumber\\ &= \infty \qquad \qquad \qquad \textrm{otherwise}, \end{align} provides a potential barrier between two phases. Here, the term $\gamma$ controls the interfacial energy density, while $W$ influences the diffuse interface width separating the precipitate and the matrix phases. Anisotropy in the interfacial energy is incorporated using the function $a\left(\mathbf{n}\right)$, which further writes as~\cite{karma1998}: \begin{align} \gamma &= \gamma_0 a(\mathbf{n}), \nonumber \\ \gamma &= \gamma_0 \left(1 - \varepsilon\left(3 -4\dfrac{{\phi}^{'4}_{x} + {\phi}^{'4}_{y}}{({\phi}^{'2}_{x} + {\phi}^{'2}_{y})^2} \right) \right), \label{Eqn_anisotropy} \end{align} where $\varepsilon$ is the strength of anisotropy in the interfacial energy, $a(\mathbf{n})$ is the anisotropy function of the interface normal, $\mathbf{n}=-\dfrac{\nabla\phi}{|\nabla\phi|}$. $\psi(\mu,\mathbf{u},\phi)$ is the grand potential density that is a function of the diffusion potential $\mu$. Finally, $f_{el}\left(\mathbf{u},\phi\right)$ is a function of the displacement field $\mathbf{u}$. By taking the variational derivative of the functional with respect to the phase-field order parameter, using Allen-Cahn dynamics, we get the evolution of order parameter $\phi$, which writes as, \begin{align} \tau W \dfrac{\partial \phi}{\partial t} &= - \dfrac{\delta F}{\delta \phi}, \end{align} \begin{align} \tau W \frac{\partial\phi}{\partial t} &= 2 \gamma W \nabla \cdot \left[a\left(\mathbf{n}\right) \left[\dfrac{\partial a\left(\mathbf{n}\right)}{\partial \nabla \phi}|\nabla \phi|^{2} + a\left(\mathbf{n}\right) \nabla \phi\right]\right] \nonumber\\ &- \dfrac{16}{\pi^2} \dfrac{\gamma}{W} \left(1-2\phi\right) - \Delta\psi \frac{\partial h(\phi)}{\partial\phi} + \dfrac{\partial f_{el}\left(\mathbf{u},\phi\right)}{\partial \phi}, \label{phi_evolve_dend} \end{align} where $\tau$ is a relaxation constant for the evolution of $\phi$. The last two terms in Eqn.~\ref{phi_evolve_dend} contributes to the driving force necessary for the evolution of the precipitate phase. Here the term $\Delta \psi$ is the difference between the grand-potential densities of the $\alpha$ and the $\beta$ phases, and the corresponding term in the update of the order parameter derives from $\psi\left(\mu, \phi\right) =\psi_\alpha\left(\mu\right)h\left(\phi\right) + \psi_\beta\left(\mu\right)(1-h\left(\phi\right))$, where $h(\phi) = \phi^2 (3 - 2\phi)$ is an interpolation polynomial. Further, the difference of the grand-potential densities between the phases may be written as a function of the departure of the diffusion potential from its equilibrium values as, $\psi_\alpha-\psi_\beta = \left(c^{\alpha}_{eq} - c^{\beta}_{eq}\right)\left(\mu-\mu_{eq}\right)$, where $c^{\alpha}_{eq}$ and $c^{\beta}_{eq}$ are the equilibrium compositions of the bulk precipitate and matrix phases respectively. We note that this is the driving force at leading order for an arbitrary description of the free-energies while it is the exact driving force for a situation where the free-energy composition relations of the matrix and precipitate are parabolic with equal curvatures. The second term in the driving force for phase transition (Eqn.~\ref{phi_evolve_dend}) comes from the derivative of the elastic free-energy density that writes as, \begin{align} f_{el}(\mathbf{u}, \phi) &= \dfrac{1}{2}C_{ijkl}(\phi)(\epsilon_{ij} - \epsilon^*_{ij}(\phi))(\epsilon_{kl} - \epsilon^*_{kl}(\phi)), \end{align} where the total strain can be computed from the displacement field $\mathbf{u}$, which writes as, \begin{align} \epsilon_{ij} &= \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \label{strain}, \end{align} while the elastic constants $C_{ijkl}$ and misfit-strain $\epsilon^*_{ij}$ can be expressed as, \begin{align} C_{ijkl}(\phi) &= C^{\alpha}_{ijkl}\phi + C^{\beta}_{ijkl}(1-\phi), \nonumber\\ \epsilon^*_{ij}(\phi) &= \epsilon^{* \alpha}_{ij}\phi + \epsilon^{* \beta}_{ij}(1-\phi). \end{align} To simplify the equations, without any loss of generality, we additionally impose that the misfit-strain exists only in the precipitate phase ($\alpha$ phase), which makes $\epsilon^{* \beta}_{ij}=0$. Thereafter, the elastic energy density can be recast as, \begin{align} f_{el}(\phi) &= Z_3 (\phi)^3 + Z_2 (\phi)^2 + Z_1 \phi + Z_0, \label{fel_phi_dend} \end{align} where, in Eqn.~\ref{fel_phi_dend}, we segregate the terms in powers of $\phi$, i.e., $Z_3$, $Z_2$, $Z_1$, and $Z_0$. Each pre-factor is a polynomial of $\phi$, elastic constants, and the misfit strains. The expansion of these pre-factors is illustrated in the~\ref{appendix_f_elast}. Therefore, the term corresponding to the elastic energy in the evolution equation of the order parameter can be computed as, \begin{align} \dfrac{\partial f_{el}\left(\mathbf{u},\phi\right)}{\partial \phi} &= 3Z_3 (\phi)^2 + 2Z_2 (\phi) + Z_1. \end{align} Eqn.~\ref{phi_evolve_dend} is solved along with the update of the diffusion potential that follows the equation, \begin{align} \dfrac{\partial \mu}{\partial t} = \left(\sum_\alpha h(\phi) \dfrac{\partial c_\alpha}{\partial \mu}\right)^{-1} \Bigg[\nabla \cdot M \nabla \mu - \sum_\alpha c_\alpha(\mu) \dfrac{\partial h}{\partial \phi} \dfrac{\partial \phi}{\partial t}\Bigg], \label{chem_pot_evolve} \end{align} where $M$ is the atomic mobility that is explicitly related to the diffusivity $D$ as $D\dfrac{dc}{d\mu}$. For our calculations, we assume a parabolic description $f\left(c\right)$ of the matrix and precipitate phases with equal leading order terms $A = (1/2)\dfrac{\partial^{2}f}{\partial c^{2}}$, thereby the mobility becomes $\dfrac{D}{2A}$. Finally, we compute the displacement field as a function of the spatial distribution of the order parameter, which is utilized to calculate the magnitude of the strain field across the domain. Thus, we solve the damped wave equation iteratively which writes as, \begin{align} \rho\frac{d^2\mathbf{u}}{dt^2} + b\frac{d\mathbf{u}}{dt} &= \nabla \cdot \boldsymbol{\sigma}. \label{mech_equilibrium} \end{align} Eqn.~\eqref{mech_equilibrium} is solved until the equilibrium is reached, i.e., $\nabla \cdot \boldsymbol{\sigma}=\mathbf{0}$. The terms $\rho$ and $b$ are chosen such that the convergence is achieved in the fastest possible time. \section{Parameter initialization} In this section, we list out the material parameters that will be used in the subsequent sections. We use a non-dimensionalization scheme where the energy density scale is derived from the magnitude of the shear modulus $1\times10^{9}\,J/m^{3}$, while the interfacial energy scale is given by $1\,J/m^{2}$. Dividing the interfacial energy scale with the energy density scale sets the length scale of $l^{*}=1\,nm$. In this work, we report all the parameters in terms of non-dimensional units. The anisotropy in the elastic energy is introduced through the Zener anisotropy parameter, i.e., $A_z$, which in turn modifies the magnitudes of the elastic constants, that are $C_{11}=C_{1111}$, $C_{12}=C_{1122}$, and $C_{44}=C_{1212}$. These elastic constants can further be elaborated in terms of the shear modulus ($\mu$), Poisson's ratio ($\nu$), and Zener anisotropy parameter ($A_z$), which controls the evolution and orientation of the instability. \begin{align} C_{44} = \mu, \quad C_{12} = 2\nu\left(\frac{\mu}{1-2\nu}\right), \quad C_{11} = C_{12} + \frac{2C_{44}}{A_z}. \end{align} In 2D, for values of $A_z$ greater than unity, $\langle10\rangle$ directions are elastically softer and $\langle11\rangle$ directions are elastically harder, whereas in 3D $\langle100\rangle$ directions are softer, while $\langle111\rangle$ directions are harder. However, if $A_z$ is less than unity, elastically soft (hard) directions are $\langle11\rangle$ ($\langle10\rangle$) in 2D, and $\langle111\rangle(\langle100\rangle)$ in 3D. For all the cases, the precipitate and the matrix have the same magnitude of $A_z$. Unless otherwise specified, all results are produced with $\mu_{mat} =\mu_{ppt} =100$, $\nu_{mat}=\nu_{ppt}=0.3$, and $A_z$ varying between 2 and 3, typically observed in Ni-based superalloys. Here, the misfit strain or eigenstrain ($\epsilon^*$) is dilatational, i.e., the magnitude of the misfit strain is the same along all the principal directions while the off-diagonal terms are zero. The magnitude of misfit strain is varied in the range from 0.5 to 1\%. The simulation is initialized with a precipitate of initial radius, $R_0=40$, that is placed at the center of the domain. The simulation domain obeys periodic boundary conditions. The size of the domain is chosen such that the diffusion fields of the neighboring precipitates do not interact with each other. The sizes are $3165d_0$ and $1099 d_0$ for 2D and 3D, respectively, where $d_0$ is the capillary length. Equilibrium compositions in the bulk precipitate $(c^{\alpha}_{eq})$ and the matrix $(c^{\beta}_{eq})$ are chosen to be 0.78125 and 0.5, respectively. In the following, we perform systematic simulation studies in both 2D and 3D in order to investigate the influence of supersaturation, misfit strain, and anisotropy strengths on the development of dendrite-like instabilities. The 2D simulations are performed using a finite-difference discretization on a regular grid and the code is written C and parallelized using MPI for running on several CPUs. Since the 3D simulations are more computationally intensive, we utilize the Fourier-spectral method that allows a quicker solution to the equations of mechanical equilibrium~\cite{SBhattacharyya2009}. The simulations are run on NVIDIA Tesla V100 GPUs using the optimized CUDA-based spectral solver for the governing equations~\eqref{phi_evolve_dend},~\eqref{chem_pot_evolve}, and~\eqref{mech_equilibrium}. \section{Evolution of the precipitate into dendrite-like shape} We begin by describing the shape evolution of the precipitate giving rise to dendrite-like shapes. Fig.~\ref{evolution_dend_time} shows such an exemplary simulation in 2D where the matrix supersaturation ($\omega$) is 53\%, the misfit at the interface is $1\%$, Zener anisotropy parameter $3.0$, and the interfacial energy is isotropic. We note that experimental observations show dendritic structures usually at a lower supersaturation, which leads to larger inter-precipitate distances allowing free development of the instabilities without overlap of the composition and the elastic fields. In the simulations, however, we are constrained by having to resolve the large elastic and composition boundary layers. Therefore, in order to derive results in a reasonable time, we have performed simulations for higher supersaturations in comparison to experimentally observed situations. However, this should not alter the results that we derive from the simulations and should be generically applicable for interpreting experimentally observed microstructures and observed trends in the change of the morphologies upon alteration of the processing conditions or the material properties. For 2D simulations, the precipitate with an initial circular shape grows into a square-like shape with rounded corners, where its faces are normal to $\langle10\rangle$ directions. In the absence of interfacial energy anisotropy, the precipitate shape and the eventual instabilities of the interface are determined by the anisotropy in the elastic energy. Further, with an increase in the size, the precipitate corners become sharper. Eventually, due to the point effect of diffusion, the corners of the precipitate grow faster (along the $\langle11\rangle$ directions) compared to its faces, which further gives rise to concavities on the precipitate faces, as shown in Fig.~\ref{phi_50k}-~\ref{phi_150k}. We note that this shape evolution is indeed due to a diffusive instability similar to the Mullins-Sekerka instability leading to the formation of dendrites during solidification for anisotropic interfacial energies, where the perturbations of the composition boundary layer ahead of the interface provide an amplifying feedback to the interface velocity leading to the development of the instability. Among the different possible perturbation modes, the elastic energy anisotropy determines the eventual shape of the precipitate during growth giving rise to dendrite-like shapes. Fig.~\ref{all_phi_profile} shows the contour plot of phase-field profiles at $\phi=0.5$, i.e., the precipitate-matrix interface, which is plotted as a function of increasing time. The composition field around the precipitate also evolves as a function of time which is captured in Fig.~\ref{phase-field-comp-field}. \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.4\linewidth]{phi1k} \label{phi_1k} } \centering \subfigure[]{\includegraphics[width=0.4\linewidth]{phi10k} \label{phi_10k} } \centering \subfigure[]{\includegraphics[width=0.4\linewidth]{phi50k} \label{phi_50k} } \centering \subfigure[]{\includegraphics[width=0.4\linewidth]{phi150k} \label{phi_150k} } \centering \subfigure[]{\includegraphics[width=0.7\linewidth]{compare_dend_evolve_with_normalized_time_az3} \label{all_phi_profile} } \caption{Evolution of the precipitate into dendritic structure for $A_z=3.0$, $\epsilon^*=1\%$ shown as time snapshots at normalized times: (a)278, (b)2780, (c)13900, (d)41700. (e) Contour plots of a one-fourth section of the evolving dendrite at different times.} \label{evolution_dend_time} \end{figure} \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{phi_az2_150k} \label{pf_dend} } \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{comp_az2_150k} \label{comp_dend} } \caption{(a) Phase field and (b) composition field of a one-fourth section of the precipitate growing into a dendritic structure. Here, supersaturation is 53\%, misfit strain is 1\%, and Zener anisotropy parameter is 3.0.} \label{phase-field-comp-field} \end{figure} In 3D, the precipitate first develops into a cube-like shape with rounded corners where the faces are normal to $\langle100\rangle$ directions. Table~\ref{tab:prec_evolve} depicts the evolution of the precipitate into a dendrite-like morphology. The left panel shows the temporal evolution of the precipitate morphology (represented using isosurfaces drawn at $\phi = 0.5$) for normalized time $t = 4609, \, 13827,\,$ and $23045$. In addition, adjacent to the isosurface representation, we present the corresponding precipitate morphology in the $(110)$ plane passing through the center of the box. As a result of the point effect of diffusion, the precipitate starts developing ears along $\langle111\rangle$ directions (see $t = 4609$) i.e. protrusions along $\langle111\rangle$ directions and depressions along $\langle110\rangle$ and $\langle100\rangle$ directions. Further ahead in time, the ears develop into prominent primary dendrite arms, whereas concavities develop on the faces of the cube. The composition evolution during the development of the dendritic structure is revealed in the last column of Table~\ref{tab:prec_evolve}. \begin{table}[htbp] \begin{tabular}{ m{2.0cm} m{2.75cm} m{4cm} m{4cm} } & Isosurface view & \hspace{0.9cm} $\phi$ map & composition map \\ $t = 4609$ & \includegraphics[width=2.8cm]{image1.png}& \includegraphics[width=3.8cm]{phi00005.png} & \includegraphics[width=3.8cm]{comp00005.png}\\ $t = 13827$ & \includegraphics[width=2.8cm]{image2.png}& \includegraphics[width=3.8cm]{phi00015.png} & \includegraphics[width=3.8cm]{comp00015.png}\\ $t = 23045$ & \includegraphics[width=2.8cm]{image3.png}& \includegraphics[width=3.8cm]{phi00025.png} & \includegraphics[width=3.8cm]{comp00025.png}\\ \end{tabular} \caption{Temporal evolution of a 3D dendritic morphology represented by time snapshots at $t = 4609$, $13827$ and $23045$. First column shows isosurfaces of the dendrites drawn at $\phi=0.5$. Second column shows the corresponding sections of the evolving dendrite on a $(110)$ plane passing through the center of the simulation box. Last column shows the corresponding composition profiles on the same plane. Here, we use misfit strain $\epsilon^{*} = 1\%$, supersaturation $\omega = 53\%$, and Zener anisotropy parameter $A_z = 3$.} \label{tab:prec_evolve} \end{table} With an increase in the amount of the supersaturation, the precipitate grows faster into a dendritic shape. This can be seen in Fig.~\ref{dend_supersaturation_compare}, where the precipitate with supersaturation $\omega=53\%$, grows its arms faster along the $\langle11\rangle$ directions giving rise to a dendrite-like morphology as compared to the one with supersaturation $\omega=40\%$. \begin{figure}[!htbp] \centering \includegraphics[width=0.5\linewidth]{dend_az3_compare_super2} \caption{Evolution of precipitate into dendritic structure for different magnitudes of the supersaturation in the matrix phase captured at the same time. At a higher supersaturation, prominent primary dendritic arms develop. Here, misfit strain is 1\% and Zener anisotropy parameter is 3.0.} \label{dend_supersaturation_compare} \end{figure} Typical solidification dendrites are characterized by a unique tip radius and velocity and a dendritic shape that is a function of the strength of the anisotropy. In the following, we perform a similar characterization of our morphologies. A usual characteristic of dendrites during solidification is that apart from the selection of a Peclet number, the product of the square of the dendrite tip radius with the velocity is also a constant. This constant is typically known as the microsolvability constant $\sigma^{*}=2d_0D/R^2_{tip}V_{tip}$, where $d_0$ is the capillary length, D is the diffusivity, and $R_{tip}$ and $V_{tip}$ are the radius and the velocity of the dendritic tip respectively, that is either estimated using the linearized microsolvability theory as presented in~\cite{barbieri1989} or using dynamical phase-field simulations~\cite{karma1996}. We follow the latter route, where we derive the value of the selection constant $\sigma^{*}$ as a function of the material parameters and supersaturation from the measured values of the dendrite-tip radius and velocity from our phase-field simulations. Fig.~\ref{sigma_star_Az3} shows the variation of the selection constant $\sigma^{*}$ as a function of time, scaled with the characteristic diffusion time. We find that the value of the selection constant has a transient where there is both an increase and decrease in the magnitude before settling into a regime where the values continue to change approximately linearly with simulation time. This linear regime initiates approximately when the primary arms have emerged as a result of the instability. This is quite different to dendrites in the presence of just interfacial energy anisotropy, where the value of $\sigma^{*}$ becomes relatively constant (see Fig.~\ref{sigma_vary_no_elasticity}) quite early in the simulation (just as the primary arms appear), even though the $R_{tip}$ (Fig.~\ref{radius_vary_no_elasticity}) and $V_{tip}$ (Fig.~\ref{velocity_vary_no_elasticity}) themselves have not yet attained their steady state values. A similar variation is also seen in the Peclet number $\dfrac{R_{tip}V_{tip}}{2D}$, where it continues to decrease approximately linearly with time, as depicted in Fig.~\ref{Pe_estrain} in the presence of elasticity, while a nearly constant value is attained for the dendrite with just interfacial energy anisotropy (see Fig.~\ref{Pe_vary_no_elasticity}). Therefore, we also do not derive any steady state with respect to the shape of the tip as well as the velocity as highlighted in Figs.~\ref{Rtip_az3} and~\ref{Vtip_az3} respectively in the presence of elasticity. \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{sigma_star_az3} \label{sigma_star_Az3} } \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{Peclet_az3_es1} \label{Pe_estrain} } \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{rtip_vary_az3_pos} \label{Rtip_az3} } \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{vtip_vary_az3} \label{Vtip_az3} } \caption{(a) Plot of the variation of $\sigma^{*}$ as a function of time in the simulations (b)Variation of the Peclet number as a function of scaled time (c) and (d) show the variation of the dendrite tip radius and velocity as a function of time.} \label{Approach_to_steady_state} \end{figure} \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{sigma-star-without-elast} \label{sigma_vary_no_elasticity} } \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{Peclet-no-without-elast} \label{Pe_vary_no_elasticity} } \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{rtip-without-elasticity} \label{radius_vary_no_elasticity} } \centering \subfigure[]{\includegraphics[width=0.45\linewidth]{vtip-without-elast} \label{velocity_vary_no_elasticity} } \caption{ Variation of (a) $\sigma^*$, (b) Peclet number, (c) $R_{tip}$, and (d) $V_{tip}$ as a function of normalized time without elasticity. Here, we choose supersatuartion of $53\%$. The selection constant $\sigma^*$ achieves a relatively steady state. In addition, the Peclet number nearly attains a steady state value. } \label{rtip_vtip_vary_no_elasticity} \end{figure} Similar trends are also noticed for the 3D simulations as well. In 3D, we measure the dendritic tip radius ($R_{tip}$) at different times by fitting the dendritic tip surface to a paraboloid of revolution. The tip position at different times is used to calculate the velocity ($V_{tip}$) along the $\langle 111 \rangle$ direction as this is the growth direction for $A_z>1$. Similar to the 2D simulations, we calculate the microsolvability constant ($\sigma^*$) and Peclet number at different times. Fig.~\ref{sigma_star_Az3_3d} shows the temporal variation of $\sigma^*$ and Fig.~\ref{Peclet_num_3d} depicts the variation of Peclet number as a function of scaled time. The values of $\sigma^*$ and Peclet number continue to vary with time without the attainment of a steady state. Figs.~\ref{Rtip_az3_3d} and~\ref{Vtip_az3_3d} represent the variation of $R_{tip}$ and $V_{tip}$ as a function of scaled time that reveals that both of these quantities also do not achieve a steady state. The $R_{tip}$ continues to increase, whereas $V_{tip}$ continues to decrease with time, and the variation is similar in nature to that observed in the 2D simulations. \begin{figure}[htbp] \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{sigma_star_evolve} \label{sigma_star_Az3_3d} } \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{Peclet_number_evolve} \label{Peclet_num_3d} } \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{rtip_evolve} \label{Rtip_az3_3d} } \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{vtip_evolve} \label{Vtip_az3_3d} } \caption{Temporal evolution of the dendritic characteristics: (a) selection constant ($\sigma^{*}$) (b) Peclet number (c) dendritic tip radius ($R_{tip}$) (d) dendritic tip velocity ($V_{tip}$). Here, supersaturation is $53\%$, misfit strain is $1\%$, and Zener anisotropy parameter is $3$. All of these dendritic characteristics do not achieve steady state values} \label{Approach_to_steady_state_3d} \end{figure} The reason for the non-attainment of a steady state is possibly linked to the variation of the jump in the value of the elastic energy at the interface. We have extracted the elastic energy along the direction normal to the dendrite-tip for different simulation times. From this, we calculate the jump in the elastic energy ($\Delta f_{el}$) by computing the difference in the values of the elastic energy on the precipitate side with the matrix side. For the 2D simulations, Fig.~\ref{elast_energy_11} highlights the variation of the elastic energy density as a function of time, while the jump in the elastic energy density at the interface is depicted in Fig.~\ref{delta_fel} that clearly shows an increase with simulation time. \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{elast_energy_along_11} \label{elast_energy_11}} \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{delta_fel_vs_time} \label{delta_fel}} \caption{(a) Variation of elastic energy density along $[11]$ direction, i.e., along the tip of the precipitate at simulation times of $5000$, $25000$, $50000$, $75000$, $100000$. The bulk elastic energy in the precipitate increases with time. (b) Temporal evolution of $\Delta f_{el}$ calculated along $[11]$ direction. The line fit to the jump in elastic energy data suggests linear increment of $\Delta f_{el}$ with time. Here, Zener anisotropy parameter is $3$, supersaturation is $53\%$, and misfit strain is $1\%$.} \end{figure} Similarly, for 3D, Fig.~\ref{fig:elast_energy_prof} depicts the variation of elastic energy density across the interface along $[111]$ direction at time $t = 800$, $1200$, $1600$, $2000$, and $2400$, while the jump of the elastic energy density at the interface in the $[111]$ direction is plotted in Fig~\ref{fig:elast_energy_jump}. \begin{figure}[htbp] \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{elast_energy_prof} \label{fig:elast_energy_prof}} \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{elast_jump_evolve} \label{fig:elast_energy_jump}} \caption{(a) Variation of elastic energy density across the interface along $[111]$ at different simulation times of $t = 800$, $1200$, $1600$, $2000$, and $2400$. Elastic energy inside precipitate increases with time. (b) Temporal evolution of $\Delta f_{el}$ along $[111]$ direction. The jump in the elastic energy along $[111]$ increases with time indicating absence of steady state. Here, supersaturation is $53\%$, misfit strain is $1\%$, and Zener anisotropy parameter is $3$. Elastic energy inside precipitate increases with time.} \end{figure} We note that although the variation is small, this will, in turn, lead to the change in the interfacial compositions as a function of time. The reason for this is as follows: as the jump in elastic energy increases with time, the dendritic tip experiences varying elastic fields ahead of interface indicating an increasing contribution of elastic energy to the interfacial equilibrium conditions during the growth of precipitate. The increasing elastic contribution to the interfacial equilibrium conditions is countered by a decreasing curvature contribution. As a result, there is continuous increase in $R_{tip}$ with increasing time. The decreasing curvature contribution possibly leads to a reduction in the point effect of diffusion, leading to slower interface dynamics that result in a continuous decrease of $V_{tip}$ with time. Therefore, the Peclet number as well as the selection constant $\sigma^{*}$ do not attain saturated values. This also means that the dendrite tip radius as well as the tip velocity never reach a steady state in contrast to the situation of dendritic growth in the presence of interfacial energy anisotropy with no elastic contribution. (e.g., dendritic growth during solidification). Therefore, in relation to the classical dendritic structures observed during solidification, the simulated structures may only be referred to as dendrite-like. Although the variation of the $\Delta f_{el}$ at the dendrite tip will lead to a change in the interfacial compositions and thereby the Peclet number, the overall volume (area for 2D) still varies linearly with time (see Fig.~\ref{ppt_volume}). The linear temporal variation of the volume of precipitate suggests the parabolic law of diffusion-controlled growth of the precipitate. \begin{figure}[!htbp] \centering \includegraphics[width=0.5\linewidth]{vol_ppt_az3} \caption{Evolution of scaled precipitate volume as a function of scaled time $tD/d_0^2$. Here, Zener anisotropy parameter id $3.0$, misfit strain is $1\%$, and supersaturation is $53\%$. The linear variation of precipitate volume suggests the diffusion-controlled growth of precipitate.} \label{ppt_volume} \end{figure} Further, we determine the effect of elastic anisotropy on the behavior of precipitate morphologies. For 2D situations, Fig.~\ref{dend_az} shows the difference in the tip shapes as a function of $A_z$ at a normalized time=41700, where we plot only one of the symmetric quadrants for simplicity. Fig.~\ref{dend_az} shows that with increase in the strength of anisotropy in the elastic energy ($A_z>1.0$), the radius of the tip of the precipitate reduces, i.e., the tip morphologies become sharper and elongated along $\langle 11 \rangle$ directions, that is also revealed in the plots showing the dendrite tip radius in Fig.~\ref{radius_vary_az_es1}. Correspondingly, the velocities at the tip are also higher for larger values of $A_z$ as highlighted in Fig.~\ref{velocity_vary_az_es1}. \begin{figure}[!htbp] \centering \includegraphics[width=0.5\linewidth]{compare_dend_tip_vs_az} \caption{Contours of $\phi=0.5$ showing one-fourth section of the dendritic structure at a normalized time of 41700 for different strengths of anisotropy in elastic energy ($A_z = 2.0$, $2.5$, and $3.0$). The higher Zener anisotropy parameter gives rise to faster growth of dendritic tip.} \label{dend_az} \end{figure} \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{rtip_vary_az_es1_pos} \label{radius_vary_az_es1} }% \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{vtip_vary_az_es1} \label{velocity_vary_az_es1} } \subfigure[]{\includegraphics[width=0.48\linewidth]{sigma_star_vs_time_vary_az} \label{sigma_vary_az} } \caption{ Effect of strength of anisotropy in elastic energy on the temporal variation of (a) $R_{tip}$ and (b) $V_{tip}$, and (c) $\sigma^*$. Here, the supersaturation is 53\% and the misfit strain is 1\%. With the increase in $A_z$, tip becomes more sharper, $V_{tip}$ increases, and $\sigma^*$ increases.} \label{rtip_vtip_vary_az_es1} \end{figure} The plot in Fig.~\ref{sigma_vary_az} shows that the magnitude of $\sigma^*$ consistently increases with normalized time $(tD/d_0^2)$ after the initial transient, whereas the magnitude of $\sigma^*$ increases with the strength of anisotropy in the elastic energy, i.e., $A_z$ at a given time. Similarly, Fig.~\ref{sigma_estrain} shows that the magnitude of $\sigma^*$ increases with time for a given misfit strain. The plot also shows that, as the magnitude of the misfit strain increases, the value of the selection constant also becomes larger. There is no clear trend observable with the change in the velocities as seen in Fig.~\ref{velocity_vary_es_az3}, while the tip radius reduces with increase in the value of the misfit strain as depicted in Fig.~\ref{radius_vary_es_az3}. Thus, there is no unique value of the selection constant $(\sigma^*)$ at a given anisotropy strength of elastic energy and the magnitude of the misfit strain. While the variation of $\sigma^{*}$ in the linear regime at a given time, is approximately linear with the variation of $A_z$, it changes approximately as $\epsilon^{*1.5}$ with the misfit strain (comparing values for a given time). \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{rtip_estrain_az3_pos} \label{radius_vary_es_az3} }% \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{vtip_estrain_az3} \label{velocity_vary_es_az3} } \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{sigma_vs_time_grand_pot_estrain_az3}} \label{sigma_estrain} \caption{ Effect of misfit strain on the temporal variation of (a) $R_{tip}$, (b) $V_{tip}$, and $\sigma^*$. Here, Zener anisotropy parameter is 3 and supersaturation is 53\%. With the increase in misfit strain, tip becomes sharper, $V_{tip}$ increases, and $\sigma^*$ increases.} \label{rtip_vtip_vary_es_az3} \end{figure} 3D simulations were also performed for different values of $A_z$ for a supersaturation $\omega=45\%$. Fig~\ref{fig:contour_t_4000} represents the contour plot of $\phi = 0.5$ in the $(110)$ plane at different levels of elastic anisotropy. The figure shows that as the anisotropy in elastic energy increases, the precipitate grows faster with a sharper dendrite tip. Figs.~\ref{fig:rtip_Az_effect} and~\ref{fig:vtip_Az_effect} reflects the same effect of decrease in $R_{tip}$ and increase in $V_{tip}$ with increase in $A_z$, respectively. Here, too, we observe that $R_{tip}$ and $V_{tip}$ do not achieve a steady state value at all levels of $A_z$. Fig.~\ref{fig:sigma_star_Az_effect} shows the variation $\sigma^*$ with the scaled time at different levels of $A_z$. The microsolvability constant $\sigma^*$ also does not achieve a steady state value over time at all levels of $A_z$. The influence of the misfit-strain on the observed microstructures is depicted in Fig.~\ref{fig:rtip_evolve_misfit_effect} where the radius of the tip becomes sharper with an increase in the value of the misfit strain. Along with this, the velocity shows an increasing trend with higher values of the misfit strain as highlighted in Fig.~\ref{fig:vtip_evolve_misfit_effect}. The selection constant on the other hand also increases with larger misfit, however no steady state is achieved as shown in Fig.~\ref{fig:sigma_star_evolve_misfit_effect}. \begin{figure}[!htbp] \centering \includegraphics[width=0.5\linewidth]{contour_map_Az_effect} \caption{Contours of $\phi = 0.5$ at a normalized time of $18436$ in the plane $(110)$ passing through the center of simulation box for different magnitudes of $A_z$ ($A_z = 2.0$, $2.5$, and $3.0$). Higher $A_z$ shows faster growth of the dendritic tip.} \label{fig:contour_t_4000} \end{figure} \begin{figure}[htbp] \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{rtip_evolve_Az_effect} \label{fig:rtip_Az_effect}}% \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{vtip_evolve_Az_effect} \label{fig:vtip_Az_effect}} \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{sigma_star_evolve_Az_effect} \label{fig:sigma_star_Az_effect}} \caption{Effect of strength of anisotropy in elastic energy on the variation of (a) $R_{tip}$ (b) $V_{tip}$, and (c) $\sigma^*$ as a function of scaled time. Here, misfit strain is $1\%$ and supersaturation is $53\%$. As $A_z$ increases, the dendritic tip becomes sharper, $V_{tip}$ increases, and $\sigma^*$ tends to increase.} \label{fig:Az_effect} \end{figure} \begin{figure}[htbp] \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{rtip_evolve_misfit_effect} \label{fig:rtip_evolve_misfit_effect}}% \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{vtip_evolve_misfit_effect} \label{fig:vtip_evolve_misfit_effect}} \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{sigma_star_evolve_misfit_effect} \label{fig:sigma_star_evolve_misfit_effect}} \caption{Effect of misfit strain on the variation of (a) $R_{tip}$ (b) $V_{tip}$, and (c) $\sigma^*$ as a function of scaled time $tD/d_0^2$. Here, the supersaturation is 53\% and Zener anisotropy parameter is 3. With the decrease in misfit strain, dendritic tip becomes more blunt, tip grows faster, and $\sigma^*$ decreases.} \label{fig:misfit_effect} \end{figure} Fig.~\ref{rtip_vtip_vary_super} depicts the effect of change in the degree of the supersaturation in the matrix, while the anisotropy in the elastic energy as well as the magnitude of the misfit strain are kept constant, i.e., $A_z=3.0$ and $\epsilon^*=1\%$. The plots show that the radius of the dendritic tip becomes sharper (see Fig.~\ref{radius_super}) with increase in the magnitude of supersaturation whereas the velocity increases (see Fig.~\ref{velocity_super}), but does not saturate. Similarly, the magnitude of the selection constant increases with time as shown in Fig.~\ref{sigma_super}.% \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{rtip_vary_az3_pos_supersat} \label{radius_super} } \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{vtip_compare_super} \label{velocity_super} } \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{sigma_star_supersat} \label{sigma_super}} \caption{ Effect of supersaturation on the temporal evolution of (a) $R_{tip}$ and (b) $V_{tip}$, (c) $\sigma^*$. Here, Zener anisotropy parameter is $3.0$ and supersaturation is 53\%. With the increase in supersaturation, tip becomes more sharper, $V_{tip}$ increases, and $\sigma^*$ decreases.} \label{rtip_vtip_vary_super} \end{figure} 3D simulations with $A_z=3.0$ and $\epsilon^*=1\%$ and different supersaturations were also performed with similar conclusions. Fig.~\ref{fig:rtip_evolve_c0_effect} reveal a decrease in the radius of the tip and a corresponding increase in the velocity Fig.~\ref{fig:vtip_evolve_c0_effect} with increasing supersaturation. The 3D simulations were limited by domain size and we could not access supersaturations that are lower. The selection constant also shows a decreasing trend with higher supersaturations as revealed in Fig.~\ref{fig:sigma_star_evolve_c0_effect}. \begin{figure}[htbp] \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{rtip_evolve_c0_effect} \label{fig:rtip_evolve_c0_effect}}% \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{vtip_evolve_c0_effect} \label{fig:vtip_evolve_c0_effect}} \centering \subfigure[]{\includegraphics[width=0.49\linewidth]{sigma_star_evolve_c0_effect} \label{fig:sigma_star_evolve_c0_effect}} \caption{Effect of supersaturation on the variation of (a) $R_{tip}$ (b) $V_{tip}$, and (c) $\sigma^*$ as a function of scaled time $tD/d_0^2$. Here, the misfit strain is 1\% and strength of elastic energy anisotropy is 3.0. As the supersaturation increases, tip becomes more sharper, tip grows faster, and $\sigma^*$ decreases.} \label{fig:c0_effect} \end{figure} \section{Competition between anisotropies in the interfacial energy and elastic energy} In the previous section, we have seen the effect of different variables such as the variation of supersaturation, misfit strain and anisotropy in elastic energy on the evolution of the dendrite-like morphologies. In this section, we investigate the competition between the influence of anisotropies in the elastic and the interfacial energies. For brevity, we will only utilize 2D simulations in this section for investigating the competition between the two forms of anisotropy and the influence on the dynamics of the instability. The anisotropy in the interfacial energy is modeled using the function~\cite{karma1998}, Eqn.~\ref{Eqn_anisotropy}, which leads to the formation of dendrites aligned along the $\langle 10 \rangle$ directions. Here, we keep the anisotropy in elastic energy $(A_z)$ constant and allow the strength in anisotropy in interfacial energy to vary in the range of 0.0 to 0.03. Also, here we have considered two broad categories with different magnitudes of elastic anisotropies, i.e., $A_z=3.0$ and $A_z=0.5$. In the first case, as the magnitude of $A_z$ is above one (similar to the case studied before), the interfacial energy and the elastic energy anisotropies lead to the formation of dendrites in $\langle 11 \rangle$ and $\langle 10 \rangle$ directions respectively, whereas, in the second case, for values of $A_z$ less than unity, the anisotropies in the elastic energy and the interfacial energy superimpose.\\ \subsection{Case A: Strength of elastic anisotropy $(A_z)>1.0$} For this case where the anisotropies in the interfacial energy and elasticity lead to dendrite-like structures in different directions, the combined influence of anisotropies in both the energies gives rise to a precipitate shape which is nearly circular in the early stage of growth as shown in Fig.~\ref{early_stage_dendrits}, at $A_z=3.0$ and $\varepsilon=0.03$. \begin{figure}[!htbp] \centering \includegraphics[width=0.3\linewidth]{early_stage_az3_varying_sf_aniso.pdf} \caption{Early stage precipitate growth under combined effect of anisotropies in the elastic energy and the interfacial energy at normalized time of 2780 for Zener anisotropy parameter of $3.0$. Here, at $A_z=3.0$ and $\varepsilon=0.03$, the precipitate acquires nearly circular shape. } \label{early_stage_dendrits} \end{figure} But, with decrease in the strength of anisotropy in the interfacial energy from 0.03 to 0.0, the precipitate not only develops sharp corners but also shows strong alignment along $\langle 11 \rangle$ directions, that is the elastically preferred direction. Fig.~\ref{dendrits_sfaniso} compares the morphologies for different strengths of anisotropies in interfacial energy $(\varepsilon=0.0-0.03)$ for a given value of $A_z=3.0$ at the normalized simulation time of $t=55600$. \begin{figure}[!htbp] \centering \includegraphics[width=0.3\linewidth]{phi_saturated_rtip_az3_varying_sf_aniso} \caption{Contours of $\phi = 0.3$ at a normalized time $t=55600$ showing one-fourth section of the symmetric dendritic structure under the combined effect of anisotropies in the elastic energy and the interfacial energy. Here, Zener anisotropy parameter is 3 and strength of anisotropy in interfacial energy varies from $\varepsilon=0.0-0.03$. The increase in $\varepsilon$ slows down the dendritic growth along $\langle 11\rangle$.} \label{dendrits_sfaniso} \end{figure} Fig.~\ref{radius_vary_inter_aniso_es1_az3} depicts that as the interfacial energy anisotropy increases, the dendrite tip radius becomes larger while the velocity of the dendrite tip reduces as highlighted in Fig.~\ref{velocity_vary_inter_aniso_es1_az3}. \begin{figure}[!htbp] \centering \subfigure[]{\centering \includegraphics[width=0.48\linewidth]{rtip_az3_es1_vary_sf_pos_correct} \label{radius_vary_inter_aniso_es1_az3} }% \subfigure[]{\centering \includegraphics[width=0.48\linewidth]{vtip_vary_az_es1_correct} \label{velocity_vary_inter_aniso_es1_az3} } \subfigure[]{\includegraphics[width=0.48\linewidth]{sigma_az3_with_sf_aniso_correct} \label{sigma_diff_az} } \caption{ Effect of anisotropy in interfacial energy on the variation of (a) $R_{tip}$, (b) $V_{tip}$ and (c) $\sigma^*$ at $A_z = 3.0$. Here, the misfit strain is 1\% and supersaturation is 53\%. The increase in $\varepsilon$ leads to wider dendritic tip with slower tip velocity and decrease in $\sigma^*$.} \label{rtip_vtip_vary_inter_aniso_es1_az3} \end{figure} Similarly, while the magnitude of $\sigma^*$ again has a linearly increasing trend with simulation time (after the initial transient), the competition between the elastic energy anisotropy and the interfacial energy anisotropy leads to a decrease in the magnitude of $\sigma^*$ at a given time of the evolution of the precipitate, which is expected as this is similar to the effective reduction of anisotropy in the system (see Fig.~\ref{sigma_diff_az}). \subsection{Case B: Strength of elastic anisotropy $(A_z)<1.0$} In the previous section, the effect of varying the anisotropy in interfacial energy at constant elastic anisotropy i.e. $A_z=3.0$ is elaborated. Here, we explore the results upon varying the interfacial energy anisotropy while keeping the elastic energy anisotropy having the magnitude below one i.e. $A_z=0.5$. An exemplary simulation with the combination of both anisotropies is depicted in Fig.~\ref{dendrits_evol_sf03}, where the arms of the dendrite are oriented along $\langle 10 \rangle$ directions. \begin{figure}[!htbp] \centering \includegraphics[width=0.5\linewidth]{dendrit_evolve_az5_with_sfinso_normalized_time} \caption{Contours of $\phi = 0.5$ in a one-fourth section at normalized times $t = 2780$, $5560$, $11120$, $16680$, $22240$ showing dendritic growth along $\langle 10 \rangle$ directions. Here, Zener anisotropy parameter is $0.5$, strength of anisotropy in interfacial energy is $0.03$, supersaturation is $53\%$, and misfit strain is $1\%$. } \label{dendrits_evol_sf03} \end{figure} The change in the dendrite shapes upon variation in the strength of anisotropy in the interfacial energy is highlighted in Fig.~\ref{dendrit_compare_az0.05_sf}. The superimposition of the two anisotropies leads to an effective increase in the anisotropy in the system. \begin{figure}[!htbp] \centering \includegraphics[width=0.3\linewidth]{dendrite_az_less_than_one_compare} \caption{Contours of $\phi=0.5$ in a one-fourth section at a same time showing dendritic structure for different strength of anisotropies in interfacial energy. Here, the Zener anisotropy parameter is $0.5$, the misfit strain is $1\%$, and the supersaturation is $53\%$. The increase in $\varepsilon$ accelerates the dendritic growth along $\langle 10 \rangle$.} \label{dendrit_compare_az0.05_sf} \end{figure} Thus, with an increase in the magnitude of interfacial energy anisotropy, the precipitate tip becomes more sharper and elongated along $\langle 10 \rangle$ directions, as highlighted in Fig.~\ref{radius_vary_inter_aniso_az_less_than_one}, while the dendrite tip velocity increases as the strength in anisotropy in the interfacial energy becomes larger (see Fig.~\ref{velocity_vary_inter_aniso_az_less_than_one}). \begin{figure}[!htbp] \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{rtip_az_less_one_es1_sfaniso_pos_correct} \label{radius_vary_inter_aniso_az_less_than_one} }% \centering \subfigure[]{\includegraphics[width=0.48\linewidth]{vtip_az_less_one_es1_sfaniso_correct} \label{velocity_vary_inter_aniso_az_less_than_one} } \subfigure[]{\includegraphics[width=0.48\linewidth]{sigma_az_less_than_one_with_sf_aniso_correct} \label{sigma_az0.5_diff_sf_aniso} } \caption{ Effect of anisotropy in interfacial energy on the temporal variation of (a) $R_{tip}$ (b) $V_{tip}$, and (c) $\sigma^*$ at $A_z=0.5$. Here, the misfit strain is 1\% and supersaturation is 53\%. The increase in $\varepsilon$ leads to sharper dendritic tip and faster tip velocity.} \label{rtip_vtip_vary_inter_aniso_az_less_than_one} \end{figure} The variation of $\sigma^*$ with a combination of different anisotropies is portrayed in Fig.~\ref{sigma_az0.5_diff_sf_aniso}, where for a given time of evolution, with increase in the strength of anisotropy in the interfacial energy the magnitude of $\sigma^*$ increases. Finally, in order to show that anisotropy in either the interfacial energy or the elastic energy is required for the formation of dendrite-like structures, we consider the case of isotropic elastic and interfacial energies with the supersaturation at $\omega=53\%$. As the precipitate grows in size, the instabilities at the interface trigger to give rise to a seaweed type structure (see Fig.~\ref{sea_weed}), without the selection of a unique tip direction or shape. This situation can also occur for cases where the influences of the elastic energy anisotropy and the interfacial energy anisotropy cancel each other for a certain combination of their respective strengths. \begin{figure}[!htbp] \centering \includegraphics[width=0.5\linewidth]{sea_weed_struct_az1} \caption{Contours of $\phi=0.5$ at different normalized times showing the development of a seaweed structure in a system with isotropic elastic energy and interfacial energy. Here, the misfit strain is 1\% and supersaturation is 53\%.} \label{sea_weed} \end{figure} \section{Conclusions and Outlook} We have systematically characterized the evolution of dendrite-like shapes as a function of elastic parameters such as the misfit strain, Zener anisotropy parameter , supersaturation in two and three dimensions. Although we notice the occurrence of solid-state morphologies that resemble dendrites typically occurring during solidification, in the presence of coherency stresses, the shapes of the tip as well the tip velocity do not achieve a steady state. Moreover, the selection constant $\sigma^{*}=2d_0D/R^{2}V$ increases linearly with simulation time for all the simulation conditions, which is in contrast to the dendrites derived just in the presence of interfacial energy anisotropy. This lack of steady state is due to a continuous change in the value of the jump in the elastic energy at the tip of the dendrite-like morphology, that increases as the shape of the tip evolves with time. Consequently, the interfacial compositions as well as the Peclet number do not saturate. Thus, in the classical sense, in the presence of coherency stresses, while the presence of anisotropy leads to the propagation of instabilities in well defined directions, there is no selection of a unique tip shape as in the case of solidification. Therefore, structures derived in solid-state in the presence of elastic anisotropy may only be referred to as dendrite-like. Future directions of study involve understanding the influence of inhomogeneity in elastic moduli, combination of anisotropies in both the misfit strain and the elastic energy on the tip dynamics and shape. Additionally, in multi-component alloys, the relative diffusivities of the different elements can lead to widely different dendritic shapes as the effective capillary length may change appreciably. Therefore, the relative ratio of the diffusivities becomes an important parameter whose influence on the instability needs to be established in alloys with three or more components. Finally, while we have discussed only precipitate growth, the model is generic and may be utilized for studies of late-stage coarsening, where an extension to multi-component alloys will again bring in exciting new possibilities. \section*{Data availability} The data that supports the results of this study are available from the corresponding author upon reasonable request. \section*{Acknowledgement} We (BB, TJ, SB, AC) thank the financial support from Department of Science and Technology (DST), Government of India (GOI), under the project TMD/CERI/Clean Coal/2017/034. TJ and SB also acknowledge financial grants and computational support under the project S\&T/15-16/DMR-309.01 from DMRL, DRDO, GOI.

proofpile-arXiv_069-9216

\section{Introduction} \begin{figure*} \centering \includegraphics[width=0.97\textwidth]{example_new.png} \caption{Examples of images with two different caption text pieces from the MS COCO caption dataset, where some captions are more fine-grained than the others that are more abstract.} \label{fig:example} \vspace{-3mm} \end{figure*} Inspired by recent development of pre-trained language models in various NLP tasks, recent studies~\citep{lu2019vilbert,su2019vl,tan2019lxmert,chenuniter,li2020oscar,yu2020ernie} on vision-language pre-training (VLP) have pushed the limits of a variety of Vision-and-Language (V+L) tasks, which learn the semantic alignment between the different modalities by harnessing from large-scale image-text pairs. The semantic gap between different modalities has always been treated as one of the most significant problems in cross-modality research. In current VLP literature, there are two mainstream architectures for bridging the cross-modal semantic gap: \textit{single-stream architecture} and \textit{two-stream architecture}. The former such as VL-BERT~\citep{su2019vl} and UNITER~\citep{chenuniter} assumes that the underlying semantics behind the two modalities is simple and clear, and thus simply concatenates image-region features and text features as input to a single Transformer~\citep{vaswani2017attention} network for early fusion in a straightforward manner. This paradigm learns the cross-modal semantic alignment from a bottom feature level by using the self-attention mechanism. Nevertheless, the design of single-stream structure treats both modality inputs equally, leaving the inherent different peculiarity of each modality not fully exploited. In contrast, the latter like LXMERT~\citep{tan2019lxmert} and ERNIE-ViL~\citep{yu2020ernie} first uses separate Transformer encoders to learn high-level abstraction of image and sentence representation respectively, and then combines the two modalities together with a cross-modal Transformer. This kind of design explicitly distinguishes between different modality inputs and aligns the cross-modal representations at a higher semantic level, but is usually parameter inefficient and may ignore certain more fundamental feature-level association. In real-world image-text data, we observe that it is easy for some of the image-text pairs to align simple semantics on both modalities, while others may be related after higher-level abstraction. As shown in Figure~\ref{fig:example}, the captions of T1 are more focused on the overview of the image with coarse-level semantics, while T2 are more detailed descriptions that emphasize on the specific parts of the images. The semantic granularity spans different levels for different captions of the same images. It is essential to explicitly consider aligning semantics at multiple levels for deeply understanding the real-world image-text data. In light of this observation, we propose a new VLP pre-training architecture SemVLP, as a fusion of single-stream and two-stream architectures, which jointly aligns the image and text representation at multiple semantic levels. We observe that both the single-stream and two-stream architectures use common Transformer module, with the main difference that the latter introduces an extra CrossAttention module to allow cross-modal alignment at higher level and the different modalities are separately encoded. To complement the advantages of different architectures, we unify the two mainstream architectures by using a shared Transformer network and a pluggable cross-modal attention module, as shown in Figure~\ref{fig:framework}. To conduct more fine-grained feature-level alignment, we choose a single-stream mode and directly concatenate image-region features and text features as input to the shared Transformer network for pre-training. To enhance high-level semantic alignment, we switch to a two-stream mode by separately encoding both the image and text modalities with the same shared Transformer, where a cross-modal attention module is further added to allow cross-modal fusion at a higher semantic level. The pre-training procedure is conducted iteratively so as to align the real-world image-text data at multiple semantic levels. During the iterative pre-training phase, the shared Transformer network is forced to align the semantics at multiple levels, which enables the trained model to adapt to diverse image-text pairs. In this way, we take advantages of both mainstream architectures for cross-modal fusion, where the parameters are shared to allow for different pre-training styles that regularize with each other. We evaluate SemVLP on a variety of representative vision-language understanding tasks, including visual question answering, natural language visual reasoning and image-text/text-image retrieval. On all these tasks, SemVLP obtains significant improvements compared to those methods that align semantics at a single fixed level, where the proposed 12-layer SemVLP model outperforms all the previous single-stream and two-stream architectures with the same model size. The main contributions of this work can be summarized as follows: (i) We introduce SemVLP, a simple and effective VLP method to learn generic image-text representations for V+L understanding tasks. (ii) We propose a new pre-training framework that aligns cross-modal semantics at multiple levels, which can take advantages of both single-stream and two-stream architectures. To the best of our knowledge, we are among the first to think about unifying the two mainstream architectures for better aligning the cross-modal semantics. (iii) We present extensive experiments and analysis to validate the effectiveness of the proposed SemVLP model, which can obtain superior performance with a 12-layer Transformer backbone on four V+L understanding tasks. \section{Related Work} Pre-training methods have substantially advanced the NLP field in both text understanding and text generation, such as BERT~\citep{devlin2018bert}, ALBERT~\cite{lan2019albert}, GPT~\citep{radford2018improving} and T5~\citep{raffel2019exploring}. Inspired by language pre-training, the research community starts to pay more attention to vision-language pre-training in multi-modal scenario, and many pre-training methods have been successfully applied to V+L tasks such as visual question answering~\citep{antol2015vqa,hudson2019gqa} and cross-modality retrieval~\citep{young2014image,lin2014microsoft}. In terms of the model architecture, there are mainly two broad directions to conduct vision-language pre-training. The first line uses a single-stream transformer architecture to model both image and text representations in a unified semantic space such as VLBERT~\citep{su2019vl}, UNITER~\citep{chenuniter} and OSCAR~\citep{li2020oscar}. In contrast, the other line adopts a two-stream Transformer architecture that first encodes the image and text modalities separately, and then fuses the cross-modal representations with another Transformer network. Furthermore, some other works focus on designing different pre-training tasks to learn better cross-modal representations such as ERNIE-ViL~\citep{yu2020ernie} and PixelBERT~\citep{huang2020pixel}. In this paper, we focus on the V+L understanding tasks with VLP method. Instead of choosing only one model architecture for VL pre-training, we introduce a pioneer work of fusing both the single-stream and two-stream architectures to better align the cross-modal semantics at multiple levels. \section{SemVLP Pre-training} \subsection{Model Architecture} \begin{figure*} \centering \includegraphics[width=0.92\textwidth]{framework_new.png} \caption{The overall framework of SemVLP. The whole model consists of a shared multi-layer Transformer encoder and an extra cross-modal attention module $\varphi_C^l$, where the Transformer encoder includes a self-attention module $\varphi_S^l$ and a FFN layer $\varphi_F^l$ (feed-forward network). In single-stream mode, we directly concatenate the image and text representation as input to the shared Transformer encoder for V+L pre-training. In two-stream mode, we reuse the same shared Transformer encoder to separately encode both the image and text representations, and add an extra cross-modal attention to align the semantics at a higher level. } \label{fig:framework} \vspace{-2mm} \end{figure*} The architecture overview of SemVLP is shown in Figure~\ref{fig:framework}. Inspired by the idea of sharing the encoder and decoder in Transformers for neural machine translation~\citep{xia2019tied}, we base the architecture of SemVLP on a shared bidirectional Transformer encoder, where a pluggable cross-modal attention module is further used to enhance high-level semantic alignment. By sharing the model parameters and adjusting the input format, SemVLP can be flexible to switch between single-stream and two-stream pre-training architectures, with the input text and image encoded in different semantic levels. In this way, we cast both the mainstream pre-training architectures into a more compact one in that there is only one copy of parameter set, which is applicable to both the low-level and high-level semantic alignment with much less parameter cost. We iteratively pre-train on the two settings towards better understanding of the real-world image-text pairs. \subsubsection{Input Embeddings} The input to SemVLP is an image and its related sentence (e.g. caption text). Each image is represented as a sequence of objects $\{o_{1},...,o_{n}\}$, and each sentence is represented as a sequence of words $\{w_{1},...,w_{m}\}$. After cross-modal fusion and alignment at multiple semantic levels, SemVLP is able to generate language representations, image representations and cross-modal representations from the image-text inputs. Given the sequence of words and objects, we first introduce the methods to embed the inputs to the feature space. \textbf{Sentence Embeddings} \quad We adopt the same method as BERT~\citep{devlin2018bert}, which uses WordPiece tokenizer to tokenize the input sentence into sub-word tokens. The sequence of input tokens is as $\{[CLS], w_{1},...,w_{m}, [SEP]\}$, where $[CLS]$ and $[SEP]$ are special tokens in BERT. The final embedding $e_i$ for each token is generated by combining the original word embedding, segment embedding and position embedding. \textbf{Image Embeddings} \quad We use a pre-trained object detector Faster R-CNN~\citep{ren2015faster} to extract the object-level image features from the image, where each object $o_j$ is represented as a 2048-dimensional feature vector $f_j$. To capture the spatial information of the object, we also encode the box-level location features for each object via a 4-dimensional vector $l_j = (\frac{x_1}{W},\frac{y_1}{H},\frac{x_2}{W},\frac{y_2}{H})$, where $(x_1, y_1)$ and $(x_2, y_2)$ denote the coordinate of the bottom-left and top-right corner while $W$ and $H$ are the width and height of the input image. We concatenate $f_j$ and $l_j$ to form a position-sensitive object feature vector, which is further transformed into $o'_j$ using a linear projection to ensure that it has the same vector dimension as that of word embeddings. Similar to special token $[CLS]$ in sentence embeddings, we also add a special feature $[IMG]$ to denote the representation of the entire image and add it to the beginning of the input object sequence. \subsection{Shared Transformer Encoder} Given the embeddings of the words for the sentence $\{e_i\}_{i=1}^m$ and the image regions $\{o'_j\}_{j=1}^n$, the full encoder is a stacked model with $L$ blocks, where the $l'$th block consists of a self-attention module $\varphi_S^l$, a nonlinear feed forward network $\varphi_F^l$ and a pluggable cross-modal attention module $\varphi_C^l$, where superscript $l$ represents the layer index. Both the self-attention and cross-modal attention modules are based on the multi-head attention~\citep{vaswani2017attention}, where the feed forward network (FFN) consists of an intermediate layer and an output layer as in BERT~\citep{devlin2018bert}. In both the single-stream and two-stream modes, the self-attention module $\varphi_S^l$ and feed forward network $\varphi_F^l$ are shared and tied to a single one copy of parameter set. The cross-modal attention module $\varphi_C^l$ is additionally used in two-stream mode to enhance the high-level semantic alignment. \subsubsection{Feature-level Semantic Alignment} To allow a fine-grained feature-level semantic alignment, we directly concatenate the image and text embedding features as input to the single-stream mode of SemVLP, which consists of the shared self-attention module and nonlinear FFN layer. Specifically, we initialize $S^0 = \{o'_1,...,o'_n,e_1,...,e_m\}$. The encoding process can be formulated as: \begin{equation*} \begin{aligned} s_i^l &= \varphi_F^l(\varphi_S^l(s_i^{l-1}, S^{l-1})) \\ S^l&=\{s_1^l, s_2^l, ..., s_{n+m}^l\}=\{o^{l}_1,...,o^{l}_n,h^{l}_1,...,h^{l}_m\} \end{aligned} \end{equation*} where $\{h^l_i\}$ and $\{o^l_j\}$ are the text and object representation of layer $l$, respectively. In this way, we can get full interaction between the image and text representations from a bottom feature-level embedding space. Eventually, we obtain $O^L=\{o^{L}_1,o^{L}_2,...,o^{L}_n\}$ and $H^L=\{h^{L}_1,h^{L}_2,...,h^{L}_m\}$, the representations of all the object outputs and text outputs of the last layer in the SemVLP encoder. The hidden representations $O^L$ and $H^L$ are then used to conduct the subsequent pre-training tasks. \subsubsection{High-level Semantic Alignment} For enhancing high-level semantic alignment, we adopt the two-stream mode of SemVLP, where text and image objects are separately encoded first and then fuse at a high-level semantic space. Therefore, we adopt a two-encoder architecture shown on bottom-right of Figure~\ref{fig:framework}. The two-stream design of SemVLP mainly derives from the Transformer encoder-decoder network, where the main difference lies in: (1) we use both the bidirectional Transformer encoders for encoding the image and text inputs, which focuses on the V+L understanding tasks, (2) except for the cross-modal attention module, the image and text encoders share the same model parameters. We find that such parameter sharing can enhance the semantic alignment at a module level, and act as a form of regularization that stabilizes the training and saves memory consumption~\citep{xia2019tied}, (3) different from previous Transformer encoder-decoder architecture which introduces the cross-attention module to all blocks of the decoder, we only introduce the cross-modal attention module at the upper parts of the blocks, so as to better fuse the cross-modal representations at high-level semantic space. The encoding process of two-stream mode can be formulated as follows: \begin{equation*} \begin{aligned} h_j^l &= \varphi_F^l(\varphi_S^l(h_j^{l-1}, H^{l-1})) \\ o_j^l &= \varphi_F^l(\varphi_S^l(o_j^{l-1}, O^{l-1})), s.t. \quad l <= L_s \\ o_{j+1}^l &= \varphi_F^l(\varphi_C^l(\varphi_S^l(o_{j+1}^{l-1}, O^{l-1}), H^L)), s.t. \quad l > L_s \end{aligned} \end{equation*} where $L_s$ indicates the layer index that cross-modal attention is introduced. The image and text feature embeddings are first separately encoded, and then the hidden states $H^L$ of text output in the last layer are used as input to the cross-modal attention for better understanding the image representations. Eventually, we can obtain the output representations of image objects and text, $O^L=\{o^{L}_1,o^{L}_2,...,o^{L}_n\}$ and $H^L=\{h^{L}_1,h^{L}_2,...,h^{L}_m\}$. With $O^L$ and $H^L$, we could use a simple network with a softmax layer to conduct the subsequent pre-training tasks. \subsection{Joint Training} \subsubsection{Pre-training Tasks} We follow LXMERT~\citep{tan2019lxmert} and use three-types of pre-training tasks: i.e., language task, vision task and cross-modality task. \textbf{Masked LM Prediction} \quad The task setup is basically the same as in BERT~\citep{devlin2018bert}, we randomly mask 15\% tokens in the text and the model is asked to predict these masked words with the output text representations $H^L$. For different pre-training modes, the masked words will be predicted either with the help of visual modality so as to resolve ambiguity (single-stream mode), or from text modality alone so as to increase task difficulty (two-stream mode). \textbf{Masked Object Prediction} \quad Similarly, we pretrain the vision side by randomly masking objects, i.e., the object features are masked with zeros. We randomly mask 15\% image objects and ask the model to predict properties of these masked objects with the output object representations $O^L$. To capture more object-level semantics, we follow the object prediction task in LXMERT~\citep{tan2019lxmert} and perform two sub-tasks: ROI-Feature Regression and Detected Label Classification. We take the detected labels output by Faster R-CNN~\citep{ren2015faster} as the ground-truth labels for prediction. \textbf{Image-Text Matching (ITM)} \quad The task setup is almost the same as in LXMERT~\citep{tan2019lxmert}, that we randomly sample 50\% mismatched image-text pairs and 50\% matched pairs, and train an classifier to predict whether an image and a sentence match each other on the representation $\textbf{h}^L_{CLS}$ (single-stream mode) and $\textbf{o}^L_{IMG}$ (two-stream mode). One difference is that we do not enforce the masked LM prediction and Object Prediction loss when sampling a mismatched image-text pair. \textbf{Image Question Answering (QA) } \quad We also cast the image question answering task as a classification problem and pre-train the model with image QA data as in LXMERT~\citep{tan2019lxmert}, which leads to a better cross-modality representation. We build the classifier on top of the representation $\textbf{h}^L_{CLS}$ for single-stream mode and on that of $\textbf{o}^L_{IMG}$ for two-stream mode. \subsubsection{Pre-training Strategy} SemVLP is pre-trained with multiple pre-training tasks and we add all these task losses with equal weights. To jointly align semantics at multiple levels, given a mini-batch of image-text pairs, 50\% of the time we update the model with single-stream mode, while 50\% of the time we update it with two-stream mode. In this way, for every update of SemVLP, the model is pre-trained at multiple semantic levels, so as to better model the diverse image-text data. \subsubsection{Fine-tuning Ingredient} After pre-training is completed, SemVLP can support fine-tuning with either a single-stream architecture or a two-stream architecture. In single-stream mode, the hidden state $h^L_{CLS}$ of the last layer is used for cross-modality calculation, while the hidden state of $o^L_{IMG}$ is used in two-stream mode. For each downstream task, we examine the performances for both the single-stream and two-stream fine-tuning. To yield a single model result, we use only the architecture mode with the optimal performance on development set for final evaluation. \section{Experiments} \begin{table*} \centering \small \begin{tabular}{cl|c|ll|lll|lll} \toprule \multicolumn{2}{c|}{\multirow{2}{*}{Models}} & \multirow{2}{*}{Params} & \multicolumn{2}{c|}{VQA} & \multicolumn{3}{c|}{IR-Flickr30K} & \multicolumn{3}{c}{TR-Flickr30K} \\ \multicolumn{2}{l|}{} & & Test-dev & Test-std & R@1 & R@5 & R@10 & R@1 & R@5 & R@10 \\ \midrule \multirow{5}{*}{Single-stream} & VisualBERT & 110M & 70.80 & 71.00 & - & - & - & - & - & - \\ & VLBERT & 110M & 71.16 & - & - & - & - & - & - & - \\ & Unicoder-VL & 110M & - & - & 71.50 & 90.90 & 94.90 & 86.20 & 96.30 & 99.00 \\ & UNITER & 110M & 72.70 & 72.91 & 72.52 & 92.36 & 96.08 & 85.90 & 97.10 & 98.80 \\ & OSCAR & 110M & 73.16 & 73.61 & - & - & - & - & - & - \\ \midrule \multirow{4}{*}{Two-stream} & ViLBERT & 221M & 70.55 & 70.92 & 58.20 & 84.90 & 91.52 & - & - & - \\ & 12-in-1 & 221M & 73.15 & - & 67.90 & - & - & - & - & - \\ & LXMERT & 183M & 72.42 & 72.54 & - & - & - & - & - & - \\ & ERNIE-ViL & ~210M & 72.62 & 72.85 & 74.44 & 92.72 & 95.94 & 86.70 & 97.80 & 99.00 \\ \midrule Our Model & SemVLP & 110M/140M & \textbf{74.52} & \textbf{74.68} & \textbf{74.80} & \textbf{93.43} & \textbf{96.12} & \textbf{87.70} & \textbf{98.20} & \textbf{99.30} \\ \bottomrule \end{tabular} \caption{Evaluation Results on VQA and Flickr30K.} \label{table:overall1} \end{table*} \subsection{Pre-training Setup} \textbf{Pre-training Data} \quad We use the same in-domain data as in LXMERT~\citep{tan2019lxmert} for pre-training. It consists of the image caption data from MS COCO~\citep{lin2014microsoft}, Visual Genome~\citep{krishna2017visual}, and image question answering data from VQA v2.0~\citep{antol2015vqa}, GQA balanced version~\citep{hudson2019gqa} and VG-QA~\citep{zhu2016visual7w}. The total amount of the dataset is 9.18M image-and-sentence pairs on 180K distinct images. Besides, we also use additional out-of-domain data from Conceptual Captions~\citep{sharma2018conceptual} and SBU Captions~\citep{ordonez2011im2text} for model pre-training, which consists of about 4M image-text pairs on 4M images. \textbf{Implementation Details} \quad The maximum sequence length for the sentence is set as 20. We use Faster R-CNN~\citep{ren2015faster} (with ResNet-101 backbone~\citep{he2016deep}) pre-trained on Visual Genome dataset~\citep{krishna2017visual} to detect the objects and extract the region features. We consistently keep 100 objects for each image to maximize the pre-training compute utilization by avoiding padding. For the model architecture, we pre-train a 12-layer SemVLP-base model with hidden size of 768, where we initialize it with the parameters from StructBERT base model~\citep{wang2019structbert}. We set $L_s=6$, which obtains the best performances on the development set of the downstream tasks, at a proper semantic level for cross-modal fusion~\footnote{\small{We only introduce the cross-modal attention from text space to image space due to the superior performance in our framework, where modeling of vision modality is emphasized.}}. We train SemVLP model with a total batch size of 256 for 40 epochs on 4 V100 GPUs. The Adam optimizer with initial learning rate of 1e-4 and a learning rate linear decay schedule is utilized. \begin{table*} \centering \small \begin{tabular}{l|ccccc|c} \toprule \multirow{2}{*}{Models} & MMN & NSM & \multirow{2}{*}{LXMERT} & \multirow{2}{*}{12-in-1} & \multirow{2}{*}{OSCAR} & \multirow{2}{*}{SemVLP} \\ & \citep{chen2019meta} & \citep{hudson2019learning} & & & & \\ \midrule Test-dev & - & - & 60.00 & - & 61.58 & \textbf{62.87} \\ Test-std & 60.83 & 63.17 & 60.33 & 60.65 & 61.62 & \textbf{63.62} \\ \bottomrule \end{tabular} \caption{Evaluation Results on GQA.} \label{table:overall2} \end{table*} \begin{table*} [!htb] \centering \small \begin{tabular}{c|cccc|c} \toprule Models (params) & VisualBERT(110M) & LXMERT(183M) & UNITER(86M) & OSCAR(110M) & SemVLP(110M/140M) \\ \midrule Dev & 67.40 & 74.90 & 77.14 & 78.07 & \textbf{79.00} \\ Test-P & 67.00 & 74.50 & 77.87 & 78.36 & \textbf{79.55} \\ \bottomrule \end{tabular} \caption{Evaluation Results on NLVR2.} \label{table:overall3} \end{table*} \subsection{Results on Downstream Tasks} We compare SemVLP model against other state-of-the-art single-stream and two-stream cross-modal pre-training models of the comparable model size~\footnote{\small{Most of the compared models have similar model size as 12-layer BERT-base.}} on the following downstream tasks. The details on these tasks and fine-tuning configurations can be found in the supplementary material. \begin{myitemize2} \itemsep0em \item \textbf{VQA v2.0}~\cite{antol2015vqa}: A visual question answering task/dataset that asks a model natural language questions on a given image. \item \textbf{Image-Text Retrieval}: We test on the popular Flickr30K dataset~\citep{young2014image}. \item \textbf{NLVR2}~\cite{suhr2018corpus}: A visual reasoning task that aims to determine whether a natural language statement is true about a pair of images. \item \textbf{GQA 2019}~\cite{hudson2019gqa}: An image question answering task/dataset that emphasizes on the reasoning capability of a model to answer a question. \end{myitemize2} The results on the four downstream V+L tasks are shown in Table~\ref{table:overall1},\ref{table:overall2},\ref{table:overall3} respectively. We can see that: (1) Among all the VLP models of similar size to BERT base, SemVLP consistently outperforms other strong single-stream and two-stream VLP methods (e.g., UNITER~\citep{chenuniter}, OSCAR~\citep{li2020oscar} and 12-in-1~\citep{lu202012}, ERNIE-ViL~\citep{yu2020ernie}) on all the examined tasks, which validates the effectiveness of SemVLP on combining single-stream and two-stream architectures to align semantics at multiple levels. (2) With much less parameters, the single-stream architecture can achieve comparable performance to the two-stream architecture, which is more parameter efficient. The proposed SemVLP model can be easily adapted to either architecture according to the typical scenario. By sharing parameters, SemVLP can also be parameter efficient while keeping superior performance. It is partially because SemVLP is pre-trained to align cross-modal semantics at multiple semantic levels, which makes the learning of semantic alignments more robust toward the diverse image-text pairs. \begin{figure} \centering \includegraphics[width=0.49\textwidth]{analysis.png} \caption{Results w.r.t different two-stream architectures for aligning high-level semantics on VQA and NLVR2 development set.} \label{fig:analysis2} \vspace{-2mm} \end{figure} \subsection{Pre-training on Different Semantic Levels} To validate the effectiveness of aligning cross-modal semantics at multiple levels, we conduct in-depth analysis on pre-training at different semantic levels with various architectures. \textbf{Analysis on Various Pre-training Architectures} \quad We first examine the importance of pre-training at multiple semantic levels by conducting ablation study on the pre-training fashions. Specifically, we pre-train the SemVLP model with only one type of model architecture each time and test the performance on the downstream tasks. All the pre-training settings are kept the same as in the original SemVLP pre-training. As shown in Table~\ref{table:analysis1}, both the two pre-training fashions play important roles in obtaining the full SemVLP model, and removing each task will consistently decrease the final downstream task performance. The single-stream architecture is used to align fair-grained feature-level semantics, while the two-stream architecture helps align semantics at a higher-level semantic space. By iterative training with a shared set of Transformer parameters, the proposed SemVLP model can take the advantage of both the single-stream architecture and two-stream architecture towards more robust vision-language pre-training. \begin{table} \centering \small \begin{tabular}{l|c|c|c} \toprule Pre-training Mode & VQA & GQA & NLVR2 \\ \midrule Baseline 1 (single-stream) & 73.72 & 61.82 & 78.02 \\ Baseline 2 (two-stream) & 73.48 & 61.68 & 77.81 \\ SemVLP & \textbf{74.52} & \textbf{62.87} & \textbf{79.00} \\ \bottomrule \end{tabular} \caption{Ablation study of different pre-training fashions on development set. Baseline 1 and Baseline 2 denote pre-training with only single-stream or only two-stream architecture, respectively.} \label{table:analysis1} \end{table} \begin{table} \centering \small \begin{tabular}{l|c|c|c} \toprule Fine-tuning Mode & VQA & GQA & NLVR2 \\ \midrule Single-stream & \textbf{74.52} & \textbf{62.87} & \textbf{79.00} \\ Two-stream & 73.92 & 62.18 & 78.48 \\ \bottomrule \end{tabular} \caption{Results w.r.t. different fine-tuning architectures after the SemVLP model is fully pre-trained.} \label{table:finetune} \vspace{-2mm} \end{table} \begin{figure*} \centering \includegraphics[width=0.85\textwidth]{semvlp_visualize_v2.png} \caption{Visualization of the Image-to-text attention example on single-stream model and SemVLP\protect~\footnotemark[3]. Object 1 and Object 2 are extracted by a pre-trained object detector Faster R-CNN.} \label{fig:visualization} \end{figure*} \footnotetext[3]{\small{We only compare the single-stream and SemVLP because the cross-attention module of two-stream model will interfere with the visualization.}} \textbf{Analysis on Different High-level Semantic Alignments} \quad There are many different ways for high-level semantic alignment, now we further analyze the advantage of our two-stream architecture and the specific ``point'' to conduct modality fusion with the cross-modal attention module. Therefore, we pre-train SemVLP with only high-level semantic alignment and examine in which layer to introduce the cross-modal attention module by setting different $L_s$. The pre-training details are kept the same as in the original SemVLP pre-training. We test the performance on VQA and NLVR2 tasks, and the results are shown in Figure~\ref{fig:analysis2}. We can see that by introducing the cross-modal attention module properly, the two-stream mode of SemVLP method obtains significantly better performance than the previous two-stream model LXMERT. The best performance is obtained when $L_s=6$, where the separated image/text encoding and cross-modal attention is equally emphasized. It again demonstrates the importance of aligning cross-modal representations at a proper semantic level. \subsection{Comparison of Fine-tuning modes} After SemVLP is fully pre-trained at multiple semantic levels, we further examine what fine-tuning architecture/mode is more appropriate for the downstream tasks. We keep the fine-tuning setting as the same with the original setting in the supplementary material, and examine the performance using different architectures. Table~\ref{table:finetune} shows that on all the examined tasks, the single-stream fine-tuning architecture gives better performance than the two-stream one does. This is due to the fact that: (1) by learning cross-modal fusion from a fine-grained feature-level, the single-stream mode can capture full association across modalities from more basic semantic granularity with powerful self-attention mechanism, which originates from the success of BERT. (2) Our design of SemVLP shows more favor of the single Transformer encoder, which is well regularized when pre-training with both training modes via parameter sharing. The pre-training strategy tends to enhance the high-level semantic alignment into single-stream training process. \subsection{Visualization} The motivation behind SemVLP is to align cross-modal semantics at multiple levels by taking advantages of both single-stream and two-stream architectures. As stated above, single-stream and two-stream models are good at feature-level and high-level semantic alignments, respectively. To verify this, we visualize the attention map on the same layer and head of both the single-stream model and SemVLP for image objects and its associated description, as shown in Figure \ref{fig:visualization}. A darker color indicates a higher attention weight. Take the image on the left of Figure \ref{fig:visualization} as an example. In order to align the objects and the description, a model needs to capture the high-level text semantics: ``Two people'' are standing and playing games, and ``a person'' is sitting on the couch. For Object 1 sitting alone, the single-stream model mis-attends to ``people'' in the description. SemVLP, on the other hand, properly pays high attention to ``person''. For Object 2 standing alongside another, our SemVLP attends to ``people'' correctly, while the single stream model fails to do so. This example shows that the single-stream cannot differentiate between the semantics of the “person” and “people” at the high level, which leads to the false alignments. The same pattern can be observed from the image on the right of Figure \ref{fig:visualization}. \section{Conclusion} In this paper, we propose a new pre-training method SemVLP to learn the joint representation of vision and language. Different from existing VLP methods relying on a fixed-level semantic alignment, we introduce to align cross-modal semantics at multiple levels, by assembling a shared Transformer encoder and a pluggable cross-modal attention module in different ways. Experiment results on various downstream V+L tasks demonstrate the effectiveness of our method for understanding the diverse semantics behind the real-world image-text data. \section*{Downstream Tasks} For all the tasks, We use the architecture mode with the optimal performance on development set for final evaluation. The hidden state of $h^L_{CLS}$ (single-stream mode) or $o^L_{IMG}$ (two-stream mode) is used to measure the cross-modality relevance. \subsection{Visual Question Answering (VQA)} The VQA task requires the model to answer natural language questions given an image. We conduct experiments on the widely-used VQA v2.0 dataset~\citep{antol2015vqa}, which contains 204K images and 1.1M questions about these images. Following~\citep{anderson2018bottom}, we treat VQA as a multi-label classification task by picking an answer from a shared set consisting of 3,129 answers. We use the hidden state of $h^L_{CLS}$ (single-stream mode) or $o^L_{IMG}$ (two-stream mode) to map the representation into 3,129 possible answers with an additional MLP layer. The model is optimized with a binary cross-entropy loss on the soft target scores. We fine-tune the SemVLP model on the VQA training data for 3 epochs with a batch size of 32, and use the BERT Adam optimizer with an initial learning rate of 5e-5. At inference, a Softmax function is used for prediction. \subsection{Image-Text Retrieval} The image-text retrieval task consists of two sub-tasks: image retrieval and text retrieval, depending on which modality is used as the retrieval target. We conduct experiments on the Flickr30K dataset~\citep{young2014image}, which contains 31,000 images collected from Flickr website, each associated with 5 captions. We follow the same split in~\citep{lee2018stacked} for training and evaluation. During fine-tuning, we follow the method in UNITER~\citep{chenuniter} and formulate it as a ranking problem. We use the hidden state of $h^L_{CLS}$ (single-stream mode) or $o^L_{IMG}$ (two-stream mode) to compute the similarity scores for the sampled positive and negative pairs, and maximize the margin between them through the circle loss~\citep{sun2020circle} as in ERNIE-Vil~\citep{yu2020ernie}. We fine-tune our model for 4 epochs with a batch size of 64 and a learning rate of 5e-5. Moreover, we use hard negatives sampling as in~\citep{chenuniter} to further improve the performance. \subsection{Natural Language Visual Reasoning for Real (NLVR2)} NLVR2~\citep{suhr2018corpus} is a challenging task for visual reasoning. The goal is to determine whether a natural language statement is true about a pair of images. It consists of 86K/7K/7K data for training/development/test. Since each data example in NLVR2 has two natural images $img_0$, $img_1$ and one language statement $s$, we use SemVLP to encode the two image-statement pairs ($img_0$, $s$) and ($img_1$, $s$), then train a classifier based on the concatenation of the two outputs as in LXMERT~\citep{tan2019lxmert}. We fine-tune SemVLP with a batch size of 32 and a learning rate of 5e-5 for 4 epochs. \subsection{Visual Reasoning in the Real World (GQA)} GQA is an image question answering task, which emphasizes on the reasoning capability of the model to answer a question. We conduct experiments on the public GQA 2019 dataset~\citep{hudson2019gqa}. For each question, the model picks a proper answer from a shared set of 1,852 candidate answers. We follow the two-stage fine-tuning method in OSCAR~\cite{li2020oscar}, where SemVLP model is first fine-tuned on unbalanced ``all-split'' for 2 epochs, and then fine-tune on the ``balanced-split'' for 2 epochs with batch size of 32 and learning rate of 5e-6.

proofpile-arXiv_069-9302

proofpile-arXiv_069-9315

\section{Introduction} \emph{Introduction}---Non-reciprocal devices are essential for protecting quantum systems from their noisy electromagnetic environments while allowing measurement and control. Superconducting circuits, one of the leading platforms for realizing quantum computers \cite{arute2019quantum}, currently uses commercial bulky isolators which utilize permanent magnets to realize uni-directional propagation of microwave signals \cite{pozar2009microwave,Fay1965OperationOT}. However, the goal of developing useful quantum computers with potentially millions of qubits \cite{national2019quantum,preskill1998reliable} demands integration of the essential components and dramatic reductions in the size of the supporting hardware. These challenges in superconducting circuits as well as other quantum systems, motivate the search for miniaturized on-chip alternatives for ferromagnetic isolators utilizing various physics such as acousto-optics \cite{kang2011reconfigurable}, electromechanics \cite{barzanjeh2017mechanical}, non-Hermitian dynamics \cite{ramezani2010unidirectional,bender2013observation} and nonlinear parametric processes \cite{kamal2011noiseless, chapman2019design,chapman2017widely, ranzani2017wideband,sounas2017non}. In particular, parametric processes in traveling wave devices are encouraging for implementing non-reciprocal dynamics as the required phase-matching conditions are usually met only in one propagation direction. Especially, parametric frequency and mode conversions are promising for implementing isolation for high efficiency and high fidelity measurements because these processes in principle can be noiseless in contrast with parametric amplification or non-Hermitian dynamics where the gain and loss introduce inevitable noise into the system \cite{haus1962quantum,caves1982quantum,clerk2010introduction}. Several theoretical and experimental works have studied the isolation using parametric frequency and mode conversions \cite{yu2009complete,lira2012electrically,ranzani2017wideband,doerr2014silicon}; however, the experimental realization of a parametric device that provides sufficient isolation over a broad instantaneous bandwidth has remained elusive. Here we present a practical scheme for broadband isolation based on adiabatic phase-matched parametric mode conversion. We propose a realization using coupled transmission lines in superconducting quantum circuits, but it is important to note that our scheme is general and applicable to a variety of platforms. Our circuit-level analysis with realistic component values suggests more than 20 dB isolation over an octave of bandwidth, comparable to commercial isolators, but, with a smaller footprint and a superconducting qubit compatible fabrication process. \begin{figure*}[t!] \centering \includegraphics[width=0.93 \linewidth]{fig1_42.pdf} \caption{ \textbf{Isolation through adiabatic mode conversion}: \textbf{a.} Modes E and O represent even and odd propagating modes in a waveguide. The coupling between two modes (shown in orange) is spatially varied both in terms of wavevector and strength via a pump mode shown in green. The conversion of mode E to mode O happens only in the forward direction. \textbf{b.} The blue (red) solid line is the dispersion relation for mode E (mode O). The dashed lines correspond to backward propagation. The green solid line shows pump dispersion relation which varies within the green-shaded region. The green arrow shows wavevectors correspond to 2 GHz pump tone which connects mode E and mode O within a highlighted range of frequency only in the forward direction but poorly phase-matched in the backward direction. \textbf{c.} Forward/backward phase mismatch for three different signal frequencies. \textbf{d.} Solid (dashed) lines show the adiabatic conversion of signal in the forward (backward) direction for three different input signal frequencies. The inset shows the coupling spatial variation (see Eq.\,\ref{quadratic_ramp}) \textbf{e.} The Blue solid line shows the isolation based on our presented scheme. The green line is the isolation performance of the scheme implemented in cQED which provides more than 20 dB isolation over 4 GHz range of frequency (shaded region in blue). The dashed line on top is the transmission of the backward signal. The red dotted line is the isolation performance without adiabatic conversion which provides only 200 MHz bandwidth of isolation (shaded region in red). \label{fig1} } \end{figure*} \emph{Adiabatic parametric mode conversion}---Parametric mode conversion provides unidirectional conversion of the signal between two modes of propagation. This nonreciprocal dynamics for the signal occurs because, with a preferred propagation direction set by pump tone, the phase-matching condition for the parametric process is met only in one direction \cite{o2014resonant}. This property naturally suggests utilizing parametric conversions to implement isolator, which has been studied in optics and microwave in traveling wave devices \cite{yu2009complete,lira2012electrically,ranzani2017wideband}. However, the major limitation of the isolation based on parametric conversions is that the full conversion only occurs around a certain frequency set by the device characteristics (e.g. device length, coupling strength, pump power). Therefore the isolation based on parametric mode conversion would not be broadband \cite{yu2009complete,lira2012electrically} and the operating frequency is sensitive to parameter variations. However, as we propose in this letter, by combining adiabatic techniques \cite{suchowski2008geometrical,suchowski2009robust,suchowski2014adiabatic} with parametric mode conversion, we can realize broadband conversion and thus broadband isolation. In Fig.\,\ref{fig1}a, we schematically illustrated the proposed adiabatic mode conversion. Here, mode E and O represent two transmission lines or two orthogonal modes of propagation in a waveguide with different dispersion relations as illustrated in Fig.\,\ref{fig1}b. The propagating signals in forward/backward direction in mode E and O can be represented as $E_{f/b}(x,t)=E_{f/b}(x) e^{-i(\omega_e t \mp k_e x)}+ \mathrm{c.c.}$ and $O_{f/b}(x,t)=O_{f/b}(x) e^{-i(\omega_o t \mp k_o x)}+\mathrm{c.c.}$, where $|E_{f/b}(x)|^2$ and $|O_{f/b}(x)|^2$ quantify the average number of photons propagating in the forward/backward direction in each mode. In the normal situation, mode E and O are decoupled and the incoming signal in each mode propagates through the medium without conversion. However, we consider a situation in which a propagating coherent pump tone in the third mode induces coupling between mode E and O in the form of \begin{equation}\label{pump_couplin_form} \kappa(x,t)=\frac{1}{2}\kappa(x) \exp[ \ -i\omega_p t +i \int_0^x k_p(x) dx \ ]+ \mathrm{c.c.} \end{equation} Here $\omega_p=\omega_o- \omega_e$ is the pump frequency which is fixed unlike the pump wavevector \begin{equation}\label{LCS} k_p(x)=K + \frac{2 \alpha}{L} (x-\frac{L}{2}), \end{equation} and the coupling strength \begin{equation}\label{quadratic_ramp} \kappa(x)= \frac{4 \kappa_0}{L^2} (L-x)x, \end{equation} which are slowly-varying functions of position. Here we choose a linear variation for pump wavevector throughout the device which is quantified by pump wavevector value at the center of the device $K$ and its variation range $\alpha$. The induced coupling strength has quadratic dependence on position and attains its maximum $\kappa_0$ at the center and vanishes at both ends of the device. The dynamics in the forward direction of propagation for signals in mode E and O can be described by the following coupled mode equations, \begin{eqnarray}\label{ode00} \frac{d}{dx} \begin{pmatrix} E_f\\ O_f \end{pmatrix}= \frac{i}{2}\begin{pmatrix} - \Delta k_f(x) & \kappa(x) \\ \kappa(x) & \Delta k_f(x) \end{pmatrix} . \begin{pmatrix} E_f\\ O_f \end{pmatrix}, \end{eqnarray} where $\Delta k_f(x)=k_o - k_e - k_p(x)$ is the instantaneous (position dependent) phase mismatch in the forward direction. In our simple model, the forward and backward dynamics are decoupled and similar equations hold for the backward process, except the mismatch is given by $\Delta k_b(x) = k_o - k_e + k_p(x)$. We note that the dynamical matrix in Eq.\,\ref{ode00} is analogous to a driven qubit Hamiltonian ($x\rightarrow t$) with variable detuning $\Delta k_f(x)$ and Rabi drive $\kappa(x)$ \cite{oliver2005mach}. Give the variation of the pump wavevector $k_p(x)$, the pump tone connects (perfectly phase matches) different frequencies in mode E to mode O at different positions for forward propagating signals. For example, as illustrated in Fig.\,\ref{fig1}b, here we consider a 2 GHz pump tone (green arrows) that couple mode E to mode O from 2.7 GHz at $x=0$ (dark green arrow) to 9.3 GHz at $x=L$ (light green arrow). Assuming linear dispersion relations, this results in a spatially linear variation for mismatch $\Delta k_{f/b}(x)$ (Fig.~\ref{fig1}c). With a proper choice of the variation parameter $K$ and $\alpha$, the mismatch $\Delta k_{f}(x)$ starts from a non-zero value and crosses zero during signal propagation in the forward direction as depicted in Fig.~\ref{fig1}c. Therefore, the dynamics in the forward direction would be ``spatial" version of rapid adiabatic passage in spin physics or Landau-Zener technique in quantum two-level systems \cite{zener1932non,oliver2005mach,shevchenko2010landau}. At the same time, the mismatch in backward direction $\Delta k_{b}(x)$ stays far from zero throughout the signal propagation in backward direction (Fig.~\ref{fig1}c). With the same analogy to two-level system, the dynamics in backward direction would be ``spatial" version of a far-detuned driven qubit (e.g. $[ \Delta k_b(x)/\kappa(x)]^2 \gg 1$). Therefore, we have adiabatic conversions between two modes only in the forward direction of propagation and negligible conversions in backward direction. Note that, the effect of the coupling strength variation is to improve the adiabatic conversion efficiency in close connection to pulse shaping techniques in NMR \cite{melinger1994adiabatic, garwood2001return, goswami2003optical}, cold atoms \cite{du2016experimental}, and qubit gate optimization \cite{martinis2014fast,guerin2011optimal,gambetta2011analytic,yang2017achieving}. Except, in our case, the variation occurs in position and the objective is to have a robust conversion for a broad range of frequencies. Both spatial variations of the pump wavevector and amplitude (strength) associated with the coupling are schematically illustrated in Fig.\,\ref{fig1}a. The adiabatic dynamics governed by Eq.\,\ref{ode00} can be solved numerically as depicted in Fig.\,\ref{fig1}d for three different incoming signals in forward and backward direction in mode E using realistic parameters (per unit length $a$) in cQED $\kappa_0=0.023 \, a^{-1}$, $\alpha=0.05\, a^{-1}$ and $L=2000 a$. As depicted in Fig.~\ref{fig1}d, the signal starting in mode E at $x=0$ is efficiently converted into mode O through an adiabatic conversion while the conversion in the backward direction is negligible. More importantly, this nonreciprocal dynamics happens over a wide bandwidth of frequency for the input signal. With terminations on both sides of mode O, this configuration acts as a two-port broadband isolator. The negligible conversion in the backward propagation contributes to the total insertion loss which is calculated to be less that 0.05\,dB with dielectric loss tangent $\delta=10^{-5}$ \cite{o2008microwave}. The broadband isolation performance of this scheme is depicted in Fig.\,\ref{fig1}e where the blue solid (dashed) line shows isolation (transmission) for the given parameters. For comparison, the red dotted line shows the corresponding isolation without performing spatial variation of the pump wavevector and coupling strength. The green line is the isolation performance of a practical implementation of the presented scheme in cQEC which is the topic of our discussion in the rest of this letter. \begin{figure}[] \centering \includegraphics[width=0.98 \linewidth]{fig2_168.pdf} \caption{\textbf{Circuit diagram:} \textbf{a.} The device consists of a pair of coupled identical transmission lines (shaded). The pump transmission line (in green) provides in-phase flux modulation $\phi_{rf}$ for SQUIDs in both transmission lines. The rf flux amplitude is spatially controlled as schematically illustrated. The dc flux (provided by the red line) has opposite direction in the upper and lower transmission line. \textbf{b.} The flux configuration in a unit cell. Here, $C_m$ and $L_m$ are geometric capacitance and inductive coupling per unit cell between two transmission lines. \textbf{c.} SQUIDs can be considered as inductors with flux-modulated inductance. The inductance is modulated in opposite directions (Eq.\,\ref{eq_ind}). \textbf{d.} The equivalent circuit of a coupled transmission line in even and odd mode basis. Even and odd modes are normally orthogonal, except, the modulated flux induces an effective inductive coupling between two modes by breaking the symmetry between upper and lower transmission lines.\label{fig2}} \end{figure} \emph{Implementation in superconducting circuit}---Now, we proceed by discussing a practical implementation of this scheme in the superconducting circuit platform. The implementation has four essential components: I. Realizing two orthogonal modes of propagation (mode E and mode O) with \textit{distinct} dispersion relations; II. Implementing the coupling between these two modes in a traveling wave fashion (Eq.\,\ref{pump_couplin_form}); III. Carrying out the pump wave vector sweep (adiabatic phase matching) (Eq.\,\ref{LCS}); IV. And finally implementing the coupling strength variation (Eq.\,\ref{quadratic_ramp}). As depicted in Fig.\,\ref{fig2}a, we consider a pair of coupled lumped element transmission lines where the inductance in each unit-cell is provided predominantly by a pair of Josephson junctions in SQUID geometry. This forms a waveguide that supports two orthogonal modes of propagation; namely \textit{even} and \textit{odd} modes. The effective inductance of a SQUID is tuned by a dc bias line (shown in magenta) and also modulated by an rf pump in a separate transmission line (shown in green). We consider the configuration in which SQUIDs in the upper and lower lines receive dc flux in opposite directions while the rf flux modulations are in-phase for both lines as depicted in Fig.\,\ref{fig2}b. Therefore the effective inductance of SQUIDs in upper and lower transmission line would be, \begin{eqnarray} L_s^{\mathrm{u,d}}=0.5 L_{J0} / \mid \cos \left(\frac{ \pm \phi_{dc} + \phi_{rf}(x,t)}{2 \phi_0}\right) \mid \, , \end{eqnarray} where $L_{J0}$ is the bare inductance of a single Josephson junction and $\phi_0=\Phi_0/2\pi$ is the reduced flux quantum and $\phi_{dc}$ is the magnitude of the dc flux threading each SQUID. The rf flux $\phi_{rf}(x,t)$ has both temporal and spatial dependence inherited from the pump transmission line. We consider the regime where the signal currents propagating in the coupled transmission lines are much smaller than the critical current of the Josephson junctions ($I \ll I_c$). In this regime, nonlinear effects due to signal propagation in the transmission lines are negligible. Therefore, SQUIDs can be considered as linear inductors whose inductance is modulated by $\phi_{rf}(x,t)$ (Fig.\,\ref{fig2}c). In the limit of small modulation, $\phi_{rf}(x,t)\ll \phi_{0}$ the effective inductance of SQUIDs in the upper and lower transmission line can be expressed as, \begin{eqnarray}\label{eq_ind} L_s^{\mathrm{u,d}}= \frac{L_{dc}}{1 \pm m(x,t)}, \end{eqnarray} where the unitless parameter $m(x,t)=\tan(\frac{\Phi_{dc}}{2 \phi_0}) \frac{\phi_{rf}(x,t)}{2 \phi_0}$ accounts for flux-pump-induced modulation and $L_{dc}=0.5 L_{J0}/\mid \cos(\frac{\phi_{dc}}{2\phi_0}) \mid $ is the dc-biased SQUID inductance \cite{zorin2019flux}. The nonlinear dynamics of the system can be described by Lagrangian approach by defining node-fluxes in each transmission line \cite{yaakobi2013parametric,zorin2019flux} as depicted in Fig.\,\ref{fig2}c. However, we are interested in the dynamics of the even and odd modes corresponding respectively to symmetric $\Phi^{(e)}_n = (\Phi^{(\mathrm{u})}_n +\Phi^{(\mathrm{d})}_n )/\sqrt{2}$ and anti-symmetric $\Phi_n^{(o)} = (\Phi^{(\mathrm{u})}_n -\Phi^{(\mathrm{d})}_n )/\sqrt{2}$ node-flux superposition in the two transmission lines \cite{orfanidis2002electromagnetic}. The equivalent circuit diagram in the even-odd basis is illustrated in Fig.\,\ref{fig2}d. Here $L_e=L_{dc}+L_m$, $L_o=L_{dc}-L_m$, $C_e=C_0$ and $C_o=C_0+2C_m$ \cite{orfanidis2002electromagnetic,supp}. In this basis, we have two otherwise orthogonal propagation modes where an effective inductive mode coupling $\mathbb{L}_m(x,t)= L_{dc}\, m(x,t)$ is induced by breaking the symmetry of upper and lower transmission lines via the rf flux. Far below the cut-off frequency of the transmission lines $\omega_{0i}=1/\sqrt{C_i L_i}$, and the Josephson plasma frequency $\omega_{Ji}=1/\sqrt{C_s L_i}\,,\, i\in\{e,o\}$, the even and odd modes' characteristic impedance has the following form $Z_i=\sqrt{L_i/C_i}$ and the corresponding dispersion relations are, \begin{eqnarray}\label{keko} k_i(\omega)&=& \frac{\omega/\omega_{0i}}{\sqrt{1- \omega^2/\omega^2_{Ji}}}\simeq \sqrt{L_i C_i} \, \omega \, , \ i\in\{o,e\}. \end{eqnarray} In Fig.\,\ref{fig1}b, we plotted the dispersion curves for even/odd modes (blue/red lines) given a set of circuit parameters listed in Table~1. Note that the distinct dispersion relation for mode E and O is crucial for inhibiting the cascading parametric processes which otherwise would degrade the device performance\cite{yu2009complete}. \begin{center} \begin{table}[] \resizebox{\linewidth}{!}{ \begin{tabular}{c|c|c|c} \multicolumn{4}{c} {Table~1: Circuit Parameters}\\ \hline $C_0=80$ fF & $C_s=40$ fF& $C_m=20$ fF& $\phi_{dc} = 2 \pi/3 \ \phi_0$ \\ $L_{J0}=250$ pH & $L_m={\text -} \, 50$ pH & $m_0=0.1$ &$\phi_{rf}^{max} = 0.1\ \phi_0$\\ \end{tabular}} \end{table} \end{center} In the continuum limit where unit cell length $a$ is much smaller than the wavelength of propagating signals ($a k_i \ll 2\pi, \ i\in\{o,e\}$), and far below the JJ plasma frequency, the dynamics of the node-flux in the even and odd modes can be described by a set of coupled PDEs \cite{supp}, \begin{eqnarray}\label{ode0} \Phi_{tt}^{(i)} - \omega_{0i}^2 \Phi_{xx}^{(i)} =\beta_{ij}^2 ( m \Phi_{xx}^{(j)} + m_x \Phi_x^{(j)}), \end{eqnarray} where we keep only the first order terms in $m=m(x,t)$ and assume $a=1$ for brevity. Subscript $t$ and $x$ represent partial derivatives of time and position respectively. Here $\beta_{ij}^2=\omega_{0i}^2 L_{dc}/L_j$, where $\{i,j\}\in\{ e,o \} , \ i \neq j$. We recast Eq.\,\ref{ode0} in terms of forward/backward traveling waves in even and odd modes $E_{f/b}(x,t) = (V_e \pm Z_e I_e)/2$ , $O_{f/b}(x,t) = (V_o \pm Z_o I_o)/2$ where $V_i= \Phi_t^{(i)}$ and $I_i= - \frac{1}{L_i} \Phi^{(i)}_{x} - \frac{1}{L_i L_j} \mathbb{L}_m(x,t) \Phi^{(j)}_x$ and consider ansatze in form of $E_{f/b}(x,t)= \mathcal{N}_e^{1/2} E(x) e^{-i(\omega_e t \mp k_e x)}+c.c$ , $O_{f/b}(x,t)=\mathcal{N}_o^{1/2} O(x) e^{-i(\omega_o t \mp k_o x)}+ c.c$. Here we conveniently choose scaling parameters $\mathcal{N}_{i}=Z_i \omega_i \, , i\in\{e,o\}$ so that $|E_{f/b}(x)|^2$ and $|O_{f/b}(x)|^2$ are proportional to the average photon number propagating in each mode. For the induced coupling parameter, we consider $m(x,t)=m(x) e^{-i(\omega_p t + \int k_p(x) dx)} + c.c$ where $\omega_p = \omega_o - \omega_e$. Substituting these relations into the coupled equations (Eq.\,\ref{ode0}), and by applying the rotating wave approximations both in time and space \footnote{Basically we ignore fast oscillatory terms either due to frequency mismatch or wavevector mismatch. With these approximations, forward and backward dynamics become effectively decoupled. In supplemental material \cite{supp} (Fig.\,S3) we show that even with including higher oscillatory terms in the dynamical matrix we get similar result.} and keeping only the first order terms in $m(x)$, we arrive at a coupled wave equations $\mathbb{U}_x=\mathbb{M}.\mathbb{U}$ where $\mathbb{U}= [ E_f(x), O_f(x), E_b(x), O_b(x)]^{T}$ and the dynamical matrix (see supplemental material for detailed derivations) \begin{eqnarray}\label{ode4by4} \mathbb{M}= \frac{i}{2}\begin{pmatrix} - \Delta k_f(x) & \kappa(x) & 0 & 0 \\ \kappa(x) & \Delta k_f(x) & 0 & 0 \\ 0 & 0 & \Delta k_b(x) & \kappa(x) \\ 0 & 0 & \kappa(x) & - \Delta k_b(x) \end{pmatrix}, \end{eqnarray} where the phase mismatches are defined as in Eq.\,\ref{ode00}. Here the effective coupling is \begin{equation}\label{kapxkeko} \kappa(x) = \sqrt{\omega_e \omega_o /(Z_e Z_o)} L_{dc} m(x)\simeq \sqrt{k_e k_o} m(x), \end{equation} which is frequency dependent unlike our simple presented model (Eq.\,\ref{ode00}). As illustrated in Fig.\,\ref{fig2}a, the spatially linear variation in pump wavevector is implemented by changing the capacitance and the inductance together in the pump transmission line as $k_p \propto\sqrt{L_p C_p}$. In this way, the impedance of the line remains constant as $Z_p\simeq\sqrt{L_p/C_p}$. With the linear variation of the pump wavevector, the phase mismatch would be a linear function of position $x$ (See Eq.\,\ref{LCS}). We choose circuit parameters so that with pump frequency $\omega_p/2\pi=$ 2 GHz, perfect phase mismatching condition occurs at the center of the device for 6 GHz input signal which means in Eq.\,\ref{LCS} we set $K=k_o(6~ \mathrm{GHz} + 2~\mathrm{GHz}) - k_e(6~\mathrm{GHz})$, and for the pump wavevector sweep range we set $\alpha=0.05\ ( a^{-1})$ which is reasonably achievable in practice. \begin{figure}[] \centering \includegraphics[width=0.98 \linewidth]{fig3_40.pdf} \caption{\textbf{Bloch sphere representation of the adiabatic conversion:} \textbf{a.} E and O represent the even and odd mode which are the natural eigenmodes in the absence of the rf pump. E' and O' are eigenmodes in presence of rf pump. $\theta_0= \tan^{-1}[ \kappa(x=0)/\Delta k_f(x=0)]$ is the angle between the even mode and system's temporal eigenmode. The red (blue) curve is the adiabatic dynamics without (with) spatial coupling variation. \textbf{b.} The corresponding conversion performance without (with) spatial coupling variation shown in red (blue). The residual signal power in the even mode at $x=L$ can be estimated by $\xi = \kappa_0^2/( \kappa_0^2 + \alpha^2)\simeq 0.19$ which sets a lower-bound of -7 dB for the isolation. \textbf{c.} The angle $\theta$ is the angle between the actual state and the instantaneous eigenmode of the system during the conversion which quantifies the deviation from the full adiabatic evolution. \textbf{d.} The deviation angle $\theta$ vs position for different signal frequencies. The deviation at the end of the device should be less than 0.2 [rad] (dashed line) to have 20\,dB isolation. \label{fig3}} \end{figure} Now we discuss the implementation of spatial coupling strength ramp (Eq.\,\ref{quadratic_ramp}) which is essential for this device to perform efficiently given practical limitations. The idea is that, to have an efficient adiabatic evolution, ideally, the system should start in an eigenmode and remain in its instantaneous eigenmode throughout the evolution. In our case, this requires that at $x=0$ the eigenmode of the system to be even/odd mode. Without spatial variation of coupling $\kappa(x)=\kappa_0$, the eigenmode of our system in the presence of flux drive would asymptotically approach to the even/odd mode in the limit of $\alpha\gg \kappa_0$. However, due to practical upper-bound limitation for the pump maximum wavevector sweep range $\alpha$ and lower-bound limitation for $\kappa_0$ due to the finite device length, this requirement would not be met (in our case $\alpha/\kappa_0\sim 2$). Therefore, as depicted in Fig.\,\ref{fig3}a and b, the signal starting at even mode at $x=0$ starts oscillating (the red path in \ref{fig3}a,b) and would not follow the ideal adiabatic path. The unwanted oscillation is translated to inefficient conversion at $x=L$, which limits the isolation performance. However, increasing the mismatch sweep range $\alpha$ is not the only way to ensure that the system starts in the eigenmode. One can slowly ramp off the coupling at both ends of the medium similar to pulse shaping techniques in the time domain \cite{martinis2014fast,guerin2011optimal,gambetta2011analytic,yang2017achieving}. With no coupling, mode E and O are eigenmodes and the signal starting at even mode at $x=0$ would not experience oscillatory evolution (e.g. the blue path in \ref{fig3}a,b). This technique of ramping the coupling drastically improves the adiabaticity of the evolution thus the efficiency of the conversion. % In our model discussed earlier, we consider a quadratic scaling for effective coupling to carry out ramp up/down (Eq.\,\ref{quadratic_ramp}). For the implementation we use slightly different ramping curve to partially compensate the frequency dependence of coupling (see Eq.\,\ref{kapxkeko}). In our adiabatic scheme, since the effective coupling for different frequency happens effectively at different position, we can partially compensate for the frequency dependency of the coupling by implementing slightly stronger coupling at lower $x$. We choose generalized Gaussian function, \begin{equation}\label{quadratic_ramp_imp} m(x)= \frac{m_0}{1-s} \left( e^{-\mid x/L-1/2\mid^p}- s \right) \end{equation} where the parameter $p=3$ for $x\leq L/2$ and $p=2$ for $x> L/2$ controls the ramp up (ramp down) steepness. The parameters $s=\exp(-1/2^{p})$ is a scaling factor to insure zero coupling at $x=0,L$ and maximum coupling at $x=L/2$. Here $m_0$ corresponds to the maximum modulated inductance. The implemented ramp function is depicted in Fig.\,\ref{fig1}c inset (green curve). Practically, the ramping implementation can be done by varying the effective mutual coupling between pump line and SQUID loops as schematically illustrated in Fig.\,\ref{fig2}a. Therefore, we have all essential components for the implementation of the proposed adiabatic scheme with a dynamics governed by Eq.\,\eqref{ode4by4}. By terminating odd mode at both ends of the device (shown in Fig.\,\ref{fig1}a), and given the circuit parameters listed in Table\,1, it gives more than 20 dB isolation over an octave of bandwidth as depicted in Fig.\,\ref{fig1}d (green curve). \emph{adiabaticity analysis}---We quantify the adiabaticity of the evolution by calculating the parameter $\theta$ which is the angle between the state of system (the magenta arrow in Fig.\,\ref{fig3}c) and the instantaneous eigenstate (red line in Fig.\,\ref{fig3}c) at any given position $x$. Ideally in a full adiabatic evolution, $\theta$ is zero throughout the conversion which means system always follows its instantaneous eigenstate. In Fig.\,\ref{fig3}d we plot $\theta$ for different signal frequencies which stays close to zero for signals near 6 GHz and deviates significantly for 4 and 8 GHz as we expected from result in Fig.\,\ref{fig1}d. The angle at the end of the device $\theta_L=\theta|_{x=L}$ determines the isolation as the probability of finding the signal at mode E at the end of the device is $P_e\simeq \theta_L^2/4$. The dashed line in Fig.\,\ref{fig3}d indicates $\theta_L=0.2\, (rad.)$ the threshold angle below which we get more than 20 dB isolation. It is worth mentioning that, the final angle $\theta_L$ can be estimated by a simple geometrical analysis which is used for error estimation in adiabatic two qubit gates \cite{martinis2014fast}, \begin{eqnarray}\label{geo_anly} \theta_{L} = -\int_0^L (\frac{ d \theta_{adi} }{d x}) dx \exp[-i \smallint g(x') dx'], \end{eqnarray} where $\theta_{adi} = \arctan[ \kappa(x)/ \Delta k_f(x)]$ and $g(x)=\sqrt{\kappa(x)^2 + \Delta k_f(x) ^2}$. For 20 dB isolation, the probability of the excitation at the end of the adiabatic evolution should be at most 1\% ($P_e<0.01$) so $\theta_L$ should be less that 0.2 radian. \begin{figure}[] \centering \includegraphics[width=0.98 \linewidth]{fig4_40.pdf} \caption{\textbf{Isolation performance vs device length:} \textbf{a.} The green (red) dotted, dashed and solid line show the isolation (transmission) performance of the device with L=800a, 2000a and 5000a respectively. \textbf{b.} The square, circular and diamond markers with dashed lines as guides to the eye, show the isolation bandwidth versus device lengths for mismatch sweep range $\alpha=$ 0.03 $a^{-1}$, 0.05 $a^{-1}$, 0.06 $a^{-1}$ respectively. \label{fig4}.} \end{figure} In Fig.\,\ref{fig4}a,we plot the isolation and transmission (insertion loss) performance for different device length $L$ (number of unit cells) given $\alpha=0.05\,a^{-1}$. Notably, the device with 800 unit cells provides over 2 GHz isolation bandwidth. The dashed lines are the estimated isolation using geometrical analysis (Eq.\,\ref{geo_anly}) which shows a great agreement with our result. The corresponding insertion loss plotted in the lower sub-panel in Fig.\,\ref{fig4}a is obtained by calculating the conversion of the mode E to mode O in backward direction and considering the typical dielectric loss in the superconducting qubit fabrications (loss tangent $ \delta=10^{-5}$). Figure~\ref{fig4}c shows the isolation bandwidth versus the device length with given parameters in table~1 for three different mismatch sweep range $\alpha$. For infinitely long device the isolation approaches to the corresponding frequency range for pump wavevector sweep (e.g. for $\alpha=0.05$ the isolation bandwidth approaches to $6.6$ GHz which is the highlighted frequency range in Fig.\,\ref{fig1}b). \emph{mode splitter and impedance considerations}--- The impedance of the even and odd modes can be independently set by circuit parameters to the desired value. In this letter, we choose the parameters so that each transmission lines have 50 $\Omega$ impedance. However, the impedance of the device is ultimately mandated by the termination configuration and the choice for the even/odd mode splitter at both ends of the device. For example, the odd mode can be blocked by a built-in broadband Wilkinson power divider. Alternatively one can use a broadband 180 hybrid couplers that split even and odd modes. The lumped element version of both types of terminations are readily available which can be naturally integrated with the device \cite{shen2009design,gao2013design}. \emph{conclusion}--- Superconducting circuits have been shown a promising capability for realizing novel quantum devices. However, the realization of a large scale quantum system requires integration of its essential components. Here we show that nonlinear processes in coupled transmission lines open new possibilities of controlling photon dispersion and engineering novel quantum devices. We proposed a practical scheme for on-chip broadband isolator using adiabatic techniques and dispersion engineering in coupled superconducting transmission lines. The presented idea can be readily applied to other platforms such as silicon photonics. \emph{acknowledgment}--- This work was funded in part by the AWS Center for Quantum Computing and by the MIT Research Support Committee from the NEC Corporation Fund for Research in Computers and Communications. Y. Ye acknowledges financial support from the MIT EECS Jin Au Kong fellowship. G. Cunningham acknowledges support from the Harvard Graduate School of Arts and Sciences Prize Fellowship. \bibliographystyle{unsrt} \section{Circuit diagram} \renewcommand{\thesection}{S\arabic{section}} \renewcommand{\thetable}{S\arabic{table}} \renewcommand{\thefigure}{S\arabic{figure}} \renewcommand{\theequation}{S\arabic{equation}} \setcounter{page}{1} \setcounter{figure}{0} \setcounter{equation}{0} Figure\,\ref{fig1s}, summarizes the details about the circuit diagram of the coupled Josephson junction transmission line that have been considered in this work. Panel a (Fig.\,\ref{fig1s}\,a) shows typical unit cell of two identical coupled transmission lines. The inductance in each line is predominantly provided by a pair of Josephson Junctions (JJ) forming a SQUID. Each JJ is characterized by its inductance $L_{J0}$ and the shunted capacitance $C_j$ as depicted in the panel (b). The JJ inductance is directly related to the critical current of the junction $L_{J0}= \phi_0/I_0$ where $\phi_0= \hbar/(2e)$ is reduced flux quantum. Here $C_0$ is the capacitance to the ground and $C_m$ and $L_m$ are mutual capacitance and inductance between two transmission lines. \begin{figure}[h] \centering \includegraphics[width=5 in]{suppl_circuit_diagram2.pdf} \caption{\textbf{Circuit diagrams} \label{fig1s}} \end{figure} In each unit-cell, a pair of identical JJs form a SQUID whose effective inductance $L_s$ depends on current $I$ passing each JJ and the external fluxes passing the SQUID loop, \begin{eqnarray}\label{squid_L_I_phi} L_s(I, \phi_{ext}) = \frac{L_{J0}}{2} \frac{1}{\sqrt{1-I^2/I_0^2}} \frac{1}{ | \cos(\frac{\phi_{ext}}{2 \phi_0}) | }. \end{eqnarray} As depicted in panel c (Fig.\,\ref{fig1s}\,c) Each SQUID receive a DC flux bias (shown in magenta) as well as a rf flux from a pump transmission line shown in green. The DC and rf pump transmission lines are linearly decoupled from the coupled transmission lines since they have different dispersion relations. In such situation, the entire effect of DC and rf transmission lines can be captured by their induced fluxes on the SQUID loops. We consider a configuration that in each unit cell the upper and lower SQUID receive DC bias in opposite direction but the rf flux is in-phase for both SQUIDs as illustrated in panel c (Fig.\,\ref{fig1s}\,c). Note that although in our illustration the pump transmission line has different distance to SQUID loops, in actuality the situation is symmetric and the effective mutual inductance between pump and SQUID loops $M_p$ are the same for both lines. Therefore the magnitude of the induces flux $\phi_{rf}$ is equal for both upper and lower SQUIDs within a unit cell. As shown in Eq.\, \ref{squid_L_I_phi}, the inductance of SQUID also depends on the current passing through the JJs. However, we consider weak signal regime where the current associated with propagating signals in upper/lower transmission lines are much smaller that critical current of JJs ($I \ll I_0 $) so that the nonlinear effects due to propagating currents are negligible. In such situation, the SQUID can be considered as an inductor whose inductance is modulated by external flux as shown in panel\,d (Fig.\,\ref{fig1s}\,d). Therefor, the effective inductance in upper and lower transmission line would be, \begin{eqnarray}\label{ind_L_m} L_s^{(u),(d)} = \frac{ 0.5 L_{J0} }{| \cos(\frac{\phi_{dc}}{2 \phi_0} \mp \frac{\phi_{rf}}{2 \phi_0}) | } \simeq \frac{ 0.5 L_J }{ \cos(\frac{\phi_{dc}}{2 \phi_0}) \pm \sin(\frac{\phi_{dc}}{2 \phi_0} ) \frac{\phi_{rf}}{2 \phi_0}} = \frac{L_{dc}}{1 \pm m}. \end{eqnarray} Where the unitless parameter $m=\tan(\frac{\Phi_{dc}}{2 \phi_0}) \frac{\phi_{ac}}{2 \phi_0}$ accounts for flux pumped induced modulation and $L_{dc}=0.5 L_{J0}/\cos(\frac{\phi_{dc}}{2\phi_0})$ is the DC-biased SQUID inductance. We consider DC flux bias in each SQUID to be $\phi_{dc}=2 \pi/3 \ \phi_0$ so we have $L_{dc}=L_{J0}$. The final circuit model of a typical unit cell is shown in panel e (Fig.\, \ref{fig1s}\,e). The proposed device can be modeled by cascading many of these unit cells. Note that all the circuit parameters are fixed throughout the device except the pump transmission line characteristics. We let $L_p$ and $C_p$ and the inductive coupling $Mp$ slowly vary versus position (unit cell number). These spatial variations are slow in the sense that the variations between two adjacent unitcell is negligible. These variations of pump characteristics are all captured in $\phi_{rf}$ in our model as shown in panel e (Fig.\, \ref{fig1s}\,e) . \section{Lagrangian analysis} The Lagrangian for a coupled transmission line can be written in terms of node-fluxes $\Phi_n^{(u)}$ and $\Phi_n^{(d)}$ (shown in Fig\,\ref{node_flux}). \begin{figure}[h] \centering \includegraphics[width=3.5 in]{suppl_lag_schem.pdf} \caption{\textbf{Node fluxes} \label{node_flux}} \end{figure} For n-th unit cell, the kinetic and potential energy would be \begin{eqnarray} T_n&=& + \frac{1}{2} C_0 \left( \dot{\Phi}_n^{(\mathrm{u})}\right)^2 + \frac{1}{2} C_s \left( \dot{\Phi}_{n+1}^{(\mathrm{u})} -\dot{\Phi}_n^{(\mathrm{u})} \right)^2 % % +\frac{1}{2} C_0 \left( \dot{\Phi}_n^{(\mathrm{d})}\right)^2 + \frac{1}{2} C_s \left( \dot{\Phi}_{n+1}^{(\mathrm{d})} -\dot{\Phi}_n^{(\mathrm{d})} \right)^2 % + \frac{1}{2} C_m \left(\dot{\Phi}_{n}^{(\mathrm{u})} -\dot{\Phi}_n^{(\mathrm{d})} \right)^2,\\ U_n&=& % +\frac{1}{2} \frac{1}{L_{s}^{\mathrm{(u)}}L_{s}^{\mathrm{(d)}} - L_m^2} \left[ L_{s}^{\mathrm{(d)}} ( \Phi_{n+1}^{(\mathrm{u})} - \Phi_{n}^{(\mathrm{u})})^2 + L_{s}^{\mathrm{(u)}} ( \Phi_{n+1}^{(\mathrm{d})} - \Phi_{n}^{(\mathrm{d})} )^2 - 2L_m ( \Phi_{n+1}^{(\mathrm{u})} -\Phi_{n}^{(\mathrm{u})} ) ( \Phi_{n+1}^{(\mathrm{d})} -\Phi_n^{(\mathrm{d})} ) \right]. \end{eqnarray} Where, for brevity, we dropped the explicit time dependencies for $\Phi_{n}^{(\mathrm{u,d})}$ and also $\phi_{ext}$ dependence for $L_s^{(\mathrm{u,d})}$. As we discussed in previous section, the inductance of SQUIDs in upper an lower transmission lines are modulated in opposite direction around $L_{dc}$ (See Eq. \ref{ind_L_m}). By considering Eq. \ref{ind_L_m} and using subscript $n$ for n-th unit cell we rewrite the potential energy as \begin{eqnarray} U_n&=& % \frac{1}{2} \frac{L_{dc}}{L_{dc}^2 - L_m^2} \left[ (1+m_n) \left( \Phi_{n+1}^{(\mathrm{u})} - \Phi_{n}^{(\mathrm{u})} \right)^2 + (1-m_n) \left( \Phi_{n+1}^{(\mathrm{d})} - \Phi_{n}^{(\mathrm{d})} \right)^2 - \frac{2 L_m}{L_{dc}} \left( \Phi_{n+1}^{(\mathrm{u})} -\Phi_{n}^{(\mathrm{u})} \right) \left( \Phi_{n+1}^{(\mathrm{d})} -\Phi_n^{(\mathrm{d})} \right) \right]. \nonumber \end{eqnarray} Where we keep terms in the first order of $m_n$ (weak modulation regime). We recast the kinetic and potential energies in the even and odd modes basis considering the following definition \begin{eqnarray} \Phi^{(i)}_n = \frac{\Phi^{(u)}_n \pm \Phi^{(d)}_n}{\sqrt{2}} \ , \ i \in \{ e, o\} \end{eqnarray} Then we have, \begin{eqnarray} T_n&=& + \frac{1}{2} C_e \left( \dot{\Phi}_n^{(e)}\right)^2 % % +\frac{1}{2} C_0 \left( \dot{\Phi}_n^{(o)}\right)^2 ,\\ U_n&=& % \frac{1}{2 L_{e}} \left( \Phi_{n+1}^{(e)} - \Phi_{n}^{(e)} \right)^2 + \frac{1}{2 L_{o}} \left( \Phi_{n+1}^{(o)} - \Phi_{n}^{(o)} \right)^2 + \frac{\mathbb{L}_m}{L_{e} L_{o}} \left( \Phi_{n+1}^{(e)} - \Phi_{n}^{(e)} \right) \left( \Phi_{n+1}^{(o)} - \Phi_{n}^{(o)} \right) . \end{eqnarray} where we keep only up to the first order in $\mathbb{L}_m = L_{dc} m_n$. Here $C_e=C_0$, $C_o= C_0+2 C_m$, $L_e=L_{dc}+L_m$, and $L_o= L_{dc}-L_m$ are corresponding capacitors and inductance in even and odd modes as depicted in the main text (Fig.\,2\,d). Note that we ignore the effect of $C_s$ as we have $\omega^2 L_{dc} C_s \ll 1 $ in the frequency range of interest. The full Lagrangian for the system would be $ \mathcal{L} = \sum_{n=1}^{N} \mathcal{L}_n $ where $ \mathcal{L}_n = T_n - U_n $. Now we use Euler-Lagrange relations \begin{eqnarray} \frac{\ \partial \,\mathcal{L}\ \ \ }{\partial \Phi_{n}^{(\mathrm{i})} }=\frac{d}{dt} \left( \frac{\ \partial \,\mathcal{L}\ \ \ }{\partial \dot{\Phi}_{n}^{(\mathrm{i})}} \right). \end{eqnarray} to arrive equations-of-motion for node fluxes, \begin{small} \begin{eqnarray} \frac{1}{L_{i}} \left( 2 \Phi_{n}^{(i)} - \Phi_{n+1}^{(i)} - \Phi_{n-1}^{(i)} \right) +\frac{L_{dc}}{L_{i} L_j} \left( - m_n ( \Phi_{n+1}^{(i)} - \Phi_{n}^{(i)}) + m_{n-1} ( \Phi_{n}^{(i)} - \Phi_{n-1}^{(i)}) \right)= C_i \ddot{\Phi}_n^{(i)}. \end{eqnarray} \end{small} Note that the conjugate momentum (node-charge) associated with the node-flux can be calculates as \begin{eqnarray}\label{nodecharges} \Pi^{(i)}_n = \frac{\partial }{\partial \dot{\Phi}^{(i)}_n} \mathcal{L} = C_i \dot{\Phi}^{(i)}_n \end{eqnarray} To proceed further, we go to the continuum limit (assuming $a k_i \ll 2 \pi,\ i=\{e,o,p\}$) and drop subscript $n$ and define derivatives (e.g. $ \Phi_n - \Phi_{n-1}\rightarrow a \Phi_{x}$ and $ \Phi_{n+1} + \Phi_{n-1} - 2 \Phi_n \rightarrow a^2 \Phi_{xx}$ ) where $a$ is the unit cell length. Hereafter, we use the convention that subscripts $x$ and $t$ represents the derivatives. In the continuum limit we have \begin{eqnarray}\label{ode0s_lag} \Phi^{(i)}_{tt} - \omega_{0i}^2 \Phi^{(i)}_{xx} =\beta_{ij}^2 ( m_x \Phi^{(j)}_x + m \Phi^{(j)}_{xx} ). \end{eqnarray} Here $\omega_{0i}=1/\sqrt{L_i C_i}$\, and $\beta^2_{ij} = \omega_{0i} L_{dc}/ L_j$ where $i \in \{ e ,o\} , i \neq j$ and we set $a=1$ for brevity. We characterize the linear property of the system by analyzing Eq.\,\ref{ode0s_lag} in the absence of the pump modulation ($m \rightarrow 0$). In this situation the right hand sides of \ref{ode0s_lag} vanishes and the remaining equations describes linear property of the system which can be solved to obtain dispersion relations of each mode. For that, we assume $\Phi^{(i)}=\Phi^{(i)}(x,t)=\phi^{(i)}(0) e^{-i(\omega_i t - k_i x))} + c.c$ and obtain, \begin{eqnarray}\label{KEKO_s} k_i= \frac{\omega_i/\omega_{0i}}{\sqrt{1- (\frac{\omega_i}{\omega_{0J}})^2 }}\simeq \omega_i/\omega_{0i} = \sqrt{L_i C_i} \, \omega_i \ , \ i \in \{ e , o ,\}. \end{eqnarray} Here $\omega_{0i}=1/\sqrt{C_i L_i}$, and $\omega_{Ji}=1/\sqrt{C_s L_i}$ are cut-off frequencies and plasma frequency respectively. In order to proceed with nonlinear analysis, it would be convenient to rearrange Eq.\ref{ode0s_lag} as, \begin{eqnarray}\label{ode0s_lag_r} \partial_t \left( \Phi^{(i)}_{t} \right)= \partial_x \left( \omega_{0i}^2 \Phi^{(i)}_{x} + \beta_{ij}^2 m(x,t) \Phi^{(j)}_x \right). \end{eqnarray} Now we define the voltage $V_i = \Phi^{(i)}_t $ and current $I_i= - \frac{1}{L_i} \Phi^{(i)}_{x} - \frac{L_{dc}}{L_i L_j} m(x,t) \Phi^{(j)}_x$ and rearrange terms (keeping terms up to the first order in $m(x,t)$) to have, \begin{eqnarray} \frac{\partial}{\partial x} V_i &=& - L_i \dot{I}_i + \frac{\partial }{\partial t} \left( \mathbb{L}_m(x,t) I_j \right),\\ \frac{\partial }{\partial x} I_i &=& - C_i \dot{V}_i , \end{eqnarray} where, $\mathbb{L}_m(x,t)= L_{dc} m(x,t)$ is the induced-inductive coupling. Therefore, we arrived at equations of motion consistent with the coupled mode theory analysis \cite{orfanidis2002electromagnetic}. For clarity we write down all four ODEs explicitly, \begin{subequations}\label{odesVI} \begin{eqnarray} \frac{d}{dx} V_e(x,t) &=& - L_{e} \frac{d}{dt} I_e(x,t) + \frac{d}{dt} \left[ \mathbb{L}_m(x,t) I_o(x,t) \right] \\ \frac{d}{dx} V_o(x,t) &=& - L_{o} \frac{d}{dt} I_o(x,t) + \frac{d}{dt} \left[ \mathbb{L}_m(x,t) I_e(x,t) \right] \\ \frac{d}{dx} I_e(x,t) &=& - C_e \frac{d}{dt} V_e(x,t) \\ \frac{d}{dx} I_o(x,t) &=& - C_o \frac{d}{dt} V_o(x,t), \end{eqnarray} \end{subequations} Therefore, in even and odd mode basis we have two otherwise decoupled transmission lines where the inductive coupling $\mathbb{L}_m(x,t)=L_{dc}m(x,t)$ is induced by external flux pump. Now we recast the voltage and current in form of forward and backward propagating signals given the following relationships, \begin{subequations} \label{FB_waves} \begin{eqnarray} E_{f/b}(x,t) = \frac{V_e(x,t) \pm Z_e I_e(x,t)}{2} , \\ O_{f/b}(x,t) = \frac{V_o(x,t) \pm Z_o I_o(x,t)}{2}, \end{eqnarray} \end{subequations} where $Z_i=\sqrt{L_i/C_i}, i \in \{e,o \}$. In order to proceed, we consider the flowing ansatze for even and mode signal propagating in both directions \begin{subequations}\label{EO_anzats} \begin{eqnarray} E_{f/b}(x,t) = E_{f/b}(x) e^{-i(\omega_e t \mp k_e x)}+ c.c.\\ O_{f/b}(x,t) = O_{f/b}(x) e^{-i(\omega_o t \mp k_o x)}+c.c., \end{eqnarray} \end{subequations} and consider a traveling wave form for the induced coupling \begin{equation}\label{mxt} \mathbb{L}_m(x,t)= L_{dc}m(x,t)=L_{dc}m(x)e^{-i( \omega_p t - \int k_p dx)} + c.c. \end{equation} where $k_p$ is the instantaneous pump wave vector and $\int k_p dx$ accounts for accumulated phase due to pump phase wavevector. Substituting Eq\,\ref{EO_anzats} and Eq.\,\ref{mxt} in Eq.\,\ref{odesVI}, we have the following form for the dynamical matrix $\mathbb{M}$, \begin{small} \begin{eqnarray} \mathbb{M} = \begin{pmatrix} 0 & \frac{i L_{dc} m(x) \omega_e e^{-i \Delta K_f(x)}}{2Z_o} & 0 & -\frac{iL_{dc} m(x) \omega_e e^{-i (\Delta K_f(x) - 2 k_o x )}}{2Z_o} \\ \frac{i L_{dc} m^*(x) \omega_o e^{i \Delta K_f(x)}}{2Z_e}& 0 &-\frac{i L_{dc} m^*(x) \omega_o e^{i (\Delta K_f(x)+2k_e)}}{2Z_e}& 0 \\ 0 & \frac{i L_{dc} m(x) \omega_e e^{-i (\Delta K_f(x) + 2 k_e x )}}{2Z_o}& 0 & -\frac{i L_{dc} m(x) \omega_e e^{-i (\Delta K_f(x) - 2 (k_o-k_e) x )}}{2Z_o} \\ % \frac{i L_{dc} m^*(x) \omega_o e^{i (\Delta K_f(x)-2k_o)}}{2Z_e} & 0 & -\frac{i L_{dc} m^*(x) \omega_o e^{i (\Delta K_f(x)-2(k_o-k_e)x)}}{2Z_e}& 0 \end{pmatrix} \end{eqnarray} \end{small} where we applied RWA in time domain. Here $\Delta K_f(x) = (k_o-k_e)x -\int{k_p dx}$ is accumulated phase mismatch. We may recast the obtained coupled equations it in a rotating frame corresponding to the following transformation $E_{f}(x) \rightarrow E_{f}(x) e^{-i \Delta K_f(x) /2}, O_{f}(x) \rightarrow O_{f}(x) e^{+i \Delta K_f(x) /2}$ and $E_{b}(x) \rightarrow E_{b}(x) e^{-i \Delta K_f(x)/2+(k_o-k_e)x}, O_{b}(x) \rightarrow O_{b}(x) e^{+i \Delta K_f(x) /2 - (k_o-k_e)x}$ to obtain, \begin{small} \begin{eqnarray} \label{full4by4} \mathbb{M} = \begin{pmatrix} -\frac{i}{2}\Delta k_f(x) & -\frac{i L_{dc} m(x) \omega_e }{2Z_o} & 0 & \frac{iL_{dc} m(x) \omega_e }{2Z_o} e^{i (k_o+k_e) x } \\ -\frac{i L_{dc} m(x) \omega_o }{2Z_e}& \frac{i}{2} \Delta k_f(x) &\frac{i L_{dc} m(x) \omega_o e^{i (k_o+k_e)}}{2Z_e}& 0 \\ 0 & -\frac{i L_{dc} m(x) \omega_e e^{-i (k_o+k_e) x }}{2Z_o}& -\frac{i}{2} \Delta k_f(x)+i(k_o-k_e) & \frac{i L_{dc} m(x) \omega_e }{2Z_o} \\ % -\frac{i L_{dc} m(x) \omega_o e^{-i (k_o+k_e)}}{2Z_e} & 0 & \frac{i L_{dc} m(x) \omega_o}{2Z_e}& \frac{i}{2}\Delta k_f(x) - i(k_o-k_e) \end{pmatrix}. \end{eqnarray} \end{small} We may also simplified the equations even further to show its correspondence to the simple model presented in the main text. For that, we perform RWA in space and ignore highly oscillatory terms in space. With this approximation the forward and backward dynamics decouple. \begin{small} \begin{eqnarray} \label{simple4by4} \mathbb{M} = \begin{pmatrix} -\frac{i}{2}\Delta k_f(x) & -\frac{i L_{dc} m(x) \omega_e }{2Z_o} & 0 & 0 \\ -\frac{i L_{dc} m(x) \omega_o }{2Z_e}& \frac{1}{2} \Delta k_f(x) &0& 0 \\ 0 & 0 & -\frac{i}{2} \Delta k_f(x)+i(k_o-k_e) & +\frac{i L_{dc} m(x) \omega_e }{2Z_o} \\ % 0 & 0 & \frac{i L_{dc} m(x) \omega_o}{2Z_e}& \frac{i}{2}\Delta k_f(x) - i(k_o-k_e) \end{pmatrix} \end{eqnarray} \end{small} It is convenient to represent signals in terms of photon number basis which is conserved during the conversion for signals (tracing out the pump). So we apply the following transformation to go from voltage signal to photon number, $E_{f/b}(x) \rightarrow 1/\sqrt{Z_e \omega_e} E(x), O_{f/b}(x) \rightarrow 1/\sqrt{Z_o \omega_o} O_{f/b}(x) $ where now $| E_{f/b}(x)|^2$ and $| O_{f/b}(x)|^2$ are proportional to average number of photon propagating in each mode. \begin{small} \begin{eqnarray} \frac{d}{dx}\begin{pmatrix} E_f(x) \\ O_f(x)\\ E_b(x)\\ O_b(x) \end{pmatrix} = \begin{pmatrix} -\frac{i}{2}\Delta k_f(x) & -\frac{i L_{dc} m(x) \sqrt{\omega_e \omega_o} }{2\sqrt{Z_e Z_o}} & 0 & 0 \\ -\frac{i L_{dc} m(x) \sqrt{\omega_e \omega_o} }{2\sqrt{Z_e Z_o}}& \frac{1}{2} \Delta k_f(x) &0& 0 \\ 0 & 0 & -\frac{i}{2} \Delta k_f(x)+i(k_o-k_e) & \frac{i L_{dc} m(x) \sqrt{\omega_e \omega_o}}{2\sqrt{Z_e Z_o}} \\ % 0 & 0 & \frac{i L_{dc} m(x) \sqrt{\omega_e \omega_o}}{2\sqrt{Z_e Z_o}}& \frac{i}{2}\Delta k_f(x) - i(k_o-k_e) \end{pmatrix} . \begin{pmatrix} E_f(x) \\ O_f(x)\\ E_b(x)\\ O_b(x) \end{pmatrix} \end{eqnarray} \end{small} The obtained equation is corresponds to the simple model introduced in the main text with $\kappa(x) = \frac{\sqrt{\omega_e \omega_o}}{\sqrt{Z_e Z_o}} L_{dc} m(x)$ and $\Delta k_b(x) = 2(k_o -k_e)-\Delta k_f(x)$ which results in a poor phase mismatch for backward traveling signals. \section{Spatial rotating wave approximation verification} In the main text we perform spatial RWA and ignore the fast spatial oscillatory terms in the dynamical matrix (Eq.\,\ref{simple4by4}, or Eq.\,9 in the main text). With this approximation the forward and backward signals decouple as introduced in the simple model. Here we verify that this approximation is indeed valid by solving the full dynamical matrix (Eq.\,\ref{full4by4}) and comparing the result with the solution of the Eq.\,\ref{simple4by4}. \begin{figure}[h] \centering \includegraphics[width=3 in]{RWA_spatial_compare.pdf} \caption{\textbf{Isolation performance with/without spatial RWA} The upper (lower) panel compare the transmission in forward (backward) direction with/without performing spatial rotating wave approximation. \label{RWA_verigication}} \end{figure}

proofpile-arXiv_069-9316

\section{Introduction} Self-dual codes over finite fields, especially binary fields, have been one of the most important and widely studied subject in algebraic coding theory. Self-dual codes over rings have been shown to have many interesting connections to invariant theory, lattice theory and the theory of modular forms, see \cite{Gebe}. Classification of binary self-dual codes of different lengths is still an ongoing research area in algebraic coding theory. In this context, various techniques have been employed to search for binary self-dual codes of different lengths with new parameters in their weight enumerators. A generator matrix of the form $[I_n \ | \ A_n],$ where $I_n$ is the $n \times n$ identity matrix and $A_n$ is some $n \times n$ matrix with entries from a finite field $\mathbb{F}_2$ is the most common technique for producing extremal binary self-dual codes. Later, this technique has been extended in \cite{Gildea1}, by replacing the matrix $A$ with $\sigma(v)$, where $\sigma(v)$ is the image of a unitary unit in a group ring under a map that sends group ring elements to matrices that are fully defined by the elements appearing in the first row. One can see \cite{Dougherty3,Dougherty4,Dougherty5} for more applications of this technique. In \cite{CompositeGCodes}, the map $\sigma(v)$ was extended to $\Omega(v)$, which gives more complex matrices over the ring $R,$ and generator matrices of the form $[I_n \ | \ \Omega(v)]$ were considered. Recently in \cite{Dougherty3}, the map $\sigma$ was extended $\tau_k$ by considering elements from the group matrix ring $M_k(R)G$ rather than elements from the group ring $RG$, and generator matrices of the form $[I_{kn} \ | \tau_k(v)]$ were considered to produce binary self-dual codes. Please see \cite{Dougherty3} for the details of this construction. The advantage of the map $\tau_k$ is that it does not only depend on the choice of the group $G,$ but it also depends on the form of the elements from the matrix ring $M_k(R),$ that is, the form of the $k \times k$ matrices over $R.$ Metaheuristic optimization algorithms have been widely used successfully for many engineering problems in which solution steps take exhaustively long time \cite{Ustun, Karaboga3, Liang, Holland, Deniz3}. In terms of algebraic coding theory, optimization algorithms were used only for the minimum distance problem \cite{Shenoy,Bland1, Bland2, Bouzkraoui}. Searching for self-dual codes with new parameters is one of the main problem in algebraic coding theory and linear search was the only search tool for this problem. Although linear search for self-dual codes achieve good results for small size search space, this is really time consuming when the search field grows. Therefore, the problem of searching for extremal self-dual codes needs some new tools for the big size search space. Recently, in \cite{Korban1}, the authors employed the genetic algorithm (GA), one of the well-known optimization algorithm, for the first time to the problem of finding new binary self-dual codes. It was proved that GA copes with large search fields significantly better and finds codes much faster then the standard linear search. In this work, a virus optimization algorithm (VOA) \cite{Cuevas} is employed for the first time to the problem of finding extremal binary self-dual codes of length 72. We consider the generator matrices of the form $[I_{36} \ | \tau_3(v)]$ together with the VOA and search for binary self-dual codes with parameters $[72,36,12].$ We find such codes with weight enumerators that were not known in the literature before. It is known by \cite{Liang} that VOA outperforms the GA in continuous problems. Therefore we compare the number of codes found by VOA and GA for the same constructions and obtain that VOA finds more binary self-dual codes than GA. The rest of the work is organized as follows. In Section~2, we give preliminary definitions and results on self-dual codes, group rings and recall the map $\tau_k(v)$ that was defined in \cite{Dougherty3}. In Section~3, we present a number of generator matrices of the form $[I_{36} \ | \ \tau_3(v)]$ and for each generator matrix, we fix the $3 \times 3$ matrices by letting them be some special matrices that we define in Section~2. In Section~4, we give a brief history of VOA and explain how to employ this algorithm to the problem of finding extremal binary self-dual codes. In Section~5, we employ the generator matrices from Section~4 and search for binary self-dual codes with parameters $[72,36,12].$ As a result we find 39 Type I and 19 Type II binary $[72,36,12]$ self-dual codes with parameters in their weight enumerators that were not previously known. We tabulate our results, stating clearly the parameters of the obtained codes and their orders of the automorphism group. We finish with concluding remarks and directions for possible future research. \section{Preliminaries} In this section we recall some well-known definitions and notions from algebraic coding theory. A code $C$ of length $n$ over a Frobenius ring $R$ is a subset of $R^n$. If the code is a submodule of $R^n$ then we say that the code is linear. Let $\mathbf{x}=(x_1,x_2,\dots,x_n)$ and $\mathbf{y}=(y_1,y_2,\dots,y_n)$ be two elements of $R^n.$ Then \begin{equation*} \langle \mathbf{x},\mathbf{y} \rangle_E=\sum x_iy_i. \end{equation*} The dual $C^{\bot}$ of the code $C$ is defined as \begin{equation*} C^{\bot}=\{\mathbf{x} \in R^n \ | \ \langle \mathbf{x},\mathbf{y} \rangle_E=0 \ \text{for all} \ \mathbf{y} \in C\}. \end{equation*} We say that $C$ is self-orthogonal if $C \subseteq C^\perp$ and is self-dual if $C=C^{\bot}.$ An upper bound on the minimum Hamming distance of a binary self-dual code was given in \cite{Rains1}. Let $d_{I}(n)$ and $d_{II}(n)$ be the minimum distance of a Type~I (singly-even) and Type~II (doubly-even) binary code of length $n$, respectively. Then \begin{equation*} d_{II}(n) \leq 4\lfloor \frac{n}{24} \rfloor+4 \end{equation*} and \begin{equation*} d_{I}(n)\leq \begin{cases} \begin{matrix} 4\lfloor \frac{n}{24} \rfloor+4 \ \ \ if \ n \not\equiv 22 \pmod{24} \\ 4\lfloor \frac{n}{24} \rfloor+6 \ \ \ if \ n \equiv 22 \pmod{24}.% \end{matrix}% \end{cases}% \end{equation*} Self-dual codes meeting these bounds are called \textsl{extremal}. A circulant matrix is one where each row is shifted one element to the right relative to the preceding row and a reverse circulant matrix is one where each row is shifted one element to the left relative to the preceding row. We label the circulant matrix as $A=circ(\alpha_1,\alpha_2\dots , \alpha_n),$ and the reverse circulant matrix as $A=revcirc(\alpha_1,\alpha_2\dots , \alpha_n),$ where $\alpha_i$ are ring elements. The transpose of a matrix $A,$ denoted by $A^T,$ is a matrix whose rows are the columns of $A,$ i.e., $A^T_{ij}=A_{ji}.$ Let $G$ be a finite group of order $n$, then the group ring $RG$ consists of $\sum_{i=1}^n \alpha_i g_i$, $\alpha_i \in R$, $g_i \in G.$ Addition in the group ring is done by coordinate addition, namely \begin{equation} \sum_{i=1}^n \alpha_i g_i +\sum_{i=1}^n \beta_i g_i =\sum_{i=1}^n (\alpha_i + \beta_i)g_i. \end{equation} The product of two elements in a group ring is given by \begin{equation} \left(\sum_{i=1}^n \alpha_i g_i \right)\left(\sum_{j=1}^n \beta_j g_j \right)= \sum_{i,j} \alpha_i \beta_j g_i g_j. \end{equation} It follows that the coefficient of $g_k$ in the product is $\sum_{g_i g_j=g_k} \alpha_i \beta_j.$ We now recall the map $\tau_k(v),$ where $v \in M_k(R)G$ and where $M_k(R)$ is a non-commutative Frobenius matrix ring and $G$ is a finite group of order $n,$ that was introduced in \cite{Dougherty3}. Let $v=A_{g_1}g_1+A_{g_2}g_2+\dots+A_{g_n}g_n \in M_k(R)G,$ that is, each $A_{g_i}$ is a $k \times k$ matrix with entries from the ring $R.$ Define the block matrix $\sigma_k(v) \in (M_{k}(R))_n$ to be \begin{equation}\label{sigmakv} \sigma_k(v)= \begin{pmatrix} A_{g_1^{-1}g_1} & A_{g_1^{-1}g_2} & A_{g_1^{-1}g_3} & \dots & A_{g_1^{-1}g_{n}} \\ A_{g_2^{-1}g_1} & A_{g_2^{-1}g_2} & A_{g_2^{-1}g_3} & \dots & A_{g_2^{-1}g_{n}} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ A_{g_{n}^{-1}g_1} & A_{g_{n}^{-1}g_2} & A_{g_{n}^{-1}g_3} & \dots & A_{g_{n}^{-1}g_{n}} \end{pmatrix} . \end{equation} We note that the element $v$ is an element of the group matrix ring $M_k(R)G.$ {\bf Construction 1} For a given element $v \in M_k(R)G,$ we define the following code over the matrix ring $M_k(R)$: \begin{equation} C_k(v)=\langle \sigma_k(v) \rangle. \end{equation} Here the code is generated by taking the all left linear combinations of the rows of the matrix with coefficients in $M_k(R).$ {\bf Construction 2} For a given element $v \in M_k(R)G,$ we define the following code over the ring $R$. Construct the matrix $\tau_k(v)$ by viewing each element in a $k$ by $k$ matrix as an element in the larger matrix. \begin{equation} B_k(v)=\langle \tau_k(v) \rangle. \end{equation} Here the code $B_k(v)$ is formed by taking all linear combinations of the rows of the matrix with coefficients in $R$. In this case the ring over which the code is defined is commutative so it is both a left linear and right linear code. Later in this work, we employ this map and consider generator matrices of the form $[I_{kn} \ | \ \tau_k(v)]$ for groups of order $12$ and for $k=3.$ That is, we consider $3 \times 3$ matrices of different forms. \section{Main Construction} Let $v \in M_3(\mathbb{F}_2)G$. In this work we consider a generator matrix of the form $$G=[I_{36} \ | \ \tau_3(v)],$$ where $G$ is a group of order $12$ and $\tau_3(v)$ is the $36 \times 36$ matrix that consists of some 12 block matrices of size $3 \times 3$. For this work, we only consider the following cases: \begin{equation}\label{nine} \begin{aligned} A_1=circ(a_1,a_2,a_3),\dots, A_{12}=circ(a_{34},a_{35},a_{36}). \end{aligned} \end{equation} \begin{equation}\label{two} A_1=revcirc(a_1,a_2,a_3), \dots, A_{12}=revcirc(a_{34},a_{35},a_{36}). \begin{aligned} \end{aligned} \end{equation} \begin{equation}\label{five} \begin{aligned} A_1=revcirc(a_1,a_2,a_3), \dots, A_6=revcirc(a_{16},a_{17}, a_{18}), \\ A_7=circ(a_{19},a_{20}, a_{21}), \dots, A_{12}=circ(a_{34},a_{35},a_{36}). \end{aligned} \end{equation} \begin{equation}\label{one} \begin{aligned} A_1=circ(a_1,a_2,a_3), \dots, A_6=circ(a_{16},a_{17}, a_{18}), \\ A_7=revcirc(a_{19},a_{20}, a_{21}), \dots A_{12}=revcirc(a_{34},a_{35},a_{36}) \end{aligned} \end{equation} \begin{equation}\label{three} \begin{aligned} A_1=circ(a_1,a_2,a_3),\dots, A_{3}=circ(a_{7},a_{8}, a_{9}),\\ A_4=revcirc(a_{10},a_{11}, a_{12}), \dots, A_{6}=revcirc(a_{16},a_{17}, a_{18}),\\ A_7=circ(a_{19},a_{20}, a_{21}),\dots, A_{9}=circ(a_{25},a_{26}, a_{27}),\\ A_{10}=revcirc(a_{28},a_{29}, a_{30}), \dots, A_{12}=revcirc(a_{34},a_{35}, a_{36}). \end{aligned} \end{equation} \begin{equation}\label{four} \begin{aligned} A_1=revcirc(a_1,a_2,a_3), A_{2}=revcirc(a_4,a_5,a_6),\\ A_{3}=circ(a_{7},a_{8}, a_{9}), A_4=circ(a_{10},a_{11}, a_{12}),\\ A_{5}=revcirc(a_{13},a_{14}, a_{15}), A_{6}=revcirc(a_{16},a_{17}, a_{18}),\\ A_7=circ(a_{19},a_{20}, a_{21}), A_{8}=circ(a_{22},a_{23}, a_{24}), \\ A_{9}=revcirc(a_{25},a_{26}, a_{27}), A_{10}=revcirc(a_{28},a_{29}, a_{30}),\\ A_{11}=circ(a_{31},a_{32}, a_{33}), A_{12}=circ(a_{34},a_{35}, a_{36}).\\ \end{aligned} \end{equation} \begin{equation}\label{six} \begin{aligned} A_1=revcirc(a_1,a_2,a_3), A_{2}=circ(a_4,a_5,a_6),\\ A_{3}=revcirc(a_{7},a_{8}, a_{9}), A_4=circ(a_{10},a_{11}, a_{12}),\\ A_{5}=revcirc(a_{13},a_{14}, a_{15}), A_{6}=circ(a_{16},a_{17}, a_{18}),\\ A_7=revcirc(a_{19},a_{20}, a_{21}), A_{8}=circ(a_{22},a_{23}, a_{24}), \\ A_{9}=revcirc(a_{25},a_{26}, a_{27}), A_{10}=circ(a_{28},a_{29}, a_{30}),\\ A_{11}=revcirc(a_{31},a_{32}, a_{33}), A_{12}=circ(a_{34},a_{35}, a_{36}).\\ \end{aligned} \end{equation} \begin{equation}\label{seven} \begin{aligned} A_1=circ(a_1,a_2,a_3), A_{2}=revcirc(a_4,a_5,a_6),\\ A_{3}=circ(a_{7},a_{8}, a_{9}), A_4=revcirc(a_{10},a_{11}, a_{12}),\\ A_{5}=circ(a_{13},a_{14}, a_{15}), A_{6}=revcirc(a_{16},a_{17}, a_{18}),\\ A_7=circ(a_{19},a_{20}, a_{21}), A_{8}=revcirc(a_{22},a_{23}, a_{24}), \\ A_{9}=circ(a_{25},a_{26}, a_{27}), A_{10}=revcirc(a_{28},a_{29}, a_{30}),\\ A_{11}=circ(a_{31},a_{32}, a_{33}), A_{12}=revcirc(a_{34},a_{35}, a_{36}).\\ \end{aligned} \end{equation} \begin{equation}\label{eight} \begin{aligned} A_1=circ(a_1,a_2,a_3), A_{2}=circ(a_4,a_5,a_6),\\ A_{3}=revcirc(a_{7},a_{8}, a_{9}), A_4=revcirc(a_{10},a_{11}, a_{12}),\\ A_{5}=circ(a_{13},a_{14}, a_{15}), A_{6}=circ(a_{16},a_{17}, a_{18}),\\ A_7=revcirc(a_{19},a_{20}, a_{21}), A_{8}=revcirc(a_{22},a_{23}, a_{24}),\\ A_{9}=circ(a_{25},a_{26}, a_{27}), A_{10}=circ(a_{28},a_{29}, a_{30}),\\ A_{11}=revcirc(a_{31},a_{32}, a_{33}), A_{12}=revcirc(a_{34},a_{35}, a_{36}).\\ \end{aligned} \end{equation} We only tried limited cases of these twelve matrices since there are a lot of choices for this. We remark that, one can also consider some other combinations of these twelve $A_i$ matrices. \subsection{The Dihedral Group $D_{12}$} Here, we consider the dihedral group of order 12 $$D_{12}=\langle a,b \ | \ b^6=a^2=1, ab=b^{-1}a \rangle$$ and 2 different ordering of the group elements. \\ \textbf{Case 1:}\\ Let $v_1=\sum_{i=0}^5 \sum_{j=0}^1 \alpha_{a^ib^j}a^ib^j \in M_3(\mathbb{F}_2)D_{12},$ and consider the ordering of the elements of the group as $$1, b^{5}, b^{4}, \dots, b^{1}, a, ab, \dots, ab^{6}, $$ then\\ \begin{equation} \tau_3(v_1)=\begin{pmatrix} A&B\\ B^T&A^T \end{pmatrix}, \end{equation} where $A=CIRC(A_1,A_2,A_3,\dots,A_6), B=CIRC(A_{7},A_{11},A_{12},\dots,A_{12})$. We define the following generator matrices; $$ \mathcal{G}_1^{1}=[I_{36} \ | \ \tau_3(v_1)], $$ where $A_i$'s come from Equation \ref{one}. $$ \mathcal{G}_1^{2}=[I_{36} \ | \ \tau_3(v_1)], $$ where $A_i$'s come from Equation \ref{two}. \textbf{Case 2:}\\ Let $v_2=\sum_{i=0}^5 \sum_{j=0}^1 \alpha_{a^ib^j}a^ib^j \in M_3(\mathbb{F}_2)D_{12}$ and consider the ordering of the elements of the group as $$1, b, b^{2}, \dots, b^{5}, a, ab, \dots, ab^{5}, $$ then\\ \begin{equation} \sigma_3(v_2)=\begin{pmatrix} A&B\\ B&A \end{pmatrix}, \end{equation} where $ A=CIRC(A_{1}, A_{2}, \dots, A_{6} ),$ $ B=REVCIRC(A_{7}, A_{11}, \dots, A_{12} ) $. We define the following generator matrices;\\ $$ \mathcal{G}_2^{1}=[I_{36} \ | \ \tau_3(v_2)], $$ where $A_i$'s come from Equation \ref{three}. $$ \mathcal{G}_2^{2}=[I_{36} \ | \ \tau_3(v_2)], $$ where $A_i$'s come from Equation \ref{four}. $$ \mathcal{G}_2^{3}=[I_{36} \ | \ \tau_3(v_2)], $$ where $A_i$'s come from Equation \ref{five}. $$ \mathcal{G}_2^{4}=[I_{36} \ | \ \tau_3(v_2)], $$ where $A_i$'s come from Equation \ref{six}. $$ \mathcal{G}_2^{5}=[I_{36} \ | \ \tau_3(v_2)], $$ where $A_i$'s come from Equation \ref{seven}. $$ \mathcal{G}_2^{6}=[I_{36} \ | \ \tau_3(v_2)], $$ where $A_i$'s come from Equation \ref{eight}. \subsection{The Cyclic Group $C_{12}$} Let $C_{12}$ be the cyclic group of order 12 and $v_3 \in M_3(\mathbb{F}_2)C_{12}$. We consider the following two matrix representations; \\ \textbf{Case 1:} \begin{equation} \sigma_3(v_3)=\begin{pmatrix} A&B\\ B'&A \end{pmatrix}, \end{equation} where $A=CIRC(A_1,A_2,A_3,\dots,A_6), B=CIRC(A_{7},A_{11},A_{12},\dots,A_{12})$ and $B'=CIRC(A_{12},A_{7},A_{8},\dots,A_{11})$. We define the following generator matrices; $$ \mathcal{G}_3^{1}=[I_{36} \ | \ \tau_3(v_3)], $$ where $A_i$'s come from Equation \ref{nine}. $$ \mathcal{G}_3^{2}=[I_{36} \ | \ \tau_3(v_3)], $$ where $A_i$'s come from Equation \ref{two}. $$ \mathcal{G}_3^{3}=[I_{36} \ | \ \tau_3(v_3)], $$ where $A_i$'s come from Equation \ref{five}. $$ \mathcal{G}_3^{4}=[I_{36} \ | \ \tau_3(v_3)],$$ where $A_i$'s come from Equation \ref{one}. \textbf{Case 2:} \begin{equation} \sigma_3(v_4)=\begin{pmatrix} A&B&C&D\\ D'&A&B&C\\ C'&D'&A&B\\ B'&C'&D'&A \end{pmatrix}, \end{equation} where $A=CIRC(A_1,A_2,A_3), B=CIRC(A_{4},A_{5},A_{6}),$ $B'=CIRC(A_{6},A_{4},A_{5})$, $C=CIRC(A_{7}, A_{8}, A_{9}), C'=CIRC(A_{9}, A_{7}, A_{8}),$ $D=CIRC(A_{10},A_{11},A_{12})$ and $D=CIRC(A_{12},A_{10},A_{11}).$ We define the following generator matrices; $$ \mathcal{G}_4^{1}=[I_{36} \ | \ \tau_3(v_4)], $$ where $A_i$'s come from Equation \ref{three}. $$ \mathcal{G}_4^{2}=[I_{36} \ | \ \tau_3(v_4)], $$ where $A_i$'s come from Equation \ref{two}. $$ \mathcal{G}_4^{3}=[I_{36} \ | \ \tau_3(v_4)], $$ where $A_i$'s come from Equation \ref{nine}. \subsection{The Group $C_{6}\times C_{2}$} Let $C_{6}\times C_{2}$ be the cross product of cyclic groups of order 6 and 2. Then for $v_5 \in M_3(\mathbb{F}_2)(C_{6}\times C_{2})_{12},$ we have\\ \begin{equation} \sigma_3(v_5)=\begin{pmatrix} A&B\\ B&A \end{pmatrix}, \end{equation} where $A=CIRC(A_1,A_2,A_3,\dots,A_6), B=CIRC(A_{7},A_{11},A_{12},\dots,A_{12})$. We now define the following generator matrices; $$ \mathcal{G}_5^{1}=[I_{36} \ | \ \tau_3(v_5)], $$ where $A_i$'s come from Equation \ref{one}. $$ \mathcal{G}_5^{2}=[I_{36} \ | \ \tau_3(v_5)], $$ where $A_i$'s come from Equation \ref{three}. $$ \mathcal{G}_5^{3}=[I_{36} \ | \ \tau_3(v_5)], $$ where $A_i$'s come from Equation \ref{five}. \subsection{The Group $C_{3}\times C_{4}$} Let $C_{3}\times C_{4}$ be the cross product of cyclic groups of order 3 and 4. Then for $v_6 \in M_3(\mathbb{F}_2)(C_{3}\times C_{4})_{12},$ we have\\ \begin{equation} \sigma_3(v_6)=\begin{pmatrix} A&B&C&D\\ D&A&B&C\\ C&D&A&B\\ B&C&D&A \end{pmatrix}, \end{equation} where $A=CIRC(A_1,A_2, A_3), B=CIRC(A_{4},A_{5}, A_6), C=CIRC(A_{7},A_{8}, A_9), D=CIRC(A_{10},A_{11},A_{12})$. We now define the following generator matrices; $$ \mathcal{G}_6^{1}=[I_{36} \ | \ \tau_3(v_6)], $$ where $A_i$'s come from Equation \ref{seven}. $$ \mathcal{G}_6^{2}=[I_{36} \ | \ \tau_3(v_6)], $$ where $A_i$'s come from Equation \ref{four}. \subsection{The Alternating Group $A_{4}$} Let $A_{4}$ be the alternating group of order 12 and $v_7 \in M_3(\mathbb{F}_2)(A_{4}),$ then we have the following generator matrix from \cite{Dougherty6}\\ \begin{equation} \sigma_3(v_7)=\left(\begin{smallmatrix} A_{1}& A_{2}& A_{3}& A_{4}& A_{5}& A_{6}& A_{7}& A_{8}& A_{9}& A_{10}& A_{11}& A_{12}\\ A_{3}& A_{1}& A_{2}& A_{12}& A_{10}& A_{11}& A_{6}& A_{4}& A_{5}& A_{9}& A_{7}& A_{8}\\ A_{2}& A_{3}& A_{1}& A_{8}& A_{9}& A_{7}& A_{11}& A_{12}& A_{10}& A_{5}& A_{6}& A_{4}\\ A_{4}& A_{5}& A_{6}& A_{1}& A_{2}& A_{3}& A_{10}& A_{11}& A_{12}& A_{7}& A_{8}& A_{9}\\ A_{12}& A_{10}& A_{11}& A_{3}& A_{1}& A_{2}& A_{9}& A_{7}& A_{8}& A_{6}& A_{4}& A_{5}\\ A_{8}& A_{9}& A_{7}& A_{2}& A_{3}& A_{1}& A_{5}& A_{6}& A_{4}& A_{11}& A_{12}& A_{10}\\ A_{7}& A_{8}& A_{9}& A_{10}& A_{11}& A_{12}& A_{1}& A_{2}& A_{3}& A_{4}& A_{5}& A_{6}\\ A_{6}& A_{4}& A_{5}& A_{9}& A_{7}& A_{8}& A_{3}& A_{1}& A_{2}& A_{12}& A_{10}& A_{11}\\ A_{11}& A_{12}& A_{10}& A_{5}& A_{6}& A_{4}& A_{2}& A_{3}& A_{1}& A_{8}& A_{9}& A_{7}\\ A_{10}& A_{11}& A_{12}& A_{7}& A_{8}& A_{9}& A_{4}& A_{5}& A_{6}& A_{1}& A_{2}& A_{3}\\ A_{9}& A_{7}& A_{6}& A_{6}& A_{4}& A_{5}& A_{12}& A_{10}& A_{11}& A_{3}& A_{1}& A_{2}\\ A_{5}& A_{6}& A_{4}& A_{10}& A_{12}& A_{10}& A_{8}& A_{9}& A_{7}& A_{2}& A_{3}& A_{1}\\ \end{smallmatrix}\right), \end{equation} We now define the following generator matrices; $$ \mathcal{G}_7^{1}=[I_{36} \ | \ \tau_3(v_7)], $$ where $A_i$'s come from Equation \ref{three}. $$ \mathcal{G}_7^{2}=[I_{36} \ | \ \tau_3(v_7)], $$ where $A_i$'s come from Equation \ref{eight}. $$ \mathcal{G}_7^{3}=[I_{36} \ | \ \tau_3(v_7)], $$ where $A_i$'s come from Equation \ref{seven}. $$ \mathcal{G}_7^{4}=[I_{36} \ | \ \tau_3(v_7)], $$ where $A_i$'s come from Equation \ref{four}. $$ \mathcal{G}_7^{5}=[I_{36} \ | \ \tau_3(v_7)], $$ where $A_i$'s come from Equation \ref{five}. \subsection{Dicyclic Group $Dic_{12}$} We consider the dicyclic group of order 12 $$Dic_{12}= \langle x,y | x^6=1, ~~y^2=x^3, x^y=x^{-1}\rangle.$$ Let $v_8 \in M_3(\mathbb{F}_2)(Dic_{12}),$ we have the following generator matrix from \cite{Dougherty7}\\ \begin{equation} \sigma_3(v_8)=\begin{pmatrix} A&B\\ C&A \end{pmatrix}, \end{equation} where $ A=CIRC(A_{1}, A_{2}, \dots, A_{6} ),$ $ B=REVCIRC(A_{7}, A_{11}, \dots, A_{12} ),$ and $C=REVCIRC(A_{10}, A_{11}, A_{12},A_{7}, A_{8}, A_{9} ).$ We define the following generator matrices;\\ $$ \mathcal{G}_8^{1}=[I_{36} \ | \ \tau_3(v_8)], $$ where $A_i$'s come from Equation \ref{two}. $$ \mathcal{G}_8^{2}=[I_{36} \ | \ \tau_3(v_8)], $$ where $A_i$'s come from Equation \ref{five}. $$ \mathcal{G}_8^{3}=[I_{36} \ | \ \tau_3(v_8)], $$ where $A_i$'s come from Equation \ref{eight}. \section{Virus Optimization Algorithm for Self-Dual Codes} Metaheuristic algorithms have been very useful and powerful tool to simulate real-world problems that were previously difficult or impossible to solve \cite{Ustun}. For algebraic coding theory, optimization algorithms were used only for the minimum distance problem, for example \cite{Shenoy,Bland1, Bland2, Bouzkraoui}. \begin{figure*}[h!] \centering \includegraphics[width=110mm]{Voa_Images.jpg} \caption{Flowchart of VOA \label{VOA}} \end{figure*} But recently in \cite{Korban1}, one of the oldest and well-known optimization algorithm, genetic algorithm, was applied for the first time to the problem of finding binary self-dual codes. VOA which is one of nature-inspired optimization approaches using iteratively population-based, mimics the behaviour of the viruses assaulting a living cells. In the VOA, the count of the viruses in the population increase at every replication process and a process called as antivirus is applied to population by immune system to prevent the population of virus uncontrollably growth. In the algorithm, there are two types of viruses called as strong and common viruses respectively for establishing the balance between its exploration and exploitation abilities. The flowchart in Figure \ref{VOA} gives the steps of the VOA. The algorithm includes three major steps that are named as initialization, replication and updating/maintenance. At initial process, the first population of virus is generated by using the VOA's parameters specified by user. The values of the objective function computed for each of the viruses are ordered and strong and common viruses in the population are chosen according to these values. In the replication process, new viruses are generated by taking account of the strong and common viruses. The number of viruses in population is controlled by using antivirus mechanism provided by Maintenance/updating. In the VOA, unless the performance of the population enhances, exploitation is intensified and antivirus is applied to population. If termination criteria has not been attained, the counter of replication is increased by one and then other replication processes are continued; otherwise the algorithm stops. \begin{figure*}[h!] \centering \includegraphics[width=130mm]{Voa_Images2.jpg} \caption{Flowchart of Code Generation Process by VOA \label{VOA2}} \end{figure*} An application of VOA to the problem of finding binary self-dual codes is given in Figure \ref{VOA2}. For the considered problem, each of the members in the population are represented by a vector with length 36 and these vectors are elements of $\mathbb{F}_2^{36}$. These vectors form the first row of the matrices $\tau_k(v)$. Then, the linear codes are generated from the generator matrices of the form $[I_{36} \ | \ \tau_3(v_i)]$. If a generated code is self-dual and has a minimum distance 12 then it is saved. The VOA generates new vectors by applying the process of replication to the old strong and common viruses, until the algorithm's termination criteria value. It was shown in \cite{Liang} that the VOA outperforms the GA in continues problems. Therefore it is natural to compare these two algorithms for the considered problem by performing a search for extremal binary self-dual codes that have generator matrices of the $[I_{36} \ | \ \tau_3(v_i)]$ form. The comparison of GA and VOA in terms of number of codes found is given in the Figure \ref{Comp2}. Calculations for the comparison of GA and VOA were done only for three generator matrices; $\mathcal{G}_6^{1}, \mathcal{G}_7^{2}$ and $\mathcal{G}_8^{1}$. We used the same software - Magma \cite{MAGMA}, for the two approaches and algorithms were run on a workstation with Intel Xeon 4.0 GHz processor and 64 GByte RAM. The size of the population and number of iteration for both GA and VOA are 500 and 100, respectively. \begin{figure*}[h!] \centering \includegraphics[width=100mm]{graph2.jpg} \caption{Comparasion of VOA and GA \label{Comp2}} \end{figure*} \section{Computational Results} In this section, we employ the generator matrices defined in Section~3 and search for binary $[72,36,12]$ self-dual codes. The possible weight enumerators for a Type~I $[72,36,12]$ codes are as follows (\cite{Dougherty4}): $$W_{72,1}=1+2\beta y^{12}+(8640-64\gamma)y^{14}+(124281-24\beta+384\gamma)y^{16}+\dots$$ $$W_{72,2}=1+2 \beta y^{12}+(7616-64 \gamma)y^{14}+(134521-24 \beta+384 \gamma)y^{16}+\dots$$ where $\beta$ and $\gamma$ are parameters. The possible weight enumerators for Type~II $[72,36,12]$ codes are (\cite{Dougherty4}): $$1+(4398+\alpha)y^{12}+(197073-12\alpha)y^{16}+(18396972+66\alpha)y^{20}+\dots $$ where $\alpha$ is a parameter. For an up-to-date list of all known Type~I and Type~II binary self-dual codes with parameters $[72,36,12]$ please see \cite{selfdual72}. We only list codes with parameters in their weight enumerators that were not known in the literature before. All the upcoming computational results were obtained by performing searches in the software package MAGMA (\cite{MAGMA}). \begin{table}[h!]\label{Type1} \caption{New Binary Type I Codes of Length 72} \resizebox{0.65\textwidth}{!}{\begin{minipage}{\textwidth} \centering \begin{tabular}{cccccccc} \hline & Generator Matrix & Type & $r_A$ & $r_B$ & $\gamma$ & $\beta$ & $|Aut(C_i)|$ \\ \hline $C_{1}$ &$\mathcal{G}_1^{1}$& $W_{72,1}$ & $(0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0)$ & $( 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0)$ & $0$ & $129$ & $72$ \\ \hline $C_{2}$ &$\mathcal{G}_1^{2}$& $W_{72,1}$ & $(1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1)$ & $( 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1)$ & $36$ & $483$ & $72$ \\ \hline $C_{3}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $( 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0)$& $( 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)$ & $0$ & $236$ & $12$ \\ \hline $C_{4}$ &$\mathcal{G}_2^{2}$& $W_{72,1}$ & $(0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1)$ & $( 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1)$ & $6$ & $341$ & $12$ \\ \hline $C_{5}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1)$ & $( 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0)$ & $6$ & $353$ & $12$ \\ \hline $C_{6}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0)$ & $( 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0)$ & $12$ & $269$ & $12$ \\ \hline $C_{7}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0)$ & $(0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1)$ & $12$ & $285$ & $12$ \\ \hline $C_{8}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1)$ & $( 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1)$ & $12$ & $314$ & $12$ \\ \hline $C_{9}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0)$ & $(0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0)$ & $18$ & $279$ & $12$ \\ \hline $C_{10}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0)$ & $( 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0)$ & $36$ & $434$ & $12$ \\ \hline $C_{11}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1)$ & $(0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1)$ & $36$ & $458$ & $12$ \\ \hline $C_{12}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0)$ & $(1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0)$ & $36$ & $470$ & $12$ \\ \hline $C_{13}$ &$\mathcal{G}_2^{1}$& $W_{72,1}$ & $(0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0)$ & $(1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0)$ & $30$ & $495$ & $12$ \\ \hline $C_{14}$ &$\mathcal{G}_2^{2}$& $W_{72,1}$ & $(0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0)$ & $(0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1)$ & $6$ & $281$ & $12$ \\ \hline $C_{15}$ &$\mathcal{G}_2^{2}$& $W_{72,1}$ & $(0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0)$ & $(1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0)$ & $6$ & $252$ & $12$ \\ \hline $C_{16}$ &$\mathcal{G}_2^{2}$& $W_{72,1}$ & $(1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1)$ & $(0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1)$ & $12$ & $357$ & $12$ \\ \hline $C_{17}$ &$\mathcal{G}_2^{2}$& $W_{72,1}$ & $(0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1)$ & $(0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0)$ & $36$ & $506$ & $12$ \\ \hline $C_{18}$ &$\mathcal{G}_2^{4}$& $W_{72,1}$ & $(0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1)$ & $(0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0 )$ & $18$ & $324$ & $26$ \\ \hline $C_{19}$ &$\mathcal{G}_2^{6}$& $W_{72,1}$ & $(0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1)$ & $(0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0)$ & $0$ & $359$ & $12$ \\ \hline $C_{20}$ &$\mathcal{G}_2^{6}$& $W_{72,1}$ & $(1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0)$ & $(1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1)$ & $6$ & $276$ & $12$ \\ \hline $C_{21}$ &$\mathcal{G}_3^{1}$ & $W_{72,1}$& $(1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1)$ & $(1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0)$ & $36$ & $504$ & $72$ \\ \hline $C_{22}$ &$\mathcal{G}_3^{1}$& $W_{72,1}$ & $(1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1)$ & $(0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0)$ & $36$ & $552$ & $72$ \\ \hline $C_{23}$ &$\mathcal{G}_3^{2}$& $W_{72,1}$ & $(0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1)$ & $(1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0)$ & $0$ & $426$ & $72$ \\ \hline $C_{24}$ &$\mathcal{G}_3^{2}$& $W_{72,1}$ & $(0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0)$ & $(0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1)$ & $36$ & $456$ & $72$ \\ \hline $C_{25}$ &$\mathcal{G}_3^{2}$& $W_{72,1}$ & $(1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1)$ & $(1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1)$ & $36$ & $633$ & $72$ \\ \hline $C_{26}$ &$\mathcal{G}_5^{1}$& $W_{72,1}$ & $(1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0)$ & $(1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1)$ & $0$ & $69$ & $136$ \\ \hline $C_{27}$ &$\mathcal{G}_5^{2}$& $W_{72,1}$ & $(0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1)$ & $(1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1)$ & $12$ & $321$ & $12$ \\ \hline $C_{28}$ &$\mathcal{G}_5^{1}$& $W_{72,1}$ & $(1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1)$ & $(1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1)$ & $0$ & $327$ & $72$ \\ \hline $C_{29}$ &$\mathcal{G}_5^{3}$& $W_{72,1}$ & $(1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1)$ & $(1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0)$ & $0$ & $75$ & $36$ \\ \hline $C_{30}$ &$\mathcal{G}_8^{1}$& $W_{72,1}$ & $(0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1)$ & $(1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1)$ & $36$ & $468$ & $72$ \\ \hline \end{tabular} \end{minipage}} \end{table} \begin{table}[h!]\label{Type1ABCD} \caption{New Binary Type I Codes of Length 72} \resizebox{0.60\textwidth}{!}{\begin{minipage}{\textwidth} \centering \begin{tabular}{cccccccccc} \hline & Generator Matrix & Type & $r_A$ & $r_B$ & $r_C$ & $r_D$ & $\gamma$ & $\beta$ & $|Aut(C_i)|$ \\ \hline $C_{31}$ &$\mathcal{G}_4^{1}$ & $W_{72,1}$ & $(1, 0, 1, 1, 0, 1, 1, 0, 1)$ & $(0, 1, 0, 1, 0, 0, 1, 0, 0)$ & $(1, 1, 0, 1, 1, 0, 1, 0, 1)$ & $(0, 0, 1, 1, 1, 1, 1,1, 0)$ & $0$ & $99$ &$36$ \\ \hline $C_{32}$ &$\mathcal{G}_6^{1}$& $W_{72,1}$ & $(1, 1, 0, 1, 1, 1, 1, 1, 1)$ & $(0, 1, 0, 1, 1, 0, 1, 0, 1)$ & $(0, 0, 0, 0, 0, 0, 1, 1, 0)$ & $(1, 1, 0, 1, 1, 1, 0, 0, 1)$& $12$ & $267$ &$24$ \\ \hline $C_{33}$ &$\mathcal{G}_6^{2}$& $W_{72,1}$ & $(1, 1, 1, 1, 1, 1, 1, 0, 0)$ & $( 0, 0, 0, 1, 0, 0, 1, 0, 0)$ & $( 0, 0, 0, 0, 1, 1, 0, 0, 0)$ & $(1, 1, 0, 0, 1, 1, 0, 1, 1)$ & $12$ & $348$ & $24$ \\ \hline \end{tabular} \end{minipage}} \end{table} \begin{table}[h!]\label{Type1-twelve} \caption{New Binary Type I Codes of Length 72 } \resizebox{0.55\textwidth}{!}{\begin{minipage}{\textwidth} \centering \begin{tabular}{ccccccccccccccccc} \hline & Generator Matrix & $r_{A_{1}}$ & $r_{A_{2}}$& $r_{A_{3}}$ & $r_{A_{4}}$ & $r_{A_{5}}$ & $r_{A_{6}}$ & $r_{A_{7}}$ & $r_{A_{8}}$ & $r_{A_{9}}$ & $r_{A_{10}}$ & $r_{A_{11}}$ &$r_{A_{12}}$ & $\gamma$ & $\beta$ &$|Aut(C_i)|$ \\ \hline $C_{34}$& $\mathcal{G}_7^{1}$ & $(0, 0, 1)$ & $(0, 1, 0)$ & $( 1, 1, 0)$ & $( 0, 1, 0)$ & $( 0, 1, 1)$ & $( 1, 0, 1)$ & $( 1, 1, 0)$ & $( 1, 0, 1)$ & $( 1, 1, 0)$ & $( 0, 0, 0)$ & $(1, 1, 1)$ & $(1, 1, 1)$ & $12$ & $373$ & $12$ \\ \hline $C_{35}$& $\mathcal{G}_7^{2}$ & $(1, 1, 1,)$ & $( 1, 1, 1)$ & $( 1, 0, 0)$ & $( 1, 1, 0)$ & $( 0, 1, 0)$ & $( 1, 1, 1)$ & $(1, 1, 0)$ & $( 0, 1, 0)$ & $( 0, 1, 0)$ & $( 0, 1, 1)$ & $( 0, 1, 0)$ & $( 1, 0, 0)$ & $0$ & $161$ & $24$ \\ \hline $C_{36}$ & $\mathcal{G}_7^{2}$ & $(1, 1, 1)$ & $( 1, 1, 0)$ & $( 1, 0, 1)$ & $( 1, 1, 0)$ & $( 0, 0, 0)$ & $( 1, 1, 0)$ & $(1, 1, 0)$ & $( 1, 0, 0)$ & $( 1, 1, 1)$ & $( 1, 0, 0)$ & $( 0, 1, 1)$ & $( 0, 1, 0)$ & $12$ & $301$ & $12$ \\ \hline $C_{37}$ & $\mathcal{G}_7^{3}$ & $(0, 0, 1)$ & $( 1, 1, 1)$ & $( 1, 1, 0)$ & $( 0, 1, 1)$ & $( 0, 1, 1)$ & $( 1, 1, 1)$ & $( 1, 1, 0)$ & $( 1, 0, 1)$ & $( 1, 1, 0)$ & $( 0, 0, 0)$ & $( 1, 0, 0)$ & $( 1, 0, 0)$ & $0$ & $289$ & $12$ \\ \hline $C_{38}$ & $\mathcal{G}_7^{4}$ & $(0, 0, 1)$ & $( 0, 1, 1)$ & $( 1, 1, 0)$ & $( 0, 1, 1)$ & $( 1, 0, 0)$ & $( 0, 1, 0)$ & $( 0, 0, 0)$ & $( 0, 0, 0)$ & $( 1, 0, 1)$ & $( 0, 0, 1)$ & $( 0, 0, 1)$ & $( 0, 0, 0)$ & $12$ & $253$ & $12$ \\ \hline $C_{39}$ & $\mathcal{G}_7^{5}$ & $(1, 1, 0)$ & $( 0, 1, 0)$ & $( 0, 0, 1)$ & $( 1, 1, 0)$ & $( 1, 0, 1)$ & $( 1, 1, 0 )$ & $( 1, 0, 0)$ & $( 1, 1, 0)$ & $( 1, 0, 0)$ & $( 1, 1, 1)$ & $( 0, 0, 0)$ & $( 0, 0, 0)$ & $0$ & $260$ & $12$ \\ \hline \end{tabular} \end{minipage}} \end{table} \begin{table}[h!]\label{Type2} \caption{New Binary Type II Codes of Length 72} \resizebox{0.65\textwidth}{!}{\begin{minipage}{\textwidth} \centering \begin{tabular}{cccccc} \hline & Generator Matrix & $r_A$ & $r_B$ & $\alpha$ & $|Aut(C_i)|$ \\ \hline $C_{40}$ &$\mathcal{G}_2^{3}$& $(1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1)$ & $( 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0)$ & $-2796$ & $72$ \\ \hline $C_{41}$ &$\mathcal{G}_2^{4}$& $(1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0)$ & $(1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1)$ & $-2598$ & $36$ \\ \hline $C_{42}$ &$\mathcal{G}_2^{5}$& $(0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0)$ & $(1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0)$ & $-2388$ & $72$ \\ \hline $C_{43}$ &$\mathcal{G}_3^{1}$ & $(1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1)$ & $( 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1 )$ & $-3948$ & $144$ \\ \hline $C_{44}$ &$\mathcal{G}_3^{2}$ & $(0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1)$ & $(0, 1, 0, 0, 0, 0, 0, 0,1, 1, 1, 1, 1, 0, 0, 1, 1, 1)$ & $-2448$ & $144$ \\ \hline $C_{45}$ &$\mathcal{G}_3^{2}$ & $(1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1)$ & $( 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1)$ & $-2352$ & $144$ \\ \hline $C_{46}$& $\mathcal{G}_3^{3}$ & $(1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1)$ & $(0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0)$ & $-4050$ & $72$ \\ \hline $C_{47}$& $\mathcal{G}_3^{3}$ & $(1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0)$ & $(1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0)$ & $-4104$ & $144$ \\ \hline $C_{48}$& $\mathcal{G}_3^{3}$& $(0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0)$ & $(1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0)$ & $-2310$ & $144$ \\ \hline $C_{49}$& $\mathcal{G}_3^{3}$ & $(0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1)$ & $(0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1 )$ & $-2292$ & $432$ \\ \hline $C_{50}$& $\mathcal{G}_3^{4}$ & $(1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0)$ & $(1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0 )$ & $-2760$ & $432$ \\ \hline $C_{51}$& $\mathcal{G}_3^{4}$ & $(0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1)$ & $(1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1)$ & $-2622$ & $72$ \\ \hline $C_{52}$& $\mathcal{G}_5^{1}$ & $(0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0)$ & $(0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0 )$ & $-4020$ & $36$ \\ \hline $C_{53}$ & $\mathcal{G}_5^{3}$ & $(1,0,0,1,0,1,1,1,1,0,0,0,0,1,0,0,1,1)$ & $(0,1,1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,1)$ & $-2688$ & $72$ \\ \hline $C_{54}$& $\mathcal{G}_8^{2}$ & $(0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1)$ & $(1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1)$ & $-3966$ & $36$ \\ \hline $C_{55}$& $\mathcal{G}_8^{3}$ & $(1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0)$ & $( 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1 )$ & $-4056$ & $72$ \\ \hline $C_{56}$& $\mathcal{G}_8^{3}$ & $(1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0)$ & $(1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1 )$ & $-3858$ & $144$ \\ \hline \end{tabular} \end{minipage}} \end{table} \begin{table}[h!]\label{Type2ABCD} \caption{New Binary Type II Codes of Length 72} \resizebox{0.65\textwidth}{!}{\begin{minipage}{\textwidth} \centering \begin{tabular}{cccccccc} \hline & Generator Matrix & $r_A$ & $r_B$ & $r_C$ & $r_D$ & $\alpha$ & $|Aut(C_i)|$ \\ \hline $C_{57}$ &$\mathcal{G}_4^{2}$& $(0, 1, 1, 0, 1, 1, 0, 1, 0)$& $(0, 0, 1, 1, 1, 1, 0, 0, 1)$ & $( 0, 1, 0, 1, 0, 1, 1, 0, 1)$ & $(1, 0, 0, 1, 0, 0, 1, 0, 1 )$ & $-4080$ & $144$ \\ \hline $C_{58}$ &$\mathcal{G}_4^{3}$& $(0, 1, 1, 0, 1, 1, 1, 1, 1)$& $(0, 1, 1, 0, 0, 1, 0, 0, 0)$ & $( 1, 0, 0, 1, 0, 1, 1, 1, 0)$ & $( 0, 0, 0, 0, 0, 0, 0, 0, 0 )$ & $-2238$ & $144$ \\ \hline \end{tabular} \end{minipage}} \end{table} \newpage \section{Conclusion} In this work, we applied the VOA to the problem of finding new binary self-dual codes by using a number of generator matrices of the form $[I_{36} \ | \ \tau_3(v)]$. We proved that the VOA outperforms the GA for the considered problem in terms of the number of codes found for the same constructions. Moreover, with the matrix construction from \cite{Dougherty3} and the VOA, we found new binary self-dual codes. In particular, we were able to construct 39 Type I binary $[72,36,12]$ self-dual codes with new weight enumerators in $W_{72,1}$: \begin{equation*} \begin{array}{l} (\gamma =0,\ \ \beta =\{69, 75, 99, 129, 161, 236, 260, 289, 327, 359, 426\}), \\ (\gamma =6,\ \ \beta =\{252, 276, 281, 341, 353 \}), \\ (\gamma =12,\ \ \beta =\{253, 267, 269, 285, 301, 314, 321, 348, 357, 373 \}), \\ (\gamma =18,\ \beta =\{279, 324 \}), \\ (\gamma =30,\ \ \beta =\{495 \}), \\ (\gamma =36,\ \beta =\{434, 456, 458, 468, 470, 483, 504, 506, 552, 633\}) \\ \end{array}% \end{equation*} and 19 Type II binary $[72,36,12]$ self-dual codes with new weight enumerators: \begin{equation*} \begin{array}{l} (\alpha =\{-2238, -2292, -2310, -2352, -2388, -2448, -2598, -2622, -2688, -2760, \\ -2796, -3858, -3948, -3966, -4020, -4050, -4056, -4080, -4104 \}). \\ \end{array}% \end{equation*}

proofpile-arXiv_069-9441

\subsubsection*{Acknowledgments.} This work is partially supported by Aut.\ Reg.\ of Sardinia project P.I.A.\ 2013 ``NOMAD''. \section{Conclusions} We have analysed the usage of smart contracts from various perspectives. In~\Cref{sec:platforms} we have examined a sample of 6 platforms for smart contracts, pinpointing some crucial technical differences between them. For the two most prominent platforms --- Bitcoin and Ethereum --- we have studied a sample of \arabic{valtotcontracts}\xspace contracts, categorizing each of them by its application domain, and measuring the relevance of each of these categories (\Cref{sec:smartcontracts}). The availability of source code for Ethereum contracts has allowed us to analyse the most common design patterns adopted when writing smart contracts (\Cref{sec:DesignPatterns}). We believe that this survey may provide valuable information to developers of new, domain-specific languages for smart contracts. In particular, measuring what are the most common use cases allows to understand which domains deserve more investments. Furthermore, our study of the correlation between design patterns and application domains can be exploited to drive the correct choice of programming primitives of domain-specific languages for smart contracts. Due to the mixed flavour of our analysis, which compares differents platforms and studies how smart contracts are interpreted on each them, our work relates to various topics. The work~\cite{Marino16ruleml} proposes design patterns for altering and undoing of smart contracts; so far, our analysis in~\Cref{sec:patterns-stats} has not still found instances of these patterns in Ethereum. Among the works which study blockchain technologies, \cite{Anderson16corr} compares four blockchains, with a special focus on the Ethereum one; \cite{Lamela16iacr} examines a larger set of blockchains, including also some which does not fit the criteria we have used in our methodology (e.g.\@\xspace, RootStock and Tezos). Many works on Bitcoin perform empirical analyses of its blockchain. For instance, \cite{reid2013analysis,ron2013quantitative} study users deanonymization, \cite{moser2015trends} measures transactions fees, and~\cite{baqerstressing} analyses Denial-of-Service attacks on Bitcoin. The work~\cite{Glaser14ecis} investigates whether Bitcoin users are interested more on digital currencies as asset or as currency, with the aim of detecting the most popular use cases of Bitcoin contracts, similarly to what we have done in~\Cref{sec:smartContracts:quantitative}. Our classification of Bitcoin protocols based on OP\_RETURN\xspace transactions is inspired from~\cite{Bartoletti16opreturn}, which also measures the space consumption and temporal trend of OP\_RETURN\xspace transactions. Recently, some authors have started to analyse the security of Ethereum smart contracts: among these, \cite{ABC17post} surveys vulnerabilities and attacks, while~\cite{Luu16ccs} and~\cite{Bhargavan16solidether} propose analysis techniques to detect them. Our study on design patterns for Ethereum smart contracts could help to improve these techniques, by targeting contracts with specific programming patterns. \section{Introduction} \label{Introduction} Since the release of Bitcoin in 2009~\cite{bitcoin}, the idea of exploiting its enabling technology to develop applications beyond currency has been receiving increasing attention~\cite{Bonneau15ieeesp}. In particular, the public and append-only ledger of transaction (the \emph{blockchain}) and the decentralized consensus protocol that Bitcoin nodes use to extend it, have revived Nick Szabo's idea of \emph{smart contracts} --- i.e.\@\xspace programs whose correct execution is automatically enforced without relying on a trusted authority~\cite{Szabo97firstmonday}. The archetypal implementation of smart contracts is Ethereum~\cite{ethereum}, a platform where they are rendered in a Turing-complete language. The consensus protocol of Ethereum ensures that all and only the valid updates to the contract states are recorded on the blockchain, so ensuring their correct execution. Besides Bitcoin and Ethereum, a remarkable number of alternative platforms have flourished over the last few years, either implementing crypto-currencies or some forms of smart contracts~\cite{Clack16corr,Luu16ccs,MultiChainGoodBadLazy,BitcoinContract,MakingSenseContracts}. For instance, the number of crypto-currencies hosted on~\href{http://coinmarketcap.com/}{coinmarketcap.com} has increased from 0 to more than 600 since 2012; the number of \href{http://github.com}{github} projects related to blockchains and smart contracts has reached, respectively, $2,715$ and $445$ units (see~\Cref{fig:StatisticsIntro}). In the meanwhile, ICT companies and some national governments have started dealing with these topics~\cite{UK16report, Japan16report}, also with significant investments. \begin{figure}[t] \hspace{-5pt} \begin{tabular}{cc} \begin{subfigure}[b]{0.5\textwidth} \begin{tikzpicture} \begin{axis}[ width = 1\linewidth, height = 4.5cm, date coordinates in=x, date ZERO=2013-05-01, xmin=2013-05-01, xmax=2017-01-01, xtick={2013-05-01, 2013-12-01, 2014-06-01, 2015-01-01, 2015-07-01, 2016-01-01, 2016-07-01, 2017-01-01}, ymin=0, ymax=650, xmajorgrids = true, ymajorgrids = true, xticklabel style={rotate=70, anchor=east, xticklabel}, xticklabel=\tiny\month.\year, yticklabel style={font=\tiny,xshift=0.5ex}, legend pos=south east, legend style={font=\tiny}, legend cell align=left, xlabel absolute, xlabel style={yshift=-0.5cm}, xlabel={Time interval}, ylabel absolute, ylabel style={yshift=-0.3cm}, ylabel={Number of currencies} ] \pgfplotstableread[col sep=comma]{results/Coinmarketcap.csv}\data \addplot [color=blue, thick] table[x index = {0}, y index = {1}]{\data}; \legend{Crypto-Currencies} \end{axis} \end{tikzpicture} \end{subfigure} \hspace{5pt} & \begin{subfigure}[b]{0.5\textwidth} \begin{tikzpicture} \begin{axis}[ width = 1\linewidth, height = 4.5cm, date coordinates in=x, date ZERO=2012-01-01, xmin=2012-01-01, xmax=2017-01-01, ymin=0, ymax=250, xmajorgrids = true, ymajorgrids = true, try min ticks=8, xticklabel style={rotate=70, anchor=east, xticklabel}, xticklabel=\tiny\month.\year, yticklabel style={font=\tiny,xshift=0.5ex}, legend pos=north west, legend style={font=\tiny}, legend cell align=left, xlabel absolute, xlabel style={yshift=-0.5cm}, xlabel={Time interval}, ylabel absolute, ylabel style={yshift=-0.3cm}, ylabel={Number of projects} ] \pgfplotstableread[col sep=comma]{results/Github.csv}\data \addplot [color=blue, thick] table[x index = {0}, y index = {1}]{\data}; \pgfplotstableread[col sep=comma]{results/Github.csv}\data \addplot [color=green, thick] table[x index = {0}, y index = {2}]{\data}; \legend{Blockchain,Smart Contract} \end{axis} \end{tikzpicture} \end{subfigure} \hspace{40pt} \end{tabular} \caption{On the left, monthly trend of the number of crypto-Currencies hosted on \href{http://coinmarketcap.com/}{coinmarketcap.com}. On the right, number of new projects related to blockchains and smart contracts which are created every month on \href{http://github.com}{github.com}.} \label{fig:StatisticsIntro} \end{figure} Despite the growing hype on blockchains and smart contracts, the understanding of the actual benefits of these technologies, and of their trustworthiness and security, has still to be assessed. In particular, the consequences of unsafe design choices for the programming languages for smart contracts can be fatal, as witnessed by the unfortunate epilogue of the DAO contract~\cite{DAO}, a crowdfunding service plundered of $\sim50M$ USD because of a programming error. Since then, many other vulnerabilities in smart contract have been reported~\cite{ABC17post,Luu16ccs,VitalikThinkingSecurity,ENSanotherbug}. Understanding how smart contracts are used and how they are implemented could help designers of smart contract platforms to create new domain-specific languages (not necessarily Turing complete~\cite{Brown16corda,Churyumov16byteball,Frantz16ecas,Popejoy16kadena}), which \emph{by-design} avoid vulnerabilities as the ones discussed above. Further, this knowledge could help to improve analysis techniques for smart contracts (like e.g.\@\xspace the ones in~\cite{Bhargavan16solidether,Luu16ccs}), by targeting contracts with specific programming patterns. \subsection{Design patterns} \label{sec:patterns-list} \begin{description} \item[Token.] This pattern is used to distribute some fungible goods (represented by tokens) to users. Tokens can represent a wide variety of goods, like e.g.\@\xspace coins, shares, outcomes or tickets, or everything else which is transferable and countable. The implications of owning a token depend on the protocol and the use case for which the token has been issued. Tokens can be used to track the ownership of physical properties (e.g.\@\xspace, \href{https://etherscan.io/address/0xe0b7927c4af23765cb51314a0e0521a9645f0e2a#code}{gold}~\cite{DGXWebsite}), or digital ones (e.g.\@\xspace, \href{https://etherscan.io/address/0x815a46107e5ee2291a76274dc879ce947a3f0850#code}{cryptocurrency}). % Some crowdfunding systems issue tokens in exchange for donations (e.g.\@\xspace, the \href{https://etherscan.io/address/0xfb6916095ca1df60bb79ce92ce3ea74c37c5d359#code}{Congress} contract). Tokens are also used to regulate user authorizations and identities. For instance, the \href{https://etherscan.io/address/0xadc46ff5434910bd17b24ffb429e585223287d7f#code}{DVIP} contract specifies rights and term of services for owners of its tokens. To vote on the poll \href{https://etherscan.io/address/0xdb6d68e1d8c3f69d32e2d83065492e502b4c67ba#code}{ETCSurvey}, users must possess a suitable token. Given the popularity of this pattern, its standardisation has been proposed~\cite{ERC20}. Notably, the majority of the analysed Ethereum contracts which issue tokens already adhere to it. \item[Authorization.] This pattern is used to restrict the execution of code according to the caller address. The majority of the analysed contracts check if the caller address is that of the contract owner, before performing critical operations (e.g.\@\xspace, sending ether, invoking \href{https://etherscan.io/address/0x3fccb426c33b1ae067115390354b968592348d05#code}{suicide} or \href{https://etherscan.io/address/0x8b4aa759d83ec43efba755fc27923e4a581bccc1#code}{selfdestruct}). % For instance, the owner of \href{https://etherscan.io/address/0xdc84953D7C6448e498Eb3C33ab0F815da5D13999#code}{Doubler} is authorized to move all funds to a new address \emph{at any time} (this may raise some concerns about the trustworthiness of the contract, as a dishonest owner can easily steal money). % \href{https://etherscan.io/address/0x684282178b1d61164febcf9609ca195bef9a33b5#code}{Corporation} checks addresses to ensure that every user can vote only once per poll. \href{https://etherscan.io/address/0x5A5eFF38DA95b0D58b6C616f2699168B480953C9#code}{CharlyLifeLog} uses a white-list of addresses to decide who can withdraw funds. \item[Oracle.] Some contracts may need to acquire data from outside the blockchain, e.g.\@\xspace from a website, to determine the winner of a bet. The Ethereum language does not allow contracts to query external sites: otherwise, the determinism of computations would be broken, as different nodes could receive different results for the same query. % Oracles are the interface between contracts and the outside. Technically, they are just contracts, and as such their state can be updated by sending them transactions. In practice, instead of querying an external service, a contract queries an oracle; and when the external service needs to update its data, it sends a suitable transaction to the oracle. Since the oracle is a contract, it can be queried from other contracts without consistency issues. % One of the most common oracles is Oraclize% \footnote{\url{http://www.oraclize.it/}}: in our sample, it is used by almost all the contracts which resort to oracles. \item[Randomness.] Dealing with randomness is not a trivial task in Ethereum. Since contract execution must be deterministic, all the nodes must obtain the same value when asking for a random number: this struggles with the randomness requirements wished. % To address this issue, several contracts (e.g.\@\xspace, \href{https://etherscan.io/address/0x76bc9e61a1904b82cbf70d1fd9c0f8a120483bbb#code}{Slot}) query oracles that generate these values off-chain. Others (e.g.\@\xspace, \href{https://etherscan.io/address/0x302fE87B56330BE266599FAB2A54747299B5aC5B#code}{Lottery}) try to generate the numbers locally, by using values not predictable \emph{a priori}, as the hash of a block not yet created. However, these techniques are not generally considered secure~\cite{ABC17post}. \item[Poll.] Polls allows users to vote on some question. Often this is a side feature in a more complex scenario. For instance, in the \href{https://etherscan.io/address/0x2AB9f67A27f606272189b307052694D3a2B158bA#code}{Dice} game, when a certain state is reached, the owner issues a poll to decide whether an emergency withdrawal is needed. To determine who can vote and to keep track of the votes, polls can use tokens, or they can check the voters' addresses. \item[Time constraint.] Many contracts implement time constraints, e.g.\@\xspace to specify when an action is permitted. % For instance, \href{https://etherscan.io/address/0x9828f591b21ee4ad4fd803fc7339588cb83a6b84#code}{BirthdayGift} allows users to collect funds, which will be redeemable only after their birthday. % In notary contracts, time constraints are used to prove that a document is owned from a certain date. In game contracts, e.g.\@\xspace \href{https://etherscan.io/address/0x302fE87B56330BE266599FAB2A54747299B5aC5B#code}{Lottery}, time constraints mark the stages of the game. \item[Termination.] Since the blockchain is immutable, a contract cannot be deleted when its use has come to an end. Hence, developers must forethink a way to disable it, so that it is still present but unresponsive. This can be done manually, by inserting ad-hoc code in the contract, or automatically, calling \code{selfdestruct} or \code{suicide}. Usually, only the contract owner is authorized to terminate a contract (e.g.\@\xspace, as in \href{https://etherscan.io/address/0xe941e5d4a66123dc74886699544fbbb942f1887a#code}{SimpleCoinFlipGame}). \item[Math.] Contracts using this pattern encode the logic which guards the execution of some critical operations. For instance, \href{https://etherscan.io/address/0x54bda709fed875224eae569bb6817d96ef7ed9ad#code}{Badge} implements a method named \code{subtractSafely} to avoid subtracting a value from a balance when there are not enough funds in an account. \item[Fork check.] The Ethereum blockchain has been forked four times, starting from July 20th, 2016, when a fork was performed to contrast the effect of the DAO attack~\cite{TheDaoHardFork}. To know whether or not the fork took place, some contracts inspect the final balance of the DAO. Other contracts use this check to detect whether they are running on the main chain or on the fork, performing different actions in the two cases. % \href{https://etherscan.io/address/0x2bd2326c993dfaef84f696526064ff22eba5b362#code}{AmIOnTheFork} is a library contract that can be used to distinguish the main chain from the forked one. \end{description} \subsection{Quantifying the usage of design patterns by category} \label{sec:patterns-stats} \begin{table}[t] \begin{center} \hspace*{0em} \small \scalebox{0.9}{% \begin{tabular}{|l|c|c|c|c|c|c|c|c|c||c|} \hline \multirow{2}{5em}{}& \textbf{Token} & \textbf{Auth.} & \textbf{Oracle} & \textbf{Random.} & \textbf{Poll} & \textbf{Time} & \textbf{Termin.} & \textbf{Fork} & \textbf{Math} & \textbf{None} \\ \hline \textbf{Financial} & 24-51 & 51-39 & 2-15 & 1-2 & 5-29 & 23-31 & 14-30 & 8-69 & 4-47 & 29-66 \\ \textbf{Notary} & 13-6 & 52-9 & 1-2 & 0-0 & 8-9 & 20-6 & 29-13 & 0-0 & 1-3 & 30-15 \\ \textbf{Game} & 3-3 & 84-27 & 25-74 & 72-93 & 25-57 & 73-43 & 21-19 & 1-3 & 2-9 & 1-1 \\ \textbf{Wallet} & 18-2 & 100-3 & 0-0 & 0-0 & 0-0 & 94-6 & 100-10 & 0-0 & 12-6 & 0-0 \\ \textbf{Library} & 0-0 & 31-2 & 0-0 & 14-3 & 0-0 & 24-3 & 24-4 & 34-24 & 21-19 & 17-3 \\ \hline\hline \textbf{Unclassified} & 43-39 & 66-21 & 3-9 & 1-1 & 3-6 & 18-10 & 28-25 & 28-25 & 1-5 & 15-15 \\ \textbf{\emph{Total}} & {\em 21-100} & {\em 61-100} & {\em 7-100} & {\em 15-100} & {\em 9-100} & {\em 33-100} & {\em 22-100} & {\em 5-100} & {\em 4-100} & {\em 20-100} \\ \hline \end{tabular} } \end{center} \caption{Relations between design patterns and contract categories. A pair $(p,q)$ at row $i$ and column $j$ means that $p\%$ of the contracts in category $i$ use the pattern of column $j$, and $q\%$ of contracts with pattern $j$ belong to category $i$.} \label{tab:patterns} \end{table} We now study how the design patterns identified in \Cref{sec:patterns-list} are used in smart contracts. Out of the \arabic{valethcontracts}\xspace analysed contracts, 648 use at least one of the 9 patterns presented, for a grand total of 1427 occurrences of usage. \Cref{tab:patterns} shows the correlation between the usage of design patterns and contract categories, as defined in~\Cref{sec:smartcontracts}. A cell at row $i$ and column $j$ shows a pair of values: the first value is the percentage of contracts of category $i$ that use the pattern of column $j$; the second one is the percentage of contracts with pattern $j$ which belongs to category $i$. So, for instance, 24\% of the contracts in the financial category use the token pattern, and 51\% of all the contracts with the token pattern are financial ones. We observe that \textit{token}, \textit{authorization}, \textit{time constraint}, and \textit{termination} are generally the most used patterns. Some patterns are spread across several categories (e.g.\@\xspace, \emph{termination} and \textit{time constraint}), while others are mainly adopted only in one. For instance, \emph{oracle} and \emph{randomness} patterns are peculiar of game contracts, while the \emph{token} pattern is mostly used in financial contracts. Although \textit{math} is the less used, it appears in each category. Some contracts do not use any pattern (29\% of financial and 30\% of notary); almost all the contracts in game and wallet categories uses at least one. Further, only 15\% of all the unclassified contracts do no use any pattern at all. The most frequent patterns in financial contracts are \textit{token} (24\%), \textit{authorization} (51\%), and \textit{time constraint} (23\%). Due to the presence of contracts which implement assets and crowdfunding services, we have that half of contracts using \textit{token} and \textit{math} patterns belong to the financial category. For instance, these services use \textit{token} for representing goods or developing polls. Moreover, a great 69\% of contracts that use the \textit{fork check} pattern is financial. This is caused by the necessity of knowing the branch of the fork before deciding to move funds. Finally, several financial applications (29\%) perform simple operations (e.g.\@\xspace sending a payment) without using any of our described patterns. The \textit{authorization} pattern is used in many notary contracts to ensure that only the owner of a document can add or modify its data, in order to avoid tampering. Most gambling games involve players who pay fees to join the game, and rewards that can be collected by the winner of the game. The \emph{authorization} pattern is used to let the owner to be the only one able to redeem participants' fees or to perform administrative operations, and to let the winner withdraw his reward. The \emph{time constraint} pattern is used to distinguish the different phases of the game. For instance, within a specific time interval players can join the game and/or bet; then, bets are over, and the game determines a winner. To choose the winner, some gambling games resort to random numbers, which are often generated through an oracle. Indeed, 25\% of games use the \textit{oracle} pattern, and the pattern itself is used 74\% of cases by a game contract. Since \emph{all} game contracts invoking an \emph{oracle} (25\%) ask for random values, and since 72\% of contracts use the \emph{random} pattern, we can deduce that 47\% of them generate random numbers without resorting to oracles. Notably, 100\% of wallet contracts adopt both \textit{authorization} and \textit{termination} design patterns. A high 94\% also uses \textit{time constraint}. On the contrary, \textit{oracle}, \textit{poll}, and \textit{randomness} patterns are of little use when developing a wallet, while \textit{math} is sometimes used for securing operations on the balance. \section{Design patterns for Ethereum smart contracts} \label{sec:DesignPatterns} In this~\namecref{sec:DesignPatterns} we study design patterns for Ethereum smart contracts. To this purpose, we consider the sample of \arabic{valethcontracts}\xspace contracts collected through the methodology described in~\Cref{sec:smartcontracts}. By manually inspecting the Solidity source code of each of these contracts, we identify some common design patterns. We start in~\Cref{sec:patterns-list} by describing these patterns. Then, in~\Cref{sec:patterns-stats} we measure the usage of the patterns in the various categories of contracts identified in~\Cref{sec:smartcontracts}. \subsection{Analysis of platforms} \label{subsection:platformanalysis} We now describe the general features of the collected platforms, focussing on: \begin{inlinelist} \item whether the platform has its own blockchain, or if it just piggy-backs on an already existing one; \item for platforms with a public blockchain, their consensus protocol, and whether the blockchain is public or private to a specific set of nodes; \item the languages used to write smart contracts. \end{inlinelist} \paragraph{Bitcoin} \cite{bitcoin} is a platform for transferring digital currency, the bitcoins (BTC). It has been the first decentralized cryptocurrency to be created, and now is the one with the largest market capitalization. % The platform relies on a public blockchain to record the complete history of currency transactions. The nodes of the Bitcoin network use a consensus algorithm based moderately hard \emph{``proof-of-work''} puzzles to establish how to append a new block of transactions to the blockchain. Nodes work in competition to generate the next block of the chain. The first node that solves the puzzle earns a reward in BTC. Although the main goal of Bitcoin is to transfer currency, the immutability and openness of its blockchain have inspired the development of protocols that implement (limited forms of) smart contracts. Bitcoin features a non-Turing complete scripting language, which allows to specify under which conditions a transaction can be redeemed. The scripting language is quite limited, as it only features some basic arithmetic, logical, and crypto operations (e.g.\@\xspace, hashing and verification of digital signatures). A further limitation to its expressiveness is the fact that only a small fraction of the nodes of the Bitcoin network processes transactions whose script is more complex than verifying a signature% \footnote{As far as we know, currently only the \emph{Eligius} mining pool accepts more general transactions (called \emph{non-standard} in the Bitcoin community). However, this pool only mines \mbox{$\sim 1\%$} of the total mined blocks~\cite{Banasik16esorics}.}. \paragraph{Ethereum}\cite{ethereum} is the second platform for market capitalization, after Bitcoin. Similarly to Bitcoin, it relies on a public blockchain, with a consensus algorithm similar to that of Bitcoin% \footnote{The consensus mechanism of Ethereum is a variant of the GHOST protocol in~\cite{Sompolinsky15fc}. }. Ethereum has its own currency, caller \emph{ether} (ETH). Smart contracts are written in a stack-based bytecode language~\cite{ethereumyellowpaper}, which is Turing-complete, unlike Bitcoin's. There also exist a few high level languages (the most prominent being \emph{Solidity}% \footnote{Solidity: \url{http://solidity.readthedocs.io/en/develop/index.html}}% ), which compile into the bytecode language. Users create contracts and invoke their functions by sending transactions to the blockchain, whose effects are validated by the network. Both users and contracts can store money and send/receive ETH to other contracts or users. \paragraph{Counterparty}\cite{counterparty} is a platform without its own blockchain; rather, it embeds its data into Bitcoin transactions. While the nodes of the Bitcoin network ignore the data embedded in these transactions, the nodes of Counterparty recognise and interpret them. Smart contracts can be written in the same language used by Ethereum. However, unlike Ethereum, no consensus protocol is used to validate the results of computations% \footnote{See FAQ: How do Smart Contracts ``form a consensus'' on \mbox{Counterparty}? \url{http://counterparty.io/docs/faq-smartcontracts/\#how-do-smart-contracts-form-a-consensus-on-counterparty}}. Counterparty has its own currency, which can be transferred between users, and be spent for executing contracts. Unlike Ethereum, nodes do not obtain fees for executing contracts; rather, the fees paid by clients are destroyed, and nodes are indirectly rewarded from the inflation of the currency. This mechanism is called \emph{proof-of-burn}. \paragraph{Stellar}\cite{Stellar} features a public blockchain with its own cryptocurrency, governed by a consensus algorithm inspired to federated Byzantine agreement~\cite{SCP}. Basically, a node agrees on a transaction if the nodes in its neighbourhood (that are considered more trusted than the others) agree as well. When the transaction has been accepted by enough nodes of the network, it becomes unfeasible for an attacker to roll it back, and it is considered as confirmed. Compared to \emph{proof-of-work}, this protocol consumes far less computing power, since it does not involve solve cryptographic puzzles. Unlike Ethereum, there is no specific language for smart contracts; still, it is possible to gather together some transactions (possibly ordered in a chain) and write them atomically in the blockchain. Since transactions in a chain can involve different addresses, this feature can be used to implement basic smart contracts. For instance, assume that a participant $A$ wants to pay $B$ only if $B$ promises to pay $C$ after receiving the payment from $A$. This behaviour can be enforced by putting these transactions in the same chain. While this specific example can be implemented on Bitcoin as well, Stellar also allows to batch operations different from payments% \footnote{\url{https://www.stellar.org/developers/guides/concepts/operations.html}}, e.g.\@\xspace creating new accounts. Stellar features special accounts, called \emph {multisignature}, which can be handled by several owners. To perform operations on these accounts, a threshold of consensus must be reached among the owners. Transaction chaining and multisignature accounts can be combined to create more complex contracts. \paragraph{Monax} \cite{Monax} % supports the execution of Ethereum contracts, without having its own currency. Monax allows users to create private blockchains, and to define authorisation policies for accessing them. Its consensus protol% \footnote{\url{https://tendermint.com/}} is organised in rounds, where a participant proposes a new block of transactions, and the others vote for it. When a block fails to be approved, the protocol moves to the next round, where another participant will be in charge of proposing blocks. A block is confirmed when it is approved by at least 2/3 of the total voting power. \paragraph{Lisk} \cite{Lisk} has its own currency, and a public blockchain with a \emph{delegated proof-of-stake} consensus mechanism% \footnote{\url{https://lisk.io/documentation?i=lisk-handbooks/DelegateHandbook}}. More specifically, 101 active delegates, each one elected by the stakeholders, have the authority to generate blocks. Stakeholders can take part to the electoral process, by placing votes for delegates in their favour, or by becoming candidates themselves. Lisk supports the execution of Turing-complete smart contracts, written either in JavaScript or in Node.js. Unlike Ethereum, determinism of executions is not ensured by the language: rather, programmers must take care of it, e.g.\@\xspace by not using functions like \textit{Math.random}. Although Lisk has a main blockchain, each smart contract is executed on a separated one. Users can deposit or withdraw currency from a contract to the main chain, while avoiding double spending. Contract owners can customise their blockchain before deploying their contracts, e.g.\@\xspace choosing which nodes can participate to the consensus mechanism. \begin{table}[t] \begin{center} \hspace*{0em} \small \scalebox{0.75}{% \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline \multirow{2}{4.5em}{\textbf{Platform}} & \multicolumn{3}{|c|}{\textbf{Blockchain}} & \multirow{2}{9.7em}{\textbf{Contract Language}} & \multirow{2}{5.5em}{\textbf{ Total Tx}} & \textbf{Volume} & \textbf{Marketcap} \\ \cline{2-4} & \textbf{Type} & \textbf{Size} & \textbf{Block int.} & & & (K USD) & (M USD) \\ \hline\hline \textbf{Bitcoin} & \multirow{2}{2.9em}{Public} & \multirow{2}{2.8em}{96 GB} & \multirow{2}{3.3em}{10 min.} & Bitcoin scripts + signatures & 184,045,240 & 83,178 & 15,482 \\ \cline{1-1} \cline{5-8} \textbf{Counterparty} & & & & EVM bytecode & 12,170,386 & 33 & 4 \\ \hline \textbf{Ethereum} & Public & 17-60 GB & 12 sec. & EVM bytecode & 14,754,984 & 10,354 & 723 \\ \hline \textbf{Stellar} & Public & ? & 3 sec. & Transaction chains + signatures & ? & 35 & 17 \\ \hline \textbf{Monax} & Private & ? & Custom & EVM bytecode + permissions & ? & n/a & n/a \\ \hline \textbf{Lisk} & Private & ? & Custom & JavaScript & ? & 45 & 15 \\ \hline \end{tabular} } \caption{General statistics of platforms for smart contracts.} \label{table:platforms} \end{center} \end{table} \medskip \Cref{table:platforms} summarizes the main features of the analysed platforms. The question mark in some of the cells indicates that we were unable to retrieve the information (e.g.\@\xspace, we have not been able to determine the size of Monax blockchains, since they are private). The first three columns next to the platform name describe features of the blockchain: whether it is public; its size; the average time between two consecutive blocks. Note that Bitcoin and Counterparty share the same cell, since the second platform uses the Bitcoin blockchain. Measuring the size of the Ethereum blockchain depends on which client and which pruning mode is used. For instance, using the \href{https://github.com/ethereum/go-ethereum/wiki/geth}{Geth} client, we obtain a measure of 17GB in ``fast sync'' mode, and of 60GB in ``archive'' mode% \footnote{\url{https://redd.it/5om2lw}}. In platforms with private blockchains, their block interval is custom. The fifth column describes the support for writing contracts. The sixth column shows the total number of transactions% \footnote{Sources: \url{https://blockchain.info/charts/n-transactions-total} (for Bitcoin), \url{https://blockscan.com} (Counterparty), and \url{https://etherscan.io} (Ethereum).}. The last two columns show the daily volume of currency tranfers, and the market capitalisation of the currency (both in USD, rounded, respectively, to thousands and millions)% \footnote{Market capitalization estimated by~\url{http://coinmarketcap.com}.}. All values reported on \Cref{table:platforms} are updated to~January 1st, 2017\xspace. \subsection{Methodology} \label{subsection:platfrommethodology} To choose the platforms subject of our study, we have drawn up a candidate list by examining all the articles of \href{http://www.coindesk.com}{\code{coindesk.com}} in the ``smart contracts'' category% \footnote{\url{http://www.coindesk.com/category/technology/smart-contracts-news}}. Starting from June 2013, when the first article appeared, up to the 15th of September 2016, 175 articles were published, describing projects, events, companies and technologies related to smart contracts and blockchains. By manually inspecting all these articles, we have found references to 12 platforms: Bitcoin, Codius, Counterparty, DAML, Dogeparty, Ethereum, Lisk, Monax, Rootstock, Symbiont, Stellar, and Tezos. We have then excluded from our sample the platforms which, at the time of writing, do not satisfy one of the following criteria: \begin{inlinelist} \item \label{item:platform-methodology:launched} have already been launched, \item \label{item:platform-methodology:running} are running and supported from a community of developers, and \item \label{item:platform-methodology:accessible} are publicly accessible. \end{inlinelist} For the last point we mean that, e.g.\@\xspace, it must be possible to write a contract and test it, or to explore the blockchain through some tools, or to run a node. We have inspected each of the candidate platforms, examining the related resources available online (e.g.\@\xspace, official websites, white-papers, forum discussions, \emph{etc}.\@\xspace) After this phase, we have removed 6 platforms from our list: Tezos and Rootstock, as they do not satisfy condition~\ref{item:platform-methodology:launched}; Codius and Dogeparty, which violate condition~\ref{item:platform-methodology:running}, DAML and Symbiont, which violate~\ref{item:platform-methodology:accessible}. Summing up, we have a sample of 6 platforms: Bitcoin, Ethereum, Counterparty, Stellar, Monax and Lisk, which we discuss in the following. \section{Platforms for smart contracts} \label{sec:platforms} In this~\namecref{sec:platforms} we analyse various platforms for smart contracts. We start by presenting the methodology we have followed to choose the candidate platforms (\Cref{subsection:platfrommethodology}). Then we describe the key features of each platform, pinpointing differences and similarities, and drawing some general statistics (\Cref{subsection:platformanalysis}). \subsection{Methodology} \label{subsection:smartcontractmethodology} We sample contracts from Bitcoin and Ethereum as follows: \begin{itemize} \item for Ethereum, we collect on January 1st, 2017\xspace all the contracts marked as ``verified'' on the blockchain explorer \href{https://etherscan.io/contractsVerified}{\code{etherscan.io}}. This means that the contract bytecode stored on the blockchain matches the source code % (generally written in a high level language, such as Solidity) submitted to the explorer. In this way, we obtain a sample of \arabic{valethcontracts}\xspace contracts. \item for Bitcoin, we start by observing that many smart contracts save their metadata on the blockchain through the OP\_RETURN\xspace instruction of the Bitcoin scripting language~\cite{BitcoinContract,MakingSenseContracts,BitcoinOpReturnDescription,Bartoletti16opreturn}. We then scan the Bitcoin blockchain on January 1st 2017\xspace, searching for transactions that embed in an OP\_RETURN\xspace some metadata attributable to a Bitcoin smart contract. To this purpose we use an explorer% \footnote{\url{https://github.com/BitcoinOpReturn/OpReturnTool}} which recognises \arabic{valbtccontracts}\xspace smart contracts, and extracts all the transactions related to them. \end{itemize} \subsection{Quantifying the usage of smart contracts by category} \label{sec:smartContracts:quantitative} \begin{table}[t] \begin{center} \small \scalebox{0.75}{% \begin{tabular}{|c|c|c|c|} \hline \textbf{Category} & \textbf{Platform} & \textbf{Contracts} & \textbf{Transactions} \\ \hline\hline \multirow{2}{8em}{\textbf{Financial}} & Bitcoin & 6 & 470,391 \\ \cline{2-4} & Ethereum & 373 & 624,046 \\ \hline \multirow{2}{8em}{\textbf{Notary}} & Bitcoin & 17 & 443,269 \\ \cline{2-4} & Ethereum & 79 & 35,253 \\ \hline \multirow{2}{8em}{\textbf{Game}} & Bitcoin & 0 & 0 \\ \cline{2-4} & Ethereum & 158 & 58,257 \\ \hline \multirow{2}{8em}{\textbf{Wallet}} & Bitcoin & 0 & 0 \\ \cline{2-4} & Ethereum & 17 & 1,342 \\ \hline \multirow{2}{8em}{\textbf{Library}} & Bitcoin & 0 & 0 \\ \cline{2-4} & Ethereum & 29 & 37,034 \\ \hline\hline \multirow{2}{8em}{\textbf{Unclassified}} & Bitcoin & 0 & 0 \\ \cline{2-4} & Ethereum & 155 & 3,679 \\ \hline\hline \multirow{3}{8em}{\textbf{Total}} & Bitcoin & \arabic{valbtccontracts}\xspace & 913,660 \\ \cline{2-4} & Ethereum & \arabic{valethcontracts}\xspace & 759,611 \\ \cline{2-4} & Overall & \arabic{valtotcontracts}\xspace & 1,673,271 \\ \hline \end{tabular} } \end{center} \caption{Transactions by category.} \label{fig:smartcontracts} \end{table} \begin{figure} \centering \begin{tikzpicture} \begin{axis} [ybar, enlargelimits=0.15, width = 13cm, height = 5.5cm, legend style={at={(0.75,0.75)},anchor=south}, scaled y ticks = false, legend columns=3, symbolic x coords={Financial,Notary,Wallet,Game,Library,Unclassified}, xlabel absolute, xlabel style={yshift=+0.2cm}, ymin=0, ymax=80, xtick pos=left, ytick pos=left, xtick={Financial,Notary,Wallet,Game,Library,Unclassified}] \addplot [fill=blue!50] coordinates {(Financial,51.5) (Notary,48.5) (Game,0) (Wallet,0) (Library,0) (Unclassified,0)}; \addplot [fill=red!50] coordinates {(Financial,82.2) (Notary,4.6) (Game,7.7) (Wallet,0.2) (Library,4.9) (Unclassified,0.5)}; \addplot [fill=violet!70] coordinates {(Financial,65.5) (Notary,28.6) (Game,3.5) (Wallet,0.1) (Library,2.2) (Unclassified,0.2)}; \legend{Bitcoin, Ethereum, Overall} \end{axis} \end{tikzpicture} \caption{Distribution of transactions by category.} \label{fig:TransactionsAndLength} \end{figure} We analyse all the transactions related to the \arabic{valtotcontracts}\xspace smart contracts in our sample. \Cref{fig:smartcontracts} displays how the transactions are distributed in the categories of~\Cref{sec:smartContracts:taxonomy}. For both Bitcoin and Ethereum, we show the number of detected contracts (third column), and the total number of transactions (fourth column). Overall, we have 1,673,271 transactions. Notably, although Bitcoin contracts are fewer than those running on Ethereum, they have a larger amount of transactions each. A clear example of this is witnessed by the financial category, where 6 Bitcoin contracts% \footnote{Bitcoin financial contracts: \href{https://www.colu.com/}{Colu}, \href{http://coinspark.org/}{CoinSpark}, \href{https://github.com/OpenAssets}{OpenAssets}, \href{http://www.omnilayer.org/}{Omni}, \href{https://www.smartbit.com.au/}{SmartBit}, \href{https://bitpos.me/}{BitPos}. } totalize two thirds of the transactions published by the 373 Ethereum contracts in the same category. While both Bitcoin and Ethereum are mainly focussed on financial contracts, we observe major differences about the other categories. For instance, the Bitcoin contracts in the Notary category% \footnote{Bitcoin notary contracts: \href{https://www.factom.com/}{Factom}, \href{https://stampery.com/}{Stampery}, \href{https://proofofexistence.com/}{Proof of Existence}, \href{https://blocksign.com/}{Blocksign}, \href{https://crypto-copyright.com/}{CryptoCopyright}, \href{https://stampd.io/}{Stampd}, \href{https://bitproof.io/}{BitProof}, \href{https://github.com/thereal1024/ProveBit}{ProveBit}, \href{https://remembr.io/}{Remembr}, \href{https://originalmy.com/}{OriginalMy}, \href{http://lapreuve.eu/explication.html}{LaPreuve}, \href{http://digitalcurrency.unic.ac.cy/free-introductory-mooc/academic-certificates-on-the-blockchain/}{Nicosia}, \href{http://www.chainpoint.org/}{Chainpoint}, \href{http://diploma.report/}{Diploma}, \href{https://monegraph.com/}{Monegraph}, \href{https://blockai.com/}{Blockai}, \href{https://www.ascribe.io}{Ascribe}, \href{https://eternitywall.it/}{Eternity Wall}, \href{https://github.com/blockstack/blockchain-id/wiki/Blockstore}{Blockstore}. } have an amount of transactions similar to that of the Financial category, unlike in Ethereum. The second most used category in Ethereum is Game. Although some games (e.g.\@\xspace, lotteries~\cite{Andrychowicz14sp,Andrychowicz16cacm,Back13coin,Bentov14crypto} and poker~\cite{Kumaresan15ccs}) which run on Bitcoin have been proposed in the last few years, the interest on them is still mainly academic, and we have no experimental evidence that these contracts are used in practice. Instead, the greater flexibility of the Ethereum programming language simplifies the development of this kind of contracts (although with some quirks~\cite{Delmolino15bw} and limitations% \footnote{% Although the Ethereum virtual machine is designed to be Turing-complete, in practice the limitations on the amount of gas which can be used to invoke contracts also limit the set of computable functions (e.g.\@\xspace, verifying checkmate exceeds the current gas limits of a transaction~\cite{Grau16chess}). }). Note that in some cases there are not enough elements to categorise a contract. This happens e.g.\@\xspace, when the contract does not link to the project webpage, and there are neither comments in online forums nor in the contract sources. \subsection{A taxonomy of smart contracts} \label{sec:smartContracts:taxonomy} We propose a taxonomy of smart contracts into five categories, which describe their intended application domain. We then classify the contracts in our sample according to the taxonomy. To this purpose, for Ethereum contracts we manually inspect the Solidity source code, while for Bitcoin contracts we search their web pages and related discussion forums. After this manual investigation, we distribute all the contracts into the five categories, that we present below. \begin{description} \item[Financial.] Contracts in this category manage, gather, or distribute money as preeminent feature. % Some contracts certify the ownership of a real-world asset, endorse its value, and keep track of trades (e.g.\@\xspace, \href{http://coloredcoins.org/explorer/}{Colu} currently tracks over 50,000 assets on Bitcoin). Other contracts implement crowdfunding services, gathering money from investors in order to fund projects (the Ethereum \href{https://forum.daohub.org/}{DAO} project was the most representative one, until its collapse due to an attack in June 2016). High-yield investment programs are a type of Ponzi schemes~\cite{Bartoletti17ponzi} that collect money from users under the promise that they will receive back their funds with interest if new investors join the scheme (e.g.\@\xspace, \href{http://governmental.github.io/GovernMental/}{Government}, \href{https://www.kingoftheether.com/}{KingOfTheEtherThrone}). Some contracts provide an insurance on setbacks which are digitally provable (e.g.\@\xspace, \href{https://fdi.etherisc.com/}{Etherisc} sells insurance policies for flights; if a flight is delayed or cancelled, one obtains a refund). Other contracts publish advertisement messages (e.g.\@\xspace, \href{http://pixelmap.io/}{PixelMap} is inspired to the \href{https://en.wikipedia.org/wiki/The_Million_Dollar_Homepage}{Million Dollar Homepage}). \item[Notary.] Contracts in this category exploit the immutability of the blockchain to store some data persistently, and in some cases to certify their ownership and provenance. Some contracts allow users to write the hash of a document on the blockchain, so that they can prove document existence and integrity (e.g.\@\xspace, \href{https://proofofexistence.com/}{Proof of Existence}). Others allow to declare copyrights on digital arts files, like photos or music (e.g.\@\xspace, \href{https://monegraph.com/}{Monegraph}). Some contracts (e.g.\@\xspace, \href{https://eternitywall.it/} {Eternity Wall}) just allow users to write down on the blockchain messages that everyone can read. Other contracts associate users to addresses (often represented as public keys), in order to certify their identity (e.g.\@\xspace, \href{https://proofofphysicaladdress.com/}{Physical Address}). \item[Game.] This category gathers contracts which implement \emph{games of chance} (e.g.\@\xspace, \href{https://etherscan.io/address/0x2ef76694fBfD691141d83F921A5ba710525De9B0#code}{LooneyLottery}, \href{https://etherscan.io/address/0x2AB9f67A27f606272189b307052694D3a2B158bA#code}{Dice}, \href{https://etherscan.io/address/0x18a672e11d637fffadccc99b152f4895da069601#code}{Roulette}, \href{https://etherscan.io/address/0x1d77340D3819007BbfD7fdD37C22BD3b5c311350#code}{RockPaperScissors}) and \emph{games of skill} (e.g.\@\xspace, \href{http://www.bspend.com/etherization}{Etherization}), as well as some games which mix chance and skill (e.g.\@\xspace, \href{https://etherscan.io/address/0x4ed65e408439a7f6459b5cfbd364f373bd6ed5f7#comments}{PRNG challenge} pays for the solution of a puzzle). \item[Wallet.] The contracts in this category handle keys, send transactions, manage money, deploy and watch contracts, in order to simplify the interaction with the blockchain. Wallets can be managed by one or many owners, in the latter case requiring multiple authorizations (like, e.g.\@\xspace in \href{https://etherscan.io/address/0xA2D4035389aae620E36Bd828144b2015564C2702#code}{Multi-owned}). \item[Library.] These contracts implement general-purpose operations (like e.g.\@\xspace, math and string transformations), to be used by other contracts. \end{description} \section{Analysing the usage of smart contracts} \label{sec:smartcontracts} In this~\namecref{sec:smartcontracts} we analyse the usage of smart contracts, proposing a classification which reflects their application domain. Then, focussing on Bitcoin and Ethereum, we quantify the usage of smart contracts in relation to their application domain. We start by presenting the methodology we have followed to sample and classify Bitcoin and Ethereum smart contracts (\Cref{subsection:smartcontractmethodology}). Then, we introduce our classification and our statistical analysis (\Cref{sec:smartContracts:taxonomy,sec:smartContracts:quantitative}).

proofpile-arXiv_069-9614

\section{Introduction} The electroweak symmetry breaking which we believe is a cornerstone in the Higgs physics could be restored at high temperature i.e. in the early universe \cite{Weinberg:1974hy,Dolan:1973qd}. The spontaneous symmetry breaking occurs by having the universe expanded and cooled and going from the symmetric phase to the broken phase through the {\it electroweak phase transition} (EWPT). An open problem in physics is the mysterious matter-antimatter asymmetry in the universe. The standard model (SM) together with the EWPT seem to be able to explain the exceed of the matter abundance against the antimatter we observe today. This is called the {\it electroweak baryogenesis} (EWBG). The necessary conditions that lead to EWBG is known as the three criteria of Sakharov \cite{Sakharov:1967dj}: Baryon number violation, C and CP violation and the thermal non-equilibrium. The standard model possesses all these conditions however at the temperature of the electroweak symmetry breaking, $T_{\text{EW}}\sim 100$ GeV, the electroweak phase transition must be of first-order type in order to support the EWBG. This can be translated into {\it washout criterion}, $v(T_c)/T_c>1$ which provides the appropriate sphaleron rate for the baryogenesis. In the standard model framework the washout condition impose an upper limit on the Higgs mass: $m_H<48$ GeV \cite{Shaposhnikov:1986jp} which is in contrast with the Higgs discovery ($m_H\sim 125$ GeV) at the LHC in 2012 \cite{Aad:2012tfa,Chatrchyan:2012xdj}. Therefore the SM alone is not enough to embed the EWBG and an extension to SM looks necessary if the electroweak baryogenesis is the right mechanism to produce the baryon asymmetry. On the other hand, almost $25\%$ of the universe is made of what we call it {\it dark matter} (DM). It does not have strong interaction with the ordinary matter and its existence can be traced only through its gravitational attraction on matter \cite{Adam:2015rua,Hinshaw:2012aka}. Again the SM is not capable of explaining this extra gravitational force in the galaxies and clusters. Inevitably one should search for an answer in a theory beyond the standard model. One of the successful DM scenarios is the {\it weakly interacting massive particle} (WIMP) which demands additional fundamental fields (particles) respect to the SM. The new particle(s) stay in thermal equilibrium with other SM particles in the early universe, but when the universe cools down while expanding, it freezes-out from the hot plasma of particles. The self-annihilation cross section of the WIMP must be of order $\braket{\sigma v}=3\times 10^{-26}~cm^3 ~s^{-1}$ to have the correct amount of DM abundance to be $\Omega h^2\sim 0.11$ measured by Planck/WMAP. In this work we try to address both the electroweak baryogenesis and the dark matter issues in one single theory. We extend the standard model by adding an extra Dirac fermion playing the role of the dark matter candidate and a (pseudo)scalar mediating between the dark sector and the SM sector \cite{Ghorbani:2014qpa}. \footnote{See \cite{Ghorbani:2017qwf} for the analysis of the renormalization group equation for the same model with a pseudoscalar mediator.} The (pseudo)scalar and the fermionic dark matter modify the thermal effective potential; hence affect the critical temperature for the electroweak phase transition. We analytically provide the critical temperature and the global minimum at the critical temperature to give the washout criterion. Many works have provided the critical temperature from the free energy by numerical computation. Some examples of the works done considering both the dark matter and the baryogenesis are \cite{Gu:2010ft,Cline:2012hg,Fairbairn:2013xaa,Lewicki:2016efe, Jiang:2015cwa,Chowdhury:2011ga,Menon:2004wv,Barger:2008jx,Li:2014wia,Addazi:2017gpt}. Here we provide an analytical expression for the critical temperature which makes it easier to impose the washout condition on the dark matter model. If the SM-DM mediator is a pseudoscalar the dark matter cross section off nucleon is negligible while if the mediator is a scalar then the direct detection experiments e.g. XENON100/LUX put a strong constraint on the space parameter. Having emphasized this point, we show in this work that the dark matter model with a pseudoscalar mediator is more successful in explaining the electroweak baryogenesis respect to when we use the scalar mediator. This paper is organized as the following: in the next section we introduce the model by extending the SM. In section \ref{ewpt} the details of the electroweak phase transition and the critical temperature are given. In section \ref{dm} the dark matter candidate is introduced. In the next section, the numerical results that shows the consistency of the model with relic density and the baryon asymmetry is presented. The appendix \ref{1loop} provides some analytical details on the effective potential. \section{The Model}\label{model} We extend the standard model by adding two new fields as follows: a Dirac fermion denoted here by $\psi$, which plays the role of the dark matter candidate, and a (pseudo)scalar denoted by $s$ which mixes with the Higgs field, $H$, and interacts with the dark matter as well through a Yukawa term. The Lagrangian can be written in its parts as, \begin{equation} {\cal L}={\cal L}_{\text{SM}}(H)+{\cal L}_{\text{dark}}(\psi)+{\cal L}_{s}(s)+{\cal L}_{\text{int}}\left(s,\psi,H\right)\,, \end{equation} where $\mathcal{L}_{\text{SM}}$ stands for the SM Lagrangian, ${\cal L}_{\text{dark}}$ for the fermionic dark matter Lagrangian, \begin{equation}\label{darkl} {\cal L}_{\text{dark}}\left(s,\psi\right)=\bar{\psi}\left(i\gamma_{\mu}\partial^{\mu}-m_{d}\right)\psi \, , \end{equation} ${\cal L}_{s}$ for the (pseudo)scalar Lagrangian, \begin{equation} {\cal L}_{s}=\cfrac{1}{2}\left(\partial_{\mu}s\right)^{2}-\frac{1}{2}\mu_{s}^{2}s^{2}-\frac{1}{4}\lambda_{s}s^{4}\, , \end{equation} and ${\cal L}_{\text{int}}$ is the (pseudo)scalar interaction with the dark and the SM sectors. When $s$ is a pseudoscalar then, \begin{equation}\label{intl1} {\cal L}_{\text{int}}\left(s,\psi,H\right)=g_{d}s\bar{\psi}\gamma^{5}\psi+\frac{1}{2}\lambda s^{2}H^{\dagger}H \, , \end{equation} and when $s$ represents the scalar, \begin{equation}\label{intl2} {\cal L}_{\text{int}}\left(s,\psi,H\right)=g_{d}s\bar{\psi}\psi+\frac{1}{2}\lambda s^{2}H^{\dagger}H \, . \end{equation} In the next sections we investigate both the scalar and the pseudoscalar cases and use the same notation for the coupling $g_d$. Note that the interaction Lagrangian does not include the odd-terms in the (pseudo)scalar-Higgs interaction terms. In order to stay in a more restricted theory with as small parameter space as possible, despite many authors we consider a less general Lagrangian for our model. The Higgs potential in the SM sector reads, \begin{equation} V_{H}=-\mu_{h}^{2}H^{\dagger}H-\lambda_{h}\left(H^{\dagger}H\right)^{2}\,. \end{equation} Both the Higgs and the (pseudo)scalar take non-zero vacuum expectation values at the low temperature. In section \ref{ewpt} to give the thermal effective potential as a function of the condensate $\braket{h}$ we begin with the tree-level potential which is given by the substitution $s \equiv \braket{s}$ and $H=\left(0\,\,\,h\equiv \braket{h} \right)^{\dagger}$ So the tree-level potential reads, \begin{equation}\label{v0} V_{0}(h,s)=-\frac{1}{2}\mu_{h}^{2}h^{2}-\frac{1}{2}\mu_{s}^{2}s^{2}+\frac{1}{4}\lambda_{h}h^{4} +\frac{1}{4}\lambda_{s}s^{4}+\frac{1}{2}\lambda s^{2}h^{2}. \end{equation} Note that we have gauged away three degrees of freedom of the Higgs doublet. \section{First-Order Phase Transition}\label{ewpt} In this section we provide the ``washout criterion'' and other necessary conditions in terms of the parameters used in the model introduced above to support the first-order electroweak phase transition. The washout criterion or $v(T_c)/T_c>1$ which provides the appropriate sphaleron rate for the phase transition to be first-order is obtained from the effective potential of the theory. In addition to the tree-level barrier we also consider the one-loop barrier. The total thermal effective potential is therefore, \begin{equation} V_{\text{eff}}=V_0(h,s)+V_{\text{1-loop}}(h,s;0)+V_{\text{1-loop}}(h,s;T)\,, \end{equation} where $V_0$ is the tree-level potential in eq. (\ref{v0}), $V^{\text{1-loop}}(h,s;0)$ is the Coleman-Weinberg one-loop correction at zero temperature \cite{Coleman:1973jx} and $V^{\text{1-loop}}(h,s;T)$ is the one-loop thermal correction \cite{Dolan:1973qd}. In the high-temperature approximation when $m_i^2/T^2 \ll 1$ for all $i$ with $m_i$ being the mass of the particle $i$ in the model, the one-loop effective potential takes the following form, \begin{equation}\label{highT} V_{\text{1-loop}}^{\text{high-T}}\left(h,s;T\right) \approx \left(\frac{1}{2}\kappa_{h}h^{2} +\frac{1}{2}\kappa_{s}s^{2}\right)T^{2}\,, \end{equation} with \begin{equation}\label{kaph} \kappa_{h}=\frac{1}{48}\left(9g^{2}+3g'^{2}+12g_{t}^{2}+24\lambda_{h}+4\lambda\right)\,, \end{equation} \begin{equation}\label{kaps} \kappa_{s}=\frac{1}{12}\left(4\lambda+3\lambda_{s} \pm 2g_{d}^{2}\right)\,, \end{equation} where in eq. (\ref{kaps}) the minus sign stands for the pseudoscalar case and the plus sign is for the scalar case. See appendix \ref{1loop} for more details. Note that we have dropped the Colman-Wienberg zero-temperature correction since at high temperature approximation (at temperature of the electroweak phase transition) only the thermal corrections are dominant. Moreover, including the zero-temperature contribution will only complicate the analytic computations. In eq. (\ref{kaph}) the parameters $g$ and $g'$ are respectively the $SU(2)_L$ and $U(1)_Y$ standard model couplings and $g_t$ is the top quark coupling. We have contributed only the heavier particles that couple stronger to the Higgs, i.e. the top quark, $t$, the gauge bosons, $W^{\pm}$ and $Z$, and the Higgs, $h$. We have ignored the lighter quarks, gluons and the leptons. In order to have a first-order phase transition the effective potential must have two degenerate minima at the critical temperature. The minima are located at \begin{equation} \frac{\partial V_{\text{eff}}}{\partial s}|_{w\left(T\right)=<s>}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ \frac{\partial V_{\text{eff}}}{\partial h}|_{v\left(T\right)=<h>}=0\,, \end{equation} where leads to \begin{equation} v_{\text{sym}}=0\,\,\,\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,\,\,\,\,v_{\text{brk}}^{2}\left(T\right) =\frac{\mu_{h}^{2}-\kappa_{h}T^{2}-\lambda w^{2}\left(T\right)}{\lambda_{h}}\label{vev1}\,, \end{equation} \begin{equation} v_{\text{brk}}^{2}\left(T\right)=\frac{\mu_{s}^{2}-\kappa_{s}T^{2}-\lambda_{s}w^{2}\left(T\right)}{\lambda}\label{vev2}\,. \end{equation} The stability conditions are obtained by setting the positivity of the second derivatives of the potential after the symmetry breaking: \begin{equation} \lambda_{s}>0,\,\,\,\,\,\lambda_{h}>0,\,\,\,\,\,4\lambda_{s}\lambda_{h}>\lambda^{2}\label{stab}\,. \end{equation} Eqs. (\ref{vev1}) and (\ref{vev2}) leads to \begin{equation} w_{\text{brk}}^{2}\left(T\right)=\alpha+\beta T^{2}\,, \end{equation} where \begin{equation} \alpha=\frac{\lambda\mu_{h}^{2}-\lambda_{h}\mu_{s}^{2}}{\lambda^{2}-\lambda_{h}\lambda_{s}}\,, \end{equation} \begin{equation} \beta=\frac{\lambda\kappa_{h}-\lambda_{h}\kappa_{s}}{\lambda^{2}-\lambda_{h}\lambda_{s}}\,. \end{equation} Using eq. (\ref{vev1}) one gets the temperature-dependent Higgs vacuum expectation value in the broken phase, \begin{equation}\label{vc} v_{\text{brk}}^{2}\left(T\right)=\left(\frac{\mu_{h}^{2}-\lambda\alpha}{\lambda_{h}}\right)-\left(\frac{\kappa_{h} +\lambda\beta}{\lambda_{h}}\right)T^{2}\,. \end{equation} The broken phase can exist up to a maximal temperature, \begin{equation} T_{\text{max}}=\sqrt{\frac{\mu_{h}^{2}-\lambda\alpha}{\kappa_{h}+\lambda\beta}},\,\,\,\,\,\,\mu_{h}^{2}>\lambda\alpha\,. \end{equation} At the critical temperature the free energy has two degenerate minima one at \\ $\left(v_{\text{sym}}=0,w_{\text{sym}}=0\right)$ and the other at $\left(v_{\text{brk}},w_{\text{brk}}\right)$, \begin{equation}\label{freee} V_{\text{eff}}^{\text{sym}}\left(T_{c}\right)=V_{\text{eff}}^{\text{brk}}\left(T_{c}\right)=0\,, \end{equation} which solves the free energy eq. (\ref{freee}) at the critical temperature as, \begin{equation}\label{Tc} T_c^{\pm}= \sqrt{x\pm y^{1/2}/z}\,, \end{equation} with \begin{equation}\label{xyz} \begin{split} &x= {\kappa_h} {\lambda_h} (\lambda {\mu^2_s}-{\lambda_s} {\mu^2_h}) +{\kappa_s} {\lambda_h} (\lambda {\mu^2_h}-{\lambda_h} {\mu^2_s})\,,\\ &y= {\kappa_h}^2 \lambda_h^2 \left(-8 \lambda ^3 {\mu^2_h} {\mu^2_s} + \lambda ^2 \left(5 {\lambda_h} {\mu_s}^4+4 {\lambda_s} {\mu_h}^4\right)-{\lambda_h}^2 {\lambda_s} {\mu_s}^4\right)\\ & +2 {\kappa_h} {\kappa_s} {\lambda_h^3} \left(7 \lambda ^2 {\mu^2_h} {\mu^2_s} -4 \lambda \left({\lambda_h} {\mu_s}^4+{\lambda_s} {\mu_ h}^4\right)+{\lambda_h} {\lambda_s} {\mu^2_h} {\mu^2_s}\right)\\ &+{\kappa_s}^2 {\lambda_h^3} \left(\lambda ^2 {\mu_h}^4-8 \lambda {\lambda_h} {\mu^2_h} {\mu^2_ s} +{\lambda_h} \left(4 {\lambda_h} {\mu_s}^4+3 {\lambda_s} {\mu_ h}^4\right)\right)\,,\\ &z={\kappa_h}^2 \left(4 \lambda ^2-{\lambda_h} {\lambda_s}\right) -6 {\kappa_h} {\kappa_s} \lambda {\lambda_h}+3 {\kappa_s}^2 {\lambda_h}^2\,. \end{split} \end{equation} The strong electroweak phase transition occurs only if \begin{equation}\label{wash} v\left(T_{c}\right)/T_{c}>1\,, \end{equation} which is a strong constraint on the parameter space of our dark matter model. Note that for each set of the couplings in eq. (\ref{xyz}) there are two critical temperatures, $T_c^{\pm}$, if $x\pm y^{1/2}/z>0$. We will see in section \ref{dm} that for both solutions there exist a viable parameter space. \section{Fermionic Dark Matter}\label{dm} We are assuming that the first-order phase transition is occurring in a temperature higher than the dark matter freeze-out temperature, i.e. $T_{c}>T_{F}$. After the symmetry breaking both Higgs particle and the scalar field undergo non-zero {\it vev}: \begin{equation} h\rightarrow v_{h}+h\label{shifth}\,, \end{equation} \begin{equation} s\rightarrow v_{s}+s\label{shifts}\,. \end{equation} This will cause to a three potential different from eq. (\ref{v0}). One can redefine the scalar fields $h$ and $s$ in order to diagonalize the mass matrix as follows, \begin{equation} \begin{split} h\rightarrow h\,\cos\theta+s\,\sin\theta \,, \\ s\rightarrow-h\,\sin\theta+s\,\cos\theta\,. \end{split} \end{equation} For simplicity we denote the new fields as $h$ and $s$ again but one should be noted that these are the eigenstates of the diagonal mass matrix now. Plugging eqs. (\ref{shifth}) and (\ref{shifts}) in eq. (\ref{v0}) and imposing the minimization condition we get \begin{equation} \mu_{s}^{2}=-\lambda_{s}v_{s}^{2}-\lambda v_{h}^{2}\label{mus}\,, \end{equation} \begin{equation} \mu_{h}^{2}=-\lambda_{h}v_{h}^{2}-\lambda v_{s}^{2} \label{muh}\,. \end{equation} The second derivative of the potential in eq. (\ref{v0}) after shifting the origin to the non-zero vacuum are, \begin{equation} \tilde{m}_{s}^{2}=2\lambda_{s}v_{s}^{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\tilde{m}_{h}^{2} =2\lambda_{h}v_{h}^{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\tilde{m}_{hs}^{2}=\lambda v_{h}v_{s} \,. \end{equation} Diagonalizing the mass matrix by a rotation by the angle $\theta$, the couplings will be related to the masses as, \begin{equation} \begin{split} \lambda_{h}=\frac{m_{s}^{2}\sin^{2}\theta+m_{h}^{2}\cos^{2}\theta}{2v_{h}^{2}}\,,\\ \lambda_{s}=\frac{m_{s}^{2}\cos^{2}\theta+m_{h}^{2}\sin^{2}\theta}{2v_{s}^{2}}\,,\\ \lambda=\frac{m_{s}^{2}-m_{h}^{2}}{2v_{h}v_{s}}\sin2\theta\,, \end{split} \end{equation} where $m_{s}^{2}$ and $m_{h}^{2}$ are the diagonalized masses and \begin{equation} \tan\theta=\frac{y}{1+\sqrt{1+y^{2}}},\,\,\,\,\,\,\,\,\,\,y=\frac{2\lambda v_{s}v_{h}}{\lambda_{h}v_{h}^{2} -\lambda_{s}v_{s}^{2}}\,. \end{equation} The independent parameters of the model can then be chosen as $m_{h}$, $m_{s}$, $m_{d}$, $g_d$ and $\sin \theta$. All the other parameters including the mass terms and the couplings in the Lagrangian are expressed in terms of these five parameters. Note that all we have said in this section is true for both scalar and pseudoscalar mediators. In the next section we see how the numerical results changes for them. \section{Numerical Results}\label{nr} Our computations are two-folded. One part is the computations for the model with a pseudoscalar mediator and the other part is the computations with a scalar mediator. We recall that the difference they make in strongly first-order phase transition, section \ref{ewpt}, stem from eq. (\ref{kaps}) where the plus sign stands for the scalar and the minus sign stands for the pseudoscalar. In dark matter side apart from the different results for the relic density for each type of mediator, one should note an important point; when choosing the mediator in our dark matter scenario to be a pseudoscalar the elastic scattering cross section for DM-nucleon is velocity suppressed \cite{Esch:2013rta,LopezHonorez:2012kv,Pospelov:2011yp,Ghorbani:2014qpa} and so the theory easily evades the direct detection bounds from LUX/XENON100/XENON1T \cite{Akerib:2016vxi,Aprile:2016swn,Aprile:2017iyp}. However this is not the case for a scalar mediator. A large region of the parameter space in the theory is cut by the direct detection bounds imposed from XENON100/LUX. Therefore, for the pseudoscalar mediator we have two type of constraints to impose: the relic density and the first-order phase transition condition. For the scalar mediator the direct detection constraint must be added to the above conditions. To compute the relic density for the dark matter one should solve the Boltzmann differential equation numerically. We have exploited the {\tt MicrOMEGAs4.3} package \cite{Belanger:2014vza} to obtain the relic abundance. The Higgs vacuum expectation value and the Higgs mass are known, $v_h=246$ GeV and $m_h=125$ GeV. The {\it vev} for the pseudoscalar and for the scalar as well is chosen to be $v_s=600$ GeV \footnote{ A caveat here is that additional (pseudo)scalars in the extended SM with non-zero changing {\it vev} during the electroweak phase transition may lead to supersonic bubble wall velocity that prevent the phase transition from being strong enough for EWBG \cite{Bodeker:2009qy,Kozaczuk:2015owa}.}. In both cases, we search for the viable parameter space bounded by the relic density value $\Omega h^2= 0.11$ and the washout condition being $v(T_c)/T_c>1$ with $v(T_c)$ and $T_c$ given in eqs. (\ref{Tc}) and (\ref{vc}). We scan the space of parameters in the ranges as GeV $1<m_s<1$ TeV, the dark matter mass GeV $10<m_d<2$ TeV, the Yukawa coupling $0<g_d<3$ and the mixing angle being fixed at $\sin \theta=0.1$. \begin{figure} \begin{minipage}[b]{.45\textwidth} \includegraphics[scale=.35, angle=270]{ewptm-pseudoscalar.eps} \end{minipage}\hspace{1.5cm} \begin{minipage}[b]{.45\textwidth} \includegraphics[scale=.35, angle=270]{ewptp-pseudoscalar.eps} \end{minipage} \caption{The allowed DM mass, $m_d$, against the Yukawa coupling, $g_d$, in the fermionic dark matter scenario with pseudoscalar mediator, bounded from the relic density and first-order phase transition conditions for {\it left)} $v^-_c/T^-_c>1$ with DM mass in the range $140-310$ GeV {\it right)} $v^+_c/T^+_c>1$ with DM mass in the range $60-320$ GeV. } \label{pseudo-plot} \end{figure} \begin{figure} \begin{minipage}[b]{.45\textwidth} \includegraphics[scale=.35, angle=270]{ewptm-scalar.eps} \end{minipage}\hspace{1.5cm} \begin{minipage}[b]{.45\textwidth} \includegraphics[scale=.35, angle=270]{ewptp-scalar.eps} \end{minipage} \caption{The allowed DM mass, $m_d$, against the Yukawa coupling, $g_d$, in the fermionic dark matter scenario with scalar mediator, bounded from the relic density, first-order phase transition condition, and the direct detection bound by XENON100/XENON1T/LUX for {\it left)} $v^-_c/T^-_c>1$ with DM mass in the range $150-300$ GeV for XENON100/LUX and no viable DM mass for the future direct detection bound by XENON1T {\it right)} $v^+_c/T^+_c>1$ with no viable DM mass.} \label{scalar-plot} \end{figure} As we have demonstrated in Figs. \ref{pseudo-plot} and \ref{scalar-plot} that despite the strong constraints on the parameter space we end up with a viable region for the theory Fig. \ref{pseudo-plot} shows the allowed DM mass against the Yukawa coupling for the dark matter model with the pseudoscalar mediator and Fig. \ref{scalar-plot} represents the same quantities for the scalar mediator. The constraints considered in two figures are the amount of the relic abundance from WMAP/Planck, the washout criterion for two critical temperature found in eq. (\ref{Tc}) and the DM-nucleon elastic scattering cross section bound from XENON100/XENON1T/LUX. It is interesting to point out the difference in two figures; the fermionic dark matter model with a pseudoscalar mediator Fig. \ref{pseudo-plot} is more successful respect to the same model with a scalar mediator Fig. \ref{scalar-plot} to accommodate the electroweak baryogenesis and the dark matter issues. The allowed DM mass in Fig. \ref{pseudo-plot} is in the range GeV $60 <m_d< 320$ Gev for the $T_c^+$ solution and in the range GeV $140 <m_d< 310$ Gev for the $T_c^-$ solution. Therefore for both critical temperatures obtained in eq. (\ref{Tc}) the pseudoscalar scenario respect the relic density and the first-order phase transition. The allowed DM mass for the scalar scenario is only in the range GeV $150 <m_d< 300$ Gev for the $T_c^-$ solution which can easily be excluded by the future direct detection experiments such as XENON1T. This point has already been reported in \cite{Li:2014wia}. For $T_c^+$ solution there is no viable region in the scalar scenario. \subsection*{Indirect Detection Bounds} The DM annihilation into SM particles is very suppressed after the thermal freeze-out in the early universe. Nevertheless, today in regions with high density of DM for instance in the Galactic Center (GC) the DM annihilation is probable. It is now well accepted from {\it Fermi} Large Area Telescope ({\it Fermi}-LAT) $6.5$ years data that the gamma rays coming from the center of the galaxies are brighter for a few GeV than expected from other known sources \cite{TheFermi-LAT:2017vmf}. The analyses of the GC gamma ray excess after considering different uncertainties puts an upper limit on the dark matter annihilation cross section in terms of the DM mass and the annihilation channel. Here we examine our model against the {\it Fermi}-LAT bounds on DM annihilation cross section for two representative channels $b\bar b$ and $\tau^+\tau^-$ which is relevant for DM masses up to $100$ GeV \cite{TheFermi-LAT:2017vmf} \footnote{Many thanks to Andrea Albert and Dmitry Malyshev for providing me with the {\it Fermi}-LAT data on Fig. 34 in \cite{TheFermi-LAT:2017vmf}}. For the current model we have computed the velocity-averaged annihilation cross section into two channels $b\bar b$ and $\tau^+ \tau^-$ for DM mass up to $100$ GeV. All other constraints considered in the last sections are imposed in this computation. It is evident from Fig. \ref{indir} that for both channels the DM mass of $\sim 61-62$ GeV obtained before for the pseudoscalar mediator case in $T_c^+$ solution (right plot in Fig. \ref{pseudo-plot}) is excluded by the {\it Fermi}-LAT limit. We have exploited the {\it Fermi}-LAT upper limit obtained from the generalized NFW dark matter density profile ($\gamma=1.25$) in the center of the Galaxy. \begin{figure} \centering \includegraphics[scale=.35, angle=270]{tautau-bb-ann.eps} \caption{The dark matter velocity-averaged annihilation cross section into $b\bar b$ (blue) and $\tau^+ \tau^-$ (red) for the DM mass up to $100$ GeV. For both channels the DM masses $61-62$ GeV shown in Fig. \ref{pseudo-plot} are excluded by the {\it Fermi}-LAT $b\bar b$ and $\tau^+ \tau^-$ upper limits.} \label{indir} \end{figure} After considering the {\it Fermi}-LAT constraint the viable DM mass with pseudoscalar mediator will be in the range $\sim 110-320$ GeV (see Fig. \ref{pseudo-plot}). Therefore the SM Higgs particle cannot decay into dark matter particle, hence the model is not bounded by invisible Higgs decay constraint. In Fig. \ref{mediator-DM} the DM mass is depicted against the pseudoscalar mediator mass considering all the aforementioned constraints. The points observed in this figure are as the following. First, for both $T_c^+$ and $T_c^-$ solutions the strongly first-order phase transition occurs in higher temperature for heavier dark matter particle. Second, the mass of the pseudoscalar mediator is in the range $290-620$ GeV. Again the LHC exotic Higgs decay constraint which requires $m_s<m_h/2$ is not applicable here. The mono-jet searches at the LHC could also restrict DM models specially for low DM mass and heavier mediators where the dark matter production cross section becomes larger. It is shown in \cite{Baek:2017vzd} that even for $m_s>2m_d$ the LHC mono-jet search limit \cite{Aaboud:2016tnv} does not constrain more the parameter space for the current model. \begin{figure} \begin{minipage}[b]{.46\textwidth} \includegraphics[scale=.35, angle=270]{mediator-DMp.eps} \end{minipage}\hspace{1.5cm} \begin{minipage}[b]{.46\textwidth} \includegraphics[scale=.35, angle=270]{mediator-DMm.eps} \end{minipage} \caption{The pseudoscalar mediator mass against the allowed DM mass for {\it left)} $T^+_c$ solution {\it right)} $T^-_c$ solution. The critical temperature is shown in the color spectrum.} \label{mediator-DM} \end{figure} \section{Conclusion} In this paper we have examined a fermionic dark matter model possessing a pseudoscalar or a scalar as the mediator whether it could explain the electroweak baryogenesis as well as evading the dark matter constraints. The (pseudo)scalar in the model is interpreted as a second Higgs-like particle beside the SM Higgs. We assumed that the scalars have zero vacuum expectation values in the symmetric phase but as the temperature comes down with the expansion of the universe, the Higgs and the (pseudo)scalar undergo a non-zero vacuum expectation value and the electroweak phase transition takes place. We have obtained the critical temperature and the washout criterion analytically. Then we have computed the relic density using the {\tt micrOMEGAs} package while we have considered all the constraints for the electroweak phase transition to be first-order. The numerical results show that there exist a viable space of parameters when the mediator is chosen to be a pseudoscalar with the dark matter mass in the range of $110-320$ GeV plus the resonance area with $m_d=61-62$ GeV. If the SM-DM mediator in the theory is taken to be a scalar we have shown that there is only a small region that can satisfy all the constraints including the direct detection bound from XENON100/LUX and that will be excluded with the new bounds from the experiments such as XENON1T. We then constrained more the model with the pseudoscalar mediator by the {\it Fermi}-LAT upper limit on DM annihilation into $b\bar b$ and $\tau^+ \tau^-$ which put a bound for DM mass up to $100$ GeV. We have shown that this bound excludes the DM mass $61-62$ GeV we obtained before and the DM mass in the viable space becomes $m_d=110-320$ GeV. The pseudoscalar mediator mass afterwards turned out to be $m_s=290-620$ GeV, that means the model cannot be constrained more from exotic and invisible Higgs decay bounds at the LHC. It has also been pointed out that the LHC mono-jet search limit does not affect the viable space of parameters.

proofpile-arXiv_069-9676

\section{Introduction} Quantum computing in the long-term promises speedup over classical algorithms for important algorithms like unordered search and integer factorization \cite{grovers}\cite{shor}. Unfortunately, we are many years away from a large-scale implementation of either of these algorithms because current quantum computers have limited number available hardware qubits \cite{ionq, ibmq, bristlecone, rigetti}, but large numbers are required to produce high quality error-correct logical qubits. In the near-term, however, there are many promising applications, like variational algorithms such as VQE and QAOA \cite{vqe}\cite{qaoa}, which aim to solve classical optimization problems using the quantum computer as a sort of accelerator in a classical-quantum optimization loop. For these types of algorithms, the size of the optimization problem we can effectively execute is bounded by device error rates, qubit coherence times, and ultimately by the number of physically available devices \cite{NISQ}. In order to extend the boundary of what is currently computable, optimizations aim to minimize gate counts, circuit duration, or physical requirements. One such optimization has been the use of multivalued quantum logic. Many quantum systems naturally have access to infinitely many discreet levels beyond the usual binary qubit levels \cite{MS-gates}. At face value, rewriting quantum algorithms in a higher radix confers the same constant advantage in terms of gate requirements and expected circuit duration as it would classically, where advantage is due strictly due to space compression. In the long term, this approach may not be enormously advantageous as the space savings may not outweigh the increase in error, especially when the number of available devices increases significantly. In the near-term, space savings are critical. Alternative approaches, such as the use of \textit{intermediate qudits} \cite{intermediate-qutrits}, rely on a mixed radix strategy where a smaller number of qudits are used, or they are used only for a short amount of time. While specialized in its use, the expected advantages are strong, for example in the generalized Toffoli decomposition (also the subject of this work) using qutrits (3 level quantum systems) temporarily enables a logarithmic depth (approximately circuit duration) decomposition with linear number of two-qudit gates and requires no additional ancilla space, scratch bits. The best known qubit-only circuits can obtain logarithmic depth with linear gate counts \cite{qubit-toffoli}. These circuits require linear amounts of additional space which makes efficient implementations infeasible on error-limited devices and bounding the program size on available hardware. While the advantage conferred from clever circuit design is clear, these prior works omit two critical considerations from their constructions. First, these circuits are not compiled to any real device. While some devices promise all-to-all connectivity, most devices have limited connections between devices meaning programs must have movement operations inserted during compilation. This introduces large amounts of communication overhead, on average, which must be minimized \cite{swap-min}. Furthermore, swapping states different dimensional qudits requires tailored communication operations which get more expensive as the dimension of the inputs increases. Second, qubit gates and qudit gates are not created equal. From a circuit point of view, they take up one unit of time each, but when implemented on device this may not be the case. To execute gates on hardware, analog pulse sequences must be generated, for example via a process called optimal control. The duration of these pulse sequences determines the length of a given gates. In some cases, converging to a high quality pulse sequence of minimal length is difficult \cite{optimal-control-bspline, optimal-control-discrete}. In this work, we consider a straight-forward compilation of intermediate qudit circuits to connectivity-limited superconducting devices. To do so, we introduce swap gates designed explicitly to communicate qudits of different dimensions, for example qubits with ququarts (4 level systems). Second, we consider realistic gate times obtained via optimal control for superconducting based quantum architectures to better study expected circuit durations beyond circuit depth. Together, we use these compilation tools to study one representative implementation of the generalized Toffoli using different levels of intermediate qudits. Our contributions are the following: \begin{itemize} \item Detail efficient decompositions of SWAP gates between qudits of different dimensions \item A simple compilation pipeline transforming intermediate qudit circuits into ones which obey connectivity-limited hardware constraints \item Introduce intermediate ququart and ququint (5 level system) implementations of the generalized Toffoli, a critical circuit component for many larger algorithms \item Evaluation of the near-term benefits conferred by using intermediate qudits using realistic gate times for superconducting devices. \end{itemize} \section{Benchmarks} We focus on the Generalized Toffoli, or N-Controlled X gate as our benchmark. This circuit is easily generalized to higher level quantum systems with similar constructions at each level. At the qubit level, this gate can be implemented with any number of ancilla qubits \cite{qubit-toffoli}. But, to achieve this circuit in logarithmic depth we need to use $n-2$ ancilla for $n$ number of controls, significantly reducing the amount of usable space available to execute a quantum circuit. An example of this circuit for 5 controls with 3 ancilla qubits is in Figure \ref{fig:qubit-cnu}. An ancilla-free log depth version of the generalized Toffoli has also been developed for qutrit capable quantum systems \cite{intermediate-qutrits}. By using intermediate qutrits, this circuit uses 1 and 2 controlled +1 gates to reduce the number of qubits used and number of gates used. The set of controls are treated as nodes in a tree, recursively stepping down the child nodes in the tree to create the circuit. The child nodes target their parent node with a controlled qudit Toffoli gate which could increment or decrease the state of a qubit rather than performing an X gate. In the case of the leaf nodes, we use a 1 controlled qudit Toffoli gates. Then, any internal nodes use a 2 control since their states will have been increased from 1 to 2 if the leaf qubits were in the 1 state. Finally, the qubit representing the root of the tree performs a 2 controlled X gate on the target qubit. We then reverse the controlled qudit Toffoli gates to return the input qutrit to their original, non-elevated state. An example of this circuit with 7 controls is seen in Figure \ref{fig:qutrit-cnu}. We expand on this idea by constructing generalized Toffoli circuits that take advantage of even higher level quantum systems. We use the same tree-like framework developed for qutrit circuits, but can use the additional computational space afforded to us by these systems. Specifically, we use this extra space to remove the use of the gates involving three qudits, replacing them with two gates that use only two qudits each. While this may seem like an increase in gate count, the three count qudit gates is decomposed into many more one and two qudit gates. For qudit Toffoli gates, 6 two qudit gates and 8 single qudit gates, which is significantly higher. In the ququart case, we use each leaf control to target its parent qudit with a +1 gate. If the controls are in the 1 state, the target will end in the 3 state after being targeted by two controlled +1 gates. Then the internal controls are 3 controlled +1 gates. Similarly, the final controlled X gate becomes a 3 controlled X gate. An example of this can be seen in Figure \ref{fig:ququart-cnu}. A similar strategy can be used for any $n$ level system. We split create the control tree into a tree with $n-2$ children rather than the original 2 children in the qutrit and ququart case. An example of this expansion into a generalized Toffoli gate for 5 level qudits (ququints) can be seen in Figure \ref{fig:ququint-cnu}. \label{sec:benchmarks} \section{Background} \label{sec:background} \begin{table} \centering \begin{tabular}{lll} \textbf{Qudit Levels} & \textbf{Interaction Time (ns)} & \textbf{Swap Time (ns)} \\ \hline \\ 1 & 30 & - \\ 2 & 50 & - \\ 3 & 50 & - \\ 4 & 50 & - \\ 0, 1* & 150 & 600 \\ 0, 2 & 500 & 1200 \\ 0, 3 & 500 & 1500 \\ 0, 4 & 600 & 1800 \\ 1, 1 & 500 & 900 \\ 1, 2 & 500 & 1200 \\ 1, 3 & 500 & 1500 \\ 1, $4^*$ & 600 & 1800 \\ 2, $2^*$ & 675 & 2950 \\ 2, 3 & 850 & 5000 \\ 2, $4^*$ & 1025 & 7050 \\ 3, 3 & 850 & 5000 \\ 3, $4^*$ & 1025 & 7050 \\ 4, $4^*$ & 1200 & 7500 \\ \end{tabular} \caption{\label{tab:times} Times used for various gates across different levels of qudits. An asterisk indicates an interpolated value.} \vspace{-1.0em} \end{table} Classically, the basic computation unit is the bit which takes the value of either 0 or 1. In the quantum setting, we often consider the quantum bit (qubit) the most fundamental unit. For many quantum technologies, such as superconducting qubits and trapped ions, the implementation naturally has access to infinitely many discrete levels which can be truncated to any dimension $d$ giving qu\textit{dits} which exist as linear superpositions superpositions of $d$ levels as $\ket{\psi} = \alpha_0\ket{0} + \alpha_1\ket{1} + ... + \alpha_{d-1}\ket{d-1}$ where if we choose $d = 2$ we recover the qubit. For a variety of reasons, such as increasing number of error channels which become harder to control, using large numbers of states is often impractical and instead we should carefully choose the computing radix based on our target applications and available hardware. For example, measurement (the process of collapsing a quantum state to a classical value) of high level systems is often challenging for trapped ions and if possible we should try to measure only qubits. To manipulate quantum states we apply \textit{gates}, can be represented as unitary matrices. For the most part, hardware supports at most gates on 1 or 2 inputs and all larger gates must be synthesized directly from smaller ones. To obtain an answer from the device, the quantum state must be measured, a non-unitary gate which collapses the state to any of the basis elements. For this work, we consider a basis set which consists of the generalized versions of the $X_{+k}$ gates and the $X_{i, j}$ gates which are classical permutations of the basis elements. The first behaves by shifting every basis element's coefficient $+k$ modulo the dimension of the system. The second behaves by swapping the coefficients of the $i$-th and $j$-th states and leaving all other coefficients the same \cite{MS-gates}. We also consider the controlled versions of these gates. In some decompositions we will use Toffoli-like gates on higher dimensions, which have constant depth decompositions into 1 and 2 qudit gates. Current quantum hardware also only supports interactions between some pairs of devices, usually indicated by the hardware \textit{connectivity graph} which for many technologies is usually sparse. Qudit states can be moved around the device with special communication operations, often SWAP gates, in order to manipulate arbitrary pairs of qudits. The number of communication operations depends both on the connectivity of the input program and the connectivity of the underlying hardware. In order to execute programs on a given hardware target, input programs must be compiled. This usually amounts to several key steps: circuit optimization and decomposition, mapping \cite{mapping}, routing \cite{swap-min}, and scheduling \cite{scheduling}. All input circuits must be decomposed into the hardware's basis gate set which is usually kept small in order to reduce calibration overheads. Efficient decompositions are important to keep gate counts, depth, and space low. Some decompositions require the use of additional qudits called \textit{ancilla} to be efficient which are scratch bits which begin and end in a known state. Quantum hardware is error-prone. Gates can fail and qudit states decay over time, so each of these steps is critical to program success - we must minimize total operations and total execution time. Prior work in quantum multivalued logic has focused primarily on gains obtained from the first of these steps: circuit optimization. Significant gate count and depth advantages can be obtained using \textit{intermediate qudits} which means that inputs and outputs of an input quantum program are binary, but are allowed to temporarily access higher level states during computation. The key observation for the advantage is \textit{localizing} the additional space which would normally take the form of ancilla \cite{intermediate-qutrits}. For the near- to intermediate- term, space is severely restricted meaning ancilla counts should be kept low to maximize the amount of computational space. While gate counts and circuit depth (the length of the circuit's critical path) are often good indicators of a circuit's execution time, it is important to consider the physical realization of such gates on hardware. Via a process called \textit{optimal control} analog pulse sequences can be produced for a given unitary. Typically, durations are chosen before hand and the goal is to produce a high fidelity (quality) implementation of the gate. Finding optimal pulse durations is challenging, and is expected to scale quadratically with the dimension of the input unitary. We consider gate times obtained via optimal control to accurately account for communication time costs and total circuit duration. While hardware supports a limited gate set, optimal control procedures can permit us to synthesize any unitary. But, we limit this to a small gate set on bounded numbers of qudits to limit classical overhead. Instead, we can interpolate to predict other gate durations or produce pulses for gates in the decomposition. For space reasons we have omitted a full discussion of optimal control, see \cite{optimal-control-bspline, optimal-control-discrete} for more information. Gate times for this work are listed in Table \ref{tab:times}. The times for qubit-qubit interactions and swaps, and single qubit interactions are known from gate times from devices such as IBM's hardware. Times for qubit-quqart, quqart-ququart, and single ququart interactions and swaps were found via optimal control. Gate durations for qutrits and ququints were found via a linear interpolation between these points. The exact times for any gate also depends on the exact Hamiltonian used to model the underlying device or other physical restrictions. The numbers obtained here are from a standard superconducting Hamiltonian, as would be found for IBM's hardware \cite{ibmq}. As seen in table \ref{tab:times}, as the radix increases, so does the time required to perform an operation on each qudit. As the highest energy level increases, the pulses required to achieved the desired result becomes more complex as it needs to satisfy more constraints and transitions, elongating the necessary pulse duration. There are several examples of how this could be achieved physically \cite{guide-superconducting-qudits}. These increased radix devices also have higher rates of decoherence, but if the circuit depth and communication this will be balanced by a shorter circuit duration. While this work does not focus on other architectures, the presented circuits would be valid, but would a new set of control experiments \cite{trapped-ion-qutrits}. However, we would expect similar scaling. \section{Conclusion} We describe a strategy for qudit communication through efficient decomposition of SWAPs, and develop a straightforward compilation pipeline for qudit circuits, using realistic gate times for derived from optimal control. Further, we were able to introduce a generalized Toffoli gate that use intermediate ququarts, ququints, following a structure that can be generalized to any number of qudit levels built on strategies developed for intermediate qutrits. Finally, we compiled these circuits for potential architectures to evaluate the benefits at each level of qudit through examination in the change in gate count, depth, execution time, and space time product. Increasing the highest possible level of a qudit offers opportunities to improve the utilization of the architecture. From qubits to qutrits we see significant decreases in space time product to due reduction in ancilla use, even with increases in time of both interactions and SWAP gates. This trend continues from qutrits to ququarts, where the extra computational space allows us to use two two qudit gates rather than a three qudit gate, reducing duration of the circuit, and the space time product of the circuit as well and improving the use of the architecture. These advantages show that there are gains to be found by implementing these higher dimensional systems and integrating qudit compilation strategies to improve the usage of near term quantum devices. \label{sec:conclusion} \section{Results and Discussion} In this work we observe the changes in gate count, depth, duration, and space time product of a given circuit as we adjust our benchmark to utilize the higher radix levels of a quantum system, up to ququint level devices for the generalized Toffoli gate. Both gate count and depth of a circuit on their own are helpful metrics to compare the runtime of two different quantum circuits. However, as mentioned, different gates have different lengths, and may use more qubits through ancilla. So, it is necessary to examine the duration of the circuit as well. The product of the duration and number of qubits is also used as the space-time product. Minimizing this metric indicates an increase efficiency of the compiled circuit on a given device. When measuring the time from these circuits, we construct a directed graph with nodes representing the operations, and edges representing dependencies based on qudits used. We are able to label these edges with the time it will take based on the current levels of the qudits at that time. This allows us to find the longest path through the graph, giving us the duration of the critical path, and the duration of the overall circuit. We examine each circuit on a 12 qudit by 12 qudit grid architecture. This provides a middle ground between the structure current quantum architectures, and devices with higher connectivity. It also is a large enough architecture to examine the scalability of a circuit as they grow in size. \begin{figure*} \centering \begin{minipage}{.45\textwidth} \centering \scalebox{\plotscale}{ \makebox[.35\linewidth][c]{ \input{figs/circuit-duration} } } \captionof{figure}{The circuit duration of the Generalized Toffoli at different controls for different qudit levels.} \label{fig:circuit-duration} \end{minipage} \quad\quad \begin{minipage}{.45\textwidth} \centering \scalebox{\plotscale}{ \makebox[.35\linewidth][c]{ \input{figs/space_time_product} } } \captionof{figure}{The space time product, or number of qudits used by duration, at different controls for different qudit levels.} \label{fig:space-time-product} \end{minipage} \vspace{-1.0em} \end{figure*} \subsection{Benefits of Qutrits} Previous examinations of the intermediate qutrit generalized Toffoli gate did not examine how it may be affected by routing the circuit on an limited connectivity architecture. In Figure \ref{fig:pre-depth}, we see that the depth of the log-depth qutrit generalized Toffoli gate is greater than the depth of the qubit generalized Toffoli gate, despite the fact that it requires more Toffoli gates to implement the qubit gate. Once SWAP gates are integrated into the circuit via the qudit compilation framework, in Figure \ref{fig:post-depth}, the depth of the qutrit generalized Toffoli gate is significantly lower than the qubit generalized Toffoli gate. However, referring to Table \ref{tab:times}, the higher the level of the operations, the longer the times. Upon measuring the duration of a circuit after inserting SWAPs, seen in Figure \ref{fig:circuit-duration} we see that at smaller numbers of controls, qutrit circuit duration is much lower than qubit circuit duration, but as the number of controls increases, these values reverse. Where the qutrit circuit was half the duration of the qubit circuit from 5 to 15 controls, they are 1.5 times the duration from 35 controls onwards. The extra time required by the SWAP gates exceeds the benefits gained from using fewer gates. However, the construction of the qutrit circuit does not require any ancilla. To take this into account, we examine the change in space time product of the qutrit versus qubit generalized Toffoli gates in Figure \ref{fig:space-time-product}. As the qutrit circuit does not require extra qubits, it can achieve a much lower space time product. This is especially evident when the qutrit circuit has a low duration, starting at a ratio of 3 to 1 from qubit to qutrit space time ratio. As the number of controls increases, this ratio converges to 1.7 to 1 space time ratio from qubits to qutrits, which matches the expected constant increase in computational space from qubits to qutrits, $\text{log}_2(3) = 1.58$. \subsection{Benefits of Ququarts over Qutrits} Moving from the qutrit circuit to the ququart circuit does not reduce the number of the qudits. However, since we do not need to decompose any three qudit gates, the circuit requires fewer gates overall. The difference in depth before and after inserting SWAPs can be seen in Figure \ref{fig:pre-depth} and Figure \ref{fig:post-depth}. After routing, the depth of the qutrit circuit is 5 to 6 times greater than the ququart circuit, following from the ququart circuit using two two qudit gates, with a depth of two, rather than a Toffoli gate, which uses eight single qudit gates and six two qudit gates and has a depth of 11. Taking the duration of these ququart gates into account in Figure \ref{fig:circuit-duration}, the reduction in gate count and depth generally outweighs the increase in time to perform the qudit interactions. Generally, the ququart circuit time will approach the qutrit duration when the number of controls approaches a power of two. Certain numbers of controls are not conducive to the tree-like structure, and will result in an excess number of gates. When we examine the space time product, ququarts more efficiently utilizes the architecture than both qubits and qutrits. In fact, the qubit to ququart space time product ratio is an average of 2.3 after the initial few increases in controls, which is inline with the expected increase of computational space from qubits to ququarts, $\text{log}_2(4) = 2$. From qutrits to ququarts the expected increase would be $\text{log}_3(4) = 1.9$, and we find the ratio in space time product from qutrits to ququarts to be 1.43, an increase in the efficiency of the use of the device. \subsection{Diminishing Benefits of Higher Levels} When we move into higher level systems, specifically levels 5, 6 and 7, we do not see the benefits of using fewer qubits. Since these generalized Toffoli gates have similar constructions to the ququart gate, and do not have the benefits of gate reduction, and do not see significant decreases in the number of gates or depth of the circuit. As seen in Figure \ref{fig:pre-depth} and Figure \ref{fig:post-depth}, the ququint depth is the same as the ququart depth before and after insertion SWAPs for routing. Interactions at the ququint level take longer than at the ququart level. In Figure \ref{fig:circuit-duration} we see that the time for a ququint circuit to execute is generally greater than the time for a ququart circuit to execute. This translates to an increased space time product when compared to ququarts. The benefits afforded by the computation space affording by this level of qudit do not outweigh the cost of the complexity and operations required to reach this level over ququart circuits. \label{sec:results} \section*{\acksname} \phantomsection\addcontentsline{toc}{section}{\acksname} } \begin{acks} Thank you to Max Siefert, Jason Chadwick and Natalia Nottingham for their helpful conversations in developing this work. \end{acks} \section{Communication Circuits} \begin{figure} \centering \centering \makebox[.2\linewidth][c]{ \input{figs/qutrit-swap-decomp.qcircuit} } \captionof{figure}{The decomposition of a qutrit based SWAP gate.} \label{fig:qutrit-swap-decomp} \vspace{-1.0em} \end{figure} Current hardware has limited connectivity and only qudits which are adjacent on hardware may interact. Communication operations, called SWAPs, are required to move qudit states around the device. Here, we present a generalized decomposition of the SWAP gate taking in qudits of any dimension. The decomposition of the qudit SWAP gate is straightforward and follows from the qubit-qubit swap. The key idea is to consider size 2 subsets of the basis elements. For qubits, this amounts to the subset $\{0, 1\}$ for which the decomposition is three 1 controlled $X_{01}$ flips. For qutrit SWAPs we now have three possible subsets $\{0, 1\}$, $\{0, 2\}$ and $\{1, 2\}$ and then perform partial SWAPs as if the qudits exist only in the subspace spanned by the elements of the subset. Here, we would do three 1 controlled $X_{01}$ flips followed by three 2 controlled $X_{02}$ flips and then three 2 controlled $X_{12}$ flips. The control value can be either of the elements of the subset. From a permutation point of view, a SWAP is decomposed into $O(n^2)$ 2-cycles where $n$ is the dimension of both qudits. For communication between qudits of different dimensions the same strategy applies but the scaling is $O(nm)$ where $n,m$ are the dimensions of the two input qudits. There have been other approaches for generalized SWAP gates by using more contrived basis elements \cite{qutrit-swaps, contrived-qudit-swaps}, but we find these to be intuitive and match the expected asymptotic scaling that optimal control predicts for gate durations. Additionally, this version of the SWAP gate will handle mixed radix inputs. That is, if one device is in a different radix from the other, the same set of gates for the higher radix SWAP can be used as would be used if they were both higher radix devices. In Figure \ref{fig:qutrit-swap-decomp} we show a swap gate decomposition for two qutrits. \label{sec:communication} \section{Compilation} \begin{figure} \centering \begin{minipage}{.2\textwidth} \centering \makebox[.2\linewidth][c]{ \input{figs/qubit-cnu.qcircuit} } \captionof{figure}{A logarithmic depth Generalized Toffoli circuit using qubits with 5 controls and 4 ancilla.} \label{fig:qubit-cnu} \end{minipage} \quad\quad \begin{minipage}{.2\textwidth} \centering \makebox[.2\linewidth][c]{ \input{figs/qutrit-cnu.qcircuit} } \captionof{figure}{A logarithmic depth Generalized Toffoli Circuit using qutrits with 7 controls.} \label{fig:qutrit-cnu} \end{minipage} \vspace{-1.0em} \end{figure} \begin{figure*} \centering \begin{minipage}{.45\textwidth} \centering \makebox[.4\linewidth][c]{ \input{figs/ququart-cnu.qcircuit} } \captionof{figure}{A logarithmic depth Generalized Toffoli Circuit using ququarts with 7 controls.} \label{fig:ququart-cnu} \end{minipage} \quad\quad \begin{minipage}{.45\textwidth} \centering \makebox[.4\linewidth][c]{ \input{figs/ququint-cnu.qcircuit} } \captionof{figure}{A logarithmic depth Generalized Toffoli Circuit using ququints with 10 controls.} \label{fig:ququint-cnu} \end{minipage} \vspace{-1.0em} \end{figure*} Since many qudits will be in an higher level state, we must use qudit SWAP gates to effectively route the circuit on a device. Determining a favorable mapping and efficient routing scheme is important to effectively utilizing a quantum architecture when executing a quantum circuit \cite{mapping}\cite{scheduling}. Placing or moving qudits that interact often far apart from one another on an architecture will require extra swap gates to move qudits within range of one another. We adapt previously developed methods used for traditional qubit mapping and routing to effectively schedule qudit operations on a high radix device. The methods described here are very similar to previously developed methods \cite{map3, map1, routing1, routing2}, but have the extra constraint of attempting to reduce time by utilizing as few high-radix communication operations as possible rather than reducing communication overall. An implementation of this compiler can be found here \cite{github-link}. We first decompose any gates that are not native to the qudit architecture. For example, the Toffoli gate is a three qubit gate that can not be executed natively, and is decomposed to 6 CNOT gates, and 14 one qubit gates. Similarly, we decompose the qudit Toffoli gate as well according to \cite{di2011elementary}. Then, we attempt to place the qudits that need to interact often in clusters to prevent extra communication costs based on the decomposed circuit. Similar to other mapping methods, we first find the interaction weights between each qudit $w(u, v)$ interaction using: \vspace{-1.0em} \begin{align*} w(u, v) = \sum_{o \in ops} \mathbbm{1}(u, v \in o.qubits) \vspace{-1.0em} \end{align*} \begin{figure*} \centering \begin{minipage}{.45\textwidth} \centering \scalebox{\plotscale}{ \makebox[.4\linewidth][c]{ \input{figs/pre-depth} } } \captionof{figure}{The depth of the Generalized Toffoli gates at different levels of qudits before communication gates are added to the circuit.} \label{fig:pre-depth} \end{minipage} \quad\quad \begin{minipage}{.45\textwidth} \centering \scalebox{\plotscale}{ \makebox[.4\linewidth][c]{ \input{figs/post-depth} } } \captionof{figure}{The depth of the Generalized Toffoli gates at different levels of qudits after communication gates are added to the circuit.} \label{fig:post-depth} \end{minipage} \vspace{-1.0em} \end{figure*} The most used qudit is placed in the center qudit of the quantum architecture. From this point, we select the qudit that maximizes the sum of weights to the already placed qubits. The selected qudit is placed in an adjacent qudit to already placed qudits according to which qudit minimizes the sum of the products of time to interaction with a qudit from the location by the interaction weight to the qudit: \vspace{-1.0em} \begin{align*} m(u) = \sum_{v \in P} w(u, v) \times d(\varphi(u), \varphi(v)) \vspace{-1.0em} \end{align*} where $\varphi$ is the mapping from virtual to physical qudits and $P$ is the already placed qudits. This keeps qudits that interact often close to one another, giving the scheduling a well placed starting point to begin routing the operations in the circuit. To route the qudit operations, we attempt to disrupt the placement of the qudits will that interact frequently as little as possible. By minimizing disruption, rather than naively moving qudits along the shortest path, we can potentially avoid extra swaps required bring these qudits back within range of one another. We have the following scoring function representing the state of the mapped qudits at a given time in circuit execution and select the best potential qudit to swap with: \vspace{-1.0em} \begin{align*} s(Q, w, d) = \sum_{u, v \in Q \times Q} w(u, v) \times d(\varphi(u), \varphi(v)) \vspace{-1.0em} \end{align*} Where $Q$ represents the qudits in the circuit, $w$ is the interaction weights of the remaining operations to execute, and $d$ is the time, including swaps for two physical qudits to interact. We find the difference between the current score, and the new mapping resulting from a SWAP. We perform whichever SWAP minimizes the difference between the current score and the new score. Once an operation is completed, we remove 1 from the weight of the interacting pair as it is no longer relevant to the placement of the current qudits. \label{sec:compilation}

proofpile-arXiv_069-10165

\section{Introduction} Ever since the coinage of the word ``entanglement" by Schr\"{o}dinger in 1935 closely following the work of Einstein, Podolsky and Rosen (EPR) \cite{EPR_35}, discussion and debate about its nature and manifestation has continued to remain one of the most engaging issues in modern physics. The paradox posed by EPR demonstrated for the first time the possibility of creating non-classical and nonlocal correlations with the help of entanglement, which Schr\"{o}dinger tried to explain in terms of quantum "steering" \cite{S_35}. Subsequently, the pioneering work of Bell \cite{B_64} paved the way for mathematically distinguishing quantum correlations from those arising through a local realist description of physical phenomena. More recently, it has been realized that quantum correlations could be classified into hierarchical categories \cite{WJD_07, JWD_07} with entanglement being the weakest, followed by steering and Bell-nonlocality. In present times, entanglement is regarded as the primary building block of quantum correlations, leading to landmark discoveries in quantum information science \cite{enatnglement_review}. Numerous protocols have already been suggested, which use these correlations as resource and result in improvements, which no classical resource could achieve \cite{teleportation,dense,1sqkd,randomness, pironio2010random}. It has been realized \cite{WJD_07, JWD_07} that the nonlocal quantum correlations responsible for steering and Bell-violation cannot exist without the presence of entanglement. As a result of this, identification and quantification of quantum correlations, have become a topic of cutting edge research in various inter-disciplinary areas of physics, mathematics and computer science, as well. In quantum information theory, the way of identifying entanglement in a given bipartite state is through the {\it separability criterion} \cite{peres_sep,HORODECKI19961}. Though this criterion is also helpful in quantification of entanglement \cite{negativity,log_negativity}, it is measurable only when full knowledge of the state is available. Such knowledge requires state tomography \cite{state_estimation_book}, which is expensive in terms of resources required. On the other hand, there are methods based on direct measurement of observables (which are single setting measurements) such as {\it entanglement witnessing} \cite{T_00,LKCH_00,bruss2002} which have been experimentally realized \cite{experimental_witness,guhne2003experimental}. In addition, other schemes have been recently proposed, such as {\it self-testing} protocols, which can identify individual entangled states giving rise to particular correlations in a given scenario \cite{MYS12,YN13,SH16,GBDSJM_18}. However, all such methods suffer from the drawback of non-universality. For instance, for every entanglement witness (EW) there exists a class of entangled states, which it cannot detect \cite{guhne2009entanglement,LKCH_00}. This prevents the use of any single EW to detect all entangled states. It is pertinent to note here that arranging a higher number of measurement settings is an expensive resource in experiments. Entanglement detection in two-qubit states has drawn renewed attention, as can be seen from several recent works \cite{recenttwoqubit3,*recenttwoqubit1,*recenttwoqubit2,*adhikari18}. Our motivation for the present study is to reduce the resources required for identifying entanglement, and here we concern ourselves with the task of identification of entanglement in an unknown state. In this context, Yu et al. \cite{ent_wit_rank2} constructed an observable acting on four copies of any two-qubit state, that could detect entanglement for certain classes of two-qubit states. Augusiak et al. \cite{ADH_08} proposed the construction of an observable which acts on four copies of a two-qubit state and results in detection of all entangled states. Therefore, universal detection of entanglement could be done through measurement in a single setting, but the cost is to supply multiple copies of the state. Further work in this direction \cite{girolami_2012,twocopy_concurrence_mixed3,dawei,heinosaari_2012}, has been performed to reduce the resources required for universal identification of entangled states. Girolami et al \cite{girolami_2012} proposed a method for identifying quantum correlations in two-qubit states through measurement of seven observables on four copies, where the observables are local in the Alice-Bob cut (the two parties sharing the bipartite state). It has been shown \cite{dawei, heinosaari_2012} that any universal entanglement detection scheme on a single copy of a state, has to be necessarily a state tomography process. Recently, the protocol in \cite{ADH_08} was extended to the completely device independent scenario \cite{MG_17}. In the present work we propose a protocol where universal detection of entanglement is possible in a single measurement setting on just {\it two} copies of any two-qubit state, using weak values. The idea of weak measurement was first proposed by Aharonov et al. in \cite{Vaidman_weak}, to show that an experimental outcome outside the eigenvalue spectrum of an observable could be obtained if a sufficiently weak coupling of the system and the apparatus along with post-selection is employed. Weak measurements have been utilized in several interesting applications such as observations of spin Hall effect \cite{HK_08}, trajectories of photons \cite{KBRSM_11}, direct measurement of the quantum wave function \cite{LSPSB_11}, and measurement of ultrasmall time delays of light \cite{SB_13}. The technique of weak measurement and reversal has also been used in the preservation of entanglement \cite{KU_99, KJ_06, KLKK_12, MXA_12}, teleportation fidelity \cite{PM_13} and steerability \cite{DGPM_17} through noisy channels. Detection of weak value has been found to be useful in observing geometric phase \cite{w_value_geometric}, non-Hermitian operators \cite{w_value-non-Herm} and quantum state \cite{w_value_state1,w_value_state2,w_value_state3}. Here we show that our protocol of entanglement detection using weak values on two copies of an arbitrary two-qubit state results in complete identification of the state, i.e., state tomography, in the similar fashion as in \cite{dawei,heinosaari_2012}. Note that, a number of attempts \cite{twocopy_concurrence_pure,twocopy_concurrence_mixed1,twocopy_concurrence_mixed2,twocopy_concurrence_mixed3} were made to measure concurrence \cite{coffman,wootters_concurrence} of two qubit states through measurement of a single observable on two copies of the state. Although, for pure states \cite{twocopy_concurrence_pure} such observables could be found, only estimates could be given for mixed states \cite{twocopy_concurrence_mixed1,twocopy_concurrence_mixed2,twocopy_concurrence_mixed3}. In this regard, our result provides a solution to this problem, as complete identification of two-qubit states, obtained through our protocol, also imply measurement of concurrence for any two-qubit state using two copies. We further show that on restricting the set of states to just pure states, the weak interaction necessary in our protocol, can be realized through local operations on each of the qubits. Finally, we also show that our protocol is robust to errors arising from inappropriate choice of weak interaction between two copies of the two-qubit states. The plan of this paper is as follows. In Sec \ref{Background} we discuss the preliminaries required for the analysis, before presenting our protocol in Sec \ref{method}. In Sec \ref{local impementation} we discuss possible implementation of our scheme through local operations. In Sec \ref{robust} we demonstrate the robustness of our protocol, before concluding in Sec \ref{conclusion}. \section{Background} \label{Background} For any Hilbert spaces $\mathcal{H}$, let the space of all linear operators be denoted by $\mathcal{L}(\mathcal{H})$, and the set of all density matrices be $\mathcal{P}_+(\mathcal{H})$. Now, consider two parties, Alice and Bob, each separately possessing a two-level quantum system (qubit) with Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, respectively. Also consider the pointer system of the measuring apparatus to be a quantum system with Hilbert space $\mathcal{H}_{E}$. Now, any bipartite quantum state, that can be written as a convex mixture of product states is called a {\it separable} state i.e. \begin{eqnarray} \rho_{sep} = \sum_{i} p_{i} \rho_{i}^{A} \otimes \rho_{i}^{B} \label{ent_state} \end{eqnarray} for any $\rho_{i}^{A}\in\mathcal{P}_+(\mathcal{H}_A)$ , $\rho_{i}^{B}\in\mathcal{P}_+(\mathcal{H}_B)$ and probability distribution $\{p_i\}_i$. Any state $\rho\in\mathcal{P}_+(\mathcal{H}_A\otimes\mathcal{H}_B)$ which is not separable is called an {\it entangled} state. Note that, as $\rho\in\mathcal{L}(\mathcal{H}_A\otimes\mathcal{H}_B)$, it can always be decomposed as, \begin{eqnarray} \rho=\sum_{ijkl} p_{kl}^{ij} \ket{i} \bra{j} \otimes \ket{k} \bra{l} \label{ppt1} \end{eqnarray} where $\{\ket{i}\}_i$ denotes an orthonormal basis in each of the subsystem Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$. Using this decomposition, we can define the partial transpose of $\rho$, with respect to the subsystem $B$, in the following way, \begin{eqnarray} \rho^{T_{B}}=\sum_{ijkl} p_{lk}^{ij} \ket{i} \bra{j} \otimes \ket{k} \bra{l} \label{ppt2} \end{eqnarray} Note that, we can similarly define $\rho^{T_A}$ and $\rho^{T_{B}}=({\rho^{T_{A}}})^T$, where $\bullet^T$ denotes transposition. Now, we can present the separability criteria, as mentioned in the previous section. Any two qubit state $\rho\in\mathcal{P}_+(\mathcal{H}_A\otimes\mathcal{H}_B)$ is separable \cite{ADH_08,sanpera_two_qubit,Verstraete_two_qubit} if and only if, \begin{eqnarray} det~ \big(\rho^{T_{B}}\big) \geq 0 \label{det_sep} \end{eqnarray} where $det(A)$ represents determinant of a matrix $A$. This criterion can also be linked to the quantification of entanglement in terms of concurrence \cite{wootters_concurrence}. Now we briefly illustrate the idea of weak measurement and weak values. In the theory of weak measurements \cite{Vaidman_weak,weak_review}, the pointer system of the measuring apparatus is kept in an initial state $\phi_{in}\in\mathcal{P}_+(\mathcal{H}_E)$ and a quantum system is {\it pre-selected} in a state $\rho\in\mathcal{P}_+(\mathcal{H})$. Then, the joint system-pointer state is evolved through a {\it weak interaction} generated by a Hamiltonian $\epsilon H\otimes P_x$, where $H$ is the Hamiltonian associated with the system, $P_x$ is the momentum operator of the pointer system, and $\epsilon$ is a small positive number representing the {\it weakness} of interaction. Following this, a strong {\it post-selective} measurement is performed on the weakly evolved state of the system in a basis $\{\ket{u_k}\}_k$, where $\ket{u_k}\in\mathcal{H}$, which results in the pointer state $\phi^k_f\in\mathcal{P}_+(\mathcal{H}_E)$ for each $k$, where \begin{eqnarray} \label{weakpointer} \phi^k_f\approx \bra{u_i}\rho\ket{u_i}e^{-i\epsilon \langle H \rangle^{(k)}_{\rho}~P_x} \phi_{in}~ e^{i\epsilon \langle H \rangle^{(k)}_{\rho}~P_x} \end{eqnarray} where $\langle H \rangle^{(k)}_{\rho}$ are the weak values, given by, \begin{eqnarray} \langle H \rangle^{(k)}_{\rho}= \frac{tr\big[H \rho \ket{u_k} \bra{u_k}\big]}{tr\big[\rho \ket{u_k} \bra{u_k}\big]}. \label{weakvalue} \end{eqnarray} Note that Eq. (\ref{weakpointer}) can be derived only under the approximation that $\epsilon$ is very small. For measuring $\langle H \rangle^{(k)}_{\rho}$ certain properties of the position and momentum wave function of $\phi^k_f$ needs to be observed. As mentioned in Ref \cite{Jozsacomplex}, these properties include shift in expectation value of the position and momentum wavefunction compared to their initial values, variance of the momentum wave function, rate of change of the position wavefunction, and strength of the weak interaction i e., $\epsilon$. A detailed analysis on this technique is provided in section II of \cite{Jozsacomplex}. Also recently, real and imaginary parts of a weak value was detected by using Laguerre-Gaussian modes \cite{w_value_detection_single} in the pointer state. For a detailed discussion on weak values, refer to \cite{weak_review}. \section{Entanglement witness via weak values using two copies of the state} \label{method} In this section, we present a technique using weak values to detect entanglement of any two-qubit state, through a single projective measurement, i.e., measurement in a single setting. It was recently shown \cite{tukia_weak}, that by suitable choice of Hamiltonian and post-selective measurement, weak values can be used to determine the concurrence of any {\it pure} two-qubit state. In this paper we generalize this idea to {\it any} two-qubit state. For this purpose, we consider only two copies of the two-qubit state in consideration. Now, let us start by considering Alice and Bob share two copies of a two-qubit state $\rho\in\mathcal{P}_+(\mathcal{H}_A\otimes\mathcal{H}_B)$. The most general form of the density matrix of a two-qubit state (mixed or pure) can be expressed in the following form, \begin{eqnarray} \label{den_matrix} \rho= \left( \begin{array}{cccc} p & u & v & w \\ u^* & q & x & y \\ v^* & x^* & r & z \\ w^* & y^* & z^* & s \\ \end{array} \right) \end{eqnarray} where, $p$, $q$, $r$ and $s$ are real, non-negative numbers summing up to $1$, and $u$, $v$, $w$, $x$, $y$ and $z$ are complex numbers in general; $u^*$ is the complex conjugate of u, etc. It should be noted that $\rho$ is Hermitian. In addition to these conditions there is another constraint of positivity of the above matrix, which has to be satisfied by $\rho$ to be a valid density matrix, but for our purpose here, we stick to the form given in Eq. (\ref{den_matrix}). \subsection{The general case} We first consider the general case, where $p$, $q$, $r$ and $s$ are nonzero. As a result, the determinant of the partially transposed matrix of $\rho$ can be written as, \begin{eqnarray} det (\rho^{T_B}) & = & p q r s \Big{(} \frac{u u^* z z^*}{p q r s}-\frac{u v y^* z^*}{p q r s}-\frac{u w^* x z}{p q r s}-\frac{u^* v^* y z}{p q r s} \nonumber \\ &&-\frac{u^* w x^* z^*}{p q r s}+\frac{v v^* y y^*}{p q r s}-\frac{v w^* x^* y}{p q r s}-\frac{v^* w x y^*}{p q r s} \nonumber \\ &&+\frac{w w^* x x^*}{p q r s}+\frac{u v w^*}{p q r}+\frac{u^* v^* w}{p q r}+\frac{u x y^*}{p q s} \nonumber \\ &&+\frac{u^* x^* y}{p q s}-\frac{u u^*}{p q}+\frac{v x^* z^*}{p r s}+\frac{v^* x z}{p r s} \nonumber \\ &&-\frac{v v^*}{p r}-\frac{x x^*}{p s}+\frac{w y^* z^*}{q r s}+\frac{w^* y z}{q r s} \nonumber \\ &&-\frac{w w^*}{q r}-\frac{y y^*}{q s}-\frac{z z^*}{r s}+1 \Big{)}. \label{det_pt} \end{eqnarray} It can be seen that the determinant in Eq. (\ref{det_pt}) is a polynomial of degree 4. In \cite{Grasslpolynomial}, it was shown that an $n$-th degree homogeneous polynomial function of the density matrix elements can be computed as the expectation value of a pair observables, which acts on $n$ copies of the density matrix. This result was later on used by Augusiak et al. \cite{ADH_08} to construct a single observable, acting on four copies of a two-qubit state, to compute the determinant in Eq. (\ref{det_pt}) for witnessing entanglement. Our aim is to reduce the number of copies of the state required, and hence reduce the resources required for the process of witnessing. For this purpose we consider the technique of using weak measurement as in \cite{tukia_weak}. Note that in Eq. (\ref{det_sep}), for detecting entanglement of the unknown state $\rho$, it is sufficient to know the sign of the determinant in Eq. (\ref{det_pt}). In other words it is enough to find the value of $(1/pqrs)~det{\rho^{T_B}}$. We also found that, finding values of the following terms (and thereby, their complex conjugates) is sufficient to determine the value of $(1/pqrs)~det{\rho^{T_B}}$: \begin{eqnarray} \frac{u^*}{p}, \frac{u}{q}, \frac{z^*}{r}, \frac{z}{s}, \frac{v^*}{p}, \frac{y^*}{q}, \frac{v}{r}, \frac{y}{s}, \frac{w^*}{p}, \frac{x^*}{q}, \frac{x}{r}, \frac{w}{s}. \label{generator} \end{eqnarray} Out of these 12 terms, it can be easily seen that 9 of them are independent. For example $\frac{u}{q}$, $\frac{z}{s}$ and $\frac{w^*}{p}$ can be expressed in terms of the remaining 9 terms. Note that this latter condition does not result in any reduction of copies required for our protocol. \begin{figure} \includegraphics[width= 8.6 cm,height= 4.0 cm]{circuit1.pdf} \caption{Circuit realization of entanglement detection through weak interaction. Here $R_X=e^{i \epsilon \sigma_x}$ represents rotation of the bloch vector about x-axis through an angle $-2\epsilon$, and $H_D=\ket{0}\bra{+}+\ket{1}\bra{-}$ represents the hadamard operation, where $\ket{\pm}=\frac{1}{\sqrt{2}}\big(\ket{0}\pm\ket{1}\big)$.} \label{circuit} \end{figure} We find that each of the terms in Eq. (\ref{generator}) can be seen as a weak value, as in Eq. (\ref{weakvalue}), if we consider two copies of the state i.e. $\rho\otimes\rho\in\mathcal{P}_+((\mathcal{H}_A\otimes\mathcal{H}_B)\otimes(\mathcal{H}_{A}\otimes\mathcal{H}_{B}))$ and choose the Hamiltonian $H\in\mathcal{L}((\mathcal{H}_A\otimes\mathcal{H}_B)\otimes(\mathcal{H}_{A}\otimes\mathcal{H}_{B}))$ in an appropriate form, along with the post-selective measurement in the computational basis i.e., $\{\ket{u_k}\}_{k=1}^{16}=\{\ket{0000},\ket{0001},\dots,\ket{1111}\}$. It turns out that a suitable form of $H$ is the following, \begin{eqnarray} H & = & \ket{00} \bra{00} \otimes H_{1} + \ket{01} \bra{01} \otimes H_{1} \nonumber\\ && + \ket{10} \bra{10} \otimes H_{2}+ \ket{11} \bra{11} \otimes H_{3} \label{obs} \end{eqnarray} where, \begin{eqnarray} H_{1}&=&\openone \otimes \sigma_x \\ H_{2}&=&\sigma_x \otimes \openone \\ H_{3}&=&\sigma_x \otimes \sigma_x \label{subobservable} \end{eqnarray} with $\sigma_x$ being the usual Pauli matrix along $x$-direction. Using the computational basis $\{\ket{u_k}\}_{k=1}^{16}$ and Eqs. (\ref{den_matrix}) and (\ref{obs}), in Eq. (\ref{weakvalue}), we find a list of weak values and the terms of Eq. (\ref{generator}), they correspond to, \begin{eqnarray} \frac{u^*}{p}=\langle H \rangle^{(1)}_{\rho\otimes\rho}~~ &;&~~ \frac{u}{q}=\langle H \rangle^{(2)}_{\rho\otimes\rho}~~ ;~~ \frac{z^*}{r}=\langle H \rangle^{(3)}_{\rho\otimes\rho}~;~~\nonumber\\ \frac{z}{s}=\langle H \rangle^{(4)}_{\rho\otimes\rho}~~ &;&~~ \frac{v^*}{p}=\langle H \rangle^{(9)}_{\rho\otimes\rho}~ ;~~ \frac{y^*}{q}=\langle H \rangle^{(10)}_{\rho\otimes\rho}~;~~\nonumber\\ \frac{v}{r}=\langle H \rangle^{(11)}_{\rho\otimes\rho}~~ &;&~~ \frac{y}{s}=\langle H \rangle^{(12)}_{\rho\otimes\rho}~~ ;~~ \frac{w^*}{p}=\langle H \rangle^{(13)}_{\rho\otimes\rho}~;~~\nonumber\\ \frac{x^*}{q}=\langle H \rangle^{(14)}_{\rho\otimes\rho}~~ &;&~~ \frac{x}{r}=\langle H \rangle^{(15)}_{\rho\otimes\rho}~~ ;~~ \frac{w}{s}=\langle H \rangle^{(16)}_{\rho\otimes\rho}~~ \label{weaklist} \end{eqnarray} Note that four weak values, generated out of the post selection measurement, are redundant. As a result they do not occur in the above equation. Therefore, it can be easily seen from Eq. (\ref{weaklist}), that our protocol leads to determination of the sign of the determinant in Eq. (\ref{det_pt}), and as a results it would lead to universal entanglement detection for two-qubit states. Note that in this protocol, detection of entanglement is made only through a single projective measurement setting, i.e., the post-selective measurement.Using the form of Hamiltonian given in Eq. (\ref{obs}), we find the unitary operator $U$, giving rise to weak interaction is given by, \begin{eqnarray} U&=& \ket{0} \bra{0}\otimes\openone \otimes\openone\otimes e^{-i\epsilon \sigma_x}+\ket{10} \bra{10} \otimes e^{-i\epsilon \sigma_x}\otimes\openone\nonumber\\ && +\ket{11} \bra{11} \otimes e^{-i\epsilon \sigma_x\otimes\sigma_x} \end{eqnarray} In the above form, we can write $e^{-i\epsilon\sigma_x\otimes\sigma_x}=\ket{+}\bra{+}\otimes e^{-i\epsilon\sigma_x}+\ket{-}\bra{-}\otimes e^{i\epsilon\sigma_x}$. Note that, this represents a conditional unitary operation, conditioned on $\{\ket{+},\ket{-}\}$ states. As a result we use Hadamard gate $H_D$, which flips states $\{\ket{+},\ket{-\}}\leftrightarrow\{\ket{0},\ket{1}\}$, to achieve the circuit realization of our protocol, as given in Fig. \ref{circuit}. Also note that, the decomposition of $H$ in Eq.(\ref{obs}) is not unique, as it can also be chosen in any form where $H_{1}$, $H_{2}$ and $H_{3}$ resides in any of the diagonal blocks of the $16 \times 16$ matrix $H$. Moreover, as mentioned earlier, there are 9 independent terms in Eq. (\ref{generator}), which leads to 9 independent linear equations. Along with these equations, the constraint $p+q+r+s=1$, gives the exact solution for all the unknown quantities in the density matrix $\rho$, and hence our protocol results in complete identification of the two-qubit state i.e state tomography. Henceforth, any standard method for finding the amount of entanglement like negativity \cite{negativity,negativity2} or concurrence \cite{coffman} can be employed, and one can calculate how much entangled the state is. In particular, the following quantity, which can be easily obtained from our protocol, can also used to estimate the amount of entanglement present in the state : \begin{equation} E(\rho)=max\{0,-det(\rho^{T_B})\}. \end{equation} Thus we see our protocol not only serves as a technique to detect arbitrary two-qubit entangled state, but also as a protocol to measure entanglement. \subsection{Special cases} Let us now consider the special cases where at least one of the diagonal elements of $\rho$ is zero. This scenario physically means receiving no signal on the pointer, for the corresponding measurement outcome, before the weak interaction is switched on. For example, if $p$ is $0$, no signal is received for outcome $\ket{0000}$, and similarly for $q,r$ or $s$ we check if no signal is received for the outcomes $\ket{0101},\ket{1010}$ or $\ket{1111}$, respectively. We will show here that even for this case the same protocol, as described in Fig. \ref{circuit}, works. We now consider each case individually, \subsubsection{Case I} When $p=0$, positivity of $\rho$ demands $u$, $v$ and $w$ must also be $0$. Similarly, when $s=0$ we must have $w=y=z=0$. As a result, in both of these cases, we have $det (\rho^{T_B})= - x x^* q r$. In both of these cases, we first check if $q$ or $r$ is zero or not. If either of them is zero, we conclude $\rho$ is separable. If not, we check if the weak value $\langle H \rangle^{(14)}_{\rho\otimes\rho}$ i e. $x^*/q$, is zero. If it is, then $\rho$ is separable, otherwise it is entangled. \subsubsection{Case II} Similarly, when $q=0$ or $r=0$ we have $u=x=y=0$ or $v=x=z=0$, respectively. In both the cases we get $det (\rho^{T_B})= - w w^* p s$. Following this, in a similar way as above, we check the values of $p$ or $s$ and subsequently, the weak value $\langle H \rangle^{(16)}_{\rho\otimes\rho}$ i e. $w/s$, to determine if $\rho$ is entangled or not. \subsection{Implementing the protocol through Local Operations} \label{local impementation} In this section, we show that if we restrict the two-qubit state to be {\it pure} states only, we can realize the weak interaction through local operations on each of the qubits. Consider $\rho=\ket{\Psi}\bra{\Psi}$, where $\ket{\Psi}=a\ket{00}+b\ket{01}+c\ket{10}+d\ket{11}$. It can be easily seen that $\rho$ is separable if and only if $ad-bc=0$. In the notations of Eq. (\ref{den_matrix}), we find $p=|a|^2,q=|b|^2,r=|c|^2,s=|d|^2,u=ab^*$ and $z=cd^*$. Therefore, for this case we modify our protocol, and choose the weak Hamiltonian in Eq (\ref{weakvalue}) to be of the form, \begin{eqnarray} H'=\openone\otimes\openone\otimes\openone\otimes\sigma_x \label{localhamil} \end{eqnarray} It can be easily checked that the unitary operator corresponding to this Hamiltonian acts locally on all of the four qubits. Now, in the same way as in the previous section, we first check if either of $p,q,r$ or $s$ is zero i.e by checking if signal is received for outcomes $\ket{0000},\ket{0101},\ket{1010}$ or $\ket{1111}$. If one, or more of $p,q,r$ and $s$ are zero, it would imply the corresponding terms among $a,b,c$ and $d$ are also zero. Using these values we can easily check if $ad-bc=0$. If none of $p,q,r$ or $s$ are zero, we check if the weak values $\langle H' \rangle^{(2)}_{\rho\otimes\rho}=\langle H' \rangle^{(4)}_{\rho\otimes\rho}$, i e. if $u/q=z/s$. If the equality holds then $\rho$ is separable, otherwise it is entangled. \section{robustness of the protocol} \label{robust} In real life experiments, errors are bound to occur. Here we show that, our protocol is robust against errors arising from inappropriate choice of weak interaction. Consider a situation, where an erroneous Hamiltonian of the form of $ H_e$ is chosen in place of the correct Hamiltonian $H$, where $||H-H_e||_1\leq\delta$. Note here, $||A||_1=tr\sqrt{A^{\dagger}A}$ represents the {\it trace norm} of a matrix $A$. As a result, the error occuring in the weak values are given by, \begin{eqnarray} \Delta_k &=&|\langle H \rangle^{(k)}_{\rho\otimes\rho}-\langle H_e \rangle^{(k)}_{\rho\otimes\rho}|=\frac{|\bra{u_k}\rho\otimes\rho (H-H_e)\ket{u_k}|}{\bra{u_k}\rho\otimes\rho\ket{u_k}}\nonumber\\ &\leq&\frac{|\bra{u_k}\rho\otimes\rho (H-H_e)\ket{u_k}|}{m} \label{robust1} \end{eqnarray} where $m$ is the minimum of $\big\{\{p,q,r,s\}\times\{p,q,r,s\}\big\}$ \cite{crossprod} and is always positive. Note that, in obtaining the above inequality we used the fact $\bra{u_k}\rho\otimes\rho\ket{u_k}\in\big\{\{p,q,r,s\}\times\{p,q,r,s\}\big\}$ and it can also be seen that in our protocol, the weak value for $k^{th}$ outcome is only measured when $\bra{u_k}\rho\otimes\rho\ket{u_k}\neq0$. As a result, the denominator never vanishes in Eq. (\ref{robust1}). Now, consider the eigenvalue decomposition $H-H_e=\sum_{i}\lambda_i\ket{i}\bra{i}$, where $\{\ket{i}\}_i$ forms an orthonormal basis in $\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_A\otimes\mathcal{H}_B$, and also note that $||H-H_e||_1=\sum_i|\lambda_i|$. As a result, \begin{eqnarray} \Delta_k &\leq&\frac{|\sum_i\lambda_i\bra{u_k}\rho\otimes\rho \ket{i}\langle{i}|{u_k}\rangle|}{m}\nonumber\\ &\leq&\frac{1}{m}\sum_i|\lambda_i|~|\bra{u_k}\rho\otimes\rho \ket{i}\langle{i}|{u_k}\rangle|. \end{eqnarray} Since $0\leq\rho\leq\openone$, it can be easily seen that $|\bra{u_k}\rho\otimes\rho \ket{i}\langle{i}|{u_k}\rangle|\leq 1$. Thus we have, \begin{eqnarray} \Delta_k \leq\frac{1}{m}\sum_i|\lambda_i|\leq \frac{||H-H_e||_1}{m}\leq\frac{\delta}{m } \end{eqnarray} Thus we see, our protocol is robust to errors arising from inappropriate choice of weak interaction.\\ \section{Conclusions} \label{conclusion} In this paper, we have proposed a universal entanglement detection protocol for two-qubit quantum states. We consider the most general form of density matrix for two-qubit states, and show that it is enough to have just two copies of the given state to identify if it is entangled or not. Our formulation is based on the determinant based separability criterion and the idea of weak values. Previously in \cite{ADH_08}, it was demonstrated that one can universally detect entanglement in two-qubit systems using four copies of the state. Our protocol therefore, leads to a clear advantage in terms of resource, as in our case it is sufficient to have just two copies of the state. Our protocol requires only a single projective measurement setting in the computational basis for the purpose of post selection in weak measurement. It is interesting to note that in our protocol the number of copies required for entanglement detection may be further reduced if some partial information about the state is known. Moreover, we have also shown that the procedure of identification is achievable by local operations, if the state in consideration is a pure state. Further, we have shown that the protocol is robust against error arising during application of the weak interaction. Before concluding, it may be noted that though our scheme reduces the number of measurement settings compared to the universal entanglement witnessing scheme of \cite{ADH_08} that requires four copies of the state at a time, this advantage comes at the expense of joint unitary actions on two copies of the state (for arbitrary mixed states). Further work involving quantitative comparison of resources used in our scheme and that employed in other schemes such as in \cite{ADH_08} would be needed to obtain a clear idea of practical viability. In this context, one may need to compare the energy cost of creating correlations \cite{energycorrelation} with the energy cost of doing measurements \cite{energymeasurement1,energymeasurement2} used in the various protocols. Finally, we note that if a similar determinant based criterion for identification of certain class of states is available for higher dimensions, we expect a similar detection protocol such as ours to work therein. \begin{acknowledgments} SC would like to acknowledge Arun K Pati for having useful discussions on weak values and weak measurement. \end{acknowledgments}

proofpile-arXiv_069-10300

\section{Introduction} \bigskip Many researchers in partial differential equations and calculus of variation are interested in Sobolev type inequalities, or Sobolev embedding theorems as some wish to call them. The borderline case when the dimension is two is sometimes known as the \emph{Moser-Trudinger inequality} or \emph{Trudinger-Moser inequality} after the work of Trudinger~\cite{trudinger} in 1967 and Moser~\cite{moser} in 1971. To this day these ideas are still used in ongoing research (see e.g.~\cite{beckner,CL,de,LL,PSSW,yuan}). In this paper we shall prove the pluricomplex counterpart to the Moser-Trudinger and Sobolev inequalities. We shall now continue with a brief discussion about the setting, and we refer the reader to Section~\ref{sec_Background} for a more detailed background. Let $\Omega$ be an open set in $\CEP{n}$. An upper semicontinuous function $u:\Omega\to\mbox{$\mathbb{R}$}\cup\{-\infty\}$ is called \emph{plurisubharmonic} if the Laplacian of $u$ is, in the sense of distributions, non-negative along each complex line that intersects $\Omega$. We shall always assume that a plurisubharmonic function is defined on a so called hyperconvex domain $\Omega\subset\CEP{n}$. This assumption is made to ensure a satisfying amount of plurisubharmonic functions with certain properties. As the abstract reveal we are interested in plurisubharmonic functions with finite pluricomplex energy. To be able to define these functions we start by defining what we recognize as the pluricomplex counterpart of test functions in the theory of distributions. We say that a plurisubharmonic function $\varphi$ defined on $\Omega$ belongs to $\mbox{$\mathcal{E}_{0}$}$ $(=\mbox{$\mathcal{E}_{0}$}(\Omega))$ if $\varphi$ is a bounded function, $\lim_{z\rightarrow\xi} \varphi (z)=0$, for every $\xi\in\partial\Omega$, and $\int_{\Omega} \ddcn{\varphi}<\infty$, where $\ddcn{\cdot}$ is the complex Monge-Amp\`{e}re operator. Finally, we say that $u\in \mbox{$\mathcal{E}_{p}$}$ $(=\mbox{$\mathcal{E}_{p}$}(\Omega))$ if $u$ is a plurisubharmonic function defined on $\Omega$ and there exists a decreasing sequence, $\{\varphi_{j}\}$, $\varphi_{j}\in\mbox{$\mathcal{E}_{0}$}$, that converges pointwise to $u$ on $\Omega$, as $j$ tends to $\infty$, and \[ \sup_{j} e_p(\varphi_j)=\sup_{j}\int_{\Omega} (-\varphi_{j})^{p}\ddcn{\varphi_{j}}< \infty\, . \] This definition implies that if $u\in\mbox{$\mathcal{E}_{p}$}$, then $e_p(u)<\infty$. This justify that we say that a function $u\in\mbox{$\mathcal{E}_{p}$}$ have finite pluricomplex $p$-energy, or simply finite pluricomplex energy. The energy cones, $\mbox{$\mathcal{E}_{p}$}$, were introduced and studied in~\cite{cegrell_pc}, and the growing use of complex Monge-Amp\`{e}re techniques in applications makes this framework of significant importance (see e.g.~\cite{ACKPZ,BBW,demailly_hiep,Dinh_et al,diller}). \bigskip The first inequality we prove is the following. \medskip \noindent \textbf{The pluricomplex Moser-Trudinger inequality.} \emph{Let $\Omega$ be a bounded hyperconvex domain in $\mbox{$\mathbb{C}$}^n$, $n\geq 2$. Then there exist constants $A(p,n,\Omega)$ and $B(p,n,\Omega)$ depending only on $p, n, \Omega$ such that for any $u\in \mathcal E_p$ we have that} \begin{equation}\label{intr_MS} \log \int_{\Omega}e^{-u}d\lambda_{2n}\leq A(p,n,\Omega)+ B(p,n,\Omega)e_p(u)^{\frac 1p}\, , \end{equation} \emph{where $d\lambda_{2n}$ is the Lebesgue measure in $\mathbb C^n$. For any given $0<\epsilon<1$ we can take the constants $A(p,n,\Omega)$ and $B(p,n,\Omega)$ to be} \begin{align*} A(p,n,\Omega)& =\log \left(\left(\pi^n+\beta(n)\frac {\epsilon n}{(n-\epsilon n)^n}\right)\operatorname{diam}(\Omega)^{2n}\right) \, \text { and } \\[2mm] B(p,n,\Omega) & =(2\epsilon n)^{-\frac np}\, , \end{align*} \emph{where $\beta(n)$ is a constant depending only on $n$. Furthermore, we have the following estimates on $B(p,n,\Omega)$} \begin{equation}\label{1.2} \frac {p}{(4\pi)^{\frac np}(n+p)^{1+\frac np}}\leq B(p,n,\Omega)\, , \end{equation} \emph{and if $\Omega=\mbox{$\mathbb{B}$}$ is the unit ball, then we have that} \begin{equation}\label{1.3} \frac {p}{(4\pi)^{\frac np}(n+p)^{1+\frac np}}\leq B(p,n,\mathbb{B})\leq \left(\frac {p^{p-1}}{(4\pi)^n(n+1)^{n+1}(n+p)^{p-1}}\right)^{\frac 1p}\, . \end{equation} \medskip The proof of this theorem is divided into parts. Inequality~(\ref{intr_MS}) is proved in Theorem~\ref{t11}. This inequality was first proved for $p=1$ in~\cite{C1}, and two months later another proof appeared~\cite{BB1} that generated slightly better estimates. In Theorem~\ref{Best} we present the proof of our estimates. In the proof of the upper bound we are using a slightly modified version of Moser's original inequality. \bigskip In Section~\ref{sec_sobolev} we shall prove with help of the pluricomplex Moser-Trudinger inequality the following inequality. \medskip \noindent \textbf{The pluricomplex Sobolev inequality.} \emph{Let $\Omega$ be a bounded hyperconvex domain in $\mbox{$\mathbb{C}$}^n$, $n\geq 2$, and let $u\in \mbox{$\mathcal{E}_{p}$}$, $p>0$. Then for all $q>0$ there exists a constant $C(p,q,n,\Omega)>0$ depending only on $p,q, n,\Omega$ such that} \begin{equation}\label{intr_lq theorem} \|u\|_{L^q}\leq C(p,q,n,\Omega)e_p(u)^{\frac {1}{n+p}}\, . \end{equation} \emph{In fact one can take} \begin{equation}\label{intr_const c} C(p,q,n,\Omega)=e^{\frac 1qA(p,n,\Omega)}\frac {(n+p)B(p,n,\Omega)^{\frac {p}{n+p}}}{n^{\frac {n}{n+p}}p^{\frac {p}{n+p}}}\, \Gamma\left(\frac {nq}{n+p}+1\right)^{\frac 1q}, \end{equation} \emph{where the constants $A(p,n,\Omega)$ and $B(p,n,\Omega)$ are given in the above Moser-Trudinger inequality. Furthmore, $\Gamma$ denotes the gamma function. In addition, inequality (\ref{intr_lq theorem}) may be written in the form} \[ \|u\|_{L^q}\leq D(p,n,\Omega)q^{\frac {n}{n+p}}e_p(u)^{\frac {1}{n+p}} \, , \] \emph{where the constant $D(p,n,\Omega)$ does not depend on $q$.} \medskip The pluricomplex Sobolev inequality is proved in Theorem~\ref{lq estimate}. It should be noted that Theorem~\ref{lq estimate} is more general than the above statement, this due to presentational reasons. Our work in Section~\ref{sec_sobolev} was inspired by~\cite{BB1,Blo96,CP2,kol}. Next let $C(p,q,n,\mathbb{B})$ denote the optimal constant in~(\ref{intr_lq theorem}), i.e. the infimum of all admissible constants. This optimal constant is classically of great importance. For example it is connected to the isoperimetric inequality and therefore classically to symmetrization of functions (see e.g.~\cite{talenti}). In pluripotential theory there have been many attempts to symmetrize plurisubharmonic functions, but few progress have been made in that direction since plurisubharmonicity might be lost during a symmetrization procedure (\cite{Baernstein}). For a positive result see~\cite{BB2}. A strong trend today is to try to prove a pluricomplex counterpart of Talenti's theorem for the Laplacean (\cite{talenti2}, see also \cite[Theorem 10.2]{Blo14}). A successful attempt would not only imply simplified proofs, but also many of the biggest unsolved problems would be conquered. For further information and details we refer to Section~10 in the excellent survey~\cite{Blo14} written by B\l ocki. With this in mind we shall in Section~\ref{sec_optimal} prove that \[ C(1,1,n,\mathbb{B})=\frac {\pi^{\frac {n^2}{n+1}}}{4^{\frac {n}{n+1}}n!(n+1)^{\frac {n}{n+1}}}\, . \] \section{Background}\label{sec_Background} In this section we shall give some necessary background on pluripotential theory. For further information we refer to~\cite{cegrell_gdm,hab,demailly_bok,klimek,kol2,PSS}. A set $\Omega\subseteq\CEP{n}$, $n\geq 1$, is called a \emph{bounded hyperconvex domain} if it is a bounded, connected, and open set such that there exists a bounded plurisubharmonic function $\varphi:\Omega\rightarrow (-\infty,0)$ such that the closure of the set $\{z\in\Omega : \varphi(z)<c\}$ is compact in $\Omega$, for every $c\in (-\infty, 0)$. We say that a plurisubharmonic function $\varphi$ defined on $\Omega$ belongs to $\mbox{$\mathcal{E}_{0}$}$ $(=\mbox{$\mathcal{E}_{0}$} (\Omega))$ if it is a bounded function, \[ \lim_{z\rightarrow\xi} \varphi (z)=0 \quad \text{ for every } \xi\in\partial\Omega\, , \] and \[ \int_{\Omega} \ddcn{\varphi}<\infty\, , \] where $\ddcn{\cdot}$ is the complex Monge-Amp\`{e}re operator. Furthermore, we say that $u\in \mbox{$\mathcal{E}_{p}$}$ $(=\mbox{$\mathcal{E}_{p}$}(\Omega))$, $p>0$, if $u$ is a plurisubharmonic function defined on $\Omega$ and there exists a decreasing sequence, $\{\varphi_{j}\}$, $\varphi_{j}\in\mbox{$\mathcal{E}_{0}$}$, that converges pointwise to $u$ on $\Omega$, as $j$ tends to $\infty$, and \[ \sup_{j} e_p(\varphi_j)=\sup_{j}\int_{\Omega} (-\varphi_{j})^{p}\ddcn{\varphi_{j}}< \infty\, . \] We shall need on several occasions the following two inequalities. The inequality in Lemma~\ref{est} follows by standard approximation techniques from the work of B\l ocki in~\cite{Blo93}. \begin{lemma}\label{est} Let $u\in \mbox{$\mathcal{E}_{0}$}$, $v\in\mbox{$\mathcal{E}_{p}$}$, $p>0$, $1\leq k\leq n$. Then \[ \int_{\Omega} (-v)^{p+k}(dd^c u)^n\leq (p+k)\cdots (p+1)\|u\|^k_{L^{\infty}}\int_{\Omega}(-v)^{p}(dd^cv)^k\wedge(dd^cu)^{n-k}. \] \end{lemma} Next theorem was proved in~\cite{persson} for $p\geq 1$, and for $0<p<1$ in~\cite{czyz_energy} (see also~\cite{cegrell_pc,CP}). \begin{theorem}\label{thm_holder} Let $p>0$ and $u_0,u_1,\ldots , u_n\in\mbox{$\mathcal{E}_{p}$}$. If $n\geq 2$, then \begin{multline*} \int_\Omega (-u_0)^p dd^c u_1\wedge\cdots\wedge dd^c u_n\\ \leq d(p,n,\Omega)\; e_p(u_0)^{p/(p+n)}e_p(u_1)^{1/(p+n)}\cdots e_p(u_n)^{1/(p+n)}\, , \end{multline*} where \[d(p,n,\Omega)= \begin{cases} \left(\frac1p\right)^{\frac {n}{n-p}} & \text{if $0<p<1$,}\\ 1 & \text{if $p=1$,}\\ p^{\frac{p\alpha (n,p)}{p-1}} & \text{if $p>1$,} \end{cases} \] and $\alpha (n,p)=(p+2)\left(\frac{p+1}{p}\right)^{n-1}-(p+1)$. \end{theorem} \section{The pluricomplex Moser-Trudinger inequality}\label{sec_Moser-Trudinger} The aim of this section is to prove the following Moser-Trudinger inequality for $\mbox{$\mathcal{E}_{p}$}$, $p>0$. \begin{theorem}\label{t11} Let $\Omega$ be a bounded hyperconvex domain in $\mathbb C^n$. Then there exist constants $A(p,n,\Omega)$ and $B(p,n,\Omega)$ depending only on $p, n, \Omega$ such that for any $u\in \mathcal E_p$ we have that \[ \log \int_{\Omega}e^{-u}d\lambda_{2n}\leq A(p,n,\Omega)+ B(p,n,\Omega)e_p(u)^{\frac 1p}\, . \] Furthermore, for any given $0<\epsilon<1$ we can take the constants $A(p,n,\Omega)$ and $B(p,n,\Omega)$ to be \[ A(p,n,\Omega)=\log \left(\left(\pi^n+\beta(n)\frac {\epsilon n}{(n-\epsilon n)^n}\right)\operatorname{diam}(\Omega)^{2n}\right) \, \text { and } \, B(p,n,\Omega)=(2\epsilon n)^{-\frac np}\, , \] where $\beta(n)$ is a constant depending only on $n$. \end{theorem} \begin{proof} First assume that $u\in \mbox{$\mathcal{E}_{0}$}(\Omega)\cap C(\bar{\Omega})$. Next, thanks to~\cite{cegrell_gdm} we can find a uniquely determined function $w\in \mbox{$\mathcal{E}_{0}$}$ that satisfies \[ (dd^cw)^n=(-u)^p(dd^cu)^n\, . \] We shall now prove that \begin{equation}\label{t11_1} u\geq t^{-\frac pn}w-t\qquad \text{for all } t>0\, . \end{equation} First notice that on the set $\{z\in \Omega: u(z)\geq -t\}$ we have that \[ u(z)\geq -t\geq t^{-\frac pn}w-t\, . \] Next, since $u\in \mbox{$\mathcal{E}_{0}$}$ and $\lim_{z\to \partial \Omega}u(z)=0$ we have that the open set $\omega=\{z\in \Omega: u(z)< -t\}$ is relatively compact in $\Omega$ and therefore \[ t^p(dd^cu)^n\leq (-u)^p(dd^cu)^n=(dd^cw)^n\, . \] Hence, \[ (dd^cu)^n\leq t^{-p}(dd^cw)^n=\Big(dd^c(t^{-\frac pn}w-t)\Big)^n\, , \] and furthermore \[ \liminf_{\omega\ni z\to \partial \omega}(u(z)-t^{-\frac pn}w(z)+t)\geq 0\, . \] Therefore, by the comparison principle (see~\cite{cegrell_pc}) we get that $u\geq t^{-\frac pn}w-t$ on $\omega$ and~(\ref{t11_1}) is valid. Fix $0<\epsilon<1$ and choose $t$ such that \[ t=\left(\frac {e_p(u)}{\epsilon^n(2n)^n}\right)^{\frac 1p}\, . \] With this choice of $t$ we have \[ \int_{\Omega}(dd^ct^{-\frac pn}w)^n=t^{-p}\int_{\Omega}(-u)^p(dd^cu)^n=t^{-p}e_p(u)=\epsilon^n (2n)^n\, . \] By using Corollary~5.2 in~\cite{ACKPZ} for the function $t^{-\frac pn}w$ we get \begin{multline*} \log \int_{\Omega}e^{-u}d\lambda_{2n}\leq \log \int_{\Omega}e^{-t^{-\frac pn}w}e^{t}d\lambda_{2n}\leq \\ \leq \log \left(\left(\pi^n+\beta(n)\frac {\epsilon n}{(n-\epsilon n)^n}\right)\operatorname{diam}(\Omega)^{2n}\right)+ \frac {e_p(u)^{\frac 1p}}{(2\epsilon n)^{\frac np}}\, . \end{multline*} By a standard procedure we can now remove the assumption that $u\in \mbox{$\mathcal{E}_{0}$}(\Omega)\cap C(\bar{\Omega})$, since for arbitrary $u\in \mbox{$\mathcal{E}_{p}$}$ there exists a sequence $u_j\in\mbox{$\mathcal{E}_{0}$}(\Omega)\cap C(\bar{\Omega})$ such that $u_j\searrow u$ and $e_p(u_j)\to e_p(u)$, $j\to \infty$ (see e.g.~\cite{CKZ}). \end{proof} Next in Corollary~\ref{t11_cor} we obtain Theorem~\ref{t11} for functions from the class $\mathcal E_{\chi}$. Let us first recall the definition of $\mathcal E_{\chi}$ (see e.g.~\cite{bgz,Guedj_Zer} for further information). \begin{definition}\label{def_Echi} Let $\Omega$ be a bounded hyperconvex domain in $\mathbb C^n$, $n\geq 2$. Let $\chi:(-\infty,0]\to (-\infty,0]$ be a continuous and nondecreasing function. Furthermore, let $\mbox{$\mathcal{E}_{\chi}$}$ contain those plurisubharmonic functions $u$ for which there exists a decreasing sequence $u_{j}\in\mbox{$\mathcal{E}_{0}$}$ that converges pointwise to $u$ on $\Omega$, as $j$ tends to $\infty$, and \[ e_{\chi}(u)=\sup_{j}\int_{\Omega} -\chi(u_j)\ddcn{u_j}< \infty\, . \] \end{definition} For example, with this notation if $\chi=-(-t)^p$, then $\mbox{$\mathcal{E}_{\chi}$}=\mbox{$\mathcal{E}_{p}$}$. It was proved in~\cite{b} and in~\cite{haihiep} that if $\chi:(-\infty,0]\to (-\infty,0]$ is continuous, and strictly increasing, then the complex Monge-Amp\`{e}re operator is well defined on $\mbox{$\mathcal{E}_{\chi}$}$. We are now in position to prove Corollary~\ref{t11_cor}. \begin{corollary}\label{t11_cor} Let $\chi:(-\infty,0]\to (-\infty,0]$ be a continuous and nondecreasing function, and let $\Omega$ be a bounded hyperconvex domain in $\mathbb C^n$, $n\geq 2$. Then for any fixed $0<\epsilon<1$ we have for all $u\in \mbox{$\mathcal{E}_{\chi}$}$ that \[ \log \int_{\Omega}e^{-u}d\lambda_{2n}\leq \log \left(\left(\pi^n+\beta(n)\frac {\epsilon n}{(n-\epsilon n)^n}\right)\operatorname{diam}(\Omega)^{2n}\right)- \chi^{-1}\left(\frac {-e_{\chi}(u)}{(2\epsilon n)^n}\right). \] \end{corollary} \begin{proof} The proof of Theorem~\ref{t11} works here as well with some changes of references. The reference~\cite{cegrell_gdm} should be replaced with~\cite{haihiep}, and~\cite{CKZ} should be replaced with~\cite{b}. \end{proof} We shall end this section with a remark about the case when the underlying space is a compact K\"ahler manifold. Let us first recall some facts. Let $(X,\omega)$ be a K\"ahler manifold of dimension $n$ with a K\"ahler form $\omega$ such that $\int_{X}\omega^n=1$. We say that $u\in \mbox{$\mathcal{E}_{p}$}(X,\omega) (=\mbox{$\mathcal{E}_{p}$})$ if there exists a sequence $u_j\in \mathcal {PSH}(X,\omega)\cap L^{\infty}(X)$ such that $u_j\leq 0$, $u_j\searrow u$, $j\to \infty$ and \[ \sup_{j}\int_X(-u_j)^p(dd^cu_j+\omega)^n<\infty\, . \] Here $\mathcal {PSH}(X,\omega)$ denote the set of $\omega$-plurisubharmonic functions. For $u\in \mbox{$\mathcal{E}_{p}$}$, set \[ e_p(u)=\int_X(-u)^p(dd^cu+\omega)^n\, . \] In the case $p=1$ , we have the following classical functional defined on $\mathcal E_1$ by \[ \mathcal E_{\omega}(u)=\frac {1}{(n+1)!}\sum_{k=0}^{n}\int_X(-u)(dd^cu+\omega)^k\wedge \omega^{n-k}\, , \] and we have the following estimation \[ \mathcal E_{\omega}(u)\leq \frac {1}{n!}\int_X(-u)(dd^cu+\omega)^n=\frac {1}{n!}\, e_1(u)\, . \] \begin{remark}\label{kahler} Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ with a K\"ahler form $\omega$ such that $\int_{X}\omega^n=1$. It was proved in~\cite{BB1} that there exist constants $a,b>0$ such that for all $u\in \mathcal E_1$ and $k>0$ it holds that \begin{equation}\label{kahler_1} \log \left (\int_Xe^{-ku}\omega^n\right)\leq ak^{n+1}\mathcal E_{\omega}(u)+b\, . \end{equation} Now let $u\in \mbox{$\mathcal{E}_{p}$}$, $p>1$. By using H\"older inequality we get that \[ e_1(u)=\int_X(-u)(dd^cu+\omega)^n\leq \left(\int_X(-u)^p(dd^cu+\omega)^n\right)^{\frac 1p}=e_p(u)^{\frac 1p}\, . \] Thus, $u\in \mathcal E_p$, and by~(\ref{kahler_1}) we arrive at \begin{multline}\label{kahler 2} \log \left (\int_Xe^{-ku}\omega^n\right)\leq ak^{n+1}\mathcal E_{\omega}(u)+b\leq \frac {ak^{n+1}}{n!}e_1(u)+b\\ \leq \frac {ak^{n+1}}{n!}e_p(u)^{\frac 1p}+b . \end{multline} This inequality shall be used on page~\pageref{kahler2}. It should be noted that the case when $0<p<1$ is at this point unknown to the authors. \end{remark} \section{Estimates of the constant $B(p,n,\Omega)$}\label{sec_Estimates} Let us introduce the following notation. For $r>0$ let $\mathbb{B} (z_0,r)=\{z\in\CEP{n}: |z-z_0|< r\}$ be the open ball with center $z_0$ and radius $r$, and to simplify the notations set $\mathbb{B}=\mathbb{B} (0,1)$. Now let $B(p,n,\Omega)$ denotes the optimal constant in the Moser-Trudinger inequality (\ref{intr_MS}), i.e. the infimum of all admissible constants. The aim of this section is to estimate the constant $B(p,n,\Omega)$ for arbitrary hyperconvex domains, inequality (\ref{1.2}), and also in the special case when $\Omega=\mbox{$\mathbb{B}$}$, inequality (\ref{1.3}). We shall arrive to the following estimates. \begin{theorem}\label{Best} Let $\Omega$ be a bounded hyperconvex domain in $\mathbb C^n$, and $B(p,n,\Omega)$ the constant in Theorem~\ref{t11}. Then we have that \[ \frac {p}{(4\pi)^{\frac np}(n+p)^{1+\frac np}}\leq B(p,n,\Omega)\, . \] \end{theorem} \begin{proof} Without loss of generality we can assume that $0\in \Omega$. Let $g_{\Omega}(z,0)$ be the pluricomplex Green function with pole at $0$, and for a parameter $\beta\leq 0$ let us define \[ u(z)=(2n+2p)\max\big(g_{\Omega}(z,0),\beta\big)\, . \] This construction yields that \[ e_p(u)=\int_{\Omega}(-u)^p(dd^cu)^n=(2\pi)^n(2n+2p)^{p+n}(-\beta)^p\, , \] and then we shall proceed by estimating the integral \begin{equation}\label{eq1} \int_{\Omega}e^{-u}d\lambda_{2n}\, . \end{equation} From the definition of the pluricomplex Green function it follows that there exist a radius $r>0$, and a constant $C>0$, such that $\mbox{$\mathbb{B}$}(0,r)\Subset\Omega$ and such that the following inequalities hold for all $z\in \mbox{$\mathbb{B}$}(0,r)$: \begin{equation}\label{eq2} \log|z|-C\leq g_{\Omega}(z,0)\leq \log |z|+C\, . \end{equation} Choose then $\beta\leq \beta_1\leq 0$ such that it holds $\{z\in \Omega:g_{\Omega}(z,0)<\beta_1\}\subset \mbox{$\mathbb{B}$}(0,r)$. From now on we shall only consider those $\beta$ with $\beta\leq \beta_1$. From~(\ref{eq2}) we now have that \[ \mbox{$\mathbb{B}$}\left(0,e^{\beta-C}\right)\subset \{z\in \Omega:g_{\Omega}(z,0)<\beta\}\subset \mbox{$\mathbb{B}$}\left(0,e^{\beta+C}\right)\, . \] We start by dividing~(\ref{eq1}) as \begin{multline*} \int_{\Omega}e^{-u}d\lambda_{2n}=\int_{\{z\in \Omega:g_{\Omega}(z,0)<\beta\}}e^{-(2n+2p)\beta}d\lambda_{2n}\\+\int_{\{z\in \Omega:g_{\Omega}(z,0)\geq \beta\}}e^{-(2n+2p) g_{\Omega}(z,0)}d\lambda_{2n}=I_1+I_2\ , \end{multline*} and notice that by~(\ref{eq2}) we have \begin{multline}\label{eq3} \frac {\pi^n}{n!}e^{-2nC}e^{-2p\beta}=e^{-(2n+2p)\beta}\lambda_{2n}(\mbox{$\mathbb{B}$}(0,e^{\beta-C}))\leq I_1\\ \leq e^{-(2n+2p)\beta}\lambda_{2n}(\mbox{$\mathbb{B}$}(0,e^{\beta+C})=\frac {\pi^n}{n!}e^{2nC}e^{-2p\beta}\, . \end{multline} Furthermore, \begin{multline*} I_2=\int_{\Omega\setminus \mathbb{B}(0,r)}e^{-(2n+2p)g_{\Omega}(z,0)}d\lambda_{2n}+\int_{\mathbb{B}(0,r)\cap \{z\in \Omega:g_{\Omega}(z,0)\geq \beta\}}e^{-(2n+2p)g_{\Omega}(z,0)}d\lambda_{2n}\\=I_3+I_4\, . \end{multline*} For $z\in \Omega\setminus \mbox{$\mathbb{B}$}(0,r)$ we have that \[ 1\leq e^{-(2n+2p)g_{\Omega}(z,0)}\leq e^{-(2n+2p)\beta_1} \] and therefore \begin{equation}\label{eq4} \lambda_{2n}(\Omega\setminus \mbox{$\mathbb{B}$}(0,r))\leq I_3\leq e^{-(2n+2p)\beta_1}\lambda_{2n}(\Omega\setminus \mbox{$\mathbb{B}$}(0,r))\, . \end{equation} We also get the estimate of $I_4$ as \begin{multline}\label{eq5} 0\leq I_4\leq \int_{\mathbb{B}(0,r)\setminus \mathbb{B}(0,e^{\beta-C})}e^{-(2n+2p)(\log|z|-C)}d\lambda_{2n}\\=e^{(2n+2p)C}\frac {2\pi^n}{(n-1)!}\int_{e^{\beta-C}}^r t^{-1-2p}dt =\frac {2\pi^ne^{(2n+2p)C}}{(n-1)!(-2p)}\left(r^{-2p}-e^{(\beta-C)(-2p)}\right)\, . \end{multline} From (\ref{eq1}), (\ref{eq3}), (\ref{eq4}) and (\ref{eq5}) it follows that there exist constant $c_1, c_2, c_3, c_4$ not depending on $\beta$ such that \[ c_1e^{-2p\beta}+c_2\leq \int_{\Omega}e^{-u}d\lambda_{2n}\leq c_3e^{-2p\beta}+c_4\, , \] and therefore \[ \lim_{\beta\to -\infty}\,\frac {\log \left(\displaystyle{\int_{\Omega}e^{-u}d\lambda_{2n}}\right)}{e_p(u)^{\frac 1p}}=\frac {p}{(4\pi)^{\frac np}(n+p)^{1+\frac np}}\, . \] Thus, \[ B(p,n,\Omega)\geq \frac {p}{(4\pi)^{\frac np}(n+p)^{1+\frac np}}\, . \] \end{proof} To prove the inequality (\ref{1.3}) we shall make use of radially symmetric plurisubharmonic functions. Let us recall some basic facts here, and we refer the reader to~\cite{AC1,monn} and the references therein for further information. Recall that a function $u:\mathbb{B}\to [-\infty,\infty)$ is said to be \emph{radially symmetric} if we have that \[ u(z)=u(|z|)\qquad \text{ for all } z\in\mathbb{B}\, . \] For each radially symmetric function $u:\mathbb{B}\to [-\infty, \infty)$ we define the function $\tilde u:[0,1)\to [-\infty, \infty)$ by \begin{equation}\label{prel_def_rad} \tilde u(t)=u(|z|)\, , \text{ where } t=|z|\, . \end{equation} On the other hand, to every function $\tilde v:[0,1)\to [-\infty, \infty)$ we can construct a radially symmetric function $v$ through~(\ref{prel_def_rad}). Furthermore, $u$ is a radially symmetric plurisubharmonic function if and only if $u(t)$ is an increasing function, and it is convex with respect to $\log t$. \bigskip Let us first show a few elementary lemmas. \begin{lemma}\label{l1} For any $\alpha >1$, and any $A>0$, there exists a constant $B$ such that for all $t\geq 0$ it holds \[ At^{\alpha}+B\geq t . \] In fact one can take \[ B=\frac {\alpha-1}{\alpha}(\alpha A)^{\frac {1}{1-\alpha}}\, . \] \end{lemma} \begin{proof} It is is enough to observe that the function \[ f(t)=At^{\alpha}-t \] attains its minimum at \[ t_0=\left(\alpha A\right)^{\frac {1}{1-\alpha}}\, , \] and that \[ \min_{[0,\infty)} f=\frac {1-\alpha}{\alpha}(\alpha A)^{\frac {1}{1-\alpha}}=-B\, . \] \end{proof} In Lemma~\ref{ep lemma} we shall make use of the following equality. For $f\in L^{p}(X,\mu)$ we have \begin{equation}\label{ep eq_1} \int_X|f|^p\,d\mu=p\int_{0}^{\infty}t^{p-1}\mu(\{x\in X: |f(x)|\geq t\})\,dt\, . \end{equation} \begin{lemma}\label{ep lemma} Let $p>0$, and let $u(z)=u(|z|)=\tilde u(t)$ be a radially symmetric plurisubharmonic function such that $\lim_{z\to \partial \mathbb{B}}u(z)=0$ and $u\in \mathcal E_p$, then we have \begin{equation}\label{ep eq} e_p(u)=(2\pi)^np\int_0^1(-\tilde u(t))^{p-1}\tilde u'(t)^{n+1}t^ndt\, . \end{equation} \end{lemma} \begin{proof} If $u(z)=u(|z|)=\tilde u(t)$ is a radially symmetric plurisubharmonic function such that $\lim_{z\to \partial \mathbb{B}}u(z)=0$, then for $t=|z|$ it holds that \[ F(t):=\frac {1}{(2\pi)^n}(dd^cu)^n(B(0,t))=t^n\tilde u'(t)^n\, , \] where $\tilde u'$ is the left derivative of a convex function $\tilde u$ (see \cite{AC1}). For $t\geq 0$ we have that \[ \{z\in \mathbb{B}:u(z)\leq -t\}=B(0,s), \, \text { where } \, s=\tilde u^{-1}(-t), \] where $\tilde u^{-1}(\inf u)=\sup\{x: \tilde u(x)=\inf \tilde u\}$. Therefore, by using~(\ref{ep eq_1}) we arrive at \begin{multline*} e_p(u)=\int_{\mathbb{B}}(-u)^p(dd^cu)^n=p\int_0^{-\inf u}t^{p-1}(dd^cu)^n(\{z\in \mathbb{B}: u(z)\leq -t\})\, dt \\ =p(2\pi)^n\int_0^{-\inf \tilde u}t^{p-1}F(\tilde u^{-1}(-t))\, dt =(2\pi)^np\int_0^1(-\tilde u(s))^{p-1}\tilde u'(s)^{n+1}s^n\,ds\, , \end{multline*} where $\tilde u(s)=t$, and this completes this proof. \end{proof} We are now in position to prove the inequality (\ref{1.3}). \begin{theorem}\label{rsep} Let $p>0$, and let $u$ be a radially symmetric plurisubharmonic function such that $\lim_{z\to \partial \mathbb{B}}u(z)=0$ and $u\in \mathcal E_p$, then we have that \[ \log \int_{\mathbb{B}}e^{-u(z)}d\lambda_{2n}\leq d+ \left(\frac {e_p(u)p^{p-1}}{(4\pi)^n(n+1)^{n+1}(n+p)^{p-1}}\right)^{\frac 1p}\, , \] where the constant $d$ does not depend on $u$. Therefore, \[ B(p,n,\mathbb{B})\leq \left(\frac {p^{p-1}}{(4\pi)^n(n+1)^{n+1}(n+p)^{p-1}}\right)^{\frac 1p}\, . \] \end{theorem} \begin{proof} By Lemma~\ref{ep lemma} we have that the pluricomplex $p$-energy of $u$ is equal to \begin{multline*} e_p(u)=(2\pi)^np\int_0^1(-\tilde u(t))^{p-1}\tilde u'(t)^{n+1}t^ndt=\\ =\frac {(2\pi)^np(n+1)^{n+1}}{(n+p)^{n+1}}\int_0^1\left(\left(-(-\tilde u(t))^{\frac {n+p}{n+1}}\right)'\right)^{n+1}t^ndt\, . \end{multline*} Therefore, if $v(t)=-(-x\tilde u(t))^{\frac {n+p}{n+1}}$, where \[ x=\left(\frac {(2\pi)^np(n+1)^{n+1}}{e_p(u)(n+p)^{n+1}}\right)^{\frac {1}{n+p}}\, , \] then $v$ be an increasing function $v:[0,1)\to (-\infty,0)$ such that $\lim_{t\to 1^-}v(t)=0$ and \[ \int_{0}^1(v'(s))^{n+1}s^nds\leq 1. \] Thanks to a slightly modified version of the classical Moser-Trudinger inequality (cf.~\cite{moser}), we arrive at \begin{equation}\label{moser3} \int_{0}^1e^{2n(-v(s))^{\frac {n+1}{n}}}s^{2n -1}ds=\int_{0}^1e^{2n(-\tilde u(s)x)^{\frac {n+p}{n}}}s^{2n -1}ds\leq \frac {c}{2n}\, . \end{equation} where the constant $c$ does not depend on $u$. Lemma~\ref{l1} yields that \[ -\tilde u(s)\leq 2n(-\tilde u(s)x)^{\frac {p+n}{n}}+\left(\frac {e_pp^{p-1}}{(4\pi)^n(n+1)^{n+1}(n+p)^{p-1}}\right)^{\frac 1p}= 2n(-\tilde u(s)x)^{\frac {p+n}{n}}+y\, . \] Hence by (\ref{moser3}), \begin{multline*} \int_{\mathbb{B}}e^{-u(z)}d\lambda_{2n}=\frac {2\pi^n}{(n-1)!}\int_0^1e^{-\tilde u(s)}s^{2n-1}ds\leq \\ \frac {2\pi^n}{(n-1)!}\int_0^1 e^{2n(-\tilde u(s)x)^{\frac {n+p}{n}}+y}s^{2n -1}ds\leq \frac {2\pi^n}{(n-1)!}\frac {c}{2n}e^y\, , \end{multline*} and finally \[ \log \int_{\mathbb{B}}e^{-u(z)}d\lambda_{2n}\leq \log \left(\frac {\pi^n c}{n!}\right)+ \left(\frac {e_pp^{p-1}}{(4\pi)^n(n+1)^{n+1}(n+p)^{p-1}}\right)^{\frac 1p}\, . \] \end{proof} A direct consequence of~(\ref{moser3}) is the following corollary which was first proved in~\cite{BB2} in the case $p=1$. \begin{corollary} Let $p>0$, and let $u$ be a radially symmetric plurisubharmonic function such that $\lim_{z\to \partial \mathbb{B}}u(z)=0$ and $u\in \mathcal E_p(\mathbb{B})$, then we have that \[ \int_{\mathbb{B}}e^{\alpha(p,n)(-u(z))^{\frac {n+p}{n}}e_p(u)^{-\frac 1n}}d\lambda_{2n}<\infty\, , \] where $\alpha(p,n)=4\pi np^{\frac 1n}\left(\frac {n+1}{n+p}\right)^{\frac {n+1}{n}}$. \end{corollary} \begin{proof} By (\ref{moser3}) we have \begin{multline*} \int_{\mathbb{B}}e^{\alpha(p,n)(-u(z))^{\frac {n+p}{n}}e_p(u)^{-\frac 1n}}d\lambda_{2n}=\frac {2\pi^n}{(n-1)!}\int_0^1e^{\alpha(p,n)(-\tilde u(s))^{\frac {n+p}{n}}e_p(u)^{-\frac 1n}}s^{2n-1}\, ds \\ < \frac {c\pi^n}{n!}<\infty. \end{multline*} \end{proof} \section{The pluricomplex Sobolev inequality}\label{sec_sobolev} In this section we shall prove the pluricomplex Sobolev inequality. We shall prove it for differences of plurisubharmonic functions with finite energy, i.e. for functions in $\delta\mbox{$\mathcal{E}_{p}$}=\mbox{$\mathcal{E}_{p}$}-\mbox{$\mathcal{E}_{p}$}$. If we for $u=u_1-u_2\in \delta\mbox{$\mathcal{E}_{p}$}$ define $\vertiii{u}_p$ by \[ \vertiii{u}_p=\inf_{u_1-u_2=u \atop u_1,u_2\in \mathcal{E}_p} e_{p}(u_1+u_2)^{\frac {1}{n+p}}\, , \] then $(\delta\mbox{$\mathcal{E}_{p}$}, \|\cdot\|_p)$ becomes a quasi-Banach space, and for $p=1$ a Banach space (see \cite{mod}). Note that in the case $u\in \mbox{$\mathcal{E}_{p}$}$ we have that $\vertiii{u}_p=e_p(u)^{\frac 1{n+p}}$. \begin{theorem}\label{lq estimate} Let $\Omega$ be a bounded hyperconvex domain in $\mbox{$\mathbb{C}$}^n$, and let $u\in \delta\mbox{$\mathcal{E}_{p}$}$, $p>0$. Then for all $q>0$ there exists a constant $C(p,q,n,\Omega)>0$ depending only on $p,q, n,\Omega$ such that \begin{equation}\label{lq theorem} \|u\|_{L^q}\leq C(p,q,n,\Omega)\vertiii{u}_p\, . \end{equation} In fact one can take \begin{equation}\label{const c} C(p,q,n,\Omega)=e^{\frac 1qA(p,n,\Omega)}\frac {(n+p)B(p,n,\Omega)^{\frac {p}{n+p}}}{n^{\frac {n}{n+p}}p^{\frac {p}{n+p}}}\, \Gamma\left(\frac {nq}{n+p}+1\right)^{\frac 1q}, \end{equation} where the constants $A(p,n,\Omega)$ and $B(p,n,\Omega)$ are given in Theorem~\ref{t11}. In addition, inequality (\ref{lq theorem}) may be written, for $q\geq 1$, in the form \begin{equation}\label{q} \|u\|_{L^q}\leq D(p,n,\Omega)q^{\frac {n}{n+p}}\vertiii{u}_p\, , \end{equation} where the constant $D(p,n,\Omega)$ does not depend on $q$. Furthermore, the identity operator $\iota:\delta\mbox{$\mathcal{E}_{p}$}\to L^q$ is compact. \end{theorem} Before we start the proof let us recall the definition of compactness in quasi-Banach spaces. \begin{definition} Let $X$, $Y$ be two quasi-Banach spaces. The operator $K:X\to Y$ is called \emph{compact} if for any sequence $\{x_n\}\subset X$ with $\|x_n\|\leq 1$, then there exists a convergent subsequence $\{y_{n_k}\}$ of $\{K(x_n)\}$. \end{definition} \begin{proof}[Proof of Theorem~\ref{lq estimate}] First assume that $u\in \mbox{$\mathcal{E}_{p}$}$, $p>0$. For $t,s>0$ define \[ f(t)=\int_{\Omega}e^{-tu}d\lambda_{2n} \, \text { and } \, \lambda(s)=\lambda_{2n}(\{z\in\Omega: u(z)<-s\}). \] Note that by Theorem~\ref{t11} there exist constants $A=A(p,n,\Omega)$ and $B=B(p,n,\Omega)$ such that \[ f(t)\leq e^Ae^{Bt^{\frac {n+p}{p}}e_p(u)^{\frac 1p}}=Ce^{g(t)}\, , \] where $g(t)=Bt^{\frac {n+p}{p}}e_p(u)^{\frac 1p}$. For $s,t>0$ we have that \begin{equation}\label{legendre transform} \lambda(s)\leq \int_{\{z\in\Omega: u(z)<-s\}}e^{-st}e^{-tu}d\lambda_{2n}\leq e^{-st}\int_{\Omega}e^{-tu}d\lambda_{2n}\leq Ce^{-st+g(t)}. \end{equation} By Lemma~\ref{l1} we now have that \[ g(t)-st=Bt^{\frac {n+p}{p}}e_p(u)^{\frac 1p}-st\geq -s^{\frac {n+p}{p}}\frac {np^{\frac pn}}{(n+p)^{1+\frac pn}}B^{-\frac pn}e_p(u)^{-\frac 1n}\, . \] Therefore, it follows from~(\ref{legendre transform}) that \begin{equation}\label{lambda} \lambda(s)\leq Ce^{-xs^{\frac {n+p}{p}}}, \, \text { where }\, x=\frac {np^{\frac pn}}{(n+p)^{1+\frac pn}}B^{-\frac pn}e_p(u)^{-\frac 1n}. \end{equation} By letting $r=xs^{\frac {n+p}{n}}$ in~(\ref{lambda}) we get that \begin{multline}\label{2} \|u\|_{L^q}^q=\int_{\Omega}(-u)^qd\lambda_{2n}=q\int_{0}^{\infty}s^{q-1}\lambda(s)ds\leq qC\int_0^{\infty}s^{q-1}e^{-xs^{\frac {n+p}{p}}}ds\\ =\frac {qnC}{(n+p)x^{\frac {nq}{n+p}}}\int_{0}^{\infty}r^{-1+\frac {nq}{n+p}}e^{-r}\,dr=\frac {qnC}{(n+p)x^{\frac {nq}{n+p}}}\, \Gamma \left (\frac {nq}{n+p}\right)\\ =Cx^{-\frac {nq}{n+p}}\, \Gamma \left (\frac {nq}{n+p}+1\right)=\frac {C(n+p)^{q}B^{\frac {pq}{n+p}}}{n^{\frac {nq}{n+p}}p^{\frac {pq}{n+p}}}\, \Gamma\left(\frac {nq}{n+p}+1\right)e_p(u)^{\frac {q}{n+p}}\, . \end{multline} Thus, (\ref{lq theorem}) holds for $u\in \mbox{$\mathcal{E}_{p}$}$. In the general case, let $u=u_1-u_2\in \delta\mbox{$\mathcal{E}_{p}$}$ it is enough to note that \[ \|u\|_{L^q}\leq \|u_1+u_2\|_{L^q}\leq C(p,q,n,\Omega)e_p(u_1+u_2)\, , \] and then taking the infimum over all possible decomposition of $u$. Next we shall prove (\ref{q}). First assume that $u\in \mbox{$\mathcal{E}_{p}$}$. Note that for $y\geq 1$ it is a fact that \[ \Gamma(y+1)\leq 2y^y\, , \] so by (\ref{2}) it holds that for $q\geq 1$ \begin{multline*} \|u\|_{L^q}\leq \left(\frac {C(n+p)^{q}B^{\frac {pq}{n+p}}}{n^{\frac {nq}{n+p}}p^{\frac {pq}{n+p}}}\Gamma\left(\frac {nq}{n+p}+1\right)e_p(u)^{\frac {q}{n+p}}\right)^{\frac 1q} \\ \leq 2^{\frac 1q}e^{\frac Aq}\frac {(n+p)B^{\frac {p}{n+p}}}{n^{\frac {n}{n+p}}p^{\frac {p}{n+p}}}\left(\frac {nq}{n+p}\right)^{\frac {n}{n+p}}e_p(u)^{\frac {1}{n+p}}\\ \leq 2e^{A}\frac {(n+p)^{\frac {p}{n+p}}B^{\frac {p}{n+p}}}{p^{\frac {p}{n+p}}}q^{\frac {n}{n+p}}e_p(u)^{\frac {1}{n+p}}\, . \end{multline*} To proceed to the general case $u=u_1-u_2\in \delta\mbox{$\mathcal{E}_{p}$}$ we follow the above procedure and arrive at \[ \|u\|_{L^q}\leq D(p,n,\Omega)q^{\frac {n}{n+p}}\vertiii{u}_p\, , \] with \[ D(p,n,\Omega)=2e^{A(p,n,\Omega)}\frac {(n+p)^{\frac {p}{n+p}}B(p,n,\Omega)^{\frac {p}{n+p}}}{p^{\frac {p}{n+p}}}\, . \] To complete this proof we shall prove that the identity operator $\iota:\delta\mbox{$\mathcal{E}_{p}$}\to L^q$ is compact. Take a sequence $\{u_n\}=\{u_n^1-u_n^2\}\subset \delta\mbox{$\mathcal{E}_{p}$}$ with $\vertiii{u}_p\leq 1$. Then by the same reasoning as above we get that \[ \|u_n^j\|_{L^q}\leq \|u_n^1+u_n^2\|_{L^q}\leq C(p,n,\Omega)\quad \text{for} \quad j=1,2\, . \] Hence, there exists a subsequence $\{u_{n_k}^j\}$ converging almost everywhere to some plurisubharmonic function $\{v^j\}$. This means that $\{u_{n_k}^1-u_{n_k}^2\}$ is a Cauchy sequence in $L^q$. Thus, $\iota$ is compact. \end{proof} The proof of Theorem~\ref{lq estimate} relies on the Moser-Trudinger inequality (Theorem~\ref{t11}). In the case when $q\leq n+p$, we can present an elementary proof only using the inequalities in Lemma~\ref{est} and Theorem~\ref{thm_holder}. \begin{proof}[Proof of Theorem~\ref{lq estimate} for $q\leq n+p$] There exists $\varphi_0\in \mbox{$\mathcal{E}_{0}$}$ such that \[ (dd^c\varphi_0)^n=d\lambda_{2n}\, , \] (see e.g.~\cite{kol2}). Let $u=u_1-u_2\in \delta\mbox{$\mathcal{E}_{p}$}$, and let $0<q\leq p+n$. Thanks to Lemma~\ref{est} and Theorem~\ref{thm_holder} we get that \begin{multline*} \|u\|_{L^q}\leq \|u_1+u_2\|_{L^q}\leq \lambda_{2n}(\Omega)^{\frac 1q-\frac{1}{p+n}}\|u_1+u_2\|_{L^{p+n}}= \\ =\lambda_{2n}(\Omega)^{\frac 1q-\frac{1}{p+n}}\left(\int_{\Omega} (-u_1-u_2)^{p+n}(dd^c \varphi_0)^n\right)^{\frac 1{n+p}}\leq \\ \leq \lambda_{2n}(\Omega)^{\frac 1q-\frac{1}{p+n}}\left((p+n)\cdots (p+1)\|\varphi_0\|^n_{L^{\infty}}\int_{\Omega}(-u_1-u_2)^{p}(dd^c(u_1+u_2))^n\right)^{\frac {1}{n+p}}\leq \\ \leq \left(\lambda_{2n}(\Omega)^{\frac{p+n-q}{q}}(p+n)\cdots (p+1)\|\varphi_0\|_{L^{\infty}}^n\right)^{\frac {1}{n+p}}e_p(u_1+u_2)^{\frac {1}{n+p}}\, . \end{multline*} Finally by taking the infimum over all possible decompositions $u=u_1-u_2$ we obtain that \[ \|u\|_{L^q}\leq C(p,q,n,\Omega)\vertiii{u}_p\, . \] \end{proof} Next we present an example that shows that it is impossible to have an estimate of the type \[ e_p(u)^{\frac {1}{n+p}}\leq C\|u\|_{L^q}\, . \] \begin{example}\label{ex1} Consider the following functions defined on the unit ball $\mathbb{B}$ in $\mbox{$\mathbb{C}$}^n$ \[ u_j(z)=\frac 1j\max\left(\log |z|,-j^{1+\frac np}\right)\, . \] Then we have that \[ u_j(z)= \begin{cases} \frac{1}{j} \log |z| & \text{ if } \exp\left(-j^{1+\frac np}\right)\leq |z|\leq 1 \\[2mm] -j^{\frac np} & \text{ if } 0\leq |z|\leq \exp\left(-j^{1+\frac np}\right)\, . \end{cases} \] Hence, $\|u_j\|_{L^q}\to 0$, as $j\to \infty$, but at the same time we have that \[ e_p(u)=\frac {1}{j^{n+p}}(2\pi)^n\left(j^{1+\frac np}\right)^p=(2\pi)^n\, , \] which is a contradiction. \hfill{$\Box$} \end{example} Example~\ref{ex2} shows that it is also impossible to have an estimate of the type \[ \|u\|_{L^{\infty}}\leq C e_p(u)^{\frac {1}{n+p}}\, . \] \begin{example}\label{ex2} Similarly as in Example~\ref{ex1} consider the following functions defined on the unit ball $\mathbb{B}$ in $\mbox{$\mathbb{C}$}^n$ \[ u_j(z)=\frac 1{j^{\frac {p}{n+p}}}\max\left(\log |z|,-j\right)\, . \] Then we have that $\|u_j\|_{L^{\infty}}=-u_j(0)=j^{\frac {n}{n+p}}\to \infty$, as $j\to \infty$, and at the same time \[ e_p(u_j)=(2\pi)^nj^p\left(\frac 1{j^{\frac {p}{n+p}}}\right)^{n+p}=(2\pi)^n \] and a contradiction is obtained. \hfill{$\Box$} \end{example} Finally we present an example that shows that it is impossible to have an estimate of the type \[ e_p(u)^{\frac {1}{n+p}}\leq C \|u\|_{L^{\infty}}\, . \] \begin{example}\label{ex3} Similarly as before we consider the following functions defined on the unit ball $\mathbb{B}$ in $\mbox{$\mathbb{C}$}^n$ \[ u_j(z)=j\max\left(\log |z|,-\frac 1j\right)\, . \] Then we have that $\|u_j\|_{L^{\infty}}=-u_j(0)=1$ and at the same time \[ e_p(u_j)=(2\pi)^nj^{n+p}\left(\frac 1j\right)^p=(2\pi)^nj^n\to \infty \] and a contradiction is obtained. \hfill{$\Box$} \end{example} Next in Corollary~\ref{t2_cor} we prove the corresponding Sobolev estimate~(\ref{lq estimate}) for functions in $\mathcal E_{\chi}$. For the definition of $\mbox{$\mathcal{E}_{\chi}$}$ see Definition~\ref{def_Echi} on page~\pageref{def_Echi}. \begin{corollary}\label{t2_cor} Let $\chi:(-\infty,0]\to (-\infty,0]$ be a continuous and nondecreasing function, let $\Omega$ be a bounded hyperconvex domain in $\mathbb C^n$, $n\geq 2$, and let $u\in \mbox{$\mathcal{E}_{\chi}$}$. Then for all $q>0$ there exists a constant $G(n,\Omega)\geq 0$ depending only on $n$ and $\Omega$ such that \begin{multline*} \|u\|_{L^q}\leq \\ G(n,\Omega)^{\frac 1q}\Gamma(q+1)^{\frac 1q}\left(\frac {e_{\chi}(u)}{(2\epsilon n)^n}\right)^{\frac 1n}\left(\inf_{s>0}s^{-q}\exp\left(-\left(\frac {e_{\chi}(u)}{(2\epsilon n)^n}\right)^{-\frac 1n}s\chi^{-1}(-s^n)\right)\right)^{\frac 1q}\, . \end{multline*} \end{corollary} \begin{proof} This is a straight forward modification of the proof of Theorem~\ref{lq estimate}. \end{proof} We shall end this section with a remark about compact K\"{a}hler manifolds. This was first proved in~\cite{BB1} for the case $p=1$. The notation and background about the K\"ahler case are stated before the remark on page~\pageref{kahler}. \begin{remark}\label{kahler2} Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ with a K\"ahler form $\omega$ such that $\int_{X}\omega^n=1$. Let $u\in \mbox{$\mathcal{E}_{p}$}$, $p>0$, and $k>0$. From (\ref{kahler 2}) we know that \[ \log \left (\int_Xe^{-ku}\omega^n\right)\leq \frac {ak^{n+1}}{n!}e_p(u)^{\frac 1p}+b\, , \] for some constants $a$ and $b$. By repeating the argument from the proof of Theorem~\ref{lq estimate} one can prove that there exists a constant $c$ depending only on $p, X$, and not on $q$, such that \[ \|u\|_{L^q}\leq cq^{\frac {n}{n+p}}e_p(u)^{\frac {1}{n+p}}\, . \] \end{remark} \section{On the Sobolev constant for the unit ball $\mathbb{B}$}\label{sec_optimal} In this section let $C(p,q,n,\mathbb{B})$ be the infimum of all admissible constants in the Sobolev type inequality given in~(\ref{lq theorem}). Our aim here is to show that \[ C(1,1,n,\mathbb{B})=\frac {\pi^{\frac {n^2}{n+1}}}{4^{\frac {n}{n+1}}n!(n+1)^{\frac {n}{n+1}}}\, . \] We shall do it in two part as follows. \begin{enumerate} \item In Example~\ref{const1}, we derive that for $q\leq n+1$ \[ C(1,q,n,\mathbb{B})\leq \frac {\pi^{\frac {n(1+n-q)}{q(n+1)}}[1\cdots(1+\lceil q-1 \rceil)]^\frac {1}{1+\lceil q-1 \rceil}}{4^{\frac {n}{n+1}}(n!)^{\frac 1q}(n+1)^{\frac {n-\lceil q-1 \rceil}{(n+1)(1+\lceil q-1 \rceil)}}}\, , \] where $\lceil \,\cdot\, \rceil$ is the ceiling function. \item In Example~\ref{const2} we prove that \[ C(p,1,n,\mathbb{B})\geq\frac {\pi^{\frac {n(n+p-1)}{n+p}}p^{\frac {p}{n+p}}}{4^{\frac n{n+p}}n!(n+p)(n\mathbf {B}(p+1,n))^{\frac {1}{n+p}}}\, , \] where $\mathbf B$ is the beta function. We shall actually obtain a bit more general result in this example. \end{enumerate} \bigskip \begin{example}\label{const1} Let $\Omega=\mathbb{B}$ be the unit ball in $\mathbb C^n$, and on $\mathbb{B}$ define \[ \varphi_0=\frac {1}{4\root n \of {n!}}(|z|^2-1)\, . \] Then $\varphi_0\in\mbox{$\mathcal{E}_{0}$}$, $(dd^c\varphi_0)^n=d\lambda_{2n}$, and \[ e_p(\varphi_0)=\frac {n\pi^n}{4^p(n!)^{1+\frac pn}}\mathbf B(p+1,n)\, , \] where $\mathbf B$ is the classical beta Euler function. Recall that if $q\leq n+p$, then the ceiling function evaluated at $q-p$ is defined by \[ \lceil q-p \rceil =\min\{k\in \mathbb N: k\geq q-p\}\, . \] Once again thanks to Lemma~\ref{est} and Theorem~\ref{thm_holder} we get that \begin{multline}\label{ineq2} \|u\|_{L^q}\leq \lambda_{2n}(\mathbb{B})^{\frac 1q-\frac {1}{\lceil q-p \rceil+p}}\|u\|_{L^{p+\lceil q-p \rceil}}\leq \\ \leq \lambda_{2n}(\mathbb{B})^{\frac 1q-\frac {1}{\lceil q-p \rceil+p}}\left((p+1)\cdots (p+\lceil q-p \rceil)\|\varphi_0\|^{\lceil q-p \rceil}_{L^{\infty}}\right. \\ \cdot\left.\int_{\Omega}(-u)^{p}(dd^cu)^{\lceil q-p \rceil}\wedge(dd^c\varphi_0)^{n-\lceil q-p \rceil}\right)^{\frac 1{p+\lceil q-p \rceil}}\\ \leq \lambda_{2n}(\mathbb{B})^{\frac 1q-\frac {1}{p+\lceil q-p \rceil}}\left((p+1)\cdots(p+\lceil q-p \rceil)\|\varphi_0\|^{\lceil q-p \rceil}_{L^{\infty}}d(p,n,\mathbb{B})\right.\\ \cdot \left. e_p(\varphi_0)^{\frac {n-\lceil q-p \rceil}{n+p}}e_p(u)^{\frac {p+\lceil q-p \rceil}{n+p}}\right)^{{\frac 1{p+\lceil q-p \rceil}}}=\\ =\frac{e_p(u)^{\frac {1}{n+p}}}{{4^{\frac {n}{n+p}}(n!)^{\frac 1q}}}\pi^{\frac {n(p+n-q)}{q(n+p)}}d(p,n,\mathbb{B})^{\frac {1}{p+\lceil q-p \rceil}}\Big(n \mathbf B(p+1,n)\Big)^{\frac {n-\lceil q-p \rceil}{(n+p)(p+\lceil q-p \rceil)}} \\ \cdot\left((p+1)\cdots(p+\lceil q-p \rceil)\right)^\frac {1}{p+\lceil q-p \rceil}\, . \end{multline} If $p=1$ we know that $d(1,n,\mathbb{B})=1$, and $n\mathbf B(2,n)=\frac {1}{n+1}$. Hence, \begin{equation}\label{ineq5} C(1,q,n,\mathbb{B})\leq \frac {\pi^{\frac {n(1+n-q)}{q(n+1)}}[(1+\lceil q-1 \rceil)!]^\frac {1}{1+\lceil q-1 \rceil}}{4^{\frac {n}{n+1}}(n!)^{\frac 1q}(n+1)^{\frac {n-\lceil q-1 \rceil}{(n+1)(1+\lceil q-1 \rceil)}}}. \end{equation} \hfill{$\Box$} \end{example} \begin{example}\label{const2} For $\alpha>0$, $k>0$, define on the unit ball $\mathbb{B}$ in $\mathbb C^n$ the following family of functions \[ u_{\alpha, k}(z)=k(|z|^{2\alpha}-1)\, . \] Then we have that \[ e_p(u_{\alpha})=\int_{B(0,1)}(-u_{\alpha})^p\ddcn{u_{\alpha}}=k^{n+p}n(4\pi)^n\alpha^n\, \mathbf {B}(p+1,n)\, , \] and \begin{multline*} \int_{\mathbb{B}}(-u_{\alpha,k}(z))^qd\lambda_{2n}=\frac {2\pi^nk^q}{(n-1)!}\int_{0}^1(1-t^{2\alpha})^qt^{2n-1}dt\\ =\frac {\pi^nk^q}{(n-1)!\alpha}\int_0^1(1-s)^qs^{\frac {n}{\alpha}-1}dr=\frac {\pi^nk^q}{(n-1)!\alpha}\mathbf B\left(q+1,\frac {n}{\alpha}\right). \end{multline*} Hence, \[ C(p,q,n,\mathbb{B})\geq\frac {\|u_{\alpha,k}\|_{L^q}}{e_p(u_{\alpha,k})^{\frac {1}{n+p}}}=\frac {\pi^{\frac {n(n+p-q)}{q(n+p)}}n^{\frac {n+p-q}{q(n+p)}}}{4^{\frac n{n+p}}(n!)^{\frac 1q}\mathbf B(p+1,n)^{\frac {1}{n+p}}}\frac {\mathbf B\left(q+1,\frac {n}{\alpha}\right)^{\frac 1q}}{\alpha^{\frac {1}{q}+\frac {n}{n+p}}}\, . \] Now set $\beta=\frac {n}{\alpha}$, and $s=\frac {n}{n+p}$. With these notation we get that \[ \frac {\mathbf B(q+1,\frac {n}{\alpha})^{\frac 1q}}{\alpha^{\frac {1}{q}+\frac {n}{n+p}}}=n^{-\frac {1}{q}-\frac {n}{n+p}}\left(\mathbf B(q+1,\beta)\beta^{1+q\frac {n}{n+p}}\right)^{\frac 1q}=n^{-\frac {1}{q}-\frac {n}{n+p}}f(\beta)^{\frac 1q}\, , \] where \[ f(\beta)=\mathbf B(q+1,\beta)\beta^{1+qs}\, . \] Next we want to find $\sup_{(0,\infty)}f(\beta)$. First we see that \[ \lim_{\beta\to 0}f(\beta)=\lim_{\beta\to \infty}f(\beta)=0\, , \] and that \[ f'(\beta)=\mathbf B(q+1,\beta)\beta^{sq}\left(\beta(\psi(\beta)-\psi(\beta+q+1))+1+sq\right)\, , \] where $\psi(x)$ is the classical digamma function. In the case when $q\in \mathbb N$, it holds that \[ \psi(\beta)-\psi(\beta+q+1)=-\sum_{j=0}^q\frac {1}{\beta+q-j}\, . \] Therefore for $q\in \mathbb N$ we have that $f'(\beta)=0$ if, and only if, \begin{equation}\label{sol} \sum_{j=0}^q\frac {\beta}{\beta+q-j}=1+qs\, . \end{equation} This implies that in the case $q=1$ the equation (\ref{sol}) have a solution given by \[ \beta_0=\frac {s}{1-s}=\frac np\, . \] Using the standard equalities \[ \mathbf B(x+1,y)=\frac {x}{x+y}\mathbf B(x,y)\, , \quad \text{ and } \quad \mathbf B(1,y)=\frac 1y\, , \] we derive that \[ \sup f(\beta)=\mathbf B(q+1,\beta_0)\beta_0^{1+s}=\frac {\beta_0^{s}}{1+\beta_0}=\frac {p^{\frac {p}{n+p}}n^{\frac {n}{n+p}}}{n+p}\, . \] Hence, \begin{equation}\label{ineq6} C(p,1,n,\mathbb{B})\geq\frac {\pi^{\frac {n(n+p-1)}{n+p}}p^{\frac {p}{n+p}}}{4^{\frac n{n+p}}n!(n+p)(n\mathbf B(p+1,n))^{\frac {1}{n+p}}}. \end{equation} In the case when $q\in \mathbb N$, $q\geq 2$, then we have that \[ \beta_0=\frac {q+1}{2}\frac {s}{1-s}=\frac {(q+1)n}{2p} \] is a good approximation of a solution to the equation (\ref{sol}), and therefore \[ \sup f(\beta)\geq \mathbf B\left(q+1,\frac {(q+1)n}{2p}\right)\left(\frac {(q+1)n}{2p}\right)^{1+\frac {n}{n+p}}\, . \] Thus, \begin{equation}\label{ineq7} C(p,q,n,\mathbb{B})\geq\frac {\pi^{\frac {n(n+p-q)}{q(n+p)}}n^{\frac {2n+p-nq}{q(n+p)}}(q+1)^{\frac {2n+p}{q(n+p)}}}{4^{\frac n{n+p}}(2p)^{\frac {2n+p}{q(n+p)}}(n!)^{\frac 1q}(nB(p+1,n))^{\frac {1}{n+p}}}\mathbf B\left(q+1,\frac {(q+1)n}{2p}\right)^{\frac 1q}\, . \end{equation} For $q\in\mbox{$\mathbb{R}$}$, $q\geq 2$, one can insert the floor function evaluated at $q$, $\lfloor q\rfloor$, in~(\ref{ineq7}). \hfill{$\Box$} \end{example}

proofpile-arXiv_069-10306

\section{Introduction} Theoretical approaches based on molecular quantum mechanics\cite{grimme_calculation_2004,Helgaker2012,correlation-szalay-cr2011-112-108,Lyakh:2012cn,Casida:2012gy} have grown into increasingly important tools to help experimentalists understand species in their electronically excited states in the gas-phase\cite{GIBSON:2006hd} or in complex environments\cite{senn_qm/mm_2009,env-gomes-arpcspc2012-108-222,Gordon:2012dk}, and with that address speciation (oxidation states of specific centers, structures)\cite{core-bagus-ssr2013-68-273,Geckeis:2013jb,Wilson:2016hb,Seidel:2016iga,Purgel:2011ds,Sures:2017hd} as well as the underlying factors driving photochemical processes (reactivity, photodissociation etc.)\cite{Liu:2009hf,SaizLopez:2012is,Xing:2014gp,Starik:2015gz,Hochlaf:2017cw}. Computational models are particularly important for species containing (a) first-row transition metals, since then one often encounters dense electronic spectra due to many low-lying quasi-degenerate states arising from the partially-filled $d$ shells which requires the treatment of both dynamical and non-dynamical electron correlation\cite{Shavitt2009}; and (b) heavy elements (e.g.\ those with atomic number Z = 31 or higher) for which the manifestations of relativistic effects\cite{Pyykko1988,saue:hamprimer,pyykko:2012bt,pyykko:2012kh,Autschbach2012,dyall-faegri-book-2007,reiher_book} significantly alter the species' electronic structure and, by extension, their excited states and other molecular properties\cite{saue:molprop,Saue2005,nrsbook}. However, as electron correlation and relativistic effects are non-additive, in order to achieve a balanced description of the electronic states of heavy element species both have to be treated on the same footing. Among the available relativistic (multireference) methods\cite{Timo_overview}, those based on the coupled-cluster ansatz\cite{bartlett:reviewcc,Lyakh:2012cn} and in particular the Fock-space coupled-cluster (FS-CC)\cite{correlation-Visscher-JCP2001-115-9720} method and its intermediate Hamiltonian (IHFS-CC) formulations\cite{correlation-Landau-JCP2000-113-9905,correlation-Landau-JCP2001-115-6862,Landau:2004dd,Eliav:2012gp}, have shown to be among the most reliable and cost-effective ones, and allowed the treatment of systems with complex, open-shell ground-states~\cite{actinide-infante-jcp2006-125-074301,actinide-Infante-JCP2007-127-124308,actinide-Gomes-PCCP2008-10-5353,ruiprez_ab_2009,ATO-Rota-JCP2011-135-114106,PereiraGomes:2014iy,Denis:2015ez,Parmar:2014kn,Figgen:2008hd,Weigand:2009dq,Nikoobakht:2015ko} as well as simpler, closed-shell ones\cite{real09,gomes10,pawel1,Tecmer:2012fc,actinide-Gomes-PCCP2013-15-15153,Tecmer:2014fs}. The appeal of (IH)FS-CC resides in the fact that they show a similar scaling to the of single-reference approaches (e.g.\ O(N$^6$) for (IH)FS-CC with coupled cluster singles doubles (CCSD)-based wavefunctions, can be implemented in a straightforward manner starting from a single-reference coupled-cluster code\cite{correlation-Visscher-JCP2001-115-9720} and yield several excited states simultaneously. Furthermore, different oxidation states belonging to different sectors of Fock space\cite{Haque:1984dk} can be obtained from a single calculation, a useful feature for actinide chemistry, where often spectra of the same species in different oxidation states are to be analyzed. There are, however, some important drawbacks in the requirement to define suitable model spaces\cite{Lindgren:1974kc,Lindgren:1987in} and the problem of intruder states\cite{Kaldor:1988wp,Kaldor:1991cd} which may prevent convergence for one of more sectors. The latter issue can be alleviated but not completely avoided by the use of the intermediate Hamiltonian formalism\cite{Malrieu:1985ce}, having as consequence that the IHFS-CC variants are the only ones that have been widely used in molecular applications employing relativistic Hamiltonians. For the transition energies of closed-shell ground states and for determining the wavefunctions and transition energies of open-shell cases (e.g.\ doublets) one often prefers a simpler ``black-box'' type method in which the definition of model spaces is not necessary and in which convergence problems due to intruder states are avoided. In those situations, the equation-of-motion coupled-cluster (EOM-CC) approach\cite{Bartlett:2011ho} is an excellent alternative to IHFS-CC: the FS-CC and EOM-CC methods are formally equivalent for single electron attachment and detachment states starting from a closed-shell reference\cite{bartlett:reviewcc,Meissner:2010fq}), though EOM-CC is a computationally more robust approach as it replaces the iterative solution of the coupled cluster equations by a diagonalization. In what follows we shall consider CCSD wavefunctions exclusively, and thus employ the shorthand CC instead of CCSD for brevity. This robusteness is also found for the (1 hole, 1 particle) sector that corresponds to singly excited states; there, however, differences in energies arising from the use of different parametrizations for the wavefunctions---linear for EOM-CC and nonlinear for IHFS-CC, with the latter having the virtue of also ensuring valence extensivity (i.e.\ with respect to active holes and particles used to define a model space)~\cite{Mukhopadhyay:1991fp,Lindgren:1987in}---become apparent and have been discussed for systems containing light\cite{Musial:2008da,fscc-vs-lrcc:musial:1,fscc-vs-lrcc:musial:2,fscc-vs-lrcc:musial:3} as well as heavy\cite{real09} elements. The appealing features of EOM-CC have made it an extremely popular method for light element systems, and its popularity is growing for heavier species as attested by the number of recent reports in the literature of EOM-CC implementations that take into account relativistic effects. Though some of the latter are based on solving the four-component (4C) Dirac equation for atomic and molecular systems\cite{Pathak:2016ck,Pathak:2015he,Pathak:2014gv,Pathak:2014dm,Pathak:2016dk} and therefore account rigorously for scalar relativistic (SR) effects and spin-orbit coupling (SOC), for reasons of computational efficiency most of them \cite{Klein:2008hx,Yang:2012gd,Wang:2015jw,Epifanovsky:2015hsa,Cao:2016fx,Cao:2017id,Zhang:2017jj,Akinaga:2017hv,Wang:2016hx} have been devised in a more approximate framework where SOC is treated to within different degrees of approximation e.g.\ starting from the spin-free exact two-component (sfX2C) Hamiltonian and including SOC via atomic mean-field (AMF) integrals\cite{so-Hes-CPL1996-251-365-371} or perturbatively. While in the aforementioned works and elsewhere in the literature\cite{gomes10,real09,ATO-Rota-JCP2011-135-114106,pawel1} one can see that approximate treatments of SOC such as in the AMF approximation can yield rather accurate excitation energies even for heavier species up to and including iodine, the situation is not as clear cut for heavier elements\cite{Zhang:2017jj}, and therefore for $5d, 5f, 6p$ elements and beyond it may be preferable to rely on approaches based on 4C Hamiltonians (that is, the Dirac--Coulomb (DC), Dirac--Coulomb--Gaunt (DCG) or Dirac--Coulomb--Breit (DCB) Hamiltonians), or on X2C approaches but using a molecular mean-field\cite{Sikkema2009} (MMF) approach, which have been shown to yield results largely indistinguishable from their 4C counterparts\cite{Tecmer:2014fs}. Our primary goal in this work is to present the implementation in the \textsc{Dirac} code\cite{DIRAC17} of the EOM-CC approach, for obtaining the energies for electron attachment (EOM-IP), detachment (EOM-EA) and singly excited states (EOM-EE) based on 4C and accurate 2C Hamiltonians (X2C-MMF), though we note the implementation can be used with single reference wavefunctions obtained with any other Hamiltonian available in \textsc{Dirac}. Here we shall place particular emphasis on the discussion of the exploitation of double point group symmetry, in contrast to other implementations which do not exploit or report the use of symmetry. Furthermore, we shall be able to perform for the first time a thorough comparison of the performance of the three EOM-CC approaches in comparison to IHFS-CC ones in the appropriate sectors and with equivalent relativistic Hamiltonians and basis sets. We recall that similar comparisons have been made by Musial and Bartlett for light-element systems using non-relativistic Hamiltonians~\cite{Musial:2008da,fscc-vs-lrcc:musial:1,fscc-vs-lrcc:musial:2,fscc-vs-lrcc:musial:3}. The calculation of transition moments and of excited state expectation values for the three EOM-CC variants considered here will be addressed in a subsequent publication. We demonstrate the use of our implementation in the study of different halogenated species: the halogen monoxides (XO, X = Cl, Br, I, At, Ts), the triiodide species (I$_3^-$) and the diiodo- (CH$_2$I$_2$) and iodobromo-methane (CH$_2$IBr) species. Our focus on halogenated species stems from the fact that these (and in particular iodine-containing ones) are of great importance to photochemical processes in the atmosphere such as ozone depletion and aerosol formation in coastal areas\cite{SaizLopez:2012is,SaizLopez:2016fx,Burkholder:2015js,SaizLopez:2014fr,GomezMartin:2013fa} -- keeping in mind aerosols are an important vector of dispersion of radioactive species in case of nuclear accidents\cite{Mehboob:2016kc,Funke:2012cc} -- and have been extensively studied theoretically and experimentally in the gas-phase. Going beyond iodine we note that species containing astatine have received considerable attention in recent years both theoretically and experimentally\cite{ato-ayed-jpc2013-117-10589,ato-ayed-jpc2013-117-5206,ato-champion-ica2009-362-2654,ato-champion-jpc2010-114-576,ato-champion-jpc2013-117-1983,PereiraGomes:2014iy,Sergentu:2016hn} due to their potential as radiotherapeutic agents so that a black-box approach such as EOM-CC may become a valuable tool to further elucidate their chemistry. Finally, by completing the monoxide series with also its heaviest member, TsO, one can investigate the growing importance of SOC on the description of the ground and excited-state wavefunctions as the charge on the halogen nucleus increases down the series. The paper is organized as follows: in the next section we briefly review the theoretical underpinnings of EOM-CC theory and discuss implementation details. This is followed by sections outlining the computational details of the EOM-CC and IHFS-CC calculations, the presentation and discussion of our results, conclusions and perspectives. In Appendix \ref{Symmetry} we provide further information on the use of double group symmetry in tensor contractions. Finally, working equations for the determination of right and left eigenvalues and eigenvectors are given in Appendix \ref{work_eqs}. Results not shown in the manuscript are available as Supplementary Information at the publisher's website and via the zenodo repository~\cite{paper:dataset}. \section{EOM-CC Theory: basic formulation} \label{sec:theory} We start from the coupled-cluster ansatz \begin{equation} |CC\rangle=\exp(\hat{T})|\Phi_0\rangle;\quad\hat{T}=\sum_{l=1}t_{l}\hat{\tau}_{l}, \end{equation} where $\Phi_0$ is the reference (Hartree--Fock) determinant and the operator $\hat{T}$ in the present work is restricted to single and double excitations \begin{equation} \hat{T}=\hat{T}_1+\hat{T}_2;\quad\hat{T}_1=\sum_{ia}t_i^aa_a^{\dagger}a_i;\quad \hat{T}_2=\frac{1}{4}\sum_{ijab}t_{ij}^{ab}a_a^{\dagger}a_b^{\dagger}a_ja_i, \end{equation} thus defining the coupled-cluster singles-and-doubles (CCSD) model. Here and in the following indices $\left\{i,j,\ldots,n,o\right\}$, $\left\{a,b,\ldots,f,g\right\}$ and $\left\{p,q,r,s\right\}$ refer to occupied (hole), virtual (particle) and general orbitals, respectively. The energy and the cluster amplitudes are found from the equations \begin{align} \langle\Phi_0|\hat{\bar{H}}|\Phi_0\rangle=&\ E\\ \langle\Phi_l|\hat{\bar{H}}|\Phi_0\rangle=&\ 0;\quad |\Phi_{l}\rangle=\hat{\tau}_{l}|\Phi_{0}\rangle,\label{eq:ampeq} \end{align} conveniently given in terms of the similarity-transformed Hamiltonian \begin{equation} \hat{\bar{H}}=\exp(-\hat{T}) \hat{H}\exp(\hat{T}). \end{equation} The equation-of-motion couped-cluster (EOM-CC) method is a robust theory for the calculation of multiple excited states on an equal footing, obtained by diagonalization of the similarity-transformed Hamiltonian $\hat{\bar{H}}$ within a selected excitation manifold. The similarity-transformed Hamiltonian is non-Hermitian, so right-handed ($R$) and left-handed ($L$) eigenvectors, determined by the solution of \begin{equation} \label{Eq:right} \hat{\bar{H}} |R_{\mu}\rangle = E_{\mu} |R_{\mu} \rangle \end{equation} \begin{equation} \langle L_{\mu}| \hat{\bar{H}} = E_{\mu} \langle L_{\mu}|, \label{Eq:lambda} \end{equation} for a given excited state ${\mu}$ with energy $E_{\mu}$, are therefore not simple adjoints of each other but obey the biorthogonality condition \begin{equation} \langle L_{\mu}|R_{\nu}\rangle = \delta_{\mu\nu}. \label{Eq:biorthogonality} \end{equation} One should note that the resolution of equations \ref{Eq:right} and \ref{Eq:lambda} closely resembles a CI-type diagonalization, where the matrix representation of $\hat{H}$ has been replaced by that of $\hat{\bar{H}}$, and the right- and left-hand wavefunctions are parametrized respectively as \begin{equation} |\Psi_{\mu}\rangle = \exp(\hat{T})\hat{R}_{\mu}|\Phi_0 \rangle \end{equation} and \begin{equation} \langle \bar {\Psi}_{\mu} | = \langle \Phi_0 | \hat{L}_{\mu}\exp(-\hat{T}), \end{equation} via $\hat{R}$ or $\hat{L}$ operators (see below), thus defining the excited states on the basis of the coupled-cluster wave function $|CC\rangle$ for the reference state. To simplify notation we will in the following no longer explicitly mention the excited state label ${\mu}$, but one should keep in mind that $\hat{R}$ or $\hat{L}$ may target one or more excited states. Different choices for the $\hat{R}$ and $\hat{L}$ operators define different EOM-CC models. In the present work we have used three most popular choices of $\hat{R}$ (of excitation kind) and $\hat{L}$ (of de-excitation kind), defining the three lowest Fock space sectors: \begin{itemize} \item excited states (1h-1p) : \begin{equation} \hat{R}^{EE} = r_0 + \sum_{ia} r^a_i \{ a^{\dag}_a a_i^{\phantom{\dag}} \} + \sum_{i>j,a>b} r^{ab}_{ij} \{ a^{\dag}_a a^{\dag}_b a_j^{\phantom{\dag}} a_i^{\phantom{\dag}} \} \end{equation} \begin{equation} \hat{L}^{EE} = l_0 + \sum_{ia} l_a^i \{ a^{\dag}_i a_a^{\phantom{\dag}} \} + \sum_{i>j,a>b} l_{ab}^{ij} \{ a^{\dag}_i a^{\dag}_j a_b^{\phantom{\dag}} a_a^{\phantom{\dag}} \} \end{equation} \item ionized states (1h) : \begin{equation} \hat{R}^{IP} = \sum_i r_i \{a_i^{\phantom{\dag}}\} + \sum_{i>j,a} r_{ij}^{a} \{a^{\dag}_a a_j^{\phantom{\dag}} a_i^{\phantom{\dag}}\} \end{equation} \begin{equation} \hat{L}^{IP} = \sum_i l^i \{a^{\dag}_i\} + \sum_{i>j,a} l^{ij}_{a} \{a^{\dag}_j a^{\dag}_i a_a\} \end{equation} \item electron-attached states (1p): \begin{equation} \hat{R}^{EA} = \sum_a r^a \{a^{\dag}_a \} + \sum_{a>b,i} r^{ab}_{i} \{a^{\dag}_a a^{\dag}_b a_i\} \end{equation} \begin{equation} \hat{L}^{EA} = \sum_a l_a \{a_a \} + \sum_{a>b,i} l_{ab}^{i} \{a^{\dag}_i a_b a_a\} , \end{equation} \end{itemize} where curly brackets refer to normal ordering with respect to the Fermi vacuum defined by the reference $\Phi_0$, and the sets $\{\mathbf{r}\}, \{\mathbf{l}\}$ to the amplitudes of the corresponding operators. We have here truncated our $\hat{R}$ and $\hat{L}$ operators at the singles-doubles level since the same truncation is used for the $\hat{T}$ operators. The equations \ref{Eq:right} and \ref{Eq:lambda}, together with the equation \ref{Eq:biorthogonality} expressing the biorthogonality of the left- and right-hand eigenvectors define the EOM-CC models under consideration. \section{Implementation details} \label{sec:implementation} The present implementation has been carried out within a development version of the DIRAC quantum chemistry package\cite{DIRAC17}. As we in this paper focus only on the determination of transition energies, we can summarize the calculation in three steps: \begin{enumerate} \item Solve closed-shell ground state CCSD equations to obtain T$_1$ and T$_2$ amplitudes. \item Construct the one and two-body intermediates based on the T$_1$ and T$_2$ amplitudes necessary for the construction of $\hat{\bar{H}}$. \item Diagonalization of $\hat{\bar{H}}$ in the full singles-doubles excitation space to obtain excitation energies and eigenvectors. An iterative matrix-free method is employed to avoid the explicit construction of $\bar{H}$, due to its generally very large size. \end{enumerate} The first step is carried out within a Kramers-unrestricted formalism\cite{CCSD-Visscher-JCP1996-105-8769-8776}, and the parallelization of the code is such that the most numerous integrals involving three or four virtual indexes are distributed over different compute nodes\cite{Pernpointner:2003fm}. As we shall see below, this scheme can be generalized to the parallelization of the EOM-EA and EOM-EE models. The intermediates in the second step are those originally defined by Bartlett, Gauss, Stanton and coworkers\cite{Gauss1991,Gauss1991b,Stanton:1993be,Gauss1995,Gwaltney:1996jw} in a spin-orbital basis, and for which the construction in a four-component formalism has been discussed in detail in our previous work\cite{Shee_JCP2016} on the calculation of ground-state properties. We have extensively made use of the previously developed tensor contraction routines compatible with double group symmetry. What follows discusses the work necessary for the third step, which required two main components: the first is a significant extension in scope of the aforementioned contraction routines, so that they were also able to carry out tensor contractions with non-totally symmetric and odd-ranked tensors while still exploiting double point group symmetry. These extensions, which are not trivial due to the handling of complex quantities, will be discussed in detail in section \autoref{subsec:extension_tc} below. The second component is a matrix diagonalizer that can handle both real and complex general matrices. It will be discussed in more detail in section \ref{sec:davidson}. \subsection{Extension to odd-ranked and non-totally symmetric tensor contractions} \label{subsec:extension_tc} In our previous work\cite{Shee_JCP2016}, we have developed general-purpose tensor contraction routines to handle tensor contractions related to relativistic coupled-cluster theory. These routines handle relativistic symmetry as expressed by double point groups, where the boson irreps of single point groups are complemented by fermion irreps, spanned by functions with half-integer spin. The routines, restricted to the highest Abelian subgroup of the full symmetry group, exploit optimal blocking and sparsity of a tensor for each contractions. They assume that the product of all tensor indices belongs to boson irreps, which is true for even-ranked tensors, but excludes odd-ranked tensors in which this product belongs to a fermion irrep. In the present work, (L)R-vectors for IP and EA are odd-ranked and hence span fermion irreps. In order to accommodate these tensors in our tensor contraction scheme we formally increase the rank by one by introducing, as a bookkeeping device, a label for each fermion irrep representing each a continuum orbital, so that electrons are considered to be ionized(attached) to(from) this orbital. This follows in spirit the well-known approach of adding a very diffuse gaussian basis function to an EOM-EE code to simulate the ionization to the continuum~\cite{stanton:1999continuumorbital}. The fundamental difference is that all actions are done at the contraction level only (similar to the EA-EOM-CCSD implementation of Nooijen and Bartlett~\cite{Nooijen_JCP95}) and do not require the definition of a basis function. Since the EOM-EE machinery (N$^6$ scaling) is not used the proper N$^5$ scaling of EOM-IP is obtained. The continuum orbital will always belong to the same fermion irrep as the orbital from where/to which ionization/electron attachment occurs. In this way, and due to our restriction to Abelian double groups at the correlated level, the (L)R-vectors become totally symmetric, and we block them according to the same scheme as used for even-ranked tensors. Since we only have one continuum orbital per irrep, the size of arrays is not increased relative to the original odd-ranked arrays. In the contraction step we ensure that continuum orbitals can only be contracted with themselves. As an illustration, we consider the following contribution to the R-sigma vector equation of EOM-IP (cf. Eq. (B2) of the Appendix) \begin{equation} \label{eq:IP-example} (\sigma)_{ij}^{ a} \leftarrow - \sum_m W^{ma}_{ij} * r_m. \end{equation} By introducing the continuum orbital this term is rewritten as \begin{equation} (\sigma)_{o1,o2}^{c1,p2} \leftarrow - \sum_{o3} W^{o3,p2}_{o1,o2} * r_{o3}^{c1}, \end{equation} where $c1$ represents the continuum orbital, and indices $o$ and $p$ refer to holes and particles, respectively. The corresponding subroutine call is: \begin {verbatim} call contraction_424((/"o3","p2","o1","o2"/),& & (/"c1","o3"/), (/"c1","p2","o1","o2"/),& & sigma2,-1.0d0,1.0d0,nrep, & & LeftTensor= \end{verbatim} where \texttt{r1} contains the trial vector coefficients. As explained above, \texttt{c1} and \texttt{o3} will belong to the same fermion irrep, thereby \texttt{r1} is blocked with respect to the symmetry of \texttt{o3}. The $\sigma$-vector is blocked according to the symmetry of \texttt{c1} as well. In this manner the operation count of IP- and EA- type contractions is reduced significantly, especially for linear molecules and other molecules with high symmetry. The second generalization of the original implementation of tensor contractions is to allow contractions for which the product of tensor indices is not totally symmetric. Such contractions occur for EOM-EE target states that belong to non-totally symmetric irreps. We have illustrated this extension schematically in Appendix \ref{Symmetry}. \subsection{Davidson diagonalization for non-Hermitian matrices} \label{sec:davidson} Diagonalization of $\hat{\bar{H}}$ in the full singles-doubles excitation space to obtain excitation energies and eigenvectors is an expensive task, since the matrix dimension is in principle huge, and we thus employ an iterative procedure of the Davidson type\cite{Davidson:1975db}. Since $\hat{\bar{H}}$ is non-Hermitian we have in this work implemented a generalized eigensolver following the algorithm of Hirao and Nakatsuji\cite{Hirao:1982fc}, which is capable of obtaining multiple roots at a time as well as handling $\bar{H}$ which can be either real or complex depending on the double point group in use -- though operating with double precision variables instead of complex ones\cite{Morgan:1992iu}. We solve the left and right eigenproblems separately, using a modified Gram-Schmidt procedure~\cite{matrix-computations:book} for orthonormalizing the trial vectors during the iterative procedure (cf. Refs.~\citenum{DAVIDSHERRILL1999143,vallet:2000epciso}). This approach is often more cost-effective for an EOM-CC implementation as excitation energies are usually the only quantity sought, requiring only the solution for one side. If both left and right calculated, the left eigenvectors are rescaled to satisfy the bi-orthogonality condition (\autoref{Eq:biorthogonality}). The first and costlier part of this algorithm is generally the formation of the left ($\boldsymbol{\sigma}^L)$ or right $(\boldsymbol{\sigma}^R)$ sigma vectors \begin{eqnarray} \boldsymbol{\sigma}^R &=& \bar{H}\mathbf{b} \label{right-sigma-vector} \\ \boldsymbol{\sigma}^L &=& \mathbf{b}^\dagger\bar{H} \label{left-sigma-vector} \end{eqnarray} where $\mathbf{b}(\mathbf{b}^\dagger)$ is the (complex conjugate) matrix of trial vectors, as it involves the contractions outlined in Appendix \ref{work_eqs}. Here we carefully avoid the possibility of generating three-body intermediates by suitably rearranging the order of contractions, which will again be reflected in the sigma vector expressions in Appendix \ref{work_eqs}. Another particularity of our implementation has to do with the parallelization of the most expensive and memory intensive step of the sigma vector construction. The terms which arise from the integrals involving four virtual orbitals are constructed with distributed-memory Message Passing Interface (MPI) parallelization. As seen in Appendix \ref{work_eqs} such terms appear in the EOM-EA and EOM-EE sigma vector equations in addition to the ground-state CCSD amplitude equations. We then synchronize final sigma vectors to the master and proceed in serial mode for the rest of the Davidson iteration steps. The choice of the initial trial vectors is of paramount importance for the final convergence of the method. We have adopted the following routes to choose our guess vectors: (a) We fully diagonalize the singles-singles block of the transformed Hamiltonian. Eigenvectors of that diagonalization are considered as guess vectors. (b) We approximate the singles-singles block of the transformed Hamiltonian by its diagonal elements, that is, ($\overline{F}_a^a $ - $\overline{F}_i^i$ - $W^{ia}_{ai}$), -$\overline{F}_i^i$ and $\overline{F}_a^a$ for EOM-EE, EOM-IP and EOM-EA, respectively, with intermediates $\overline{F} $ and $W$ defined in Appendix \ref{work_eqs}. The corresponding unit vectors are considered as guess vectors. Since they are selected according to energy and the number of roots requested, we refer to these as pivoted unit vectors. Further computational savings for IP and EA calculations can be achieved by using the fact that for real and complex double groups the states are doubly degenerate due to time-reversal symmetry and each span a different irrep. This means we only need consider one of the two degenerate Kramers pair as our guess vectors for each irrep, and thus may calculate only half of the total number of $\sigma$-vectors (for excitation energy calculations similar considerations are ungainly, since symmetry-adaptation requires constructing multideterminant reference states). However, for the quaternion double groups this scheme cannot be employed in a straightforward manner, and we must request twice as many roots as we want states, irrespective of the nature of the calculations. Finally, the implementation allows for the use of root following using the overlap between initial and generated trial vectors\cite{Butscher:1976ku,Zuev:2014fc} during the procedure, in the case one wishes to target states with dominant (1h1p), (1h0p) or (1p0h) character, which may turn out to be higher in energy than states with (2h2p), (2h1p) or (2p1h) character. \section{Computational Details} All coupled-cluster calculations were carried out with a developmental version of the \textsc{Dirac} electronic structure code\cite{DIRAC17} (revisions \texttt{e25ea49} and \texttt{7c8174a}), employing Dyall's basis sets\cite{basis-Dyall-TCA2002-108-335,basis-Dyall-TCA2003-109-284,basis-Dyall-TCA2012-131-3962,dyallbasis} of triple-zeta quality (dyall.av3z) for the halogens, and Dunning's aug-cc-pVTZ sets\cite{basis-Kendall-JCP1992-96-6796-6806} for oxygen, all of which are left uncontracted. In these calculations we employed the molecular mean-field\cite{Sikkema2009} approximations to the Dirac--Coulomb ($^2$DC$^M$) and Dirac--Coulomb--Gaunt ($^2$DCG$^M$) Hamiltonians -- where in the latter the Gaunt-type integrals are explicitly taken into account only during the SCF step -- along with the usual approximation of the energy contribution from $\left(SS|SS \right )$-type two-electron integrals by a point-charge model~\cite{relat-Visscher-TCA1997-98-68}. Apart from the EOM-CC method, we have employed the intermediate Hamiltonian Fock-Space (IHFS-CC) method\cite{correlation-Visscher-JCP2001-115-9720,Landau:2004dd}. Details of the main ($P_m$) and intermediate ($P_i$) model and complement ($Q$) spaces used will be given below for each system. To further simplify the notation, in what follows we abbreviate EOM-CC and IHFS-CC to EOM and IHFS respectively, adding whenever appropriate the qualifiers EE/IP/EA for the first and (1h1p)/(1h0p)/(0h1p) for the second to denote the Fock-space sector under consideration. We also note that in the cases of known doubly degenerate electronic states (e.g.\ the $\Omega=1/2_{(g/u)},\ldots$ for electron attachement/detachment or the $\Omega= \pm1_{(g/u)},\ldots$ for excitation energies in linear symmetry), in the EOM calculations only one has been explicitly calculated. Furthermore, in EOM calculations, unless otherwise noted, we have used pivoted unit vectors as initial trial vectors and new solution vectors were generated : a) using the root following procedure for EOM-IP; b) not using the root following procedure for EOM-EE/EA. \subsection{Halogen monoxides radicals (XO, X = Cl -- Ts)} The electronic states of halogen monoxide radicals have been obtained starting from the anions (XO$^-$) in order to provide a closed-shell reference determinant for electron detachment calculations for both IHFS(1h0p) and EOM-IP calculations. In all calculations $C_{{\infty}v}$ symmetry was used. All spinors with energies between -10.0 and 100.0$\,E_h$ have been correlated, which corresponds to considering, respectively: (a) 20 electrons and 206 virtuals for the systems containing Cl; (b) 32 electrons and 246 virtuals for the systems containing Br; (c) 32 electrons and 248 virtuals for the systems containing I; (d) 46 electrons and 340 virtual spinors for the systems containing At; and (e) 46 electrons and 306 virtuals for the systems containing Ts. In terms of the nature of the occupied atomic spinors correlated, the spaces above correspond to including the $2s2p$ oxygen and the $(n)sp (n-1)spd (n-2)f$ halogen atomic shells ($n$ denoting the valence shell; $f$ shells are obviously available only for At and Ts). In the IHFS(1h0p) case, the $P_m$ space for all species but TsO comprises the five highest occupied molecular spinors of the anion which arise from the valence ($p$-$p$) manifold ($\pi^{(2)}_{1/2} \pi_{3/2}^{(2)} \sigma_{1/2}^{(2)} \pi_{1/2}^{*(2)} \pi_{3/2}^{*(2)}$), thus placing the $\sigma_{1/2}^{*(0)}$ and all remaining virtuals in the $Q$ space, whereas the $P_i$ space included all other occupied spinors. For TsO, we encountered convergence problems due to intruder states with the aforementioned $P_m$ space, and had to move the lowest-lying $\pi_{1/2}^{(2)}$ into $P_i$. In the EOM-IP case, the number of roots requested was 3 and 2 for $\Omega=1/2,\ 3/2$, respectively, which allows us to obtain the ground and low-lying states, which correspond to those obtained with IHFS(1,0) for the model space above. Spectroscopic constants were obtained by constructing potential energy curves for each species and performing polynomial fits to energies calculated for X-O internuclear distances ranging from (a) 1.46~\AA~to 1.98~\AA~for ClO; (b) 1.58~\AA~to 2.14~\AA~for BrO; (c) 1.66~\AA~and 2.16~\AA, for IO; (d) 1.84 and 2.40~\AA, for AtO and (e) 1.86 and 2.44~\AA, for TsO, respectively, with spacings no smaller than 0.01~\AA~between points. For most of the calculated points, the Hartree-Fock self-consistent field (SCF) procedure converged to the correct state with the default start potential. For TsO with an elongated bond length (beyond 2.32~\AA) this procedure lead to a wrong SCF solution, and we needed to adjust the start potential to ensure occupation of the correct orbitals in the early stage of the SCF iterations. \subsection{Triiodide} We investigated the excitation energies, electron attachment and electron detachment for I$_3^-$ with EOM and IHFS starting from the closed-shell ground state of I$_3^-$ in all cases. The geometry used is $R = 2.93$~\AA, which was used previously to compare different electronic structure methods for excitation energies~\cite{gomes10}. All spinors with energies between -3.0 and 12.0$\,E_h$ have been correlated, which corresponds to considering 52 electrons and 332 virtual spinors. This choice is slightly different from that of Ref.~\citenum{gomes10}, since there we employed the augmented core-valence triple-zeta basis and with that included additional virtual spinors in the complement space $Q$. In all calculations the $D_{{\infty}h}$ point group was used. In the case of IHFS calculations we considered the same active spaces used in Ref.~\citenum{gomes10}: for the IHFS(1h0p) calculations, the $P_m$ space contained the 16 highest-lying occupied spinors ($4, 2, 6, 4$ for $\omega={1/2}_g, {3/2}_g, {1/2}_u, {3/2}_u$, respectively), with the remaining 6 occupied spinors ($4, 0, 2, 0$ for $\omega={1/2}_g, {3/2}_g, {1/2}_u, {3/2}_u$, respectively) being included in $P_i$. For the IHFS(0h1p) calculations, the $P_m$ space contained the 20 lowest-lying virtual spinors ($6, 2, 8, 4$ for $\omega={1/2}_g, {3/2}_g, {1/2}_u, {3/2}_u$, respectively), with the subsequent 24 virtual spinors ($6, 6, 4, 4, 2, 2$ for $\omega={1/2}_g, {3/2}_g, {5/2}_g, {1/2}_u, {3/2}_u, {5/2}_u$, respectively) making up the $P_i$ space. For the IHFS(1h1p) calculations the $P_m$ and $P_i$ spaces for both calculations are constructed as the direct product of the respective spaces from the IHFS(0h1p) and IHFS(1h0p). The number of roots requested in the EOM calculations was: $3, 1, 3, 2$ in $\Omega={1/2}_g, {3/2}_g, {1/2}_u, {3/2}_u$ symmetries for EOM-IP; $6, 4, 2, 6, 3, 1$ in $\Omega={1/2}_g, {3/2}_g, {5/2}_g, {1/2}_u, {3/2}_u, {5/2}_u$ symmetries for EOM-EA; and $10$ in each of the $\Omega={0}_g, 1_g, 2_g, 1_u, 2_u$ symmetries for EOM-EE. In the EOM-IP calculations the root following procedure was not used. \subsection{Dihalomethanes} We investigated the excitation energies, electron attachment and electron detachment for the CH$_2$IBr and CH$_2$I$_2$ systems with EOM-EE and IHFS, starting from the closed-shell ground state in both cases. All calculations were performed on a single structure obtained by a geometry optimization performed with the ADF code\cite{FonsecaGuerra1998,ADF2001,ADF2017authors}, using TZ2P basis sets and the scalar relativistic Zeroth-Order Regular Approximation (ZORA) Hamiltonian. The corresponding Cartesian coordinates and structural parameters can be found in the Supplementary Information. All calculations were performed in $C_{2v}$ (CH$_2$I$_2$) and $C_{s}$ (CH$_2$IBr) symmetries. As C$_{2v}$ is not an Abelian double group\cite{luuk:sym}, the C$_2$ subgroup was employed in the coupled cluster calculation and defines the symmetry labels for the states that were calculated. All spinors with energies between -3.0 and 6.0$\,E_h$ for CH$_2$I$_2$ and -4.0 and 6.0$\,E_h$ CH$_2$IBr have been correlated, which corresponds to considering, respectively: (a) 40 electrons and 364 virtual spinors for CH$_2$IBr; and (b) 40 electrons and 374 virtual spinors for CH$_2$I$_2$. In terms of the nature of the occupied atomic spinors correlated, the spaces above correspond to including the $1s$ of hydrogen, the $2s2p$ of carbon, and the $(n)sp (n-1)spd$ halogen atomic shells ($n$ denoting the valence shell). The number of roots requested for the EOM calculations is as follows: 7 and 6 of $^1E$ symmetry for EOM-IP and EOM-EA, respectively, for each of the species; and for EOM-EE, 12 of $A, B$ symmetries, respectively, for each of the species. For the IHFS(1h0p) calculations, the $P_m$ spaces for both species contained the 12 highest-lying spinors (6 in each of the $^1e$, $^2e$ representation), with the remaining 28 spinors (14 in each of the $^1e$, $^2e$ representations) being included in $P_i$. For the IHFS(0h1p) calculations, the $P_m$ space for CH$_2$I$_2$ contained the 26 (13 in each of the $^1e$, $^2e$ representations) lowest-lying spinors, with 30 additional spinors (15 in each of the $^1e$, $^2e$ representations) making up the $P_i$ space, whereas for CH$_2$IBr 20 and 30 spinors make up the $P_m$ and $P_i$ spaces, respectively, and, as was the case for the (1h0p) sector, these are evenly divided between the $^1e$ and $^2e$ representations. For the IHFS(1h1p) calculations on the CH$_2$I$_2$ and CH$_2$IBr, the $P_m$ and $P_i$ spaces for both calculations are constructed as the direct product of the respective spaces from the IHFS(0h1p) and IHFS(1h0p). Unfortunately, IHFS(1h1p) calculations with the corresponding model space did not converge, and attempts with larger model spaces were not practically feasible due to technical constraints (the MPI implementation did not fully support the 64-bit integers needed to address a larger memory space) which occur due to the increase in storage requirements caused by the use of complex algebra in C$_s$ symmetry. \section{Results} \label{sec:results} \subsection{Halogen monoxides} We begin the discussion by analyzing our results for the halogen monoxide radicals. In Table~\ref{tab:so-monoxide-radicals}, we present the spin-orbit splitting of the $^2\Pi$ ground states, \begin{equation} T_\text{so} = E(X ^2\Pi_{1/2}) - E(X ^2\Pi_{3/2}), \end{equation} calculated at the ground-state ($X ^2\Pi_{3/2}$) EOM-IP equilibrium structure. Comparing first the EOM-IP and IHFS(1h,0p) $T_\text{so}$ values in Table~\ref{tab:so-monoxide-radicals}, we see for both $^2$DCG$^M$ and $^2$DC$^M$ the expected close agreement between the two methods along the series: up to IO differences are of the order of 0.001 eV or better for both Hamiltonians, for AtO differences are slightly larger (about 0.01 eV and 0.005 eV for $^2$DC$^M$ and $^2$DCG$^M$, respectively), while for TsO discrepancies of around 0.023 eV and 0.025 eV for the $^2$DC$^M$ and $^2$DCG$^M$ Hamiltonians, respectively, are found. These differences between the two methods are due to use of the Intermediate Hamiltonian formalism: whereas EOM-IP and FS(1h0p) should yield exactly the same results for singly ionized states this does not hold for EOM-IP and IHFS(1h0p). The pronounced differences between EOM-IP and IHFS(1h0p) for TsO can either be due to missing contributions from higher sectors (in particular 2h1p) or the division into $P_m$ and $P_i$ spaces in IHFS calculations. The latter is a sensitive point for our IHFS calculations since states belonging to the $P_i$ space are not dressed and therefore treated in a CI-like way~\cite{Landau:2004dd}. To better understand the observed trends in spin-orbit splittings, it is illustrative to look at the composition of the electronic states of XO in terms of the molecular spinors of XO$^-$ at the EOM-IP equilibrium structures (listed in the Supplementary Information). In all cases, the $X ^2\Pi_{3/2}$ and $X ^2\Pi_{1/2}$ states are dominated by ionizations from the $\pi^*_{3/2}$ and $\pi^*_{1/2}$ orbitals of the anions. For the ClO molecule these $\pi^*$ orbitals has most weight on the less electronegative oxygen atom and their energies are only slightly split by spin--orbit coupling. As the electronegativity of the halogen atom decreases along the series, the bonding $\pi$ orbital overall becomes centered on the oxygen whereas the antibonding $\pi^*$ orbital moves to the halogen. For IO the spin--orbit splitting is considerable but one may still interpret the highest $\omega=1/2$ and $\omega=3/2$ orbitals as two $\pi^*$ orbitals, now with the dominant weight on iodine. This simple picture starts to break down for AtO in which the $\omega=1/2$ orbital also contain a significant $\sigma$ contribution and has an increased oxygen participation. For the TsO molecule, spin-orbit coupling is so strong that the notion of $\sigma$ and $\pi$ orbitals is better avoided. The lowest orbital in the p-orbital valence space is the Ts $7p_{1/2}$ orbital which is relatively compact and hardly participates in chemical bonding (cf. Ref.\citenum{Faegri_JCP2001}). The $\omega=3/2$ orbitals are also virtually non-bonding and centered either on the O or the Ts. The bonding is provided by the $\omega=1/2$ component of the Ts $7p_{3/2}$ orbital which combines with the O $2p_\sigma$. These changes induced by the increasing importance of spin-orbit coupling on the bonding orbitals can be visualized by means of plotting the spinor magnetization densities (Supplementary Information). For TsO, the qualitatively different orbital structure is also reflected in the composition of the two lowest states. While the $X ^2\Pi_{3/2}$ still shows contributions from configurations where also the lower energy $\omega=3/2$ is unoccupied, the $X ^2\Pi_{1/2}$ state is nearly completely dominated by a configuration in which the $\pi^*_{1/2}$ is singly occupied. For the EOM-IP calculations we see also a small contribution from a configuration in the 2h1p sector in which this ionization is accompanied by an excitation from the $\sigma_{1/2}$ to a high-lying $\sigma^*_{1/2}$ spinor. For the higher lying $\Omega = 1/2$ and $\Omega = 3/2$ states that arise by excitation from the bonding $\pi$ orbitals, we largely observe the same patterns with respect to their composition (see Supplementary Information) as discussed for the spin-orbit split ground state. Again there is one dominant singly ionized configuration for both EOM-IP and IHFS(1h0p) and a small 2h1p contribution for the former. The exception is again TsO, where for the third $\Omega = 1/2$ state of TsO we find the largest discrepancy in energy (1.63 eV) between EOM-IP and IHFS(1h0p) among all states considered. This is due to the fact that the lowest $\pi_{1/2}$ orbital had to be put in the $P_i$ space, which leads to a poor description of states that are dominated by ionization from this orbital. We note that while EOM-IP and IHFS(1h0p) perform in very nearly the same way for states dominated by singly ionizations (and belonging to $P_m$) irrespective of the Hamiltonian used, the choice of Hamiltonian does have important implications for the value of $T_\text{so}$: the $^2$DC$^M$ are larger than the $^2$DCG$^M$ ones, and this difference grows slowly along the series (0.003 eV for ClO, 0.0035 eV for BrO, 0.0041 eV for IO, 0.0047 eV for AtO), culminating in the largest difference (0.0216 eV) for TsO. However, since $T_\text{so}$ itself increases still faster as the halogen becomes heavier, in absolute terms the importance of the Gaunt interaction diminishes along the series. The effect of the Gaunt term corresponds to about 7\% of $T_\text{so}$ for ClO (and therefore must be taken into account), whereas for AtO it corresponds to less than half percent (and therefore can be safely ignored), only to become important again for TsO, for which its contribution being just short of 2\%. We see a rather good agreement between our results and experimental ones~\cite{IO-Gilles-JCP1992-96-8012} based on photodetachment measurements on the XO$^-$ species (and therefore similar to our computational approach), with the $^2$DCG$^M$ results being in general closer to experiment. Since our calculations do not take into account corrections for zero-point vibrations, basis set incompleteness and higher excitations, we are not in a position to make definitive quantitative statements on the performance of the different methods. The only theoretical study that has accounted for vibrational corrections is that of Peterson and coworkers~\cite{IO-Peterson-JPC2006-110-13877-13883}, though using single-reference CCSD(T) calculations. These, however, yield $T_\text{so}$ values which underestimate the experimental values, in particular for IO. Other theoretical calculations have corrected for the single-reference approach by using multireference CI~\cite{Zhou:2017ky} or EOM-IP but using different approximate Hamiltonians, that differ both in the treatment of scalar relativistic effects and SOC: Akinaga and Nakajima~\cite{Akinaga:2017hv} used a two-component method combining the third-order Douglas--Kroll--Hess Hamiltonian (DKH3) for scalar relativistic effects and screened nuclear SOC integrals, Epifanovsky and coworkers~\cite{Epifanovsky:2015hsa} used the one and two-electronic Breit--Pauli (BP) Hamiltonian with atomic mean-field SOC integrals, and no scalar relativistic effects, whereas Cheng and coworkers~\cite{Cheng:2018ji} have used spin-free exact two-component Hamiltonian (SFX2C-1e) for scalar relativity in conjunction with BP, atomic and molecular mean-field SOC integrals (the Hamiltonian used is the SO part of modified Dirac Hamiltonian by Dyall\cite{Dyall1994}) for the SO part. Even though in all of the above mentioned methods the EOM-CC framework have been used, they differ in terms of their treatment of SOC - Akinaga and Nakajima~\cite{Akinaga:2017hv} considers SOC from the start with their approximated Hamiltonian, Epifanovsky and coworkers~\cite{Epifanovsky:2015hsa} and Cheng and coworkers~\cite{Cheng:2018ji} both treat SOC as a perturbation to their choice of spin-free/scalar relativistic Hamiltonians. Afterwards an effective Hamiltonian is constructed with the SOC integrals at the EOM level. However, the latter two approaches differ from one another because Cheng and coworkers~\cite{Cheng:2018ji} treat amplitude relaxation due to SOC perturbation while Epifanovsky and coworkers~\cite{Epifanovsky:2015hsa} do not. As not all calculations use strictly the same basis sets (some use triple-zeta bases with core-valence correlating functions or quadruple-zeta bases, but in both without adding diffuse functions as done here), the differences we observe between the different EOM-IP results are not exclusively due to the different Hamiltonians used. With some precaution we, however, believe we can affirm that, first, all EOM-IP approaches show more or less the same performance for ClO and get quite close to experiment. Second, for BrO, apart from the X2C-based approach of Cheng and coworkers~\cite{Cheng:2018ji}, which shows results comparable to our $^2$DCG$^M$ ones, all others seem to underestimate the experimental $T_\text{so}$. Finally, for IO, at a first glance the BP-based approach of Cheng and coworkers~\cite{Cheng:2018ji} yields the closest results to experiment, but the good agreement between our results and their X2C-based ones indicates that this may be somewhat fortuitous and due to cancellation of errors in the Hamiltonian and the EOM-CC method. It is quite likely, though, that approximate treatments of spin-orbit interaction such as proposed by Cheng and coworkers~\cite{Cheng:2018ji} are sufficiently accurate to treat molecules containing up to iodine. As Cheng and coworkers~\cite{Cheng:2018ji} have not explicitly explored their approach for heavier species, we can only speculate as to their general applicability for species containing $6p$ elements and beyond, where strong second-order SOC effects are expected. We end the comparison of theoretical results for $T_\text{so}$ by noting that our results for IO and AtO show $^2$DC$^M$ faithfully reproduce the IHFS(1h0p) results of Gomes and coworkers~\cite{PereiraGomes:2014iy} using the 4-component DC Hamiltonian, in line with the findings of Tecmer and coworkers~\cite{Tecmer:2014fs} for small actinide species. We decided to extend this comparison here and show in Table~\ref{tab:so-monoxide-radicals} $T_\text{so}$ values for the DC Hamiltonian, along with a more detailed comparison in the Supplementary Information. We see differences in energy between the DC and $^2$DC$^M$ Hamiltonians of less than a tenth of a milli-electron volt. A closer inspection of the results in the Supplementary Information reveals that, in absolute terms, the differences between Hamiltonians are very systematic and of the order of milli-electron volts. Thus, the differences between Hamiltonians for relative energies (whether between the anion and the radical, or between the states of the radical) are in effect about two order of magnitude lower than the absolute ones, so that the molecular mean-field approach shows errors of less than a wavenumber for all excitation energies considered, as well as ionization energies. \begin{table}[htb] \caption{Spin-orbit splittings ($\mathrm{T_{so}}$, in eV) between the $\Omega=3/2$ and the $\Omega=1/2$ components of the $^2\Pi$ ground state for the halogen monoxides for the EOM-IP and IHFS(1h0p) approaches employing augmented triple zeta bases and the molecular mean-field Hamiltonians with ($^2$DCG$^M$) and without ($^2$DC$^M$) the Gaunt interaction, calculated at the EOM-IP equilibrium structures for the $\Omega=3/2$ ground-state. For comparison we present EOM-IP results with the DC Hamiltonian, experimental values~\cite{IO-Gilles-JCP1992-96-8012} and prior theoretical results: the SO-icMRCI of Zhou {\em et al.}~\cite{Zhou:2017ky}, the CCSD(T) with SOC corrections of Peterson {\em et al.}~\cite{IO-Peterson-JPC2006-110-13877-13883}, and the EOM-IP results of : (a) Akinaga {\em et al.}~\cite{Akinaga:2017hv}, using the DKH3 Hamiltonian and screened nucleus SOC; (b) Epifanovsky {\em et al.}~\cite{Epifanovsky:2015hsa}, using the Breit-Pauli Hamiltonian and AMF-SOC integrals; (c) Cheng {\em et al.}\cite{Cheng:2018ji}, using the Breit-Pauli Hamiltonian with T-relaxed SOC; and (d) Cheng {\em et al.}\cite{Cheng:2018ji}, using the one-component X2C Hamiltonian with T-relaxed SOC.} \begin{center} \begin{tabular}{l c c c c c} \hline \hline & ClO & BrO & IO & AtO & TsO \\ \hline EOM DC & 0.0422 & 0.1294 & 0.2776 & 0.7232 & 1.1953 \\ EOM $^2$DC$^M$ & 0.0423 & 0.1295 & 0.2777 & 0.7233 & 1.1954 \\ IHFS $^2$DC$^M$ & 0.0423 & 0.1294 & 0.2777 & 0.7238 & 1.2194 \\ EOM $^2$DCG$^M$ & 0.0396 & 0.1260 & 0.2737 & 0.7189 & 1.1915 \\ IHFS $^2$DCG$^M$ & 0.0396 & 0.1260 & 0.2737 & 0.7194 & 1.2165 \\ & & & & & \\ IHFS DC~\cite{PereiraGomes:2014iy} & & & 0.28 & 0.72 & \\ MRCI ~\cite{Zhou:2017ky} & & 0.1189 & & & \\ EOM(a)~\cite{Akinaga:2017hv} & 0.0422 & 0.1151 & & & \\ EOM(b)~\cite{Epifanovsky:2015hsa} & 0.0382 & 0.1123 & & & \\ EOM(c)~\cite{Cheng:2018ji} & 0.0394 & 0.1196 & 0.2533 & & \\ EOM(d)~\cite{Cheng:2018ji} & 0.0395 & 0.1220 & 0.2658 & & \\ CCSD(T)~\cite{IO-Peterson-JPC2006-110-13877-13883} & 0.0388 & 0.1061 & 0.2201 & & \\ & & & & & \\ Exp.~\cite{IO-Gilles-JCP1992-96-8012} & 0.0397 & 0.1270 & 0.2593 & & \\ \hline \hline \end{tabular} \end{center} \label{tab:so-monoxide-radicals} \end{table}% We will now consider the characteristics of the potential energy curves of the lowest electronic states of the halogen monoxides presented in Figure~\ref{potential-energy-curves-xo-all}. Due to the similarities in results between the $^2$DCG$^M$ and $^2$DC$^M$ Hamiltonians and between EOM-IP and IHFS(1h0p), we have opted to only present $^2$DCG$^M$ EOM-IP results since they seem to better represent the spin-orbit splitting. Additional data for these states (equilibrium structures, harmonic vibrational frequencies and vertical as well as adiabatic excitation energies) can be found in the Supplementary Information. In Figure~\ref{potential-energy-curves-xo-all} the changing nature of the $\pi$ and $\pi^*$ orbitals that we discussed before is clearly visible in the curves. For the lightest two halides (ClO, BrO) the splitting of the $A ^2\Pi$ is more pronounced than that of the $X ^2\Pi$ curves as it is the bonding orbital that contains the largest halogen fraction. For iodine and for heavier halides the splitting is more pronounced for the $X ^2\Pi$ than for the $A ^2\Pi$ because for these molecules the antibonding orbital contains the largest halogen fraction. The SOC does not have a large influence on the difference in equilibrium bond lengths between the $X ^2\Pi$ and $A ^2\Pi$ states, which is relatively stable at about 0.15~\AA. Another trend we observe from Figure~\ref{potential-energy-curves-xo-all} that is further evinced by the data in the Supplementary Information is the decrease of the ground-state harmonic frequencies along the series for $X ^2\Pi$, from about 900 cm$^{-1}$ for ClO to 573 and 457 cm$^{-1}$ for the $X_{3/2} {^2\Pi}$ and $X_{1/2} {^2\Pi}$ of TsO. While this is partly due to the increasing reduced mass, also the decrease in force constants indicates a reduction of the bond strength by almost a factor of two going from ClO to TsO. For the excited $A_{3/2} {^2\Pi}$ and $A_{1/2} {^2\Pi}$ states, the trend is less clear. While having significantly smaller force constants overall due to double occupancy of the $\pi^*$ instead of $\pi$ orbitals, the force constants of the $A$ states in IO are larger than these of the lighter ClO and BrO, as well as these of the heavier AtO. TsO is again special, with the two A states crossing each other and having rather different force constants. This shows that for such heavy atoms, the treatment of SOC as a minor perturbation can not be justified. \begin{figure}[h] \centering \begin{minipage}[b]{0.57\linewidth} \centering \includegraphics[width=\textwidth]{clo_eomcc_x2cmmf+gaunt_first_four_states_avtz_curves_landscape.png} \label{fig:potential-energy-curves-clo} \end{minipage} \begin{minipage}[b]{0.57\linewidth} \centering \includegraphics[width=\textwidth]{bro_eomcc_x2cmmf+gaunt_first_four_states_avtz_curves_landscape.png} \label{fig:potential-energy-curves-bro} \end{minipage} \begin{minipage}[b]{0.57\linewidth} \centering \includegraphics[width=\textwidth]{io_eomcc_x2cmmf+gaunt_first_four_states_avtz_curves_landscape.png} \label{fig:potential-energy-curves-io} \end{minipage} \begin{minipage}[b]{0.57\linewidth} \centering \includegraphics[width=\textwidth]{ato_eomcc_x2cmmf+gaunt_first_four_states_avtz_curves_landscape.png} \label{fig:potential-energy-curves-ato} \end{minipage} \centering \begin{minipage}[b]{0.57\linewidth} \includegraphics[width=\textwidth]{tso_eomcc_x2cmmf+gaunt_first_four_states_avtz_curves_landscape.png} \label{fig:potential-energy-curves-tso} \end{minipage} \caption{Potential energy curves of the spin-orbit split $X ^2\Pi$ and $A ^2\Pi$ states of the XO molecules, obtained with EOM-IP and the $^2$DCG$^M$ Hamiltonian.~\cite{paper:figures}} \label{potential-energy-curves-xo-all} \end{figure} \subsection{Triiodide} We discuss now our results for the triiodide molecule, previously investigated by some of us~\cite{gomes10} with a large array of relativistic correlated electronic structure methods (CASPT2, TD-DFT, MRCI and IHFS-CC), and more recently by Wang and coworkers~\cite{Wang:2014es} with EOM approaches which take SOC into account in an approximate manner. As the experimental interest on this species has to do with its complex dissociation behavior upon electronic excitations, our main interest here is in comparing our EOM-EE approach to IHFS(1h1p), which we know accurately describes the two absorbing $0_u^+$ states in the ranges of 3.43--3.45 eV and 4.25--4.28 eV, respectively (these excitation energies having been determined by photofragment yield spectra~\cite{Choi:2000jv,Zhu:2001fy}), as well as how EOM-EE with molecular mean-field Hamiltonians compares to the approximate schemes proposed by Wang and coworkers~\cite{Wang:2014es}. \begin{table}[htb] \caption{Vertical excitation energies (T$_v$, in eV) for I$_3^-$ with the EOM-EE and IHFS(1h1p) approaches employing the molecular mean-field Hamiltonians with ($^2$DCG$^M$) and without ($^2$DC$^M$) the Gaunt interaction at R = 2.93 \AA. We also present SO-CASPT2 results by Gomes and coworkers\cite{gomes10}, approximate EOM-EE schemes by Wang and coworkers~\cite{Wang:2014es}((a) EOM-SOC-CCSD; (b) SOC-EOM-CCSD; (c) cSOC-EOM-CCSD) and experimental results~\cite{Choi:2000jv,Zhu:2001fy}, along with the mean absolute differences (MAD) and its standard deviation ($\sigma$) between EOM-EE and IHFS(1h1p) calculations and and those between IHFS(1h1p)-$^2$DCG$^M$ and the SO-CASPT2 and approximate EOMCC methods.} \begin{center} \begin{tabular}{c c cc p{0.1cm} cc p{0.1cm} ccc p{0.1cm} c} \hline \hline & & \multicolumn{2}{c}{$^2$DC$^M$ } && \multicolumn{2}{c}{$^2$DCG$^M$ } && \multicolumn{3}{c}{EOM~\cite{Wang:2014es}} && CAS\\ \cline{3-4} \cline{6-7} \cline{9-11} State & $\Omega$ & EOM &IHFS && EOM &IHFS && (a) & (b) & (c) && PT2\cite{gomes10}\\ \hline 1 & 2$_g$ & 2.24 & 2.07 && 2.25 & 2.08 && 2.22 & 2.36 & 2.16 && 2.24 \\ 2 & 1$_g$ & 2.37 & 2.20 && 2.38 & 2.21 && 2.35 & 2.50 & 2.29 && 2.32 \\ 3 & 0$_u^-$ & 2.37 & 2.22 && 2.38 & 2.23 && 2.34 & 2.49 & 2.29 && 2.47 \\ 4 & 1$_u$ & 2.38 & 2.23 && 2.38 & 2.24 && 2.34 & 2.47 & 2.30 && 2.47 \\ 5 & 0$_g^-$ & 2.84 & 2.66 && 2.84 & 2.66 && 2.81 & 2.96 & 2.72 && 2.76 \\ 6 & 0$_g^+$ & 2.89 & 2.71 && 2.89 & 2.71 && 2.86 & 2.99 & 2.75 && 2.82 \\ 7 & 1$_g$ & 3.07 & 2.88 && 3.07 & 2.89 && 3.04 & 3.20 & 2.96 && 2.85 \\ 8 & 2$_u$ & 3.32 & 3.19 && 3.33 & 3.20 && 3.30 & 3.47 & 3.25 && 3.10 \\ 9 & 1$_u$ & 3.41 & 3.27 && 3.42 & 3.27 && 3.39 & 3.55 & 3.34 && 3.11 \\ 10 & 0$_u^+$ & 3.66 & 3.51 && 3.67 & 3.52 && 3.65 & 3.79 & 3.56 && 3.52 \\ 11 & 2$_g$ & 4.09 & 3.92 && 4.10 & 3.93 && 4.04 & 4.19 & 3.98 && 3.98 \\ 12 & 0$_u^-$ & 4.08 & 3.93 && 4.08 & 3.93 && 4.05 & 4.18 & 3.91 && 3.79 \\ 13 & 1$_u$ & 4.18 & 4.02 && 4.18 & 4.02 && 4.15 & 4.29 & 4.01 && 3.80 \\ 14 & 1$_g$ & 4.21 & 4.03 && 4.22 & 4.04 && 4.17 & 4.32 & 4.10 && 4.06 \\ 15 & 0$_u^+$ & 4.49 & 4.33 && 4.49 & 4.33 && 4.50 & 4.67 & 4.42 && 4.51 \\ 16 & 0$_g^-$ & 4.69 & 4.51 && 4.69 & 4.51 && 4.65 & 4.76 & 4.51 && 4.51 \\ 17 & 0$_g^+$ & 4.70 & 4.51 && 4.70 & 4.51 && 4.65 & 4.82 & 4.51 && 4.53 \\ 18 & 1$_g$ & 4.90 & 4.71 && 4.90 & 4.71 && 4.86 & 4.99 & 4.73 && 4.60 \\ MAD & & 0.17 & && 0.17 & && 0.13 & 0.28 & 0.05 && 0.11 \\ $\sigma$& & 0.02 & && 0.02 & && 0.02 & 0.03 & 0.03 && 0.08 \\ \hline \hline \end{tabular} \end{center} \flushleft Exp.: Photofragment yield spectra~\cite{Choi:2000jv,Zhu:2001fy}: 3.43--3.45, 4.25--4.28. \label{tab:triiodide-ee} \end{table}% From our results, presented in Table~\ref{tab:triiodide-ee}, we observe that EOM-EE excitation energies systematically overestimate the IHFS(1h1p) ones; we find a mean absolute deviation (MAD) of 0.17 eV for both the $^2 $DC$^M$ and $^2 $DCG$^M$ Hamiltonians, with small standard deviations ($\sigma$) of 0.02 eV in both cases. On other words, spectra obtained with EOM will show a shift in the origin with respect to IHFS, but will otherwise look the same. This behavior has been discussed previously by Musial and Bartlett for light-element systems~\cite{fscc-vs-lrcc:musial:1,fscc-vs-lrcc:musial:2,fscc-vs-lrcc:musial:3}, and has to do with the differences in parametrization for the excited-state wavefunctions in the two approaches (linear for EOM-EE and non-linear for IHFS(1h1p)). In the IHFS(1h1p) sector, the nonlinear parametrization of the wave operator, apart from the single excitation operators contains the product of the electron attachment and detachment operators whose presence assures valence extensivity~\cite{Mukhopadhyay:1991fp}, in contrast to the linear operator in EOM-EE which does not. Some of us had already observed the same systematic behavior of EOM in comparison to IHFS for another heavy element species (UO$_2^{2+}$), but since in that study a two-step SO-LRCC (thus analogous to EOM-EE) calculation~\cite{real09} was used, there could still be doubts as to whether the observed differences arose solely from the difference in parametrization. Here, as we have used exactly the same Hamiltonians for both methods, we can affirm that the differences indeed come from the parametrization. The SO-CASPT2 results of Ref.~\citenum{gomes10}, reproduced here for the convenience of the readers, show by comparison a slightly smaller MAD but at the same time much less systematic behavior than EOM, with some states being very close to the IHFS ones and others quite far apart, as reflected by the larger $\sigma$ value of 0.08 eV. Our view is that this underscores the lesser reliability of CASPT2 with respect to coupled-cluster approaches since it can result, for instance, in spurious inversions between states: for EOM we observe one such inversion for states 11 ($2_g$) and 12 ($0_u^{-}$), which are very close in energy in both EOM and IHFS calculations, whereas for CASPT2 we see two such inversions between states 11, 12 and 13 ($1_u$); Furthermore, CASPT2 places states such as 8 ($2_u$) and 9 ($1_u$), or 15 ($0_u^{+}$) and 16 ($0_g^{-}$) much close together than both coupled-cluster approaches. In line with what has been established in our prior investigation~\cite{gomes10}, the EOM-EE and IHFS(1h1p) wavefunctions for the absorbing $0_u^+$ states are predominantly made up of transitions from the $\sigma_{1/2g}$ to $\sigma^*_{1/2u}$ and $\pi_{1/2g}$ to $\sigma^*_{1/2u}$ spinors. For IHFS(1h1p) the excited determinants making up the $0_u^+$ wavefunctions correspond to excitations within the main ($P_m$) model space (essentially from $\sigma_{1/2g}$ and $\pi_{1/2g}$ to the LUMO $\sigma^*_{1/2u}$), with small contributions from excitations falling within the intermediate ($P_i$) model space. For EOM the picture is very much the same as that of IHFS(1h1p), though we note there are also contributions from doubly excited (2h2p) determinants, as well as singly excited (1h1p) determinants containing high-lying virtuals, both of which fall outside of what is the IHFS(1h1p) $P$ space; individually these are all negligible contributions, but taken as a whole they represent a minor but non-negligible ($\simeq$ 4\%) contribution. The excitation energies of triiodide have also been investigated with the three approximate EOM schemes introduced by Wang and coworkers~\cite{Wang:2014es}, which are based on the inclusion of SOC at the post-SCF step using a one-electron SOC operator ($\hat{h}^{SO}$) originating from the relativistic effective core potential (RECP) operator. In the first scheme (EOM-SOC-CCSD), $\hat{h}^{SO}$ is introduced for the solution of the ground-state coupled-cluster equations (so SOC is directly included in the $F_{ae}, F_{mi}, F_{me}, W_{mbij}$, and $W_{abej}$ intermediates and indirectly in the other intermediates through the cluster amplitudes), and the excited states are obtained by diagonalizing the corresponding similarity-transformed Hamiltonian (including $\hat{h}^{SO}$) in the space of singly and doubly excited determinants. In the second scheme (SOC-EOM-CCSD), $h^{SO}$ is included in the Hamiltonian for the EOM step only, and thus neither the Hartree-Fock not the ground-state coupled-cluster wavefunctions incorporate any SOC effects. The SOC-EOM-CCSD scheme is computationally less expensive since only the $F_{ae}, F_{mi}, F_{me}, W_{mbij}$, and $W_{abej}$ intermediates can be complex, but at the cost of having to determine the EOM excitation energies in the space of the ground-state, singly and doubly excited determinants while at the same time introducing unlinked terms involving matrix elements of the SOC Hamiltonian which make the excited state energies not size-intensive and the ground-state energy not size-extensive. Finally, the third scheme (cSOC-EOM-CCSD) approximates SOC-EOM-CCSD by neglecting the unlinked terms in SOC-EOM-CCSD. SOC effects on the ground state are not taken into account with this approach, and interactions between double-excitation determinants and single-excitation determinants through SOC are not fully considered because of the neglect of the term where the SOC operator coupled singly and doubly excited determinants in the EOM equations. We note these schemes exploit time-reversal symmetry, and that due to the use of orbitals not including SOC, it was possible to use single-point group symmetry. From Table~\ref{tab:triiodide-ee} we see that EOM-SOC-CCSD performs rather consistently with the four-component-based approaches presented here, with small deviations (from 0.02 to 0.05 eV) from our results which are likely due to difference in basis sets and the truncation of the correlating space in our case. As a consequence, EOM-SOC-CCSD are similarly very systematic in their deviation from IHFS(1h1p), showing a slightly better MAD value for EOM-SOC-CCSD than ours that is likely to be a fortuitous result, as the corresponding $\sigma$ value is the same as ours. A comparison to SOC-EOM-CCSD, on the other hand, shows that the latter is a rather poor approximate scheme, as not only individual excitation energies are quite different from ours (generally shifted upwards by over 0.1 eV, and with a MAD value nearly twice as large as ours for EOM-SOC-CCSD), though the systematic nature of the difference between EOM and IHFS is still roughly intact, as there is only a small increase of the $\sigma$ value to 0.03 eV. Finally, the correction to SOC-EOM-CCSD (cSOC-EOM-CCSD) does reduce the MAD value and is therefore on average close to IHFS than the more rigorous EOM schemes, but as was the case for SOC-EOM-CCSD, this obscures the fact that the errors are less uniformly distributed than for EOM-SOC-CCSD or the four-component-based EOM since one also has a $\sigma$ value of 0.03 eV. This makes the method less reliable in practice. In addition to the excitation energies, we present in Table~\ref{tab:triiodide-ip-ea} some of the lowest ionization energies and electron affinities of the triiodide (thus yielding the I$_3$ and I$_3^{2-}$ radicals). Our ionization energies show a rather good agreement with experiment for the states under consideration -- and particularly for the first -- which is not surprising, in the light of the good performance shown by EOM-IP for the XO species, and taking into account that the experimental results are for vertical electron detachment and the chosen bond lenght is quite close to the equilibrium structure of I$_3^{-}$. As for IO, we see that the Gaunt interaction plays a negligible role in the ionization energies, and also that the IHFS model space is sufficiently flexible that the EOM and IHFS results are essentially identical. \begin{table}[htb] \caption{Vertical electron detachment ($\mathrm{IP^{\Omega}_{n}}$, in eV) and attachment ($\mathrm{EA^{\Omega}_{n}}$, in eV) energies for the I$_3^{-}$ species, calculated at R = 2.93 {\AA} with the EOM-IP/EA and IHFS(1h0p)/(0h1p) approaches, respectively, employing the molecular mean-field Hamiltonians with ($^2$DCG$^M$) and without ($^2$DC$^M$) the Gaunt interaction, as well as experimental values for the ionization energies~\cite{Choi:2000eo}}. \begin{center} \begin{tabular}{l cccc} \hline \hline Method & IP$^{3/2u}_1$ & IP$^{1/2g}_2$ & IP$^{1/2u}_3$ & IP$^{3/2g}_4$ \\ \hline EOM $^2$DC$^M$ & 4.28 & 4.47 & 4.92 & 4.99 \\ IHFS $^2$DC$^M$ & 4.28 & 4.47 & 4.92 & 4.99 \\ EOM $^2$DCG$^M$ & 4.28 & 4.47 & 4.91 & 5.00 \\ IHFS $^2$DCG$^M$ & 4.28 & 4.47 & 4.91 & 5.00 \\ Exp.~\cite{Choi:2000eo} & 4.25 & 4.53 & 4.87 & 4.93 \\ \hline & EA$^{1/2u}_1$ & EA$^{1/2u}_2$ & EA$^{1/2g}_3$ & EA$^{1/2u}_4$ \\ \hline EOM $^2$DC$^M$ & 2.51 & 3.64 & 3.88 & 4.39 \\ IHFS $^2$DC$^M$ & 2.51 & 3.66 & 3.89 & 4.39 \\ EOM $^2$DCG$^M$ & 2.51 & 3.65 & 3.88 & 4.39 \\ IHFS $^2$DCG$^M$ & 2.52 & 3.67 & 3.89 & 4.39 \\ \hline \hline \end{tabular} \end{center} \label{tab:triiodide-ip-ea} \end{table}% For the electron affinities the same trends as for ionization energies are observed with respect to the importance of the Gaunt interactions and the similarity of EOM and IHFS results. Unfortunately, we are unable to compare the calculated values to experiment since, to the best of our knowledge, such results are not available in the literature. \subsection{Dihalomethanes} We now turn our attention to the diiodo- (CH$_2$I$_2$) and iodobromo-methane (CH$_2$IBr) species which, apart from their experimental interest, are examples of species with lower symmetry than those discussed before and therefore more costly to treat from a computational standpoint (CH$_2$IBr requiring the use of complex algebra). We begin with their ionisation energies shown in Table~\ref{tab:ip-halomethanes}. We see there is hardly any difference between $^2$DCG$^M$ and $^2$DC$^M$ results (at most differences of 0.01 eV) for both CH$_2$I$_2$ and CH$_2$IBr. In all cases, the ionizations are determined to be single-particle processes, with the absence of important (2h1p) amplitudes in the EOM-IP case. The ionization energies are in good agreement with experiment~\cite{Potts:79nRiJUm,Cartoni:2015ex,Satta:2015gp,Tsal:1975er,Lago:2005kr,Zhao:2014cu,Sandor:2016eq,Xing:2014gp,Lee:2005jt}, with typical differences being of the order of 0.1 eV. Such differences are quite far from what are the best experimental error bars avaliable~\cite{Lago:2005kr}, which are well under 0.05 eV for both species, but given that our calculations have not been performed at the experimental structures\cite{Kudchadker:1975cp} but rather on PBE-optimized ones, and still lack corrections due to basis set completeness--and probably more importantly, of higher-order electron correlation effects--we consider this accuracy to be sufficient for the purposes of this paper. \begin{table}[htb] \caption{Electron detachment energies ($\mathrm{IP_{n}}$, in eV) for the CH$_2$I$_2$ and CH$_2$IBr species with the EOM-IP and IHFS(1h0p) approaches employing the molecular mean-field Hamiltonians with ($^2$DCG$^M$) and without ($^2$DC$^M$) the Gaunt interaction and augmented triple zeta bases at the ZORA-SO/PBE/TZ2P geometry. Experimental results and prior theoretical calculations are shown when available.} \begin{center} \begin{tabular}{l cccccc} \hline \hline Method & IP$_1$ & IP$_2$ & IP$_3$ & IP$_4$ & IP$_5$ & IP$_6$ \\ \hline \multicolumn{7}{c}{CH$_2$I$_2$}\\ \hline EOM $^2$DC$^M$ & 9.37 & 9.69 & 10.12 & 10.46 & 12.75 & 13.52 \\ IHFS $^2$DC$^M$ & 9.37 & 9.69 & 10.12 & 10.46 & 12.75 & 13.53 \\ EOM $^2$DCG$^M$ & 9.36 & 9.68 & 10.12 & 10.45 & 12.74 & 13.52 \\ IHFS $^2$DCG$^M$ & 9.36 & 9.68 & 10.12 & 10.45 & 12.74 & 13.52 \\ & & & & & & \\ TD-B3LYP~\cite{Satta:2015gp} & 9.46 & 9.57 & 9.74 & 10.29 & 12.65 & 13.32 \\ Exp.~\cite{Potts:79nRiJUm} & 9.46 & 9.76 & 10.21 & 10.56 & 12.75 & 13.67 \\ Exp.~\cite{vonNiessen:1982fz} & 9.46 & 9.76 & 10.2 & 10.6 & 12.8 & 13.7 \\ Exp.~\cite{Lago:2005kr} & 9.42 & & & & & \\ \hline \multicolumn{7}{c}{CH$_2$IBr}\\ \hline EOM $^2$DC$^M$ & 9.65 & 10.17 & 10.79 & 10.98 & 13.33 & 14.24 \\ IHFS $^2$DC$^M$ & 9.65 & 10.16 & 10.76 & 10.95 & 13.32 & 14.25 \\ EOM $^2$DCG$^M$ & 9.65 & 10.16 & 10.79 & 10.98 & 13.33 & 14.24 \\ IHFS $^2$DCG$^M$ & 9.65 & 10.16 & 10.76 & 10.95 & 13.32 & 14.25 \\ & & & & & & \\ SO-MRCI~\cite{Sandor:2014cu} & 9.69 & 10.26 & 10.91 & 11.12 & 13.62 & \\ Exp.~\cite{Lago:2005kr} & 9.69 & & & & & \\ \hline \hline \end{tabular} \end{center} \label{tab:ip-halomethanes} \end{table}% There have not been many other theoretical studies of ionization energies in the literature: we are aware of the TD-B3LYP~\cite{Satta:2015gp} study of Satta and coworkers for CH$_2$I$_2$ and the SO-MRCI studies of Weinacht and coworkers~\cite{GonzalezVazquez:2010ki,Geissler:2011dy,Sandor:2014cu} for CH$_2$IBr. With respect to TD-B3LYP, we observe our results are of similar quality for the first ionization energy, with B3LYP overestimating the experimental value by slightly less (0.04 eV) than we underestimate it (0.06 eV). However, for the higher ionizations we see a consistent underestimation of the experimental results by TD-B3LYP, while EOM-IP results are often 0.1 eV closer to experiment. In the case of CH$_2$IBr, EOM-IP again underestimates the experimental result (by 0.04 eV) whereas the SO-MRCI results more closely match experiment, something that may reflect additional orbital relaxation in the SO-MRCI calculations since state-averaged (spin-free) orbitals were used. For the other ionizations the two methods yield results which are apart by more than 0.1 eV, with a notable difference of 0.29 eV for IP$_5$. Given that for CH$_2$I$_2$ the higher ionizations by EOM-IP have followed the experimental ones rather well, we wonder the extent to which the SO-MRCI results are biased towards the description of the low-lying states. Unfortunately, to our knowledge there are no experimental results for these ionizations to shed further light on the performance of the methods. \begin{table}[htb] \caption{Electron attachment energies ($\mathrm{EA_{n}}$, in eV) for the CH$_2$I$_2$ and CH$_2$IBr species with the EOM-EA and IHFS(0h1p) approaches employing the molecular mean-field Hamiltonians with ($^2$DCG$^M$) and without ($^2$DC$^M$) the Gaunt interaction and augmented triple zeta bases at the ZORA-SO/PBE/TZ2P geometry. Experimental EAs~\cite{Modelli:1992jp} obtained by electron transmission spectroscopy (ET) or dissociative attachment spectra (DA) are available for CH$_2$I$_2$ only.} \begin{center} \begin{tabular}{l l cccccc} \hline \hline Method & EA$_1$ & EA$_2$ & EA$_3$ & EA$_4$ & EA$_5$ & EA$_6$ \\ \hline \multicolumn{7}{c}{CH$_2$I$_2$}\\ \hline EOM $^2$DC$^M$ & -0.32 & 0.50 & 0.76 & 1.01 & 1.51 & 1.71 \\ IHFS $^2$DC$^M$ & -0.32 & 0.52 & 0.77 & 1.03 & 1.54 & 1.74 \\ EOM $^2$DCG$^M$ & -0.32 & 0.50 & 0.76 & 1.01 & 1.51 & 1.71 \\ IHFS $^2$DCG$^M$& -0.32 & 0.52 & 0.77 & 1.03 & 1.52 & 1.74 \\ & & & & & & \\ Exp., ET~\cite{Modelli:1992jp} & $<0$ & 0.68 & & & & \\ Exp., DA~\cite{Modelli:1992jp} & $<0$ & 0.46 & & & & \\ \hline \multicolumn{7}{c}{CH$_2$IBr}\\ \hline EOM $^2$DC$^M$ & -0.02 & 0.54 & 0.99 & 1.02 & 1.64 & 1.93 \\ IHFS $^2$DC$^M$ & 0.01 & 0.59 & 1.04 & 1.11 & 1.66 & 2.02 \\ EOM $^2$DCG$^M$ & -0.02 & 0.54 & 0.99 & 1.02 & 1.64 & 1.93 \\ IHFS $^2$DCG$^M$ & 0.01 & 0.59 & 1.04 & 1.12 & 1.66 & 2.02 \\ \hline \hline \end{tabular} \end{center} \label{tab:ea-halomethanes} \end{table}% Our results for electron affinities are summarized in Table~\ref{tab:ea-halomethanes}. From these we see that for CH$_2$I$_2$ we have both EOM-EA and IHFS(0h1p) predicting a bound electron attachment state (corresponding to the CH$_2$I$_2^-$ species), in line with the experimental results for Modelli and coworkers~\cite{Modelli:1992jp}, who measured a bound first electron attachment state, and a second attachment energy. In passing we note that Modelli and coworkers found bound states for CHI$_3^-$ and CI$_4^-$ as well. Our calculations have placed the first electron attachment state (EA$_1$) at 0.32 eV below the ground state for the neutral species, and the second state (EA$_2$) at 0.50 eV, which agrees well with the value obtained via the dissociative attachment spectra (0.46 eV). We agree less in the interpretation of the process: with the help of MS-X$\alpha$ calculations, Modelli and coworkers have modelled it as the addition of an electron to a single virtual orbital. In our calculations, all electronic states but EA$_4$ correspond to multideterminantal wavefunctions--for the first two states, which are the most relevant ones for the comparison to the experimental results, we have that a determinant with an electron attached to the LUMO and another with an electron attached to the LUMO+1 contribute to the wavefunctions of EA$_1$ to about 56\% and 30\%, respectively; in the case of EA$_2$, these contribute by about 37\% and 59\%, respectively. The EOM-EA calculations for CH$_2$IBr show a similar trend, but with the first electron attachment state (EA$_1$) being only slightly bound, at 0.02 eV below the CH$_2$IBr ground state energy and with a second attachment state (EA$_2$) at around 0.54 eV above the CH$_2$IBr ground state energy. We also observe that wavefunctions for the electron attachment states are made up of more than a single determinant. The IHFS(0h1p) calculations yield results not far from EOM-EA for all the electron attachment states considered, but that show instead a weakly unbound (0.01 eV) EA$_1$. We believe this is an artifact of the calculations, due to the impossibility of using a larger model space. There are unfortunately no experimental results to which compare our calculations for CH$_2$IBr, but we note that in the work of Guerra and coworkers~\cite{Guerra:1991jp}, for the series of chloromethanes (from CH$_3$Cl to CCl$_4$) only CCl$_4^-$ shows a bound state; for the series of bromomethanes (from CH$_3$Br to CBr$_4$) the CH$_2$Br$_2^-$ species is not bound (though it shows a state slightly above zero energy~\cite{Modelli:1992jp}), but further substituting hydrogens by bromines yields stable anions. These findings, taken together with those for the iodine-substituted species, indicate that the heavier halogens help stabilize the first electron attachment state for the same degree of substitution. This makes it plausible that CH$_2$IBr$^-$, by the substitution of bromine by iodine, would have its first electron attachment stable stabilized with respect to CH$_2$Br$_2^-$ and have it become (weakly) bound. Finally, we present results for excitation energies in Tables~\ref{tab:excitations-ch2i2} and~\ref{tab:excitations-ch2ibr} for CH$_2$I$_2$ and CH$_2$IBr, respectively. We are only aware of the works of Liu and coworkers~\cite{Liu:2007cg,Liu:2006cp}, who considered SOC for these systems with the SO-CASPT2 approach. The electronic spectra of CH$_2$I$_2$ has been well-studied experimentally, in gas-phase~\cite{Ito:1961eh,Kawasaki:1975gb} and in organic solvents~\cite{Ito:1961eh,Gedanken:1979fa}. In the gas-phase, two main features at 4.29 eV and 4.98 eV have been first identified~\cite{Ito:1961eh}, with later photodissociation studies~\cite{Kawasaki:1975gb} revealing additional transitions. For CH$_2$IBr we are aware of studies in the gas phase~\cite{Butler:1987il}, which yield valence transitions at 4.58 eV and 5.79 eV, which are slightly changed in the presence of a organic solvent~\cite{Lee:1982ko}. As for I$_3^-$, we observe in Table~\ref{tab:excitations-ch2i2} a tendency of EOM-EE results to systematically overestimate the IHFS(1h1p) ones, with a mean deviation of 0.18 eV for both Hamiltonians considered (with differences between $^2$DCG$^M$ and $^2$DC$^M$ of the order of 0.01 eV or less) and similarly low $\sigma$ values (0.03 eV), which is quite close to what is obtained for I$_3^-$ (MAD of 0.17 eV and $\sigma$ of 0.02 eV). The states considered have been found to be of a singly excited nature and the EOM and IHFS wavefunctions are dominated by the same excited determinants. We observe once more the tendency of SO-CASPT2 to show somewhat higher deviations to IHFS(1h,1p) than EOM-EE, with much more uneven errors for the different excitations than EOM-EE (MAD of 0.43 eV and $\sigma$ of 0.09 eV). Without transition moments it is difficult to comment on the accuracy with respect to the experimental values, since our results show a number of close-lying states with energies close to the experimental peak values, but at the same time this give us confidence that we shall be able to reproduce the peak positions to a tenth of an eV or less. \begin{table}[htb] \caption{Vertical excitation energies (T$_v$, in eV) for CH$_2$I$_2$ with the EOM-EE and IHFS(1h1p) approaches employing the molecular mean-field Hamiltonians with ($^2$DCG$^M$) and without ($^2$DC$^M$) the Gaunt interaction. We also present the SO-CASPT2 results with ANO-RCC bases of Liu and coworkers\cite{Liu:2007cg} (at CASPT2 equilibrium geometries in C$_1$ symmetry) and experimental results, along with the mean absolute difference (MAD) and its standard deviation ($\sigma$) between the EOM-EE and IHFS(1h1p) approaches and between IHFS(1h1p)-$^2$DCG$^M$ and SO-CASPT2.} \begin{center} \begin{tabular}{c c cc p{0.2cm} cc p{0.2cm} c} \hline \hline & & \multicolumn{2}{c}{$^2$DC$^M$ } && \multicolumn{2}{c}{$^2$DCG$^M$ } && \\ \cline{3-4} \cline{6-7} State & Symmetry & EOM &IHFS && EOM &IHFS && CASPT2\cite{Liu:2007cg}\\ \hline 1 & a & 3.60 & 3.44 && 3.60 & 3.44 && 3.76 \\ 2 & b & 3.62 & 3.46 && 3.62 & 3.45 && 3.78 \\ 3 & a & 3.63 & 3.47 && 3.63 & 3.46 && 3.78 \\ 4 & b & 3.85 & 3.68 && 3.85 & 3.68 && 4.03 \\ 5 & a & 3.87 & 3.70 && 3.87 & 3.70 && 4.27 \\ 6 & b & 3.94 & 3.79 && 3.94 & 3.79 && 4.27 \\ 7 & a & 3.99 & 3.83 && 3.99 & 3.83 && 4.31 \\ 8 & b & 4.06 & 3.90 && 4.06 & 3.90 && 4.38 \\ 9 & b & 4.22 & 4.04 && 4.22 & 4.03 && 4.50 \\ 10 & a & 4.32 & 4.15 && 4.32 & 4.14 && 4.60 \\ 11 & b & 4.35 & 4.17 && 4.35 & 4.16 && 4.62 \\ 12 & a & 4.49 & 4.32 && 4.49 & 4.31 && \\ 13 & b & 4.64 & 4.48 && 4.63 & 4.47 && \\ 14 & a & 4.68 & 4.52 && 4.68 & 4.52 && \\ 15 & b & 4.75 & 4.58 && 4.74 & 4.58 && \\ 16 & a & 4.91 & 4.75 && 4.91 & 4.74 && \\ 17 & a & 5.59 & 5.33 && 5.59 & 5.33 && \\ 18 & b & 5.61 & 5.35 && 5.61 & 5.35 && \\ MAD & & 0.18 & && 0.18 & && 0.43 \\ $\sigma$& & 0.03 & && 0.03 & && 0.09 \\ \hline \hline \end{tabular} \end{center} \flushleft Exp., photodissociation of molecular beams, model\cite{Kawasaki:1975gb}: 3.98, 4.34, 4.42, 4.89, 4.97, 5.79. \\ Exp., absorption in vapour, maxima\cite{Ito:1961eh}: 4.29, 4.98. \\ Exp., absorption in iso-octane, maxima\cite{Ito:1961eh}: 4.25, 4.94, 5.85. \\ Exp., absorption in iso-octane, resolved\cite{Ito:1961eh}: 4.02, 4.39, 4.91, 5.88. \\ Exp., MCD in cyclohexane, resolved\cite{Gedanken:1979fa}: 3.97, 4.19, 4.49, 4.91, 5.48, 5.88. \\ \label{tab:excitations-ch2i2} \end{table}% For the excitation energies of CH$_2$IBr, the general observations with respect to a comparison to experiment made for CH$_2$I$_2$ apply, as we see from Table~\ref{tab:excitations-ch2ibr} that we obtain EOM-EE excitation energies that match well the energies of the experimental peak maxima, and that there's little differences between Hamiltonians (in general, differences are smaller than 0.01 eV). We note from the EOM-EE results that once more there's little difference between Hamiltonians. Unfortunately, we were unable to perform a detailed comparison to IHFS(1h1p) due to the impossibility of converging the latter calculations, and therefore only make a comparison to SO-CASPT2~\cite{Liu:2006cp}. As was the case for CH$_2$I$_2$, there is a general shift to higher energies in the SO-CASPT2 results compared to the EOM-EE $^2$DCG$^M$ ones, but which is not very systematic: shifts for the low-end of the spectrum (up to 4.8 eV) are of $\simeq$0.15-0.21 eV range, with the high-end (from about 5.8 eV onwards) showing large variations (from 0.11 to 0.42 eV) while the variations in the mid-range of the spectrum are of about 0.1 eV, which make the $\sigma$ value go up to 0.17 eV. \begin{table}[htb] \caption{Vertical excitation energies (T$_v$, in eV) for CH$_2$IBr with the EOM-EE approach employing the molecular mean-field Hamiltonians with ($^2$DCG$^M$) and without ($^2$DC$^M$) the Gaunt interaction, using the correlating space which does not include the Br $3d$ spinors. We also present the SO-CASPT2 results with ANO-RCC bases of Liu and coworkers\cite{Liu:2006cp} (obtained at CASPT2 equilibrium geometries, and for C$_1$ symmetry) and experimental results, along with the mean absolute difference (MAD) and its standard deviation ($\sigma$) between EOM-EE-$^2$DCG$^M$ and SO-CASPT2.} \begin{center} \begin{tabular}{c c c p{0.2cm} c p{0.2cm} c} \hline \hline State & Symmetry & $^2$DC$^M$ && $^2$DCG$^M$ && CASPT2\cite{Liu:2007cg}\\ \hline 1 & b & 3.86 && 3.86 && 4.07 \\ 2 & a & 3.86 && 3.86 && 4.07 \\ 3 & b & 3.98 && 3.98 && 4.16 \\ 4 & a & 4.04 && 4.04 && 4.25 \\ 5 & b & 4.34 && 4.34 && 4.49 \\ 6 & a & 4.45 && 4.44 && 4.61 \\ 7 & b & 4.66 && 4.66 && 4.75 \\ 8 & a & 4.67 && 4.67 && 4.77 \\ 9 & b & 5.08 && 5.07 && 5.17 \\ 10 & a & 5.13 && 5.13 && 5.21 \\ 11 & b & 5.16 && 5.16 && 5.25 \\ 12 & a & 5.30 && 5.30 && 5.39 \\ 13 & b & 5.37 && 5.37 && 5.44 \\ 14 & a & 5.39 && 5.38 && 5.81 \\ 15 & b & 5.80 && 5.80 && 5.91 \\ 16 & a & 5.82 && 5.82 && 6.24 \\ 17 & b & 6.09 && 6.09 && 6.24 \\ 18 & b & 6.10 && 6.10 && 6.28 \\ MAD & & && && 0.17 \\ $\sigma$& & && && 0.17 \\ \hline \hline \end{tabular} \end{center} \flushleft Exp., absorption on hexane\cite{Lee:1982ko}: 4.63, 5.82 \\ Exp., absorption in gas phase\cite{Butler:1987il}: 4.58, 5.79; 6.52 (Rydberg) \label{tab:excitations-ch2ibr} \end{table}% \section{Conclusions} In this work we have described the formulation and implementation in the \textsc{Dirac} code of the EOM-CCSD method for electron attachment (EOM-EA-CCSD), electron detachment (EOM-IP-CCSD) and excitation energies (EOM-EE-CCSD) based on four-component Hamiltonians. This implementation, which can be used with any of the Hamiltonians available in \textsc{Dirac}, exploits double point group symmetry for all of the above EOM variants, and yields both left and right eigenvectors. We have proceeded in validating the implementation by a careful comparison to intermediate Hamiltonian Fock-space (IHFS-CCSD) calculations on different classes of systems: the series of halogen monoxide radicals (XO, X = Cl--Ts); the triiodide (I$_3^-$) species and some of its electron detached (the I$_3$ radical) and attached (the I$_3^{2-}$ species) states; as well as the neutral, cationic and anionic forms of the diiodo- and iodobromethane molecules. In all of the electron attachment and detachment cases considered (XO, I$_3$/I$_3^{2-}$, CH$_2$I$_2^{-/+}$, CH$_2$IBr$^{-/+}$) we have found the EOM-CCSD and IHFS-CCSD methods to differ only slightly, as expected from formal considerations. Whenever more significant discrepancies were found, we believe to have shown these are due to the shortcomings on the main model spaces employed in the IHFS-CCSD case, which was not sufficiently large. In the cases where experimental data are available, our calculations are in very good agreement with them. For the excited states, in the cases where we compared singly excited states (I$^-_3$ and CH$_2$I$_2$) and had a ground state reference described by a single determinant, the agreement between EOM-EE-CCSD and IHFS-CCSD is quite good, though EOM-EE-CCSD shows a tendency to systematically overestimate the IHFS-CCSD values. From formal considerations the two methods are in general not expected to yield the same results due to the lack of valence extensivity for EOM-EE-CCSD. The differences found here are in line with those found in other comparisons of EOM-EE-CCSD and IHFS-CCSD for other small systems (containing or not heavy centers). We believe the tradeoff in EOM-EE-CCSD between losing some accuracy (in the absolute sense) by forsaking valence extensivity and the gain in robusteness in the calculations (due to the diagonalization-based approach used, as opposed to the iterative one used in IHFS-CCSD) is well worth taking in practical applications. This is exemplified in the case of CH$_2$IBr, for which we have not been able to converge the IHFS-CCSD(1h1p) calculations. Though in this case we have been unable to compare the two coupled cluster approaches, we note the performance of EOM-EE-CCSD relative to the SO-CASPT2 results is similar to that for CH$_2$I$_2$ and I$^-_3$. We believe a more stringent comparison to SO-CASPT2 and experiments requires the availability of transition moments for the EOM-EE-CCSD case, which we shall describe in a subsequent publication. Another important aspect not addressed in this work is that of going beyond the CCSD model for transition energies, since it is known for instance that molecular properties and energetics can be quite sensitive to details of the electron correlation. In subsequent publications we envisage to explore the inclusion of triple excitations, notably via a perturbative approach, on the different EOM variants~\cite{Watts:1996bi,Stanton:1996hw,Manohar:2009kza,Matthews:2016bi}. Finally, as far as Hamiltonians are concerned we have shown first that, based on single-point comparisons of our EOM-CC results for the XO systems and triiodide calculations performed with the Dirac--Coulomb Hamiltonian, that the corresponding two-component molecular mean-field approach ($^2$DC$^M$) does indeed yield results which are nearly indistinguishable from the former, and for this reason we strongly recommend the use of the molecular mean-field approach. Concerning the inclusion of the Gaunt interaction, we have determined it to have in general a small effect on the energetics of the species under consideration (and on the spectroscopic constants for the case of halogen monoxides), though for ClO, BrO and TsO it is necessary to include it as it corresponds to roughly between 7\% and 2\% of that of the spin-orbit splitting of the $^2\Pi$ ground-state. \section{Supplementary Material} See supplementary information for results discussed but not shown in the manuscript: (a) spectroscopic constants, vertical and adiabatic excitation energies, projection and wavefunction analysis for halogen monoxide radicals, along with spinor magnetization plots for the halogen monoxide anions; (b) DC and ${^2}$DC$^M$ energies for halogen monoxides (anions and radicals), triiodide radical and anions; (c) optimized structures for the halomethanes. These resources are also available via the zenodo repository~\cite{paper:dataset}, along with the outputs for the respective calculations. \section{Acknowledgements} The members of the PhLAM laboratory acknowledge support from the CaPPA project (Chemical and Physical Properties of the Atmosphere), funded by the French National Research Agency (ANR) through the PIA (Programme d'Investissement d'Avenir) under contract ``ANR-11-LABX-0005-01'' as well as by the Ministry of Higher Education and Research, Hauts de France council and European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER) CLIMBIO (Changement climatique, dynamique de l'atmosph\`ere, impacts sur la biodiversit\'e et la sant\'e humaine). Furthermore, ASPG acknowledges funding from the CNRS Institute of Physics (INP) via the PICS program (grant 6386), and computational time provided by the French national supercomputing facilities (grants DARI x2016081859, A0010801859, A0030801859), and both AS and ASPG acknowledge many illuminating discussions on the implementation of the matrix-free diagonalization method with Dr.\ Jean-Pierre Flament (PhLAM). \section{Halogen monoxides} In Table~\ref{tab:spectro-monoxide-radicals} we present the vertical and adiabatic electronic spectra of the halogen monoxide radicals, as well as equilibrium geometries and vibrational frequencies of the individual states obtained with EOM-IP and IHFS(1h0p) with the $^2$DC$^M$ and $^2$DCG$^M$ Hamiltonians, whereas in Table~\ref{tab:spectro-monoxide-radical-dc-vs-2dcm} we compare the results for the Dirac-Coulomb and $^2$DC$^M$ Hamiltonians. In Figure~\ref{relativistic-effects-xo} the trends along the series for the internuclear distances, vibrational frequencies and vertical $\Omega = 3/2 - 1/2$ energy difference for the $X ^2\Pi$ and $A ^2\Pi$ states, based on the data in Table~\ref{tab:spectro-monoxide-radicals} for the $^2$DCG$^M$ Hamiltonian, are also presented. In complement to that, in Tables~\ref{tab:clo-state-wfs} to~\ref{tab:tso-state-wfs} we present the states' wavefunctions for the different states of each species, in terms of the dominant configurations and their expansion coefficients in the respective wavefunction parametrizations ($r_i$ and eventually $r_{ij}^a$ for EOM-IP, as well as $c_i$ IHFS(1h0p), which has the same role as $r_i$). We note we have chosen, for the wavefunctions, to present only the $^2$DC$^M$ results as there are no qualitative differences between the Hamiltonians. The $^2$DCG$^M$ values can be retrieved from the output files provided as part of the supplementary information. In Tables~\ref{tab:clo-state-prjana}, \ref{tab:bro-state-prjana}, \ref{tab:io-state-prjana}, \ref{tab:ato-state-prjana} and \ref{tab:tso-state-prjana} we present a projection analysis \cite{Faegri_JCP2001} as well as the $\sigma$- and $\pi$-character of the valence orbitals of the anions (XO$^-$) from 4-component Hartree--Fock calculations based on the Dirac--Coulomb--Gaunt Hamiltonian. The dominant configuration of the corresponding XO radicals are given in Tables~\ref{tab:clo-state-wfs}, \ref{tab:bro-state-wfs}, \ref{tab:io-state-wfs}, \ref{tab:ato-state-wfs} and \ref{tab:tso-state-wfs}. In all cases the $\pi^{*}_{3/2}$, $\pi^{*}_{1/2}$, $\sigma_{1/2}$, $\pi_{3/2}$ and $\pi_{1/2}$ spinors are occupied in the SCF reference wavefunctions for the XO$^-$ species. In Figure~\ref{spinor-magnetization-xo-all}, through the plots of the spinor spin magnetization, we provide a graphical representation not of the aforementioned four-component spinors but the nearly equivalent SO-ZORA/Hartree-Fock/QZ4P two-component ones obtained with ADF. As indicated in the ADF documentation (\url{http://www.scm.com}), these two-component spinors \begin{equation} \psi(\mathbf{r}) = \begin{bmatrix} \phi^{Re}_\alpha(\mathbf{r}) + i\phi^{Im}_\alpha(\mathbf{r}) \\ \phi^{Re}_\beta(\mathbf{r}) + i\phi^{Im}_\beta(\mathbf{r}) \end{bmatrix} \end{equation} with $\{\phi^{Re}_a, \phi^{Im}_a\}$ the real and imaginary parts of the $a$($=\alpha/\beta$)-spin functions, are represented by arrows showing the direction of the spin magnetization \begin{equation} \mathbf{m}(\mathbf{r}) = \psi^\dagger(\mathbf{r}) \boldsymbol{\Sigma} \psi(\mathbf{r}), \qquad \boldsymbol{\Sigma} = (\sigma_x, \sigma_y, \sigma_z) \end{equation} ($\sigma_x, \sigma_y, \sigma_z$ are the Pauli matrices) over space, drawn starting from points in space where the square root of the density $\rho(\mathbf{r}) = \psi^\dagger(\mathbf{r}) \psi(\mathbf{r})$ is equal to a given isosurface value. The arrow colors represent the sign (red: minus; blue: plus) of the phase factor $e^{i\theta}$ which, along with $\mathbf{m}$, fully determine the spinor. In this representation, the direction of the arrows indicate, for instance, to which extent the spinor is an eigenfunction of spin: when this is verified the arrows all point to the same direction. \section{Triiodide} In Table~\ref{tab:spectro-i3-dc-vs-2dcm} we compare the results for the Dirac-Coulomb and $^2$DC$^M$ Hamiltonians for the I$_3^-$, I$_3$ and I$_3^{2-}$ species. \section{Halomethanes} In Table~\ref{tab:xyz-halomethanes} we provide the Cartesian coordinates of the CH$_2$I$_2$ and CH$_2$IBr obtained by geometry optimization at the SO-ZORA/PBE/TZP level of theory. In Table~\ref{tab:bonds-angles-halomethanes} we provide the interatomic distances and angles for the same species, alongside the experimental numbers for CH$_2$I$_2$.

proofpile-arXiv_069-10361

\section{Introduction}\label{sec-1} \del{ In the field of fluid dynamics, electromagnetics, and acoustics, the physical phenomena such as mechanical vibrations and wave motions are commonly observed. These phenomena are usually modelled by hyperbolic partial differential equations (PDEs). One of the examples of hyperbolic PDEs is the wave equation, which describes the propagation of variety of waves (e.g. water waves, sound waves, and seismic waves). The wave equation subject to random perturbations gained a lot of attention during 1960's due to their applications in various fields such as oceanography, quantum mechanics and gene therapy, to name a few; we give several engineering examples that we aim to numerically access in Section \ref{sec-appli}. For an introduction to the theory of stochastic wave equation (SWE) and its applications, we refer to \cite[Chapter 5]{Chow_2015}, and references therein. } Let ${\mathcal O} \subset {\mathbb R}^d$, for $1 \leq d \leq 3$ be a bounded domain. We consider the numerical approximation of the following stochastic semilinear wave equation perturbed by multiplicative noise of It\^o type: \begin{align}\label{stoch-wave1:1} \begin{cases} \partial_t^2 u + A u = F(u,\partial_t u) + \sigma(u, \partial_t u)\,\partial_t W \qquad &\mbox{\rm in}\ (0,T) \times \mathcal{O}\,, \\ u(0,\cdot) = u_0\, , \quad \partial_t u(0,\cdot) = v_0 \qquad &\mbox{\rm in}\ \mathcal{O}\,,\\ u(t,\cdot) =0 \qquad &\mbox{\rm on}\ \partial \mathcal{O}, \,\forall \, t \in (0,T)\,, \end{cases} \end{align} where $A$ is a strongly elliptic second order differential operator of the form \begin{equation}\label{stoch-wave1:1b} A \varphi(x)= - \sum_{i, j=1}^d \frac{\partial} {\partial x_i} \Bigl( a_{ij}(x) \frac{\partial}{\partial x_j} \varphi(x)\Bigr) \qquad \forall\, x \in {\mathcal O}\,, \end{equation} with suitably smooth coefficients $a_{ij}(x)$, where $a^{ij} = a^{ji} \ \forall i, j,$ and for every non-zero $\xi \in {\mathbb R}^d$, $\sum_{i, j}^d a_{ij}(x) \xi_i \xi_j \geq \gamma |\xi|^2$, for some constant $\gamma > 0$. Here, $F$ and $\sigma$ are Lipschitz in both arguments. Let $\mathfrak{P}:= (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, and $\{W(t)\}_{t\geq0}$ be a finite dimensional Wiener process defined on it; the initial data $u_0$ and $v_0$ are given $\mathcal{F}_0-$measurable random variables. A strong variational solution to (\ref{stoch-wave1:1}) exists, see {\em e.g.}~\cite[Sec.~6.8]{Chow_2015}, and is usually constructed via the reformulation of (\ref{stoch-wave1:1})$_1$ as a first order system by setting $v = \partial_t u$, \begin{align}\label{stoch-wave1:1a} \begin{cases} &{\rm d} u = v\, {\rm d} t\, \\ &{\rm d} v = \bigl(-A u + F (u , v) \bigr)\, {\rm d} t + \sigma(u,v) \,{\rm d} W(t)\,, \end{cases} \end{align} and then using a Faedo-Galerkin method, related uniform energy bounds, and a compactness argument; see Definition \ref{def-Strong_sol}, and Appendix \ref{Lem-EE} below. A prototype example is $A = -\Delta$, for which we associate the following energy functional \begin{equation}\label{energ1} {\mathcal E}(u,v) := {\mathcal E}_{\tt kin}(v) + {\mathcal E}_{\tt ela}(u) = \frac{1}{2} \int_{\mathcal O} \vert v(x)\vert^2 {\rm d} x + \frac{1}{2} \int_{\mathcal O} \vert \nabla u(x)\vert^2 {\rm d} x\,, \end{equation} where the first term represents the kinetic energy, and the second the elastic energy of the propagating wave with pointwise elongation $u: [0,T] \times \overline{\mathcal O} \times \Omega \rightarrow {\mathbb R}$. --- We begin the further discussion with an example to motivate the effect of noise. \begin{example}\label{noise-effect} Let $\mathcal{O} = (0, 1),$ $T=1,$ $A = -\Delta$, $F\equiv0$ in \eqref{stoch-wave1:1}, and $W$ be of the form \eqref{w-form}, with $M =3$, and $e_j(x) = \sqrt{2} \sin(j\pi x)$. --- The first line in Fig.~\ref{energy} displays single trajectories of $u$ for different $\sigma \equiv \sigma(u, v).$ For $\sigma \equiv 0$ both, the amplitude and wavelength remain constant over time in snapshot {\rm (A)}, as does $\mathcal{E}(u, v)$ in {\rm ({D})}. For $\sigma(u, v)=\frac{1}{2}u$, the amplitude of a single wave realization in snapshot {\rm (B)} changes --- as do the trajectory-wise energy parts in {\rm (E)} ---, while the wavelength remains constant over time. The computation of the (approximate) total energy uses ${\tt MC}=10^3$ Monte-Carlo simulations in snapshot {\rm (G)}: it is conserved, and close to {\rm (D)}. For $\sigma(u, v)=\frac{1}{2}v$ both, the wavelength and frequency of a single trajectory are heavily affected, see snapshot {\rm (C)}, and {\rm (F)}, where only $t \mapsto \mathcal{E}_{\tt ela}(u(t, \omega))$ is smooth. In contrast, the dynamics of ${\mathbb E}_{\tt MC}[{\mathcal E}(u,v)]$ in {\rm (H)} recovers the exchange of elastic and kinetic energy parts, but the total energy is not conserved any more. The proper resolution of snapshot {\rm (H)} required $5$ times more Monte-Carlo simulations than for {\rm (G)}. \del{ The components of the energy of a wave are kinetic and potential, which depends on the amplitude and the frequency of the wave. The total mechanical energy of the wave is the sum of its potential energy and kinetic energy, which is defined as \[\mathcal{E}(u, v):= \frac{1}{2} \int_0^1 (\nabla u)^2 \,dx + \frac{1}{2} \int_0^1 (v)^2 \,dx.\] Time-changes of energy profiles are depicted in Figure \ref{energy}. The three different panels of this figure display the evolution of the above mentioned energies for different functions $\sigma=\sigma(u,v)$. } \begin{figure}[htbp!] \centering \subfloat[$\sigma(u, v)=0$]{\includegraphics[width=0.33\textwidth]{s0-sol}\label{fig:f4}} \hfill \subfloat[$\sigma(u, v)= \frac{1}{2} u$]{\includegraphics[width=0.33\textwidth]{su-sol}\label{fig:f5}} \hfill \subfloat[$\sigma(u, v)= \frac{1}{2} v$]{\includegraphics[width=0.33\textwidth]{sv-sol}\label{fig:f6}} \par \subfloat[$\sigma(u, v)=0$]{\includegraphics[width=0.33\textwidth]{s0-ene}\label{f-f1}} \hfill \subfloat[$\sigma(u, v)= \frac{1}{2} u$]{\includegraphics[width=0.33\textwidth]{su-ene}\label{f-f2}} \hfill \subfloat[$\sigma(u, v)= \frac{1}{2} v, \beta=\frac{1}{4}$]{\includegraphics[width=0.33\textwidth]{sigma_v_alpha_1by4}\label{f-f3}} \par \subfloat[$\sigma(u, v)= \frac{1}{2} u,$ with ${\tt MC}=10^3$]{\includegraphics[width=0.33\textwidth]{su-ene-ave}\label{f-f4}} \subfloat[$\sigma(u, v)=\frac{1}{2} v,$ with ${\tt MC}=5 \times 10^3$]{\includegraphics[width=0.33\textwidth]{sv-ene-ave}\label{f-f5}} \caption{{\bf (Example \ref{noise-effect})} (1st line) Single trajectory of $u$ from \eqref{stoch-wave1:1}, simulated via $(\widehat{\alpha}, \beta)-$scheme ($\widehat{\alpha} = 1$). (2nd line) Corresponding elastic ($\mathcal{E}_{\tt ela}$), kinetic ($\mathcal{E}_{\tt kin}$), total energy ($\mathcal{E}$), for mesh sizes $h=2^{-7}$ and $k=2^{-10}$. (3rd line) Plots $t \mapsto \mathbb{E}_{\tt MC} [\mathcal{E}(u(t), v(t))]$, with ${\tt MC}=10^3$ in snapshot (G) and ${\tt MC}=5 \times 10^3$ in (H).} \label{energy} \end{figure} \end{example} The first works to numerically solve (\ref{stoch-wave1:1}) are \cite{Walsh_2006} and \cite{Sar+San_2006}, where (semi-)discrete schemes were constructed based on the solution concept of a mild solution for (\ref{stoch-wave1:1}): in \cite{Walsh_2006}, which considered ${\mathcal O}= {\mathbb R}$, $A=-\Delta$, Lipschitz nonlinearities $F \equiv F(u)$ and $\sigma \equiv \sigma(u)$, and white noise, a strong convergence rate ${\mathcal O}(k^{1/2})$ was shown for an explicit finite difference scheme, where the temporal step size $k$ is equal to the mesh size $h$ of the Cartesian spatial mesh; the error analysis uses the Green's function, which is explicitly known in this case, and hence used the mild solution concept for this Cauchy problem. A further development in this direction is \cite{CQ_2016}, where ${\mathcal O}=(0,1), A= -\Delta$ in (\ref{stoch-wave1:1}), and the authors used the explicit representation of (discrete) Green's function, such that its implementation crucially hinges on the availability of eigenvalues and eigenfunctions of the Laplacian; see also \cite[Sec.~5.3]{Chow_2015}, and \cite{Hochbruck_2010}. The stable scheme then allows independent choices of $k$ and $h$, and the proof of \cite[Thm.~4.1]{CQ_2016} provides convergence rates both, in terms of spatial and temporal discretization. We also mention \cite{CLS_2013}, where ${\mathcal O}$ is a bounded convex domain with polygonal boundary, and $A= -\Delta$ in (\ref{stoch-wave1:1}); the space-time discretization was proposed with the explicit knowledge of the related (discrete) semigroup, whose efficient implementation again hinges on the knowledge of the related eigenvalues and eigenfunctions. Later, in \cite{ACLW_2016} the authors addressed the multiplicative noise case with $\sigma \equiv \sigma(u)$, where $\sigma$ and also the nonlinearity $F \equiv F(u)$ were assumed to be zero on the boundary. The above mentioned works did not address the case when $F \equiv F(u, v)$ and $\sigma \equiv \sigma(u, v)$. In engineering applications for elastic and acoustic wave propagations which may be described by (\ref{stoch-wave1:1}), the considered domains ${\mathcal O} \subset {\mathbb R}^d$ are typically complicated, and/or the propagating medium is heterogeneous, with layers, anisotropies, cavities ({\em e.g.} in seismology, or material testing), or may even be random. Moreover, models of type (\ref{stoch-wave1:1}) often require non-constant and non-self-adjoint operators, such as those in (\ref{stoch-wave1:1b}), which may even have random coefficients. Therefore, such engineering problems often exclude the efficient use of semigroup based methods through spectral theory as discussed above. This motivates us to aim for the following goals in this work: \begin{itemize} \item[1)] Use an implicit method in time (below referred to as $(\widehat{\alpha}, \beta)-$method, where $\widehat{\alpha}=0$; see Scheme \ref{Hat-scheme}) to approximate \eqref{stoch-wave1:1} with $F \equiv F(u, \partial_t u)$ and $\sigma \equiv \sigma(u, \partial_t u)$, and employ variational methods for its error analysis. This part is motivated by \cite{Dup_1973} for the deterministic linear wave equation, {\em i.e.}, $F \equiv \sigma \equiv 0$. For {finite dimensional noise} of type \eqref{w-form}, we use energy arguments to obtain ${\mathcal O}(k)$ for the temporal error --- which coincides with the order obtained in \cite[Thm.~4]{ACLW_2016} and \cite[Thm.~4.1]{CQ_2016} for an exponential integrator, in the case $\sigma \equiv \sigma(u)$, $F \equiv F(u)$ and trace-class noise in (\ref{stoch-wave1:1}). We obtain ${\mathcal O}(k^{\frac 12})$ for the temporal error in the general case $\sigma \equiv \sigma(u, v)$ and $F \equiv F(u, v)$, which has not been addressed in the existing literature. \item[2)] For $\sigma \equiv \sigma(u)$ and $F \equiv F(u)$, in fact, we improve the $(\widehat{\alpha}, \beta)-$method to a higher-order method which yields improved convergence order ${\mathcal O}(k^{3/2})$ for approximates of $u$ in ${\mathbb L}^2$; see Theorem \ref{lem:scheme1:con}. The additional term that arises for $\widehat{\alpha} = 1$ is {\em motivated} by It\^o's formula, and uses increments \begin{equation}\label{w-tilde} \widetilde{\Delta_n W}:= \int_{t_n}^{t_{n+1}} (s-t_n) \,\mathrm{d} W(s) = \int_{t_n}^{t_{n+1}} s \, {\mathrm d}W(s) - t_n \Delta_n W\, . \end{equation} \del{ We show that this term approximates the explicit increment $\widehat{\Delta_n W} := k \Delta_n W$ in such a way that the overall order ${\mathcal O}(k^{3/2})$ is not affected. } \item[3)] Computational experiments in Section~\ref{sec-6} show that these results are sharp {\em w.r.t.} the used noise, {\em i.e.}, there are examples for $\sigma \equiv \sigma(u,v)$ where the error converges only in order ${\mathcal O}(k)$ --- rather than ${\mathcal O}(k^{3/2})$ in the case $\sigma \equiv \sigma(u)$. \end{itemize} In this work, we focus on proper time discretizations for (\ref{stoch-wave1:1}), which we consider to be the essential part of an overall discretization, and leave a related finite element error analysis for future work. The results will be derived for (\ref{stoch-wave1:1}) with $A = -\Delta$ to simplify the technical setup, but easily generalize to $A$ in (\ref{stoch-wave1:1b}), even with random coefficients there. Moreover, the $(\widehat{\alpha}, \beta)-$method is neither a spectral Galerkin method nor does its implementation hinge on related semigroups. \smallskip While being inspired by the second order time-stepping scheme of \cite{Dup_1973} for the deterministic wave equation, where $u^{n, \frac 12} := \frac{1}{2} (u^{n+1} + u^{n-1})$, we propose the following scheme for (\ref{stoch-wave1:1}): \begin{scheme}\label{scheme--2} {\bf ($(\widetilde{\alpha},\beta)-$scheme)} Fix $\widetilde{\alpha} \in \{0,1\}$ and $\beta \in [0, 1/2)$. Let $\{ t_n\}_{n=0}^N$ be a mesh of size $k>0$ covering $[0, T],$ and $\{ (u^n, v^n)_{n=0, 1}\}$ be given $\mathcal{F}_{t_n}$-measurable, $[{\mathbb H}^1_0]^2$-valued r.v's. For every $n \ge 1$, find $[{\mathbb H}^1_0]^2$-valued, ${\mathcal F}_{t_{n+1}}$-measurable r.v's~$(u^{n+1}, v^{n+1})$ such that ${\mathbb P}$-a.s. \begin{eqnarray} \label{scheme2--1} \quad &&(u^{n+1}-u^{n}, \phi) = k(v^{n+1},\phi) \quad \forall \phi \in \mathbb{L}^2\, , \\ \nonumber \quad &&(v^{n+1}-v^{n}, \psi) = - k \big(\nabla \widetilde{u}^{n,\frac 12}, \nabla\psi \big) + \Bigl(\sigma(u^{n}, v^{n- \frac 12})\, \Delta_n W, \psi\Bigr) \\ \label{scheme2--2} \quad &&\qquad\qquad\qquad\quad+ \widetilde{\alpha} \,\Bigl( D_u\sigma(u^{n}, v^{n-\frac 12})\, v^{n}\, \widetilde{\Delta_n W}, \psi\Bigr) \\ \nonumber \quad &&\qquad\qquad\qquad\quad+ \frac{k}{2}\Bigl(3F(u^n, v^n) - F(u^{n-1}, v^{n-1}), \psi\Bigr) \quad \forall \psi \in \mathbb{H}^1_0\, , \end{eqnarray} where \begin{align}\label{tilde1/2} \widetilde{u}^{n,\frac 12} \equiv {{\widetilde{u}^{n,\frac 12}_\beta}} := \frac{1+ \beta\,k^{\beta}}{2} u^{n+1} + \frac{1- \beta\,k^{\beta}}{2} u^{n-1}\,, \end{align} and \begin{align*} \Delta_{n} W := W(t_{n+1}) - W(t_n) \ \ \mbox{ and } \ \ v^{n- \frac 12} := \frac 12 (v^n + v^{n-1})\, . \end{align*} \end{scheme} Note that $\widetilde{u}^{n,\frac 12} = u^{n,\frac 12}$ for $\beta=0$. Also, in the case when $F \equiv F(u)$, $\sigma \equiv \sigma(u)$ and $\beta=0$, the $(\widetilde{\alpha},\beta)-$scheme simplifies to (for $n \geq 1$) \begin{eqnarray} \label{scheme2:1} \quad &&(u^{n+1}-u^{n}, \phi) = k(v^{n+1},\phi) \quad \forall \phi \in \mathbb{L}^2\, , \\ \quad &&(v^{n+1}-v^{n}, \psi) = - k \big(\nabla u^{n, \frac 12}, \nabla\psi \big) + \Bigl(\sigma(u^{n}) \Delta_n W, \psi\Bigr)+ \widetilde{\alpha} \,\Bigl(D_u\sigma(u^{n}) v^{n} \widetilde{\Delta_n W}, \psi\Bigr) \label{scheme2:2}\\ \quad &&\qquad\qquad\qquad\quad+ \frac{k}{2}\Bigl(3F(u^n) - F(u^{n-1}), \psi\Bigr) \quad \forall \psi \in \mathbb{H}^1_0\, \nonumber . \end{eqnarray} Scheme~\ref{scheme--2} involves the increment $\widetilde{\Delta_n W}$ from \eqref{w-tilde}. For their implementation, we approximate it by {\em $\widehat{\Delta_n W}$} defined as \begin{align}\label{W-hat} \widehat{\Delta_n W} := k W(t_{n+1}) - k^2 \sum_{\ell=1}^{k^{-1}} W(t_{n,\ell})\,, \end{align} where $\big\{ W(t_{n, \ell}) \big\}_{\ell = 1}^{k^{-1}}$ is the piecewise affine approximation of $W$ on $[t_{n},t_{n+1}]$ on an equidistant mesh $\{ t_{n,\ell}\}_{\ell=1}^{k^{-1}},$ of size $k^2:= t_{n, \ell+1} - t_{n, \ell}$. To motivate this approximation, we first use It\^o's formula to restate $\widetilde{\Delta_n W}$ as \begin{equation}\label{DnW-tilde} \begin{split} \widetilde{\Delta_n W} &= \Bigl(t_{n+1} W(t_{n+1}) - t_{n} W(t_{n})\Bigr) - \int_{t_n}^{t_{n+1}} W(s)\, {\mathrm d}s - t_n \Delta_n W \\ &= \int_{t_n}^{t_{n+1}} \bigl[W(t_{n+1}) - W(s)\bigr]\, {\rm d}s = k W(t_{n+1}) - \int_{t_n}^{t_{n+1}} W(s)\, {\rm d}s \end{split} \end{equation} and we approximate the last integral on the right-hand side of \eqref{DnW-tilde} by $k^2 \sum_{\ell=1}^{k^{-1}} W(t_{n,\ell})$. Thus, we have the following implementable scheme: \begin{scheme}\label{Hat-scheme} {\bf ($(\widehat{\alpha}, \beta)-$scheme)} Consider Scheme \ref{scheme--2}. We refer to \eqref{scheme2--1}--\eqref{scheme2--2} as $(\widehat{\alpha}, \beta)-$scheme, when $\widetilde{\alpha}$ and $\widetilde{\Delta_n W}$ are replaced by $\widehat{\alpha}$ and $\widehat{\Delta_n W}$, respectively. \end{scheme} \medskip The following example motivates that the convergence rate for the $(1, 0)-$scheme is boosted from ${\mathcal O}(k)$ to ${\mathcal O}(k^{3/2})$, in case $\sigma \equiv \sigma(u)$ and $F \equiv F(u)$. \begin{example}\label{exm-alp0} Let $\mathcal{O}=(0, 1)$, $T=1$, $A = -\Delta$, $F \equiv 0$, $\sigma(u)=2 \sin(u)$ in \eqref{stoch-wave1:1a}. Let $$u_0(x) = \sin(2 \pi x) \qquad \mbox{and} \qquad v_0(x) = \sin(3 \pi x)\,,$$ and $W$ as in Example \ref{noise-effect}. Fig. \ref{5.1} displays convergence studies for the scheme \eqref{scheme2:1}--\eqref{scheme2:2}$:$ for $\widehat{\alpha} = 0$, the plots {\rm (A) -- (C)} show $\mathbb{L}^2$-errors in $u$, $\nabla u$, evidencing convergence order ${\mathcal O}(k)$, and those for $v$ evidence convergence order ${\mathcal O}(k^{1/2})$. For $\widehat{\alpha} = 1$, the convergence order improves to ${\mathcal O}(k^{3/2})$ for $u$, $\nabla u$, and ${\mathcal O}(k)$ for $v$; see plots {\rm (D) -- (F)}. See Section \ref{sec-6} for more details. \begin{figure}[h!] \centering \subfloat[$\mathbb{L}^2$-error for $u$ ($\widehat{\alpha}=0$ case)]{\includegraphics[width=0.33\textwidth]{ou-er-su-1k}\label{fig:f1}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$ ($\widehat{\alpha}=0$ case)]{\includegraphics[width=0.33\textwidth]{ogu-er-su-1k}\label{fig:f2}} \hfill \subfloat[$\mathbb{L}^2$-error for $v$ ($\widehat{\alpha}=0$ case)]{\includegraphics[width=0.33\textwidth]{ov-er-su-1k}}\label{fig:f3} \par \subfloat[$\mathbb{L}^2$-error for $u$ ($\widehat{\alpha}=1$ case)]{\includegraphics[width=0.33\textwidth]{u-err-su-MC1k}\label{fig4}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$ ($\widehat{\alpha}=1$ case)]{\includegraphics[width=0.33\textwidth]{gu-err-su-MC1k}\label{fig5}} \hfill \subfloat[$\mathbb{L}^2$-error for $v$ ($\widehat{\alpha}=1$ case)]{\includegraphics[width=0.33\textwidth]{v-err-su-MC1k}\label{fig6}} \caption{{\bf (Example \ref{exm-alp0})} Temporal rates of convergence for the scheme \eqref{scheme2:1}--\eqref{scheme2:2} with $F\equiv 0$ and $\sigma(u) = 2 \sin(u)$; $\widehat{\alpha}=0$ in {\rm (A)}, {\rm (B)}, {\rm (C)}, and $\widehat{\alpha}=1$ in {\rm (D)}, {\rm (E)}, {\rm (F)}; discretization parameters: $h= 2^{-7}, k=\{2^{-3}, \cdots, 2^{-6}\}$, ${\tt MC}=3000$.} \label{5.1} \end{figure} \end{example} \medskip The rest of the paper is organized as follows. In Section \ref{Nota+Prel}, we precise the data requirements in \eqref{stoch-wave1:1a} with $A = -\Delta$ and provide the structure assumptions on $F$ and $\sigma$. In Section~\ref{Mat-fra}, we recall the concept of a strong variational solution for the problem \eqref{stoch-wave1:1a} and discuss its regularity. In Section~\ref{sec-3}, we prove stability results for the $(\widehat{\alpha}, \beta)-$scheme. In Section~\ref{sec-4}, we prove strong rates of convergence for the above mentioned schemes. In Section~\ref{sec-6}, we present comparative computational studies which evidence the role of noise in various cases and validates the proved error estimate results. \bigskip \section{Preliminaries and Assumptions}\label{Nota+Prel} \subsection{Notation and useful results} Let $(L^p(\mathcal{O}), \|\cdot\|_{L^p})$ and $(W^{m, p}(\mathcal{O}), \|\cdot\|_{W^{m, p}})$ be the Lebesgue and Sobolev spaces respectively, endowed with usual norms for $m \in \mathbb{N}$ and $1 \leq p \leq \infty$. We denote ${\mathbb L}^p := L^p(\mathcal{O})$ and ${\mathbb W}^{m,p} := W^{m, p}(\mathcal{O})$. For $p=2$, let $(\cdot, \cdot)$ be the inner-product in ${\mathbb L}^2$, and ${\mathbb H}^m := {\mathbb W}^{m, 2}$. We define ${\mathbb H}^1_0 := \{u\in {\mathbb H}^1: u |_{\partial \mathcal{O}} = 0\}$. Let $\mathbb{X}, \mathbb{Y}$ be two separable Hilbert spaces. Let $\mathcal{L}(\mathbb{X}, \mathbb{Y})$ denote the space of all linear maps from $\mathbb{X}$ to $\mathbb{Y}$, and $\mathcal{L}_m(\mathbb{X}, \mathbb{Y})$ denotes the space of all multi-linear maps from $\mathbb{X} \times \cdots \times \mathbb{X}$ ($m$-times) to $\mathbb{Y}$ for $m \geq 2$. Throughout this paper, for some $\Phi: \mathbb{H}^1_0 \times \mathbb{H}^1_0 \to \mathbb{L}^2$, we use the notation $D_u \Phi(u, v) \in \mathcal{L}(\mathbb{H}^1_0, \mathbb{L}^2)$ for the Gateaux derivative w.r.t $u$, whose action is seen as \[h \mapsto D_u \Phi(u, v)(h), \quad \mbox{for } h \in \mathbb{H}^1_0\, .\] We denote the second derivative w.r.t. $u$ by $D_u^2 \Phi(u, v) \in \mathcal{L}_2(\mathbb{H}^1_0, \mathbb{L}^2)$, whose action can be seen as \[(h, k) \mapsto D_u^2 \Phi(u, v)(h, k) := (D_u^2 \Phi(u, v) h)(k) \quad \mbox{for } (h, k) \in [\mathbb{H}^1_0]^2\, .\] Similarly, we define $D_v \Phi(u, v), D_v^2 \Phi(u, v)$. \smallskip \subsubsection{A quadrature formula} \del{ In our paper, we use an important trapezoid inequality in order to find a relation between the It\^o integral $\widetilde{\Delta_n W}$ and $\widehat{\Delta_n W}.$ In \cite{Dragomir_2011}, we have the generalisation of the classical trapezoid inequality for integrators of bounded variation and H\"older continuous integrands. We state the result here. \begin{lemma}\label{Dra-thm} Let $f:[a, b] \to \mathbb{C}$ be a $p-\eta$-H\"older type function, that is, it satisfies the condition \[|f(x) - f(y)| \leq \eta\,|x-y|^p, \quad \forall x, y\in [a, b],\] where $\eta>0$ and $p\in (0, 1]$ are given, and $z:[a, b] \to \mathbb{C}$ is a function of bounded variation on $[a, b]$. Then we have the inequality: \begin{align} \Big| \frac{f(a)+f(b)}{2} \cdot [z(a)-z(b)] - \int_a^b f(t)\,dz(t)\Big| \leq \frac{1}{2^{p}} \,\eta\,(b-a)^p \,TV_{[a,b]}(z)\, , \end{align} where $TV_{[a,b]}(z)$ denotes the total variation of $z$ on the interval $[a, b].$ \end{lemma} } \noindent The following quadrature formula will be crucially used in our error analysis (see \cite[Thm. 2]{DM_2000}). \begin{lemma}\label{quadrature1} Let $f \in {C}^{1,\gamma}([0,T]; {\mathbb R})$, for some $\gamma \in (0,1]$. Then there holds \begin{align*} \Bigl|\frac{f(0) + f(T)}{2} -\frac{1}{T}\int_0^T f(\xi) \, {\rm d} \xi\Bigr| \le \frac{\widetilde{C}}{(\gamma+2)(\gamma + 3)} T^{1+\gamma}, \end{align*} where $\widetilde{C}>0$ satisfies \begin{equation}\label{cond1} \bigl|f'(t)-f'(s)\bigr| \le \widetilde{C} |t-s|^{\gamma} \qquad \forall \, s,t \in [0,T]\, . \end{equation} \end{lemma} \smallskip \subsection{Assumptions} In this section, we list all the assumptions and hypotheses that are imposed throughout this paper. \subsubsection{Domain and initial data} We make the following assumptions.\\ \noindent {\bf (A1)} \label{Domain} Let $\mathcal{O}\subset {\mathbb R}^d,$ for $1 \leq d\leq 3$, be a bounded domain \begin{itemize} \item[$(i)$] with $\partial \mathcal{O}$ of class $\mathrm{C}^1,$ and $(u_0, v_0) \in \mathbb{H}^1_0 \times \mathbb{L}^2\, ,$ \item[$(ii)$] with $\partial \mathcal{O}$ of class $\mathrm{C}^2,$ and $(u_0, v_0) \in ( \mathbb{H}^1_0 \cap \mathbb{H}^2) \times \mathbb{H}^1_0\, ,$ \item[$(iii)$] with $\partial \mathcal{O}$ of class $\mathrm{C}^3,$ and $(u_0, v_0) \in (\mathbb{H}^1_0 \cap \mathbb{H}^3) \times (\mathbb{H}^1_0 \cap \mathbb{H}^2)\, .$ \item[$(iv)$] with $\partial \mathcal{O}$ of class $\mathrm{C}^4,$ and $(u_0, v_0) \in (\mathbb{H}^1_0 \cap \mathbb{H}^4) \times (\mathbb{H}^1_0 \cap \mathbb{H}^3)\, .$ \end{itemize} \subsubsection{Probability set-up} For simplicity, let $W$ be a finite-dimensional Wiener process.\\ \noindent {\bf (A2)} \label{pro-space} Let $\mathfrak{P}:=\big( \Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geq 0}, \mathbb{P}\big)$ be a stochastic basis with a complete filtration $\{\mathcal{F}_t\}_{t \geq 0} \subseteq \mathcal{F}$. For some $M \in \mathbb{N}$, let $W$ be a $\mathbb{K}$-valued Wiener process on $\mathfrak{P}$ of the form \begin{align}\label{w-form} W(t, x, \omega) := \sum_{j = 1}^M \beta_j(t, \omega) e_j(x)\,, \end{align} where $\mathbb{K} \subseteq \mathbb{H}^1_0 \cap \mathbb{H}^3$ is a Hilbert space, and $\big\{\beta_j(t, \omega) ; t \geq 0 \big\}$ are mutually independent Brownian motions relative to $\{\mathcal{F}_t\}_{t \geq 0}$, and $\{e_j\}_{j=1}^M$ be an orthonormal basis of $\mathbb{K}$. \del{ Note that the series does not converge in $\mathbb{L}^2,$ but in a larger Hilbert space $U$ such that $\mathbb{L}^2 \hookrightarrow U$ is Hilbert-Schmidt. For all $s, t\in [0, T]$ and $g, h \in \mathbb{L}^2$ we have \[\mathbb{E} \big[ (W(s)g \cdot W(t)h) \big] = (s \wedge t) (g, h)_{\mathbb{L}^2}\,.\] This means that the covariance operator of $W$ is $\mbox{Id}: \mathbb{L}^2 \to \mathbb{L}^2$. We shall always assume that $W(t)$ is adapted to the given filtration $(\mathcal{F}_t)_{t \geq 0}$. For a given linear operator $\Phi$ from $\mathbb{L}^2$ to some Hilbert space $\mathbb{K},$ the Wiener process $\Phi\,{\rm d} W = \sum_{k \in \mathbb{N}} \beta_k \Phi \,e_k$ is well-defined in $\mathbb{K}$ provided we have $\Phi \in \mathcal{L}_2(\mathbb{L}^2, \mathbb{K}).$ } \subsubsection{The nonlinearity of the model} Let $F: [{\mathbb H}^1_0]^2 \rightarrow \mathbb{L}^2$ and $\sigma: [{\mathbb H}^1_0]^2 \rightarrow {\mathbb H}^1_0$. \\ \noindent {\bf (A3)} \label{as-Fs} Assume $F(u, v) = F_1(u) + F_2(v)$ and $\sigma(u, v) = \sigma_1(u) + \sigma_2(v)$, such that $F_2(v)$ and $\sigma_2(v)$ are affine in $v$. For any $u, \tilde{u} \in \mathbb{H}^1_0$, there is a constant $C_{\tt L}>0$ such that the Lipschitz condition holds: \begin{align*} &\|F_1(u) - F_1(\tilde{u}) \|_{\mathbb{L}^2} + \|\sigma_1(u) - \sigma_1(\tilde{u}) \|_{\mathbb{L}^2} \leq C_{\tt L} \,\| u - \tilde{u}\|_{\mathbb{L}^2}\,. \end{align*} \smallskip \noindent {\bf (A4)} \label{as-bd-Fs} There exists a constant $C_g>0$ such that \begin{align*} &\|D_u^m F_1(\cdot)\|_{L^{\infty}(\mathbb{H}^1_0; \mathcal{L}_m(\mathbb{H}^1_0, \mathbb{L}^2))} + \|D_u^m \sigma_1(\cdot)\|_{L^{\infty}(\mathbb{H}^1_0; \mathcal{L}_m(\mathbb{H}^1_0, \mathbb{H}^1_0))} \le C_g \quad (m=1, 2, 3)\,. \end{align*} By the assumption {\bf (A4)}, we deduce that $F(u, v) \in \mathbb{H}^1$. Since $F(u, v)$ is not assumed to be zero on the boundary, we introduce the following notation. \smallskip \noindent {\bf (A5)} Let $\widehat{F}(u, v) := F(u, v) - F(0, 0) = F_1(u) + F_2(v) - F(0, 0)$, and assume $F(0, 0) \in L^2(0, T; \mathbb{H}^m)$ for $m=1, 2, 3$. \bigskip \section{Definition and properties of solution}\label{Mat-fra} \noindent We recall the concept of a strong variational solution for \eqref{stoch-wave1:1a} with $A = -\Delta$ and establish stability results in higher spatial norms, and bounds in temporal H\"older norms. \begin{definition}\label{def-Strong_sol} Assume {\bf (A1)$_{i}$}, {\bf (A2)} and {\bf (A3)}. We call the tuple $(u, v)$ a strong variational solution of \eqref{stoch-wave1:1a} with $A = -\Delta$ on the interval $[0, T]$ if \begin{itemize} \item[$(i)$] \ $(u, v) \ \mbox{is an} \ \mathbb{H}^1_0 \times \mathbb{L}^2\mbox{-valued}, \{\mathcal{F}_t \}\mbox{-adapted process};$ \item[$(ii)$] \ $(u, v) \in L^2\big(\Omega; C([0, T]; \mathbb{H}^1_0)\big) \times L^2\big(\Omega; C([0, T]; \mathbb{L}^2)\big);$ and {\small{ \begin{align}\label{strong1} (u(t),\phi) &= \int_0^t (v ,\phi)\, {\rm d}s + (u_0,\phi)\, \quad \forall \phi \in \mathbb{L}^2\, ,\\ (v(t),\psi) &= -\int_0^t \Bigl[(\nabla u , \nabla \psi) +\bigl(F(u,v), \psi \bigr)\Bigr] {\rm d}s +\int_0^t \bigl(\psi,\sigma (u,v)\, {\rm d}W(s) \bigr) + (v_0,\psi)\, \quad \forall \psi \in \mathbb{H}^1_0\, , \label{strong2} \end{align} }} holds for each $t \in [0, T]$ $\mathbb{P}$-a.s.. \item[$(iii)$] There exists a constant $C>0$, depending on $T, C_{\tt L}$ and initial data such that there holds $\mathbb P$-a.s. \begin{align*} \mathbb{E}\Bigl[\mathop {\sup} \limits_{0 \le t \le T} \mathcal{E}(u(t), v(t)) \Bigr] \le C\, . \end{align*} \end{itemize} \end{definition} \noindent The existence of a unique strong variational solution was shown in \cite[Sec. 6.8, Thm. 8.4]{Chow_2015}. \smallskip \begin{lemma}\label{lem:L2} Let $(u, v)$ be the strong (variational) solution to the problem \eqref{strong1}--\eqref{strong2}. For $p \in \mathbb{N}$, there holds $\mathbb P$-a.s. \begin{itemize} \item[$(i)$] under the hypotheses {\bf (A1)$_{i}$}, {\bf (A2)}, and {\bf (A3)}, the $\{\mathcal{F}_t \}_{t \geq 0}-$adapted process \\$(u, v) \in L^{2p} \big(\Omega; L^{\infty}(0, T; \mathbb{H}^{1} \times \mathbb{L}^{2}) \big)$, and there exists $K_1 \equiv K_1(p)>0$, such that \begin{align} \label{lem:L2:1} &\mathbb{E}\Bigl[\mathop {\sup} \limits_{0 \le t \le T} \Big( \|u(t)\|_{{\mathbb H}^1}^{2p} + \|v(t)\|_{{\mathbb L}^2}^{2p} \Big) \Bigr] \le K_1\, ; \end{align} \item[$(ii)$] under the hypotheses {\bf (A1)$_{ii}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1$, the $\{\mathcal{F}_t \}_{t \geq 0}-$adapted process $(u, v) \in L^{2p} \big(\Omega; L^{\infty}(0, T; \mathbb{H}^{2} \times \mathbb{H}^{1}) \big)$, and there exists $K_2 \equiv K_2(p)>0$, such that \begin{align} \label{lem:H1:1} &\mathbb{E} \Bigl[\mathop {\sup} \limits_{0 \le t \le T} \Big( \| u(t)\|_{\mathbb{H}^2}^{2p} + \| v(t)\|_{\mathbb{H}^1}^{2p} \Bigr) \Bigr] \le K_2\, ; \end{align} \item[$(iii)$] under the hypotheses {\bf (A1)$_{iii}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1, 2$, the $\{\mathcal{F}_t \}_{t \geq 0}-$ adapted process $(u, v) \in L^{2p} \big(\Omega; L^{\infty}(0, T; \mathbb{H}^{3} \times \mathbb{H}^{2}) \big)$, and there exists $K_3 \equiv K_3(p)>0$, such that \begin{align} \label{lem:H2:1} &\mathbb{E} \Bigl[ \mathop {\sup} \limits_{0 \le t \le T} \Big( \|u(t)\|_{\mathbb{H}^3}^{2p} + \|v(t)\|_{\mathbb{H}^2}^{2p} \Big) \Bigr] \le K_3\, ; \end{align} \item[$(iv)$] under the hypotheses {\bf (A1)$_{iv}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1, 2, 3$, the $\{\mathcal{F}_t \}_{t \geq 0}-$ adapted process $(u, v) \in L^{2p} \big(\Omega; L^{\infty}(0, T; \mathbb{H}^{4} \times \mathbb{H}^{3}) \big)$, and there exists $K_4 \equiv K_4(p)>0$, such that \begin{align} \label{lem:H3:1} &\mathbb{E} \Bigl[ \mathop {\sup} \limits_{0 \le t \le T} \Big( \|u(t)\|_{\mathbb{H}^4}^{2p} + \|v(t)\|_{\mathbb{H}^3}^{2p} \Big) \Bigr] \le K_4\, . \end{align} \end{itemize} \end{lemma} \begin{proof} The proof is given in Appendix \ref{Lem-EE}. \end{proof} \subsection{H\"older continuity in time}\label{Hoe-cont} In this subsection, we derive temporal H\"older continuity estimates for the solution pair $(u, v)$ of the problem \eqref{strong1}--\eqref{strong2} with respect to different norms, which will be useful in the error analysis in later section. \begin{lemma} \label{lem:Holder} Let $(u, v)$ be the strong (variational) solution to the problem \eqref{strong1}--\eqref{strong2}. Then, for any $s,t \in [0,T]$, we have for $p \ge 1$ \begin{itemize \item[$(i)$] under the hypotheses {\bf (A1)$_{i}$}, {\bf (A2)}, and {\bf (A3)}, there holds $\mathbb P$-a.s. \[\left( \mathbb{E} \Bigl[ \sup_{s \leq r \leq t} \|u(r)-u(s)\|_{{\mathbb L}^2}^{2p} \Bigr] \right)^{1/2p} \le C(K_1)\, |t-s| \, ;\] \item[$(ii)$] under the hypotheses {\bf (A1)$_{ii}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1$, there holds $\mathbb P$-a.s. \[ \left( \mathbb{E} \Bigl[ \sup_{s \leq r \leq t} \|u(r)- u(s)\|_{{\mathbb H}^1}^{2p} \Bigr] \right)^{1/2p} + \mathbb{E} \Bigl[ \sup_{s \leq r \leq t} \|v(r)-v(s)\|_{{\mathbb L}^2}^2 \Bigr] \le C(K_2)\, |t-s|\, ;\] \item[$(iii)$] under the hypotheses {\bf (A1)$_{iii}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1, 2$, there holds $\mathbb P$-a.s. \[ \left( \mathbb{E} \Bigl[ \sup_{s \leq r \leq t} \| u(r)- u(s) \|_{{\mathbb H}^2}^{2p} \Bigr] \right)^{1/2p} + \mathbb{E} \Bigl[ \sup_{s \leq r \leq t} \| v(r)- v(s)\|_{{\mathbb H}^1}^2 \Bigr] \le C(K_3)\, |t-s| \, ,\] \item[$(iv)$] under the hypotheses {\bf (A1)$_{iv}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1, 2, 3$, there holds $\mathbb P$-a.s. \[\left( \mathbb{E} \Bigl[ \sup_{s \leq r \leq t} \|u(r)- u(s)\|_{{\mathbb H}^3}^{2p} \Bigr] \right)^{1/2p} +\mathbb{E} \Bigl[ \sup_{s \leq r \leq t} \|v(r)-v(s)\|_{{\mathbb H}^2}^2 \Bigr] \le C(K_4)\, |t-s|\, ,\] \end{itemize} where the positive constants $C(K_i)$ for $i=1, \cdots, 4$, depend on the constants $K_i$, defined in Lemma \ref{lem:L2}. \end{lemma} \begin{proof} The proof is given in Appendix \ref{Hoe-con}. \end{proof} \bigskip \section{Discrete Stability Analysis for the $(\widehat{\alpha}, \beta)-$scheme}\label{sec-3} If compared to the term $-\Delta {u}^{n,\frac{1}{2}}$, the term $-\Delta\widetilde{u}^{n,\frac{1}{2}}$ in the $(\widehat{\alpha}, \beta)-$scheme fortifies stability properties of the method: in fact, the identity \begin{equation}\label{tidt1} \widetilde{u}^{n, \frac 12} = {u}^{n, \frac 12} + \beta \frac{k^{\beta}}{2} \big(u^{n+1} - u^{n-1} \big) = {u}^{n, \frac 12} + \beta k^{1+\beta} v^{n+\frac{1}{2}} \end{equation} creates an additional `numerical dissipation' term scaled by $\beta k^{2+\beta}$ in \eqref{stoch-wave1:1a}, which suffices to control general noises $\sigma \equiv \sigma(u,v)$, in case $0 < \beta <\frac{1}{2}$ (see Lemma \ref{lem:scheme1:stab} below); for $\sigma \equiv \sigma(u)$ only, the scheme \eqref{scheme2:1}--\eqref{scheme2:2} yields a stable scheme. \smallskip In this section, we discuss the discrete stability analysis for the $(\widehat{\alpha}, \beta)-$scheme and we make a remark on the stability results of the scheme \eqref{scheme2:1}--\eqref{scheme2:2} as this is a sub-case of the $(\widehat{\alpha}, \beta)-$scheme. We recall \eqref{energ1}, where the energy functional is stated. \smallskip \noindent {\bf (B1)} For the stability results, we need the following assumptions on the iterates $(u^1, v^1)$: \begin{itemize} \item[$(i)$] Along with {\bf (A1)}$_{i}$, assume $(u^1, v^1) \in L^{2^{p}}\big(\Omega; [\mathbb{H}^1_0 ]^2 \big)$ for $p \geq 1$. \item[$(ii)$] Along with {\bf (A1)}$_{ii}$, assume $(u^1, v^1) \in L^{2^{p}}\big(\Omega; [\mathbb{H}^1_0 \cap \mathbb{H}^2 ]^2 \big)$ for $p \geq 1$. \end{itemize} {{ \begin{lemma}\label{lem:scheme1:stab} Let $\widehat{\alpha} \in \{0, 1\}$. Assume {\bf (A1)$_{ii}$}, {\bf (A2), (A3)}, {\bf (A4)} for $m=1$, and {\bf (B1)$_{i}$}. Then, there exists an $\big[ \mathbb{H}^1_0 \big]^2$-valued $\{ (\mathcal{F}_{t_{n}})_{0 \le n \le N} \}$-adapted solution $\{(u^{n}, v^{n}); \, 0 \leq n \leq N\}$ of the $(\widehat{\alpha}, \beta)-$scheme. Moreover, for {\color{black}{$0 < \beta < \frac{1}{2}$}}, and $k \leq k_0(C_{\tt L}, C_g)$ sufficiently small, there exists a constant ${\color{black}{C_1 \equiv C_1(\beta)}} > 0$ that does not depend on $k>0$ such that \begin{equation}\label{energy1} \max_{1 \le n \le N-1} \mathbb{E} \Bigl[ \mathcal{E}(u^{n+1},v^{n+1}) \Bigr] + {\beta {k^{2+\beta}\, \sum_{n=1}^{N-1} \mathbb{E} \Big[ \| \nabla v^{n+ \frac 12} \|_{\mathbb{L}^2}^2 \Big]}} \le C_1\, . \end{equation} In addition, there exists a further constant $C_{2,p} \equiv C_{2,p}(\beta) >0$ such that we have \begin{equation}\label{high-moment} \max_{1 \le n \le N-1} \mathbb{E}\bigl[ {\mathcal E}^{2^p}(u^{n+1},v^{n+1}) \bigr] \leq C_{2, p} \qquad (p \geq 1)\, . \end{equation} Additionally, assume $\sigma_2(v) \equiv 0 \equiv F_2(v)$ in {\bf (A3)} and $\beta=0$. For $k \leq k_0(C_{\tt L}, C_g)$ sufficiently small, there exists a constant $C_3 > 0$ independent of $k>0$ such that \begin{align}\label{energy2} \max_{1 \le n \le N} \mathbb{E}\Bigl[ \| u^{n}\|^2_{\mathbb{L}^2}\Bigr] + \frac 14 \mathbb{E} \Big[ k \sum_{j=1}^n \bigl\Vert \nabla u^{j} \bigr\Vert^2_{{\mathbb L}^2} \Big] \le C_3\, . \end{align} There exists further constant $C_{4, p}>0$ such that we have \begin{align}\label{energy2-himo} \max_{1 \le n \le N} \mathbb{E}\Bigl[ \| u^{n}\|^{2^{p}}_{\mathbb{L}^2}\Bigr] \leq C_{4, p} \qquad (p \geq 1)\, . \end{align} \end{lemma} \noindent The following remark discusses specific problems to derive this stability result. \begin{remark}\label{rema2} {\bf 1.}~The derivation of (discrete) stability estimates for a (temporal) discretization for the stochastic wave equation \eqref{stoch-wave1:1a} --- like Scheme \ref{scheme--2} --- differs from corresponding tasks for parabolic SPDEs, which is due to the {\em conservation of energy} in the deterministic case. In this case (where $\sigma \equiv 0$), the test function $v^{n+1/2}$ is `natural' to deduce \eqref{energy1}; it is used in \cite{Dup_1973} as well, and exploits the (third) binomial formula, such that the first term in \eqref{scheme2--2} becomes $$\bigl(v^{n+1} - v^{n}, \frac{1}{2} [v^{n+1} + v^{n}] \bigr) = \frac{1}{2} \bigl( \Vert v^{n+1} \Vert^2_{{\mathbb L}^2} - \Vert v^{n}\Vert^2_{{\mathbb L}^2}\bigr)\, ;$$ see also \eqref{S1S:1} below. Conceptional difficulties now appear in the stochastic case where $\sigma \neq 0$ --- see {\em e.g.} the term $\mathscr{J}^n_1$ in \eqref{S1S:1}. A well-known strategy in a setting of parabolic SPDEs would be to employ ${\mathbb E}\big[\Delta_n W \big] = 0$ to conclude \begin{eqnarray}\label{probl1}\mathscr{J}^n_1 &=& {\mathbb E}\Bigl[\Bigl(\sigma (u^n, v^{n- \frac 12}) \,\Delta_nW, \frac{1}{2}[v^{n+1} - v^{n}] \Bigr)\Bigr] \\ \nonumber &\leq& C_{\delta}\, {\mathbb E}\Bigl[\Vert\sigma (u^n, v^{n- \frac 12}) \,\Delta_nW \Vert_{{\mathbb L}^2}^2\Bigr] + \delta \, {\mathbb E}\bigl[ \Vert v^{n+1} - v^n\Vert^2_{{\mathbb L}^2}\bigr]\qquad (\delta >0)\, , \end{eqnarray} and to then absorb the last term by a corresponding one on the left-hand side --- which would arise if $v^{n+1}$ instead would have been chosen as test function. We avoid the estimation in \eqref{probl1} by using the equation \eqref{scheme2--2} to replace $v^{n+1} - v^{n}$ in $\mathscr{J}^n_1$; see \eqref{sig-v-n+1} below. \smallskip \noindent {\bf 2.}~The last term in \eqref{tidt1} is the reason to evaluate $\sigma$ resp. $D_u \sigma$ at $(u^{n},v^{n-\frac 12})$ in \eqref{scheme2--2} --- instead of {\em e.g.}~at $(u^n, v^n)$; see also the left-hand side of \eqref{test-mult}, and the estimation of the terms $\mathscr{J}^{n, 1}_{1,2}$, $\mathscr{J}^{n, 2}_{1,1}$, and $\mathscr{J}^{n, 2}_{1,2}$ in the proof of Lemma \ref{lem:scheme1:stab}. \smallskip \noindent {\bf 3.}~The inequality \eqref{high-moment} assembles higher moment estimates, which will be used in Section \ref{sec-4} to derive improved rates of convergence; see Theorem \ref{lem:scheme1:con}. \smallskip \noindent {\bf 4.}~The following two inequalities will be used frequently in the proof of discrete stability; see parts {\bf 1a)} and {\bf 1b)} below. The identity \eqref{DnW-tilde} leads to \begin{align*} {\mathbb E}\left[ \vert \widetilde{\Delta_n W}\vert^2 \right] \leq k \int_{t_n}^{t_{n+1}} \mathbb{E} \Big[ |W(t_{n+1}) - W(s)|^2 \Big] \,{\rm d} s \leq C k^3\, , \end{align*} and by the identity \eqref{W-hat}, for $q=1, 2$ we infer \begin{align*} {\mathbb E}\left[ \vert \widehat{\Delta_n W}\vert^{2q} \right] &\leq C k^{2q} \,\mathbb{E} \big[ |W(t_{n+1})|^{2q} \big] + C k^{2q+1} \,\sum_{\ell =1}^{k^{-1}} \mathbb{E} \big[ |W(t_{n, \ell})|^{2q} \big] \leq C k^{3q} + C k^{4q} \leq C k^{3q}\, . \end{align*} The estimation of the distance between $\widetilde{\Delta_n W}$ and $\widehat{\Delta_n W}$ is useful in the convergence analysis in the next section, which is derived in \eqref{dist-tilde-hat}. \smallskip \noindent {\bf 5.}~If $\sigma_2(v) \equiv 0 \equiv F_2(v)$ in {\bf (A3)} and $\beta=0$ hold, we consider the scheme \eqref{scheme2:1}--\eqref{scheme2:2}, where to verify discrete stability is easier; in fact, the identity (\ref{S1S:1}) simplifies considerably since both, $F$ and $\sigma$ do not depend on $v$ any more; see {\em e.g.} the estimate for $\mathscr{J}^{n, 1}_{1, 1}$ in \eqref{413a}. For the higher moment estimates, we multiply the corresponding \eqref{frakE} of the scheme \eqref{scheme2:1}--\eqref{scheme2:2} with $\mathfrak{E}(u^{n+1}, v^{n+1})$ only, which is different from the general case; see step {\bf 2)} below. \smallskip \noindent {\bf 6.}~In the proof of \eqref{energy2}, under the hypotheses $\sigma_2(v) \equiv 0 \equiv F_2(v)$ in {\bf (A3)}, $\beta=0$ and {\bf (A4)} for $m=1$, we combine both equations of the scheme \eqref{scheme2:1}--\eqref{scheme2:2} to write a single equation for $u^{\ell}$ and sum over the first $n$ steps; see \eqref{sum_ul}. If we would apply the same approach for the general $\sigma \equiv \sigma(u^{\ell}, v^{\ell - 1/2})$, we could use the first equation of the $(\widehat{\alpha}, \beta)-$scheme to replace $v^{\ell - 1/2}$ by $\frac{1}{2k} \big( u^{\ell}- u^{\ell-2} \big)$. Thus, to estimate \eqref{kln2-sigmau} in general case, we use the growth condition to write \begin{align*} k^2 \sum^n_{\ell=1} \mathbb{E} \Big[\|\sigma(u^{\ell}, (1/2k) ( u^{\ell}- u^{\ell-2}))\|_{\mathbb{L}^2}^2 \Big] \leq C_{\tt L} k^2 \sum^n_{\ell=1} \mathbb{E}\Big[ 1+ \|\nabla u^{\ell} \|_{\mathbb{L}^2}^2 + \frac{1}{4 k^2} \big(\|u^{\ell} \|_{\mathbb{L}^2}^2 + \|u^{\ell-2} \|_{\mathbb{L}^2}^2 \big) \Big]\, , \end{align*} which, by to the last term, is not in suitable form to apply the discrete Gronwall lemma. Thus, the approach to prove \eqref{energy2} is not useful for the general $\sigma \equiv \sigma(u, v)$. \end{remark} \begin{proof}[Proof of Lemma \ref{lem:scheme1:stab}] The ${\mathbb P}$-a.s.~solvability easily follows from Lax-Milgram lemma, and {\bf (A3)}. Using the $\mathbb{L}^2$-regularity theory for elliptic equations on regular domains (see \cite[Sec.~15.5]{Gil+Tru}), the system in Scheme \ref{scheme--2} holds strongly $\mbox{Leb} \otimes \mathbb{P}-$a.s. The proof of Lemma \ref{lem:scheme1:stab} is split into the following three steps {\bf 1)} -- {\bf 3)}. \smallskip \noindent {\bf 1) Proof of $(\ref{energy1})$.} We use the test function $2k v^{n+1/2} = u^{n+1} - u^{n-1}$ in (\ref{scheme2--2}), and identity (\ref{tidt1}) to get \begin{equation}\label{S1S:1} \begin{split} &\frac{1}{2} {\mathbb E}\Bigl[\|v^{n+1}\|^2_{\mathbb{L}^2} -\|v^{n}\|^2_{\mathbb{L}^2} \Bigr] + k {\mathbb E}\Bigl[ \big(\nabla u^{n, \frac 12}, \nabla v^{n+\frac 12} \big)\Bigr] + \beta\,k^{2+\beta} {\mathbb E}\bigl[\| \nabla v^{n+ \frac 12} \|_{\mathbb{L}^2}^2\bigr] \\ &= {\mathbb E}\Bigl[\Bigl(\sigma (u^n, v^{n- \frac 12}) \,\Delta_nW, v^{n+\frac 12} \Bigr) + \,\widehat{\alpha} \,\Bigl(D_u\sigma(u^n, v^{n-\frac 12}) v^n \,\widehat{\Delta_n W}, v^{n+\frac 12} \Bigr) \\ &\qquad+ \frac{k}{2}\Bigl( 3F(u^n, v^n) - F(u^{n-1}, v^{n-1}), v^{n+\frac 12}\Bigr)\Bigr] =: \sum_{i=1}^3 \mathbb{E} \big[ \mathscr{J}^n_i \big]\, . \end{split} \end{equation} Next, we use (\ref{scheme2--1}) in strong form, sum it for two subsequent steps, and multiply this equation with $-\Delta u^{n, \frac 12}$; we then arrive at \begin{align}\label{1eq-test} \frac 14 \Big[ \|\nabla u^{n+1}\|_{\mathbb{L}^2}^2 - \|\nabla u^{n-1}\|_{\mathbb{L}^2}^2 \Big] = k \Bigl(\nabla u^{n, \frac 12}, \nabla v^{n+\frac 12} \Bigr)\,. \end{align} Since the right-hand side of \eqref{1eq-test} is equal to the second term on the left-hand side of \eqref{S1S:1} we conclude that \begin{equation}\label{test-mult} \begin{split} &\frac{1}{2} {\mathbb E}\Bigl[\|v^{n+1}\|^2_{\mathbb{L}^2} -\|v^{n}\|^2_{\mathbb{L}^2} \Bigr] + \frac 14 {\mathbb E}\Big[ \|\nabla u^{n+1}\|_{\mathbb{L}^2}^2 - \|\nabla u^{n-1}\|_{\mathbb{L}^2}^2 \Big] \\ &\quad+ \beta \,k^{2+\beta} {\mathbb E} \big[ \| \nabla v^{n+ \frac 12} \|_{\mathbb{L}^2}^2 \big] = \sum_{i=1}^3 \mathbb{E} \big[ \mathscr{J}^n_i \big]\, . \end{split} \end{equation} Now we estimate each term on the right-hand side of \eqref{test-mult}. Using properties of the increments $\Delta_n W$, we get $\mathbb{E}\bigl[\bigl(\sigma (u^n, v^{n-\frac 12})\Delta_n W, v^{n} \bigr)\bigr]=0$. Using this we infer \begin{align*} \mathbb{E} \big[ \mathscr{J}^n_1 \big] =\mathbb{E}\Bigl[\Bigl(\sigma (u^n, v^{n-\frac 12})\Delta_n W, v^{n+ \frac 12} \Bigr)\Bigr] &= \mathbb{E}\Bigl[\Bigl(\sigma (u^n, v^{n- \frac 12})\Delta_n W, \frac{v^{n+1} + v^n}{2} \Bigr) \Bigr] \\ &=\frac 12 \mathbb{E}\Bigl[\Bigl(\sigma (u^n, v^{n- \frac 12})\Delta_n W, v^{n+1} - v^n \Bigr) \Bigr] \, . \end{align*} We use (\ref{scheme2--2}) --- in modified form as stated in Scheme \ref{Hat-scheme} --- to replace $v^{n+1} -v^n$. Hence {\small{ \begin{equation}\label{sig-v-n+1} \begin{split} \mathbb{E}\big[ \mathscr{J}^n_1 \big] &= \frac 12 \mathbb{E}\Bigl[\Bigl(\sigma (u^n, v^{n- \frac 12})\Delta_n W, k \Delta u^{n, \frac 12} \Bigr) \Bigr] + \frac 12 \mathbb{E}\Bigl[\Bigl(\sigma (u^n, v^{n- \frac 12}) \Delta_n W, \beta k^{2+\beta} \Delta v^{n+ \frac 12} \Bigr) \Bigr] \\ &\quad+ \frac 12 \mathbb{E} \Big[ \|\sigma (u^n, v^{n- \frac 12})\|_{\mathbb{L}^2}^2 |\Delta_n W|^2 \Big] \\ &\quad+ \frac{\widehat{\alpha}}{2}\, \mathbb{E}\Bigl[\Bigl(\sigma (u^n, v^{n- \frac 12})\Delta_n W, D_u\sigma(u^{n}, v^{n- \frac 12}) v^{n} \,\widehat{\Delta_n W} \Bigr) \Bigr] \\ &\quad+ \frac{k}{4}\,\mathbb{E}\Bigl[\Bigl(\sigma (u^n, v^{n- \frac 12})\Delta_n W, \bigl[ 3F(u^n, v^n) - F(u^{n-1}, v^{n-1}) \bigr] \Big) \Bigr] =: \sum_{i=1}^5 \mathscr{J}^{n, i}_1\, . \end{split} \end{equation} }} In the following parts {\bf a)}--{\bf c)}, we independently bound $\mathbb{E} \big[ \mathscr{J}^{n}_1 \big]$ through $\mathbb{E} \big[ \mathscr{J}^{n}_3 \big]$ in (\ref{S1S:1}). \smallskip \noindent {\bf a) Estimation of $\mathbb{E}\big[ \mathscr{J}^{n}_1 \big]$ in (\ref{sig-v-n+1}).} We estimate the five terms $\mathscr{J}^{n, i}_1, \,i=1, \cdots, 5,$ on the right-hand side of \eqref{sig-v-n+1}. Let $\underline{D_u \sigma} \equiv D_u \sigma(u^n, v^{n- \frac 12}) \in \mathcal{L}(\mathbb{H}^1_0, \mathbb{H}^1_0)$ and $\underline{D_v \sigma} \equiv D_v \sigma(u^n, v^{n- \frac 12}) \in \mathcal{L}(\mathbb{H}^1_0, \mathbb{H}^1_0)$. By integration by parts and using $\sigma (u^n, v^{n-\frac 12}) = 0$ on $\partial {\mathcal O}$ we infer \begin{equation}\label{j^n,1_1est} \begin{split} \mathscr{J}^{n, 1}_1 &= \frac 12 \mathbb{E}\Bigl[-\Bigl( \underline{D_u \sigma}\, \nabla u^n \Delta_n W, k \nabla u^{n, \frac 12} \Bigr) \Bigr] + \frac 12 \mathbb{E}\Bigl[-\Bigl( \underline{D_v \sigma} \, \nabla v^{n- \frac 12} \Delta_n W, k \nabla u^{n, \frac 12} \Bigr) \Bigr] \\ &= \mathscr{J}^{n, 1}_{1, 1} +\mathscr{J}^{n, 1}_{1, 2}\, . \end{split} \end{equation} \noindent Using {\bf (A4)} for $m=1$ and the It\^o isometry we get \begin{equation}\label{413a} \begin{split} \mathscr{J}^{n, 1}_{1, 1} &\leq C_g^2\, \mathbb{E} \Big[ \|\nabla u^n\|_{\mathbb{L}^2}^2 |\Delta_n W|^2\Big] + C k^2 \,\mathbb{E} \Big[ \|\nabla u^{n, \frac 12}\|_{\mathbb{L}^2}^2 \Big] \\ &\leq C_g^2\, k\, \mathbb{E} \big[ \|\nabla u^n\|_{\mathbb{L}^2}^2 \big] + C k^2\, \mathbb{E} \Big[ \|\nabla u^{n+1}\|_{\mathbb{L}^2}^2 +\|\nabla u^{n-1}\|_{\mathbb{L}^2}^2 \Big]\, . \end{split} \end{equation} Using {\bf (A4)} for $m=1$, the independence property of the increment $\Delta_n W$, the It\^o isometry and the identity $2k v^{n+1/2} = u^{n+1}-u^{n-1}$, we estimate \begin{equation}\label{delv-sigma-1/2} \begin{split} \mathscr{J}^{n, 1}_{1, 2} &= \frac 12 \mathbb{E}\Bigl[-\Bigl( \underline{D_v \sigma} \, \nabla v^{n- \frac 12} \Delta_n W, \frac k2\, \nabla \big[ u^{n+1} - u^{n-1} \big] \Bigr) \Bigr] \\ &= \frac 12 \mathbb{E}\Bigl[-\Bigl( k^{1- \frac{\beta}{2}}\underline{D_v \sigma} \,\nabla v^{n- \frac 12} \Delta_n W, k^{1+ \frac{\beta}{2}} \,\nabla v^{n+ \frac 12}\Bigr) \Bigr] \\ &\leq \frac{3}{8 \beta} \,C_g^2\, \,k^{1+(2- \beta)}\, \mathbb{E} \Big[ \|\nabla v^{n- \frac 12}\|_{\mathbb{L}^2}^2 \Big] + \frac{\beta}{6} \,k^{2+ \beta}\, \mathbb{E} \Big[ \|\nabla v^{n+\frac 12}\|_{\mathbb{L}^2}^2 \Big]\,. \end{split} \end{equation} The terms on the right-hand sides of (\ref{413a}) and (\ref{delv-sigma-1/2}) may now be controlled by those on the left-hand side of (\ref{test-mult}) after summation over $1 \le n \le N-1$, provided that $k \leq \big( \frac{4}{3 C_g^2}\big)^{\frac{1}{1-2\beta}}$ for $\beta < \frac 12$ is sufficiently small. \smallskip \noindent Now we turn to $\mathscr{J}^{n, 2}_{1}$ in (\ref{sig-v-n+1}): integration by parts and using the fact that $\sigma (u^n, v^{n-\frac 12}) = 0$ on $\partial {\mathcal O}$ we get \begin{equation}\label{sig-v-n+1_} \begin{split} \mathscr{J}^{n, 2}_{1} &= \frac 12\, \mathbb{E}\Bigl[-\Bigl( k^{1+\frac{\beta}{2}}\underline{D_u \sigma} \, \nabla u^n \,\Delta_n W, \beta\,k^{1+\frac{\beta}{2}}\, \nabla v^{n+\frac 12} \Bigr) \Bigr] \\ &\quad+ \frac 12 \,\mathbb{E}\Bigl[-\Bigl( k^{1+\frac{\beta}{2}} \underline{D_v \sigma} \, \nabla v^{n- \frac 12} \Delta_n W, \beta\,k^{1+\frac{\beta}{2}}\,\nabla v^{n+\frac 12} \Bigr) \Bigr] =: \mathscr{J}^{n, 2}_{1, 1} +\mathscr{J}^{n, 2}_{1, 2}\, . \end{split} \end{equation} Using {\bf (A4)} for $m=1$ and the independence property of $\Delta_n W$ we estimate \begin{align*} \mathscr{J}^{n, 2}_{1, 1} \leq C \beta \,k^{3+\beta}\,\mathbb{E} \big[ \|\nabla u^n\|_{\mathbb{L}^2}^2 \big] + \frac \beta6\, k^{2+ \beta}\, \mathbb{E} \Big[ \|\nabla v^{n+\frac 12}\|_{\mathbb{L}^2}^2 \Big]\,, \end{align*} where the second term in the right-hand side can be be controlled by the corresponding term on the left-hand side of (\ref{test-mult}). Again, using {\bf (A4)} for $m=1$ we obtain for the second term in (\ref{sig-v-n+1_}) \begin{align*} \mathscr{J}^{n, 2}_{1, 2} \leq \beta \frac 38 C_g^2\,k\,k^{2+\beta}\,\mathbb{E} \big[ \|\nabla v^{n-\frac 12}\|_{\mathbb{L}^2}^2 \big] + \frac \beta6\, k^{2+ \beta}\, \mathbb{E} \Big[ \|\nabla v^{n+\frac 12}\|_{\mathbb{L}^2}^2 \Big]\,, \end{align*} where the right-hand side can be managed with the left-hand side of \eqref{test-mult} for $\beta <1/2$. \smallskip \noindent We continue with the next term $\mathscr{J}^{n, 3}_1$ in \eqref{sig-v-n+1}: by It\^o isometry and {\bf (A3)}, \begin{align*} \mathscr{J}^{n, 3}_1 &= \frac 12 \mathbb{E} \Big[ \|\sigma (u^n, v^{n- \frac 12})\|_{\mathbb{L}^2}^2 |\Delta_n W|^2 \Big] \leq C k \, \mathbb{E} \Big[ 1+ \|\nabla u^{n}\|_{\mathbb{L}^2}^2 + \|v^{n}\|_{\mathbb{L}^2}^2 + \|v^{n-1}\|_{\mathbb{L}^2}^2 \Big]\, , \end{align*} where $C>0$ depends on $C_{\tt L}$. Next comes $\mathscr{J}^{n, 4}_1$: using {\bf (A3)}, {\bf (A4)} for $m=1$ and item {\bf 4.}~of Remark \ref{rema2}, we infer \begin{equation} \begin{split} \mathscr{J}^{n, 4}_1 &\leq C \mathbb{E} \Big[ \|\sigma (u^n, v^{n- \frac 12})\|_{\mathbb{L}^2}^2 |\Delta_n W|^2 \Big] + \frac{\widehat{\alpha}^2}{4} C_g^2 \, \mathbb{E} \Big[ \|v^{n} \|_{\mathbb{L}^2}^2 \big|\widehat{\Delta_n W} \big|^2 \Big] \\ &\leq C k \, \mathbb{E} \Big[ 1+ \|\nabla u^{n}\|_{\mathbb{L}^2}^2 + \|v^{n}\|_{\mathbb{L}^2}^2 + \|v^{n-1}\|_{\mathbb{L}^2}^2 \Big] + \frac{\widehat{\alpha}^2}{4} C_g^2 \, k^3\,\mathbb{E} \big[ \|v^{n} \|_{\mathbb{L}^2}^2 \big]\,, \end{split} \end{equation} where $C>0$ depends on $C_{\tt L}$. The last term is $\mathscr{J}^{n, 5}_1$: by {\bf (A3)} we obtain \begin{equation}\label{j^n,1_5est} \begin{split} \mathscr{J}^{n, 5}_1 &\leq C \mathbb{E} \Big[ \|\sigma (u^n, v^{n- \frac 12})\|_{\mathbb{L}^2}^2 |\Delta_n W|^2 \Big] + C k^2 \, \mathbb{E} \Big[ \|F(u^n, v^n)\|_{\mathbb{L}^2}^2 + \|F(u^{n-1}, v^{n-1})\|_{\mathbb{L}^2}^2 \Big] \\ &\leq C k \, \mathbb{E} \Big[ 1+ \|\nabla u^{n}\|_{\mathbb{L}^2}^2 + \|\nabla u^{n-1}\|_{\mathbb{L}^2}^2 + \|v^{n}\|_{\mathbb{L}^2}^2 + \|v^{n-1}\|_{\mathbb{L}^2}^2 \Big] \,, \end{split} \end{equation} where the constant $C>0$ depends on $C_{\tt L}$. Thus, the estimate of $\mathscr{J}^n_1$ through those of $\mathscr{J}^{n, 1}_1$ through $\mathscr{J}^{n, 5}_1$ is complete. \smallskip \noindent {\bf b) Estimation of $\mathbb{E}\big[ \mathscr{J}^{n}_2 \big]$ in (\ref{S1S:1}).} By {\bf (A4)} for $m=1$, item {\bf 4.}~of Remark \ref{rema2}, and the independence property of $\widehat{\Delta_n W}$, \begin{equation*} \begin{split} \mathscr{J}^n_2 &\leq \widehat{\alpha}^2 \frac 1k \,\mathbb{E}\Bigl[ \| D_u\sigma(u^n, v^{n-\frac 12}) v^n \|_{\mathbb{L}^2}^2 \big|\widehat{\Delta_n W} \big|^2 \Bigr] + C k\,\mathbb{E} \Big[ \| v^{n+ \frac 12}\|_{\mathbb{L}^2}^2\Bigr] \\ &\le C_g^2 \,\widehat{\alpha}^2\,k^2 \,\mathbb{E} \big[ \| v^{n}\|_{\mathbb{L}^2}^2\bigr] + C k\,\mathbb{E} \Big[ \| v^{n+1}\|_{\mathbb{L}^2}^2 + \| v^{n}\|_{\mathbb{L}^2}^2\Bigr]\, . \end{split} \end{equation*} \smallskip \noindent {\bf c) Estimation of $\mathbb{E}\big[ \mathscr{J}^{n}_3 \big]$ in (\ref{S1S:1}).} By {\bf (A3)} we estimate \begin{equation}\label{S1S:4} \begin{split \mathscr{J}^n_3 &= k \mathbb{E} \Big[ \Bigl(F(u^n, v^n) , v^{n+\frac 12}\Bigr) \Big] +\frac{k}{2}\, \mathbb{E} \Big[\Bigl(F(u^n, v^n) - F(u^{n-1}, v^{n-1}), v^{n+ \frac 12}\Bigr) \Big] \\ & \le C k\, \mathbb{E} \big[ \|v^{n+\frac 12}\|^2_{\mathbb{L}^2} \big] + C \,k\, \mathbb{E} \Bigl[ 1+ \| \nabla u^{n}\|_{\mathbb{L}^2}^2 + \| \nabla u^{n-1}\|_{\mathbb{L}^2}^2 + \| v^{n}\|_{\mathbb{L}^2}^2 + \| v^{n-1}\|_{\mathbb{L}^2}^2 \Bigr]\, . \end{split} \end{equation} Now, we may use the parts {\bf a)} through {\bf c)} to bound the terms on the right-hand side of \eqref{test-mult}. Summation over all $1 \le n \le N-1$, for $k \leq k_0 \equiv k_0(C_{\tt L}, C_g)$ sufficiently small, leads to {{ \begin{equation}\label{accu-all-1} \begin{split} &\frac{1}{4} \mathbb{E} \Big[ \mathcal{E}(u^N, v^N) \Big] + \beta \frac 14 \,k^{2+\beta}\, \mathbb{E} \big[ \| \nabla v^{N+ \frac 12} \|_{\mathbb{L}^2}^2\big] \\ &\leq \frac{1}{4} \mathbb{E} \big[ \mathcal{E}(u_0, v^1) \big] + \beta \frac{k^{2+\beta}}{4} {\mathbb E}\bigl[ \Vert \nabla v^{1/2}\Vert_{\mathbb{L}^2}^2\bigr] + \frac 14 \mathbb{E} \big[ \|\nabla u^1\|_{\mathbb{L}^2}^2 \big] + C k \sum_{n=1}^{N-1} \mathbb{E}\big[ \mathcal{E}(u^n, v^n) \big] \, . \end{split} \end{equation} }} By {\bf (B1)$_{i}$}, the implicit version of the discrete Gronwall lemma then shows (\ref{energy1}) for $\beta\in(0, \frac{1}{2})$. \del{ \noindent Now taking the sum over $n=1, \cdots, N-1$, and rearranging we infer that \begin{equation}\label{eq-4.9} \begin{split} &\mathbb{E} \big[ \mathcal{E}(u^{N},v^{N}) \big] + \frac{1}{2} \sum_{n=1}^{N-1} \mathbb{E}\big[ \|v^{n+1}-v^{n}\|^2_{\mathbb{L}^2} \big] \\ &\leq (1+C\,k)\,\mathbb{E} \big[ \mathcal{E}(u^{1},v^{1}) \big] + C \,k \sum_{n=1}^{N-1} \mathbb{E} \big[ \mathcal{E}(u^{n+1},v^{n+1}) \big] + C\,k\, \mathbb{E} \big[ \mathcal{E}(u^{0},v^{0}) \big] \, , \end{split} \end{equation} } \medskip \noindent {\bf 2) Proof of \eqref{high-moment} for $p=1$.} To simplify technicalities, we put $F \equiv 0$. Let us denote $\mathfrak{E}(u^{n+1}, v^{n+1}) := \frac 14 \big[ \|\nabla u^{n+1}\|^2_{\mathbb{L}^2} + 2 \|v^{n+1}\|^2_{\mathbb{L}^2}\big]$. Arguing as before \eqref{test-mult} then leads to \begin{eqnarray}\label{frakE} &&\big[ \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big] + \beta k^{2+\beta} \Vert \nabla v^{n+\frac 12}\Vert^2_{{\mathbb L}^2} \\ \nonumber &&\qquad = \Big(\sigma(u^{n}, v^{n- \frac 12}) \Delta_n W, v^{n + \frac 12} \Big) + \widehat{\alpha}\,\Big( D_u\sigma(u^{n}, v^{n- \frac 12}) v^{n} \,\widehat{\Delta_n W}, v^{n + \frac 12} \Big)\, . \end{eqnarray} Now fix $\frac{1}{4} \leq \delta_1, \delta_2 \leq 1$, then multiply (\ref{frakE}) with $$\delta_1 \mathfrak{E}(u^{n+1}, v^{n+1}) + \delta_2 \Bigl( \mathfrak{E}(u^{n+1}, v^{n+1}) + \mathfrak{E}(u^{n-1}, v^{n})\Bigr) \, ,$$ and take the expectation to get {\small{ \begin{equation}\label{sum-mult-ene} \begin{split} &\frac{\delta_1 + 2\delta_2}{2} \mathbb{E} \Big[ \mathfrak{E}^2(u^{n+1}, v^{n+1}) - \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] + \frac{\delta_1}{2} \mathbb{E} \Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big] \\ &\quad+ \beta k^{2+\beta} {\mathbb E}\Bigl[ \Vert \nabla v^{n+\frac 12}\Vert^2_{{\mathbb L}^2} \big[ (\delta_1 + \delta_2) \mathfrak{E}(u^{n+1}, v^{n+1})+\delta_2\mathfrak{E}(u^{n-1}, v^{n}) \big] \Bigr] \\ &= \mathbb{E} \Big[ \Big(\sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, v^{n + \frac 12} \Big) \cdot \big[(\delta_1 + \delta_2) \mathfrak{E}(u^{n+1}, v^{n+1})+\delta_2\mathfrak{E}(u^{n-1}, v^{n}) \big] \Big] \\ &\quad+ \widehat{\alpha}\,\mathbb{E} \Big[ \Big( D_u\sigma(u^{n}, v^{n-\frac 12}) v^{n} \,\widehat{\Delta_n W}, v^{n + \frac 12} \Big) \cdot \big[ (\delta_1 + \delta_2) \mathfrak{E}(u^{n+1}, v^{n+1})+\delta_2 \mathfrak{E}(u^{n-1}, v^{n}) \big] \Big] \\ & =: \mathscr{K}^{n,1} + \mathscr{K}^{n,2}\,. \end{split} \end{equation} }} We independently estimate the terms $\mathscr{K}^{n,1}$ and $\mathscr{K}^{n,2}$. \smallskip \noindent {\bf a) Estimation of $\mathscr{K}^{n,1}$ in (\ref{sum-mult-ene}).} This term may be written as the sum of two others: \begin{equation}\label{1.21-eq} \begin{split} \mathscr{K}^{n,1}&= (\delta_1 + \delta_2) \mathbb{E} \Big[ \Big(\sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, v^{n + \frac 12} \Big) \cdot \big(\mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big) \Big] \\ &\qquad+ (\delta_1+2\delta_2) \mathbb{E} \Big[ \Big(\sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, v^{n + \frac 12} \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] := \mathscr{K}^{n,1}_1 + \mathscr{K}^{n,1}_2\,. \end{split} \end{equation} We consider $\mathscr{K}^{n,1}_1$ first. By ${\mathbb E}\bigl[\vert \Delta_n W\vert^4 \bigr] = {\mathcal O}(k^2)$, and {\bf (A3)} we find {\small{ \begin{equation*} \begin{split} \mathscr{K}^{n,1}_1 &\leq C_{\delta_1}\, \mathbb{E}\Big[ \|\sigma(u^{n}, v^{n-\frac 12}) \Delta_n W \|_{\mathbb{L}^2}^2 \|v^{n + \frac 12} \|_{\mathbb{L}^2}^2 \Big] + \frac{\delta_1}{4} \mathbb{E}\Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big] \\ & \leq \frac{C_{\delta_1}}{k} \mathbb{E}\Big[ \|\sigma(u^{n}, v^{n-\frac 12})\|_{\mathbb{L}^2}^4 |\Delta_n W |^4 \Big] + C_{\delta_1} k\, \mathbb{E}\Big[ \|v^{n + \frac 12} \|_{\mathbb{L}^2}^4 \Big] + \frac{\delta_1}{4} \mathbb{E}\Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big] \\ &\leq C_{\delta_1}k\, \mathbb{E}\Big[ 1 + \sum_{\ell=-1}^1\mathfrak{E}^2(u^{n+\ell}, v^{n+\ell})\Big] + \frac{\delta_1}{4} \mathbb{E}\Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big] \, , \end{split} \end{equation*} }} where the last term on the right-hand side can be absorbed on the left-hand side of \eqref{sum-mult-ene}. \noindent We continue with $\mathscr{K}^{n,1}_2$: on using ithe ndependence property of $\Delta_n W$, and equation \eqref{scheme2--2}, {\small{ \begin{equation*} \begin{split} \mathscr{K}^{n,1}_2 &\leq 3 \,\Bigl\vert\mathbb{E} \Big[ \Big(\sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, v^{n+1} - v^n \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] \Bigr\vert \\ &= 3 \, \Bigl\vert \mathbb{E} \Big[ \Big(\sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, k \Delta \widetilde{u}^{n, \frac 12} \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] \Bigr\vert + 3\, \Bigl\vert \mathbb{E} \Big[ \|\sigma(u^{n}, v^{n-\frac 12}) \Delta_n W \|_{\mathbb{L}^2}^2\, \mathfrak{E}(u^{n-1}, v^{n}) \Big]\Bigr\vert \\ &\quad+ 3 \, \Bigl\vert\mathbb{E} \Big[ \Big(\sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, D_u \sigma(u^n, v^{n-\frac 12}) v^n \widehat{\Delta_n W} \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big]\Bigr\vert =: \mathscr{K}^{n,1}_{2, 1} + \mathscr{K}^{n,1}_{2, 2} + \mathscr{K}^{n,1}_{2, 3} \, . \end{split} \end{equation*} }} We split $\mathscr{K}^{n,1}_{2, 1} := \mathscr{K}^{n,1, {\tt A}}_{2, 1} + \mathscr{K}^{n,1, {\tt B}}_{2, 1}$ because of (\ref{tidt1}); here, $\mathscr{K}^{n,1, {\tt A}}_{2, 1}$ is as $\mathscr{K}^{n,1}_{2, 1}$, where $\widetilde{u}^{n, \frac{1}{2}}$ is replaced by ${u}^{n, \frac{1}{2}}$. We use integration by parts, and the fact that $\sigma (u^n, v^{n-\frac 12}) = 0$ on $\partial {\mathcal O}$, {\bf (A4)} for $m=1$, the independence property of $\Delta_n W$ and that ${\mathbb E}\bigl[\vert \Delta_n W\vert^4 \bigr] = {\mathcal O}(k^2)$ to conclude \begin{equation*} \begin{split} \mathscr{K}^{n,1,{\tt A}}_{2, 1} &= \frac 92 \, \Bigl\vert \mathbb{E} \Big[ -\Big(\nabla \sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, k \nabla [u^{n+1} - u^{n-1}] \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] \Bigr\vert \\ &= 9 \,\Bigl\vert\mathbb{E} \Big[ -\Big(\nabla \sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, k^{1- \frac{\beta}{2}} k^{1+ \frac{\beta}{2}}\, \nabla v^{n+\frac 12} \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] \Bigr\vert \\ &\leq \frac{C_{\delta_2}}{\beta} k^{3-\beta}\,\mathbb{E} \Big[\| \underline{D_u \sigma}\, \nabla u^n \|_{\mathbb{L}^2}^2 \mathfrak{E}(u^{n-1}, v^{n}) \Big] + \frac{C_{\delta_2}}{\beta} k^{3-\beta}\,\mathbb{E} \Big[\|\underline{D_v \sigma}\, \nabla v^{n-\frac 12}\|_{\mathbb{L}^2}^2 \mathfrak{E}(u^{n-1}, v^{n}) \Big] \\ &\quad+ \frac{\beta \delta_2}{4} k^{2+\beta} \,\mathbb{E} \Big[ \|\nabla v^{n+\frac 12}\|_{\mathbb{L}^2}^2 \,\mathfrak{E}(u^{n-1}, v^{n}) \Big] \\ &\leq C_{\delta_2} C_{\beta} C_g^2 k^{3-\beta} \mathbb{E}\Big[ \mathfrak{E}^2(u^{n}, v^{n}) + \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] + C_{\delta_2} C_{\beta} C_g^2 \,k^{3-\beta} \,\mathbb{E} \Big[ \|\nabla v^{n-\frac 12}\|_{\mathbb{L}^2}^2 \,\mathfrak{E}(u^{n-1}, v^{n}) \Big] \\ &\quad+ \frac{\beta \delta_2}{4} k^{2+\beta} \,\mathbb{E} \Big[ \|\nabla v^{n+\frac 12}\|_{\mathbb{L}^2}^2 \,\mathfrak{E}(u^{n-1}, v^{n}) \Big]\,, \end{split} \end{equation*} \noindent where $C_{\beta} >0$ is a constant dependent on $\beta$ for $0<\beta<\frac{1}{2}$. We use a similar idea to estimate $\mathscr{K}^{n,1,{\tt B}}_{2, 1}$, \begin{equation*} \begin{split} \mathscr{K}^{n,1,{\tt B}}_{2, 1} &= \frac{9}{2} \,\Bigl\vert \mathbb{E} \Big[ -\Big(\nabla \sigma(u^{n}, v^{n-\frac 12}) \Delta_n W, \beta k^{2+\beta} \nabla v^{n+\frac{1}{2}} \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \big)\Bigr] \Bigr\vert \\ &\leq \frac{C_{\delta_2}}{\beta} k^{3+\beta}\,\mathbb{E} \Big[\| \underline{D_u \sigma}\, \nabla u^n \|_{\mathbb{L}^2}^2 \mathfrak{E}(u^{n-1}, v^{n}) \Big] + \frac{C_{\delta_2}}{\beta} k^{3+\beta}\,\mathbb{E} \Big[\|\underline{D_v \sigma}\, \nabla v^{n-\frac 12}\|_{\mathbb{L}^2}^2 \mathfrak{E}(u^{n-1}, v^{n}) \Big] \\ &\quad+ \beta \frac{\delta_2}{4} k^{2+\beta} \,\mathbb{E} \Big[ \|\nabla v^{n+\frac 12}\|_{\mathbb{L}^2}^2 \,\mathfrak{E}(u^{n-1}, v^{n}) \Big] \\ &\leq C_{\delta_2} C_{\beta} C_g^2\, k^{3+\beta}\,\mathbb{E} \Big[ \mathfrak{E}^2(u^{n}, v^{n}) + \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] + C_{\delta_2} C_{\beta} C_g^2\, k^{3+\beta}\,\mathbb{E} \Big[\| \nabla v^{n-\frac 12}\|_{\mathbb{L}^2}^2 \mathfrak{E}(u^{n-1}, v^{n}) \Big] \\ &\quad+ \beta \frac{\delta_2}{4} k^{2+\beta} \,\mathbb{E} \Big[ \|\nabla v^{n+\frac 12}\|_{\mathbb{L}^2}^2 \,\mathfrak{E}(u^{n-1}, v^{n}) \Big]\, , \end{split} \end{equation*} where the last two terms in the right-hand sides of $\mathscr{K}^{n,1,{\tt A}}_{2, 1}$ and $\mathscr{K}^{n,1,{\tt B}}_{2, 1}$ may be controlled by those on the left-hand side of \eqref{sum-mult-ene} after summation over $1 \le n \le N-1$, provided that $k$ is sufficiently small and $\beta < \frac 12$ . \noindent Similarly, using {\bf (A3)}, and ${\mathbb E}\bigl[\vert \Delta_n W\vert^4 \bigr] = {\mathcal O}(k^2)$ we estimate \begin{equation*} \mathscr{K}^{n,1}_{2, 2} \leq \frac Ck\, \mathbb{E} \Big[ \|\sigma(u^{n}, v^{n-\frac 12})\|_{\mathbb{L}^2}^4 |\Delta_n W|^4 \Big] + C k\, \mathbb{E}\Big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] \leq C k\, \mathbb{E} \Big[ 1+ \sum_{\ell=-1}^0\mathfrak{E}^2(u^{n+\ell}, v^{n+\ell})\Big]\,. \end{equation*} Using ${\mathbb E}\bigl[\vert \widehat{\Delta_n W}\vert^4 \bigr] = {\mathcal O}(k^6)$, and {\bf (A3)} gives \begin{equation*} \begin{split} \mathscr{K}^{n,1}_{2, 3} &\leq \frac Ck\, \mathbb{E} \Big[ \| \sigma(u^{n}, v^{n-\frac 12}) \Delta_n W\|_{\mathbb{L}^2}^2 \,\|D_u \sigma(u^n, v^{n-\frac 12}) v^n \widehat{\Delta_n W}\|_{\mathbb{L}^2}^2 \Big] + C k\, \mathbb{E}\Big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] \\ &\leq \frac Ck\, \mathbb{E} \Big[ \| \sigma(u^{n}, v^{n-\frac 12})\|_{\mathbb{L}^2}^4 |\Delta_n W|^4 \Big] + \frac{C_g^4}{k} \mathbb{E} \Big[ \| v^n\|_{\mathbb{L}^2}^4 \big| \widehat{\Delta_n W} \big|^4 \Big] + C k\, \mathbb{E}\Big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] \\ &\leq C k\, \mathbb{E}\Big[1+ \sum_{\ell=-1}^0\mathfrak{E}^2(u^{n+\ell}, v^{n+\ell})\Big]\,. \end{split} \end{equation*} {\bf b) Estimation of $\mathscr{K}^{n,2}$ in (\ref{sum-mult-ene}).} By {\bf (A4)} for $m=1$ and using the fact that ${\mathbb E}\bigl[\vert \widehat{\Delta_n W}\vert^4 \bigr] = {\mathcal O}(k^6)$, we infer \begin{equation*} \begin{split} \mathscr{K}^{n,2} &\leq \frac{\widehat{\alpha}^2}{k} \mathbb{E} \Big[ \|D_u\sigma(u^{n}, v^{n-\frac 12}) v^{n} \,\widehat{\Delta_n W} \|_{\mathbb{L}^2}^2\, \|v^{n + \frac 12}\|_{\mathbb{L}^2}^2 \Big] \\ &\quad+ Ck \,\mathbb{E}\Big[ \mathfrak{E}^2(u^{n+1}, v^{n+1}) \Big] + Ck \,\mathbb{E}\Big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] \\ &\leq \frac{\widehat{\alpha}^4}{k^3} \mathbb{E} \Big[ \|D_u\sigma(u^{n}, v^{n-\frac 12}) v^{n}\|_{\mathbb{L}^2}^4 \big| \widehat{\Delta_n W}\big|^4 \Big] + Ck\, \mathbb{E} \Big[ \|v^{n + \frac 12}\|_{\mathbb{L}^2}^4 \Big] + Ck \,\mathbb{E}\Big[ \sum_{\ell=-1}^1\mathfrak{E}^2(u^{n+\ell}, v^{n+\ell}) \Big] \\ &\leq \widehat{\alpha}^4\,C_g^4\,k^3 \,\mathbb{E} \Big[ \mathfrak{E}^2(u^{n}, v^{n}) \Big] + Ck \,\mathbb{E}\Big[ \sum_{\ell=-1}^1\mathfrak{E}^2(u^{n+\ell}, v^{n+\ell}) \Big]\,. \end{split} \end{equation*} Now we insert the estimates from parts {\bf a)} and {\bf b)} into (\ref{sum-mult-ene}), and sum over $1 \leq n \leq N-1$. Then, for all $k \leq k_0 \equiv k_0(C_{\tt L}, C_g)$ there exists $C \equiv C(\beta)>0$ for $\beta\in (0, \frac{1}{2})$ such that the assertion \eqref{high-moment} for $p=1$ follows from the implicit version of the discrete Gronwall lemma. \medskip \noindent {\bf 3) Proof of \eqref{high-moment} for $p \geq 2$.} Starting from the identity (\ref{sum-mult-ene}), we multiply $\delta_1 {\mathfrak E}^{2^{p-1}}(u^{n+1},v^{n+1}) + \delta_2 \big[{\mathfrak E}^{2^{p-1}}(u^{n+1},v^{n+1}) + {\mathfrak E}^{2^{p-1}}(u^{n-1},v^{n}) \big]$ on both sides, and then take expectations. We may then follow the same argument as in {\bf 2)} to settle the assertion. \del{ {\small{ \begin{equation}\label{high-est-2} \begin{split} &\mathbb{E} \bigl[{\mathcal E}^4(u^{n+1}, v^{n+1}) - {\mathcal E}^4(u^{n}, v^{n})\bigr] + \mathbb{E} \Big[ \bigl\vert \bigl[{\mathcal E}^2(u^{n+1}, v^{n+1}) - {\mathcal E}^2(u^{n}, v^{n})\bigr]\bigr\vert^2 \Big]\\ &\quad + 2\,\mathbb{E} \Big[ {\mathcal E}^2(u^{n+1},v^{n+1})\bigl\vert {\mathcal E}(u^{n+1}, v^{n+1}) - {\mathcal E}(u^{n}, v^{n})\bigr\vert^2 \Big] + 2\,\mathbb{E} \Big[ {\mathcal E}^3(u^{n+1}, v^{n+1}) \|v^{n+1}-v^{n}\|^2_{\mathbb{L}^2} \Big] \\ & \leq 2\,\mathbb{E} \Big[ {\mathcal E}^2(u^{n+1},v^{n+1}) \bigl(I^{(n)}_1 + I^{(n)}_2 + I^{(n)}_3 \bigr) \Big] \, . \end{split} \end{equation} }} We consider the first term in the right-hand side $\mathbb{E} \big[ {\mathcal E}^2(u^{n+1},v^{n+1}) I^{(n)}_1 \big]$ to estimate. Recall the splitting of $I^{(n)}_1$ in \eqref{spl-In1}. Using Young's inequality we estimate the term \begin{equation} \begin{split} {\mathbb E} \Big[{\mathcal E}^2(u^{n+1},v^{n+1}) (I^{(n)}_{1;1} + I^{(n)}_{1;2}) \Big] &\leq \frac{1}{4} {\mathbb E}\bigl[{\mathcal E}^3(u^{n+1}, v^{n+1}) \Vert v^{n+1} - v^n\Vert^2_{{\mathbb L}^2}\bigr] \\ &\quad + {\mathbb E}\bigl[\Vert \sigma(u^n, v^n)\Vert^2_{\mathbb{L}^2} \vert \Delta_nW\vert^2 {\mathcal E}^3(u^{n+1},v^{n+1})\bigr] \, . \end{split} \end{equation} The first term in the right-hand side above can be managed with the fourth term in the left-hand side of \eqref{high-est-2}. For the second term above, we use {\bf (A3)}, \eqref{E-bound}, It\^o isometry, and Young's inequality repeatedly to obtain, \begin{equation} \begin{split} {\mathbb E}\bigl[\Vert \sigma(u^n, v^n)\Vert^2_{\mathbb{L}^2} \vert \Delta_nW\vert^2 {\mathcal E}^3(u^{n+1},v^{n+1})\bigr] \leq C_{\tt L}\,k \,\mathbb{E} \big[ {\mathcal E}^4(u^{n+1},v^{n+1}) +{\mathcal E}^4(u^{n},v^{n})\big]\, . \end{split} \end{equation} Now we aim to bound ${\mathbb E}\bigl[{\mathcal E}^2(u^{n+1}, v^{n+1})(I^{(n)}_{2;1}+ I^{(n)}_{2;2})\bigr]$ by splitting it into four terms as \begin{equation} \begin{split} &{\mathbb E}\bigl[{\mathcal E}^2(u^{n+1}, v^{n+1}) (I^{(n)}_{2;1}+ I^{(n)}_{2;2})\bigr] \\ &= {\mathbb E}\Bigl[ \bigl(\sigma (u^n, v^n) \Delta_nW ,v^n \bigr){\mathcal E}^2(u^{n+1}, v^{n+1})\bigl( {\mathcal E}(u^{n+1}, v^{n+1}) \pm {\mathcal E}(u^{n}, v^{n})\bigr) \Bigr] \\ &\leq \frac{1}{8}{\mathbb E}\Bigl[ {\mathcal E}^2(u^{n+1}, v^{n+1}) \,\bigl\vert {\mathcal E}(u^{n+1}, v^{n+1}) - {\mathcal E}(u^{n}, v^{n}) \bigr\vert^2 \Bigr] \\ &\quad + C\,{\mathbb E} \Bigl[ \bigl\vert {\mathcal E}^2(u^{n+1}, v^{n+1}) - {\mathcal E}^2(u^{n}, v^{n})\bigr\vert \bigl\vert \bigl(\sigma (u^n, v^n) \Delta_nW ,v^n \bigr)\bigr\vert^2 \Bigr] \\ &\quad + C\,{\mathbb E}\Bigl[ {\mathcal E}^2(u^n,v^n) \bigl\vert \bigl( \sigma(u^n, v^n)\Delta_n W, v^n\bigr)\bigr\vert^2\Bigr] + \frac{1}{12}{\mathbb E}\Bigl[ \bigl\vert {\mathcal E}^2(u^{n+1}, v^{n+1}) - {\mathcal E}^2(u^{n}, v^{n}) \bigr\vert^2\Bigr] \\ &=: \mathscr{J}^{(n)}_1+ \mathscr{J}^{(n)}_2 + \mathscr{J}^{(n)}_3 + \mathscr{J}^{(n)}_4\, . \end{split} \end{equation} The terms $ \mathscr{J}^{(n)}_1$ and $ \mathscr{J}^{(n)}_4$ can be managed in the left-hand side of \eqref{high-est-2}. We only need to estimate $ \mathscr{J}^{(n)}_2$ and $ \mathscr{J}^{(n)}_3$. We use \eqref{E-bound}, It\^o isometry, and Young's inequality repeatedly to obtain, \[ \mathscr{J}^{(n)}_2 \leq \frac{1}{12} {\mathbb E} \Bigl[ \bigl\vert {\mathcal E}^2(u^{n+1}, v^{n+1}) - {\mathcal E}^2(u^{n}, v^{n})\bigr\vert^2 \Big] + C_{\tt L}\,k \,\mathbb{E} \big[ {\mathcal E}^4(u^{n}, v^{n}) \big] ,\] and \[ \mathscr{J}^{(n)}_3 \leq C_{\tt L}\,k \,\mathbb{E} \big[ {\mathcal E}^4(u^{n}, v^{n}) \big] . \] Now we estimate the term $\mathbb{E} \big[ {\mathcal E}^2(u^{n+1},v^{n+1}) I^{(n)}_2\big]$ by using the Cauchy-Schwarz inequality as \begin{equation} \begin{split} &2\, \widetilde{\alpha}\, {\mathbb E}\Bigl[\bigl( \partial_u\sigma(u^n,v^n)v^n \widetilde{\Delta_n W}, v^{n+1} \bigr) {\mathcal E}^2(u^{n+1}, v^{n+1})[{\mathcal E}(u^{n+1}, v^{n+1}) \pm {\mathcal E}(u^{n}, v^{n})] \Bigr] \\ & \leq C \,\widetilde{\alpha}^2\,{\mathbb E}\bigl[ \Vert v^{n}\Vert^2_{{\mathbb L}^2} \,\Vert v^{n+1} \Vert^2_{{\mathbb L}^2} \,\vert \widetilde{\Delta_n W}\vert^2 \,{\mathcal E}^2(u^{n+1}, v^{n+1})\bigr] \\ &\quad +\frac{1}{8} {\mathbb E}\bigl[ {\mathcal E}^2(u^{n+1}, v^{n+1})\vert{\mathcal E}(u^{n+1}, v^{n+1}) - {\mathcal E}(u^{n}, v^{n})\vert^2 \bigr] \\ &\quad + 2 \,\widetilde{\alpha}\,{\mathbb E}\Bigl[ \bigl( \partial_u\sigma(u^n,v^n)v^n \widetilde{\Delta_n W}, v^{n+1} \bigr) \bigl[{\mathcal E}^2(u^{n+1}, v^{n+1}) \pm {\mathcal E}^2(u^n,v^n)\bigr]{\mathcal E}(u^n, v^n)\Bigr]\\ & =: \mathcal{J}^n_1 + \mathcal{J}^n_2 + \mathcal{J}^n_3\, . \end{split} \end{equation} We treat the above three terms separately. To handle the term $\mathcal{J}^n_1,$ we dominate the term $\Vert v^{n+1}\Vert^2_{{\mathbb L}^2}$ by $ {\mathcal E}(u^{n+1},v^{n+1})$, then we use the generalised Young's inequality (with conjugates $3/4$ and $1/4$) to obtain \begin{equation}\label{eq-5.9} \begin{split} \mathcal{J}^n_1 &= C \,\widetilde{\alpha}^2\,{\mathbb E}\bigl[ k^{3/4}\,\Vert v^{n} \Vert^2_{{\mathbb L}^2} \,\vert \widetilde{\Delta_n W}\vert^2 \cdot k^{-3/4}\,{\mathcal E}^3(u^{n+1}, v^{n+1})\bigr] \\ &\leq k\, {\mathbb E}\big[{\mathcal E}^4(u^{n+1},v^{n+1}) \big] + Ck^{-3}\, {\mathbb E} \big[\Vert v^n\Vert^8_{{\mathbb L}^2} \vert \widetilde{\Delta_n W}\vert^8 \big] \\ &\leq k\, {\mathbb E} \big[{\mathcal E}^4(u^{n+1},v^{n+1}) \big] + k^9\, \widetilde{\alpha}^8\,{\mathbb E} \big[{\mathcal E}^4(u^n, v^n) \big]\, . \end{split} \end{equation} The second term $\mathcal{J}^n_2$ can be managed in the left-hand side of \eqref{high-est-2}. For the last term we use the Cauchy-Schwarz inequality to obtain \begin{equation}\label{eq-5.10--} \begin{split} \mathcal{J}^n_3 &\leq \frac{1}{12} {\mathbb E}\bigl[ \bigl\vert {\mathcal E}^2(u^{n+1}, v^{n+1}) - {\mathcal E}^2(u^n,v^n)\bigr\vert^2\bigr] \\ &\quad + C \,\widetilde{\alpha}^2\,\mathbb{E} \Big[ \big| \bigl( \partial_u\sigma(u^n,v^n)v^n \widetilde{\Delta_n W}, v^{n+1} \bigr)\big|^2 {\mathcal E}^2(u^n, v^n)\Big] \\ &\quad + C\, \widetilde{\alpha}^2\,\mathbb{E} \Big[ \bigl( \partial_u\sigma(u^n,v^n)v^n \widetilde{\Delta_n W}, v^{n+1} \bigr) {\mathcal E}^3(u^n, v^n)\Big] \, . \end{split} \end{equation} The first term in the right-hand side of \eqref{eq-5.10--} can be managed in the left-hand side of \eqref{high-est-2}. To handle the second and third terms, we use the same technique as used for the term $\mathcal{J}^n_1$ in \eqref{eq-5.9}. \noindent The term $\mathbb{E} \big[ {\mathcal E}^2(u^{n+1},v^{n+1}) I^{(n)}_3 \big]$ can be estimated similarly as in \eqref{F_sums} to get, {\small{ \begin{equation} \begin{split} &\mathbb{E} \big[ {\mathcal E}^2(u^{n+1},v^{n+1}) I^{(n)}_3 \big] \leq C_{\tt L} \,k \mathbb{E} \big[ \bigl({\mathcal E}^4(u^n, v^n) + {\mathcal E}^4(u^{n-1}, v^{n-1})\bigr) \big] + C_{\tt L}\,k \mathbb{E} \big[{\mathcal E}^4(u^{n+1}, v^{n+1}) \big]. \end{split} \end{equation} }} Accumulating all the above estimates in \eqref{high-est-2}, rearranging the terms, taking sum from $n=1$ to $N-1$ and proceeding similarly as before we get the assertion $(ii)$ for $p=2$. The case $p=3$ can be proved similarly. Note that if $\widetilde{\alpha}=0,$ then we will also get the assertion $(ii)$. } \medskip \noindent {\bf 4) Proof of \eqref{energy2}.} Let $\widehat{\alpha} = 1$. Suppose $\sigma_2(v) \equiv 0 \equiv F_2(v)$ in {\bf (A3)} and $\beta=0$. We combine both equations in the scheme \eqref{scheme2:1}--\eqref{scheme2:2} to get \begin{equation}\label{ul-comb} \begin{split} \bigl[u^{\ell+1} - u^\ell\bigr] - \bigl[u^\ell - u^{\ell-1}\bigr] &= k^2 \Delta u^{\ell,1/2} + k \,\sigma(u^\ell)\Delta_\ell W + \widehat{\alpha} k \, D_u\sigma(u^\ell)v^{\ell} \widehat{\Delta_{\ell} W} \\ &\quad + \frac{k^2}{2}\, \bigl[ 3F(u^n) -F(u^{n-1})\bigr] \end{split} \end{equation} for all $1 \leq \ell \leq N$. Now sum over the first $n$ steps, and define $\overline{u}^{n+1} := \sum_{\ell=1}^n {u}^{\ell+1}$ to get \begin{equation}\label{sum_ul} \begin{split} \bigl[u^{n+1} - u^n\bigr]- k^2 \Delta \overline{u}^{n,1/2} &= \bigl[ u^1 - u^0\bigr] + k \sum^n_{\ell=1} \sigma(u^\ell) \Delta_{\ell} W + \widehat{\alpha} k \sum^n_{\ell=1} D_u\sigma(u^\ell)v^{\ell} \widehat{\Delta_{\ell} W} \\ &\quad +\frac{k^2}{2} \sum^n_{\ell=1} \bigl[3F(u^\ell) - F(u^{\ell-1})\bigr] \, . \end{split} \end{equation} Multiply both sides with $u^{n+1/2}$ and use integration by parts to get {\small{ \begin{equation}\label{uL2-energy} \begin{split} &\frac 12 \Big[ \|u^{n+1}\|_{\mathbb{L}^2}^2 - \|u^{n}\|_{\mathbb{L}^2}^2 \Big] + k^2 \big( \nabla \overline{u}^{n,1/2}, \nabla u^{n+1/2} \big) \\ &= \big( u^1 - u^0, u^{n+1/2} \big) + k \Big(\sum^n_{\ell=1} \sigma(u^\ell) \Delta_{\ell} W, u^{n+1/2} \Big) + \widehat{\alpha} k \Big(\sum^n_{\ell=1}\ D_u\sigma(u^\ell)v^{\ell} \widehat{\Delta_{\ell} W}, u^{n+1/2} \Big) \\ &\quad+ \frac{k^2}{2} \sum^n_{\ell=1} \Big( 3F(u^\ell) - F(u^{\ell-1}), u^{n+1/2} \Big) =: \mathfrak{K}^n_{1} + \mathfrak{K}^n_{2} + \mathfrak{K}^n_{3} + \mathfrak{K}^n_{4}\, . \end{split} \end{equation} }} We observe that the last term in the left-hand side may be written as {\small{ \begin{align*} k^2 \big( \nabla \overline{u}^{n,1/2}, \nabla u^{n+1/2} \big) = \frac{k^2}{4} \Big( \nabla [ \overline{u}^{n+1} + \overline{u}^{n-1}], \nabla [ \overline{u}^{n+1} - \overline{u}^{n-1}] \Big) = \frac{k^2}{4} \Big[ \|\nabla \overline{u}^{n+1}\|_{\mathbb{L}^2}^2 - \|\nabla \overline{u}^{n-1}\|_{\mathbb{L}^2}^2 \Big]\,. \end{align*} }} Taking expectation on both sides leads to \begin{align}\label{uL2-dest} \frac 12 \mathbb{E}\Big[ \|u^{n+1}\|_{\mathbb{L}^2}^2 - \|u^{n}\|_{\mathbb{L}^2}^2 \Big] + \frac{k^2}{4} \mathbb{E} \Big[ \|\nabla \overline{u}^{n+1}\|_{\mathbb{L}^2}^2 - \|\nabla \overline{u}^{n-1}\|_{\mathbb{L}^2}^2 \Big] = \sum_{j=1}^4 \mathbb{E}\big[\mathfrak{K}^n_{j} \big]\, . \end{align} Since $u^1 - u_0 = k v^1$, by {\bf (B1)$_{i}$} we infer \begin{align}\label{uL2-desta} \mathbb{E}\big[\mathfrak{K}^n_{1} \big] \leq \frac 1k \mathbb{E} \big[ \|u^1 - u_0\|_{\mathbb{L}^2}^2 \big] + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^2 \big] \leq Ck + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^2 \big] \, . \end{align} Using the It\^o isometry and {\bf (A3)} we infer \begin{equation}\label{kln2-sigmau} \begin{split} \mathbb{E}\big[\mathfrak{K}^n_{2} \big] &\leq k \sum^n_{\ell=1} \mathbb{E} \Big[ \|\sigma(u^{\ell}) \|_{\mathbb{L}^2}^2 |\Delta_{\ell} W|^2 \Big] + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^2 \big] \\ &\leq C k^2 C_{\tt L}^2 \sum^n_{\ell=1}\, \mathbb{E} \Big[ 1 + \|u^{\ell} \|_{\mathbb{L}^2}^2 \Big] + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^2 \big] \, . \end{split} \end{equation} Using item {\bf 4.} of Remark \ref{rema2}, and {\bf (A4)} for $m=1$ we infer \begin{equation}\label{kn3-hat} \begin{split} \mathbb{E}\big[\mathfrak{K}^n_{3} \big] &\leq k \sum^n_{\ell=1} \mathbb{E} \Big[ \|D_u \sigma(u^{\ell}) v^{\ell}\|_{\mathbb{L}^2}^2 \big|\widehat{\Delta_{\ell} W} \big|^2 \Big] + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^2 \big] \\ &\leq k^4 C_{g}^2 \sum^n_{\ell=1}\, \mathbb{E} \big[ \|v^{\ell} \|_{\mathbb{L}^2}^2 \big] + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^2 \big] \, . \end{split} \end{equation} Since $v^{\ell} = \frac 1k \big[ u^{\ell} - u^{\ell-1}\big]$, we further estimate \eqref{kn3-hat} by \begin{align*} &\leq k^2 C_{g}^2 \sum^n_{\ell=1}\, \mathbb{E} \big[ \|u^{\ell}\|_{\mathbb{L}^2}^2 + \| u^{\ell-1} \|_{\mathbb{L}^2}^2\big] + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^2 \big]\,. \end{align*} Using {\bf (A3)} we estimate $\mathbb{E}\big[\mathfrak{K}^n_{4} \big]$ by \begin{equation} \begin{split} \mathbb{E}\big[\mathfrak{K}^n_{4} \big] &\leq C k^2 C_{\tt L}^2 \sum^n_{\ell=1} \mathbb{E} \Big[ 1 +\|u^{\ell} \|_{\mathbb{L}^2}^2 + \| u^{\ell-1} \|_{\mathbb{L}^2}^2 \Big] + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^2 \big] \, . \end{split} \end{equation} We insert these estimates into \eqref{uL2-dest} and sum over $1 \leq n \leq N-1$. Then, for all $k \leq k_0 \equiv k_0(C_{\tt L}, C_{g})$ and by the implicit version of the discrete Gronwall lemma, there exists a constant $C>0$ such that the assertion \eqref{energy2} holds. \medskip \noindent {\bf 5) Proof of \eqref{energy2-himo} for $p=1$.} To simplify technicalities, we put $F \equiv 0$. Let us denote $\widetilde{\mathfrak{E}}(u^{n}, \overline{u}^n) := \Big[ \frac 12 \|u^{n}\|^2_{\mathbb{L}^2} + \frac{k^2}{4} \|\nabla \overline{u}^{n}\|_{\mathbb{L}^2}^2 \Big]$. Then we can rewrite \eqref{uL2-dest} as \begin{align}\label{uL2-dest-E} \widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1}) - \widetilde{\mathfrak{E}}(u^{n}, \overline{u}^{n-1}) = \mathfrak{K}^n_{1} + \mathfrak{K}^n_{2} + \mathfrak{K}^n_{3}\, . \end{align} Multiply both sides with $\widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1})$, using binomial formula and taking expectation we obtain \begin{equation}\label{E-tilde-u} \begin{split} &\frac 12 \mathbb{E}\Big[\widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) - \widetilde{\mathfrak{E}}^2(u^{n}, \overline{u}^{n-1}) \Big] + \frac 12 \mathbb{E} \Big[\big| \widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1}) - \widetilde{\mathfrak{E}}^2(u^{n}, \overline{u}^{n-1}) \big|^2 \Big] \\ &= \mathbb{E} \Big[ \mathfrak{K}^n_{1}\, \widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1}) \Big] + \mathbb{E} \Big[ \mathfrak{K}^n_{2}\, \widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1}) \Big] +\mathbb{E} \Big[ \mathfrak{K}^n_{3} \,\widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1}) \Big] \,. \end{split} \end{equation} Using Young's inequality, and arguing similarly to (\ref{uL2-desta}) shows \begin{equation} \begin{split} \mathbb{E} \Big[ \mathfrak{K}^n_{1}\, \widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1}) \Big] &\leq \frac{1}{k^3} \mathbb{E} \Big[ \|u^1 - u_0\|_{\mathbb{L}^2}^4 \Big] + k^2 \mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^4 \big] + k\,\mathbb{E} \Big[ \widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) \Big] \\ &\leq Ck + Ck\, \mathbb{E} \Big[ \widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) + \widetilde{\mathfrak{E}}^2(u^{n}, \overline{u}^{n}) \Big]\,. \end{split} \end{equation} By adding and subtracting $\widetilde{\mathfrak{E}}(u^{n}, \overline{u}^{n-1})$, and using {\bf (A3)}, we estimate the second term on the right-hand side of \eqref{E-tilde-u} by \begin{equation} \begin{split} &\mathbb{E} \Big[ \mathfrak{K}^n_{2}\, \big( \widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1}) - \widetilde{\mathfrak{E}}(u^{n}, \overline{u}^{n-1}) \big) \Big] + \mathbb{E} \Big[ \mathfrak{K}^n_{2}\, \widetilde{\mathfrak{E}}(u^{n}, \overline{u}^{n-1}) \Big] \\ &\leq \mathbb{E} \big[ |\mathfrak{K}^n_{2}|^2 \big] + \frac 14 \,\mathbb{E} \Big[ \big| \widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) - \widetilde{\mathfrak{E}}(u^{n}, \overline{u}^{n-1}) \big|^2 \Big] \\ &\quad+ C k^2 C_{\tt L}^2 \sum^n_{\ell=1}\, \mathbb{E} \Big[ 1 + \|u^{\ell} \|_{\mathbb{L}^2}^4 \Big] + C k\, \mathbb{E} \big[\|u^{n+1/2} \|^4_{\mathbb{L}^2} \big] +C k\, \mathbb{E} \big[ \widetilde{\mathfrak{E}}^2(u^{n}, \overline{u}^{n-1}) \big] \\ &\leq C k^2 C_{\tt L}^2 \sum^n_{\ell=1}\, \mathbb{E} \Big[ 1 + \widetilde{\mathfrak{E}}^2(u^{\ell}, \overline{u}^{\ell}) \Big] + C k\, \mathbb{E} \big[ \widetilde{\mathfrak{E}}^2(u^{n}, \overline{u}^{n-1}) \big] \\ &\quad+ \frac 14 \,\mathbb{E} \Big[ \big| \widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) - \widetilde{\mathfrak{E}}(u^{n}, \overline{u}^{n-1}) \big|^2 \Big] \,, \end{split} \end{equation} where the last term in the right-hand side may be absorbed on the left-hand side of \eqref{E-tilde-u}. \noindent By item {\bf 4.} of Remark \ref{rema2}, and {\bf (A4)} for $m=1$ we estimate \begin{equation} \begin{split} &\mathbb{E} \Big[ \mathfrak{K}^n_{3}\, \widetilde{\mathfrak{E}}(u^{n+1}, \overline{u}^{n+1}) \Big] \leq \frac 1k \,\mathbb{E} \big[ |\mathfrak{K}^n_{3}|^2 \big] + Ck\, \mathbb{E} \Big[ \widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) \Big] \\ &\leq \tilde{C}_g^4 \sum^n_{\ell=1} \mathbb{E} \Big[ \|v^{\ell}\|_{\mathbb{L}^2}^4 \big|\widehat{\Delta_{\ell} W} \big|^4 \Big] + C k \,\mathbb{E} \big[ \|u^{n+1/2} \|_{\mathbb{L}^2}^4 \big] + Ck\, \mathbb{E} \Big[ \widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) \Big] \\ &\leq C_g^4 k^6 \frac{1}{k^4} \sum^n_{\ell=1} \mathbb{E} \Big[ \|u^{\ell}\|_{\mathbb{L}^2}^4 + \|u^{\ell-1}\|_{\mathbb{L}^2}^4 \Big] + Ck\, \mathbb{E} \Big[ \widetilde{\mathfrak{E}}^2(u^{n}, \overline{u}^{n}) + \widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) \Big] \\ &\leq C k^2 \sum^n_{\ell=1} \mathbb{E} \Big[ \widetilde{\mathfrak{E}}^2(u^{\ell}, \overline{u}^{\ell}) + \widetilde{\mathfrak{E}}^2(u^{\ell-1}, \overline{u}^{\ell-1}) \Big] + Ck\, \mathbb{E} \Big[ \widetilde{\mathfrak{E}}^2(u^{n}, \overline{u}^{n}) + \widetilde{\mathfrak{E}}^2(u^{n+1}, \overline{u}^{n+1}) \Big]\,. \end{split} \end{equation} Now we insert the estimates into (\ref{E-tilde-u}), and sum over $1 \leq n \leq N-1$. Then, for all $k \leq k_0 \equiv k_0(C_{\tt L}, C_g)$ assertion \eqref{energy2-himo} for $p=1$ follows from the implicit version of the discrete Gronwall lemma. \medskip \noindent {\bf 6) Proof of \eqref{energy2-himo} for $p \geq 2$.} Starting from the identity (\ref{E-tilde-u}), we multiply ${\widetilde{\mathfrak E}}^{2^{p-1}}(u^{n+1},v^{n+1})$ in both sides, and then take the expectation. We may then follow the same argument as in {\bf 5)} to settle the assertion. \end{proof} }} \medskip \del{ {\color{magenta}{ \noindent Now, we briefly discuss the discrete stability analysis for $\widehat{\alpha}-$scheme. \begin{lemma}\label{lem:alpha:stab} Let the assumptions of Lemma \ref{lem:scheme1:stab} hold. Additionally, assume {\bf (A5)}. Then, for $k \leq k_0(C_{\tt L}, C_g)$, there exists an $\big[ \mathbb{H}^1_0 \big]^2$-valued $\{ (\mathcal{F}_{t_{n}})_{0 \le n \le N} \}$-adapted solution $\{(u^{n}, v^{n}); \, 0 \leq n \leq N\}$ of $\widehat{\alpha}-$scheme, and \begin{itemize} \item[$(i)$] there exists a constant $\widetilde{C}_1 > 0$ that does not depend on $k>0$ such that \begin{equation* \max_{1 \le n \le N-1} \mathbb{E} \Bigl[ \mathcal{E}(u^{n+1},v^{n+1}) \Bigr] \le \widetilde{C}_1\, . \end{equation*} \item[$(ii)$] there exist further constant $\widetilde{C}_{2,p}>0$ such that \[ \max_{1 \le n \le N-1} \mathbb{E}\bigl[ {\mathcal E}^{2^p}(u^{n+1},v^{n+1}) \bigr] \leq \widetilde{C}_{2, p} \qquad (p \geq 1)\, . \] \end{itemize} \end{lemma} \begin{proof} {\bf 1) Proof of $(i)$.} Consider the $\widehat{\alpha}-$scheme. We use the same test functions as in the proof of $(\ref{energy1})$ in Lemma \ref{lem:scheme1:stab} to arrive at \begin{equation}\label{tes-mul} \begin{split} &\frac{1}{2} {\mathbb E}\bigl[\|v^{n+1}\|^2_{\mathbb{L}^2} -\|v^{n}\|^2_{\mathbb{L}^2} \bigr] + \frac 14 {\mathbb E}\big[ \|\nabla u^{n+1}\|_{\mathbb{L}^2}^2 - \|\nabla u^{n-1}\|_{\mathbb{L}^2}^2 \big] \\ &= {\mathbb E}\Bigl[\Bigl(\sigma (u^n) \,\Delta_nW, v^{n+\frac 12} \Bigr) \Bigr] + \,\widehat{\alpha} \,{\mathbb E}\Bigl[\Bigl(\partial_u\sigma(u^n) v^n \,\widehat{\Delta_n W}, v^{n+\frac 12} \Bigr) \Bigr] \\ &\qquad+ \frac{k}{2}\, {\mathbb E}\Bigl[ \Bigl( 3F(u^n) - F(u^{n-1}), v^{n+\frac 12}\Bigr)\Bigr] = \sum_{i=1}^3 \mathscr{M}^n_i . \end{split} \end{equation} Using properties of the increments $\Delta_n W$ and that $\sigma (u^n) = 0$ on $\partial {\mathcal O}$, and replacing $v^{n+1} -v^n$ by using \eqref{scheme2:2} we infer {\small{ \begin{equation* \begin{split} \mathbb{E} \big[ \mathscr{M}^n_1 \big] = \frac 12 \mathbb{E}\Bigl[\Bigl(\sigma (u^n)\Delta_n W, v^{n+1} -v^n \Bigr) \Bigr] &=\frac 12 \mathbb{E}\Bigl[\Bigl(\sigma (u^n)\Delta_n W, k \Delta u^{n, \frac 12} \Bigr) \Bigr] + \frac 12 \mathbb{E} \Big[ \|\sigma (u^n)\|_{\mathbb{L}^2}^2 |\Delta_n W|^2 \Big] \\ &\quad+ \frac{\widehat{\alpha}}{2}\, \mathbb{E}\Bigl[\Bigl(\sigma (u^n)\Delta_n W, \partial_u\sigma(u^{n}) v^{n} \,\widehat{\Delta_n W} \Bigr) \Bigr] \\ &\quad+ \frac{k}{4}\,\mathbb{E}\Bigl[\Bigl(\sigma (u^n)\Delta_n W, \bigl(3F(u^n) - F(u^{n-1}) \bigr) \Big) \Bigr] := \sum_{i=1}^4 \mathscr{M}^{n, i}_1\, . \end{split} \end{equation*} }} The term $\mathscr{M}^{n, 1}_1$ is the same as the term $\mathscr{J}^{n, 1}_{1, 1}$ defined in \eqref{j^n,1_1est} in the proof of Lemma \ref{lem:scheme1:stab}. We estimate $\mathscr{M}^{n, 1}_1$ as in \eqref{413a} to get \begin{equation*} \begin{split} \mathscr{M}^{n, 1}_1 &\leq C_g^2 \,k \,\mathbb{E} \big[ \| \nabla u^n \|_{\mathbb{L}^2}^2\big] + C k\, \mathbb{E} \Big[ \| \nabla u^{n+1} \|_{\mathbb{L}^2}^2 +\| \nabla u^{n-1} \|_{\mathbb{L}^2}^2\Big]\,. \end{split} \end{equation*} Using the It\^o isometry we infer \begin{align*} \mathscr{M}^{n, 2}_1 &\leq C_{\tt L} k \,\mathbb{E} \big[ 1 +\| \nabla u^n \|_{\mathbb{L}^2}^2\big] \,. \end{align*} Using {\bf (A4)$_{i}$}, and the fact that ${\mathbb E}\bigl[ \vert \widehat{\Delta_n W}\vert^2 \bigr] \leq Ck^3$, we get \begin{equation*} \begin{split} \mathscr{M}^{n, 3}_1 &\leq \frac 12 \mathbb{E} \Big[ \|\sigma (u^n)\|_{\mathbb{L}^2}^2 |\Delta_n W|^2 \Big] + \frac{\widehat{\alpha}^2}{8}\mathbb{E} \Big[ \|\partial_u \sigma (u^n) v^n \|_{\mathbb{L}^2}^2 |\widehat{\Delta_n W}|^2 \Big] \\ &\leq C_{\tt L} k \,\mathbb{E} \big[ 1+\| \nabla u^n \|_{\mathbb{L}^2}^2\big] + \frac{\widehat{\alpha}^2 C_g^2}{8} k^3 \,\mathbb{E} \big[ \|v^n \|_{\mathbb{L}^2}^2\big]\,. \end{split} \end{equation*} Again, using the It\^o isometry we infer \begin{equation*} \begin{split} \mathscr{M}^{n, 4}_1 &\leq \mathbb{E} \Big[ \|\sigma (u^n)\|_{\mathbb{L}^2}^2 |\Delta_n W|^2 \Big] + C k^2 \mathbb{E} \Big[ \|F(u^n)\|_{\mathbb{L}^2}^2 + \|F(u^{n-1})\|_{\mathbb{L}^2}^2 \Big] \\ &\leq C_{\tt L} k \,\mathbb{E} \big[ 1+\| \nabla u^n \|_{\mathbb{L}^2}^2\big] + C(C_{\tt L}) k^2 \,\mathbb{E} \Big[ 1+\| \nabla u^n \|_{\mathbb{L}^2}^2 + \| \nabla u^{n-1} \|_{\mathbb{L}^2}^2 \Big]\, . \end{split} \end{equation*} Now, we estimate the term $\mathscr{M}^{n}_2$ in \eqref{tes-mul}. Using {\bf (A4)$_{i}$}, and the fact that ${\mathbb E}\bigl[ \vert \widehat{\Delta_n W}\vert^2 \bigr] \leq Ck^3$, we obtain, \begin{equation*} \begin{split} \mathscr{M}^{n}_2 &\leq \frac{\widehat{\alpha}^2}{2} \frac{1}{k} \,\mathbb{E} \Big[ \|\partial_u \sigma (u^n) v^n \|_{\mathbb{L}^2}^2 |\widehat{\Delta_n W}|^2 \Big] + \frac{k}{2} \,\mathbb{E} \big[ \|v^{n+\frac 12} \|_{\mathbb{L}^2}^2\big] \\ &\leq \frac{\widehat{\alpha}^2 C_g^2}{2} k^2 \,\mathbb{E} \big[ \|v^n \|_{\mathbb{L}^2}^2 \big] + C k\,\mathbb{E} \Big[ \|v^n \|_{\mathbb{L}^2}^2 + \|v^{n+1} \|_{\mathbb{L}^2}^2 \Big]\, . \end{split} \end{equation*} Finally, we estimate the term $\mathscr{M}^{n}_3$ as \begin{equation*} \begin{split} \mathscr{M}^{n}_3 &\leq C k\, \mathbb{E} \Big[ \|F(u^n)\|_{\mathbb{L}^2}^2 + \|F(u^{n-1})\|_{\mathbb{L}^2}^2 \Big] + C k\,\mathbb{E} \big[ \|v^{n+\frac 12} \|_{\mathbb{L}^2}^2\big] \\ &\leq C k \,\mathbb{E} \Big[ 1+\| \nabla u^n \|_{\mathbb{L}^2}^2 + \| \nabla u^{n-1} \|_{\mathbb{L}^2}^2 +\|v^n \|_{\mathbb{L}^2}^2 + \|v^{n+1} \|_{\mathbb{L}^2}^2 \Big]\,. \end{split} \end{equation*} Finally, insert the above estimates into (\ref{tes-mul}), and sum over $1 \leq n \leq N-1$. Then, for all $k \leq k_0 \equiv k_0(C_{\tt L}, C_g)$, the assertion $(i)$ follows from the implicit version of the discrete Gronwall lemma. \smallskip \noindent {\bf 2) Proof of $(ii)$ for $p=1$.} Consider the $\widehat{\alpha}-$scheme. For simplify technicalities, we put $F \equiv 0$. Recall the notation $\mathfrak{E}$ in step {\bf 2)} of Lemma \ref{lem:scheme1:stab}. Arguing as before \eqref{tes-mul} then leads to ${\mathbb P}$-a.s. \begin{equation}\label{frakE-u} \begin{split} &\mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n})= \Big(\sigma(u^{n}) \Delta_n W, v^{n + \frac 12} \Big)+ \widehat{\alpha}\,\Big( \partial_u\sigma(u^{n}) v^{n} \,\widehat{\Delta_n W}, v^{n + \frac 12} \Big)\, . \end{split} \end{equation} Multiplying both sides with $\mathfrak{E}(u^{n+1}, v^{n+1})$, using binomial formula and taking expectation we infer \begin{equation}\label{bE-frakE} \begin{split} &\frac 12 \mathbb{E} \Big[ \mathfrak{E}^2(u^{n+1}, v^{n+1}) - \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] + \frac12 \mathbb{E} \Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big] \\ &= \mathbb{E} \Big[ \Big(\sigma(u^{n}) \Delta_n W, v^{n + \frac 12} \Big) \cdot \mathfrak{E}(u^{n+1}, v^{n+1}) \Big] \\ &\quad+ \widehat{\alpha}\,\mathbb{E} \Big[ \Big( \partial_u\sigma(u^{n}) v^{n} \,\widehat{\Delta_n W}, v^{n + \frac 12} \Big) \cdot \mathfrak{E}(u^{n+1}, v^{n+1}) \Big] =: \mathfrak{N}_1 + \mathfrak{N}_2\, . \end{split} \end{equation} We can split $\mathfrak{N}_1$ into two terms as \begin{align*} \mathfrak{N}_1 &= \mathbb{E} \Big[ \Big(\sigma(u^{n}) \Delta_n W, v^{n + \frac 12} \Big) \cdot \big\{ \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big\} \Big] \\ &\quad+ \mathbb{E} \Big[ \Big(\sigma(u^{n}) \Delta_n W, v^{n + \frac 12} \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] =: \mathfrak{N}_{1, 1} + \mathfrak{N}_{1, 2}\, . \end{align*} We first estimate $\mathfrak{N}_{1, 1}$. Using Young's inequality and ${\mathbb E}\bigl[\vert \Delta_n W\vert^4 \bigr] = {\mathcal O}(k^2)$ we infer, \begin{align*} \mathfrak{N}_{1, 1} &\leq \mathbb{E} \Big[ \big\|\sigma(u^{n}) \Delta_n W \big\|_{\mathbb{L}^2}^2 \big\| v^{n + \frac 12} \big\|_{\mathbb{L}^2}^2 \Big] + \frac 14 \mathbb{E} \Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big] \\ &\leq \frac 1k \mathbb{E} \Big[ \|\sigma(u^{n}) \|_{\mathbb{L}^2}^4\, |\Delta_n W|^4 \Big] + k\, \mathbb{E} \big[ \| v^{n + \frac 12} \|_{\mathbb{L}^2}^4 \big] + \frac 14 \mathbb{E} \Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big] \\ &\leq C k \,\mathbb{E} \big[ \| \nabla u^{n} \|_{\mathbb{L}^2}^4 \big] + C k\,\mathbb{E} \Big[ \| v^{n+1} \|_{\mathbb{L}^2}^4 + \| v^{n} \|_{\mathbb{L}^2}^4 \Big] + \frac 14 \mathbb{E} \Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big] \\ &\leq C k \,\mathbb{E} \Big[ \mathfrak{E}^2(u^{n+1}, v^{n+1}) + \mathfrak{E}^2(u^{n}, v^{n}) \Big] + \frac 14 \mathbb{E} \Big[ \big| \mathfrak{E}(u^{n+1}, v^{n+1}) - \mathfrak{E}(u^{n-1}, v^{n}) \big|^2 \Big]\, , \end{align*} where the last term in the right-hand side may be managed in the left-hand side of \eqref{bE-frakE}. Using independence properties of $\Delta_n W$ and then using \eqref{scheme2:2} we infer \begin{align*} \mathfrak{N}_{1, 2} &= \mathbb{E} \Big[ \Big(\sigma(u^{n}) \Delta_n W, v^{n + 1} - v^n \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] \\ &= \mathbb{E} \Big[ \Big(\sigma(u^{n}) \Delta_n W, k \Delta u^{n, \frac 12} \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] + \mathbb{E} \Big[ \|\sigma(u^{n}) \|_{\mathbb{L}^2}^2 |\Delta_n W|^2 \,\mathfrak{E}(u^{n-1}, v^{n}) \Big] \\ &\quad+ \widehat{\alpha}\,\mathbb{E} \Big[ \Big(\sigma(u^{n}) \Delta_n W, \partial_u \sigma(u^n) v^n \widehat{\Delta_n W}\Big) \cdot \mathfrak{E}(u^{n-1}, v^{n}) \Big] =: \mathfrak{N}_{1, 2}^{\tt A} + \mathfrak{N}_{1, 2}^{\tt B} + \mathfrak{N}_{1, 2}^{\tt C}\, . \end{align*} Using integration by parts, {\bf (A4)$_{i}$} and ${\mathbb E}\bigl[\vert \Delta_n W\vert^4 \bigr] = {\mathcal O}(k^2)$ we get \begin{align*} \mathfrak{N}_{1, 2}^{\tt A} &= -\mathbb{E} \Big[ \Big(\partial_u \sigma(u^{n}) \nabla u^n \Delta_n W, k \nabla u^{n, \frac 12} \Big) \cdot \mathfrak{E}(u^{n-1}, v^{n})\Big] \\ &\leq \frac 1k \mathbb{E} \Big[ \big\|\partial_u \sigma(u^{n}) \nabla u^n \Delta_n W \big\|_{\mathbb{L}^2}^2 \,k^2\, \|\nabla u^{n, \frac 12}\|_{\mathbb{L}^2}^2 \Big] + k\,\mathbb{E} \Big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] \\ &\leq \frac{C_g^4}{k} \mathbb{E} \Big[ \|\nabla u^n \|_{\mathbb{L}^2}^4 \,|\Delta_n W|^4 \Big] +C k^3\, \mathbb{E} \Big[ \|\nabla u^{n+1}\|_{\mathbb{L}^2}^2 + \|\nabla u^{n-1}\|_{\mathbb{L}^2}^2 \Big] + k\,\mathbb{E} \Big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \Big] \\ &\leq C k\,\mathbb{E} \Big[ \mathfrak{E}^2(u^{n}, v^{n}) + \mathfrak{E}^2(u^{n+1}, v^{n+1}) +\mathfrak{E}^2(u^{n-1}, v^{n}) \Big]\, . \end{align*} Using ${\mathbb E}\bigl[\vert \Delta_n W\vert^4 \bigr] = {\mathcal O}(k^2)$ we infer, \begin{align*} \mathfrak{N}_{1, 2}^{\tt B} &\leq C k \,\mathbb{E} \big[ \|\nabla u^n \|_{\mathbb{L}^2}^4 \big] + C k \,\mathbb{E} \big[\mathfrak{E}^2(u^{n-1}, v^{n}) \big] \leq C k \,\mathbb{E} \Big[ \mathfrak{E}^2(u^{n}, v^{n}) + \mathfrak{E}^2(u^{n-1}, v^{n}) \Big]\, . \end{align*} Using {\bf (A4)$_{i}$}, ${\mathbb E}\bigl[\vert \Delta_n W\vert^4 \bigr] = {\mathcal O}(k^2)$ and ${\mathbb E}\bigl[\vert \widehat{\Delta_n W}\vert^4 \bigr] = {\mathcal O}(k^6)$ we obtain \begin{align*} \mathfrak{N}_{1, 2}^{\tt C} &\leq \frac{\widehat{\alpha}^2}{k} \mathbb{E} \Big[ \| \sigma(u^n) \Delta_n W\|_{\mathbb{L}^2}^2\, \|\partial_u \sigma(u^n) v^n \widehat{\Delta_n W}\|_{\mathbb{L}^2}^2 \Big] + C k \,\mathbb{E} \big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \big] \\ &\leq \frac 1k \mathbb{E} \Big[ \|\nabla u^n \|_{\mathbb{L}^2}^4 |\Delta_n W|^4 \Big] + \frac{\widehat{\alpha}^4 C_g^4}{k} \mathbb{E} \Big[ \|v^n\|_{\mathbb{L}^2}^2 |\widehat{\Delta_n W}|^4 \Big] + C k \,\mathbb{E} \big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \big] \\ &\leq C k \,\mathbb{E} \big[ \mathfrak{E}^2(u^{n}, v^{n}) \big] + \widehat{\alpha}^4 C_g^4 k^5 \,\mathbb{E} \big[ \mathfrak{E}^2(u^{n}, v^{n}) \big] + C k \,\mathbb{E} \big[ \mathfrak{E}^2(u^{n-1}, v^{n}) \big]\, . \end{align*} Similarly, we estimate $\mathfrak{N}_2$ as \begin{align*} \mathfrak{N}_{2} &\leq \frac{\widehat{\alpha}^2}{k} \mathbb{E} \Big[ \big\| \partial_u \sigma(u^n) v^n \widehat{\Delta_n W} \big\|_{\mathbb{L}^2}^2 \|v^{n+\frac 12}\|_{\mathbb{L}^2}^2 \Big] + C k\,\mathbb{E} \big[ \mathfrak{E}^2(u^{n+1}, v^{n+1}) \big] \\ &\leq \frac{\widehat{\alpha}^2}{k^2} \mathbb{E} \Big[ \big\| \partial_u \sigma(u^n) v^n \big\|_{\mathbb{L}^2}^4 |\widehat{\Delta_n W} |^4 \Big] + C k\,\mathbb{E} \big[ \|v^{n+\frac 12}\|_{\mathbb{L}^2}^4 \big] + C k\,\mathbb{E} \big[ \mathfrak{E}^2(u^{n+1}, v^{n+1}) \big] \\ &\leq C_g^4\, \widehat{\alpha}^4\, k^4\, \mathbb{E} \big[ \|v^n\|_{\mathbb{L}^2}^4 \big] + C k\,\mathbb{E} \Big[ \|v^{n+1}\|_{\mathbb{L}^2}^4 + \|v^{n}\|_{\mathbb{L}^2}^4 \Big] + C k\,\mathbb{E} \big[ \mathfrak{E}^2(u^{n+1}, v^{n+1}) \big] \\ &\leq C k \,\mathbb{E} \Big[ \mathfrak{E}^2(u^{n+1}, v^{n+1}) + \mathfrak{E}^2(u^{n}, v^{n}) \Big] \, . \end{align*} Inserting the above estimates into (\ref{bE-frakE}), and sum over $1 \leq n \leq N-1$. Then, for all $k \leq k_0 \equiv k_0(C_{\tt L}, C_g)$ assertion $(ii)$ for $p=1$ follows from the implicit version of the discrete Gronwall lemma. For $p \geq 2$, we start from the identity (\ref{bE-frakE}), then multiply $ {\mathfrak E}^{2^{p-1}}(u^{n+1},v^{n+1})$ in both sides, then take the expectation and follow the same argument as in {\bf 2)} to settle the assertion. \end{proof} }} } \medskip \section{Strong Rates of Convergence for $(\widehat{\alpha}, \beta)-$scheme}\label{sec-4} We prove convergence rate ${\mathcal O}(k^{1/2})$ for the iterates $\{ (u^n, v^n)\}_{n\geq1}$ of the $(\widehat{\alpha}, \beta)-$scheme for $\widehat{\alpha} \in \{0, 1\}$; if additionally $\sigma_2(v) \equiv 0 \equiv F_2(v)$ in {\bf (A3)} holds, we may put $\beta = 0$, and \begin{itemize} \item[a)] the convergence rate improves to ${\mathcal O}(k)$ for iterates $\{ u^n\}_{n\geq 1}$ in case $\widehat{\alpha}=0$, and \item[b)] to ${\mathcal O}(k^{3/2})$ in case $\widehat{\alpha}=1$. \end{itemize} For the convergence analysis, we need the following assumption on $(u^1, v^1)$.\\ \noindent {\bf (B2)} Along with {\bf (A1)}$_{ii}$ and {\bf (B1)}$_{ii}$, let $u^0 = u(0)$ and $v^0 = v(0)$, and $(u^1, v^1)$ satisfy \begin{align* \Big( \mathbb{E} \Big[ \|u(t_1) - u^1\|_{\mathbb{H}^1}^2 + \|v(t_1) - v^1\|_{\mathbb{L}^2}^2 \Big] \Big)^{1/2} = \mathcal{O}\big(k^{1/2} \big). \end{align*} \smallskip \begin{theorem}\label{lem:scheme1:con} Let $(u,v)$ be the strong solution of (\ref{stoch-wave1:1a}) with $A=-\Delta$. Let $\{ (u^n, v^n)\}_{n\geq1}$ be the iterates from $(\widehat{\alpha}, \beta)-$scheme for $k \leq k_0(C_{\tt L}, C_g)$ sufficiently small, $\widehat{\alpha} \in \{0, 1\}$, and {\color{black}{$0 < \beta < \frac{1}{2}$}}. Then, under the hypotheses {\bf (A1)$_{iii}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1, 2$, and {\bf (B2)}, there exists {\color{black}{$C \equiv C(\beta) >0$}} such that \begin{align}\label{eq-5.32} \max_{1 \le n \le N} \Big( \mathbb{E}\Bigl[ \| u(t_n) - u^n\|^2_{\mathbb{H}^1} + \|v(t_n) - v^n\|^2_{\mathbb{L}^2} \Bigr] \Big)^{1/2} \le C k^{1/2}\,. \end{align} For the following, additionally suppose $\sigma_2(v) \equiv 0 \equiv F_2(v)$ in {\bf (A3)} and that the initial data $u^0, u^1, v^0$ satisfy {{ \begin{equation}\label{ini-1} \begin{split} &\bigg( {\mathbb E}\Bigl[ \Vert u(t_1) - u^1\Vert^2_{{\mathbb L}^2} \Bigr] \\ &\quad+ \frac{1}{k^2}\, {\color{black}{{\mathbb E} \Bigl[ \big\Vert k v_0 - (u^1 - u^0) + \frac{k^2}{2} \Delta u_0 + k^2 F(u_0) + k \sigma(u_0) \Delta_0 W \big\Vert_{{\mathbb L}^2}^2 \Bigr] }} \bigg)^{1/2} = \mathcal{O}\big(k^{3/2} \big)\, . \end{split} \end{equation} }} \begin{itemize} \item[$(i)$] Consider the $(0, 0)-$scheme and assume {\bf (A1)$_{iii}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1, 2$, and {\bf (B2)}. Then there exists $C >0$ such that \begin{align}\label{pi} \max_{1 \le n \le N} \Big( \mathbb{E}\Bigl[ \| u(t_n, \cdot) - u^{n}\|^2_{\mathbb{L}^2}\Bigr] \Big)^{1/2} + \frac 12 \bigg(\mathbb{E} \bigg[ k \sum_{j=1}^n \bigl\Vert \nabla \big[ u(t_{j}, \cdot)- u^{j} \big] \bigr\Vert^2_{{\mathbb L}^2} \bigg] \bigg)^{1/2} \le C k\, . \end{align} \end{itemize} \begin{itemize} \item[$(ii)$] Consider the $(1, 0)-$scheme and assume {\bf (A1)$_{iv}$}, {\bf (A2), (A3)}, and {\bf (A4), (A5)} for $m=1, 2, 3$, and {\bf (B2)}. Then, there exists $C>0$ such that \begin{align}\label{pii} \max_{1 \le n \le N} \Big( \mathbb{E}\Bigl[ \| u(t_n, \cdot) - u^{n}\|^2_{\mathbb{L}^2}\Bigr] \Big)^{1/2} + \frac 12 \bigg(\mathbb{E} \bigg[ k \sum_{j=1}^n \bigl\Vert \nabla \big[ u(t_{j}, \cdot)- u^{j} \big] \bigr\Vert^2_{{\mathbb L}^2} \bigg] \bigg)^{1/2} \le C k^{3/2}\, . \end{align} \end{itemize} \end{theorem} \smallskip \noindent The following remark discusses the realizability of (\ref{ini-1}), and key tools to verify this theorem. \begin{remark} {{ {\bf 1.}~{{In Section \ref{sec-6}, we choose $(u^0, v^0) = (u(0), v(0))$, together with \begin{align}\label{u1v1choice} \begin{cases} {\color{black}{u^1 = u_0 +k\,v_0 + \frac{k^2}{2} \Delta u_0 + k^2 F(u_0) + (k+k^2) \,\sigma(u_0) \Delta_0 W}}\,,\\ v^1 = v_0 + k \,\sigma(u_0) \Delta_0 W\, . \end{cases} \end{align} By the choice of $v^1$, the verification of {\bf (B2)} is straight-forward. We now prove that \eqref{ini-1} holds in this case: first, we consider \eqref{stoch-wave1:1a} in integral form on $[0,t_1]$, \begin{align}\label{sto-wave1:2a} \begin{cases} &u(t_1) = u_0 + \int_0^{t_1} v(s)\,{\rm d} s \\ &v(s) = v_0+\int_0^{s} \Delta u(\tau) {\rm d} \tau + \int_0^{s} F(u(\tau)) {\rm d} \tau + \int_0^{s} \sigma(u(\tau)) {\rm d} W(\tau)\,, \quad 0 \leq \tau \leq s\, , \end{cases} \end{align} and insert (\ref{sto-wave1:2a})$_2$ into (\ref{sto-wave1:2a})$_1$; a change of order of integration then gives {\small{ \begin{equation}\label{stoch-wave1:3a} \begin{split} u(t_1) &= u_0 + t_1 v_0 + \int_0^{t_1} \int_0^s \Delta u(\tau)\, {\rm d} \tau {\rm d} s + \int_0^{t_1} \int_0^s F(u(\tau))\, {\rm d} \tau {\rm d} s + \int_0^{t_1} \int_0^s \sigma(u(\tau)) {\rm d} W(\tau) \, {\rm d} s \\ & = u_0 + k v_0 + \int_0^{t_1} \int_{\tau}^{t_1} {\rm d} s \,\Delta u(\tau)\, {\rm d} \tau + \int_0^{t_1} \int_{\tau}^{t_1} {\rm d} s \,F(u(\tau)) {\rm d} \tau + \int_0^{t_1} \int_{\tau}^{t_1} {\rm d} s \,\sigma(u(\tau)) {\rm d} W(\tau)\, . \end{split} \end{equation} }} \smallskip \noindent Thus, \begin{equation}\label{u(t_1)} \begin{split} u(t_1) &= u_0 + k v_0 + \int_0^{t_1} (t_1 - \tau) \Delta u(\tau)\,{\rm d} \tau + \int_0^{t_1} (t_1 - \tau) F(u(\tau))\,{\rm d} \tau \\ &\quad+ \int_0^{t_1} (t_1 - \tau) \sigma(u(\tau)) {\rm d} W(\tau)\, . \end{split} \end{equation} \noindent Subtracting \eqref{u1v1choice}$_{1}$ from \eqref{u(t_1)} we infer \begin{align*} u(t_1) - u^1 &= \int_0^{t_1} (t_1 - \tau) \Delta u(\tau)\,{\rm d} \tau - \frac{k^2}{2} \Delta u_0 - k^2 F(u_0) + \int_0^{t_1} (t_1 - \tau) F(u(\tau))\,{\rm d} \tau \\ &\quad+ \int_0^{t_1} (t_1 - \tau) \sigma(u(\tau)) {\rm d} W(\tau) - (k+k^2) \sigma(u_0) \big(W(t_1) - W(0)\big)\,. \end{align*} By {\bf (A1)$_{iii}$, (A3)}, It\^o isometry, Lemma \ref{lem:L2} $(i), (ii)$ and Lemma \ref{lem:Holder} $(i)$ we infer \begin{align} \mathbb{E} \Big[ \|u(t_1) - u^1\|_{\mathbb{L}^2}^2 \Big] &\leq C k^2\, \int_0^{t_1} \mathbb{E} \Big[ \| \Delta u(\tau)\|_{\mathbb{L}^2}^2 \Big] \,{\rm d} \tau + C k^4 \,\mathbb{E} \big[ \|u_0 \|_{\mathbb{H}^2}^2 \big] + C k^4 \,\mathbb{E} \big[ \|F(u_0) \|_{\mathbb{L}^2}^2 \big] \nonumber \\ &\quad+ C k^2\, \int_0^{t_1} \mathbb{E} \Big[ \| F(u(\tau))\|_{\mathbb{L}^2}^2 \Big] \,{\rm d} \tau + k^3\, \mathbb{E} \big[ \| \sigma(u_0)\|_{\mathbb{L}^2}^2 \big] \label{eq_1.7} \\ &\quad + C k^2\, \int_0^{t_1} \mathbb{E} \Big[ \| \sigma(u(\tau)) - \sigma(u_0) \|_{\mathbb{L}^2}^2 \Big]\, {\rm d} s \leq C k^3 \, .\nonumber \end{align} Similarly, by \eqref{u1v1choice}$_{1}$, {\bf (A1)$_{i}$} and It\^o isometry we get \begin{equation}\label{eq_1.8} \begin{split} &{\mathbb E} \Bigl[ \big\Vert k v_0 - (u^1 - u^0)+ \frac{k^2}{2} \Delta u_0 +k^2 F(u_0)+ k \,\sigma(u_0) \Delta_0 W \big\Vert_{{\mathbb L}^2}^2 \Bigr] \\ &\leq k^4\, \mathbb{E} \Big[ \|\sigma(u_0) \|_{\mathbb{L}^2}^2 |\Delta_0 W|^2 \Big] \leq C k^5 \, . \end{split} \end{equation} Thus, combining \eqref{eq_1.7} and \eqref{eq_1.8} we get the assertion \eqref{ini-1} for $u^1$. \del{ To validate the choice of $v^1$ in \eqref{u1v1choice}$_{2}$, we use {\bf (A1)$_{i}$} and It\^o isometry to get \begin{align*} &{\mathbb E} \Bigl[ \big\Vert k v_0 - (u^1 - u^0) + \frac{k^2}{2} \Delta u_0 + k^2 F(u_0) + k \,\sigma(u_0) \Delta_0 W \big\Vert_{{\mathbb L}^2}^2 \Bigr] \\ &= {\mathbb E} \Bigl[\Vert k v_0 - k v^1 \Vert_{{\mathbb L}^2}^2 \Bigr] = k^2 {\mathbb E} \Bigl[\Vert v^1 - v_0 \Vert_{{\mathbb L}^2}^2 \Bigr] \leq k^4\, \mathbb{E} \Big[ \|\sigma(u_0) \|_{\mathbb{L}^2}^2 |\Delta_0 W|^2 \Big] \leq C k^5 \, , \end{align*} which settles the assertion \eqref{ini-1}. } }} }} \smallskip \noindent {\bf 2.}~For $\widetilde{\alpha} \neq 0$, the additional noise term in \eqref{scheme2:2} improves the accuracy of the $(\widehat{\alpha}, 0)-$scheme, where $\widetilde{\Delta_n W}$ is approximated by $\widehat{\Delta_n W}\,.$ By \eqref{W-hat}, \eqref{DnW-tilde}, and the fact that $t_{n,\ell+1} - t_{n,\ell}=k^2$, we estimate the distance between $\widetilde{\Delta_n W}$ and $\widehat{\Delta_n W}$ as \begin{equation*} \begin{split} {\mathbb E}\Bigl[ \big\vert \widetilde{\Delta_n W} - \widehat{\Delta_n W} \big\vert^2 \Bigr] &= {\mathbb E}\biggl[ \Big| - \int_{t_n}^{t_{n+1}} W(s)\,{\rm d}s + k^2 \sum_{\ell=1}^{k^{-1}} W(t_{n,\ell}) \Big|^2 \biggr] \\ &= {\mathbb E}\biggl[ \Big| \sum_{\ell=1}^{k^{-1}} \int_{t_{n, \ell}}^{t_{n, \ell+1}} \big( W(s)- W(t_{n,\ell}) \big)\, {\rm d}s\Big|^2 \biggr]\,. \end{split} \end{equation*} By the independence property of the increment $\Delta_n W$, we further estimate \begin{equation}\label{dist-tilde-hat} \begin{split} &\leq k \sum_{\ell=1}^{k^{-1}} \int_{t_{n, \ell}}^{t_{n, \ell+1}} {\mathbb E}\Bigl[ \big| W(s)- W(t_{n,\ell}) \big|^2 \Bigr] \,{\rm d}s \leq k \sum_{\ell=1}^{k^{-1}} \int_{t_{n, \ell}}^{t_{n, \ell+1}} (s-t_{n,\ell}) \,{\rm d}s \leq C k^4\,. \end{split} \end{equation} \smallskip \noindent {\bf 3.}~The basic estimate is \eqref{eq-5.32}, which will be given in part {\bf 1)} in the proof below. Its derivation uses the H\"older estimates in Lemma \ref{lem:Holder} for $(u,v)$ in {\em strong} norms. The strategy of proof is similar to the one used in the stability analysis for $(\widehat{\alpha}, \beta)-$scheme in Section \ref{sec-3}; see item {\bf 1.}~in Remark \ref{rema2}: the central term to estimate is $T_4^{(n)}$ in \eqref{e1_1:9}, in which we replace the increments $e^{n+1}_v-e^{n}_v$ via the {\em error equation \eqref{e1_1:4}} to obtain terms which are scaled by $k$, or the stochastic increments ${\Delta_n W}$ and $\widetilde{\Delta_n W}$. The order limiting term then is $T_{4, 1}^{(n,4)}$ in \eqref{limiti}, which may be traced back to the noise term $\sigma$, which may depend on $v$ as well. In this case (only), the {\em additional term $-k^{2+\beta} \Delta v^{n+1/2}$} in Scheme \ref{scheme--2} is needed to control the effect of noise: see the additional term on the left-hand side of \eqref{e1_1:6} to {\em e.g.}~bound the corresponding term in \eqref{er-T4,1}. The verification of assertions \eqref{pi} and \eqref{pii} differs completely from this strategy: it starts with the reformulation \eqref{scheme2:3} that leads to the error identity \eqref{error-eq}, which then is tested with $e_u^{n+1/2}$; the noise part may here be estimated in a straight manner. \smallskip \noindent {\bf 4.}~Part {\bf 2)} in the proof below is conceptually motivated from arguments in \cite{Bak_76}; however, their realization in the stochastic setting differs considerably. We remark that estimate \eqref{eq-5.32} is needed to verify assertion \eqref{pi} --- next to Lemma \ref{quadrature1} to bound the quadrature error of the trapezoidal rule for integrands with limited regularity; see term $I^{\ell, n}_7$ in \eqref{e1_2:5}. \smallskip \noindent {\bf 5.}~If $\widetilde{\alpha} = 0$, the estimate \eqref{restri} for term $I_3^{\ell,n}$ in \eqref{e1_2:5} restricts the order, and assertion \eqref{pi} follows; the improvement \eqref{pii} uses $\widetilde{\alpha} = 1$, {\em s.t.} this term $I_3^{\ell,n}$ gives way to the sum of new terms in \eqref{e2_2:3}, which are of higher order; see {\bf a)-- e)} in part {\bf 3)} in the proof below. \smallskip \noindent {\bf 6.}~For $\sigma \equiv \sigma(u, v)$ {\em or} $F \equiv F(u, v)$, neither assertion \eqref{pi} nor \eqref{pii} in Theorem \ref{lem:scheme1:con} may be concluded, due to the restricted H\"older regularity properties of $v$ opposed to $u$. In this setting, either $\sigma$ {\em or} $F$ in \eqref{scheme2:3} in the proof below would depend on $v$ as well, and thus would modify corresponding terms in \eqref{error-eq}. For $\sigma \equiv \sigma(u,v)$, (a modified version of) {\bf (A3)} would additionally create a term $Ck^2 \sum_{\ell=1}^n {\mathbb E}\bigl[ \Vert e^{\ell}_v\Vert^2_{{\mathbb L}^2}\bigr]$ on the right-hand side of \eqref{dazu1}, which may not be handled via Gronwall's lemma to lift the order. For $F \equiv F(u,v)$, the argument in \eqref{dazu2} fails, which rests on Lemma~\ref{quadrature1}, and the H\"older continuity of $v = \partial_t u$. \smallskip \noindent {\bf 7.}~In the proof of \eqref{eq-5.32}, where $\sigma \equiv \sigma(u, v)$ and $F \equiv F(u, v)$, we do not require the discrete energy bounds proved in Lemma \ref{lem:scheme1:stab}. We only require the energy bounds proved in Lemma \ref{lem:L2}. This is possible, since we can add and subtract $\nabla u(t_n)$ or $v(t_n)$ whenever $L^2$-norm of $\nabla u^n$ or $v^n$ appears. However, the higher moment bounds in energy norm (proved in Lemma \ref{lem:scheme1:stab}) are required to show the improved convergence order $\mathcal{O}(k^{3/2})$ in the proof of \eqref{pii} of Theorem \ref{lem:scheme1:con}. \end{remark} \begin{proof}[Proof of Theorem \ref{lem:scheme1:con}] {\bf 1) Proof of \eqref{eq-5.32}}. For simplicity, we here give the proof for $F \equiv 0$. Correspondingly, let $(u,v)$ solve \eqref{stoch-wave1:1a}, and $\{(u^n, v^n) \}_{n\in \mathbb{N}}$ solves $(\widehat{\alpha}, \beta)-$scheme. We denote by $e_{u}^{n}:= u(t_n)-u^n$ and $e_v^n:= v(t_n)-v^n$ error iterates, which are zero on the boundary and solve \del{ By \eqref{strong1}--\eqref{strong2}, for all $t_n \in [0,T], n \ge 0$, by regularity theory we have \begin{eqnarray} \label{e1_1:1} \bigl( \nabla [u(t_{n+1}) - u(t_n)], \phi\bigr) &=& \Bigl(\int_{t_n}^{t_{n+1}} \nabla [v - v(t_{n+1}) + v(t_{n+1})]\,{\rm d}s, \phi\Bigr)\, , \\ \nonumber \bigl(v(t_{n+1})-v(t_n), \psi\bigr) &=& -\int_{t_n}^{t_{n+1}} (\nabla u, \nabla \psi ) \, {\rm d}s + \int_{t_n}^{t_{n+1}} \bigl( F(u,v), \psi\bigr) \, {\rm d}s \\ \label{e1_1:2} && +\Bigl(\int_{t_n}^{t_{n+1}} \sigma (u, v) \, {\rm d}W(s),\psi\Bigr)\, . \end{eqnarray} Subtracting \eqref{scheme2:1}--\eqref{scheme2:2} from \eqref{e1_1:1}--\eqref{e1_1:2} and } \begin{eqnarray} \label{e1_1:3} \quad &&e^{n+1}_u-e^n_u = k \,e^{n+1}_v + \int_{t_n}^{t_{n+1}} \bigl(v(s)-v(t_{n+1})\bigr)\, {\rm d}s\, ,\\ \nonumber \quad &&e^{n+1}_v - e^n_v = k \,\Delta e^{n,1/2}_u + \int_{t_n}^{t_{n+1}} \Delta \bigg[ \frac{ 2 u(s) - [u(t_{n+1})+u(t_{n-1})]}{2} \bigg]\, {\rm d}s \\ \nonumber \quad &&\qquad \qquad \quad - \beta k^{2+\beta}\,\Delta e_v^{n+\frac 12} + \beta \frac{k^{2+\beta}}{2}\,\Delta \bigl[v(t_{n+1}) + v(t_n)\bigr]\\ \label{e1_1:4} \quad && \qquad \qquad \quad + \int_{t_n}^{t_{n+1}} \bigl[\sigma\bigl(u(s), v(s)\bigr) - \sigma\bigl(u^n, v^{n-\frac 12} \bigr) \bigr] \, {\rm d}W(s) - \widehat{\alpha}\,D_u\sigma(u^n, v^{n-\frac 12})\,v^n\, \widehat{\Delta_n W}\,. \end{eqnarray} We multiply \eqref{e1_1:4} with $e^{n+\frac 12}_v$ and use \eqref{e1_1:3} to get \begin{equation}\label{e1_1:6} \begin{split} \frac 12 \Big[ \|e^{n+1}_v \|_{\mathbb{L}^2}^2 - \|e^{n}_v \|_{\mathbb{L}^2}^2 \Big] + \frac 14 \Big[ \|\nabla e^{n+1}_u \|_{\mathbb{L}^2}^2 - \|\nabla e^{n-1}_u \|_{\mathbb{L}^2}^2 \Big] + \beta k^{2+\beta} \Vert \nabla e^{n+\frac 12}_v\Vert^2_{{\mathbb L}^2}\leq \sum_{j=1}^5 T_j^{(n)}\, , \end{split} \end{equation} where \begin{align*} T_1^{(n)} &:= \int_{t_n}^{t_{n+1}} \Bigl(\nabla \bigl[v(s) -v(t_{n+1})\bigr], \nabla e^{n,\frac 12}_u \Bigr) \,{\rm d}s + \int_{t_{n-1}}^{t_{n}} \Bigl(\nabla \bigl[v(s) -v(t_{n})\bigr], \nabla e^{n, \frac 12}_u \Bigr)\,{\rm d}s\, , \\ T_2^{(n)} &:= -\int_{t_n}^{t_{n+1}} \Bigl(\nabla \Big[ \frac{2 u(s) - [u(t_{n+1})+u(t_{n-1})]}{2} \Big], \nabla e^{n+ \frac 12}_v\Bigr)\, {\rm d}s \, , \\ T_3^{(n)} &:= \beta \frac{k^{2+\beta}}{2} \Big(\nabla \bigl[v(t_{n+1}) + v(t_{n})\bigr], \nabla e_v^{n+ \frac 12} \Big)\, , \\ T_4^{(n)} &:= \Bigl(\int_{t_n}^{t_{n+1}} \Bigl[\sigma\bigl(u(s),v(s)\bigr) - \sigma(u^n, v^{n-\frac 12}) \Bigr] \, {\rm d}W(s), e^{n+\frac 12}_v\Bigr) \, , \\ T_5^{(n)} &:= -\,\widehat{\alpha}\,\Bigl(D_u\sigma(u^n, v^{n-\frac 12})v^n \,\widehat{\Delta_n W}, e^{n+\frac 12}_v \Bigr)\, . \end{align*} \noindent We estimate the expectation of each term on the right-hand side of \eqref{e1_1:6}. By Lemma~\ref{lem:Holder} $(iii)$, we infer \begin{align*}\label{e1_1:7} \mathbb{E} \bigl[T_1^{(n)} + T_2^{(n)} \bigr] &\le Ck^2+ C k \, \mathbb{E} \Bigl[\| \nabla e^{n+1}_u\|^2_{\mathbb{L}^2} + \| \nabla e^{n-1}_u\|^2_{\mathbb{L}^2} \Bigr] + C k \, \mathbb{E} \Bigl[ \|e^{n+1}_v\|^2_{\mathbb{L}^2} + \|e^{n}_v\|^2_{\mathbb{L}^2}\Bigr]\, . \end{align*} We use Lemma \ref{lem:L2} $(ii)$ to estimate \begin{align* \mathbb{E} \bigl[T_3^{(n)} \bigr] &\leq \beta \frac{k^{2+\beta}}{6} {\mathbb E}\bigl[\Vert \nabla e^{n+\frac 12}_v\Vert^2_{{\mathbb L}^2}\bigr] + \frac{C}{\beta} k^{2+\beta}\, . \end{align*} \noindent By properties of $\Delta_n W$ we rewrite the term $T_4^{(n)}$ as \begin{equation}\label{e1_1:9} \begin{split} \mathbb{E} \bigl[T_4^{(n)} \bigr] &= \frac 12 \,\mathbb{E} \Bigl[ \Bigl( \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, e^{n+1}_v - e^{n}_v\Bigr)\Bigr] \\ &\quad + \frac 12 \,\mathbb{E} \Bigl[ \Bigl(\int_{t_n}^{t_{n+1}} \bigl[\sigma \bigl(u(s),v(s) \bigr) - \sigma\bigl(u(t_n),v(t_{n-1/2})\bigr)\bigr] {\rm d}W(s),e^{n+1}_v-e^{n}_v\Bigr)\Bigr] \\ &:= T_{4, 1}^{(n)} + T_{4, 2}^{(n)}\, . \end{split} \end{equation} In order to estimate $T_{4, 1}^{(n)}$, we use equation \eqref{e1_1:4} to write {\small{ \begin{equation*} \begin{split} T_{4, 1}^{(n)} &= \frac 12 \,\mathbb{E} \Bigl[ \Bigl( \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, k \Delta e_u^{n, 1/2}\Bigr)\Bigr] \\ \nonumber &\quad + \frac 12 \,\mathbb{E} \biggl[ \biggl( \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, \int_{t_n}^{t_{n+1}} \Delta \Big[ \frac{2u - [u(t_{n+1})+u(t_{n-1})]}{2} \Big] \, {\rm d}s \biggr)\biggr] \\ \nonumber &\quad + \frac 12 \,\mathbb{E} \Bigl[ \Bigl( \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, - k^{2+\beta}\,\Delta v^{n+\frac 12}\Bigr)\Bigr] \\ &\quad + \frac 12 \,\mathbb{E} \biggl[ \biggl( \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, \int_{t_n}^{t_{n+1}} [\sigma(u,v) - \sigma(u^n, v^{n-\frac 12})] \, {\rm d}W(s)\biggr)\biggr] \\ \nonumber &\quad + \frac{\widehat{\alpha}}{2} \,\mathbb{E} \biggl[ \biggl( \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, D_u\sigma(u^n, v^{n-\frac 12})\,v^n\, \widehat{\Delta_n W}\biggr] \\ \nonumber &:= T_{4, 1}^{(n,1)} + T_{4, 1}^{(n, 2)} + T_{4, 1}^{(n, 3)} + T_{4, 1}^{(n, 4)} + T_{4, 1}^{(n, 5)}\, . \end{split} \end{equation*} }} We consider $T_{4, 1}^{(n,1)}$ first; to properly address the dependence of $\sigma$ on $v$, we first restate it with the help of (\ref{e1_1:3}) and use the fact that $\sigma\bigl(u(t_n), v(t_{n-1/2})\bigr)=\sigma(u^n, v^{n-1/2})=0$ on $\partial \mathcal{O}$ to obtain {\small{ \begin{eqnarray*}T_{4, 1}^{(n,1)} &=& \frac 12 \,\mathbb{E} \Bigl[ \Bigl( \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, k \Delta [e_u^{n+1} - e^{n-1}_u]\Bigr)\Bigr] \\ &=&-\frac 12 \,\mathbb{E} \Bigl[ \Bigl( \nabla \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, 2k^2 \nabla e_v^{n+1/2} \Bigr)\Bigr] \\ && -\frac 12 \,\mathbb{E} \Bigl[ \Bigl( \nabla \bigl[ \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2})\bigr]\Delta_{n}W, k\nabla {\mathcal R}^{n+1/2}_v\Bigr)\Bigr] =: T_{4, 1, {\tt A}}^{(n,1)} + T_{4, 1, {\tt B}}^{(n,1)} \,, \end{eqnarray*} }} where ${\mathcal R}^{n+1/2}_v := \int_{t_n}^{t_{n+1}} \bigl(v(s)-v(t_{n+1})\bigr)\, {\rm d}s + \int_{t_{n-1}}^{t_{n}} \bigl(v(s)-v(t_{n})\bigr)\, {\rm d}s$. By chain rule, and {\bf (A4)} for $m=1$ we obtain \begin{equation*} \begin{split} T_{4, 1, {\tt A}}^{(n,1)} &\leq - \mathbb{E} \Big[ C_g \big\{ 2\|\nabla u(t_n)\|_{\mathbb{L}^2} + 2\|\nabla v(t_{n-1/2})\|_{\mathbb{L}^2} + \|\nabla e_u^n\|_{\mathbb{L}^2} + \|\nabla e_v^{n-1/2}\|_{\mathbb{L}^2} \big\} |\Delta_n W| \\ &\qquad \quad \cdot k^{1-\frac{\beta}{2}} k^{1+\frac{\beta}{2}} \,\|\nabla e_v^{n+1/2}\|_{\mathbb{L}^2} \Big]\, . \end{split} \end{equation*} We apply Young's inequality, Ito isometry, \eqref{energy1} and Lemma \ref{lem:L2} $(i), (ii)$ to further bound $T_{4, 1, {\tt A}}^{(n,1)}$ by \begin{equation}\label{er-T4,1} \begin{split} T_{4, 1, {\tt A}}^{(n,1)} &\leq \frac{C_g^2}{\beta}\,k^{2- \beta} \,\mathbb{E} \Big[ \big\{ 2 \|\nabla u(t_n)\|^2_{\mathbb{L}^2} +2 \|\nabla v(t_{n-1/2})\|^2_{\mathbb{L}^2} + \|\nabla e_u^n\|^2_{\mathbb{L}^2} \\ &\qquad \qquad \qquad+ \|\nabla e_v^{n-1/2}\|^2_{\mathbb{L}^2} \big\} |\Delta_n W|^2 \Big] + \frac{\beta}{6} k^{2+\beta} \,\mathbb{E} \Big[ \|\nabla e_v^{n+1/2}\|^2_{\mathbb{L}^2} \Big] \\ &\leq C_{\beta} k^{3- \beta} + C_{\beta} k^{3- \beta} {\mathbb E}[\|\nabla e_u^n\|^2_{\mathbb{L}^2}] + C_g^2 C_{\beta}\,k^{3- \beta} \,\mathbb{E} \Big[ \|\nabla e_v^{n-1/2}\|^2_{\mathbb{L}^2} \Big] \\ &\qquad+ \frac{\beta}{6} k^{2+\beta} \,\mathbb{E} \Big[ \|\nabla e_v^{n+1/2}\|^2_{\mathbb{L}^2} \Big]\, , \end{split} \end{equation} where $C_{\beta} \equiv C(\beta)>0$ is a constant for $\beta\in(0, \frac{1}{2})$ and the last two terms on the right-hand side may be absorbed on the left-hand side of (\ref{e1_1:6}) for $k \leq k_0$ sufficiently small, and $\beta < \frac{1}{2}$. Arguing similarly and by Lemma \ref{lem:Holder} $(iii)$ we infer \begin{equation}\label{er-T4,1,B} \begin{split} T_{4, 1, {\tt B}}^{(n,1)} &\leq C k^{3- \beta} +C_g^2\,k^{3- \beta} \,\mathbb{E} \big[ \|\nabla e_v^{n-1/2}\|^2_{\mathbb{L}^2} \big] + C k^{2+\beta}\, , \end{split} \end{equation} where the second term on right-hand side may be absorbed on the left-hand side of (\ref{e1_1:6}) for $k \leq k_0$ sufficiently small, and $\beta < \frac{1}{2}$. \noindent We now estimate $T_{4, 1}^{(n,2)}$: by properties of $\Delta_n W$, {\bf (A3)} and Lemma \ref{lem:Holder} $(iii)$ we get \begin{equation*} \begin{split} T_{4, 1}^{(n,2)} &\leq C_{\tt L} k\, \mathbb{E} \Big[ \big( \|\nabla u(t_{n}) - \nabla u^n\|_{\mathbb{L}^2}^2 + \|v(t_{n-1/2}) - v^{n-1/2}\|_{\mathbb{L}^2}^2 \big) |\Delta_n W|^2 \Big] \\ &\quad+ C \int_{t_n}^{t_{n+1}} \mathbb{E} \Big[ \Big\| \Delta \Big[ u(s) -\frac{u(t_{n+1})+u(t_{n-1})}{2} \Big] \Big\|_{\mathbb{L}^2}^2 \Big] \\ &\leq Ck\,\mathbb{E} \Big[ \|\nabla e_u^{n}\|_{\mathbb{L}^2}^2 + \|e_v^{n}\|_{\mathbb{L}^2}^2 + \|e_v^{n-1}\|_{\mathbb{L}^2}^2 \Big] +C k^3\, . \end{split} \end{equation*} Using similar arguments as for the estimate of \eqref{er-T4,1} we infer \begin{equation*} \begin{split} T_{4, 1}^{(n,3)} &\leq \frac{C_g^2}{\beta} k^{3+\beta} \mathbb{E} \Big[ \|\nabla u(t_{n})\|_{\mathbb{L}^2}^2 + 2 \|\nabla v(t_{n-1/2})\|_{\mathbb{L}^2}^2 + \|\nabla u^{n}\|_{\mathbb{L}^2}^2 + \|\nabla e_v^{n-1/2}\|_{\mathbb{L}^2}^2 \Big] \\ &\quad + \frac{\beta}{6} k^{2+\beta}\, \mathbb{E} \Big[ \|\nabla e_v^{n+1/2}\|_{\mathbb{L}^2}^2 + \|\nabla v(t_{n+1/2})\|_{\mathbb{L}^2}^2 \Big] \\ &\leq \frac{C_g^2}{\beta}\,k^{3+ \beta} \,\mathbb{E} \Big[ \|\nabla e_v^{n-1/2}\|^2_{\mathbb{L}^2} \Big] + \frac{\beta}{6} k^{2+\beta} \,\mathbb{E} \Big[ \|\nabla e_v^{n+1/2}\|^2_{\mathbb{L}^2} \Big] + C_{\beta} k^{2+\beta}\, . \end{split} \end{equation*} \noindent Using {\bf (A3)}, Lemma \ref{lem:Holder} $(ii)$, and properties of $\Delta_n W$, we estimate \begin{equation} \begin{split} T_{4, 1}^{(n,4)} &\leq C\, \mathbb{E} \Big[ \| \sigma(u(t_n), v(t_{n-1/2})) - \sigma(u^n, v^{n-1/2})\|_{\mathbb{L}^2}^2\,|\Delta_n W|^2 \Big] \\ &\quad+ C \int_{t_n}^{t_{n+1}} \mathbb{E} \Big[ \| \sigma\bigl(u(s), v(s)\bigr) - \sigma\bigl(u(t_n), v(t_{n-1/2})\bigr)\|_{\mathbb{L}^2}^2 \Big]\,\rm{d}s \\ \label{limiti}&\leq Ck\,\mathbb{E} \Big[ \|\nabla e_u^{n}\|_{\mathbb{L}^2}^2 + \|e_v^{n}\|_{\mathbb{L}^2}^2 + \|e_v^{n-1}\|_{\mathbb{L}^2}^2 \Big] \\ &\quad+ C_{\tt L} \int_{t_n}^{t_{n+1}} \mathbb{E} \Big[ \|\nabla [u-u(t_n)]\|_{\mathbb{L}^2}^2 + \|v-v(t_{n-1/2})]\|_{\mathbb{L}^2}^2 \Big] \\ &\leq Ck\,\mathbb{E} \Big[ \|\nabla e_u^{n}\|_{\mathbb{L}^2}^2 + \|e_v^{n}\|_{\mathbb{L}^2}^2 + \|e_v^{n-1}\|_{\mathbb{L}^2}^2 \Big] + C k^2\, . \end{split} \end{equation} Using {\bf (A3)}, {\bf (A4)} for $m=1$, item {\bf 4.}~of Remark \ref{rema2}, and using Lemma \ref{lem:L2} $(i)$ (due to addition and subtraction of $v(t_n)$ term to $v^n$) we estimate \begin{equation*} \begin{split} T_{4, 1}^{(n,5)} &\leq k\, \mathbb{E} \Big[ \|\sigma\bigl(u(t_n), v(t_{n-1/2})\bigr) - \sigma(u^n, v^{n-1/2}) \|_{\mathbb{L}^2}^2 \Big] +\frac{\widehat{\alpha}^2}{4} k^3\,\mathbb{E} \Big[ \|D_u\sigma(u^n, v^{n-\frac 12})\,v^n \|_{\mathbb{L}^2}^2 \Big] \\ &\leq C_{\tt L} k\,\mathbb{E} \Big[ \|\nabla e_u^{n}\|_{\mathbb{L}^2}^2 + \|e_v^{n- 1/2}\|_{\mathbb{L}^2}^2 \Big] +\frac{\widehat{\alpha}^2}{4} C_g^2\, k^3\, \mathbb{E}\big[ \|e_v^{n}\|_{\mathbb{L}^2}^2 \big] + C k^3\, . \end{split} \end{equation*} Similar arguments, in combination with the H\"older estimates in Section \ref{Hoe-cont} may be used to estimate $T_{4, 2}^{(n)}$ in \eqref{e1_1:9}. Now, we estimate the last term in the right-hand side of \eqref{e1_1:6}. \noindent Using {\bf (A4)} for $m=1$, It\^o isometry, and Lemma \ref{lem:L2} $(i)$ (due to addition and subtraction of $v(t_n)$ term to $v^n$), we obtain \begin{equation*} \begin{split} \mathbb{E}\big[ T_5^{(n)} \big] &\leq \frac{\widehat{\alpha}^2}{k} \mathbb{E} \Big[ \|D_u\sigma(u^n, v^{n-\frac 12}) v^n \|_{\mathbb{L}^2}^2 \big|\widehat{\Delta_n W} \big|^2 \Big] + C k\,\mathbb{E} \big[ \|e^{n+1}_v \|_{\mathbb{L}^2}^2+\|e^{n}_v \|_{\mathbb{L}^2}^2 \big] \\ &\leq \widehat{\alpha}^2\,C_g^2\,k^2\, \mathbb{E} \big[ \|v^n \|_{\mathbb{L}^2}^2 \big] + C k\,\mathbb{E} \big[ \|e^{n+1}_v \|_{\mathbb{L}^2}^2+\|e^{n}_v \|_{\mathbb{L}^2}^2 \big] \leq C k^2+ C k\,\mathbb{E} \big[ \|e^{n+1}_v \|_{\mathbb{L}^2}^2+\|e^{n}_v \|_{\mathbb{L}^2}^2 \big] \, . \end{split} \end{equation*} We now insert these estimates into (\ref{e1_1:6}), for which we apply expectations, and sum over iteration steps. The implicit version of the discrete Gronwall lemma along with {\bf (B2)} then yields the assertion, again provided $k \leq k_0$ is sufficiently small and $\beta \in (0, 1/2)$. \medskip \noindent {\bf 2) Proof of \eqref{pi}.} Suppose $\sigma_2(v) \equiv 0 \equiv F_2(v)$ in {\bf (A3)}, and $\widehat{\alpha} = 0$. We combine both equations in the $(0, 0)-$scheme, \begin{equation}\label{scheme2:3} \bigl[u^{\ell+1} - u^\ell\bigr] - \bigl[u^\ell - u^{\ell-1}\bigr] = k^2 \Delta u^{\ell,1/2} + \frac{k^2}{2} \bigl[ 3F(u^n) -F(u^{n-1})\bigr] + k \sigma(u^\ell)\Delta_\ell W \end{equation} for all $1 \leq \ell \leq N$. Now sum over the first $n$ steps, and define $\overline{u}^{n+1} := \sum_{\ell=1}^n {u}^{\ell+1}$. We arrive at \begin{equation}\label{e1_2:2} \bigl[u^{n+1} - u^n\bigr] - k^2 \Delta \overline{u}^{n,1/2} = \bigl[ u^1 - u^0\bigr] +\frac{k^2}{2} \sum^n_{\ell=1} \bigl[3F(u^\ell) - F(u^{\ell-1})\bigr] + k \sum^n_{\ell=1} \sigma(u^\ell) \Delta_{\ell} W\, . \end{equation} We proceed correspondingly with \eqref{strong2}, which we integrate in time: thanks to \eqref{strong1}, we get ($0 \le \lambda \le \mu \le T$) \begin{equation}\label{e1_2:3} \begin{split} &\bigl[u(\mu) - u(\lambda)\bigr] - \int_\lambda^\mu \int^s_0 \Delta u(\xi) \, {\rm d} \xi{\rm d}s \\ &\quad = [\mu-\lambda] v_0 + \int_\lambda^\mu \int^s_0 F \bigl(u(\xi)\bigr) \, {\rm d} \xi{\rm d}s + \int_{\lambda}^{\mu} \int_0^s \sigma \bigl( u(\xi)\bigr)\, {\rm d}W(\xi){\rm d}s \, , \end{split} \end{equation} where $s \in [t_n, t_{n+1}]$. Setting $\mu=t_{n+1}$, $\lambda=t_n$ in \eqref{e1_2:3}, subtracting \eqref{e1_2:2} from \eqref{e1_2:3} then leads to \begin{align}\label{error-eq} \bigl[e^{n+1}_u -e^n_u\bigr] - k^2 \Delta \overline{e}^{n,1/2}_u &= k v_0 - (u^1 - u^0) + \underbrace{\int_{t_n}^{t_{n+1}} {\color{black}{\int_0^s}} \sigma \bigl( u(\xi)\bigr)\, {\rm d}W(\xi)\,{\rm d}s - k \sum^n_{\ell=1} \sigma(u^\ell) \Delta_{\ell} W}_{{\tt := I}} \nonumber \\ &\quad+\underbrace{\int_{t_n}^{t_{n+1}} {\color{black}{\int_{0}^{s} }} \Delta u(\xi) \,{\rm d} \xi \,{\rm d} s - k^2 \sum_{\ell =1}^n \frac{\Delta \big[ u(t_{\ell+1}) + u(t_{\ell-1}) \big]}{2}}_{{\tt := II}} \\ &\quad+ \underbrace{\int_{t_n}^{t_{n+1}} {\color{black}{\int^s_0}} F \bigl(u(\xi)\bigr) \, {\rm d} \xi\,{\rm d}s - \frac{k^2}{2} \sum^n_{\ell=1} \bigl[3F(u^\ell) - F(u^{\ell-1})\bigr]}_{{\tt := III}}\,. \nonumber \end{align} We first rewrite the term ${\tt I}$ in a form which is more suitable to obtain error estimates. Consider the term {\color{black}{ \begin{align} \int_0^s \sigma \bigl( u(\xi)\bigr)\, {\rm d}W(\xi)\,{\rm d}s = \bigg( \sum_{\ell=0}^{n-1} \int_{t_{\ell}}^{t_{\ell+1}} + \int_{t_n}^{s} \bigg) \sigma \bigl( u(\xi)\bigr)\, {\rm d}W(\xi)\,{\rm d}s\,. \end{align} }} By rearranging the terms, we can rewrite ${\tt I}$ as the sum of five terms which are suitable to obtain error estimates \begin{equation}\label{sigma-rearr} \begin{split} {\tt I} &= k \sum^n_{\ell=1} \bigl[\sigma\bigl( u(t_\ell)\bigr) - \sigma(u^\ell) \bigr] \Delta_{\ell} W + \int_{t_n}^{t_{n+1}} \sum^{n-1}_{\ell=1} \int_{t_\ell}^{t_{\ell+1}} \bigl[\sigma \bigl( u(\xi) \bigr) - \sigma\bigl(u(t_\ell)\bigr)\bigr] {\rm d}W(\xi){\rm d}s \\ &\quad + \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} \big[ \sigma \bigl( u(\xi)\bigr) - \sigma \bigl( u(t_n)\bigr) \big] \, {\rm d}W(\xi)\,{\rm d}s + \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} \sigma \bigl( u(t_n)\bigr)\, {\rm d}W(\xi)\,{\rm d}s \\ &\quad + \int_{t_n}^{t_{n+1}} \int_{t_0}^{t_1} \sigma \bigl( u(\xi)\bigr) \, {\rm d}W(\xi)\,{\rm d}s \,. \end{split} \end{equation} Term ${\tt II}$ gives the quadrature error, for which we aim to apply Lemma \ref{quadrature1}. This result can not directly be applied here as the second term involves the evaluation of $u$ at times $t_{\ell+1}$ and $t_{\ell-1}$, which are at distance $2k$. Thus, we rewrite the following integral as ($t_0 =0$) {\color{black}{ \begin{equation}\label{split-deltau} \begin{split} \int_0^s \Delta u(\xi)\,{\rm d} \xi &= \bigg(\frac{1}{2} \sum_{\ell=1}^n \int_{t_{\ell-1}}^{t_{\ell+1}} + \frac{1}{2} \int_{t_{0}}^{t_{1}} + \frac{1}{2} \int_{t_{n}}^{t_{n+1}} - \int_s^{t_{n+1}} \bigg) \Delta u(\xi)\,{\rm d} \xi\,, \end{split} \end{equation} }} where the first term on the right-hand side is now suitable to use Lemma \ref{quadrature1}. \medskip We are now ready for the error analysis. We multiply both sides of \eqref{error-eq} with ${e}_u^{n+1/2}$, and observe that \begin{equation*} \begin{split} k^2 \Big( \nabla \overline{e}^{n,1/2}_u, \nabla e_u^{n+1/2} \Big) &= \frac{k^2}{4} \Bigl( \nabla \overline{e}^{n +1}_u + \nabla \overline{e}^{n -1}_u, \nabla [\overline{e}_u^{n+1} - \overline{e}_u^{n-1}] \Bigl) = \frac{k^2}{4} \Bigl[ \| \nabla \overline{e}^{n+1}_u\|^2_{\mathbb{L}^2} - \| \nabla \overline{e}^{n-1}_u\|^2_{\mathbb{L}^2} \Bigr]\, . \end{split} \end{equation*} Using this, and rearranging the terms on the right-hand side of \eqref{error-eq} leads to the following error equation \begin{eqnarray}\nonumber &&\frac{1}{2} \Bigl[ \|e_u^{n+1}\|^2_{\mathbb{L}^2} - \|e_u^{n}\|^2_{\mathbb{L}^2}\Bigr] +\frac{k^2}{4} \Bigl[ \| \nabla \overline{e}^{n+1}_u\|^2_{\mathbb{L}^2} - \| \nabla \overline{e}^{n-1}_u\|^2_{\mathbb{L}^2} \Bigr] \\ \nonumber &&\quad = \big( k v_0 - [u^1 - u^0], e^{n+1/2}_u \big) + k\Bigl( \sum^n_{\ell=1} \bigl[\sigma\bigl( u(t_\ell)\bigr) - \sigma(u^\ell) \bigr] \Delta_{\ell} W,e_u^{n+1/2}\Bigr) \\ \nonumber &&\qquad + \Bigl( \int_{t_n}^{t_{n+1}} \sum^{n-1}_{\ell=1} \int_{t_\ell}^{t_{\ell+1}} \bigl[\sigma \bigl( u(\xi) \bigr) - \sigma\bigl(u(t_\ell)\bigr)\bigr] {\rm d}W(\xi){\rm d}s, e^{n+1/2}_u \Bigr) \\ \nonumber &&\qquad + \Bigl( \int_{t_n}^{t_{n+1}} {\color{black}{\int_{t_n}^{s}}} \big[ \sigma \bigl(u(\xi)\bigr) - \sigma \bigl(u(t_n)\bigr) \big] \, {\rm d}W(\xi)\,{\rm d}s, e^{n+1/2}_u \Bigr) \\ \label{e1_2:5}&&\qquad + \Bigl( \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} \sigma \bigl(u(t_n)\bigr) \, {\rm d}W(\xi)\,{\rm d}s, e^{n+1/2}_u \Bigr) + \Bigl( \int_{t_n}^{t_{n+1}} {\color{black}{\int_{t_0}^{t_1}}} \sigma \bigl(u(\xi)\bigr) \, {\rm d}W(\xi)\,{\rm d}s, e^{n+1/2}_u \Bigr) \\ \nonumber &&\qquad - \bigg( \int_{t_n}^{t_{n+1}} {\color{black}{\int_{0}^{s}}} \nabla u(\xi) \, {\rm d} \xi \, {\rm d}s - k^2 \sum_{\ell=1}^n \frac{\nabla \big[ u(t_{\ell+1}) + u(t_{\ell-1}) \big]}{2} , \nabla e^{n+1/2}_u \bigg) \\ \nonumber &&\qquad + \int_{t_n}^{t_{n+1}} \sum^{n-1}_{\ell=1} \int_{t_\ell}^{t_{\ell+1}} \Bigl(F (u) - \frac{1}{2} \bigl[3F\bigl(u(t_\ell)\bigr) - F\bigl(u(t_{\ell-1})\bigr)\bigr], e^{n+1/2}_u\Bigr) \, {\rm d} \xi{\rm d}s \\ \nonumber &&\qquad + \frac{k^2}{2} \sum^{n-1}_{\ell=1} \Bigl( 3\bigl[F\bigl(u(t_\ell)\bigr) - F(u^\ell) \bigr] - \bigl[F(u(t_{\ell-1})) - F(u^{\ell-1})\bigr] , e_u^{n+1/2}\Bigr) \\ \nonumber &&\qquad + \int_{t_n}^{t_{n+1}} {\color{black}{\int_{t_0}^{t_1}}} \Big( F\big(u(\xi) \big) , e^{n+1/2}_u\Bigr) \, {\rm d} \xi{\rm d}s + \int_{t_n}^{t_{n+1}} {\color{black}{\int_{t_n}^{s}}} \Big( F\big(u(\xi) \big), e^{n+1/2}_u\Bigr) \, {\rm d} \xi{\rm d}s \\ \nonumber &&\qquad - \frac{k^2}{2} \Bigl( \big[3F(u^n) - F(u^{n-1})\bigr] , e_u^{n+1/2}\Bigr) \\ \nonumber &&\quad =:I^{n}_1+ I^{\ell, n}_2 +\ldots + I^{\ell, n}_{12}\, . \end{eqnarray} We take the expectation on both sides and estimate all the terms on the right-hand side of \eqref{e1_2:5} separately. We begin with the term $I^{\ell, n}_2$. \smallskip \noindent {\bf a) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_2\bigr]$.} By It\^o isometry, and {\bf (A3)}, we have \begin{eqnarray}\label{dazu1} \mathbb{E} \bigl[I^{\ell, n}_2\bigr] &\le& Ck \, \mathbb{E} \bigl[ \|e_u^{n+1/2}\|^2_{\mathbb{L}^2} \bigr] + k\, \mathbb{E}\Bigl[\Bigl\| \sum_{\ell=1}^n \bigl[\sigma\bigl( u(t_\ell)\bigr) - \sigma(u^\ell) \bigr] \Delta_{\ell} W\Bigr\|^2_{\mathbb{L}^2}\Bigr] \\ \nonumber &\le&Ck\, \mathbb{E} \bigl[ \|e_u^{n+1/2}\|^2_{\mathbb{L}^2} \bigr] + \tilde{C}_{\tt L} k^2 \sum_{\ell=1}^n \mathbb{E}\bigl[ \|e_u^\ell\|^2_{\mathbb{L}^2}\bigr]\, . \end{eqnarray} \smallskip \noindent {\bf b) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_3\bigr]$.} The term $I^{\ell, n}_3$ can be controlled by It\^o isometry, {\bf (A3)}, and Lemma~\ref{lem:Holder} $(i)$ as \begin{eqnarray}\nonumber \mathbb{E} \bigl[I^{\ell, n}_3\bigr] &=& \mathbb{E} \Bigl[\Bigl( \int_{t_n}^{t_{n+1}} \sum_{\ell=1}^{n-1} \int_{t_\ell}^{t_{\ell+1}} \bigl[\sigma \bigl(u(\xi)\bigr) - \sigma\bigl(u(t_\ell)\bigr)\bigr] \, {\rm d}W(\xi){\rm d}s, e^{n+1/2}_u \Bigr)\Bigr] \\ \label{restri} &\le& \tilde{C}_{\tt L} \int_{t_n}^{t_{n+1}} \sum_{\ell=1}^{n-1} \int_{t_\ell}^{t_{\ell+1}} \mathbb{E} \Bigl[ \bigl\| u(\xi) - u(t_\ell)\bigr\|^2_{\mathbb{L}^2} \Bigr] {\rm d} \xi \,{\rm d} s + Ck \, \mathbb{E} \bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\bigr] \\ \nonumber &\le& \tilde{C}_{\tt L} k^3 +Ck \, \mathbb{E} \bigl[\|e_u^{n+1}\|^2_{\mathbb{L}^2} + \|e_u^n\|^2_{\mathbb{L}^2}\bigr]\, . \end{eqnarray} \smallskip \noindent {\bf c) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_4\bigr]$.} We use the similar arguments as for $\mathbb{E} \bigl[I^{\ell, n}_3\bigr]$ to get \begin{align*} \mathbb{E} \bigl[I^{\ell, n}_4\bigr] &= \mathbb{E} \Bigl[\Bigl( \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} \bigl[\sigma \bigl(u(\xi)\bigr) - \sigma\bigl(u(t_n)\bigr)\bigr] \, {\rm d}W(\xi){\rm d}s, e^{n+1/2}_u \Bigr)\Bigr] \\ &\leq \tilde{C}_{\tt L} \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} \mathbb{E} \Bigl[ \bigl\| u(\xi) - u(t_n)\bigr\|^2_{\mathbb{L}^2} \Bigr] {\rm d} \xi \,{\rm d} s + Ck \, \mathbb{E} \bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\bigr] \\ &\leq C k^4 +Ck \, \mathbb{E} \bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\bigr] \,. \end{align*} \smallskip \noindent {\bf d) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_5\bigr]$.} By independence of stochastic increments, $$ \mathbb{E} \bigl[I^{\ell, n}_5\bigr] = k \mathbb{E} \Big[ \Big( \sigma \big(u(t_n) \big) \big( W(s) - W(t_n)\big), e_u^{n+1/2} \Big)\Big] = \frac{k}{2} \mathbb{E} \Big[ \Big( \sigma \big(u(t_n) \big) \big( W(s) - W(t_n)\big), e_u^{n+1} - e_u^n \Big)\Big]\,. $$ Using \eqref{e1_1:3} we rewrite \begin{align*} \mathbb{E} \bigl[I^{\ell, n}_5\bigr] &= \frac{k}{2} \mathbb{E} \Big[ \Big( \sigma \big(u(t_n) \big) \big( W(s) - W(t_n)\big), k e_v^{n+1} \Big)\Big] \\ &\quad+ \frac{k}{2} \mathbb{E} \Big[ \Big( \sigma \big(u(t_n) \big) \big( W(s) - W(t_n)\big), \int_{t_n}^{t_{n+1}} \big( v(s) - v(t_{n+1})\big)\,{\rm d} s \Big)\Big] = \mathbb{E} \bigl[I^{\ell, n}_{5; 1}\bigr] + \mathbb{E} \bigl[I^{\ell, n}_{5; 2}\bigr] \end{align*} Using {\bf (A3)}, \eqref{eq-5.32} and the It\^o isometry, \begin{align*} \mathbb{E} \bigl[I^{\ell, n}_{5; 1}\bigr] \leq C k^2 \,\mathbb{E}\Big[ \big\| \sigma \big(u(t_n) \big) \big\|_{\mathbb{L}^2}^2 \Big] \,\mathbb{E}\Big[ |W(s) - W(t_n)|^2 \Big] + C k^2 \mathbb{E}\big[ \|e_v^{n+1}\|_{\mathbb{L}^2}^2 \big] \leq C k^3\,. \end{align*} Using the similar arguments and by Lemma \ref{lem:Holder} $(ii)$ we infer \begin{align*} \mathbb{E} \bigl[I^{\ell, n}_{5; 2}\bigr] \leq C k^2 \,\mathbb{E}\Big[ \big\| \sigma \big(u(t_n) \big) \big\|_{\mathbb{L}^2}^2 \Big] \,\mathbb{E}\Big[ |W(s) - W(t_n)|^2 \Big] + C k^2 \mathbb{E}\Big[ \|v(s) - v(t_{n+1})\|_{\mathbb{L}^2}^2 \Big] \leq C k^3\,. \end{align*} \smallskip \noindent {\bf e) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_6\bigr]$.} We add and subtract the term $\sigma(u_0)$ to get \begin{align*} \mathbb{E} \bigl[I^{\ell, n}_6\bigr] &= \mathbb{E} \bigg[ \Bigl( \int_{t_n}^{t_{n+1}} \int_{t_0}^{t_1} \big( \sigma \bigl(u(\xi)\bigr) - \sigma \bigl(u_0 \bigr) \big) \, {\rm d}W(\xi)\,{\rm d}s, e^{n+1/2}_u \Bigr) \bigg] \\ &\quad+ \mathbb{E} \bigg[ \Bigl( \int_{t_n}^{t_{n+1}} \int_{t_0}^{t_1} \sigma \bigl(u_0\bigr) \, {\rm d}W(\xi)\,{\rm d}s, e^{n+1/2}_u \Bigr) \bigg] := \mathbb{E} \bigl[I^{\ell, n}_{6; 1}\bigr] + \mathbb{E} \bigl[I^{\ell, n}_{6; 2}\bigr] \,, \end{align*} where the term $\mathbb{E} \bigl[I^{\ell, n}_{6; 1}\bigr]$ will follow the similar arguments as for $\mathbb{E} \bigl[I^{\ell, n}_4\bigr]$ in step {\bf c)} to yield \begin{align*} \mathbb{E} \bigl[I^{\ell, n}_{6; 1}\bigr] \leq C k^4 +Ck \, \mathbb{E} \bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\bigr] \,. \end{align*} The term $ \mathbb{E} \bigl[I^{\ell, n}_{6; 2}\bigr] = k \mathbb{E} \Big[ \big( \sigma \bigl(u_0\bigr) \Delta_0 W, e^{n+1/2}_u \big) \Big] $ will be merged with $I^n_1$\,. \smallskip \noindent {\bf f) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_7\bigr]$.} We will apply the quadrature formula (Lemma \ref{quadrature1}) to handle this term as mentioned above for the term ${\tt II}$. To get suitable terms for the error bounds, we use the identity \eqref{split-deltau} and again rearrange the terms by splitting the integral $\int_{t_{n}}^{t_{n+1}} \cdot \,{\rm d} s$ into $\int_{t_{n}}^{s} \cdot \,{\rm d} s + \int_{s}^{t_{n+1}} \cdot \,{\rm d} s$, to get \begin{equation}\label{nab-xi} \begin{split} \int_0^s \nabla u(\xi)\,{\rm d} \xi &= \frac{1}{2} \Bigg( \overbrace{\sum_{\ell=1}^n \int_{t_{\ell-1}}^{t_{\ell+1}} \nabla u(\xi)\,{\rm d} \xi}^{:= {\tt I}_7} + \Big( \overbrace{\int_{t_{0}}^{t_{1}} + \int_{t_{n}}^{s} - \int_{s}^{t_{n+1}} \Big) \nabla u(\xi)\,{\rm d} \xi}^{:= {\tt II}_7} \Bigg) \,. \end{split} \end{equation} Below, we handle the terms separately. \smallskip \noindent ${\bf f}_1$) Lemma~\ref{quadrature1} (with $T=2k$) now applies to handle ${\tt I}_7$. For this, we choose $f(\xi) = \mathbb{E}\bigl[ (\nabla u(\xi), \nabla e_u^{n+1/2})\bigr]$ for all $\xi \in [t_n, t_{n+1}]$. Using integration by parts, by Lemma~\ref{lem:Holder} $(iv)$, we have $\gamma = \frac{1}{2}$ in (\ref{cond1}), \begin{equation*}\label{e1_2:8} \Bigl|\mathbb{E}\Bigl[\Bigl(\nabla [v(t)- v(s)], \nabla e_u^{n+1/2} \Bigr)\Bigr]\Bigr| \le C \Bigl(\mathbb{E}\bigl[ \| \nabla e_u^{n+1/2} \|^2_{\mathbb{L}^2}\bigr]\Bigr)^{1/2} \vert t-s \vert^{\frac{1}{2}}\, . \end{equation*} We know that \begin{align*} \frac{1}{2k} \int_{t_{\ell-1}}^{t_{\ell+1}} f(\xi)\,{\rm d} \xi - \frac{f(t_{\ell+1})+f(t_{\ell-1})}{2} = \frac{1}{2k} \bigg[ \int_{t_{\ell-1}}^{t_{\ell+1}} \Big\{ f(\xi) - \frac{f(t_{\ell+1})+f(t_{\ell-1})}{2} \Big\}\,{\rm d} \xi \bigg]\,. \end{align*} As a consequence \begin{align*}\label{e1_2:9} &\frac{1}{2k} \bigg| \int_{t_{\ell-1}}^{t_{\ell+1}} \mathbb{E}\Bigl[\Bigl(\nabla \Big[u(\xi) - \frac{u(t_{\ell+1}) + u(t_{\ell-1})}{2} \Big], \nabla e_u^{n+1/2} \Bigr)\Bigr] \, {\rm d} \xi \bigg| \\ &= \bigg| \frac{1}{2k} \int_{t_{\ell-1}}^{t_{\ell+1}} f(\xi)\,{\rm d} \xi - \frac{f(t_{\ell+1})+f(t_{\ell-1})}{2} \bigg| \leq \frac{\tilde{C}}{(\frac{1}{2}+2)(\frac{1}{2}+3)} (2k)^{3/2} \Big( \mathbb{E}\big[ \|\nabla e_u^{n+1/2}\|_{\mathbb{L}^2}^2 \big] \Big)^{1/2}\,. \end{align*} Using \eqref{eq-5.32}, we can finally bound the term with the integral from $t_n$ to $t_{n+1}$ by \begin{align*} \leq C k^{5/2} \Big( \mathbb{E}\big[ \|\nabla e_u^{n+1/2}\|_{\mathbb{L}^2}^2 \big] \Big)^{1/2} \leq C k^2 \mathbb{E}\big[ \|\nabla e_u^{n+1/2}\|_{\mathbb{L}^2}^2 \big] + C k^3\, \leq Ck^3\,. \end{align*} \smallskip \noindent ${\bf f}_2$) Consider the difference of the last two terms in ${\tt II}_7$. By adding and subtracting the terms $\frac{1}{2}\int_{t_n}^{t_{n+1}} \int_{t_n}^s \nabla u(t_n) \,{\rm d} \xi \,{\rm d} s$ and $\frac{1}{2} \int_{t_n}^{t_{n+1}} \int^{t_{n+1}}_s \nabla u(t_n) \,{\rm d} \xi \,{\rm d} s$ we get \begin{align*} &\frac{1}{2} \int_{t_n}^{t_{n+1}} \bigg( \int_{t_{n}}^{s} \nabla u(\xi)\,{\rm d} \xi - \int_{s}^{t_{n+1}} \nabla u(\xi) \,{\rm d} \xi \bigg)\,{\rm d} s \\ &= \overbrace{\frac{1}{2} \int_{t_n}^{t_{n+1}} \int_{t_n}^s \nabla \Big( u(\xi) - u(t_n) \Big)\,{\rm d} \xi \,{\rm d} s}^{{\tt II}_{7, a}} - \overbrace{\frac{1}{2} \int_{t_n}^{t_{n+1}} \int_{s}^{t_{n+1}} \nabla \Big( u(\xi) - u(t_n) \Big)\,{\rm d} \xi \,{\rm d} s}^{{\tt II}_{7, b}} \\ &\quad+ \frac{1}{2} \nabla u(t_n) \bigg( \int_{t_n}^{t_{n+1}} \int_{t_n}^s {\rm d} \xi \,{\rm d} s - \int_{t_n}^{t_{n+1}} \int_{s}^{t_{n+1}} {\rm d} \xi \,{\rm d} s \bigg)\,, \end{align*} where the last term on the right-hand side vanishes as $\int_{t_n}^{t_{n+1}} \big[ (s- t_n) - (t_{n+1}-s)\big] \,{\rm d} s = 0$. So, we estimate the first two terms only. By standard estimation, \begin{align*} &\int_{t_n}^{t_{n+1}} \mathbb{E} \Big[ \Big({\tt II}_{7, a}, \nabla e_u^{n+1/2} \Big) \Big]\,{\rm d} s \\ &\leq C\,k^2 \int_{t_n}^{t_{n+1}} \mathbb{E} \Big[ \big\|\nabla \big[ u(\xi) - u(t_n)\big] \big\|_{\mathbb{L}^2}^2 \Big]\,{\rm d} \xi + C k \mathbb{E} \big[ \|\nabla e_u^{n+1/2}\|_{\mathbb{L}^2}^2 \big] \\ &\leq C\,k^5 + C k \mathbb{E} \big[ \|\nabla e_u^{n+1/2}\|_{\mathbb{L}^2}^2 \big] \end{align*} thanks to Lemma \ref{lem:Holder} $(ii)$. Similarly, one may handle the term that involves ${\tt II}_{7, b}$. The only term left to handle from the right-hand side of \eqref{nab-xi} is $\int_{t_{0}}^{t_{1}} \nabla u(\xi)\,{\rm d} \xi$. We can rewrite the term $\int_{t_n}^{t_{n+1}} \int_{t_{0}}^{t_{1}} \nabla u(\xi)\,{\rm d} \xi\,{\rm d} s$ as $\int_{t_n}^{t_{n+1}} \int_{t_{0}}^{t_{1}} \nabla \big[ u(\xi) - u_0 \big] \,{\rm d} \xi\,{\rm d} s + k^2 \nabla u_0\,.$ To estimate the first term, we proceed as before to conclude \begin{align*} &\Bigg| \int_{t_n}^{t_{n+1}} \mathbb{E} \bigg[ \Big(\frac{1}{2} \int_{t_{0}}^{t_{1}} \nabla \big[ u(\xi) - u_0 \big] \,{\rm d} \xi, \nabla e_u^{n+1/2} \Big) \bigg]\,{\rm d} s \Bigg| \\ &\leq C \int_{t_n}^{t_{n+1}} \int_{t_{0}}^{t_{1}} \xi^2\,{\rm d} \xi \,{\rm d} s + C k \mathbb{E}\big[ \|\nabla e_u^{n+1/2}\|_{\mathbb{L}^2}^2\big] \leq C k^4 + C k \mathbb{E}\big[ \|\nabla e_u^{n+1/2}\|_{\mathbb{L}^2}^2\big]\,. \end{align*} Now, we only need to handle the term $\mathbb{E} \big[ \big( \frac{1}{2} k^2 \Delta u_0, e_u^{n+1/2} \big)\big]\,.$ This term will now be combined with the term $I^n_1$. \smallskip \noindent {\bf g) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_8\bigr]$.} As a first step, we split it into two parts, \begin{equation*} \begin{split} {\mathbb E}[I^{\ell, n}_8] &= \int_{t_n}^{t_{n+1}} \sum^{n-1}_{\ell=1} \int_{t_\ell}^{t_{\ell+1}} {\mathbb E}\Bigl[ \Bigl(F \bigl(u(\xi)\bigr) - \frac{1}{2} \bigl[F\bigl(u(t_{\ell})\bigr) + F\bigl(u(t_{\ell+1})\bigr) \bigr], e^{n+1/2}_u \Bigr)\Bigr] \, {\rm d} \xi{\rm d}s \\ & \quad+ \frac{k^2}{2}\sum_{\ell=1}^n {\mathbb E}\Bigl[\Bigl( F\bigl( u(t_{\ell+1})\bigr) - 2 F\bigl(u(t_{\ell})\bigr) + F \bigl( u(t_{\ell-1})\bigr),e^{n+1/2}_u\Bigr)\Bigr] =: {\mathbb E} \big[ I^{\ell, n}_{8;1} + I^{\ell, n}_{8;2} \big]\, . \end{split} \end{equation*} To handle these two terms, we use Lemma~\ref{quadrature1} with $f(\xi) = \mathbb{E}\bigl[ \bigl( F\bigl(u(\xi)\bigr), e_u^{n+1/2} \bigr)\bigr]$ where $\xi \in [t_n, t_{n+1}]$, and verify $\gamma = \frac{1}{2}$ in (\ref{cond1}): by {\bf (A4)} for $m=1, 2$, the chain-rule and the mean-value theorem \begin{equation}\label{dazu2} \begin{split} &\Bigl\vert \bigl(D_t F\bigl(u(t)\bigr) - D_t F\bigl(u(s)\bigr), e^{n+1/2}_u\bigr)\Bigr\vert \\ &= \Bigl\vert \Bigl(D_u F\bigl(u(t)\bigr) v(t) - D_u F\bigl(u(s)\bigr) v(s), e^{n+1/2}_u\Bigr) \Bigr\vert \\ &=\Bigl\vert \Big( \big(D_u F(u(t)) - D_u F(u(s)) \big) v(t) + D_u F(u(s)) (v(t)-v(s)), e^{n+1/2}_u \Big)\Bigr\vert \\ &= \Bigl\vert \Big( \big(\underline{D_u^2 F}(u(t)-u(s)) \big) (v(t)) + D_u F(u(s)) (v(t)-v(s)), e^{n+1/2}_u \Big) \Bigr\vert \\ &\leq \tilde{C}_g \,\Vert u(t) - u(s)\Vert_{{\mathbb H}^1} \Vert v(t)\Vert_{{\mathbb H}^1} \Vert e^{n+1/2}_u\Vert_{{\mathbb L}^2}+ \tilde{C}_g\, \Vert v(t) - v(s)\Vert_{{\mathbb L}^2} \Vert e^{n+1/2}_u\Vert_{{\mathbb L}^2}\, . \end{split} \end{equation} where $\underline{D_u^2 F} := D_u^2 F(\widetilde{u}_{\rho})$ and $\widetilde{u}_{\rho} := \rho u(t) + (1-\rho) u(s)$, for some $\rho \in [0,1]$. Lemma~\ref{lem:Holder} $(ii)$ then establishes $\gamma = \frac{1}{2}$ in (\ref{cond1}), and so Lemma~\ref{quadrature1} yields $${\mathbb E}\bigl[ I^{\ell, n}_{8;1} \bigr] \leq C k^{\frac{5}{2}}C \Bigl(\mathbb{E}\bigl[ \bigl\| e_u^{n+1/2} \bigr\|^2_{\mathbb{L}^2}\bigr]\Bigr)^{\frac{1}{2}} \leq Ck^4 + k\, \mathbb{E}\bigl[ \bigl\| e_u^{n+1/2} \bigr\|^2_{\mathbb{L}^2}\bigr]\, .$$ {{ In order to estimate ${\mathbb E}[I^{\ell, n}_{8;2}]$, we may write for some $\theta \in (0, 1)$ \begin{align*} F(u(t_{\ell+1})) &= F(u(t_{\ell})) + D_u F(u(t_{\ell})) (u(t_{\ell+1}) - u(t_{\ell})) \\ &\quad+ \frac{1}{2} \Big( D_u^2 F\big(u(t_{\ell})+ \theta (u(t_{\ell+1}) - u(t_{\ell})) \big) (u(t_{\ell+1}) - u(t_{\ell})) \Big) \big(u(t_{\ell+1}) - u(t_{\ell}) \big) \,, \end{align*} and \begin{align*} F(u(t_{\ell-1})) &= F(u(t_{\ell})) + D_u F(u(t_{\ell})) (u(t_{\ell-1}) - u(t_{\ell})) \\ &\quad+ \frac{1}{2} \Big( D_u^2 F\big(u(t_{\ell})+ \theta (u(t_{\ell-1}) - u(t_{\ell})) \big) (u(t_{\ell-1}) - u(t_{\ell})) \Big) \big(u(t_{\ell-1}) - u(t_{\ell}) \big) \,. \end{align*} Then, adding the above two terms we get \begin{align*} &F\bigl( u(t_{\ell+1})\bigr) - 2 F\bigl( u(t_{\ell})\bigr) + F\bigl( u(t_{\ell-1}) \bigr) \\ &= \underline{D_u F} \big(u(t_{\ell+1}) - 2 u(t_{\ell})+ u(t_{\ell-1}) \big) + \frac 12 \big(\underline{D_u^2 F} \,( u(t_{\ell+1}) - u(t_{\ell})) \big) (u(t_{\ell+1}) - u(t_{\ell})) \\ &\quad+ \frac 12 \big(\overline{D_u^2 F} \,( u(t_{\ell-1}) - u(t_{\ell}))\big)(u(t_{\ell-1}) - u(t_{\ell}))\, , \end{align*} where $\underline{D_u F} := D_u F(u(t_{\ell}))$, $\underline{D_u^2 F} := D_u^2 F\big(u(t_{\ell})+ \theta (u(t_{\ell+1}) - u(t_{\ell})) \big)$ and $\overline{D_u^2 F} := D_u^2 F\big(u(t_{\ell})+ \theta (u(t_{\ell-1}) - u(t_{\ell})) \big)$. We begin with the first term on the right-hand side: first, by the mean value theorem, there exist $\zeta_1, \zeta_2 \in [0,1]$, such that $$ u(t_{\ell+1}) - u(t_{\ell}) = k v\bigl( \zeta_1 t_{\ell}+[1-\zeta_1]t_{\ell+1}\bigr)\,, \qquad - \bigl[u(t_{\ell}) - u(t_{\ell-1})\bigr] = - k v\bigl( \zeta_2 t_{\ell-1}+[1-\zeta_2]t_{\ell}\bigr)\,. $$ Hence, Lemma \ref{lem:Holder} $(ii)$ settles ${\mathcal O}(k^{\frac{3}{2}})$ for this term. If combined with Lemma \ref{lem:Holder} $(i)$, {\bf (A4)} for $m=1, 2$, we can conclude \[{\mathbb E}[I^{\ell, n}_{8;2}] \leq C {k^4} + k\, \mathbb{E}\big[ \|e^{n+1/2}_u\big\|^2_{\mathbb{L}^2} \big] \, .\] }} \del{ We use the mean-value theorem twice to conclude $${\color{black}{{\mathbb E}[I^{\ell, n}_{5;2}] \leq Ck^2 \sum_{\ell=1}^n {\mathbb E}\Bigl[ \Bigl(\Vert u (t_{\ell+1} ) - u (t_{\ell} )\Vert^2_{{\mathbb L}^2} + k\, \Vert d_t u (t_{\ell+1} ) - d_t u (t_{\ell} )\Vert_{{\mathbb L}^2} \Bigr)\Vert e^{n+1/2}_u\Vert_{{\mathbb L}^2}\Bigr].}} $$ Using Lemma \ref{lem:Holder_L2_u} $(i)$, and the mean-value theorem one more time for the second summand in combination with Lemma \ref{lem:Holder_L2_u} $(i)$ and \ref{lem:Holder_L2} $(i)$ we estimate further, $$ \leq Ck^2 \sum_{\ell=1}^n \bigl(k^2 + k^{\frac{3}{2}}\bigr) \Bigl({\mathbb E}\Bigl[ \Vert e^{n+1/2}_u\Vert^2_{{\mathbb L}^2}\Bigr]\Bigr)^{\frac{1}{2}} \leq Ck^4 + k\, \mathbb{E}\bigl[ \bigl\| e_u^{n+1/2} \bigr\|^2_{\mathbb{L}^2}\bigr]\, .$$ } \noindent {\bf h) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_9\bigr]$.} Finally, by {\bf (A3)} we infer \begin{align*} {\mathbb E}[I^{\ell, n}_9] \leq Ck^2 \biggl(\sum_{\ell=1}^n {\mathbb E}\Bigl[ \Vert e_u^{\ell}\Vert^2_{{\mathbb L}^2} + \Vert e_u^{\ell-1}\Vert^2_{{\mathbb L}^2}\Bigr] + {\mathbb E}\bigl[\Vert e^{n+1/2}_u\Vert_{{\mathbb L}^2}^2\bigr]\biggr)\, . \end{align*} \noindent {\bf i) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_{10}\bigr]$.} Adding and subtracting $F(u_0)$ to the term we get, \begin{align*} \mathbb{E} \bigl[I^{\ell, n}_{10}\bigr] = \mathbb{E}\bigg[\int_{t_n}^{t_{n+1}} \int_{t_0}^{t_1} \Big( \big[ F(u(\xi)) - F(u_0)\big], e^{n+1/2}_u \Big) \bigg] \,{\rm d} \xi \,{\rm d} s + \mathbb{E}\Big[ \big( k^2 F(u_0), e^{n+1/2}_u\big) \Big]\,. \end{align*} Using previous arguments as before, by {\bf (A3)} and Lemma \ref{lem:Holder} $(i)$ we bound the first term on the right-hand side by $C k^4 + C k\, {\mathbb E}\bigl[\Vert e^{n+1/2}_u\Vert_{{\mathbb L}^2}^2\bigr]\,.$ The second term on the right-hand side will now be combined with the term $I^n_1$. \smallskip \noindent {\bf j) Estimation of $\mathbb{E} \bigl[I^{\ell, n}_{11} + I^{\ell, n}_{12}\bigr]$.} Here, we consider the estimation of $I^{\ell, n}_{11}$ and $I^{\ell, n}_{12}$ together. We subtract the term $\frac{1}{2} \big[ 3 F(u(t_n)) - F(u(t_{n-1})) \bigr]$ from $I^{\ell, n}_{11}$ and add the corresponding term with $I^{\ell, n}_{12}$. Then, the new decomposition can be estimated similarly as $\mathbb{E} \big[I^{\ell, n}_{8}\big]$ and $\mathbb{E} \big[I^{\ell, n}_{9} \big]$ (see steps {\bf g)} and {\bf h)}, respectively) to finally get \begin{align*} \mathbb{E} \bigl[I^{\ell, n}_{11} + I^{\ell, n}_{12}\bigr] \leq C k^4 + Ck \bigg({\mathbb E}\Bigl[ \Vert e_u^{n}\Vert^2_{{\mathbb L}^2} + \Vert e_u^{n-1}\Vert^2_{{\mathbb L}^2}\Bigr] + {\mathbb E}\bigl[\Vert e^{n+1/2}_u\Vert_{{\mathbb L}^2}^2\bigr]\biggr)\,. \end{align*} \smallskip \noindent {\bf k) Estimation of $\mathbb{E} \bigl[I^{n}_1\bigr]$.} To estimate this term, we need to combine the terms $k \sigma \bigl(u_0\bigr) \Delta_0 W, \,\frac{k^2}{2} \Delta u_0$ and $k^2 F(u_0)$, which are coming from the steps ${\bf e)}, {\bf f}_2)$ and ${\bf i)}$, respectively. By the assumption (\ref{ini-1}) we infer \begin{align*} &{\mathbb E} \Big[ \Big( k v_0 - [u^1 - u^0] + \frac{k^2}{2} \Delta u_0 + k^2 F(u_0)+ k \sigma \bigl(u_0\bigr) \Delta_0 W, e_u^{n+1/2} \Big) \Big] \\ & \leq \frac Ck\, {\mathbb E} \Big[ \big\| k v_0 - [u^1 - u^0] + \frac{k^2}{2} \Delta u_0 + k^2 F(u_0) + k \sigma \bigl(u_0\bigr) \Delta_0 W \big\|_{\mathbb{L}^2}^2 \Big] + C k\, {\mathbb E} \Big[ \Vert e^{n+1/2}_u \Vert^2_{{\mathbb L}^2} \Big] \\ &\leq C k^4 +Ck\, {\mathbb E} \Big[ \Vert e^{n+1/2}_u \Vert^2_{{\mathbb L}^2} \Big]\,. \end{align*} \smallskip Now we combine all the above estimates in (\ref{e1_2:5}) in summarized form, then the implicit version of the discrete Gronwall lemma yields assertion \eqref{pi}. \medskip \noindent {\bf 3) Proof of \eqref{pii}.} Similar to (\ref{e1_2:2}), we have for $\widehat{\alpha} = 1$ \begin{equation}\label{tilde-comb-sum} \begin{split} &\big[u^{n+1} - u^n \big] - k^2 \Delta \overline{u}^{n,1/2} - \big[ u^1 - u^0 \big] \\ &= k \sum^n_{\ell=1} \sigma(u^\ell) \Delta_{\ell} W + \widehat{\alpha} k \sum^n_{\ell=1} D_u\sigma(u^\ell)v^{\ell} \widehat{\Delta_{\ell} W}+\frac{k^2}{2} \sum^n_{\ell=1} \bigl[3F(u^\ell) - F(u^{\ell-1})\bigr]\, . \end{split} \end{equation} So the additional term on the right-hand side of the error equation \eqref{error-eq} is \begin{align}\label{dist-th} \widehat{\alpha} k\sum_{\ell=1}^n - D_u\sigma(u^\ell) v^{\ell} \widetilde{\Delta_{\ell} W} + \widehat{\alpha} k\sum_{\ell=1}^n - D_u\sigma(u^\ell) v^{\ell} \bigl[ (\widehat{\Delta_{\ell} W}- \widetilde{\Delta_{\ell} W})\bigr] := \mathfrak{I}^{\ell,n}_{3,{\tt A}} + \mathfrak{I}^{\ell,n}_{3,{\tt B}}\,. \end{align} We now follow the argumentation in {\bf 2)}: multiplication with ${e}_u^{n+1/2}$ of the modified error equation (\ref{error-eq}) then leads to (\ref{e1_2:5}), where $\mathfrak{I}^{\ell,n}_{3,{\tt A}}$ is merged with $I^{\ell, n}_3$. Then, the sum $I^{\ell, n}_3+I^{\ell, n}_4 + I^{\ell, n}_5$ may be rewritten as the following sums {\small{ \begin{equation}\label{e2_2:3} \begin{split} &\underbrace{\Bigl( \int_{t_n}^{t_{n+1}} \sum^{n-1}_{\ell=1} \int_{t_\ell}^{t_{\ell+1}} \Bigl[ D_u \sigma \bigl(u(t_\ell) \bigr)v(t_\ell) - D_u \sigma(u^\ell) v^{\ell} \Bigr](\xi - t_\ell) \, {\rm d}W(\xi){\rm d}s, e_u^{n+1/2}\Bigr)}_{:= {I}^{\ell,n}_{3,{\tt A}_1}}\\ &\quad + \underbrace{\Bigl( \int_{t_n}^{t_{n+1}} \sum^{n-1}_{\ell=1} \int_{t_\ell}^{t_{\ell+1}} \Bigl[\sigma \bigl( u(\xi)\bigr) - \sigma\bigl(u(t_\ell)\bigr) - D_u \sigma\bigl(u(t_\ell)\bigr) v(t_\ell)(\xi - t_\ell)\Bigr] \, {\rm d}W(\xi){\rm d}s, e_u^{n+1/2} \Bigr)}_{:= {I}^{\ell,n}_{3,{\tt A}_2}} \\ &\quad + \underbrace{\Bigl( \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} \Bigl[ D_u \sigma \bigl(u(t_n) \bigr)v(t_n) - D_u \sigma(u^n) v^{n} \Bigr](\xi - t_n) \, {\rm d}W(\xi){\rm d}s, e_u^{n+1/2}\Bigr)}_{:= {I}^{\ell,n}_{3,{\tt A}_3}} \\ &\quad + \underbrace{\Bigl( \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} \Bigl[\sigma \bigl( u(\xi)\bigr) - \sigma\bigl(u(t_n)\bigr) - D_u \sigma\bigl(u(t_n)\bigr) v(t_n)(\xi - t_n)\Bigr] \, {\rm d}W(\xi){\rm d}s, e_u^{n+1/2} \Bigr)}_{:= {I}^{\ell,n}_{3,{\tt A}_4}}\,. \end{split} \end{equation} }} We independently bound the other error terms in (\ref{e1_2:5}) in this modified setting: \smallskip \noindent {\bf a)} To bound ${\mathbb E}[I^{\ell, n}_{3;{\tt A}_1}]$ in \eqref{e2_2:3}, we use It\^o isometry, the mean-value theorem, {\bf (A4)} for $m=1, 2$, to get \begin{equation}\label{I3;1_Itois} \begin{split} \mathbb{E}\bigl[I^{\ell, n}_{3;{\tt A}_1}\bigr] &\leq k \mathbb{E} \Bigl[ \frac{1}{k} \cdot \Big\| \int_{t_{n}}^{t_{n+1}} \sum^{n-1}_{\ell=1} \int_{t_{\ell}}^{t_{\ell+1}} \bigl[ D_u \sigma \bigl(u(t_{\ell}) \bigr)v(t_{\ell}) - D_u\sigma(u^{\ell}) v^{\ell} \bigr] \\ &\qquad \qquad \times (\xi - t_{\ell}) \, {\rm d}W(\xi)\, {\rm d}s \Big\|_{\mathbb{L}^2}^2 \Big] + k \, \mathbb{E}\bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\bigr] \\ &\leq \int_{t_{n}}^{t_{n+1}} \sum^{n-1}_{\ell=1} \mathbb{E} \Big[ \int_{t_{\ell}}^{t_{\ell+1}} \big\| D_u \sigma \bigl(u(t_{\ell}) \bigr)v(t_{\ell}) - D_u\sigma(u^{\ell}) v^{\ell} \big\|^2_{\mathbb{L}^2} \\ &\qquad \qquad \qquad \times (\xi - t_{\ell})^2 \,d \xi \,{\rm d}s \Big] + k \, \mathbb{E}\bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\bigr] \\ &\le C k^4 \,\sum_{\ell=1}^{n-1} \mathbb{E}\Bigl[\bigl\| D_u\sigma\bigl( u(t_{\ell})\bigr) v( t_{\ell}) - D_u\sigma(u^{\ell}) v^{\ell} \bigr\|^2_{\mathbb{L}^2} \Bigr] + k \, \mathbb{E}\bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\bigr] \\ &\le Ck^4\,\sum_{\ell=1}^{n-1} \Bigl(\mathbb{E} \Big[ \|e_v^{\ell} \|^2_{\mathbb{L}^2} \Big] +{\mathbb E} \Big[\widetilde{I^{\ell, n}_{3;{\tt A}_1}} \Big] \Bigr) + k \, \mathbb{E}\Bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\Bigr]\,, \end{split} \end{equation} where $\widetilde{I^{\ell, n}_{3;{\tt A}_1}} := \bigl\|[ D_u\sigma\bigl( u(t_{\ell})\bigr) - D_u\sigma(u^{\ell})] v\bigl( t_{\ell}\bigr)\bigr\|^2_{\mathbb{L}^2}$. In order to handle the first term in the right-hand side, we estimate \eqref{I3;1_Itois} further by \begin{align*} \le Ck^2 \sum_{\ell=1}^{n-1} \Bigl(\mathbb{E} \Big[ \|e_u^{\ell} \|^2_{\mathbb{L}^2}+\|e_u^{\ell-1} \|^2_{\mathbb{L}^2} \Big] + k^2 {\mathbb E} \Big[\widetilde{I^{\ell, n}_{3;{\tt A}_1}} \Big] \Bigr)+ k \, \mathbb{E}\Bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2}\Bigr]\,. \end{align*} % We estimate the second term in the right-hand side as \begin{equation}\label{eq-embe} \begin{split} C k^4 \sum_{\ell=1}^{n-1} {\mathbb E}\Big[\widetilde{I^{\ell, n}_{3;{\tt A}_1}} \Big] &\leq C k^4 \sum_{\ell=1}^{n-1} \mathbb{E}\Bigl[ \|e_u^{\ell}\|_{\mathbb{L}^2}^2 \|v(t_{\ell})\|_{\mathbb{L}^{\infty}}^2 \Bigr] \\ &\leq C k \cdot k^3 \sum_{\ell=1}^{n-1} \mathbb{E}\Bigl[ \|e_u^{\ell}\|_{\mathbb{L}^2} \| e_u^{\ell}\|_{\mathbb{L}^2} \|v(t_{\ell})\|_{\mathbb{L}^{\infty}}^2 \Bigr] \\ &\leq Ck^2 \sum_{\ell=1}^{n-1} {\mathbb E} \Bigl[ \Vert e^{\ell}_u\Vert^2_{{\mathbb L}^2} \Bigr] + C k^6 \sum_{\ell=1}^{n-1} {\mathbb E} \Bigl[ \| e^{\ell}_u\Vert^4_{{\mathbb L}^2} \Bigr] + C k^6 \sum_{\ell=1}^{n-1} {\mathbb E} \Bigl[ \Vert v(t_{\ell})\Vert^8_{{\mathbb L}^{\infty}} \Bigr]\, , \end{split} \end{equation} where the last term on the right-hand side is bounded by $Ck^5$ due to Lemma \ref{lem:L2} $(iii)$. The second term on the right-hand side is bounded further by $C k^6 \sum_{\ell=1}^{n-1} {\mathbb E} \Bigl[ \Vert u(t_{\ell})\Vert^4_{{\mathbb L}^2} + \Vert u^{\ell}\Vert^4_{{\mathbb L}^2} \Bigr]$, which may be bounded by $Ck^5$, thanks to Lemma \ref{lem:L2} $(i)$ for $p=2$, and \eqref{energy2-himo}. \smallskip \noindent {\bf b)} Now consider ${\mathbb E}[I^{\ell, n}_{3;{\tt A}_2}]$. Let $\xi \in [t_n, t_{n+1}]$; we use the mean-value theorem twice, {\bf (A4)} for $m=1, 2$, to conclude \begin{equation}\label{sig-u-esti} \begin{split} &\bigl\Vert \sigma \bigl( u(\xi) \bigr) - \sigma\bigl(u(t_{\ell})\bigr) -D_u \sigma\bigl(u(t_{\ell})\bigr)v(t_{\ell})(\xi - t_{\ell}) \bigr\Vert^2_{{\mathbb L}^2} \\ & = \Bigl\Vert [ D_u \sigma(\widetilde{u}_\zeta) - D_u \sigma\bigl(u(t_{\ell})\bigr)]\int_{t_{\ell}}^{\xi} v(\eta)\, {\rm d}\eta + D_u \sigma\bigl(u(t_{\ell})\bigr)\int_{t_{\ell}}^{\xi} \bigl[v(\eta)- v(t_{\ell})\bigr]\, {\rm d}\eta \Bigr\Vert^2_{{\mathbb L}^2} \\ & \leq C \Vert \nabla u(\xi) - \nabla u(t_{\ell}) \Vert_{{\mathbb L}^2}^4 + C k^2\, \sup_{t_{\ell} \leq \xi \leq t_{\ell+1}} \Vert v(\xi) - v(t_{\ell}) \Vert^2_{{\mathbb L}^2}\, , \end{split} \end{equation} where $\widetilde{u}_{\zeta} = \zeta u(\xi) + (1-\zeta) u(t_{\ell})$, for some $\zeta \in [0,1]$. Thus, we have \begin{equation*} \begin{split} {\mathbb E}[I^{\ell, n}_{3;{\tt A}_2}] &\leq Ck \, {\mathbb E}\Bigl[ \sup_{t_{\ell} \leq \xi \leq t_{\ell+1}} \bigl( \Vert \nabla u(\xi) - \nabla u(t_{\ell}) \Vert_{{\mathbb L}^2}^4 + k^2 \, \Vert v(\xi) - v(t_{\ell}) \Vert^2_{{\mathbb L}^2}\bigr) \Bigr] + k \, {\mathbb E} \big[ \Vert e^{n+1/2}_u\Vert^2_{{\mathbb L}^2} \big] \\ &\leq C k^4 + k \, {\mathbb E} \big[\Vert e^{n+1/2}_u\Vert^2_{{\mathbb L}^2} \big]\, . \end{split} \end{equation*} \noindent {\bf c)} We now consider ${\mathbb E}[I^{\ell, n}_{3;{\tt A}_3}]$. This term can be estimated by using the same arguments as for ${\mathbb E}[I^{\ell, n}_{3; {\tt A}_1}]$; see \eqref{I3;1_Itois} and \eqref{eq-embe}. Since we do not have the summation in this term, we will get \[ {\mathbb E}[I^{\ell, n}_{3; {\tt A}_3}] \leq C k^6 + C k \, \mathbb{E}\Bigl[\|e_u^{n+1}\|^2_{\mathbb{L}^2} + \|e_u^{n}\|^2_{\mathbb{L}^2}\Bigr]\,. \] \noindent {\bf d)} Then, we consider ${\mathbb E}[I^{\ell, n}_{3;{\tt A}_4}]$, which will follow the same arguments as for ${\mathbb E}[I^{\ell, n}_{3; {\tt A}_2}]$; see \eqref{sig-u-esti}. We get the estimate \[ {\mathbb E}[I^{\ell, n}_{3; {\tt A}_4}] \leq C k^5 + C k \, \mathbb{E}\bigl[\|e_u^{n+1/2}\|^2_{\mathbb{L}^2} \bigl]\,. \] \noindent {\bf e)} To estimate the term involving $\mathfrak{I}^{\ell,n}_{3,{\tt B}}$, which is defined in \eqref{dist-th}, we use Young's inequality to write \begin{equation* \begin{split} {\mathbb E}\bigl[\bigl(\mathfrak{I}^{\ell,n}_{3,{\tt B}}, {e}_u^{n+1/2} \bigr)\bigr] &\leq \frac{\widehat{\alpha}^2}{k} k^2 \,\mathbb{E} \Big[ \Big\| \sum_{\ell =1}^n D_u \sigma(u^{\ell}) v^{\ell} \big[ \widehat{\Delta_{\ell} W} - \widetilde{\Delta_{\ell} W} \big] \Big\|_{\mathbb{L}^2}^2 \Big] + k\, \mathbb{E} \big[ \Vert {e}_u^{n+1/2}\Vert^2_{{\mathbb L}^2} \big]\,. \end{split} \end{equation*} Then, we use {\bf (A4)} for $m=1$, and independence of increments $\Delta_nW$ to get \begin{equation*} \begin{split} &\leq \widehat{\alpha}^2\,C_g^2\, k \,\sum_{\ell =1}^n \mathbb{E} \Big[ \| D_u \sigma(u^{\ell}) v^{\ell} \|_{\mathbb{L}^2}^2 \big| \widehat{\Delta_{\ell} W} - \widetilde{\Delta_{\ell} W} \big|^2 \Big] + k\, \mathbb{E} \big[ \Vert {e}_u^{n+1/2}\Vert^2_{{\mathbb L}^2} \big]\,. \end{split} \end{equation*} Finally, we use \eqref{dist-tilde-hat} and \eqref{energy1} of Lemma \ref{lem:scheme1:stab} to obtain \begin{equation}\label{untersch1} \begin{split} &\leq \widehat{\alpha}^2\,C_g^2\, k^5 \,\sum_{\ell =1}^n \mathbb{E} \big[ \|v^{\ell} \|_{\mathbb{L}^2}^2 \big] + k\, \mathbb{E} \big[ \Vert {e}_u^{n+1/2}\Vert^2_{{\mathbb L}^2} \big] \leq C k^4 + k\, \mathbb{E} \big[ \Vert {e}_u^{n+1/2}\Vert^2_{{\mathbb L}^2} \big]\, . \end{split} \end{equation} \smallskip \noindent {\bf f)} We may modify the argument in part {\bf 2)} to improve the order for the for the order limiting term ${\tt I}_7$ in $I^{\ell, n}_7$; see step {\bf f}$_1$). Using integration by parts and using Lemma~\ref{lem:Holder} $(iv)$ instead, we verify (\ref{cond1}) of Lemma \ref{quadrature1} for $\gamma=1/2$ (by choosing $f(\xi) = \mathbb{E}\bigl[ (\nabla u(\xi), \nabla e_u^{n+1/2})\bigr]$ for all $\xi \in [t_n, t_{n+1}]$ ) to get \begin{equation*} \Bigl|\mathbb{E}\Bigl[\bigl(\nabla [v(t)- v(s)], \nabla e_u^{n+1/2} \bigr)\Bigr]\Bigr| \le C \Bigl(\mathbb{E}\bigl[ \| e_u^{n+1/2} \|^2_{\mathbb{L}^2}\bigr]\Bigr)^{1/2} \vert t-s \vert^{\frac{1}{2}}\, . \end{equation*} Using this estimate we may bound the term in ${\bf f}_1)$ by \begin{equation*} \le Ck^{\frac{5}{2}} \Bigl(\mathbb{E}\bigl[ \bigl\| e_u^{n+1/2} \bigr\|^2_{\mathbb{L}^2}\bigr]\Bigr)^{1/2} \leq Ck\, {\mathbb E}\bigl[ \Vert e^{n+1/2}_u\Vert^2_{{\mathbb L}^2} \bigr] + Ck^4\, . \end{equation*} Thanks to the above estimates, and after summation over all iteration steps in (\ref{e1_2:5}) we may then conclude assertion \eqref{pii}. \end{proof} \bigskip \section{Computational experiments}\label{sec-6} In this section, we provide computational studies to check \begin{itemize} \item how essential the assumptions {\bf (A1)--(A5)} and {\bf (B1)--(B2)} ({\em i.e.,} needed in Sections \ref{Mat-fra}--\ref{sec-4}) are in actual computations. In this respect, we computationally study the impact of rough initial data $(u_0, v_0)$ on the discrete dynamics, as well as of drift nonlinearities $F$ (see Example \ref{exm-alp7}). \item If the diffusion $\sigma \equiv \sigma(v)$ and the drift $F \equiv 0$, then there is a reduction of convergence order as proved in \eqref{eq-5.32} of Theorem \ref{lem:scheme1:con}; see Example \ref{ex-sv}. \item The diffusion $\sigma \equiv \sigma(u,v) = 0$ on the boundary, and satisfies {\bf (A3)}. Example \ref{exm-alp8} discusses the effect that noise has, which is non-homogeneous on the boundary, or violates {\bf (A3)}. \item By Theorem \ref{lem:scheme1:stab}, $\beta$ in the $(\widehat{\alpha}, \beta)$-scheme needs be chosen from $(0, 1/2)$ to ensure stable, accurate simulation of \eqref{stoch-wave1:1a} with $\sigma \equiv \sigma(u,v)$ and $F \equiv F(u,v)$. The simulations in Example \ref{exa-bet1} evidence a small choice for $\beta$ for faster Monte Carlo approximation. \end{itemize} \medskip We use the lowest order conforming finite element method to simulate the $(\widehat{\alpha}, \beta)-$scheme on a regular triangulation $\mathcal{T}_h$ of $\mathcal{O}$; see \cite{BS_2008}. Let the finite element space be \begin{align*} {\mathbb V}_h := \bigl\{u_h \in \mathbb{H}^1_0: \ u_h \bigl\vert_K \in \mathcal{P}_1(K) \quad \forall \, K \in \mathcal{T}_h \bigr\}\, , \end{align*} where $\mathcal{P}_1(K)$ denotes the space of polynomials of degree one on $K \in \mathcal{T}_h$. \\ \noindent As initial data, we choose $u^1$ and $v^1$ as \begin{align}\label{u1+v1} \begin{cases} u^1 &= u_0 +k\,v_0 + \frac{k^2}{2} \Delta u_0 + k^2 F(u_0) + (k+k^2) \,\sigma(u_0) \Delta_0 W\,, \\ v^1 &= v_0 +k \sigma(u_0) W(t_1)\,, \end{cases} \end{align} where $u_0, v_0$ (not finite element valued) satisfy assumptions {\bf (A1)$_{iv}$} and {\bf (B2)}. Recall the definitions for $\widetilde{u}^{n, 1/2}$ and $\widehat{\Delta_n W}$ in \eqref{tidt1} and \eqref{W-hat}, respectively. We implement the following scheme: \begin{scheme} Let $\widehat{\alpha} \in \{0, 1\}$, and $0 \leq \beta < \frac{1}{2}$. Let $\{ t_n\}_{n=0}^N$ be a mesh of size $k>0$ covering $[0, T]$, and \eqref{u1+v1}. For every $n \ge 1$, find a $[{\mathbb V}_h]^2$-valued, ${\mathcal F}_{t_{n+1}}$-measurable random variable~$(u_h^{n+1}, v_h^{n+1})$ such that \begin{align} \label{scheme1:1FEM} \bigl(u_h^{n+1} - u_h^n, \phi_h) &= k(v_h^{n+1},\phi_h) &\forall \phi_h \in {\mathbb V}_h\, , \\ \notag (v_h^{n+1}-v_h^n, \psi_h) &= -k \big(\nabla \widetilde{u}_h^{n, 1/2}, \nabla \psi_h \big) + \Bigl(\sigma(u^n_h, v^{n-\frac{1}{2}}_h)\Delta_n W, \psi_h \Bigr) &\\ &\quad + \widehat{\alpha} \,\Bigl(D_{u} \sigma(u_h^{n}, v_h^{n-\frac{1}{2}}) v_h^{n} \,\widehat{\Delta_n W}, \psi_h \Bigr) \label{scheme1:2FEM} &\\ &\quad +\frac{k}{2} \Bigl( 3 F(u_h^n, v_h^n) - F(u_h^{n-1}, v_h^{n-1}), \psi_h \Bigr) &\forall \psi_h \in {\mathbb V}_h\, . \notag \end{align} \end{scheme} \smallskip \subsection{Convergence rates} The numerical experiments are performed using MATLAB. In this section, for all the examples we choose $\mathcal{O} = (0, 1)$, $T=1$, $A = -\Delta$ in \eqref{stoch-wave1:1a}. We choose $u_0(x) = \sin(2 \pi x)$ and $v_0(x) = \sin(3 \pi x)$, and $u^1, v^1$ are chosen as in \eqref{u1+v1}. A reference solution is computed with a step size $k_{\tt ref}=2^{-7}$ and $h_{\tt ref}=2^{-7}$ to approximate the exact solution and the sample Wiener processes $W$. The expected values are approximated by computing averages over ${\tt MC} = 3000$ number of samples. The plots are shown for the time steps $k= \{2^{-3}, \cdots, 2^{-6}\}$. Example \ref{exm-alp0} in Section 1 provides computational evidence for the improved convergence rate $\mathcal{O}(k^{3/2})$ for the scheme \eqref{scheme2:1}--\eqref{scheme2:2} with $\widehat{\alpha}=1$ in the situations where $\sigma \equiv \sigma(u)$. In the following example, we consider $\sigma \equiv \sigma(v)$, and find a convergence rates of ${\mathcal O}(k^{1/2})$ in simulations {\rm (A)--(C)} of Fig. \ref{conv-F-sv}, which validates \eqref{eq-5.32} of Theorem \ref{lem:scheme1:con}. So we observe a reduction of convergence order if compared to Example \ref{exm-alp0}, where $\sigma \equiv \sigma(u)$. \smallskip \del{ \begin{example}[{\bf Additive noise}]\label{ex-additive-s} In Example \ref{exm-alp0}, take $\sigma(x)= 4x(1-x), \ x \in [0, 1]$. Let $u_0(x) = \sin(2 \pi x) $ and $ v_0(x) = \sin(3 \pi x)$ and $u^1, v^1$ are chosen as \eqref{u1+v1}. The $\mathbb{L}^2$-error for $u$ and $\nabla u$ are computed with the scheme \eqref{scheme2:1}--\eqref{scheme2:2} and Fig. \ref{additive} confirm the convergence order $\mathcal{O}(k^{3/2})$ and $\mathbb{L}^2$-error for $v$ confirms $\mathcal{O}(k)$. \smallskip \begin{figure}[h!] \centering \subfloat[$\mathbb{L}^2$-error for $u$]{\includegraphics[width=0.33\textwidth]{u-err-4x1-xMC1k}\label{fig:y1}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$]{\includegraphics[width=0.33\textwidth]{gu-err-4x1-xMC1k}\label{fig:y2}} \hfill \subfloat[$\mathbb{L}^2$-error for $v$]{\includegraphics[width=0.33\textwidth]{v-err-4x1-xMC1k}}\label{fig:y3} \caption{{\bf (Example \ref{ex-additive-s})} Temporal rates of convergence for $(\widehat{\alpha}, 0)-$scheme for $\widehat{\alpha}=1$ with $\sigma(x)= 4x(1-x)$; discretization parameters: $h=2^{-7}, k=\{2^{-3}, \cdots, 2^{-6}\}$, ${\tt MC}=3000$.} \label{additive} \end{figure} \end{example} } \begin{example}\label{ex-sv} Consider $\sigma(v) = \frac{3}{2} v$ and $F \equiv 0$. Fig. \ref{conv-F-sv} displays convergence studies for the $(\widehat{\alpha}, \beta)-$scheme for $\widehat{\alpha}=1$ and $\beta=1/4:$ the plots {\rm (A)--(C)} of $\mathbb{L}^2$-errors in $u, \nabla u$ and $v$, respectively, confirm convergence order ${\mathcal O}(k^{1/2})$; see \eqref{eq-5.32} of Theorem \ref{lem:scheme1:con}. \begin{figure}[h!] \centering \subfloat[$\mathbb{L}^2$-error for $u$]{\includegraphics[width=0.33\textwidth]{u-err-sv-MC1k}\label{fig--1}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$]{\includegraphics[width=0.33\textwidth]{gu-err-sv-MC1k}\label{fig--2}} \hfill \subfloat[$\mathbb{L}^2$-error for $v$]{\includegraphics[width=0.33\textwidth]{v-err-sv-MC1k}\label{fig--2}} \caption{{\bf (Example \ref{ex-sv})} Rates of convergence of the $\big(1, \frac{1}{4} \big)-$scheme with $\sigma(v) = \frac{3}{2} v$ and $F \equiv 0$.} \label{conv-F-sv} \end{figure} \end{example} \del{ \begin{example}\label{exm-alp6} {\color{blue}{In Example \ref{ex-additive-s}, take $\sigma(x) = 2(x+1)(x+2), \ x \in [0, 1]$. The errors are computed via the $(\widehat{\alpha}, 0)-$scheme. We compare the Fig. \ref{additive} with Fig. \ref{no+bou} to observe the reduction of convergence order for $\nabla u$ to $\mathcal{O}(k)$ in plot {\rm (B)}; in plots {\rm (A)} and {\rm (C)}, $\mathbb{L}^2$-errors in $u$ and $v$ remain the same as $\mathcal{O}(k^{3/2})$ and $\mathcal{O}(k)$, respectively. }} \smallskip \begin{figure}[h!] \subfloat[$\mathbb{L}^2$-error for $u$]{\includegraphics[width=0.33\textwidth]{n0-u-er-po-1k}\label{fig:-x--1}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$]{\includegraphics[width=0.33\textwidth]{n0-gu-er-po-1k}\label{fig:-x--2}} \hfill \subfloat[$\mathbb{L}^2$-error for $v$]{\includegraphics[width=0.33\textwidth]{n0-v-er-po-1k}}\label{fig:-x--3} \label{fig:--y-3} \caption{{\bf (Example \ref{exm-alp6})} {\color{blue}{Temporal rates of convergence for $(\widehat{\alpha}, 0)-$scheme with $\sigma(x) = 2(x+1)(x+2)$; discretization parameters: $h=1/100, k=\{2^{-3}, \cdots, 2^{-6}\}$, ${\tt MC}=1200$.}}} \label{no+bou} \end{figure} \end{example} } \smallskip \noindent In the following example, we discuss four different cases where \begin{itemize} \item[$(i)$] $F \equiv F(u,v)$ is non-zero on the boundary, but Lipschitz and $\sigma \equiv \sigma(u)$; \item[$(ii)$] $F \equiv F(u,v)$ only H\"older continuous, and $\sigma \equiv \sigma(u)$; \item[$(iii)$] $F \equiv F(u,v)$ is same as $(i)$, and $\sigma \equiv \sigma(u, v)$ satisfying {\bf (A3)}; \item[$(iv)$] $F \equiv F(u,v)$ is same as $(ii)$, and $\sigma \equiv \sigma(u, v)$ satisfying {\bf (A3)}. \end{itemize} We observe that although $F \equiv F(u,v)$ violates {\bf (A3)} in $(ii)$, we still get improved convergence rates, but if $\sigma \equiv \sigma(u, v)$, we get the convergence order $\mathcal{O}(k^{1/2})$ as shown in \eqref{eq-5.32} of Theorem \ref{lem:scheme1:con}. \begin{example}\label{exm-alp7} We consider the following cases: \begin{itemize} \item[$(i)$] $\sigma(u) = u$ and $F(u, v) = \cos(u)+2v$; \item[$(ii)$] $\sigma(u) = u$ and $F(u, v) = \sqrt{u} + \sqrt{v+2}$; \item[$(iii)$] $\sigma(u, v) = \frac{u}{1+u^2} + v$ and $F(u, v) = \cos(u)+2v$; \item[$(iv)$] $\sigma(u, v) = \frac{u}{1+u^2} + v$ and $F(u, v) = \sqrt{u} + \sqrt{v+2}$; \end{itemize} The errors are computed via the $(\widehat{\alpha}, \beta)-$scheme with $\widehat{\alpha}=1$ for $\beta=1/4 :$ the plots {\rm (A)--(C)} for the problem $(i)$ evidence the convergence order ${\mathcal O}(k^{3/2})$ for $u, \nabla u$, and ${\mathcal O}(k)$ for $v$. We observe the same convergence rates for the problem $(ii)$ despite the lack of Lipschitzness of $F$ which violates {\bf (A3)}; see plots {\rm (D)--(F)} of Fig. \ref{5.7}. The plots {\rm (G)--(I)} of $\mathbb{L}^2$-errors in $u, \nabla u$ and $v$, respectively, for the problem $(iii)$ and evidence the convergence order ${\mathcal O}(k^{1/2})$ as shown in \eqref{eq-5.32} of Theorem \ref{lem:scheme1:con}. We observe the same order of convergence for the problem $(iv)$; see plots {\rm (J)--(L)} of Fig. \ref{5.7}. Thus, the above two examples verify that the estimate \eqref{eq-5.32} is sharp in the case of diffusion $\sigma \equiv \sigma(u,v)$. \smallskip \begin{figure}[h!] \subfloat[$\mathbb{L}^2$-error for $u$ in $(i)$]{\includegraphics[width=0.24\textwidth]{u-er-FSuv-12h}\label{fig:x_-1}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$ in $(i)$]{\includegraphics[width=0.24\textwidth]{gu-er-FSuv-12h}\label{fig:x_-2}} \hfill \subfloat[$\mathbb{L}^2$-error for $v$ in $(i)$]{\includegraphics[width=0.24\textwidth]{v-er-FSuv-12h}}\label{fig:x_-3} \hfill \subfloat[$\mathbb{L}^2$-error for $u$ in $(ii)$]{\includegraphics[width=0.24\textwidth]{u-er-FSuv-sqrt}}\label{figu:x_-1} \par \subfloat[$\mathbb{L}^2$-error for $\nabla u$ in $(ii)$]{\includegraphics[width=0.24\textwidth]{gu-er-FSuv-sqrt}}\label{figu:x_-2} \hfill \subfloat[$\mathbb{L}^2$-error for $v$ in $(ii)$]{\includegraphics[width=0.24\textwidth]{v-er-FSuv-sqrt}}\label{figu:x_-1} \hfill \subfloat[$\mathbb{L}^2$-error for $u$ in $(iii)$]{\includegraphics[width=0.24\textwidth]{u-err-FSuv-1k}\label{fig:x_-1-}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$ in $(iii)$]{\includegraphics[width=0.24\textwidth]{gu-err-FSuv-1k}\label{fig:x_-2-}} \par \subfloat[$\mathbb{L}^2$-error for $v$ in $(iii)$]{\includegraphics[width=0.24\textwidth]{v-err-FSuv-1k}}\label{fig:x_-3-} \hfill \subfloat[$\mathbb{L}^2$-error for $u$ in $(iv)$]{\includegraphics[width=0.24\textwidth]{u-err-FS-sqrt}}\label{fi:y_-1-} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$ in $(iv)$]{\includegraphics[width=0.24\textwidth]{gu-err-FS-sqrt}}\label{fi:y_-2-} \hfill \subfloat[$\mathbb{L}^2$-error for $v$ in $(iv)$]{\includegraphics[width=0.24\textwidth]{v-err-FS-sqrt}}\label{fi:y_-3-} \label{fig:--y-3} \caption{{\bf (Example \ref{exm-alp7})} Rates of convergence of the $\big(1, \frac{1}{4} \big)-$scheme.} \label{5.7} \end{figure} \end{example} \smallskip In the next example, we drop the assumption on $\sigma \equiv \sigma(u)$ to be Lipschitz and zero on the boundary to see which of these violations spot the reduction of the convergence order of scheme \eqref{scheme2:1}-\eqref{scheme2:2}. \begin{example}\label{exm-alp8} Let $F \equiv 0$. Consider the following cases: \begin{itemize} \item[$(i)$] $\sigma(u) = \frac{1}{1+ u^2}$; \item[$(ii)$] $\sigma(u) = \sqrt{|u|}$. \end{itemize} In Fig. \ref{nln0}, the errors are computed via the scheme \eqref{scheme2:1}--\eqref{scheme2:2} with $\widehat{\alpha}=1.$ For problem $(i)$ (nonzero boundary), the plots {\rm (A)--(B)} for $\mathbb{L}^2$-errors in $u, \nabla u$, respectively, show the convergence order ${\mathcal O}(k^{3/2})$ and the plot {\rm (C)} for $\mathbb{L}^2$-error in $v$ shows ${\mathcal O}(k)$. For the problem $(ii)$ (non-Lipschitz), the convergence rates for $\mathbb{L}^2$-errors in $u, \nabla u$ are reduced to ${\mathcal O}(k)$; see plots {\rm (D)--(E)}, but $\mathbb{L}^2$-error in $v$ remains same as ${\mathcal O}(k)$; see plot {\rm (F)}. \smallskip \begin{figure}[h!] \subfloat[$\mathbb{L}^2$-error for $u$ in $(i)$]{\includegraphics[width=0.33\textwidth]{u-not0-1k}\label{f:x1}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$ in $(i)$]{\includegraphics[width=0.33\textwidth]{gu-not0-1k}\label{f:x2}} \hfill \subfloat[$\mathbb{L}^2$-error for $v$ in $(i)$]{\includegraphics[width=0.33\textwidth]{v-not0-1k}}\label{f:x3} \par \subfloat[$\mathbb{L}^2$-error for $u$ in $(ii)$]{\includegraphics[width=0.33\textwidth]{u-notL-1k}\label{f-:-x1}} \hfill \subfloat[$\mathbb{L}^2$-error for $\nabla u$ in $(ii)$]{\includegraphics[width=0.33\textwidth]{gu-notL-1k}\label{f-:-x2}} \hfill \subfloat[$\mathbb{L}^2$-error for $v$ in $(ii)$]{\includegraphics[width=0.33\textwidth]{v-notL-1k}}\label{f-:-x3} \label{f:y3} \caption{{\bf (Example \ref{exm-alp8})} Rates of convergence of the the scheme \eqref{scheme2:1}--\eqref{scheme2:2} for $\widehat{\alpha}=1$.} \label{nln0} \end{figure} \end{example} \subsection{Choice of $\beta$ and required number of {\tt MC}} \begin{example}\label{exa-bet1} Let $\mathcal{O}=(0, 1)$, $T=0.5$, $A = -\Delta$, $F \equiv 0$, $\sigma(v)= 5v$. We compute $W$ on the mesh of size $k = 2^{-12}$ covering $[0, 0.5]$. In the $(\widehat{\alpha}, \beta)$-scheme, the term $\widetilde{u}^{n, \frac 12} = {u}^{n, \frac 12} + \beta k^{1+\beta} v^{n+\frac{1}{2}}$ involves $\beta$, where the last term creates an additional numerical dissipation term in \eqref{stoch-wave1:1a} to control discretization effect of the noise. For $\beta=0$ with $\sigma \equiv \sigma(u)$ and $F \equiv F(u)$, the scheme \eqref{scheme2:1}--\eqref{scheme2:2} is stable, but for general case we require $\beta \in (0, 1/2)$ for the stability of the $(\widehat{\alpha}, \beta)$-scheme; see Lemma \ref{lem:scheme1:stab}. For increased value of $\beta$, stabilization effect vanishes for small $k$. Thus, a smaller choice of $\beta$ is preferred to have the stability of the scheme. The snapshot $(A)$ in Fig. \ref{beta-MC} shows for $\beta = 0, \frac 14, \frac 12, \frac 34, 1$, that at least ${\tt MC}= 400, 600, 800, 1000,1400$, are needed to have a steady of the energy $\mathcal{E}$ at time $T=0.5$. The snapshot $(B)$ evidence a higher number of {\tt MC} as we increase $\beta$ to have a steady energy curve. \begin{figure}[h!] \centering \subfloat[Energy for different $\beta$]{\includegraphics[width=0.33\textwidth]{all-ene-beta}\label{fig:f1}} \subfloat[$\beta$ vs number of {\tt MC}]{\includegraphics[width=0.33\textwidth]{betavsMCplots}\label{fig:x3}} \caption{{\bf (Example \ref{exa-bet1})} $(\widehat{\alpha}, \beta)$-scheme with $\sigma(v)= 5v$, and $F \equiv 0$. } \label{beta-MC} \end{figure} \end{example} \del{ \begin{example}\label{nl-F} In this example, we discuss the possible blow-up of energy by considering the non-zero $F=F(u, v)$ in the system \eqref{stoch-wave1:1}. Figs. \ref{energy-F} $(A)-(F)$ below display the energy profiles (as introduced before) for different $F$ and $\sigma.$ Figs. \ref{energy-F} $(A), (B)$ and $(C)$ suggest that there is no blow-up of energy when we consider $F(u, v)=u^2$ for different $\sigma.$ In fact, this holds true for any $F$ that is polynomial function in $u.$ \begin{figure}[h!] \centering \del{ \subfloat[$\sigma(u, v)=0,$ $F(u, v)=v^2$]{\includegraphics[width=0.32\textwidth]{s0-sol-F}\label{fig--f1}} \hfill \subfloat[$\sigma(u, v)=u,$ $F(u, v)=v^2$]{\includegraphics[width=0.32\textwidth]{su-sol-F}\label{fig--f2}} \hfill \subfloat[$\sigma(u, v)=v,$ $F(u, v)=v^2$]{\includegraphics[width=0.32\textwidth]{sv-sol-F}\label{fig--f3}} \par } \subfloat[$\sigma(u, v)=0,$ $F(u, v)=u^2$]{\includegraphics[width=0.32\textwidth]{s0-ene-F}\label{fi-f1}} \hfill \subfloat[$\sigma(u, v)=u,$ $F(u, v)=u^2$]{\includegraphics[width=0.32\textwidth]{su-ene-F}\label{fi-f2}} \hfill \subfloat[$\sigma(u, v)=v,$ $F(u, v)=u^2$]{\includegraphics[width=0.32\textwidth]{sv-ene-F}\label{fi-f3}} \par \subfloat[$\sigma(u, v)=0,$ $F(u, v)=v^2$]{\includegraphics[width=0.32\textwidth]{s0-ene-Fb}\label{fi-f4}} \hfill \subfloat[$\sigma(u, v)=u,$ $F(u, v)=v^2$]{\includegraphics[width=0.32\textwidth]{su-ene-Fb}\label{fi-f5}} \hfill \subfloat[$\sigma(u, v)=v,$ $F(u, v)=v^2$]{\includegraphics[width=0.32\textwidth]{sv-ene-Fb}\label{fi-f6}} \del{ \par \subfloat[$\sigma(u, v)=0,$ $F(u, v)=v^5$]{\includegraphics[width=0.32\textwidth]{s0-ene-Fv5}\label{fi-f10}} \hfill \subfloat[$\sigma(u, v)=u,$ $F(u, v)=v^5$]{\includegraphics[width=0.32\textwidth]{su-ene-Fv5}\label{fi-f11}} \hfill \subfloat[$\sigma(u, v)=v,$ $F(u, v)=v^5$]{\includegraphics[width=0.32\textwidth]{sv-ene-Fv5}\label{fi-f12}} } \caption{{\em The $\widetilde{\alpha}$-scheme \eqref{scheme2:1}-\eqref{scheme2:2} is implemented with $h=2^{-6}$ and $k=2^{-9}$.}} \label{energy-F} \end{figure} For $F(u, v)=v^2$, we observe a blow-up of the energy at different times for different $\sigma$; see Figures \ref{energy-F} $(D), (E)$ and $(F)$. We observe that the blow-up occurs at an earlier time in the case when $\sigma(u, v)=v$. \end{example} }

proofpile-arXiv_069-10550

\section*{Introduction}} \label{intro} \IEEEPARstart{D}{ynamic \dMa{s} were already investigated by the authors in \cite{dem}. The reached theory is good enough for a description of tense operators in a logic satisfying the double negation law when a frame is given as well for the task to determine a frame provided tense operators are given.} When studying partial dynamic \dMa{}s, we are given a De Morgan poset and we solve both the questions mentioned above. For this, we have to modify our original definition by axioms which are formulated in the language of ordered sets with involution only. On the other hand, we have an advantage of using the algebraic tools introduced already in \cite{dem} which can essentially shorten our paper. For the reader convenience we repeat that tense operators are introduced for to incorporate the time dimension in the logic under consideration. It means that our given logic is enriched by the operators $G$ and $H$, see e.g. \cite{1} for the classical logic and \cite{chirita}, \cite{2}, \cite{chajda} for several non-classical logics. It is worth noticing that the operators $G$ and $H$ can be considered as certain kind of modal operators which were already studied for intuitionistic calculus by D. Wijesekera \cite{wijesekera} and in a general setting by W.B.~Ewald \cite{Ewald}. For the logic of quantum mechanics (see e.g. \cite{dvurec} for details of the so-called quantum structures), the underlying algebraic structure is e.g. an orthomodular lattice or the so-called effect algebra (see \cite{dvurec},\cite{FoBe}) and the corresponding tense logic was treated in \cite{chajdakolarik,dyn,dynpos,dyn2}, in a bit more general setting also in \cite{botur}. The paper is organized as follows. After introducing several necessary algebraic concepts, we introduce tense operators in a \dMp, i.e., in an arbitrary logic satisfying double negation law without regards what another logical connectives are considered. Moreover, in this logic neither the principle of contradiction, nor the principle of excluded middle are valid for the negation, but all De Morgan laws hold. Also we get a simple construction of tense operators which uses lattice theoretical properties of the underlying ordered set. In Section \ref{setreppartDMa} we outline the problem of a representation of partial dynamic \dMa{}s and we solve it for partial dynamic \dMa{}s satisfying natural assumptions. This means that we get a procedure how to construct a corresponding frame to be in accordance with the construction from Section \ref{prelim}. In particular, any dynamic \dMa{} is set representable. \medskip \section{{Preliminaries} and basic facts} \label{prelim} We refer the reader to \cite{Balbes} for standard definitions and notations for lattice structures. \begin{definition} A structure ${\mathbf A}=(A;\leq,',0,1)$ (${\mathbf A}=(A;\wedge,\vee,',0,1)$) is called a {\bfseries De Morgan poset} ({\bfseries De Morgan lattice}) if $(A;\leq,0,1)$ is a poset ($(A;\wedge,\vee,0,1)$ is a lattice) with the top element $1$ and the bottom element $0$ and $'$ is a unary operation called {\bfseries negation} with properties\ $a\leq b \Rightarrow b'\leq a'$\ and $a=a''$. \end{definition} In fact in a De Morgan poset $A$ we have $a\leq b$ iff $b'\leq a'$, because $a\leq b\Rightarrow b'\leq a'\Rightarrow a''\leq b''\Rightarrow a\leq b$. A \textbf{ morphism} $f\colon A\to B$ \textbf{ of bounded posets} (\dMp{}s) is an order, (negation), top element and bottom element preserving map. A morphism $f\colon A\to B$ of bounded posets is \textbf{order reflecting} if ($f(a)\leq f(b)$ if and only if $a\leq b$) for all $a, b\in A$. Let $h\colon A \to B$ be a partial mapping of \dMp{}s. We say that the partial mapping $h^{\partial}\colon A\to B$ is the {\bfseries dual of $h$} if $h^{\partial}(a)$ is defined for all $a\in A$ such that $a'\in \mathop{dom} h$ in which case $$ h^{\partial}(a)=h(a')'. $$ \begin{figure}[h] \centering \begin{tikzpicture}[scale=0.71654976583] \coordinate [label=right:\phantom{ll}\hbox{$1=0'$}] (1) at (0,-1.2); \coordinate [label=left:\phantom{lll}\hbox{$c=b'$}\phantom{lll}] (c) at (-1.4,-2.6); \coordinate [label=left:\phantom{lll}\hbox{$a=d'$}\phantom{lll}] (a) at (-1.4,-4.6); \coordinate [label=right:\phantom{l}\hbox{$d=a'$}\phantom{lll}] (d) at (1.4,-2.6); \coordinate [label=right:\phantom{l}\hbox{$b=c'$}\phantom{lll}] (b) at (1.4,-4.6); \coordinate [label=right:\phantom{lll}\hbox{$0=1'$}] (0) at (0,-6); \draw (0) -- (a) -- (c) -- (1); \draw (0) -- (b) -- (d) -- (1); \draw (a) -- (d); \draw (b) -- (c); \fill[black,opacity=.75] (0) circle (4pt); \fill[black,opacity=.75] (1) circle (4pt); \fill[black,opacity=.75] (a) circle (4pt); \fill[black,opacity=.75] (b) circle (4pt); \fill[black,opacity=.75] (c) circle (4pt); \fill[black,opacity=.75] (d) circle (4pt); \end{tikzpicture} \caption{The poset of Example \ref{benotbeJP}}\label{benzene} \end{figure} \begin{example}\label{benotbeJP} {\upshape{}The \dMp{} $\mathbf{M}=({M};\leq,$ $', 0,1)$, $M=\{0, a, b, c, d, 1\}$ displayed by the Hasse diagram in Figure \ref{benzene} is the smallest non-lattice \dMp. } \end{example} \begin{observation}[\cite{dyn}]\label{obsik} Let $\mathbf A, \mathbf B$ be bounded posets (\dMp{s}), $T$ a set of morphisms from $\mathbf A$ to $\mathbf B$ of bounded posets (\dMp{s}). The following conditions are equivalent: \begin{enumerate} \item[{\rm(i)}] \(((\forall t \in T)\, {t}(a)\leq {t}(b))\implies a\leq b\) for any elements \(a,b\in A\); \item[{\rm(ii)}] The morphism $i_{\mathbf A}^{T}\colon{}A \to B^{T}$ defined by $i_{\mathbf A}^{T}(a)=(t(a))_{t\in T}$ for all $a\in A$ is order reflecting. \end{enumerate} \end{observation} We then say that $T$ is a {\bfseries full set of order preserving maps with respect to} ${\mathbf B}$. We may in this case identify ${\mathbf A}$ with a subposet (sub-\dMp{}) of ${\mathbf B}^{T}$ since $i_{\mathbf A}^{T}$ is an order reflecting morphism of bounded posets (\dMp{s}). A pair $(f,g)$ of order-preserving mappings $f\colon{}A\to B$ and $g\colon{}B \to A$ between posets ${\mathbf A}$ and ${\mathbf B}$ is a {\bfseries Galois connection} or an {\bfseries adjunction} between ${\mathbf A}$ and ${\mathbf B}$ provided that $f(a)\leq b\ \text{if and only if}\ a\leq g(b)$ {for all}\ $a\in A, b\in B.$ In an adjunction $(f, g)$ the mapping $f$ is called the {\bfseries left adjoint} and the mapping $g$ is called the {\bfseries right adjoint}. The pair $(f,g)$ of order-preserving mappings $f\colon{}A\to B$ and $g\colon{}B \to A$ is an adjunction if and only if $$a\leq g(f(a)) \, \text{and}\, f(g(b))\leq b\ \text{for all}\, a\in A, b\in B.$$ The second concept which will be used are so-called tense operators. They are in certain sense quantifiers which quantify over the time dimension of the logic under consideration. These tense operators were firstly introduced as operators on Boolean algebras (see \cite{1} for an overview). Chajda and Paseka introduced in \cite{dem} the notion of a dynamic \dMa. \label{orto} The following notion of a partial dynamic \dMa\ is stronger than the notion introduced in \cite{dem} but for dynamic \dMa{s}\ both notions coincide. Note only that our condition (P1) combined with the condition (T4) for tense De Morgan algebras in the sense of \cite{figallo} yields our condition (P4). \begin{definition}\label{dadM} By a \textbf{partial dynamic \dMa{}} is meant a triple $\mathbf{D}=(\mathbf{A};G,H)$ such that $\mathbf{A}=(A;\leq,',0,1)$ is a \dMp{} with negation $'$ and $G, H$ are partial mappings of $A$ into itself satisfying \begin{itemize} \item[(P1)] $G(0)=0$, $G(1)=1$, $H(0)=0$ and $H(1)=1$. \item[(P2)] $x\leq y$ implies $G(x)\leq G(y)$ whenever $G(x), G(y)$ exist, and $H(x)\leq H(y)$ whenever $H(x), H(y)$ exist. \item[(P3)] $x\leq GP(x)$ whenever $H(x')$ exists, $P(x)=H^{\partial}(x)$ and $GP(x)$ exists, and $x\leq HF(x)$ whenever $G(x')$ exists, $F(x)=G^{\partial}(a)$ and $HF(x)$ exists. \item[(P4)] $x\leq y$ implies $G(x)\leq F(y)$ whenever $G(y')$ and $G(x)$ exist, and $H(x)\leq P(y)$ whenever $H(y')$ and $H(x)$ exist. \end{itemize} Just defined $G$ and $H$ will be called \textbf{tense operators} of a partial dynamic \dMa{} $\mathbf{D}$. If both $G$ and $H$ are total we will speak about a \textbf{dynamic \dMa{}}. If we omit the condition (P3), i.e., only the conditions (P1), (P2) and (P4) are satisfied we say that $G$ and $H$ are \textbf{semi-tense operators on} $\mathbf{A}$. If $(\mathbf{A}_1;G_1, H_1)$ and $(\mathbf{A}_2;G_2, H_2)$ are partial dynamic algebras, then a {\bfseries morphism of partial dynamic algebras} $f\colon (\mathbf{A}_1;G_1, H_1)\to (\mathbf{A}_2;G_2, H_2)$ is a morphism of \dMp{}s such that $f(G_1(a))=G_2(f(a))$, for any $a\in A_1$ such $G_1(a)$ is defined and $f(H_1(b))=H_2(f(b))$, for any $b\in A_1$ such $H_1(b)$ is defined. Partial dynamic \dMa{} $\mathbf{D}=(\mathbf{A};G,H)$ is called {\bfseries complete} if its reduct $(A;\leq,$ $0,1)$ is a complete lattice. \end{definition} The semantical interpretation of these \textbf{tense operators} $G$ and $H$ is as follows. Consider a pair $(T,\leq)$ where $T$ is a non-void set and $\leq$ is a partial order on $T$. Let $s\in T$ and $f(s)$ be a formula of a given logical calculus. We say that $G\bigl( f(t)\bigr)$ \textbf{is valid} if for any $s\geq t$ the formula $f(s)$ is valid. Analogously, $H\bigl( f(t)\bigr)$ is valid if $f(s)$ is valid for each $s\leq t$. Thus the unary operators $G$ and $H$ constitute an algebraic counterpart of the tense operations ``it is always going to be the case that" and ``it has always been the case that", respectively. Similarly, the operators $F$ and $P$ can be considered in certain sense as existential quantifiers ``it will at some time be the case that" and ``it has at some time been the case that". \medskip In what follows we want to provide a meaningful procedure giving tense operators on every \dMp\ which will be in accordance with an intuitive idea of time dependency. By a \textbf{frame} (see e.g. \cite{2}) is meant a couple $(T,R)$ where $T$ is a non-void set and $R$ is a binary relation on $T$. Furthermore, we say that $R$ is {\bfseries serial} for all $x\in T$ there is $y\in T$ such that $x\mathrel{R}y$. In particular, every reflexive relation is serial. The set $T$ is considered to be a \textbf{time scale}, the relation $R$ expresses a relationship ``to be before" and ``to be after". Having a \dMp{} $\mathbf{A}=(A;\leq,{}', 0, 1)$ and a non-void set $T$, we can produce the direct power $\mathbf{A}^T=(A^T;\leq,{}', o,j)$ where the relation $\leq$ and the operation $'$ are defined and evaluated on $p,q\in A^T$ componentwise, i.e. $p\leq q$ if $p(t)\leq q(t)$ for each $t\in T$ and $p'(t)=p(t)'$ for each $t\in T$. Moreover, $o, j$ are such elements of $A^T$ that $o(t)=0$ and $j(t)=1$ for all $t\in T$. \begin{theorem}{\em\cite[Theorem II.7,Corollary II.9]{dem}} {}\label{upcomplate} Let $\mathbf{M}=(M;\leq,',0,1)$ be a complete \dMa{} and let $(T,R)$ be a frame. Define mappings $\widehat{G}, \widehat{H}$ of $M^T$ into itself as follows: For all $p\in M^T$ and all $x\in T$, $$\begin{array}{r c l} \mbox{$\widehat{G}(p)(x)$}&=&\mbox{$\bigwedge_{M}\{p(y)\mid x \mathrel{R} y\} $}\phantom{.}\quad \text{and}\\ \mbox{$\widehat{H}(p)(x)$}&=&\mbox{$\bigwedge_{M}\{p(y)\mid y \mathrel{R} x\} $}. \end{array}$$ Then \begin{enumerate}[{\rm (a)}] \item $\widehat{G},\widehat{H}$ are total operators on $\mathbf{M}^T$. \item If $R$ is serial then $\widehat{G}$ is a semi-tense operator. \item If $R^{-1}$ is serial then $\widehat{H}$ is a semi-tense operator. \item If $R$ and $R^{-1}$ are serial then $\mathbf{D}=(\mathbf{M}^T;\widehat{G},\widehat{H})$ is a dynamic \dMa{}. \end{enumerate} \end{theorem} We say that the {\bfseries operators $\widehat{G}$ and $\widehat{H}$ on ${M}^T$ are constructed by means of $(T,R)$}. \section{Set representation of partial dynamic De Morgan algebras}\label{setreppartDMa} In Theorem \ref{upcomplate}, we presented a construction of natural tense operators when a \dMp{} and a frame are given. However, we can ask, for a given partial dynamic \dMa\ $(\mathbf{A};G,H)$, whether there exist a frame $(T,R)$ and a complete de Morgan lattice $\mathbf{M}=(M;\leq,',0,1)$ such that the tense operators $G, H$ can be derived by this construction where $(\mathbf{A};G,H)$ is embedded into the power algebra $(\mathbf{M}^T;\widehat{G},\widehat{H})$. Hence, we ask, that there exists a suitable set $T$ and a binary relation $R$ on $T$ such that if every element $p$ of $A$ is in the form $(p(t))_{t\in T}$ in $M^T$ then $G(p)(s)= \bigwedge_M\{p(t)\mid s\mathrel{R}t\}$ for all $p\in \mathop{dom} G$ and $s\in T$, and $H(p)(s) = \bigwedge_M\{p(t)\mid t\mathrel{R}s\}$ for all $p\in \mathop{dom} H$ and $s\in T$. If such a representation exists then one can recognize the time variability of elements of $A$ expressed as time dependent functions $p\colon{}T \to M$ and $(\mathbf{A};G,H)$ is said to be {\bfseries representable in} $\mathbf{M}$ {\bfseries with respect to} $T$. In what follows, we will show that there is a set representation theorem for (partial) dynamic De Morgan algebras. Let us start with the following example. \begin{example} \label{TDDMAboolupcomplate} \upshape Let $\mathbf{2}=(\{0, 1\};\vee, \wedge, ', 0, 1)$ be a two-element Boolean algebra. We will denote by $\mathbf{M}_{2}=({M}_{2};\leq, ', 0, 1)$ a complete De Morgan lattice such that ${M}_{2}=\{0, 1\}\times \{0, 1\}$, $({M}_{2};\vee, \wedge, 0, 1)$ is a lattice reduct of the Boolean algebra $\mathbf{2}\times \mathbf{2}$ with the induced order $\leq$ and the negation on ${M}_{2}$ is defined by $(a,b)'=(b', a')$. Let $(T,R)$ be a frame. Let the operators $\widehat{G}$ and $\widehat{H}$ on ${\mathbf M}_{2}^T$ be constructed by means of $(T,R)$. Then by Theorem \ref{upcomplate} we have that $(\mathbf{M}_2^T;\widehat{G},\widehat{H})$ is a complete dynamic De Morgan algebra. Moreover, ${\mathbf M}_{2}^T$ is isomorphic as a lattice to a De Morgan algebra of sets. \end{example} \begin{figure}[h] \centering \begin{tikzpicture}[scale=0.7976583] \coordinate [label=left:\phantom{lll}\hbox{$(1,0)=(1,0)'$}\phantom{lll}] (a) at (-1.4,-4.6); \coordinate [label=right:\phantom{llll}\hbox{$(1,1)=(0,0)'$}\phantom{lll}] (e) at (0,-3.2); \coordinate [label=right:\phantom{llll}\hbox{$(0,1)=(0,1)'$}\phantom{lll}] (b) at (1.4,-4.6); \coordinate [label=right:\phantom{llll}\hbox{$(0,0)=(1,1)'$}] (0) at (0,-6); \draw (0) -- (a); \draw (0) -- (b) ; \draw (a) -- (e); \draw (b) -- (e); \fill[black,opacity=.75] (0) circle (4pt); \fill[black,opacity=.75] (a) circle (4pt); \fill[black,opacity=.75] (b) circle (4pt); \fill[black,opacity=.75] (e) circle (4pt); \end{tikzpicture} \caption{Figure of the underlying poset of the complete De Morgan lattice $\mathbf{M}_{2}$ from Example \ref{TDDMAboolupcomplate}}\label{FigM2v} \end{figure} Recall that the four-element De Morgan poset $\mathbf{M}_{2}$, considered as a distributive De Morgan lattice, generates the variety of all distributive De Morgan lattices (see e.g. \cite{Balbes}, \cite{Petrovich}). This result was a motivation for our study of the representation theorem of De Morgan posets. For any De Morgan poset ${\mathbf A}=(A;\leq, ', 0, 1)$, we will denote by $T_{\mathbf A}^{\text{DMP}}$ a set of morphisms of De Morgan poset into the four-element De Morgan poset $\mathbf{M}_{2}$. The elements $\kappa_D\colon{} A\to M_2$ of $T_{\mathbf A}^{\text{DMP}}$ (indexed by proper down-sets $D$ of $\mathbf{A}$ which correspond to morphisms of bounded posets $h_D\colon{} A\to \{0, 1\}$ such that $h_D(a)=0$ iff $a\in D$) are morphisms of De Morgan posets defined by the prescription ${{\kappa}_{D}}(a)=({h_{D}}(a), {h_{D}}^{\partial}(a))$ for all $a\in A$. As we will see later, this set $T_{\mathbf A}^{\text{DMP}}$ will serve as our time scale. Hence, our next task is to determine a binary relation $R$ on $T_{\mathbf A}^{\text{DMP}}$ such that the couple $(T_{\mathbf A}^{\text{DMP}}, R)$ will be our appropriate frame. Let us denote, for any proper down-set $D$ of $\mathbf{A}$, by $\partial(D)$ the set $A\setminus \{d'\mid d\in D$\}. Then $\partial(D)$ is again a proper down-set of $\mathbf{A}$ such that $ {h_{D}}^{\partial}={h_{{\partial}(D)}}$ and we have $ {h_{D}}={{h_{{\partial}(D)}}^{\partial}}$. To simplify the notation we will use for elements of $T_{\mathbf A}^{\text{DMP}}$ letters $s$ and $t$ whenever we will need not their concrete representation via down-sets. \begin{proposition}\label{distDMPvloyeni} Let ${\mathbf A}=(A;\leq, ', 0, 1)$ be a De Morgan poset{}. Then the map $i_{{}{\mathbf A}}\colon{}A\to M_2^{T_{\mathbf A}^{\text{DMP}}}$ given by $i_{{}{\mathbf A}}(a)(s)=s(a)$ for all $a\in A$ and all $s\in T_{\mathbf A}^{\text{DMP}}$ is an order-reflecting morphism of De Morgan posets such that $i_{{}{\mathbf A}}(A)$ is a De Morgan subposet of \/ ${\mathbf M}_2^{T_{\mathbf A}^{\text{DMP}}}$. \end{proposition} \begin{proof} First, let us show that, for any proper down-sets $D$ of $\mathbf{A}$, the mapping $\kappa_D\colon{} A\to M_2$ is a morphism of De Morgan posets. Since both ${h_{D}}$ and ${h_{D}}^{\partial}$ are order-preserving we have that $\kappa_D$ is order-preserving. Now, let us compute $$\begin{array}{l} {h_{D}}^{\partial}(0)=({h_{D}}(0'))'=({h_{D}}(1))'=1'=0,\\ {h_{D}}^{\partial}(1)=({h_{D}}(1'))'=({h_{D}}(0))'=0'=1. \end{array} $$ It follows that both ${h_{D}}$ and ${h_{D}}^{\partial}$ preserve $0$ and $1$, i.e., they are morphisms of bounded posets. Hence $\kappa_D$ is a morphism of bounded posets. Let $a\in A$. Let us check that ${{\kappa}_{D}}(a')={{\kappa}_{D}}(a)'$. We have $$ \begin{array}{r c l} {{\kappa}_{D}}(a)'&=&({h_{D}}(a), {h_{D}}^{\partial}(a))'\\ &= ({h_{D}}^{\partial}(a)', {h_{D}}(a)')= ({h_{D}}(a'), {h_{D}}(a'')')\\ &=&({h_{D}}(a'), {h_{D}}^{\partial}(a')) {{\kappa}_{D}}(a'). \end{array} $$ This yields that $\kappa_D$ is a morphism of De Morgan posets. It remains to check that $T_{\mathbf A}^{\text{DMP}}$ a full set of morphisms. Let $a, b\in A$ such that $s(a)\leq s(b)$ for all $s\in T_{\mathbf A}^{\text{DMP}}$. In particular, ${h_{D}}(a)\leq {h_{D}}(b)$ for all proper down-sets $D$ of $\mathbf{A}$. Put $D=\{x\in A\mid x\leq b\}$. Since $ {h_{D}}(b)=0$ we have $ {h_{D}}(a)=0$, i.e., $a\in D$ and hence $a\leq b$. \end{proof} The next theorem solves our problem of finding the binary relation $R$ on $T_{\mathbf A}^{\text{DMP}}$. In fact, we are restricted here on a semi-tense operator $G$ only. \begin{theorem}\label{semiDMPdreprest} Let ${\mathbf A}=(A;\leq, ', 0, 1)$ be a De Morgan poset, $G\colon{}A\to A$ a semi-tense operator on ${\mathbf A}$. Let us put $$ \begin{array}{r@{\,}c@{\,}l} R_{G}&=&\{(s, t)\in T_{{\mathbf A}}^{\text{DMP}}\times T_{{\mathbf A}}^{\text{DMP}}\mid (\forall x\in \mathop{dom}(G))\\ & &\phantom{\{(s, t)\in T_{{\mathbf A}}^{\text{DMP}}\times T_{{\mathbf A}}^{\text{DMP}}\mid\ (s(G(x))\leq t(x))\}.\\ \end{array}$$ Then $(T_{{\mathbf A}}^{\text{DMP}},R_{G})$ is a frame with $R$ being serial. Let $\widehat{G}$ be the operator constructed by means of the frame $(T_{{\mathbf A}}^{\text{DMP}},R_{G})$ and let us put $\widehat{F}={\widehat{G}}^{\partial}$. Then the mapping $i_{{}{{\mathbf A}}}$ is an order-reflecting morphism of De Morgan posets into the complete De Morgan lattice ${\mathbf M}_2^{T_{{\mathbf A}}^{\text{DMP}}}$ such that the following diagram commutes: $$ \begin{diagram} {A} \lTo(2,0)^{{}{F}}&{A} \rTo(2,0)^{{}{G}}&{A}&&\\ \dTo(0,3)^{i_{{}{{\mathbf A}}}}&&\dTo(0,3)^{i_{{}{{\mathbf A}}}}& \dTo(0,3)_{i_{{}{{\mathbf A}}}}&&\\ {\mathbf M}_2^{T_{{\mathbf A}}^{\text{DMP}}}&\lTo(2,0)_{\widehat{F}}& {\mathbf M}_2^{T_{{\mathbf A}}^{\text{DMP}}}&\rTo(2,0)_{\widehat{G}} {\mathbf M}_2^{T_{{\mathbf A}}^{\text{DMP}}}&& \end{diagram} $$ \end{theorem} \begin{proof} First, let us verify that the following holds: \begin{enumerate} \item for all $b\in \mathop{dom}(G)$ and for all $s\in T_{{\mathbf A}}^{\text{DMP}}$, $s(G(b))=\bigwedge_{M_2}\{t(b)\mid s \mathrel{R_{G}} t\}$, \item for all $b\in \mathop{dom}(F)$ and for all $s\in T_{{\mathbf A}}^{\text{DMP}}$, $s(F(b))=\bigvee_{M_2}\{s(b)\mid s \mathrel{R_{G}} t\}$. \end{enumerate} Let us first check statement 1. Assume that $b\in \mathop{dom}(G)$ and $s\in T_{{\mathbf A}}^{\text{DMP}}$, $s=\kappa_D$ where $D$ is a proper down-set of $\mathbf{A}$. Then, for all $t\in T_{{\mathbf B}}^{\text{DMP}}$ such that $s \mathrel{R_G} t$, $s(G(b)) \leq t(b)$. Hence $$(h_D(G(b)), h_D^{\partial}(G(b)) \leq \mbox{$\bigwedge_{M_2}$}\{t(b)\mid s \mathrel{R_G} t\}.$$ To get the other inequality assume that $h_D(G(b))=0$ or $h_D^{\partial}(G(b))=0$. Assume that $h_D(G(b))=0$. Put $V=\{z\in A\mid$ $(\exists x\in \mathop{dom(G)})(z\geq x\ \text{and}\ h_D(G(x))=1)\}$ and $X=\{z\in A\mid (\exists y\in \mathop{dom(F)})(z\leq y$ $\text{and}\ h_D(F(y))=0)\ \text{or}\ z\leq b\}$. Then $V$ is a proper upper subset of ${\mathbf A}$, $1\in V$, and $X$ is a proper downset of ${\mathbf A}$, $0\in X$ such that $X\cap V=\emptyset$ and $b\in X$. To verify this, assume that there is an element $z\in X\cap V$. Then there is $x\in \mathop{dom(G)}$, $x\leq z$ such that $h_D(G(x))=1$. Also, we have that there is $y\in \mathop{dom(F)}$, $z\leq y$ such that $ h_D(F(y))=0$ or $x\leq z\leq b$). It follows that $x\leq y$ and $1=h_D(G(x))\leq h_D(F(y))=0$ or $h_D(G(b))=1$, a contradiction. Let $U$ be a maximal down-set of ${\mathbf A}$ including $X$ such that $V\cap U=\emptyset$. Hence $U$ determines a morphism $h_U\colon{}A\to \{ 0,1\}$ of bounded posets such that ${h_{U}}(z)=0$ for all $z\in X$ and ${h_{U}}(z)=1$ for all $z\in V$, i.e., $h_D(G(x))\leq {h_{U}}(x)$ for all $x\in \mathop{dom(G)}$. Let us check that ${h_{D}}^{\partial}(G(x))\leq {h_{U}}^{\partial}(x)$ for all $x\in \mathop{dom(G)}$. Assume that $1={h_{D}}^{\partial}(G(x))=h_D(F(x'))'$. Then $h_D(F(x'))=0$, i.e., $x'\in X$. It follows that ${h_{U}}(x')=0$, i.e., ${h_{U}}^{\partial}(x)=1$. But this yields that $(\forall x\in \mathop{dom}(G)) \phantom{\{}(s(G(x))\leq \kappa_U(x))$, i.e., $s \mathrel{R_G} \kappa_U=(h_U(-), {h_{U}}^{\partial}(-))$ and ${h_{U}}(b)=0$. Assume now that $h_D^{\partial}(G(b))=h_{{\partial}(D)}(G(b))=0$. As above, there is a maximal downset $W$ of ${\mathbf A}$ such that ${h_{\partial(W)}}^{\partial}(b)= {h_{W}}(b)=0$, $$h_{{\partial}(D)}(G(x))\leq {h_{W}}(x)\ \text{and }\ h_{{\partial}(D)}^{\partial}(G(x))\leq {h_{W}}^{\partial}(x)$$ for all $x\in \mathop{dom}(G)$. It follows that $h_D(G(x))\leq {h_{\partial(W)}}(x)$ and ${h_{D}}^{\partial}(G(x))\leq {h_{\partial(W)}}^{\partial}(x)$ for all $x\in B$, i.e., $s \mathrel{R_{G}} \kappa_{\partial(W)}$ and ${h_{\partial(W)}}^{\partial}(b)=0$. Consequently, $s(G(b))= \bigwedge_{M_2}\{t(b)\mid s \mathrel{R_G} t\}$. Let us check Statement 2. We have $$ \begin{array}{@{}r@{}l} s(F(&b))=s(G(b')')=s(G(b'))'\\[0.15cm &=(\bigwedge_{M_2}\{t(b')\mid s \mathrel{R_{G}} t\})'\\[0.15cm &=\bigvee_{M_2}\{t(b')'\mid s \mathrel{R_{G}} t\ =\bigvee_{M_2}\{t(b)\mid s \mathrel{R_{G}} t\}. \end{array} $$ It remains to verify that $R_G$ is serial. Let $s\in T_{{\mathbf A}}^{\text{DMP}}$. We know from $(P1)$ that $0=s(G(0))= \bigwedge_{M_2}\{t(0)\mid s \mathrel{R_G} t\}$. The set $\{t\in T_{{\mathbf A}}^{\text{DMP}}\mid s \mathrel{R_G} t\}$ is non-empty (otherwise one has $0=s(G(0))=1$, a contradiction). \end{proof} In what follows, we show that if $G$ and $H$ are semi-tense operators such that the induced relations $R_G$ and $R_H$ satisfy a natural condition $R_G=(R_H)^{-1}$ then the obtained frame is just the one we asked for. \begin{theorem}\label{seDMPdreprest} Let ${\mathbf A}=(A;\leq, ', 0, 1)$ be a De Morgan poset, $G, H\colon{}A\to A$ be semi-tense operators on ${\mathbf A}$ such that $R_G=(R_H)^{-1}$. Then $(T_{{\mathbf A}}^{\text{DMP}},R_{G})$ is a frame with $R_G$ and $ (R_G)^{-1}$ serial. Let $(\mathbf{M}_2^{T_{{\mathbf A}}^{\text{DMP}}};\widehat{G},\widehat{H})$ be the dynamic De Morgan algebra constructed by means of the frame $(T_{{\mathbf A}}^{\text{DMP}},R_{G})$. Then the mapping $i_{{}{{\mathbf A}}}$ is an order-reflecting morphism of De Morgan posets into the complete De Morgan lattice ${\mathbf M}_2^{T_{{\mathbf A}}^{\text{DMP}}}$ such that the following diagram commutes: $$ \begin{diagram} {A} \lTo(2,0)^{{}{H}}&{A} \rTo(2,0)^{{}{G}}&{A}&&\\ \dTo(0,3)^{i_{{}{{\mathbf A}}}}&&\dTo(0,3)^{i_{{}{{\mathbf A}}}}& \dTo(0,3)_{i_{{}{{\mathbf A}}}}&&\\ {\mathbf M}_2^{T_{{\mathbf A}}^{\text{DMP}}}&\lTo(2,0)_{\widehat{H}}& {\mathbf M}_2^{T_{{\mathbf A}}^{\text{DMP}}}&\rTo(2,0)_{\widehat{G}} {\mathbf M}_2^{T_{{\mathbf A}}^{\text{DMP}}}&& \end{diagram} $$ \end{theorem} \begin{proof} It immediately follows from Theorem \ref{upcomplate} and Theorem \ref{semiDMPdreprest}. \end{proof} The following theorem gives us a complete solution of our problem established in the beginning of this section. This is a new result showing that the partial dynamic \dMa{} can be equipped with the corresponding frame similarly as it is known for Boolean algebras in \cite{1} at least in a case when the tense operators $G$ and $H$ are interrelated. In particular, any dynamic \dMa{} has such a frame. \begin{theorem}\label{sercDMPdreprest} Let $({\mathbf A};G,H)$ be a partial dynamic De Morgan algebra such that \begin{enumerate}[(a) ] \item $x\in \mathop{dom}(G)$ implies $G(x)'\in \mathop{dom}(H)$, \item $x\in \mathop{dom}(H)$ implies $H(x)'\in \mathop{dom}(G)$. \end{enumerate} Then $(T_{{\mathbf A}}^{\text{DMP}},R_{G})$ is a frame with $R_G$ and $ (R_G)^{-1}$ serial such that $({\mathbf A};G,H)$ can be embedded into $(\mathbf{M}_2^{T_{{\mathbf A}}^{\text{DMP}}};\widehat{G},\widehat{H})$. \end{theorem} \begin{proof} It is enough to check that $R_G=(R_H)^{-1}$. Let $(s,t)\in R_G$, i.e., $(\forall x\in \mathop{dom}(G))(s(G(x))\leq t(x))$. We have to check that $(t,s)\in R_H$. Let $y\in \mathop{dom}(H)$. Then by assumption (b) we obtain that $H(y)'\in \mathop{dom}(G)$. It follows that $G(H(y)')'=F(H(y)$ is defined and by axiom (P3) we get that $G(H(y)')'\leq y$, i.e., $y'\leq G(H(y)')$. Since $(s,t)\in R_G$ we obtain that $s(y')\leq s(G(H(y)'))\leq t(H(y)')$. But $s$ and $t$ are morphisms of bounded \dMp{s} which yields that $ t(H(y))\leq s(y)$. Hence $R_G\subseteq (R_H)^{-1}$. A symmetry argument gives us that $R_H\subseteq (R_G)^{-1}$, i.e., $R_G=(R_H)^{-1}$. \end{proof} From Theorem \ref{sercDMPdreprest} we obtain the following. \begin{corollary}\label{totsercDMPdreprest} Let $({\mathbf A};G,H)$ be a dynamic De Morgan algebra. Then $(T_{{\mathbf A}}^{\text{DMP}},R_{G})$ is a frame with $R_G$ and $ (R_G)^{-1}$ serial such that $({\mathbf A};G,H)$ can be embedded into $(\mathbf{M}_2^{T_{{\mathbf A}}^{\text{DMP}}};\widehat{G},\widehat{H})$. \end{corollary} \begin{remark} Usually, one uses complex algebras associated with the given model to establish a discrete duality between algebraic and relational models. Figallo and Pelaitay in \cite{figallo} established a discrete duality between tense distributive De Morgan algebras and so-called tense De Morgan spaces. Since we are only interested in the representation of tense operators we use relational models without any additional structure. \end{remark} \section*{Acknowledgements} This is a pre-print of an article published as \newline I. Chajda, J. Paseka, Set representation of partial dynamic De Morgan algebras, In: Proceedings of the 46th IEEE International Symposium on Multiple-Valued Logic, Springer, (2016), 119--124, doi: 10.1109/ISMVL.2013.56. The final authenticated version of the article is available online at: \newline https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=\&\-arnumber=6524667. Both authors acknowledge the support by a bilateral project New Perspectives on Residuated Posets financed by Austrian Science Fund (FWF): project I 1923-N25, and the Czech Science Foundation (GA\v CR): project 15-34697L. J.~Paseka acknowledges the financial support of the Czech Science Foundation (GA\v CR) un\-der the grant Algebraic, Many-valued and Quantum Structures for Uncertainty Modelling: project 15-15286S.

proofpile-arXiv_069-10669

\section{INTRODUCTION} Wireless sensor networks have used for a wide range of applications such as environmental monitoring, military monitoring, health care and crisis management \cite{1}. In these environments without supervision, the sensors can be easily replaced or recharged \cite{2}; therefore, energy conservation in the networks is one of the most important research challenges. One method commonly used to prolong the network lifetime is through collecting data at the cluster heads \cite{3}. In cluster-based approaches, sensors do not need to communicate directly with the base station. Instead, cluster heads are responsible to organize cluster members and send the data collected within the cluster to the base station. This process leads to a significant reduction in the amount of data transmitted, resulting in lower energy consumption and longer network lifetime \cite{3}. LEACH \cite{4} is the first hierarchical routing protocol that is more efficient over traditional routing protocols. In LEACH, the cluster head is selected with a probabilistic method and attempts to distribute the network load at each sensor node \cite{5}. This protocol does not guarantee the number and position of cluster heads. LEACH-C \cite{6} is another routing protocol that follows a centralized approach to elect a cluster head using base station and location information of each node. This will produce better number of clusters and distributes the CHs evenly among the clusters. This method increases the network overhead, since all sensor nodes must send their location information to base station in each round \cite{5}. The K-means clustering algorithm in addition to extensive applications in various domains, such as data mining and image processing, has been used in \cite{7} to cluster and extend the lifetime of wireless sensor networks. The K-means algorithm is based largely on Euclidean distances and selection of clusters dependent on the remaining energy of the nodes \cite{7}. In this algorithm, the parameter K is chosen by the user, which in practice cannot be assumed that the actual number of clusters is predetermined. Hierarchical clustering (such as LEACH and LEACH-C) and distance-based clustering (such as K-means) are methods employed to optimize energy consumption and increase the lifetime of wireless sensor networks. The combination of these two methods of clustering is the main motive for this study. Approximate Rank-Order is a hierarchical clustering based on the distance criterion used in image processing to clustering millions of images \cite{8}. From here on, we use the ARO shortcut name to refer to the Approximate Rank-Order clustering. In this paper, a clustering algorithm in wireless sensor networks is provided using ARO clustering, in which the clustering step is performed before selecting the cluster head. The idea that the clustering stage should first be done is based on the work done in \cite{9}. This allows energy consumption in the network to be reduced by eliminating overhead advertisements (which is in all four other algorithms) per round and thus increasing the lifetime of the network. Comparing the time complexity of the proposed approach with other algorithms, the time complexity of this approach in the setup phase and cluster formation is equal to the time complexity of the Approximate Rank-Order algorithm in \cite{8} and equal to O(n). The time complexity of the LEACH algorithm according to \cite{10} is equal to O (1), and LEACH-C and LEACH with fuzzy descriptors algorithms are also O (1). The time complexity of the K-means algorithm is also equal to O (ncdi), in which n is the number of nodes, c is the number of clusters, d is the number of dimensions, and i is the number of repetitions \cite{11}. It is important to note that in the proposed approach, the clustering stage is performed only once, and the complexity of the time O (n) does not have a significant effect on the overall execution time. Because the total execution time for other algorithms is proportional to the number of rounds, in these experiments the number of rounds is greater than the number of nodes. The remainder of the article is as follows: Section 2 provides a summary of related work in this field. Section 3 introduces the radio model used in this paper. Section 4 describes the proposed approach. Simulation results and comparison of proposed approach with LEACH, LEACH-C, K-means and LEACH with fuzzy descriptors algorithms are presented in Section 5. \section{Related Work} Yarinezhad, Hashemi WSN \cite{yarinezhad2019routing} LBCH \cite{al2018lbch} CRDP \cite{wang2018crpd} Cluster head selection \cite{sarkar2019cluster} BPA-CRP \cite{darabkh2019bpa} Fuzzy \cite{hamzah2019energy} secure routing \cite{maitra2019cluster} unequal cluster \cite{yang2018unequal} cluster routing \cite{tuna2018clustering} mixed integer \cite{li2018clustering} pareto \cite{elhabyan2018pareto} qos aware \cite{yahiaoui2018energy} lifetime \cite{arjunan2018lifetime} integrated clustering \cite{gupta2018integrated} CREEP \cite{dutt2018cluster} neuro fuzzy \cite{thangaramya2019energy} Bayesian \cite{tyagi2015bayesian} LA-MHR \cite{tanwar2018mhr} EEMHR \cite{zhang2019energy} systematic \cite{tanwar2015systematic} lifte extended \cite{tyagi2015lifetime} cognitive \cite{tyagi2015cognitive} learning automata \cite{tyagi2015learning} EHE-LEACH \cite{tyagi2013ehe} Over the past decade, research activities in the field of clustering wireless sensor networks have grown significantly. A large number of clustering algorithms are proposed in wireless sensor networks; Such as LEACH \cite{4}, LEACH-C \cite{6}, K-means \cite{7}, LEACH with fuzzy descriptors \cite{5}, and SCRC-WS \cite{9}. In this section, we provide a brief review on the most popular WSN clustering protocols to get deeper understanding from their functions. This analysis helps us to analyze the pros and cons of state-of-the-art clustering protocols for WSN. \subsection{LEACH Protocol} Low Power Adaptive Hierarchy Clustering (LEACH) \cite{4} is a self-organizing adaptive protocol that uses a distributed clustering algorithm \cite{2}. The purpose of LEACH is to randomly select sensor nodes as cluster heads so that energy dissipation in relation to the base station spreads across all sensor nodes in the network. Each clustering cycle consists of the setup phase (cluster formation) and the steady phase (data transfer). During the setup phase, a sensor node selects a random number between 0 and 1. If this random number is less than the threshold T(n), the sensor node is selected as the cluster head. T(n) is calculated as follows \cite{12}: \begin{align}\label{vequ1} T(n)=\left\{\begin{array}{lc}\frac P{1-P\;\ast\;\lbrack r\;mod\;({\displaystyle\frac1P})\rbrack}&if\;n\in G\\0&otherwise\end{array}\right. \end{align} In equation (1), P is the desired percentage of nodes to become a cluster head, r is the number of rounds for the election, and G is the set of nodes that have not been selected yet as a cluster head in the last round of 1/P. The cluster head nodes notify other nodes after they have been selected. When the sensor nodes receive an advertisement message, they determine the cluster to which they want to belong based on the signal strength of the received message. Then, the cluster heads allocate the time on which the sensor nodes can send data to the cluster heads based on a TDMA approach. During the steady-state phase, the sensor nodes can send data to cluster heads. The cluster heads collect data from the nodes in their cluster before sending these data to the base station. After a certain period of time spent on the steady-state phase, the network re-enters the setup phase and enters another round of selection of cluster heads \cite{12}. \subsection{LEACH-C Protocol} While using LEACH has the benefits of a random, adaptive, and self-organized clustering algorithm, this protocol does not guarantee the number and placement of cluster head nodes. Therefore, using a central control algorithm to form clusters will help to produce better clusters by dispersing cluster head nodes throughout the network. This is the core idea behind the LEACH-C protocol which, unlike LEACH, uses a centralized clustering algorithm \cite{6}. The purpose of LEACH-C protocol is to combine energy efficient clustering and routing for application-specific data aggregation to increase network lifetime and reduce latency in data access \cite{13}. In the setup phase of LEACH-C, each node sends information about its current position (possibly determined by the GPS receiver) and current energy level to the base station. In addition to forming good clusters (for example, the proper physical location and the number of nodes that are roughly the same), base station must ensure that the energy load is uniformly distributed over all nodes. To this end, the base station calculates the average energy level of the network nodes so as nodes whose energy levels are below the average cannot be selected as cluster heads for the current round. This algorithm attempts to minimize the amount of energy for nodes that are not cluster heads to transmit their data to the cluster head, by minimizing the total sum of squared distances between all non-cluster head nodes and closest to the cluster head. After determining the cluster heads of the current round, the base station broadcasts a message containing the cluster head ID to each node. If a node’s cluster head ID matches its own ID, the node is a cluster head; otherwise, it is a normal node and determines its TDMA slot for data transmission and goes to sleep until it is time to transmit data \cite{6}. This method increases the network overhead because all sensor nodes must send their location information to each base station at a time in every setup phase \cite{5}. \subsection{K-means Algorithm in WSN} Due to the widespread use of the K-means algorithm in various domains, such as data mining and image processing, the authors in \cite{7} have used this method to cluster the sensor nodes and extend the lifetime of wireless sensor networks. The K-means algorithm is mainly based on the Euclidean distance of nodes and selection of cluster heads based on the residual energy of the nodes. In this method, the sink node collects information about the identifier, position, and residual energy of all nodes and stores this information in a list in the central node. After receiving this information from all the nodes, it begins to cluster the sensors using the K-means algorithm. The steps in the K-means clustering algorithm in wireless sensor networks are as follows \cite{7}: \begin{enumerate} \item \relax To form the cluster ’k’ of the sensor nodes, the ’k’ number of centroids is initially selected at random locations; \item \relax Euclidian distance is calculated from each node to all centroids and allocated to the centroid nearest. Thus the ’k’ initial clusters are formed; \item \relax The positions of the centroids in each cluster are recalculated and the position change from the previous one is examined; \item \relax If there is a change in the position of each centroid, all steps are repeated from step 2, otherwise, the clusters will be finalized and the clustering process will end. \end{enumerate} \subsection{Introducing the FUZZY-LEACH algorithm} LEACH Clustering is the most famous hierarchical protocol in which the cluster head is selected based on the threshold value, and only the cluster heads can send the information to the base station. But in this approach, a privileged cluster is selected among the cluster heads, which can send information to the mobile base station by choosing appropriate fuzzy descriptors such as residual energy, mobility, and centrality of clusters. Fuzzy inference engine (Mamdani's rule) is used to select the chance to be superior cluster head \cite{5}. Fuzzy Logic is used to model human experience and human decision-making behavior. In addition, it can handle the uncertainty of real-time applications more accurately than probabilistic models. FL is used in this method to control the uncertainty for choosing the superior cluster head. The main advantage of using FL is to overcome the overhead of collecting and calculating the energy and location information of each node. Most FL-based clustering algorithms are considered constant, sink node / base station \cite{5}. \section{Energy Model} In this section, we will introduce the radio model used in this paper to compare clustering algorithms. We consider the radio model used in \cite{4} for the LEACH protocol. The energy dissipation model is shown in \ref{vfig1}. In Figure \ref{vfig1}, $E_{Tx}(d) $ is the energy spent to transmit k bits over d, and $E_{Rx}$ represents the energy spent to process k bit message. The parameter $E_{elec} $ shows the energy consumed in each bit to run both transmitter and receiver circuits. This parameter depends on many factors such as digital coding, modulation, filtering, and signal propagation \cite{9}. The energy consumed by the radio transmitter is defined by the following equation \cite{6}: \begin{align}\label{vequ2} E_{Tx}(k,\;d)=\left\{\begin{array}{lc}k.E_{elec}+k.E_{fs}.d^2&if\;d<d_o\\k.E_{elec}+k.E_{mp}.d^4&if\;d\geq d_o\end{array}\right. \end{align} Where $E_{fs}$ and $E_{mp}$ represent the amplifier energy respectively in the model of free space channel (energy loss $d^2$) and in the model of multipath fading channel (energy loss $d^4$) multipath. They are dependent on d, where d is the distance between the sender and the receiver. If the distance d is less than the threshold $d_{o}$, then the free space model is used; otherwise, the multimode model is used \cite{9}. The threshold value $d_{o}$ is presented by Heinzelman et al. in \cite{6}. $d_{o}$ is defined as follows: \begin{align}\label{vequ3} d_o=\sqrt{\frac{E_{fs}}{E_{mp}}} \end{align} \begin{figure}[!ht] \centering \includegraphics[width=8cm]{1.png} \caption{The radio model used to calculate energy consumption \cite{4}}\label{vfig1} \end{figure} \section{The Proposed Approach} In this paper, we adopt the Approximate Rank-Order (ARO) clustering algorithm, which is recently proposed in \cite{8}, as an energy-efficient scheme to cluster nodes in wireless sensor networks. In this section, we first introduce Rank-Order and Approximate Rank-Order algorithms and then, we present our proposed clustering approach. \subsection{Introduction of Rank-Order Clustering Algorithm} The Rank-Order clustering algorithm was proposed by Zhu et al. \cite{14}, which is an agglomerative hierarchical clustering algorithm based on nearest neighbor distance measure. The algorithm works as follows: initially, all the samples are considered as separate clusters, then the distance vector consisting the distance between all the pairs of clusters is computed and the pairs whose distance is below the threshold are merged together. This way onwards, new vector of cluster-to-cluster distance is repeatedly recomputed and merges are performed based on the new distance vector. This main point in Rank-Order clustering is the definition of cluster-to-cluster distance metric. In this algorithm, the distance between two clusters is defined as the minimum distance between any two samples in the clusters. The first measurement of the distances used in Rank-Order clustering is obtained by the following equation \cite{8}: \begin{align}\label{vequ4} d(a,\;b)=\sum_{i=1}^{O_a(b)}O_b(f_a(i)) \end{align} Where $f_a (i)$ is the $i$-th face in the neighbor list of a, and $O_b (f_a (i))$ is the rank of face $f_a (i)$ in the neighbor list of b, which the nearest neighbor lists are generated according to some underlying distance measure (e.g., the Euclidean distance). This asymmetric distance function is used to define a symmetric distance between two faces a and b. as \cite{8}: \begin{align}\label{vequ5} D(a,\;b)=\frac{d(a,\;b)+d(b,\;a)}{min(O_a(b),\;O_b(a))} \end{align} A cluster-level normalized distance measure is effectively used to restrict mergers to local neighborhoods. Specifically, the minimum distance between the two points in a pair of clusters is calculated and divided by the mean distance \cite{7873333}: \begin{align}\label{vequ6} D^N(C_i,\;C_j)=\frac1{\phi(C_i,\;C_j)}\times d(C_i,\;C_j) \end{align} Where \begin{align}\label{vequ7} \varphi(C_i,\;C_j)=\frac1{\mid C_i\mid +\mid C_j\mid }\times\sum_{a\in C_i\cup C_j}\frac1K\sum_{k=1}^Kd(a,\;f_a(k)) \end{align} Where $D^N (C_i,C_j )$, is the minimum distance between any two points in the cluster i and j divided by the mean distance of each point in $C_i$ or $C_j$ to their K nearest neighbors. The threshold 1 is always used for this function, and it is tested whether or not the minimum distance between any two points in the clusters is below the average distance of all points in both clusters to their K nearest neighbors \cite{8}. The symmetric rank-order distance function returns low values if the two samples are close to each other and have neighbors in common. After calculating the distances, clustering phase begins where each sample is considered as a distinct cluster, then the symmetric distances between each pair of clusters are calculated, and finally, clusters with the distance below the specified threshold are merged together. Afterwards, the nearest neighbor lists are updated for each newly formed cluster, and the distances between the remaining clusters are recalculated; this approach continues until no further clusters can be merged. In Rank-Order algorithm, instead of determining the optimal number of clusters C, a distance threshold is used to measure the rank-order distance along with a neighborhood size for the cluster-level normalized distance; these parameters set the specified number of clusters for a particular dataset, and their effective values are determined empirically \cite{8}. In terms of runtime, calculation the full nearest neighbor lists for each sample affords $O(n^2)$ cost. Additionally, the actual clustering step used here is iterative, with cost per iteration proportional to the current number of clusters squared, so both the nearest neighbor computation and the clustering step itself are costly with increasing dataset size \cite{8}. \subsection{Review of ARO algorithm} The Rank-Order clustering method has an obvious scalability problem that requires calculating the nearest neighbor lists for each sample in the data set, which if calculated directly, has an $O(n^2)$ cost. Although there are various approximate methods for calculating the nearest neighbors, they are usually only able to calculate a short list of the top k nearest neighbors rather than a comprehensive ranking the dataset \cite{8}. Using approximate methods for faster calculation of the nearest neighbor requires some modification in the original Rank-Order clustering algorithm. In particular, instead of taking into account all the neighbors in the summation equation (\ref{vequ4}), we can sum up to at most the top k neighbors. In addition, instead of using the cluster-level normalized distance from the original algorithm, it is possible to approximate the distance between the pairs of samples for which both are within the other sample’s top-200 nearest neighbors. It should be noted that if only one short list of the top-k neighbors is considered, the presence or absence of a particular instance in this list may be more important than the sample’s numerical rank. Thus, a distance measure based on the sum of the presence/absence of the shared nearest neighbors, instead of the ranks, can be used which results in the following distance function \cite{8}: \begin{align}\label{vequ8} d_m(a,\;b)=\sum_{i=1}^{min(O_a(b),\;k)}I_b(O_b(f_a(i)),\;k) \end{align} Where $I_b (x,k)$ is an indicator function with the value of 0 if face x is in face b’s top k nearest neighbors, otherwise it is equal to 1. In practice, this modification leads to better clustering accuracy compared to the sum of ranks in a direct way, as in the original formula. Effectively, this distance function indicates that the presence or absence of shared neighbors towards the top of the nearest neighbor list (within the top-200 ranks) is important, while the numerical values of ranks themselves are not \cite{8}. The normalization method used in the original algorithm is still effective and contributes to more accurate clustering results even with this modification to the original algorithm. The modified distance measure is defined as follows \cite{8}: \begin{align}\label{vequ9} D_m(a,\;b)=\frac{d_m(a,\;b)+d_m(b,\;a)}{min(O_a(b),\;O_b(a))} \end{align} In addition, in order to improve the runtime of the clustering process, as mentioned earlier, 1) only the distances between samples which appear in each other’s nearest neighbor lists are calculated, and 2) instead of multiple merge iterations as in the original algorithm, only one round of merges of individual faces into clusters are done. This means that only one clustering repetition is performed compared to the original algorithm that has runtime $C^2$ in each cluster iteration, and additionally only check for merges on a subset of all possible pairs (because only 200 nearest neighbors for each sample Is considered). This results in the runtime of $O(n)$ from the final clustering step (assuming that the nearest neighbors have been pre-calculated) \cite{8}. Finally, the clustering approach in ARO algorithm is as follows \cite{8}: \begin{enumerate} \item \relax Extract dataset; \item \relax Calculate a set of top-k nearest neighbors for each face in the dataset; \item \relax Calculate pairwise distances between each face, and those faces in its top-k nearest neighbor list for which the face is also on the neighbor’s nearest neighbor list, following equation (\ref{vequ9}); \item \relax Merge all pairs of faces with distances below a threshold. \end{enumerate} Choosing the threshold to determine the number of clusters, C in a dataset, is one of the most difficult issues of data clustering. In practical applications, it cannot be assumed that the actual number of clusters is predetermined, so this algorithm is evaluated in several effective values of C and the best result is reported \cite{8}. \subsection{Proposed ARO-WSN approach} In this paper, we intend to use the ARO algorithm in order to cluster nodes in a wireless sensor environment. In addition to applying ARO algorithm in the clustering step, we will also take advantage of the idea employed in \cite{9} where the clustering step is performed before selecting cluster heads. Our proposed approach considers the residual energy of nodes when selecting the cluster heads. In the following, three steps of the proposed algorithm including clustering, cluster head selection and steady-state will be described. \subsubsection{Clustering step } At this point, network sensors will cluster based on the ARO clustering algorithm. The algorithm uses a distance measurement based on the sum of the presence/absence of shared nearest neighbors, instead of the ranks that are formulated in equation (8). The final clustering method we use is the same as the ARO clustering steps, with the difference that in the third step, instead of using the cluster-level normalized symmetric distance, we use the same asymmetric equation (8) to calculate the distance between the sensor nodes. Therefore, the clustering step of the proposed approach is as follows: \begin{enumerate} \item \relax Collect the location of the sensor nodes by the base station. Each sensor node knows its location (equipped with a GPS) and transmits it to the base station; \item \relax Calculate a collection of k nearest neighbors for each sensor node in the network. It should be noted that the meanings of neighboring nodes are nodes that are closer to the node in terms of Euclidean distance; \item \relax Calculate pairwise distances between each node, and those nodes in its top-k nearest neighbor list for which the node is also on the neighbor’s nearest neighbor list, following equation (\ref{vequ8}); \item \relax Merge all pairs of nodes with distances below a threshold. \end{enumerate} The pseudo code of the clustering step of our proposed approach, which is referred to below as ARO-WSN, is depicted in Figure \ref{vfig2}. In our proposed approach, for the N set of sensor nodes (the value of N varies from 100 to 300 nodes in experiments), the k parameter is considered to be 20. This value has been selected experimentally after several experiments. The value of the threshold for equation (\ref{vequ8}) is also set to 1.5, according to the test in Section 5-4. \begin{figure}[!ht] \centering \includegraphics[width=8cm]{2.png} \caption{The pseudo code of the ARO-WSN clustering step}\label{vfig2} \end{figure} \subsubsection{Cluster head selection step} After the cluster is set, the next step is to select the cluster head. As previously emphasized, the clustering step is performed only once before executing repetitions, and is the only cluster head to be replaced in each round. In the proposed approach, the selection of cluster heads is random and dependent on the residual energy of the nodes. The selection of cluster heads is such that in each round, it is first tested whether or not the current cluster head energy is greater than the average total energy of the network. If the energy of the cluster head is greater than the average energy of all nodes, it can also be in the next round and it does not give its role to the other node. But if its energy is less than the average energy of the nodes, then randomly one of the other nodes is candidate, and after checking its energy level, if it has the necessary energy, is selected as the cluster head. This node should send an advertisement message with a node identifier in order to be new cluster head in the r round. As a result, each cluster head will be able to collect data from cluster nodes and transfer the collected data to the base station. The pseudo code of the cluster head selection step in the ARO-WSN algorithm is shown in Figure \ref{vfig3}. \begin{figure}[!ht] \centering \includegraphics[width=8cm]{3.png} \caption{The pseudo code of the ARO-WSN cluster head selection step}\label{vfig3} \end{figure} \subsubsection{Data transmission} Once clusters and cluster heads are created, each normal node sends the sensed data to its designated cluster head. After that all data is received, each cluster head sends the collected data directly to the base station. The pseudo code of the data transmission step in ARO-WSN algorithm is shown in Figure \ref{vfig4}. \begin{figure}[!ht] \centering \includegraphics[width=8cm]{4.png} \caption{The pseudo code of the ARO-WSN data transmission step}\label{vfig4} \end{figure} \section{SIMULATION RESULTS} We have done extensive simulations based on the following setup to evaluate our proposed solution. We consider a dense wireless sensor networks with N nodes which are randomly distributed in a $100m\times100m$ field. The base station is located outside of the sensing area at a location with coordinates $(x_{sink}=0.5*x,y_{sink}=0.75*y+y)$. The number of nodes in all experiments is constant and equal to 500, except for scenario in which we test the effect of node density. Since nodes have limited energy, their energy levels is decreasing during the simulation period. When the energy storage of a node is evacuated, it is considered as a dead node and so, it cannot transmit or receive data \cite{9}. In all simulation scenarios, transmission or reception of data makes energy consumption. The parameters used for the energy model of all three algorithms in the simulations can be found in Table \ref{vtab1}. \begin{table}[!ht] \centering \caption{The parameters of the radio model used in this simulation}\label{vtab1} \begin{tabular}{|c|c|c|} \hline Parameter & Value \\ \hline $E_{elec}=E_{Tx}=E_{Rx}$ & 50 nJ/bit\\ $E_{fs}$ & 10 pJ/bit/$m^2$\\ $E_{mp}$ & 0.0013 pJ/bit/$m^4$\\ $E_{DA}$ & 5 nJ/bit/msg\\ $d_o$ & 88 m\\ Message size & 4000 bits\\ \hline \end{tabular} \end{table} In the following, our proposed ARO-WSN algorithm is compared with the following three algorithms: \begin{itemize} \item \relax The K-means protocol is a distance-based clustering algorithm. Since our proposed approach is also a distance-based algorithm, K-means is a good choice for the purpose of comparison. \item \relax The LEACH protocol is one of the most famous hierarchical clustering protocols in WSNs. Since our proposed approach adopts a hierarchical approach, we will compare ARO-WSN with LEACH. \item \relax The LEACH-C protocol is a centralized, energy-aware extension of the LEACH protocol. Since the ARO-WSN algorithm is also a centralized and energy-aware approach, LEACH-C is a good candidate for the evaluations. \item \relax LEACH protocol with fuzzy descriptors. \end{itemize} Our criteria for the network lifetime is First Node Dead (FND); the moment when the battery storage of first sensor node in the network is completely discharged and Last Node Dead (LND); the moment when the battery of all sensor nodes in the network is completely depleted. Simulation results are the multi-run average of a random, constant size topology for all the algorithms. \subsection{Impact of node density} In this section, we evaluate the lifetime of the wireless sensor network for the different values of N based on FND and LND metrics. Figure \ref{vfig5} and Figure \ref{vfig6} show the effects of node density on the network lifetime, with FND and LND metrics in comparable clustering methods, respectively. \begin{figure}[!ht] \centering \includegraphics[width=10cm]{5.png} \caption{Impact of node density on performance of compared algorithms with FND criterion}\label{vfig5} \end{figure} As shown in Figure \ref{vfig5}, for different values of N from 100 to 500, the ARO-WSN algorithm shows a performance improvement compared to the LEACH and LEACH-C algorithms. But the ARO-WSN algorithm has failed to improve its performance compared to the K-means algorithm. Despite the better performance of k-means in this case, the proposed algorithm is a better option for clustering sensor networks due to the flexibility in the number of clusters and less time overhead. From the value N = 200, the ARO-WSN algorithm has a greater FND value than FUZZY-LEACH. Unlike the ARO-WSN, it seems that the FUZZY-LEACH algorithm loses its efficiency at higher densities. This also applies to the LEACH and LEACH-C algorithms. We will further analyze the details of the chart. In Figure \ref{vfig5}, the LEACH-C protocol has shown worse performance than LEACH. This is due to the fact that, in homogeneous environments, the formation of the clusters and the selection of the cluster heads in a decentralized way is more reasonable. In homogeneous network settings, all nodes have the same energy levels and if the cluster formation is done centralized, cluster heads must transfer all the new clustering information to the base station in each round with the same capabilities as the normal nodes. Although the ARO-WSN algorithm is also a centralized algorithm, the clustering phase in this algorithm is performed only once just before choosing the cluster heads. In ARO-WSN, the location of all nodes and members of each cluster is sent only once to the base station, and this information will not change during the period. Only once the cluster head is updated, and if the energy level of current cluster head is lower than the average total energy of the network, the advertisement message will be broadcast. Generally, the ARO-WSN and K-mean algorithms have better performance than LEACH and LEACH-C. Compared to LEACH, ARO-WSN and K-means algorithms, in addition to having the advantages of hierarchical clustering, perform based on distance measure. As shown in the energy consumption model in equation (\ref{vequ2}), energy consumption depends directly on the distance between the nodes and the cluster head and the distance among the cluster heads and the base station. Therefore, distance-based approaches consume less energy than LEACH, because one of the disadvantages of LEACH is the possibility to select inappropriate cluster heads in terms of location. If the LEACH-C algorithm is compared in a heterogeneous environment with other algorithms, it will perform better; but in the homogeneous environment, it has poor performance for the reasons explained. The ARO-WSN algorithm performs better at higher densities than FUZZY-LEACH and experiences the first node death in a longer time period than FUZZY-LEACH. Unlike the ARO-WSN, FUZZY-LEACH creates a new cluster in each round, which requires more energy. The disadvantages of this work are not shown at lower density, but in higher densities requiring more information to be transmitted, the advantage of forming clusters before the start of repetitions is clearly seen. The ARO-WSN with the first node death criterion has improved about \%60 compared to LEACH, about \%85 compared to LEACH-C and about \%22 compared to FUZZY-LEACH in all densities. In Figure \ref{vfig6}, we investigate the impacts of node density on network lifetime with LND criterion in compared clustering methods. \begin{figure}[!ht] \centering \includegraphics[width=9.5cm]{6.png} \caption{Impact of node density on performance of compared algorithms with LND criterion}\label{vfig6} \end{figure} Although in the FND criterion, the ARO-WSN algorithm failed to the K-means algorithm and in the low density to the FUZZY-LEACH algorithm, with the LND criterion, it was able to achieve good improvement over all four other algorithms. In more number of rounds, the advantage of performing clustering before selecting the cluster head and fixing the clusters more visible. The proposed approach with the last node death criterion has improved by \%42, \%67, \%64 and \%24, respectively, against the K-means, LEACH, LEACH-C and FUZZY-LEACH algorithms. As shown in Figure \ref{vfig6}, the ARO-WSN algorithm has almost constant LND value at different densities of the node, indicating that this approach has good stability and scalability at different densities of the node. \subsection{Energy and lifetime} In these experiments, we run compared algorithms under many networks with N = 500 nodes. The primary energy of all nodes is equal to 0.5J. The amount of data to send to the base station is unlimited. Figure \ref{vfig7} shows the number of active nodes in time for the ARO-WSN, K-means, LEACH, LEACH-C and FUZZY-LEACH algorithms. \begin{figure}[!ht] \centering \includegraphics[width=10cm]{7.png} \caption{Number of alive nodes over time of the compared protocols}\label{vfig7} \end{figure} As can be seen clearly in Figure \ref{vfig7}, the ARO-WSN algorithm with both FND and LND criterion has significantly improved in relation to the LEACH, LEACH-C and FUZZY-LEACH algorithms. In comparing two K-means and ARO-WSN algorithms with the FND criterion, the ARO-WSN algorithm has a relatively large distance to the K-means curve. In Figure \ref{vfig8}, we observe the total residual energy of the network in each round of transmission using compared approaches. This chart clearly and simply depicts the increasing network lifetime. In this diagram, the LEACH algorithm is the first approach in which the total energy of the network ends, and then the LEACH-C algorithm has the second rank in the discharge of network energy. The K-means algorithm, although with a great distance from the LEACH-C algorithm approaches the completion of the entire network energy, about 1,000 rounds faster than the ARO-WSN algorithm experiences the death of the last node and the total energy depletion of the network. The death of the last node with the FUZZY-LEACH approach is also 600 rounds faster than the proposed approach. Figure \ref{vfig8} illustrates the fact that the ARO-WSN algorithm has been successful in increasing network lifetime. The ARO-WSN algorithm has consumed only \%38 of the total network energy in round 500. While K-means, LEACH, LEACH-C and FUZZY-LEACH algorithms have consumed \%44, \%100, \%95.6 and \%42 of total energy, respectively. At round 1500, the K-means, LEACH, and LEACH-C algorithms have consumed all of the network's energy, but the ARO-WSN algorithm has stored about \%10 of the total energy, and can still continue to collect information from the environment around the network. In the 1900s, the FUZZY-LEACH algorithm lost all of the network's energy, and the ARO-WSN algorithm remained at \%3 of the energy in the network. \begin{figure}[!ht] \centering \includegraphics[width=10cm]{8.png} \caption{The evolution of the remaining energy in the network}\label{vfig8} \end{figure} Figure \ref{vfig9} shows the number of messages received by the base station for the compared approaches. \begin{figure}[!ht] \centering \includegraphics[width=10cm]{9.png} \caption{Number of messages received at the base station over time}\label{vfig9} \end{figure} In Figure \ref{vfig9}, it has been shown that in each algorithm, the incremental trend of transmission of messages in the corresponding curve is fixed at a specific time moment. This means that at that moment, the last node in the network failed and the network lifetime has been over. In this diagram, the LEACH and LEACH-C algorithms initially transmitted more messages to the base station. These messages are a collection of advertisement messages and data collected by cluster heads. In the implementation of these algorithms, channel failure issues are not considered; therefore, all nodes can transfer data collected from the environment to the cluster head. It can be concluded that the difference in the number of messages transmitted to the base station is related to the advertisement messages broadcasted by the LEACH and LEACH-C algorithms. According to equation (\ref{vequ2}), the amount of energy consumed depends on the size of the messages; therefore, the greater the number of messages sent and received in the sensors, the greater the power consumption in the network. The faster transmission of the message in the LEACH and LEACH-C algorithms has led to faster energy consumption in both of these approaches, resulting network lifetime faster has finished. Here, the lower number of advertisement messages has caused superiority of the ARO-WSN algorithm, which indeed is the result of the lack of repeating the clustering in each round. The K-means curve is close to the ARO-WSN message transmission curve. This shows that the algorithm acts more cautiously in transmission of messages than two LEACH and LEACH-C algorithms; but as shown in Figure \ref{vfig9}, the moment of the death of the last node in this algorithm is far from the ARO-WSN algorithm; therefore, there is no more opportunity to transfer message packs to the base station. Figure \ref{vfig10} and Figure \ref{vfig11} show the performance of the compared algorithms with different primary energies with FND criterion and LND criterion, respectively. \begin{figure}[!ht] \centering \includegraphics[width=10cm]{10.png} \caption{Impact of the initial energy amount on the performance of compared algorithms with FDN criterion}\label{vfig10} \end{figure} \begin{figure}[!ht] \centering \includegraphics[width=10cm]{11.png} \caption{Impact of the initial energy amount on the performance of compared algorithms with LDN criterion}\label{vfig11} \end{figure} According to Figure \ref{vfig10}, we see that the performance of all the compared algorithms is improved by increasing the initial energy. The ARO-WSN algorithm works better than other algorithms with increasing initial energy. Therefore, it can be concluded that distance partnership in the formation of a cluster can be much more effective than the initial energy level. With increasing primary energy, the difference in FND value between K-means and ARO-WSN distance-based algorithms increases with the two LEACH and LEACH-based algorithms. With the initial increase in primary energy, the proposed approach, which did not have the FND score superior to the K-means algorithm, experiences the first death later than the K-means algorithm. Figure \ref{vfig11} gives similar results to Figure \ref{vfig10}, with the difference that here the difference the LND value between the two ARO-WSN and K-means algorithms is also significant. \subsection{Impact of message size} In all experiments that have been done so far, the size of the messages sent from the nodes to the cluster head and from the cluster head to the base station is constant and equal to 4000 bits. According to Equation (\ref{vequ2}), the size of the messages is effective in the energy consumption model for sending and receiving messages; therefore, the larger the size of messages sent and received in the sensor nodes, the more energy is consumed and the network lifetime decreases. In this experiment, we want to show the effect of the size of received and sent messages from sensor nodes in the performance of the compared approaches. \begin{figure}[!ht] \centering \includegraphics[width=10cm]{12.png} \caption{Impact of message size on the performance of compared algorithms with FDN criterion}\label{vfig12} \end{figure} \begin{figure}[!ht] \centering \includegraphics[width=10cm]{13.png} \caption{Impact of message size on the performance of compared algorithms with LND criterion}\label{vfig13} \end{figure} Regarding Figure \ref{vfig12} and Figure \ref{vfig13}, with increasing message size, the first and last node death occurs in all approaches earlier, but the ARO-WSN algorithm still maintains its improvement over other algorithms. Figure \ref{vfig12} shows that in the message size of 4000 bits, the ARO-WSN algorithm has a lower FND value than K-means. This issue is also present in all the graphs that have been presented so far and the proposed algorithm has not been able to improve with the FND criterion toward to the K-means algorithm. As previously mentioned, the clustering step in the ARO-WSN algorithm is performed before selecting the cluster head. Also, cluster head update is only performed if the cluster head energy is less than the average total energy of the network; but in the other three approaches, clustering is performed in each round. The messages exchanged in each round to create new clustering and cluster head updates in these three approaches are greater than the messages sent to the base station in the ARO-WSN algorithm for updating cluster heads (see Fig. \ref{vfig9}); therefore, according to equation (\ref{vequ2}), by increasing the size and number of messages sent and received, more energy is consumed and the lifetime of the network decreases. \subsection{Effect of parameter C value} As indicated in Section 4.2, selecting the threshold C for determining the number of clusters in a dataset is one of the most difficult issues in data clustering. In practice, it cannot be assumed that the actual number of clusters is predetermined, so this algorithm is evaluated in several effective values of C and the best result is reported. In Section 4.3.1 this value is considered for the proposed approach to 1.5. Figure \ref{vfig14} shows the effect of the value of the parameter C on the proposed approach with the death of the first node criterion. As shown in Figure \ref{vfig14}, the ARO-WSN algorithm has a greater FND value in C = 1.5 and for all experiments, 1.5 is selected for this parameter in the proposed approach. \begin{figure}[!ht] \centering \includegraphics[width=10cm]{14.png} \caption{The Impact of parameter C on the performance of the proposed approach with the FND criterion}\label{vfig14} \end{figure} \section{CONCLUSION} The purpose of this paper is to increase the lifetime of wireless sensor networks through clustering. Given the hierarchical and distance-based clustering features, the ARO algorithm, which is a hierarchical clustering based on distance and has been used in the image processing to clustering millions of images, was selected in order to clustering network sensors. In addition to using the ARO algorithm at the clustering step, the idea of performing the clustering step before choosing the cluster head was also used in the proposed ARO-WSN approach. This idea will reduce the network power consumption by eliminating overhead advertisements per round and thus increase network lifetime. The simulation results show that in the ARO-WSN, the death of the last node compared to the LEACH, LEACH-C, K-means and FUZZY-LEACH algorithms, and the death of the first node compared to the LEACH and LEACH-C algorithms, and in the higher densities than the FUZZY-LEACH algorithms occurs later. The ARO-WSN with the first node death criterion has improved about \%60 compared to LEACH, about \%85 compared to LEACH-C and about \%22 compared to FUZZY-LEACH in all densities. In addition, with the last node death criterion was improved by \%42, \%67, \%64 and \%24, respectively, with the K-means, LEACH, LEACH-C and FUZZY-LEACH algorithms. In terms of energy consumption, when the K-means, LEACH and LEACH-C algorithms consume all the network energy, the proposed approach saves \%10 of the total network energy and can still continue to collect information from the surrounding environment. Therefore, the proposed ARO-WSN approach has succeeded in reducing energy consumption through clustering and, consequently, increasing the lifespan of wireless sensor networks. \nocite{*

proofpile-arXiv_069-10725

\section{Introduction}\label{sec:intro} \setcounter{equation}{0} In the present paper we focus on the Cauchy problem % \begin{equation} \label{cpintro} \left\{ \begin{array}{ll} L u(t) = f(t), & t\in (0,T] \\[1ex] D_{t}^k u(0) = g_{k}, & k=0,\dots,m-1, \end{array} \right. \end{equation} where $L=L(t,D_t;x,D_x)$ is a linear partial differential operator of the form \begin{equation}\label{Lintro} L(t,D_t;x,D_x)= D_{t}^m + \sum _{j=1}^{m}\sum_{|\alpha|\le j}c_{j\alpha}(t;x)D_x^\alpha D_t^{m-j}, \end{equation} $D=-i\partial$, with $(t,x)-$smoothly depending coefficients $c_{j\alpha}$, possibly admitting a polynomial growth, namely, \begin{equation}\label{grw} |\partial_t^k\partial_x^\beta c_{j\alpha}(t;x)|\le C_{k\alpha }\langle x\rangle^{j-|\beta|},\qquad \alpha,\beta\in\mathbb{Z}^n_+,\ |\alpha|\leq j,\ j=1,\ldots,m, \end{equation} for $\langle x \rangle:=(1+|x|^2)^{1/2}$, some $C_{k\alpha}>0$ and all $t\in[0,T]$, $x\in\R^n$. The hypothesis of smoothness with respect to $t$ is assumed only for the sake of simplicity, since here we will not deal with questions concerning low regularity in time for the coefficients of the Cauchy problem. A very wide literature concerning this topic has been developed along the past 40 years, starting with the paper \cite{cdgs}. The assumption \eqref{grw} suggests to set the problem within the framework of the so-called SG calculus (see \cite{CO,PA72}), defined through symbols satisfying global estimates on $\R^n\times\R^n$. Explicitly, given $(m,\mu)\in\R^2$, the class of SG symbols $S^{m,\mu}$ consists of all symbols $a\in C^\infty(\R^n\times\R^n)$ such that % $$ | \partial_x^{{\alpha}} \partial_{\xi}^{{\beta}} a(x,\xi)| \leq C_{{\alpha}{\beta}}\,\jb{x}^{m-|{\alpha}|}\jb{\xi}^{\mu-|{\beta}|} $$ for all $ x, \xi \in \R^n$, $\alpha,\beta\in \mathbb{Z}^n_+$, and some $C_{{\alpha}{\beta}}>0$, see Section \ref{sec:sgcalc} below for the precise definitions and some basic properties of the corresponding calculus. Notice that for $m=0$ we recover the class $S^\mu={S}^{0,\mu}$ of (H\"ormander's) symbols uniformly bounded with respect to the space variable $x$. We are interested in existence and uniqueness of the solution to \eqref{cpintro} in suitable weighted Sobolev-type spaces of functions or distributions, according to the regularity of the Cauchy data and of the operator $L$. More precisely, it is well-estabilished (see, e.g., \cite{M2} for PDEs with uniformly bounded coefficients, and \cite{CO} for PDEs in the SG framework) that, to have existence of a unique solution to \eqref{cpintro} in Sobolev-type spaces, an hyperbolicity assumption is needed. The operator $L$ is said to be \emph{hyperbolic} if its principal symbol $$ L_m(t,\tau;x,\xi):=\tau^m + \sum _{j=1}^{m}\sum_{|\alpha|= j}c_{j\alpha}(t;x)\xi^\alpha \tau^{m-j} $$ factorizes as \begin{equation} \label{hypintro}L_m(t,\tau;x,\xi)=\prod_{j=1}^m (\tau-\tau_j(t;x,\xi)), \end{equation} with real-valued and smooth roots $\tau_j$, usually called \emph{characteristic roots} of the operator $L$. A sufficient condition for existence of a unique solution is the \textit{separation between the roots}, which reads either as \begin{align} \tag{H}\quad |\tau_j(t;x,\xi)-\tau_k(t;x,\xi)|\geq & C\langle\xi\rangle &&\text{(uniformly bounded coefficients) or}, \\ \tag{SGH}\quad |\tau_j(t;x,\xi)-\tau_k(t;x,\xi)|\geq & C\langle x\rangle\langle\xi\rangle &&\text{(SG type coefficients)}, \end{align} for a suitable $C>0$ and every $(t;x,\xi)\in [0,T]\times\R^{2n}$. Conditions (H) or (SGH), respectively, are assumed to hold true either for every $j\neq k$ (\emph{strict hyperbolicity}), or at least for every $\tau_j$, $\tau_k$ belonging to different groups of coinciding roots (\emph{weak hyperbolic with roots of constant multiplicity}). The strict hyperbolicity condition (H) (respectively, (SGH)) allows to prove existence and uniqueness in Sobolev spaces (respectively, in Sobolev-Kato spaces) of a solution to \eqref{cpintro} for every $f,g_k$, $k=0,\dots,m-1$, in Sobolev spaces (respectivey, Sobolev-Kato spaces). A weak hyperbolicity condition with roots of constant multiplicity, together with a condition on the lower order terms of the operator $L$ (a so called \emph{Levi condition}) allows to prove a similar result, and the phenomena of loss of derivatives and modification of the behavior as $|x|\to\infty$ with respect to the initial data appear. The first phenomenon has been observed for the first time in \cite{cdgs}, the second one in \cite{scncpp}. If there is no separation condition on the roots, then the only available results in literature that we are aware of are those due to Morimoto \cite{Morimoto} and Taniguchi \cite{Taniguchi02}, dealing with \emph{involutive roots} in the uniformly bounded coefficients case. The condition of involutiveness, which is weaker than the separation condition (H), requires that, for all $j,k\in\N$, $t\in[0,T]$, $x,\xi\in\R^n$, the Poisson brackets $$\{\tau-\tau_j,\tau-\tau_{k}\}:=\partial_t\tau_j-\partial_t\tau_{k}+ \tau'_{j,\xi}\cdot \tau'_{k,x}-\tau'_{j,x}\cdot \tau'_{k,\xi}$$ may be written as \begin{equation}\label{invintro} \{\tau-\tau_j,\tau-\tau_{k}\} = b_{jk}\cdot(\tau_j-\tau_{k})+d_{jk}, \end{equation} for suitable parameter-dependent, real-valued symbols $b_{jk},d_{jk}\in C^\infty([0,T];S^{0}(\R^{2n}))$, $j,k\in\N$. Under condition \eqref{invintro} and adding a Levi condition, the authors reduced \eqref{cpintro} to an equivalent first order system \begin{equation}\label{sysint}\begin{cases} \mathbf{L} U(t) = F(t), & 0\leq t\leq T, \qquad \mathbf{L} = D_t + \Lambda(t,x,D_x) +R(t,x,D_x), \\[1ex] U(0) = G, & \end{cases} \end{equation} with $\Lambda$ a diagonal matrix having entries given by operators with symbol coinciding with the $\tau_j$, $j=1,\dots, m$, and $R$ a (full) matrix of operators of order zero. They constructed the fundamental solution to \eqref{sysint}, that is, a smooth family $\{E(t,s)\}_{0\leq s\leq t\leq T}$ of operators such that \begin{equation*}\begin{cases} \mathbf{L} E(t,s) = 0 & 0\leq s\leq t\leq T, \\[1ex] E(s,s) = I, & s\in[0,T), \end{cases} \end{equation*} so obtaining by Duhamel's formula the solution $$U(t)=E(t,0)G+i\int_0^t E(t,s)F(s)ds$$ to the system and then the solution $u(t)$ to the corresponding higher order Cauchy problem \eqref{cpintro} by the reduction procedure. \vskip+0.5cm In the present paper we consider the Cauchy problem \eqref{cpintro} in the SG setting, that is, under the growth condition \eqref{grw}, the hyperbolicity condition \eqref{hypintro} and the involutiveness assumption \eqref{invintro} for suitable parameter-dependent, real-valued symbols $b_{jk},d_{jk}\in C^\infty([0,T];S^{0,0}(\R^{2n}))$, $j,k\in\N$. The involutiveness of the characteristic roots of the operator, or, equivalently, of the eigenvalues of the (diagonal) principal part of the corresponding first order system is here the main assumption (see Assumption \ref{assumpt:invo} in Section \ref{subsec:commlaw} for the precise statement of such condition). We obtain several results concerning the Cauchy problem \eqref{cpintro}: \begin{itemize} \item we prove, in Theorem \ref{thm:mainLastSecThm}, the existence, for every $f\in C^\infty([0,T]; H^{r,\rho}(\R^n))$ and $g_k\in H^{r+m-1-k,\rho+m-1-k}(\R^n)$, $k=0,\ldots,m-1$, of a unique $$ u\in \bigcap_{k\in\mathbb{Z}^+}C^k([0,T']; H^{r-k,\rho-k}(\R^n)), $$ for a suitably small $0< T'\leq T$, solution of \eqref{cpintro}; we also provide, in \eqref{eq:soleqordm}, the explicit expression of $u$, using SG Fourier integral operators (SG FIOs); \item we give a precise description, in Theorem \ref{thm:propwfs}, of a global wave-front set of the solution, under a (mild) additional condition on the operator $L$; namely, we prove that, when $L$ is SG-classical, the (smoothness and decay) singularities of the solution of \eqref{cpintro} with $f\equiv0$ are points of unions of arcs of bicharacteristics, generated by the phase functions of the SG FIOs appearing in the expression \eqref{eq:soleqordm}, and emanating from (smoothness and decay) singularities of the Cauchy data $g_k$, $k=0,\dots,m-1$; \item we deal, in Theorem \ref{thm:linearin}, with a stochastic version of the Cauchy problem \eqref{cpintro}, namely, with the case when $$f(t;x)=\gamma(t;x)+\sigma(t;x)\dot\Xi(t;x),$$ and $\Xi$ is a random noise of Gaussian type, white in time and with possible correlation in space; we give conditions on the noise $\Xi$, on the coefficients $\gamma,\sigma$, and on the data $g_k$, $k=0,\dots,m-1$, such that there exists a so-called \emph{random field solution} of \eqref{cpintro}, that is, a stochastic process which gives, for every fixed $(t,x)\in [0,T]\times\R^n$, a well-defined random variable (see Section \ref{sec:sghypspde} below for details). \end{itemize} All the results described above are achieved through the use of Fourier integral operators of SG-type, which are recalled in Section \ref{sec:fioprelim}. The \emph{involutiveness assumption} is the key to prove that the fundamental solution of the first order system \eqref{sysint} may be written as a \emph{finite sum} of (iterated integrals of smooth parameter-dependent families of) SG FIOs (modulo a regularizing term). This is proved in Theorem \ref{thm:mainthm}, and follows from the analysis performed in Section \ref{sec:sgphfcomlaw}. The (rather technical) results proved in Section \ref{sec:sgphfcomlaw} are indeed crucial to achieve our main results. Namely, we prove that, under the involutiveness assumption, multi-products of regular SG-phase functions given as solutions of eikonal equations satisfy a \emph{commutative law} (Theorem \ref{cruthm}). Similar to the classical situation, this extends to the SG case the possibility to reduce the fundamental solution to a finite sum of terms (modulo a regularizing operator). This also implies commutativity properties of the flows \textit{at infinity} generated by the corresponding (classical) SG-phase functions. Their relationship with the symplectic structures studied in \cite{CS17} will be investigated elsewhere. The paper is organized as follows. In Section \ref{sec:sgcalc} we recall basic elements of the SG calculus of pseudo-differential and Fourier integral operators. In Section \ref{sec:sgphfcomlaw} we prove our first main theorem, namely, a commutative property for multi-products of suitable families of regular SG-phase functions, determined by involutive families of SG-symbols. In Section \ref{sec:fundsol} we prove our second main theorem, showing how such commutative property can be employed to obtain the fundamental solution for certain $N\times N$ first order linear systems with diagonal principal part and involutive characteristics. In Section \ref{sec:sginvcp} we study the Cauchy problem for SG-hyperbolic, involutive linear differential operators. Here two more main theorems are proved: a well-posedness result (with loss of smoothness and decay) and a propagation of singularity result, for wave-front sets of global type, whose definition and basic properties are also recalled. Finally, in Section \ref{sec:sghypspde} we prove our final main theorem, namely, the existence of random field solutions for a stochastic version of the Cauchy problems studied in Section \ref{sec:sginvcp}. \section{Symbolic calculus for pseudo-differential and Fourier integral operators of SG type}\label{sec:sgcalc} \setcounter{equation}{0} In this section we recall some properties of SG pseudo-differential operators, an operator class defined through symbols satisfying global estimates on $\R^n\times\R^n$. Moreover, we state several fundamental theorems for SG Fourier integral operators. Further details can be found, e.g., in \cite{AsCor,CO,EgorovSchulze,Kumano-go:1,SH87} and the references therein, from which we took most of the materials included in this section. Here and in what follows, $A\asymp B$ means that $A\lesssim B$ and $B\lesssim A$, where $A\lesssim B$ means that $A\le c\cdot B$, for a suitable constant $c>0$. \subsection{Pseudodifferential operators of SG type} \par The class ${S}^{m,\mu}(\R^{2n})$ of SG symbols of order $(m,\mu) \in \R^2$, is given by all the functions $a(x,\xi) \in C^\infty(\R^{2n})$ with the property that, for any multiindices ${\alpha},{\beta} \in \mathbb{Z}_+^n$, there exist constants $C_{{\alpha}{\beta}}>0$ such that the conditions % \begin{equation}\label{eq:disSG} | D_{\xi}^{{\alpha}}D_x^{{\beta}} a(x,\xi)| \leq C_{{\alpha}{\beta}}\,\jb{x}^{m-|{\beta}|}\jb{\xi}^{\mu-|{\alpha}|} \end{equation} % hold, for all $ x, \xi \in \R^n$. Here $\langle x \rangle=(1+|x|^2)^{1/2}$ when $x\in\R^n$, and $\mathbb{Z}_+$ is the set of non-negative integers. For $a(x,\xi)\in{S}^{m,\mu}(\R^{2n})$ , $m,\mu\in\R$, we define the semi-norms $\vvvert a\vvvert _l^{m,\mu}$ by % \begin{align}\label{semi_norm_MSG} \vvvert a\vvvert_l^{m,\mu} =\max_{|{\alpha}+{\beta}|\leq l} \sup_{x,\xi\in\R^n}\jb{x}^{-m+|{\beta}|}\jb{\xi}^{-\mu+|{\alpha}|}| D_{\xi}^{{\alpha}}D_x^{{\beta}} a(x,\xi)|, \end{align} % where $l\in\mathbb{Z}_+$. The quantities~\eqref{semi_norm_MSG} define a Fr\'echet topology of ${S}^{m,\mu}(\R^{2n})$. Moreover, let $$ {S}^{\infty,\infty}(\R^{2n})=\bigcup_{m,\mu\in\R}{S}^{m,\mu}(\R^{2n}), \qquad {S}^{-\infty,-\infty}(\R^{2n})=\bigcap_{m,\mu\in\R}{S}^{m,\mu}(\R^{2n}). $$ \par The functions $a\in{S}^{m,\mu}(\R^{2n})$ can be $(\nu\times\nu)$-matrix-valued. In such case the estimate \eqref{eq:disSG}~~must be valid for each entry of the matrix. The next technical lemma is useful when dealing with compositions of SG symbols. % \begin{lemma} \label{compositionwithe1symbol} Let $f\in{S}^{m,\mu}(\R^{2n})$, $m,\mu\in\R$, and $g$ vector-valued in $\R^n$ such that $g\in{S}^{0,1}(\R^{2n})\otimes\R^n$ and $\jb{g(x,\xi)}\sim\jb{\xi}$. Then $f(x,g(x,\xi))$ belongs to ${S}^{m,\mu}(\R^{2n})$. \end{lemma} % \par % The previous result can be found in \cite{Cothesis}, and can of course be extended to the other composition case, namely $h(x,\xi)$ vector valued in $\R^n$ such that it belongs to ${S}^{1,0}(\R^{2n})\otimes\R^n$ and $\jb{h(x,\xi)}\sim \jb{x}$, implying that $f(h(x,\xi),\xi)$ belongs to ${S}^{m,\mu}(\R^{2n})$. \par We now recall definition and properties of the pseudo-differential operators $a(x,D)= {\operatorname{Op}} (a)$ where $a\in{S}^{m,\mu}$. % % Given $a\in{S}^{m,\mu}$, we define $ {\operatorname{Op}} (a)$ through the left-quantization (cf. Chapter XVIII in \cite {Ho1}) \begin{equation}\label{eq:psidos} ( {\operatorname{Op}} (a)u)(x)= (2 \pi)^{-n}\int_{\R^n} e^{i x \cdot \xi} a(x, \xi) \widehat{u}(\xi) \,d\xi, \quad u\in {\mathcal{S}} , \end{equation} with $\widehat{u}$ the Fourier transform of $u\in {\mathcal{S}} $, given by \begin{equation}\label{eq:tfu} \widehat{u}(\xi)=\int_{\R^n} e^{-ix\cdot\xi}u(x)\,dx. \end{equation} \par The operators in \eqref{eq:psidos} form a graded algebra with respect to composition, i.e., % $$ \operatorname{Op} ({S} ^{m_1,\mu _1})\circ \operatorname{Op} ({S} ^{m_2,\mu _2}) \subseteq \operatorname{Op} ({S} ^{m_1+m_2,\mu _1+\mu _2}). $$ % The symbol $c\in {S} ^{m_1+m_2,\mu _1+\mu _2}$ of the composed operator $ {\operatorname{Op}} (a)\circ\operatorname{Op}(b)$, where $a\in {S} ^{m_1,\mu _1}$, $b\in {S} ^{m_2,\mu _2}$, admits the asymptotic expansion % \begin{equation} \label{eq:comp} c(x,\xi)\sim \sum_{\alpha}\frac{i^{|\alpha|}}{\alpha!} \,D^\alpha_\xi a(x,\xi)\, D^\alpha_x b(x,\xi), \end{equation} % which implies that the symbol $c$ equals $a\cdot b$ modulo $S ^{m_1+m_2-1,\mu _1+\mu _2-1}$. The residual elements of the calculus are operators with symbols in % \[ {S} ^{-\infty,-\infty}={S} ^{-\infty,-\infty} = \bigcap_{(m,\mu) \in \R^2} {S} ^{m,\mu} = {\mathcal{S}} , \] % that is, those having kernel in $ {\mathcal{S}} $, continuously mapping $ {\mathcal{S}} ^\prime$ to $ {\mathcal{S}} $. \par An operator $A= {\operatorname{Op}} (a)$, is called \emph{elliptic} (or ${S} ^{m,\mu}$-\emph{elliptic}) if $a\in {S} ^{m,\mu}$ and there exists $R\geq0$ such that % \begin{equation}\label{eq:ellcond} C\jb{x}^{m} \jb{\xi}^{\mu}\le |a(x,\xi)|,\qquad |x|+|\xi|\geq R, \end{equation} % for some constant $C>0$. An elliptic SG operator $A \in \operatorname{Op} ({S} ^{m,\mu})$ admits a parametrix $P\in \operatorname{Op} ({S} ^{-m,-\mu})$ such that % \[ PA=I + K_1, \quad AP= I+ K_2, \] % for suitable $K_1, K_2\in {\operatorname{Op}} ({S}^{-\infty,-\infty})$, where $I$ denotes the identity operator. \par It is a well-known fact that SG-operators give rise to linear continuous mappings from $ {\mathcal{S}} $ to itself, extendable as linear continuous mappings from $ {\mathcal{S}} '$ to itself. They also act continuously between the so-called Sobolev-Kato (or weighted Sobolev) space, that is from $H^{r,\varrho}(\R^n)$ to $H^{r-m,\varrho-\mu}(\R^n)$, where $H^{r,\varrho}(\R^n)$, $r,\varrho \in \R$, is defined as \begin{equation*} H^{r,\varrho}(\R^n)= \left\{u \in {\mathcal S}^\prime(\R^{n}) \colon \|u\|_{r,\varrho}= \|{\jb .}^r\jb D^\varrho u\|_{L^2}< \infty\right\}. \end{equation*} \par The so-called SG manifolds were introduced in 1987 by E.~Schrohe \cite{Sc}. This is a class of non-compact manifolds, on which a version of SG calculus can be defined. % Such manifolds admit a finite atlas, whose changes of coordinates behave like symbols of order $(0,1)$ (see \cite{Sc} for details and additional technical hypotheses). A relevant example of SG manifolds are the manifolds with cylindrical ends, where also the concept of classical SG operator makes sense, % see, e.{\,}g. \cite{BC10a,MP02}. \par We close this section with the next result, which deals with the composition (or multi-product) of $(M+1)$ SG pseudo-differential operators, where $M\ge1$. The proof of Theorem \ref{thm:mpsgpdosymestim} can be found in \cite{Abdeljawadthesis}. \begin{theorem}\label{thm:mpsgpdosymestim} % Let $M\geq 1$, $p_j(x,\xi)\in{S}^{m_j,\mu_j}(\R^{2n})$, $m_j,\mu_j\in\R$, $j=1,\dots,M+1$. Consider the multi-product % \begin{equation}\label{eq:qm} Q_{M+1}=P_1 \cdots P_{M+1} \end{equation} % of the operators $P_j= {\operatorname{Op}} (p_j)$, $j=1,\dots,M+1$, and denote by $q_{M+1}(x,\xi)\in{S}^{m,\mu}(\R^{2n})$ the symbol of $Q_{M+1}$, where $m=m_1+\dots+m_{M+1}$, $\mu=\mu_1+\dots+\mu_{M+1}$. Then, the boundedness of $\displaystyle\oplus_{j=1}^{M+1} p_j$ in $\displaystyle\oplus_{j=1}^{M+1}{S}^{m_j,\mu_j}(\R^{2n})$, implies the boundedness of $q_{M+1}(x,\xi)$ in ${S}^{m,\mu}(\R^{2n})$. % \end{theorem} \par \subsection{Fourier integral operators of SG type}\label{sec:fioprelim} % % \par Here we recall the class of Fourier integral operators we are interested in. Also, we show their compositions with the SG pseudo-differential operators, and the compositions between Type I and Type II operators. \par The simplest \text{Fourier integral operators}\ are defined, for $u\in {\mathcal{S}} $, as % \begin{align} \label{eq:typei} u\mapsto ( {\operatorname{Op}} _\varphi (a)u)(x)&= (2\pi )^{-n}\int_{\R^n} e^{i\varphi (x,\xi )} a(x,\xi ) \widehat u(\xi )\, d\xi , \end{align} % and % \begin{align} \label{eq:typeii} u\mapsto ( {\operatorname{Op}} ^*_\varphi (a)u)(x)&= (2\pi )^{-n}\iint_{\R^{2n}} e^{i(x\cdot \xi -\varphi (y,\xi ))} \overline {a(y,\xi )} u(y)\, dyd\xi, \end{align} % with suitable choices of phase function $\varphi$ and symbols $a$ and $b$. \par The operators $\operatorname{Op} _\varphi (a)$ and $\operatorname{Op} _\varphi ^*(a)$ are sometimes called SG \text{Fourier integral operators}\ of type I and type II, respectively, with symbol $a$ and phase function $\varphi$. Note that a type II operator satisfies $\operatorname{Op}^*_\varphi(a)=\operatorname{Op} _\varphi (a)^*$, that is, it is the formal $L^2$-adjoint of the type I operator $\operatorname{Op}_\varphi(a)$. We now introduce the definition of the class of admissible phase functions in the SG context, as it was given in \cite{Coriasco:998.1}. % \begin{definition}[SG phase function]\label{def:phase} A real valued function $\varphi\in C^\infty(\R^{2n})$ belongs to the class $\mathcal P$ of SG phase functions if it satisfies the following conditions: \begin{enumerate} \item $\varphi\in {S}^{1,1}(\R^{2n})$; \item $\<\varphi'_x(x,\xi)\>\asymp\<\xi\>$ as $|(x,\xi)|\to\infty$; \item $\<\varphi'_\xi(x,\xi)\>\asymp\<x\>$ as $|(x,\xi)|\to\infty$. \end{enumerate} \end{definition} % The SG \text{Fourier integral operators}\ of type I and type II, $ {\operatorname{Op}} _\varphi(a)$ and $ {\operatorname{Op}} ^*_\varphi(b)$, are defined as in \eqref{eq:typei} and \eqref{eq:typeii}, respectively, with $\varphi\in {\cal{P}} $ and $a,b\in {S}^{m,\mu}$. Notice that we do not assume any homogeneity hypothesis on the phase function $\varphi$. The next theorem, treating composition between SG pseudo-differential operators and SG \text{Fourier integral operators}\, was originally proved in \cite{Coriasco:998.1}, see also \cite{CJT4, CoPa, CT14}. % \begin{theorem}\label{thm:compi} Let $\varphi\in {\cal{P}} $ and assume $p\in {S}^{t,\tau}(\R^{2n})$, $a,b\in {S}^{m,\mu}(\R^{2n})$. Then, % \begin{align*} {\operatorname{Op}} (p)\circ {\operatorname{Op}} _\varphi (a) &= {\operatorname{Op}} _\varphi (c_1+r_1) = {\operatorname{Op}} _\varphi (c_1)\ \mathrm{mod} \ {\operatorname{Op}} ({S}^{-\infty,-\infty}(\R^{2n})), \\[1ex] {\operatorname{Op}} (p)\circ {\operatorname{Op}} ^* _\varphi(b) &= {\operatorname{Op}} ^* _\varphi(c_2+r_2) = {\operatorname{Op}} ^* _\varphi (c_2)\ \mathrm{mod} \ {\operatorname{Op}} ({S}^{-\infty,-\infty}(\R^{2n})), \\[1ex] {\operatorname{Op}} _\varphi (a) \circ {\operatorname{Op}} (p) &= {\operatorname{Op}} _\varphi (c_3+r_3) = {\operatorname{Op}} _\varphi (c_3)\ \mathrm{mod} \ {\operatorname{Op}} ({S}^{-\infty,-\infty}(\R^{2n})), \\[1ex] {\operatorname{Op}} _\varphi ^*(b) \circ {\operatorname{Op}} (p) &= {\operatorname{Op}} ^* _\varphi(c_4+r_4) = {\operatorname{Op}} ^* _\varphi (c_4)\ \mathrm{mod} \ {\operatorname{Op}} ({S}^{-\infty,-\infty}(\R^{2n})), \end{align*} for some $c_j\in {S}^{m+t,\mu+\tau}(\R^{2n})$, $r_j\in {S}^{-\infty,-\infty}(\R^{2n})$, $j=1,\dots ,4$. % \end{theorem} % In order to obtain the composition of SG \text{Fourier integral operators}\ of type I and type II, some more hypotheses are needed, leading to the definition of the classes $ {\cal{P}}^\epsilon $ and $ {\cal{P}}^\epsilon (\tau)$ of regular SG phase functions, cf. \cite{Kumano-go:1}. % \begin{definition}[Regular SG phase function]\label{def:phaser} Let $\tau\in [0,1)$ and $r>0$. A function $\varphi\in\mathcal P$ belongs to the class $\mathcal P_r(\tau)$ if it satisfies the following conditions: \begin{description} \item[1\label{1}] $\vert\det(\varphi''_{x\xi})(x,\xi)\vert\geq r$, $\text{ for any } x,\xi\, \in \mathbb{R}^{n}$; \item[2\label{2}] the function $J(x,\xi):=\varphi(x,\xi)-x\cdot\xi$ is such that \arraycolsep1.5pt\begin{eqnarray}\label{hyp} \displaystyle\sup_{\afrac{x,\xi\in\R^n}{|\alpha+\beta|\leq 2}} \frac{|D_\xi^\alpha D_x^\beta J(x,\xi)|}{\langle x\rangle^{1-|\beta|}\<\xi\>^{1-|\alpha|}}\leq \tau. \end{eqnarray}\arraycolsep5pt \end{description} If only condition (1) holds, we write $\varphi\in {\cal{P}}^\epsilon $. \end{definition} % \begin{remark} Notice that if condition \eqref{hyp} actually holds true for any ${\alpha},\,{\beta}\, \in \mathbb{Z}_+^n$, then $J(x,\xi)/\tau$ is bounded with constant $1$ in ${S}^{1,1}(\R^{2n})$. Notice also that condition (1) in Definition \ref{def:phaser} is authomatically fulfilled when condition (2) holds true for a sufficiently small $\tau\in[0,1)$. \end{remark} \par For $\ell\in\N$, we also introduce the semi-norms % \[ \|J\|_{2,\ell}:= \displaystyle\sum_{2\leq |\alpha+\beta|\leq 2+\ell}\sup_{(x,\xi)\in \R^{2n}} \displaystyle\frac{|D_\xi^\alpha D_x^\beta J(x,\xi)|}{\langle x\rangle^{1-|\beta|}\<\xi\>^{1-|\alpha|}}, \] % and \[ \|J\|_\ell:=\displaystyle\sup_{\afrac{x,\xi\in\R^n}{|\alpha+\beta|\leq 1}}\frac{|D_\xi^\alpha D_x^\beta J(x,\xi)|}{\langle x\rangle^{1-|\beta|}\<\xi\>^{1-|\alpha|}}+\|J\|_{2,\ell}. \] % We notice that $\varphi\in\mathcal P_r(\tau)$ means that \eqref{1} of Definition \ref{def:phaser} and $\|J\|_0\leq \tau$ hold, and then we define the following subclass of the class of regular SG phase functions: \begin{definition}\label{def:phaserell} Let $\tau\in [0,1)$, $r>0$, $\ell \geq 0$. A function $\varphi$ belongs to the class $\mathcal P_r(\tau,\ell)$ if $\varphi\in\mathcal P_r(\tau)$ and $\|J\|_\ell\leq \tau$ for the corresponding $J$. \end{definition} % Theorem \ref{thm:compii} below shows that the composition of SG \text{Fourier integral operators}\ of type I and type II with the same regular SG phase functions is a SG pseudo-differential operator. % \begin{theorem}\label{thm:compii} Let $\varphi\in {\cal{P}}^\epsilon $ and assume $a\in {S}^{m,\mu}(\R^{2n})$, $b\in {S}^{t,\tau}(\R^{2n})$. Then, % \begin{align*} {\operatorname{Op}} _\varphi(a)\circ {\operatorname{Op}} _\varphi^*(b)= {\operatorname{Op}} (c_5+r_5)= {\operatorname{Op}} (c_5)\ \mathrm{mod} \ {\operatorname{Op}} ({S}^{-\infty,-\infty}(\R^{2n})), \\[1ex] {\operatorname{Op}} _\varphi ^*(b)\circ {\operatorname{Op}} _\varphi (a)= {\operatorname{Op}} (c_6+r_6)= {\operatorname{Op}} (c_6)\ \mathrm{mod} \ {\operatorname{Op}} ({S}^{-\infty,-\infty}(\R^{2n})), \end{align*} % for some $c_j\in {S}^{m+t,\mu+\tau}(\R^{2n})$, $r_j\in {S}^{-\infty,-\infty}(\R^{2n})$, $j=5,6$. \end{theorem} \par Furthermore, asymptotic formulae can be given for $c_j$, $j=1,\dots ,6$, in terms of $\varphi$, $p$, $a$ and $b$, see \cite{Coriasco:998.1}. Finally, when $a\in {S}^{m,\mu}$ is elliptic and $\varphi\in {\cal{P}}^\epsilon $, the corresponding SG \text{Fourier integral operators}\ admit a parametrix, that is, there exist $b_1,b_2\in {S}^{-m,-\mu}$ such that \begin{align} \label{eq:parami} {\operatorname{Op}} _\varphi(a)\circ {\operatorname{Op}} _\varphi^*(b_1)= {\operatorname{Op}} _\varphi^*(b_1)\circ {\operatorname{Op}} _\varphi(a) &= I\ \mathrm{mod} \ {\operatorname{Op}} ({S}^{-\infty,-\infty}), \\[1ex] \label{eq:paramii} {\operatorname{Op}} _\varphi ^*(a)\circ {\operatorname{Op}} _\varphi (b_2)= {\operatorname{Op}} _\varphi(b_2)\circ {\operatorname{Op}} ^*_\varphi(a) &= I\ \mathrm{mod} \ {\operatorname{Op}} ({S}^{-\infty,-\infty}), \end{align} where $I$ is the identity operator. For all these results, as well as for Theorem \ref{thm:fiocont} below, see again \cite{Coriasco:998.1,CJT4,CT14}. \par \begin{theorem}\label{thm:fiocont} % Let $\varphi\in {\cal{P}}^\epsilon $ and $a\in{S}^{m,\mu}(\R^{2n})$, $m,\mu\in\R$. Then, for any $r,\varrho\in\R$, $ {\operatorname{Op}} _\varphi(a)$ and $ {\operatorname{Op}} ^*_\varphi(a)$ continuously map $H^{r,\varrho}(\R^n)$ to $H^{r-m,\varrho-\mu}(\R^n)$. % \end{theorem} \par % \subsection{Multi-products of regular SG phase functions and of regular SG Fourier integral operators}\label{sec:mpsgphf} For the convenience of the reader, here we recall a few results from \cite{AsCor}. Let us consider a sequence $\{\varphi_j\}_{j\in\N}$ of regular SG phase functions $\varphi_j(x,\xi)\in \mathcal P_r(\tau_j)$, $j\in\N$, with \arraycolsep1.5pt\begin{eqnarray}\label{1/4} \displaystyle\sum_{j=1}^\infty \tau_j=:\tau_0<1/4. \end{eqnarray}\arraycolsep5pt By \eqref{1/4} we have that for every $\ell\in\N$ there exists a constant $c_\ell>0$ such that \arraycolsep1.5pt\begin{eqnarray}\label{marr} \|J_k\|_{2,\ell}\leq c_\ell\tau_k\quad\ {\rm and} \quad \displaystyle\sum_{k=1}^\infty\|J_k\|_{2,\ell}\leq c_\ell\tau_0. \end{eqnarray}\arraycolsep5pt We set $\overline{\tau}_M=\sum_{j=1}^{M}\tau_j.$ \par With a fixed integer $M\geq 1$, we denote % \begin{align}\label{multivardef}\begin{split} (X,\Xi)&=(x_0,x_1,\ldots,x_M,\xi_1,\ldots,\xi_M,\xi_{M+1}):=(x,T,\Theta,\xi), \\ (T,\Theta)&=(x_1,\ldots,x_M,\xi_1,\ldots,\xi_M), \end{split} \end{align} % and define the function of $2(M+1)n$ real variables \arraycolsep1.5pt\begin{eqnarray}\label{serveilnome} \psi(X,\Xi):=\displaystyle\sum_{j=1}^M\left(\varphi_j(x_{j-1},\xi_j)-x_j\cdot\xi_j\right) +\varphi_{M+1}(x_M,\xi_{M+1}). \end{eqnarray}\arraycolsep5pt For every fixed $(x,\xi)\in\R^{2n}$, the critical points $(Y,N)=(Y,N)(x,\xi)$ of the function of $2Mn$ variables $\widetilde{\psi}(T,\Theta)=\psi(x,T,\Theta,\xi)$ are the solutions to the system \begin{align}\label{critpntprob} \begin{cases} \psi'_{\xi_j}(X,\Xi)=\varphi'_{j, \xi}(x_{j-1},\xi_{j})-x_j=0 & j=1,\ldots,M, \\ \psi'_{x_j}(X,\Xi)=\varphi'_{j+1, x}(x_j,\xi_{j+1})-\xi_j=0 & j=1,\ldots,M, \end{cases} \end{align} % in the unknowns $(T,\Theta)$. That is $(Y,N)=(Y_1,\ldots,Y_M,N_1,\ldots,N_M)(x,\xi)$ satisfies, if $M=1$, % \arraycolsep1.5pt\begin{eqnarray}\label{(C1)}\begin{cases} Y_1(x,\xi)=\varphi'_{1, \xi}(x,N_{1}(x,\xi)) \\ N_1(x,\xi)=\varphi'_{2, x}(Y_1(x,\xi),\xi), \end{cases} \end{eqnarray}\arraycolsep5pt % or, if $M\ge2$, % \arraycolsep1.5pt\begin{eqnarray}\label{(C)}\begin{cases} Y_1(x,\xi)=\varphi'_{1,\xi}(x,N_{1}(x,\xi)) \\ Y_j(x,\xi)=\varphi'_{j,\xi}(Y_{j-1}(x,\xi),N_{j}(x,\xi)), & j=2,\ldots,M \\ N_j(x,\xi)=\varphi'_{j+1, x}(Y_j(x,\xi),N_{j+1}(x,\xi)), & j=1,\ldots,M-1 \\ N_M(x,\xi)=\varphi'_{M+1, x}(Y_M(x,\xi),\xi) . \end{cases} \end{eqnarray}\arraycolsep5pt % In the sequel we will only refer to the system \eqref{(C)}, tacitly meaning \eqref{(C1)} when $M=1$. \begin{definition}[Multi-product of SG phase functions]\label{mprod} If, for every fixed $(x,\xi)\in \R^{2n}$, the system \eqref{(C)} admits a unique solution $(Y,N)=(Y,N)(x,\xi)$, we define \arraycolsep1.5pt\begin{eqnarray}\label{3.4} \phi(x,\xi)=(\varphi_1\ \sharp\ \cdots\ \sharp\ \varphi_{M+1})(x,\xi):=\psi(x,Y(x,\xi),N(x,\xi),\xi). \end{eqnarray}\arraycolsep5pt The function $\phi$ is called multi-product of the SG phase functions $\varphi_1,\ldots,\varphi_{M+1}.$ \end{definition} \par The following properties of the multi-product SG phase functions can be found in \cite{AsCor}. \begin{proposition}\label{stime} Under the assumptions \eqref{hyp} and \eqref{1/4}, the system \eqref{(C)} admits a unique solution $(Y,N)$, satisfying % \arraycolsep1.5pt\begin{eqnarray}\label{324} \{(Y_j-Y_{j-1})/\tau_j\}_{j\in\N}\ is\ bounded\ in\ {S}^{1,0}(\R^{2n}), \\ \{(N_j-N_{j+1})/\tau_{j+1}\}_{j\in\N}\ is\ bounded\ in\ {S}^{0,1}(\R^{2n}). \end{eqnarray}\arraycolsep5pt \end{proposition} \par \begin{proposition}\label{properties} Under the assumptions \eqref{hyp} and \eqref{1/4}, the multi-product $\phi(x,\xi)$ in Definition \ref{mprod} is well defined for every $M\geq 1$ and has the following properties. \begin{enumerate} \item There exists $k\geq 1$ such that $\phi(x,\xi)=(\varphi_1\ \sharp\ \cdots\ \sharp\ \varphi_{M+1})(x,\xi)\in\mathcal P_r(k\bar\tau_{M+1})$ and, setting $$ J_{M+1}(x,\xi):=(\varphi_1\ \sharp\ \cdots\ \sharp\ \varphi_{M+1})(x,\xi)-x\cdot\xi, $$ the sequence $\{J_{M+1}/\bar\tau_{M+1}\}_{M\geq1}$ is bounded in ${S}^{1,1}(\R^{2n})$. \item The following relations hold: \[\begin{cases} \phi'_x(x,\xi)=\varphi'_{1, x}(x,N_1(x,\xi)) \\ \phi'_\xi(x,\xi)=\varphi'_{M+1, \xi}(Y_M(x,\xi),\xi), \end{cases} \] where $(Y,N)$ are the critical points \eqref{(C)}. \item The associative law holds: $\varphi_1\ \sharp\ (\varphi_2\ \sharp\ \cdots\ \sharp\ \varphi_{M+1}) =(\varphi_1\ \sharp\ \cdots\ \sharp\ \varphi_{M})\ \sharp\ \varphi_{M+1}$. \item For any $\ell\geq 0$ there exist $0<\tau^\ast<1/4$ and $c^\ast\geq 1$ such that, if $\varphi_j\in \mathcal P_r(\tau_j,\ell)$ for all $j$ and $\tau_0\leq \tau^\ast$, then $\phi\in \mathcal P_r(c^\ast\bar\tau_{M+1},\ell)$. \end{enumerate} \end{proposition} \par Passing to regular SG Fourier integral operators, one can prove the following algebra properties. \begin{proposition}\label{thm:main} Let $\varphi_j\in {\cal{P}}^\epsilon (\tau_j)$, $j=1,2, \dots, M$, $M\ge 2$, be such that $\tau_1+\cdots+\tau_M\leq\tau\leq\frac{1}{4}$ for some sufficiently small $\tau>0$, and set % \begin{align*} \Phi_0(x,\xi)&=x\cdot\xi, \\ \Phi_1&=\varphi_1, \\ \Phi_j&=\varphi_1\sharp\cdots\sharp\varphi_j, \; j= 2, \dots, M \\ \Phi_{M,j}&=\varphi_j\sharp\varphi_{j+1}\sharp\cdots\sharp\varphi_{M}, j=1,\dots,M-1, \\ \Phi_{M,M}&=\varphi_{M}, \\ \Phi_{M,M+1}(x,\xi)&=x\cdot\xi. \end{align*} % Assume also $a_j\in {S}^{m_j,\mu_j}(\R^{2n})$, and set $A_j= {\operatorname{Op}} _{\varphi_j}(a_j)$, $j=1,\dots,M$. Then, the following properties hold true. % \begin{enumerate} % \item Given $q_j, q_{M,j}\in {S}^{0,0}(\R^{2n})$, $j=1,\dots, M$, such that \[ {\operatorname{Op}} ^*_{\Phi_j}(q_j)\circ I_{\Phi_j}=I, \quad I^*_{\Phi_{M,j}}\circ {\operatorname{Op}} _{\Phi_{M,j}}(q_{M,j})=I, \] % set $Q_j^*= {\operatorname{Op}} ^*_{\Phi_j}(q_j)$, $Q_{M,j}= {\operatorname{Op}} _{\Phi_{M,j}}(q_{M,j})$, and % \[ R_j=I_{\Phi_{j-1}}\circ A_j\circ Q^*_j, \quad R_{M,j}=Q_{M,j}\circ A_j\circ I^*_{\Phi_{M,j+1}}, \quad j=1,\dots,M. \] % Then, $R_j,R_{M,j}\in {\operatorname{Op}} ({S}^{0,0}(\R^{2n}))$, $j=1,\dots,M$, and % \begin{equation}\label{eq:prodAj} A=A_1\circ\cdots\circ A_M=R_1\circ\cdots\circ R_{M}\circ I_{\Phi_M} =I^*_{\Phi_{M,1}}\circ R_{M,1}\circ\cdots\circ R_{M,M}. \end{equation} % \item There exists $a\in {S}^{m,\mu}(\R^{2n})$, $m=m_1+\cdots+m_M$, $\mu=\mu_1+\cdots+\mu_M$ such that, setting $\phi=\varphi_1\sharp\cdots\sharp\varphi_M$, % \[ A=A_1\circ\cdots\circ A_M= {\operatorname{Op}} _{\phi}(a). \] % \item For any $l\in\mathbb{Z}_+$ there exist $l^\prime\in\mathbb{Z}_+$, $C_l>0$ such that % \begin{equation}\label{eq:estsna} \vvvert a \vvvert_l^{m,\mu} \le C_l\prod_{j=1}^M \vvvert a_j \vvvert_{l^\prime}^{m_j,\mu_j}. \end{equation} % \end{enumerate} % \end{proposition} % \par % \subsection{Eikonal equations and Hamilton-Jacobi systems in SG classes}\label{subsec:eik} \par To study evolution equations within the SG environment, we need first to introduce parameter-dependent symbols, where the parameters give rise to bounded families in ${S}^{m,\mu}$. The proof of the next technical Lemma \ref{lem:invariantcomsymbol} can be found in \cite{Abdeljawadthesis}. \par \begin{definition} Let $\Omega\subseteq\R^N$. We write $f\in C^k\left(\Omega;{S}^{m,\mu}(\R^{2n})\right)$, with $m,\,\mu\in \R$ and $k\in\mathbb{Z}_+$ or $k=\infty$, if \begin{enumerate}[(i)] \item $f=f(\omega;x,\xi)$, $\omega\in\Omega,\,x,\,\xi\in\R^n$; \item for any $\omega\in \Omega$, $\partial^{\alpha}_\omega f(\omega)\in {S}^{m,\mu}(\R^{2n})$, for all ${\alpha}\in\mathbb{Z}_+^N,\,|{\alpha}|\leq k$; \item $\{\partial^{\alpha}_\omega f(\omega)\}_{\omega\in\Omega}$ is bounded in ${S}^{m,\mu}(\R^{2n})$, for all ${\alpha}\in\mathbb{Z}_+^N,\,|{\alpha}|\leq k$. \end{enumerate} \end{definition} \begin{lemma}\label{lem:invariantcomsymbol} Let $\Omega\subseteq\R^N$, $a\in C^k\left(\Omega;{S}^{m,\mu}(\R^{2n})\right)$ and $h\in C^k\left(\Omega;{S}^{0,0}(\R^{2n})\otimes\R^N\right)$ such that $k\in\mathbb{Z}_+$ or $k=\infty$. Assume also that, for any $\omega\in \Omega$, $x,\xi\in\R^n$, the function $h(\omega;x,\xi)$ takes value in $\Omega$. Then, setting $b(\omega)=a(h(\omega))$, that is, $b(\omega;x,\xi)=a(h(\omega;x,\xi);x,\xi)$, we find $b\in C^k\left(\Omega;{S}^{m,\mu}(\R^{2n})\right)$. \end{lemma} \par Given a real-valued symbol $a\in C([0,T]; {S}^{1,1}(\R^{2n}))$, consider the so-called \textit{eikonal equation} % \arraycolsep1.5pt\begin{eqnarray}\label{eik} \begin{cases} \partial_t\varphi(t,s;x,\xi)=a(t;x,\varphi'_x(t,s;x,\xi)),& t\in [0,T'], \\ \varphi(s,s;x,\xi)=x\cdot\xi,& s\in [0,T'], \end{cases} \end{eqnarray}\arraycolsep5pt % with $0<T'\leq T$. By an extension of the theory developed in \cite{Coriasco:998.2}, it is possible to prove that the following Proposition \ref{trovala!} holds true. % \begin{proposition}\label{trovala!} Let $a\in C([0,T]; {S}^{1,1}(\R^{2n}))$ be real-valued. Then, for a small enough $T'\in(0,T]$, equation \eqref{eik} admits a unique solution $\varphi\in C^1([0,T']^2;{S}^{1,1}(\R^{2n}))$, satisfying $J\in C^1([0,T']^2;{S}^{1,1}(\R^{2n}))$ and \beq \partial_s\varphi(t,s;x,\xi)=-a(s;\varphi'_\xi(t,s;x,\xi),\xi), \end{eqnarray}\arraycolsep5pt for any $t,s\in[0,T']$. Moreover, for every $h\geq 0$ there exists $c_h\geq 1$ and $T_h\in[0,T']$ such that $\varphi(t,s;x,\xi)\in\mathcal P_r(c_h|t-s|)$, with $\| J\|_{2,h}\leq c_h |t-s|$ for all $0\leq s\leq t\leq T_h$. \end{proposition} \noindent In the sequel we will sometimes write $\varphi_{ts}(x,\xi):=\varphi(t,s;x,\xi)$, for a solution $\varphi$ of \eqref{eik}. \par \begin{remark} Note that the eikonal equation \eqref{eik} appears in the so-called geometric optics approach to the solution of ${\mathcal L} u=f,\, u(0)=u_0$ for the hyperbolic operator % \begin{equation}\label{hypoprcortoeikeq} {\mathcal L}=D_t-a(t;x,D_x) \qquad \text{ on } [0,T]. \end{equation} % \end{remark} % \par As a simple realization of a sequence of phase functions satisfying \eqref{hyp} and \eqref{1/4}, we recall the following example, see \cite{AsCor} and \cite{Kumano-go:1}. \begin{example} Let $\varphi(t,s;x,\xi)$ be the solution of the eikonal equation \eqref{eik}. Choose the partition $\Delta_{M+1}(T_1)\equiv\Delta(T_1)$ of the interval $[s,t]$, $0\leq s\leq t\leq T_1$, given by \begin{equation}\label{partitionof[s,t]} s=t_{M+1}\leq t_M\leq\cdots\leq t_1\leq t_0=t, \end{equation} and define the sequence of phase functions \[\chi_j(x,\xi)= \begin{cases} \varphi(t_{j-1}, t_j;x,\xi) & 1\leq j\leq M+1 \\ x\cdot\xi & j\geq M+2. \end{cases}\] From Proposition \ref{mainprop} we know that $\chi_j\in\mathcal P_r(\tau_j)$ with $\tau_j=c_0(t_{j-1}-t_j)$ for $1\leq j\leq M+1$ and with $\tau_j=0$ for $j\geq M+2$. Condition \eqref{1/4} is fulfilled by the choice of a small enough positive constant $T_1$, since \[\displaystyle\sum_{j=1}^\infty \tau_j=\displaystyle\sum_{j=1}^{M+1} c_0(t_{j-1}-t_j)=c_0(t-s)\leq c_0T_1<\frac14\] if $T_1<(4c_0)^{-1}$. Moreover, again from Proposition \ref{mainprop}, we know that $\| J_j\|_{2,0}\leq c_0 |t_j-t_{j-1}|=\tau_j$ for all $1\leq j\leq M+1$ and $J_j=0$ for $j\geq M+2,$ so each one of the $J_j$ satisfies \eqref{hyp}. \end{example} % \par % We now focus on the Hamilton-Jacobi system corresponding to the real-valued Hamiltonian $a\in C([0,T];{S}^{1,1}(\R^{2n}))$, namely, % \par % \begin{eqnarray}\label{hameq} \left\{\begin{array}{ll} \partial_t{q}(t,s;y,\eta)&=- a^\prime_\xi\left(t;q(t,s;y,\eta),p(t,s;y,\eta)\right), \\ \partial_t{p}(t,s;y,\eta)&=\phantom{-} a^\prime_x(t;q(t,s;y,\eta),p(t,s;y,\eta)), \end{array} \right. \end{eqnarray} where $t,s\in[0,T]$, $T>0$, and the Cauchy data \begin{eqnarray} \label{hameqCauchyData}\left\{\begin{array}{cc} q(s,s;y,\eta)=y,\\ p(s,s;y,\eta)=\eta. \end{array}\right. \end{eqnarray} \par \par We will now recall how the solution of \eqref{hameq}, \eqref{hameqCauchyData} is related with the solution of \eqref{eik} in the SG context. We here mainly refer to known results from \cite[Ch. 6]{CO} and \cite{Coriasco:998.2}. \par \begin{proposition}\label{prop:equizeroder} Let $a\in C\left([0,T];{S}^{1,1}(\R^{2n})\right)$ be real-valued. Then, the solution $(q,p)(t,s;y,\eta)$ of the Hamilton-Jacobi system \eqref{hameq} with the Cauchy data \eqref{hameqCauchyData} satisfies \par \begin{equation}\label{eq:equiv} \jb{q(t,s;y,\eta)}\sim \jb{y},\qquad \jb{p(t,s;y,\eta)}\sim \jb{\eta}. \end{equation} \end{proposition} \par \begin{proposition} Under the same hypotheses of Proposition \ref{prop:equizeroder}, the maximal solution of the Hamilton-Jacobi system \eqref{hameq} with the Cauchy data \eqref{hameqCauchyData} is defined on the whole product of intervals $[0,T]\times [0,T]$. \end{proposition} % \par \begin{proposition}\label{prop:classofqandp} The solution $(q,p)$ of the Hamilton-Jacobi system \eqref{hameq} with $a\in C^\infty([0,T];{S}^{1,1}(\R^{2n}))$ real-valued, and the Cauchy data \eqref{hameqCauchyData}, satisfies \begin{description} \item[i] {$q$ belongs to $C^\infty([0,T]^2;{S}^{1,0}(\R^{2n}))$,} \item[ii] {$p$ belongs to $C^\infty([0,T]^2;{S}^{0,1}(\R^{2n}))$.} \end{description} \end{proposition} \par \begin{lemma}\label{lem:bndedlemhamsol} The following statements hold true. \begin{description} \item[i\label{lem:classofpandqfirstpart}] {Let $a\in C([0,T];{S}^{1,1}(\R^{2n}))$ be real-valued. Then, the solution $(q,p)(t,s;y,\eta)$ of the Hamilton-Jacobi system \eqref{hameq} with Cauchy data \eqref{hameqCauchyData} satisfies, for a sufficently small $T'\in (0,T]$ and a fixed $t$ such that $0\leq s,t\leq T',\ s\neq t$,} \begin{eqnarray}\label{SGboundednessOfHamiltonSolution} \left\{\begin{array}{ccc} {(q(t,s;y,\eta)-y)/(t-s)} &\text{ is bounded in }&{S}^{1,0}(\R^{2n}) \\[1ex] {(p(t,s;y,\eta)-\eta)/(t-s)} &\text{ is bounded in }&{S}^{0,1}(\R^{2n}) \end{array} \right. \end{eqnarray} and \begin{eqnarray}\label{continuityofp-yandpinbanachspace} \left\{\begin{array}{cc} q(t,s;y,\eta),q(t,s;y,\eta)-y \in C^1(I(T');{S}^{1,0}(\R^{2n})), \\[1ex] p(t,s;y,\eta),p(t,s;y,\eta)-\eta \in C^1(I(T');{S}^{0,1}(\R^{2n})), \end{array} \right. \end{eqnarray} where, for $T>0$, $I(T)=\{(t,s)\colon 0\leq t,s\leq T\}$. \item[ii\label{lem:classofpandqsecondpart}] { Furthermore, if, additionally, $a$ belongs to $C^\infty([0,T];{S}^{1,1}(\R^{2n}))$, then, $q(t,s;y,\eta)-y\in C^\infty(I(T');{S}^{1,0}(\R^{2n}))$ and $p(t,s;y,\eta)-\eta\in C^\infty(I(T');{S}^{0,1}(\R^{2n}))$. } \end{description} \end{lemma} % \par % The proof of Lemma \ref{lem:bndedlemhamsol} combines techniques and results similar to those used in \cite{CO}, \cite{Coriasco:998.2} and \cite{Kumano-go:1}, so we omit the details. % \par Now, we observe that there exists a constant $T_1\in( 0, T']$ such that $q(t,s;y,\eta)$ is invertible with respect to $y$ for any $(t,s)\in I(T_1)$ and any $\eta\in \R^n$. Indeed, this holds by continuity and the fact that $$ q(s,s;y,\eta)=y\Rightarrow\displaystyle\frac{\partial q}{\partial y}(s,s;y,\eta)=I_n. $$ \par We denote the inverse function by $\overline{q}$, that is $$ y=\overline{q}(t,s)=\overline{q}(t,s;x,\eta) \Leftrightarrow x=q(t,s;y,\eta), $$ which exists on $I(T_1)$. Moreover, $\overline{q}\in C^\infty(I(T_1);{S}^{1,0}(\R^{2n}))$, cf. \cite{CO,Coriasco:998.2}. \par Observe now that, in view of Lemma \ref{lem:bndedlemhamsol}, $\|\frac{\partial q}{\partial y}-I_n\|\rightarrow 0$ when $t\rightarrow s$, uniformly on $I(T_1)$. Then, one can deduce the following result, which is an extension to the analogous ones that can be found e.g., in \cite{CO,Kumano-go:1}. % \par % \begin{lemma}\label{lem:bndedleminversehamsol} Assume that $a\in C([0,T];{S}^{1,1}(\R^{2n}))$ is real-valued, and let $T_1\in (0,T']$, $\epsilon_1\in(0,1]$ be constants such that on $I(T_1)$ we have \begin{equation}\label{contractioncont} \left\|\frac{\partial q}{\partial y}-I_n\right\|\leq 1- {\mathrm{e}} _1. \end{equation} Then, the mapping $x=q(t,s;y,\xi)\,:\,\R_y^n\ni y\longmapsto x\in\R_x^n$ with $(t,s,\xi)$ understood as parameter, has the inverse function $y=\overline{q}(t,s;x,\xi)$ satisfying \begin{align}\label{bndoftheinvfunc} \left\{\begin{array}{ll} \overline{q}(t,s;x,\xi)-x\; \text{belongs to } C^1(I(T_1);{S}^{1,0}(\R^{2n})), \\ \{(\overline{q}(t,s;x,\xi)-x)/|t-s|\}\text{ is bounded in } {S}^{1,0}(\R^{2n}), \text{ whenever } 0\leq s,t\leq T_1; s\neq t. \end{array} \right. \end{align} Moreover, if, additionally, $a\in C^\infty([0,T];{S}^{1,1}(\R^{2n}))$, we also have $\overline{q},\overline{q}(t,s;x,\xi)-x\in C^\infty(I(T_1);{S}^{1,0}(\R^{2n}))$. \end{lemma} % \par % The proof of the next Proposition \ref{mainprop} follows by slight modifications of the classical arguments given, e.g., in \cite{Kumano-go:1} (see \cite{Abdeljawadthesis} for details). % \begin{proposition}\label{mainprop} Let $a\in C([0,T];{S}^{1,1}(\R^{2n}))$ be real-valued, $q(t,s;y,\eta),\,p(t,s;y,\eta)$ and $\overline{q}(t,s;x,\xi)$ be the symbols constructed in the previous Lemmas \ref{lem:bndedlemhamsol} and \ref{lem:bndedleminversehamsol}. We define $u(t,s;y,\eta)$ by % % \begin{multline}\label{integralsolform} u(t,s;y,\eta)=y\cdot\eta + \\ \int_s^t \left(a(\tau;q(\tau,s;y,\eta),p(\tau,s;y,\eta))-a'_\xi(\tau;q(\tau,s;y,\eta),p(\tau,s;y,\eta))\cdot p(\tau,s;y,\eta)\right)\,d\tau, \end{multline} and \begin{equation}\label{eq:phfundefbyhamsyssol} \varphi(t,s;x,\xi)=u(t,s;\overline{q}(t,s;x,\xi),\xi). \end{equation} Then, $\varphi(t,s;x,\xi)$ is a solution of the eikonal equation \eqref{eik} and satisfies \begin{align} \label{xivarphiequal}\varphi'_\xi(t,s;x,\xi)&=\phantom{-}\overline{q}(t,s;x,\xi), \\ \label{xvarphiequal} \varphi'_x(t,s;x,\xi)&=\phantom{-}p(t,s;\overline{q}(t,s;x,\xi),\xi), \\ \label{svarphiequal}\partial_s\varphi(t,s;x,\xi)&=-a(s;\varphi'_\xi(t,s;x,\xi),\xi), \\ \label{equivonderivofvarphi}\jb{\varphi'_x(t,s;x,\xi)}&\sim\jb{\xi} \text{ and }\jb{\varphi'_\xi(t,s;x,\xi)}\sim\jb{x}. \end{align} % % Moreover, for any $l\geq0$ there exists a constant $c_l\geq 1$ and $T_l\in(0,T_1]$ such that $c_lT_l<1,\, \varphi(t,s;x,\xi)$ belongs to $ {\cal{P}}^\epsilon (c_l|t-s|,l)$ and $\{J(t,s)/|t-s|\}$ is bounded in ${S}^{1,1}(\R^{2n})$ for $0\leq t,s\leq T_l\leq T_1,\,t\neq s,$ where $J(t,s;x,\xi)=\varphi(t,s;x,\xi)-x\cdot\xi$. \end{proposition} % \par % \begin{corollary}\label{CinftyclassofJ} Let $a\in C([0,T];{S}^{1,1}(\R^{2n}))$ be real-valued, and let $q,\,p$ and $\overline{q}$ be the symbols constructed in Lemma \ref{lem:bndedlemhamsol} and Lemma \ref{lem:bndedleminversehamsol}, respectively. Then, $J\in C^1(I(T_1);{S}^{1,1}(\R^{2n}))$. Moreover, if, additionally, $a\in C^\infty([0,T];{S}^{1,1}(\R^{2n}))$, we find $J\in C^\infty(I(T_1);{S}^{1,1}(\R^{2n}))$. \end{corollary} \subsection{Classical symbols of SG type}\label{subsec:zpower} In the last part of Section \ref{sec:sginvcp} we will focus on the subclass of symbols and operators which are SG-classical, that is, $a\in {S}^{m,\mu}_\mathrm{cl}\subset {S}^{m,\mu}$. In this subsection we recall their definition and summarize some of their main properties, using materials coming from \cite{BC10a} (see, e.g., \cite{EgorovSchulze} for additional details and proofs). In the sequel of this subsection we denote by $\widetilde{\mathscr{H}}_\xi^{m}(\R^n)$ the space of homogeneous functions of order $m$ with respect to the variable $\xi$, smooth with respect to the variable $x$, and by $\widetilde{\mathscr{H}}_x^{\mu}(\R^n)$ the space of homogeneous functions of order $\mu$ with respect to the variable $x$, smooth with respect to the variable $\xi$. \begin{definition} \label{def:sgclass-a} \begin{itemize} \item[i)]A symbol $a(x, \xi)$ belongs to the class ${S}^{m,\mu}_{\mathrm{cl}(\xi)}(\R^{2n})$ if there exist $a_{m-i, \cdot} (x, \xi)\in \widetilde{\mathscr{H}}_\xi^{m-i}(\R^n)$, $i=0,1,\dots$, such that, for a $0$-excision function $\omega$, \[ a(x, \xi) - \sum_{i=0}^{N-1}\omega(\xi) \, a_{m-i, \cdot} (x, \xi)\in {S}^{m-N, \mu}(\R^n), \quad N=1,2, \ldots; \] \item[ii)]A symbol $a(x, \xi) $ belongs to the class ${S}_{\mathrm{cl}(x)}^{m,\mu}(\R^{2n})$ if there exist $a_{\cdot, \mu-k}(x, \xi)\in \widetilde{\mathscr{H}}_x^{\mu-k}(\R^n)$, $k=0,\,\dots$, such that, for a $0$-excision function $\omega$, \[ a(x, \xi)- \sum_{k=0}^{N-1}\omega(x) \, a_{\cdot, \mu-k}(x,\xi) \in {S}^{m, \mu-N}(\R^n), \quad N=1,2, \ldots \] \end{itemize} \end{definition} \begin{definition} \label{def:sgclass-b} A symbol $a(x,\xi)$ is SG-classical, and we write $a \in {S}_{\mathrm{cl}(x,\xi)}^{m,\mu}(\R^{2n})={S}_{\mathrm{cl}}^{m,\mu}(\R^{2n})={S}_{\mathrm{cl}}^{m,\mu}$, if \begin{itemize} \item[i)] there exist $a_{m-j, \cdot} (x, \xi)\in \widetilde{\mathscr{H}}_\xi^{m-j}(\R^n)$ such that, for a $0$-excision function $\omega$, $\omega(\xi) \, a_{m-j, \cdot} (x, \xi)\in {S}_{\mathrm{cl}(x)}^{m-j, \mu}(\R^n)$ and \[ a(x, \xi)- \sum_{j=0}^{N-1} \omega(\xi) \, a_{m-j, \cdot}(x, \xi) \in {S}^{m-N, \mu}(\R^n), \quad N=1,2,\dots; \] \item[ii)] there exist $a_{\cdot, \mu-k}(x, \xi)\in \widetilde{\mathscr{H}}_x^{\mu-k}(\R^n)$ such that, for a $0$-excision function $\omega$, $\omega(x)\,a_{\cdot, \mu-k}(x, \xi)\in {S}_{\mathrm{cl}(\xi)}^{m, \mu-k}(\R^n)$ and \[ a(x, \xi) - \sum_{k=0}^{N-1} \omega(x) \, a_{\cdot, \mu-k} \in {S}^{m, \mu-N}(\R^n), \quad N=1,2,\dots \] \end{itemize} We set $L_{\mathrm{cl}(x, \xi)}^{m,\mu}(\R^{2n})=L_{\mathrm{cl}}^{m,\mu}(\R^{2n})= {\operatorname{Op}} ({S}^{m,\mu}_{\mathrm{cl}}(\R^{2n}))$. \end{definition} \noindent The next two results are especially useful when dealing with SG-classical symbols. \begin{theorem} \label{thm4.6} Let $a_{k} \in {S}_{\mathrm{cl}}^{m-k,\mu-k}(\R^{2n})$, $k =0,1,\dots$, be a sequence of SG-classical symbols and $a \sim \sum_{k=0}^\infty a_{k}$ its asymptotic sum in the general SG-calculus. Then, $a \in {S}_{\mathrm{cl}}^{m,\mu}(\R^{2n})$. \end{theorem} \begin{theorem} \label{thm4.6.1.1} Let $\mathbb{B}^n= \{ x \in \R^n: |x| \le 1 \}$ and let $\chi$ be a diffeomorphism from the interior of $\mathbb{B}^n$ to $\R^{n}$ such that \[ \chi(x) = \displaystyle \frac{x}{|x|(1-|x|)} \quad \mbox{for} \quad |x| > 2/3. \] Choosing a smooth function $[x]$ on $\R^n$ such that $1 - [x] \not= 0$ for all $x$ in the interior of $\mathbb{B}^n$ and $|x| > 2/3 \Rightarrow [x] = |x|$, for any $a \in {S}^{m,\mu}_{\mathrm{cl}}(\R^{2n})$ denote by $(D^\mathbf{m} a)(y, \eta)$, $\mathbf{m}=(m,\mu)$, the function % \begin{equation} \label{eq4.26.1} b(y,\eta) = (1-[\eta])^{m_{1}} (1-[y])^{m_{2}} a(\chi(y), \chi(\eta)). \end{equation} % Then, $D^\mathbf{m}$ extends to a homeomorphism from ${S}^{m,\mu}_{\mathrm{cl}}(\R^{2n})$ to $C^\infty(\mathbb{B}^n \times \mathbb{B}^n)$. % \end{theorem} \noindent Note that the definition of SG-classical symbol implies a condition of compatibility for the terms of the expansions with respect to $x$ and $\xi$. In fact, defining $\sigma_\psi^{m-j}$ and $\sigma_e^{\mu-i}$ on ${S}_{\mathrm{cl}(\xi)}^{m,\mu}$ and ${S}_{\mathrm{cl}(x)}^{m,\mu}$, respectively, as \begin{align*} \sigma_\psi^{m-j}(a)(x, \xi) &= a_{m-j, \cdot}(x, \xi),\quad j=0, 1, \ldots, \\ \sigma_e^{\mu-i}(a)(x, \xi) &= a_{\cdot, \mu-i}(x, \xi),\quad i=0, 1, \ldots, \end{align*} it possibile to prove that \[ \begin{split} a_{m-j,\mu-i}=\sigma_{\psi e}^{m-j,\mu-i}(a)=\sigma_\psi^{m-j}(\sigma_e^{\mu-i}(a))= \sigma_e^{\mu-i}(\sigma_\psi^{m-j}(a)), \\ j=0,1, \ldots, \; i=0,1, \ldots \end{split} \] Moreover, the algebra property of SG-operators and Theorem \ref{thm4.6} imply that the composition of two SG-classical operators is still classical. For $A= {\operatorname{Op}} (a)\in L^{m,\mu}_\mathrm{cl}$ the triple $\sigma(A)=(\sigma_\psi(A),\sigma_e(A),\sigma_{\psi e}(A))=(a_{m,\cdot}\,,\, a_{\cdot,\mu}\,,\, a_{m,\mu})$ is called the \textit{principal symbol of $A$}. This definition keeps the usual multiplicative behaviour, that is, for any $A\in L^{r,\varrho}_\mathrm{cl}$, $B\in L^{s,\sigma}_\mathrm{cl}$, $r,\varrho,s,\sigma\in\R$, $\sigma(AB)=\sigma(A)\,\sigma(B)$, with componentwise product in the right-hand side. We also set \begin{align*} \Symp{A}(x,\xi) = &\;\,\Symp{a}(x,\xi) =\\ = &\;\;\omega(\xi) a_{m,\cdot}(x,\xi) + \omega(x)(a_{\cdot,\mu}(x,\xi) - \omega(\xi) a_{m,\mu}(x,\xi)) \end{align*} for a $0$-excision function $\omega$. Theorem \ref{thm:ellclass} below allows to express the ellipticity of SG-classical operators in terms of their principal symbol. \begin{theorem} \label{thm:ellclass} An operator $A\in L^{m,\mu}_\mathrm{cl}(\R^{2n})$ is elliptic if and only if each element of the triple $\sigma(A)$ is everywhere non-vanishing on its domain of definition. \end{theorem} \begin{remark} % The composition results in the previous Subsection \ref{sec:fioprelim} have \textit{classical counterparts}. Namely, when all the involved starting elements are SG-classical, the resulting objects (multi-products, amplitudes, etc.) are SG-classical as well. % \end{remark} \section{Commutative law for multi-products of regular SG-phase functions}\label{sec:sgphfcomlaw} \setcounter{equation}{0} % % \par % In this section we prove a commutative law for the multi-products of regular SG phase functions. Through this result, we further expand the theory of SG Fourier integral operators, which we will be able to apply to obtain the solution of Cauchy problems for weakly hyperbolic linear differential operators, with polynomially bounded coefficients, variable multiplicities and involutive characteristics. More precisely, we focus on the $\sharp$-product of regular SG phase functions obtained as solutions to eikonal equations. Namely, let $[\varphi_j(t,s)](x,\xi)=\varphi_j(t,s;x,\xi)$ be the phase functions defined by the eikonal equations \eqref{eik}, with $\varphi_j$ in place of $\varphi$ and $a_j$ in place of $a$, where $a_j \in C([0,T];{S}^{1,1})$, $a_j$ real-valued, $j\in\N$. Moreover, let $I_{\varphi_j}(t,s)=I_{\varphi_j(t,s)}$ be the SG Fourier integral operator with phase function $\varphi_j(t,s)$ and symbol identically equal to $1$. \par Assume that $\{a_j\}_{j\in\N}$ is bounded in $C([0,T];{S}^{1,1})$. Then, by the above Proposition \ref{mainprop}, there exists a constant $c$, independent of $j$, such that \begin{equation*} \varphi_j(t,s)\in {\cal{P}}^\epsilon (c|t-s|),\, j\in\N. \end{equation*} We make a choice of $T_1$, once and for all, such that $cT_1\leq\tau_0$ for the constant $\tau_0$ defined in \eqref{1/4}. Moreover, for convenience below, we define, for $M\in\mathbb{Z}_+$, % \begin{equation}\label{eq:bft} \begin{aligned} \mathbf{t}_{M+1}&=(t_0,\dots,t_{M+1})\in\Delta(T_1), \\[1ex] \mathbf{t}_{M+1,j}(\tau)&=(t_0,\dots,t_{j-1},\tau,t_{j+1},\dots,t_{M+1}), \; 0\le j \le M+1, \end{aligned} \end{equation} where $\Delta(T_1) =\Delta_{M+1}(T_1)= \{(t_0,\dots,t_{M+1})\colon 0\leq t_{M+1}\leq t_{M}\leq\dots\leq t_0\leq T_1\}$. % \par % \subsection{Parameter-dependent multi-products of regular SG phase functions}\label{subsec:parpdepphase} % \par % Let $M\geq 1$ be a fixed integer. We have the following well defined multi-product \begin{equation}\label{sharpprodofphi} \phi(\mathbf{t}_{M+1};x,\xi)=[\varphi_1(t_0,t_1)\sharp\varphi_2(t_1,t_2) \sharp\dots\sharp\varphi_{M+1}(t_M,t_{M+1})](x,\xi), \end{equation} where we will often set $t_0=t$, $t_{M+1}=s$, for $\mathbf{t}_{M+1}\in\Delta(T_1)$ from \eqref{eq:bft}. Explicitly, $\phi$ is defined as in \eqref{3.4}, by means of the critical points $(Y,N)=(Y,N)(\mathbf{t}_{M+1};x,\xi)$, obtained, when $M\ge2$, as solutions of the system \begin{equation}\label{Ct}\begin{cases} Y_1(\mathbf{t}_{M+1};x,\xi) =\varphi'_{1,\xi}(t_0,t_1;x,N_{1}(\mathbf{t}_{M+1};x,\xi)) \\ Y_j(\mathbf{t}_{M+1};x,\xi) =\varphi'_{j,\xi}(t_{j-1},t_j;Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j}(\mathbf{t}_{M+1};x,\xi)), & j=2,\ldots,M \\ N_j(\mathbf{t}_{M+1};x,\xi) =\varphi'_{j+1, x}(t_j,t_{j+1};Y_j(\mathbf{t}_{M+1};x,\xi),N_{j+1}(\mathbf{t}_{M+1};x,\xi)), & j=1,\ldots,M-1 \\ N_M(\mathbf{t}_{M+1};x,\xi) =\varphi'_{M+1, x}(t_M,t_{M+1}; Y_M(\mathbf{t}_{M+1};x,\xi),\xi), \end{cases} \end{equation} namely, \begin{equation}\label{theM+1sharpproductphi} \begin{aligned} \phi&(\mathbf{t}_{M+1};x,\xi):=\\ &\sum_{j=1}^M\left[ \varphi_j(t_{j-1},t_j;Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_j(\mathbf{t}_{M+1};x,\xi)) -Y_j(\mathbf{t}_{M+1};x,\xi)\cdot N_j(\mathbf{t}_{M+1};x,\xi) \right] \\ +&\phantom{\sum_{j=1}^M}\varphi_{M+1}(t_M,t_{M+1};Y_M(\mathbf{t}_{M+1};x,\xi),\xi). \end{aligned} \end{equation} Next, we give some properties of the multi-product $\phi$, whose proofs can be found in \cite{Abdeljawadthesis}. \begin{proposition}\label{prop:tderivofmultiph} Let $\phi$ be the multi-product \eqref{theM+1sharpproductphi}, with real-valued $a_j\in C([0,T];{S}^{1,1}(\R^{2n}))$, $j=1,\dots,M+1$. Then, the following properties hold true. \begin{description} \item[i\label{i.propofsharprod}] \begin{align}\label{tderofmultprodphasefunc} \begin{split} \left\{\begin{array}{cl} \partial_{t_0}\phi(\mathbf{t}_{M+1};x,\xi)&=\phantom{-}a_1(t_0;x,\phi'_x(\mathbf{t}_{M+1};x,\xi))\\ \partial_{t_j}\phi(\mathbf{t}_{M+1};x,\xi)&=\phantom{-}a_{j+1}(t_j;Y_j(\mathbf{t}_{M+1};x,\xi),N_j(\mathbf{t}_{M+1};x,\xi)) \\ & \phantom{=}-a_j(t_j;Y_j(\mathbf{t}_{M+1};x,\xi),N_j(\mathbf{t}_{M+1};x,\xi)),\; j=1,\dots,M\\ \partial_{t_{M+1}}\phi(\mathbf{t}_{M+1};x,\xi)&=-a_{M+1}(t_{M+1},\phi'_\xi(\mathbf{t}_{M+1};x,\xi),\xi). \end{array}\right. \end{split} \end{align} \item[ii\label{ii.propofsharprod}] For any $\varphi_j$ solution to the eikonal equation associated with the Hamiltonian $a_j$, we have \begin{align*} \begin{split} \left\{\begin{array}{cc} \varphi_i(t,s)\sharp\varphi_j(s,s)&=\varphi_i(t,s)\\ \varphi_i(s,s)\sharp\varphi_j(t,s)&=\varphi_j(t,s), \end{array}\right. \end{split} \end{align*} for all $i,j=1,\dots,M+1$. \end{description} \end{proposition} % \begin{proposition}\label{propcritpntbound} Let $\{a_j\}_{j\in\N}$ be a family of parameter-dependent, real-valued symbols, bounded in $C^\infty([0,T];{S}^{1,1}(\R^{2n}))$, and $(Y,N)$ be the solution of \eqref{Ct}. Then, for $\gamma_k\in\mathbb{Z}_+$, $k=0,1\dots,M+1$, the following properties hold true. \begin{description} \item[i\label{bndoncritpnts}] \begin{align}\label{eq:bndcritpnts} \begin{cases} \{\partial_{t_0}^{\gamma_0}\dots\partial_{t_{M+1}}^{\gamma_{M+1}} \left(Y_j-Y_{j-1}\right)\}_{j\in\N}\text{ is bounded in }{S}^{1,0}(\R^{2n}), \\ \{\partial_{t_0}^{\gamma_0}\dots\partial_{t_{M+1}}^{\gamma_{M+1}} \left(N_j-N_{j+1}\right)\}_{j\in\N}\text{ is bounded in }{S}^{0,1}(\R^{2n}). \end{cases} \end{align} \item[ii\label{bndonJfunc}] For $J_{M+1}(\mathbf{t}_{M+1};x,\xi)=\phi(\mathbf{t}_{M+1};x,\xi)-x\cdot\xi$, we have \begin{equation*} \{\partial_{t_0}^{\gamma_0}\dots\partial_{t_{M+1}}^{\gamma_{M+1}}J_{M+1}\} \text{ is bounded in }{S}^{1,1}(\R^{2n}), \end{equation*} \end{description} \end{proposition} \par \begin{corollary} Under the hypotheses of Proposition \ref{propcritpntbound}, we have, for some $T_1\in(0,T]$ as in Proposition \ref{mainprop}, and $j=0,\dots,M+1$, \begin{align*} \begin{cases} Y_j\text{ belongs to }C^\infty\left(\Delta(T_1);{S}^{1,0}(\R^{2n})\right), \\ N_j\text{ belongs to } C^\infty\big(\Delta(T_1);{S}^{0,1}(\R^{2n})\big). \end{cases} \end{align*} \end{corollary} \par For any parameter-dependent, real-valued family $\{a_j\}_{j\in\N}\subset C^\infty([0,T];{S}^{1,1})$, we consider the solutions $(q_j,p_j)(t,s;y,\eta)$ of the Hamilton-Jacobi system \eqref{hameq}, with the Hamiltonian $a_j$ in place of $a$, $j\in\N$. We define the trajectory $(\tilde{q}_{j},\tilde{p}_{j})(\mathbf{t}_{j-1},\sigma;y,\eta)$, for $(y,\eta)\in\R^n\times\R^n$, $\mathbf{t}_{j-1}\in\Delta_{j-1}(T_1)$, $\sigma\in[t_j, t_{j-1}]$, $j\in\N$, by \begin{equation}\label{trajectory} \begin{alignedat}[t]{3} (\tilde{q}_1,\tilde{p}_1)(t_0,\sigma;y,\eta)&=(q_1,p_1)(\sigma,t_0;y,\eta), && t_1\leq\sigma\leq t_0, \\ (\tilde{q}_{j},\tilde{p}_{j})(\mathbf{t}_{j-1},\sigma;y,\eta)&=(q_j,p_j)(\sigma,t_{j-1};(\tilde{q}_{j-1},\tilde{p}_{j-1})(\mathbf{t}_{j-1};y,\eta)), \; && t_j\leq\sigma\leq t_{j-1}, j\geq 2. \end{alignedat} \end{equation} \par \begin{proposition}\label{prop:trajec} Let $(Y,N)=(Y,N)(\mathbf{t}_{M+1};x,\xi)$ be the solution of \eqref{Ct} under the hypotheses of Proposition \ref{propcritpntbound}. Then, we have \arraycolsep1.5pt\begin{eqnarray}\label{trajequal1} \begin{cases} q_1(t_1,t_0;x,\phi'_x(\mathbf{t}_{M+1};x,\xi))=Y_1(\mathbf{t}_{M+1};x,\xi)\\ p_1(t_1,t_0;x,\phi'_x(\mathbf{t}_{M+1};x,\xi))=N_1(\mathbf{t}_{M+1};x,\xi), \end{cases} \end{eqnarray}\arraycolsep5pt \begin{equation}\label{trajequal2} \begin{cases} q_j(t_j,t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1} (\mathbf{t}_{M+1};x,\xi))=Y_j(\mathbf{t}_{M+1};x,\xi) \\[1ex] p_j(t_j,t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1} (\mathbf{t}_{M+1};x,\xi))=N_j(\mathbf{t}_{M+1};x,\xi),\qquad 2\leq j\leq M, \end{cases} \end{equation} and, for any $j\leq M$, \begin{equation}\label{subtaged} (\tilde{q}_{j},\tilde{p}_{j}) (\mathbf{t}_{j};x,\phi'_x(\mathbf{t}_{M+1};x,\xi)) \tag{\theequation$_j$} =(Y_j,N_j)(\mathbf{t}_{M+1};x,\xi). \end{equation} \end{proposition} \par Notice that, from the Propositions \ref{propcritpntbound} and \ref{prop:trajec}, we obtain the following one. \par \begin{proposition}\label{trajclass} Let $\{a_j\}_{j\in\N}$ be a family of parameter-dependent, real-valued symbols, bounded in $C^\infty([0,T];{S}^{1,1}(\R^{2n}))$. Then we have, for the trajectory $\left(\tilde{q}_{j},\tilde{p}_{j}\right) (\mathbf{t}_{j-1},\sigma;y,\eta)$ defined in \eqref{trajectory}, \begin{align*} \begin{cases} \{\partial_{t_0}^{\gamma_0}\dots\partial_{t_{j-1}}^{\gamma_{j-1}} \partial_\sigma^{\gamma_{j}}\tilde{q}_{j}\}_{j\in\N,\,\mathbf{t}_{j-1}\in\Delta_{j-1}(T_1),\, \sigma\in[t_j, t_{j-1}]} \text{ is bounded in }{S}^{1,0}(\R^{2n}), \\[1ex] \{\partial_{t_0}^{\gamma_0}\dots\partial_{t_{j-1}}^{\gamma_{j-1}} \partial_\sigma^{\gamma_{j}}\tilde{p}_{j}\}_{j\in\N,\,\mathbf{t}_{j-1}\in\Delta_{j-1}(T_1),\, \sigma\in[t_j, t_{j-1}]} \text{ is bounded in }{S}^{0,1}(\R^{2n}), \end{cases} \end{align*} where $\gamma_k\in\mathbb{Z}_+$ for $0\leq k\leq j$. \end{proposition} \par \subsection{A useful quasilinear auxiliary PDE}\label{subs:auxpde} Consider the following quasi-linear partial differential equation \begin{align}\label{quasi} \begin{cases} \partial_{t_{j-1}} \Upsilon(\mathbf{t}_{M+1})-H(\Upsilon(\mathbf{t}_{M+1}),\mathbf{t}_{M+1}) \cdot \Upsilon^\prime_x(\mathbf{t}_{M+1})-G(\mathbf{t}_{M+1,j}(\Upsilon(\mathbf{t}_{M+1})))=0, \\[1ex] \Upsilon|_{t_{j-1}=t_j}=t_{j+1} \end{cases} \end{align} where, for $s\in \R$, $\mathbf{t}_{M+1,j}(s)$ is defined in \eqref{eq:bft}, $\Upsilon(\mathbf{t}_{M+1})=\Upsilon(\textbf{t}_{M+1};x,\xi)\in C^\infty(\Delta(T_1);{S}^{0,0})$, and $H(\tau,\mathbf{t}_{M+1})=L(\tau,\textbf{t}_{M+1};x,\xi)$ is a vector-valued family of symbols of order $(1,0)$ such that $H\in C^\infty([t_{j+1},t_{j-1}]\times\Delta({T_1});{S}^{1,0}\otimes\R^n)$. We also assume that $G(\mathbf{t}_{M+1,j}(\tau))=G(\textbf{t}_{M+1,j}(\tau);x,\xi)$ belongs to $C^\infty(\Delta(T_1);{S}^{0,0})$, and satisfies \[ G(\mathbf{t}_{M+1};x,\xi)>0, \quad G(\mathbf{t}_{M+1};x,\xi)|_{t_{j}=t_{j-1}}\equiv 1, \] for any $\mathbf{t}_{M+1}\in\Delta(T_1)$, $(x,\xi)\in\R^{2n}$. \par The following Lemma \ref{lem:exitOfSolForOrdin} (cf. \cite{Taniguchi02}) is the key step to prove our first main result, Theorem \ref{cruthm}. In fact, it gives the solution of the characteristics system \begin{align}\label{ordin} \left\{\begin{array}{ll} &\partial_{t_{j-1}} R(\mathbf{t}_{M+1})= -H(K(\mathbf{t}_{M+1}),\mathbf{t}_{M+1};R(\mathbf{t}_{M+1}),\xi) \\[1ex] &\partial_{t_{j-1}}K(\mathbf{t}_{M+1})=\phantom{-}G(\mathbf{t}_{M+1,j} (K(\mathbf{t}_{M+1}));R(\mathbf{t}_{M+1}),\xi) \\[1ex] &R|_{t_{j-1}=t_j}=y,\,K|_{t_{j-1}=t_j}=t_{j+1}, \end{array} \right. \end{align} which then easily provides the solution to the quasi-linear equation \eqref{quasi}. The latter, in turn, is useful to simplify the computations in the proof ot Theorem \ref{cruthm}. In view of this, we then give a mostly complete proof of Lemma \ref{lem:exitOfSolForOrdin}. \par \begin{lemma}\label{lem:exitOfSolForOrdin} There exists a constant $T_2\in(0, T_1]$ such that \eqref{ordin} admits a unique solution $(R,K)=(R,K)(\mathbf{t}_{M+1};y,\xi)\in C^\infty(\Delta(T_2);({S}^{1,0}(\R^{2n})\otimes\R^n)\times {S}^{0,0}(\R^{2n}))$, $t_{j-1}\in[t_j,T_2]$, which satisfies, for any $\mathbf{t}_{M+1}\in\Delta(T_2)$, $(y,\xi)\in\R^{2n}$, \begin{equation}\label{estODE} \left\|\frac{\partial R}{\partial y}(\mathbf{t}_{M+1};y,\xi)-I \right\| \leq C(t_{j-1}-t_{j+1}), \end{equation} for a suitable constant $C>0$ independent of $M$, and \begin{align}\label{equivODE} \left\{\begin{array}{ll} &t_{j+1}\leq K(\mathbf{t}_{M+1};y,\xi) \leq t_{j-1} \\[1ex] &K_{|t_{j-1}=t_{j}}=t_{j+1}. \end{array} \right. \end{align} \end{lemma} \par \begin{proof} First, we notice that, as a consequence of Lemma \ref{lem:invariantcomsymbol}, the compositions in the right-hand side of \eqref{ordin} are well defined, and produce symbols of order $(1,0)$ and $(0,0)$, respectively, provided that $(R,K)\in C^\infty(\Delta(T_2);{S}^{1,0}(\R^{2n})\times {S}^{0,0}(\R^{2n}))$ and $K(\mathbf{t}_{M+1})\in[t_{j+1},t_{j-1}]$ for any $\mathbf{t}_{M+1}\in\Delta(T_2)$. \par We focus only on the variables $(t_{j-1},t_j,t_{j+1};y,\xi)$, since all the others here play the role of (fixed) parameters, on which the solution clearly depends smoothly. We then omit them in the next computations. We will also write, to shorten some of the formulae, $(R,K)(s)=(R,K)(\mathbf{t}_{{M+1},{j-1}}(s);y,\xi)$, $s\in[t_j,T_2]$, $T_2\in(0,T_1]$ sufficiently small, to be determined below, $\mathbf{t}_{M+1}\in\Delta(T_2)$, $(y,\xi)\in\R^{2n}$. We rewrite \eqref{ordin} in integral form, namely \begin{align}\label{equiordin} \left\{\begin{array}{ll} & R(s)=y-{\displaystyle \int_{t_j}^{s}}H(K(\sigma);\sigma,t_j,t_{j+1};R(\sigma),\xi)\,d\sigma \\[1ex] &K(s)=t_{j+1}+{\displaystyle\int_{t_j}^{s}}G(\sigma,K(\sigma),t_{j+1};R(\sigma),\xi)\,d\sigma, \\[1ex] \end{array} \right. \end{align} $s\in[t_j,T_2]$, $\mathbf{t}_{M+1}\in\Delta(T_2)$, $(y,\xi)\in\R^{2n}$, and solve \eqref{equiordin} by the customary Picard method of successive approximations. That is, we define the sequences \begin{align}\label{PicSequence} \begin{cases} &R_{l+1}(s)=y-{\displaystyle \int _{t_j}^{s}}H(K_l(\sigma);\sigma,t_j,t_{j+1};R_l(\sigma),\xi)\,d\sigma \\[1ex] &K_{l+1}(s)=t_{j+1}+{\displaystyle \int _{t_j}^{s}}G(\sigma,K_l(\sigma),t_{j+1};R_l(\sigma),\xi)\,d\sigma, \end{cases} \end{align} for $l=1,2,\dots$, $s\in[t_j,T_2]$, $\mathbf{t}_{M+1}\in\Delta(T_2)$, $(y,\xi)\in\R^{2n}$, with \begin{align* R_0(s)=y,\qquad K_0(s)=s-t_{j}+t_{j+1}. \end{align*} \par We start by showing that $\{R_{l}\}_l$ and $\{K_l\}_l$ are bounded in $ C^\infty(\Delta(T_2);{S}^{1,0})$ and $ C^\infty(\Delta(T_2);{S}^{0,0})$, respectively, and that \eqref{estODE} and \eqref{equivODE} hold true for $R_l$ and $K_l$ in place of $R$ and $K$, respectively, for any $l\in\mathbb{Z}_+$, uniformly with respect to $l,j,M$. This follows by induction. Namely, notice that all the stated properties are true for $l=0$. Indeed, it is clear that $R_0\in C^\infty(\Delta(T_2);{S}^{1,0})$ and $K_0\in C^\infty(\Delta(T_2);{S}^{0,0})$, with seminorms bounded by $\max\{1,2T_2\}$. \eqref{estODE} with $R_0$ in place of $R$ is trivial, while \eqref{equivODE} with $K_0$ in place of $K$ follows immediatly, by inserting $s=t_{j-1}$ in $K_0(s)$ and recalling that $\mathbf{t}_{M+1}\in\Delta(T_2)\Rightarrow t_{j-1}-t_{j}+t_{j+1}\in[t_{j+1},t_{j-1}]$. Assume now that \eqref{estODE} and \eqref{equivODE} hold true for $(R_\ell,K_\ell)$ for all the values of the index $\ell$ up to $l\ge0$. We then find, by the same composition argument mentioned above, $R_{l+1}\in C^\infty(\Delta(T_2);{S}^{1,0})$ and $K_{l+1}\in C^\infty(\Delta(T_2);{S}^{0,0})$, with seminorms uniformly bounded with respect to $l$, since they depend only on the seminorms of $L$, $H$, $R_l$, $K_l$, and $T_2$. It follows that \eqref{estODE} holds true also for $R_{l+1}$ in place of $R$, since \begin{align*} \left\|\frac{\partial R_{l+1}}{\partial y}(\mathbf{t}_{M+1};y,\xi)-I \right\| &= \left\| \int _{t_j}^{t_{j-1}}\frac{\partial}{\partial y} \left[H(K_l(\sigma);\sigma,t_j,t_{j+1};R_l(\sigma),\xi)\right]d\sigma \right\| \\ &\le C(t_{j-1}-t_{j}) \le C(t_{j-1}-t_{j+1}), \end{align*} for a suitable constant $C>0$ independent of $l$. Of course, $K_{l+1}(\mathbf{t}_{M+1};y,\xi)|_{t_{j-1}=t_{j}}=t_{j+1}$ is clear, by the definition of $K_{l+1}$, $l\ge0$. It is also immediate that the hypotheses and the definition of $K_{l+1}$, $l\ge0$, imply that $K_{l+1}(\mathbf{t}_{M+1};y,\xi)$ is, for any fixed $(y,\xi)\in\R^{2n}$, $0\le t_{M+1}\le \cdots t_{j+1}\le t_{j-1}\le \cdots t_0\le T_2\le T_1$, a monotonically decreasing function with respect to $t_j\in[t_{j+1},t_{j-1}]$. \eqref{equivODE} with $K_{l+1}$ in place of $K$, $l\ge0$, follows by such property and the hypotheses on $G$, observing that, for any $l\ge0$, $\mathbf{t}_{M+1}\in\Delta(T_2)$, $(y,\xi)\in\R^{2n}$, $s\in[t_{j+1},t_{j-1}]$, \begin{equation}\label{eq:extrKl} K_{l+1}(\mathbf{t}_{M+1,j-1}(s);y,\xi)|_{t_j=t_{j+1}}=s, \quad K_{l+1}(\mathbf{t}_{M+1,j-1}(s);y,\xi)|_{t_j=s}=t_{j+1}, \end{equation} which, in particular, also shows \begin{equation}\label{eq:extrKlbis} K_{l+1}(\mathbf{t}_{M+1};y,\xi)|_{t_j=t_{j+1}}=t_{j-1}, \quad K_{l+1}(\mathbf{t}_{M+1};y,\xi)|_{t_j=t_{j-1}}=t_{j+1}. \end{equation} Notice that \eqref{eq:extrKlbis}, together with the monotonicity property of $K_{l+1}(\mathbf{t}_{M+1};y,\xi)$ with respect to $t_j\in[t_{j+1},t_{j-1}]$, $l\ge0$, $\mathbf{t}_{M+1}\in\Delta(T_2)$, $(y,\xi)\in\R^{2n}$, completes the proof of \eqref{equivODE} with $K_{l+1}$ in place of $K$ and the argument. Then, it just remains to prove \eqref{eq:extrKl}. We again proceed by induction. \eqref{eq:extrKl} is manifestly true for $K_0$. Assume then that it holds true for $K_\ell$ for all the values of the index $\ell$ up to $l\ge0$. We find, in view of the hypotheses on $G$ and the inductive hypothesis, \begin{align*} K_{l+1}(\mathbf{t}_{M+1,j-1}(s);&y,\xi)|_{t_j=t_{j+1}} \\ &=t_{j+1}+\int_{t_{j+1}}^s G(\sigma, K_l(\mathbf{t}_{M+1,j-1}(\sigma),t_{j+1},t_{j+1};y,\xi),R_l(\sigma),\xi)\,d\sigma \\ &=t_{j+1}+\int_{t_{j+1}}^s G(\sigma,\sigma,t_{j+1};R_l(\sigma),\xi)\,d\sigma=t_{j+1}+\int_{t_{j+1}}^s d\sigma=s, \\ K_{l+1}(\mathbf{t}_{M+1,j-1}(s);&y,\xi)|_{t_j=s} \\ &=t_{j+1}+\int_{s}^s G(\sigma, K_l(\mathbf{t}_{M+1,j-1}(\sigma),s,t_{j+1};y,\xi),R_l(\sigma),\xi)\,d\sigma=t_{j+1}, \end{align*} which completes the proof of \eqref{eq:extrKl}. \par In order to show that $\{R_l\}$ and $\{K_l\}$ converge, we employ Taylor formula with respect to the variable $t_{j-1}$. For an arbitrary $N\in\mathbb{Z}_+$ we can write \begin{align}\label{tayforKl} K_{l+1}(t_{j-1})-K_l(t_{j-1})&= \sum\limits_{k<N}\frac{\left(\left(\partial_{t_{j-1}}^kK_{l+1}\right)(t_j)- \left(\partial_{t_{j-1}}^kK_{l}\right)(t_j)\right)(t_{j-1}-t_j)^k}{k!} \\[1ex] &+\frac{1}{N!}\int_{t_j}^{t_{j-1}}(t_{j-1}-\sigma)^N \left(\left(\partial_{t_{j-1}}^{N+1}K_{l+1}\right)(\sigma)- \left(\partial_{t_{j-1}}^{N+1}K_{l}\right)(\sigma)\right)d\sigma\nonumber \end{align} and \begin{align}\label{tayforRl} R_{l+1}(t_{j-1})-R_l(t_{j-1})&= \sum\limits_{k<N}\frac{\left(\left(\partial_{t_{j-1}}^kR_{l+1}\right)(t_j)- \left(\partial_{t_{j-1}}^kR_{l}\right)(t_j)\right)(t_{j-1}-t_j)^k}{k!} \\[1ex] &+\frac{1}{N!}\int_{t_j}^{t_{j-1}}(t_{j-1}-\sigma)^N \left(\left(\partial_{t_{j-1}}^{N+1}R_{l+1}\right)(\sigma)- \left(\partial_{t_{j-1}}^{N+1}R_{l}\right)(\sigma)\right)d\sigma,\nonumber \end{align} respectively. The summations in the above equalities \eqref{tayforKl} and \eqref{tayforRl} are actually identically vanishing. To prove this assertion, one proceeds by induction on $N$, employing a standard argument involving the Fa\'a di Bruno formula for the derivatives of composed functions (see \cite{Taniguchi02} and the references quoted therein for more details). \par Now, using standard inequalities for the remainder, together with the fact that $\{R_l\}_l$ is bounded in $C^\infty(\Delta(T_2);{S}^{1,0}\otimes\R^n)$, while $\{K_l\}_l$ is bounded in $C^\infty(\Delta(T_2);{S}^{0,0})$, from \eqref{tayforKl} and \eqref{tayforRl} we get, for any ${\alpha},{\beta}\in \mathbb{Z}_+$, \begin{multline}\label{K_lEstim} \sup_{(y,\xi)\in\R^{2n}}\left|\partial_y^{\alpha} \partial_\xi^{\beta} \left(K_{l+1}(\mathbf{t}_{{M+1},{j-1}}(t_{j-1});y,\xi)-K_l(\mathbf{t}_{{M+1},{j-1}} (t_{j-1});y,\xi)\right) \jb{y}^{|{\alpha}|}\jb{\xi}^{|{\beta}|}\right| \\[1ex] \leq C_{\alpha\beta}\frac{\left(t_{j-1}-t_j\right)^{N+1}}{(N+1)!}, \end{multline} with $C_{\alpha\beta}$ independent of $j,\,N$, and, similarly, \begin{multline}\label{R_lEstim} \sup_{(y,\xi)\in\R^{2n}}\left|\partial_y^{\alpha} \partial_\xi^{\beta} \left(R_{l+1}(\mathbf{t}_{{M+1},{j-1}}(t_{j-1});y,\xi)-R_l(\mathbf{t}_{{M+1},{j-1}}(t_{j-1});y,\xi)\right) \jb{y}^{-1+|{\alpha}|}\jb{\xi}^{|{\beta}|}\right| \\[1ex] \leq \widetilde{C}_{\alpha\beta}\frac{\left(t_{j-1}-t_j\right)^{N+1}}{(N+1)!}. \end{multline} where $\widetilde{C}_{\alpha\beta}$ is independent of $j,\,N$. Writing $l$ in place of $N$ in the right-hand side of \eqref{K_lEstim} and \eqref{R_lEstim}, it easily follows that $(R_l,K_l)$ converges, for $l\to+\infty$, to a unique fixed point $(R,K)$, which satisfies the stated symbol estimates. Since, as we showed above, the properties \eqref{estODE} and \eqref{equivODE} hold true for $(R_l,K_l)$ in place of $(R,K)$, $l\ge0$, uniformly with respect to $M,j,l$, they also hold true for the limit $(R,K)$. The proof is complete. \end{proof} The next Corollary \ref{cor:qlcpsol} is a standard result in the theory of Cauchy problems for quasilinear PDEs of the form \eqref{quasi}. Its proof is based on the hypotheses on $L$ and $H$, and the properties of the solution of \eqref{ordin}. \begin{corollary}\label{cor:qlcpsol} % Under the same hypotheses of Lemma \ref{lem:exitOfSolForOrdin}, denoting by $\bar{R}(\mathbf{t}_{M+1};x,\xi)$ the solution of the equation % \[ R(\mathbf{t}_{M+1};y,\xi)=x, \quad \mathbf{t}_{M+1}\in\Delta(T_2),x,\xi\in\R^n, \] % the function % \[ \Upsilon(\mathbf{t}_{M+1}; x,\xi)=K(\mathbf{t}_{M+1};\bar{R}(\mathbf{t}_{M+1};x,\xi),\xi) \] % solves the Cauchy problem \eqref{quasi} for $x,\xi\in\R^n$, $\mathbf{t}_{M+1}\in\Delta(T_2)$, for a sufficiently small $T_2\in(0,T_1]$. % \end{corollary} \begin{remark}\label{rem:invJR} % Notice that \eqref{estODE} implies that, for all $y,\xi\in\R^n$, $\mathbf{t}_{M+1}\in\Delta(T_2)$, $T_2\in(0,T_1]$ suitably small, the Jacobian matrix $\dfrac{\partial R}{\partial y}(\mathbf{t}_{M+1};y,\xi)$ belongs to a suitably small open neighbourhood of the identity matrix, so it is invertible, with norm in an interval of the form $[1-\varepsilon,1+\varepsilon]$, for positive, arbitrarily small $\varepsilon$. A standard argument in the SG symbol theory (see, e.g., \cite{CO, Coriasco:998.1, Cothesis}), then shows $\bar{R}\in C^\infty(\Delta(T_2),{S}^{1,0}(\R^{2n})\otimes\R^n)$ and % \[ \jb{{R}(\mathbf{t}_{M+1};y,\xi)}\sim\jb{y}, \quad \jb{\bar{R}(\mathbf{t}_{M+1};x,\xi)}\sim\langle x\rangle, \] % with constants independent of $\mathbf{t}_{M+1}\in\Delta(T_2)$, $\xi\in\R^n$. This also implies that $\Upsilon$ satisfies $t_{j+1}\le \Upsilon(\mathbf{t}_{M+1}; x,\xi)\le t_{j-1}$, $\mathbf{t}_{M+1}\in\Delta(T_2)$, $x,\xi\in\R^n$ and $\Upsilon\in C^\infty(\Delta(T_2); {S}^{0,0}(\R^{2n}))$. % \end{remark} \subsection{Commutative law for multi-products of SG phase functions given by solutions of eikonal equations}\label{subsec:commlaw} Let $\{a_j\}_{j\in\N}$ be a bounded family of parameter-dependent, real-valued symbols in $C^\infty([0,T];{S}^{1,1})$ and let $\{\varphi_j\}_{j\in\N}$ be the corresponding family of phase functions in $ {\cal{P}}^\epsilon (c|t-s|)$, obtained as solutions to the eikonal equations associated with $a_j$, $j\in\N$. In the aforementioned multi-product \eqref{sharpprodofphi}, we commute $\varphi_j$ and $\varphi_{j+1}$, defining a new multi-product $\phi_j$, namely \begin{equation}\label{comsharpmultprod} \begin{aligned} \phi_j(\mathbf{t}_{M+1};x,\xi)=\left(\varphi_1(t_0,t_1)\right.&\!\!\left.\sharp\varphi_2(t_1,t_2) \sharp\dots\sharp\varphi_{j-1}(t_{j-2},t_{j-1})\sharp\right. \\[1ex] &\!\!\left.\sharp\varphi_{j+1}(t_{j-1},t_j)\sharp\varphi_j(t_j,t_{j+1})\sharp\right. \\ &\!\!\left. \sharp\varphi_{j+2}(t_{j+1},t_{j+2})\sharp\dots\sharp \varphi_{M+1}(t_M,t_{M+1})\right)(x,\xi), \end{aligned} \end{equation} where $\mathbf{t}_{M+1}=(t_0,t_1,\dots,\dots t_{M+1})\in \Delta(T_1)$. \par \begin{assumption}[Involutiveness of symbol families]\label{assumpt:invo} Given the family of parameter-dependent, real-valued symbols $\{a_j\}_{j\in\N}\subset C^\infty([0,T];{S}^{1,1}(\R^{2n})$, there exist families of parameter-dependent, real-valued symbols $\{b_{jk}\}_{j,k\in\N}$ and $\{d_{jk}\}_{j,k\in\N}$, such that $b_{jk},d_{jk}\in C^\infty([0,T];{S}^{0,0}(\R^{2n}))$, $j,k\in\N$, and the Poisson brackets \begin{align*} \{\tau-a_j(t;x,\xi),\tau-a_{k}(t;x,\xi)\} &:=\partial_ta_j(t;x,\xi)-\partial_ta_{k}(t;x,\xi) \\ &\phantom{:}+ a'_{j,\xi}(t;x,\xi) \cdot a'_{k,x}(t;x,\xi)-a'_{j,x}(t;x,\xi)\cdot a'_{k,\xi}(t;x,\xi) \end{align*} satisfy \begin{equation}\label{pbrack} \{\tau-a_j(t;x,\xi),\tau-a_{k}(t;x,\xi)\} = b_{jk}(t;x,\xi)\cdot(a_j-a_{k})(t;x,\xi)+d_{jk}(t;x,\xi), \end{equation} for all $j,k\in\N$, $t\in[0,T]$, $x,\xi\in\R^n$. \end{assumption} We can now state our first main theorem. \par \begin{theorem}\label{cruthm} Let $\{a_j\}_{j\in\N}$ be a family of parameter-dependent, real-valued symbols, bounded in $C^\infty\left([0,T];{S}^{1,1}(\R^{2n})\right)$, and let $\varphi_j\in {\cal{P}}^\epsilon (c|t-s|)$, for some $c>0$, be the phase functions obtained as solutions to \eqref{eik} with $a_j$ in place of $a$, $j\in\N$. Consider $\phi(\mathbf{t}_{M+1})$ and $\phi_j(\mathbf{t}_{M+1})$ defined in \eqref{sharpprodofphi} and \eqref{comsharpmultprod}, respectively, for any $M\ge2$ and $j\leq M$. Then, Assumption \ref{assumpt:invo} implies that there exists $T^\prime\in(0,T_2]$, independent of $M$, such that we can find a symbol family $Z_j\in C^\infty(\Delta(T');{S}^{0,0}(\R^{2n}))$ satisfying, for all $\mathbf{t}_{M+1}\in\Delta(T^\prime)$, $x,\xi\in\R^n$, \begin{equation}\label{eq:Zprop1} \begin{aligned} &t_{j+1}\leq Z_j(\mathbf{t}_{M+1};x,\xi)\leq t_{j-1}, \\[1ex] &Z_j|_{{t_j=t_{j-1}}}=t_{j+1},\text{ and } Z_j|_{{t_j=t_{j+1}}}=t_{j-1}, \\[1ex] &\partial_{t_j}Z_j(\mathbf{t}_{M+1};x,\xi)\asymp -1. \end{aligned} \end{equation} Moreover, we have \begin{equation* \phi_j(\mathbf{t}_{M+1};x,\xi)=\phi(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi) +\Psi_j(\mathbf{t}_{M+1};x,\xi), \mathbf{t}_{M+1}\in\Delta(T^\prime),x,\xi\in\R^n, \end{equation*} where $\Psi_j \in C^\infty(\Delta(T');{S}^{0,0}(\R^{2n}))$ satisfies \begin{equation* \Psi_j\equiv 0\quad\text{ if }\quad d_j\equiv 0. \end{equation*} \end{theorem} \par The argument originally given in \cite{Taniguchi02} extends to the SG setting, in view of Lemma \ref{lem:exitOfSolForOrdin} above, and allows to prove Theorem \ref{cruthm}. For the sake of completeness, we include it here (see \cite{Abdeljawadthesis} for additional details). \par \begin{proof}[Proof of Theorem \ref{cruthm}] Let $\{(Y_1,\dots,Y_M,N_1,\dots,N_M)\}(\mathbf{t}_{M+1};x,\xi)$ be the solution of the critical point system \begin{align*}\begin{cases} x_j&=\varphi'_{j,\xi}(t_{j-1},t_j;x_{j-1},\xi_j) \\[1ex] \xi_j&=\varphi'_{j+1,x}(t_j,t_{j+1};x_j,\xi_{j+1}), \end{cases} \end{align*} such that $x_j,\xi_j\in\R^n$, $x_0=x$, and $\xi_{M+1}=\xi$ (cf. \cite{Abdeljawadthesis,AsCor,Kumano-go:1}). \par In view of \eqref{comsharpmultprod}, let $\left(\tilde{Y}_1, \dots,\tilde{Y}_M,\tilde{N}_1,\dots,\tilde{N}_M\right)(\mathbf{t}_{M+1};x,\xi)$ be the solution to the critical points problem, for the phase functions in modified order, namely, \begin{align*} \begin{cases} x_k&=\varphi'_{k,\xi}(t_{k-1},t_k;x_{k-1},\xi_k)\qquad\text{ if }k\in\{1,\dots,j-1,j+2,\dots,M\} \\[1ex] x_j&=\varphi'_{j+1,\xi}(t_{j-1},t_j;x_j,\xi_j), \\[1ex] x_{j+1}&=\varphi'_{j,\xi}(t_j,t_{j+1};x_j,\xi_{j+1}), \end{cases} \end{align*} \par \begin{align*} \begin{cases} \xi_k&=\varphi'_{k+1,x}(t_k,t_{k+1};x_k,\xi_{k+1})\qquad\text{ if }k\in\{1,\dots,j-2,j+1,\dots,M\} \\[1ex] \xi_{j-1}&=\varphi'_{j+1,x}(t_{j-1},t_j;x_{j-1},\xi_j), \\[1ex] \xi_j&=\varphi'_{j,x}(t_j,t_{j+1};x_j,\xi_{j+1}), \end{cases} \end{align*} where $x_0=x$ and $\xi_{M+1}=\xi$. For convenience below, we also set \begin{align*} \begin{cases} a_0(t;x,\xi)&\equiv 0, \\[1ex] Y_0=\tilde{Y}_0&=x, \\[1ex] N_0=\phi'_x,\;&\tilde{N}_0=\phi'_{j,x}. \end{cases} \end{align*} \par Let $\Psi_j$ be defined as \begin{equation}\label{Psidef} \Psi_j(\mathbf{t}_{M+1};x,\xi)=\phi_j(\mathbf{t}_{M+1};x,\xi) -\phi(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi). \end{equation} Here we look for a symbol $Z_j = Z_j(\mathbf{t}_{M+1};x,\xi)$ satisfying \eqref{eq:Zprop1} such that $\Psi_j\in C^\infty(\Delta(T^\prime);{S}^{0,0})$, $T^\prime\in(0,T_2]$. In view of Proposition \ref{prop:tderivofmultiph} and \eqref{comsharpmultprod}, we find \begin{equation}\label{PsiTDer} \begin{aligned} \partial_{t_{j-1}}\Psi_j(\mathbf{t}_{M+1};x,\xi)&=(\partial_{t_{j-1}}\phi_j)(\mathbf{t}_{M+1};x,\xi) -(\partial_{t_{j-1}}\phi)(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi) \\[1ex] &-(\partial_{t_j}\phi)(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi)\cdot\partial_{t_{j-1}}Z_j(\mathbf{t}_{M+1};x,\xi) \\[1ex] &=a_{j+1}(t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi)) \\[1ex] &-a_{j-1}(t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi)) \\[1ex] &-\left[a_j(t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi))\right]_{t_{j}=Z_j(\mathbf{t}_{M+1};x,\xi)} \\[1ex] &+\left[a_{j-1}(t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi))\right]_{t_{j}=Z_j(\mathbf{t}_{M+1};x,\xi)} \\[1ex] &-(\partial_{t_j}\phi)(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi)\cdot\partial_{t_{j-1}}Z_j(\mathbf{t}_{M+1};x,\xi). \end{aligned} \end{equation} % \par % When $j\geq 2$, we use the trajectory $\left(\tilde{q}_{j-1},\tilde{p}_{j-1}\right)(\mathbf{t}_{j-2},\sigma;y,\eta)$ defined in \eqref{trajectory}. Then (cf. Proposition \ref{prop:trajec}), we have, for $\sigma=t_{j-1}$, the equalities \begin{align*} \begin{cases} (\tilde{q}_{j-1},\tilde{p}_{j-1})(\mathbf{t}_{j-1};x,\phi'_x(\mathbf{t}_{M+1};x,\xi))&=(Y_{j-1},N_{j-1})(\mathbf{t}_{M+1};x,\xi), \\[1ex] (\tilde{q}_{j-1},\tilde{p}_{j-1})(\mathbf{t}_{j-1};x,\phi'_{j,x}(\mathbf{t}_{M+1};x,\xi))&=(\tilde{Y}_{j-1},\tilde{N}_{j-1})(\mathbf{t}_{M+1};x,\xi). \end{cases} \end{align*} % \par % Next, we set \begin{align}\label{eq:alphaTrajectory} \left\{\begin{array}{rl} {\alpha}_1(\sigma;z,\zeta)&=a_2(\sigma;z,\zeta), \\[1ex] {\alpha}_j(\sigma;\mathbf{t}_{j-2};z,\zeta)&=\left(a_{j+1}-a_{j-1}\right) \big(\sigma;\left(\tilde{q}_{j-1},\tilde{p}_{j-1}\right) (\mathbf{t}_{j-2},\sigma;z,\zeta)\big),\; j\geq 2. \end{array} \right. \end{align} In \eqref{eq:alphaTrajectory} the compositions are well defined, in view of the properties of the symbols $(\tilde{q}_{j-1},\tilde{p}_{j-1})$ in Proposition \ref{trajclass}, which imply that the conditions of Lemma \ref{lem:invariantcomsymbol} are satisfied. Thus, ${\alpha}_j\in C^\infty(\Delta_{j-1}(T_1);{S}^{1,1})$, $j=1,\dots,M$. Moreover, ${\alpha}_j$ satisfies \begin{equation}\label{eq:alphaDef} \begin{cases} {\alpha}_j(\mathbf{t}_{j-2},\sigma;x,\phi'_x(\mathbf{t}_{M+1};x,\xi))\!\!\!\!\!&=(a_{j+1}-a_{j-1})\left(\sigma;Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi)\right), \\[1ex] {\alpha}_j(\mathbf{t}_{j-2},\sigma;x,\phi'_{j,x}(\mathbf{t}_{M+1};x,\xi))\!\!\!\!\!&=(a_{j+1}-a_{j-1})(\sigma;\tilde{Y}_{j-1}(\mathbf{t}_{M+1};x,\xi),\tilde{N}_{j-1}(\mathbf{t}_{M+1};x,\xi)), \end{cases} \end{equation} and when $j=1$ the variables $(\mathbf{t}_{j-2},\sigma)$ reduce to $\sigma$. % \par % Finally, let $[T_j(\tau)](\mathbf{t}_{M+1};x,\xi)= T_j(\tau,\mathbf{t}_{M+1};x,\xi)$ be defined as \[ T_j(\tau)=\int_{0}^{1}{\alpha}'_{j,\xi}\bigg(\mathbf{t}_{j-1};x,\rho\phi'_{j,x} (\mathbf{t}_{M+1};x,\xi)+(1-\rho) \phi'_x(\mathbf{t}_{M+1,j} (\tau);x,\xi)\bigg)\,d\rho. \] Notice that, again by Lemma \ref{lem:invariantcomsymbol} and the properties of the involved symbols, we find $T_j\in C^\infty([t_{j+1},t_{j-1}]\times\Delta(T_1);{S}^{1,0}\otimes\R^n)$. Indeed, in view of the fact that both $\phi$ and $\phi_j$ are regular SG phase functions, for all $\rho\in[0,1]$, $\tau\in[t_{j+1},t_{j-1}]$, % \[ \jb{ \rho\phi'_{j,x} (\mathbf{t}_{M+1};x,\xi)+(1-\rho) \phi'_x(\mathbf{t}_{M+1,j} (\tau);x,\xi)}\sim\langle \xi \rangle, \] % uniformly with respect to all the involved parameters. % \par % We now show that $\phi_j$ satisfies a certain PDE, whose form we will simplify using the results in Subsection \ref{subs:auxpde}. First of all, we observe that \begin{equation}\label{eq:alphaDifference} \begin{aligned} {\alpha}_j(\mathbf t_{j-1};&x,\phi'_{j,x}(\mathbf t_{M+1};x,\xi))- {\alpha}_j(\mathbf t_{j-1};x,\phi'_x(\mathbf t_{M+1,j}(\tau);x,\xi)) \\[1ex] &=T_j(\tau)\cdot\bigg(\phi'_{j,x}(\mathbf t_{M+1};x,\xi)- \phi'_x(\mathbf t_{M+1,j}(\tau);x,\xi)\bigg). \end{aligned} \end{equation} From \eqref{Psidef} it follows \begin{equation}\label{PsiXDer} \Psi'_{j,x}(\mathbf{t}_{M+1};x,\xi)=\phi'_{j,x}(\mathbf t_{M+1};x,\xi)- (\partial_{t_j}\phi)(\mathbf t_{M+1,j}(Z_j);x,\xi)\cdot Z'_{j,x}(\mathbf{t}_{M+1};x,\xi). \end{equation} % \par % Now, we rewrite $\partial_{t_{j-1}}\Psi_j$ from \eqref{PsiTDer}, using \eqref{eq:alphaDef}, \eqref{eq:alphaDifference} and \eqref{PsiXDer}: \begin{equation}\label{newPsiTDer} \begin{aligned} \partial_{t_{j-1}}\Psi_j(\mathbf{t}_{M+1};x,\xi)&={\alpha}_j\left(\mathbf t_{j-1};x,\phi'_{j,x}(\mathbf t_{M+1};x,\xi)\right) \\[1ex] &-{\alpha}_j\left(\mathbf t_{j-1};x,\phi'_x(\mathbf t_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi)\right) \\[1ex] &-(\partial_{t_{j}}\phi)(\mathbf t_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi)\cdot(\partial_{t_{j-1}}Z_j)(\mathbf{t}_{M+1};x,\xi) \\[1ex] &-\left[\left(a_j-a_{j+1}\right)(t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi))\right]_{t_j=Z_j(\mathbf{t}_{M+1};x,\xi)} \\[1ex] &=T_j(Z_j(\mathbf{t}_{M+1};x,\xi))\cdot\Psi'_{j,x}(\mathbf{t}_{M+1};x,\xi) \\[1ex] &-(\partial_{t_j}\phi)(\mathbf t_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi)\cdot \\[1ex] &\cdot\left(\partial_{t_{j-1}}Z_j(\mathbf{t}_{M+1};x,\xi)-T_j(Z_j(\mathbf{t}_{M+1};x,\xi))\cdot Z'_{j,x}(\mathbf{t}_{M+1};x,\xi))\right) \\[1ex] &- \left[\left(a_j-a_{j+1}\right)(t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi))\right]_{t_j=Z_j(\mathbf{t}_{M+1};x,\xi)}. \end{aligned} \end{equation} % \par % Once more, we use the solution $\left(q_j,p_j\right)(t,s;y,\eta)$ of the Hamilton-Jacobi system \eqref{hameq}, with $a$ replaced by $a_j$, and define $\tilde{{\alpha}}_j$ as \[ \tilde{{\alpha}}_j(\sigma,t;y,\eta)=\left(a_j-a_{j+1}\right)(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)). \] Then, after a differentiation with respect to $\sigma$, we have \begin{align*} \partial_\sigma\tilde{{\alpha}}_j&(\sigma,t;y,\eta)= \\[1ex] &=\partial_\sigma\left(\left(a_j-a_{j+1}\right) (\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)\right) \\[1ex] &=(\partial_ta_j)\left(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)\right)- (\partial_ta_{j+1})(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)) \\[1ex] &+\bigg (a'_{j,x}(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta))-a'_{j+1,x} (\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta))\bigg)\cdot\partial_tq_j(\sigma,t;y,\eta) \\[1ex] &+\bigg( a'_{j,\xi}(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta))- a'_{j+1,\xi} (\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta))\bigg)\cdot\partial_tp_j(\sigma,t;y,\eta), \end{align*} and we use \eqref{hameq} to write \begin{align*} \partial_\sigma\tilde{{\alpha}}_j(\sigma,t;y,\eta)&=\left[\partial_\sigma(a_j-a_{j+1})\right] (\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)) \\[1ex] &+a'_{j,\xi}(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta))\cdot a'_{j+1,x} (\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)) \\[1ex] &- a'_{j,x}(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta))\cdot a'_{j+1,\xi} (\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)) \\[1ex] &=\{\tau-a_j,\tau-a_{j+1}\}(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)). \end{align*} Assumption \ref{assumpt:invo} then implies \begin{equation}\label{tildeAlphaEqu} \partial_\sigma\tilde{{\alpha}}_j(\sigma,t;y,\eta)=b_{j,j+1}(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta))\cdot\tilde{{\alpha}}_j(\sigma,t;y,\eta)+d_{j,j+1}(\sigma;\left(q_j,p_j\right)(\sigma,t;y,\eta)). \end{equation} Solving \eqref{tildeAlphaEqu} as a first order linear ODE in $\sigma$ with unknown $\tilde{{\alpha}}_j(\sigma,t;y,\eta)$, and writing $b_j$ in place of $b_{j,j+1}$, $d_j$ in place of $d_{j,j+1}$, respectively, we see that \begin{align*} \tilde{{\alpha}}_j(\sigma,t;y,\eta)&=\exp\left(\int_{t_j}^{\sigma} b_j (\tau;\left(q_j,p_j\right)(\tau,t;y,\eta))d\tau\right) \cdot\left[\tilde{{\alpha}}_j(t_{j},t;y,\eta)+ \phantom{ \int_{t_j}^\sigma d_j\left(\nu;\left(q_j,p_j\right)(\nu,t;y,\eta)\right) \cdot \exp\left(\int_{\nu}^{\sigma}b_j\left(\varsigma;\left(q_j,p_j\right) (\varsigma,t;y,\eta)\right)d\varsigma\right)d\nu } \right. \\[1ex] &\left.+{\displaystyle{\int_{t_j}^\sigma}}d_j\left(\nu;\left(q_j,p_j\right)(\nu,t;y,\eta)\right) \cdot \exp\left(-\int_{\nu}^{\sigma}b_j\left(\varsigma;\left(q_j,p_j\right) (\varsigma,t;y,\eta)\right)d\varsigma\right)d\nu.\right] \end{align*} Once again, notice that all the composition performed so far are well defined, and produce SG symbols, in view of Lemma \ref{lem:invariantcomsymbol}, \eqref{eq:equiv}, and recalling that $h\in{S}^{0,0}\Rightarrow \exp(h)\in{S}^{0,0}$ (see, e.g., \cite{Coriasco:998.1,Cothesis}). % \par As stated in Proposition \ref{prop:trajec}, we can write $\tilde{{\alpha}}_j$ in terms of the solution to the critical points problem \eqref{critpntprob}. Indeed, by \eqref{trajequal1}, \eqref{trajequal2} we get \begin{align*} \tilde{{\alpha}}_j(t_j,t_{j-1};&\left(q_j,p_j\right)(t_j,t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi)))= \\[1ex] &=\tilde{{\alpha}}_j(t_j,t_{j-1};Y_{j}(\mathbf{t}_{M+1};x,\xi),N_{j}(\mathbf{t}_{M+1};x,\xi)) \\[1ex] &=\left(a_j-a_{j+1}\right)(t_j;Y_j(\mathbf{t}_{M+1};x,\xi),N_j(\mathbf{t}_{M+1};x,\xi)). \end{align*} Moreover, using Proposition \ref{prop:tderivofmultiph}, we obtain the equality \begin{align}\label{aphaTildaPartialPhi} \tilde{{\alpha}}_j(t_j,t_{j-1};\left(q_j,p_j\right)(t_j,t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi)))= -\partial_{t_{j}}\phi(\mathbf{t}_{M+1};x,\xi). \end{align} Define also \begin{align*}\label{GDef} G_j(t_j)&\equiv G_j(\mathbf{t}_{M+1};x,\xi)= \\[1ex] &=\exp\left[\int_{t_j}^{t_{j-1}} b_j(\tau;\left(q_j,p_j\right) (\tau,t_{j-1};Y_{j-1}(\mathbf{t}_{M+1};x,\xi),N_{j-1}(\mathbf{t}_{M+1};x,\xi)))d\tau\right] \end{align*} % and % \begin{align* F_j(t_j)&\equiv F_j(\mathbf{t}_{M+1};x,\xi)= \\[1ex] &=\left[\int_{t_j}^{t_{j-1}}d_j\left(\nu;\left(q_j,p_j\right) (\nu,t_{j-1};y,\eta)\right) \cdot \right. \\[1ex] &\phantom{=(\int_{t_j}^{t_{j-1}}}\cdot\left.\exp\left(-\int_{\nu}^{t_{j-1}}b_j \left(\varsigma;\left(q_j,p_j\right)(\varsigma,t_{j-1};y,\eta)\right) d\varsigma\right)d\nu\right]_{(y,\eta)=(Y_{j-1},N_{j-1})(\mathbf{t}_{M+1};x,\xi)}, \end{align*} where both $G_j$ and $F_j$, as a consequence of Lemma \ref{lem:invariantcomsymbol} and the properties of $b_j$, $d_j$, $(q_j,p_j)$ and $(Y_{j-1},N_{j-1})$, are symbols belonging to $C^\infty(\Delta(T_1);{S}^{0,0})$. % \par % Then, using the formulae \eqref{newPsiTDer} and \eqref{aphaTildaPartialPhi} above, we find that $\Psi_j$ must fullfill \begin{equation}\label{psiComp} \partial_{t_{j-1}}\Psi_j=T_j(Z_j)\cdot \Psi'_{j,x}-F_j(Z_j) -(\partial_{t_j}\phi)(Z_j)\bigg(\partial_{t_{j-1}}Z_j-T_j(Z_j) \cdot Z'_{j,x}-G_j(Z_j)\bigg), \end{equation} where we omitted everywhere the dependence on $(\mathbf{t}_{M+1};x,\xi)$, $(\partial_{t_j}\phi)(Z_j)$ stands for $(\partial_{t_j}\phi)(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi)$, and % \[ F_j(Z_j)=F_j(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi), G_j(Z_j)=G_j(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi). \] % \par % Now, in order to simplify \eqref{psiComp}, we choose $Z_j$ as solution to the quasilinear Cauchy problem, \begin{align}\label{quasiZj} \begin{cases} \partial_{t_{j-1}}Z_j&=T_j(Z_j)\cdot Z'_{j,x}+G_j(Z_j) \\[1ex] Z|_{t_j={t_{j-1}}}&=t_{j+1}. \end{cases} \end{align} % \par % It is easy to see that \eqref{quasiZj} is a quasilinear Cauhcy problem of the type considered in Subsection \ref{subs:auxpde}. In view of Lemma \ref{lem:exitOfSolForOrdin}, we can solve \eqref{quasiZj} through its characteristic system \eqref{ordin}, with $T_j$ in place of $L$ and $G_j$ in place of $H$, choosing a sufficiently small parameter interval $[0,T^\prime]$, $T^\prime\in(0,T_2]$. Indeed, by Corollary \ref{cor:qlcpsol} and Remark \ref{rem:invJR} (cfr. \cite{Taniguchi02}), defining % \[ Z_j(\mathbf{t}_{M+1};x,\xi)=K(\mathbf{t}_{M+1};\bar{R}(\mathbf{t}_{M+1};x,\xi),\xi), \quad \mathbf{t}_{M+1}\in\Delta(T^\prime), x,\xi\in\R^n, \] % gives a solution of \eqref{quasiZj} with all the desired properties. With such choice of $Z_j$, \eqref{psiComp} is then reduced to the linear, non-homohgeneous PDE \begin{align}\label{newPsiComp} \partial_{t_{j-1}}\Psi_j=T_j(Z_j)\cdot\Psi'_{j,x}-F_j(Z_j), \end{align} with the initial condition \begin{align}\label{inConNewPsi} {\Psi_j}|_{t_{j-1}=t_j}=0. \end{align} Notice that \eqref{inConNewPsi} holds true since we have $Z_j|_{{t_j=t_{j-1}}}=t_{j+1}$, that, together with \eqref{ii.propofsharprod} in Proposition \ref{prop:tderivofmultiph}, gives \begin{multline*} \Psi_j(\mathbf{t}_{M+1,j-1}(t_j);x,\xi)= \\[1ex] \phi_j(\mathbf{t}_{M+1,j-1}(t_j);x,\xi) -\phi(t_0,\dots,t_{j-2},t_j,{Z_j(\mathbf{t}_{M+1};x,\xi)}|_{t_j=t_{j-1}},t_{j+1},\dots,t_{M+1};x,\xi) \\[1ex] =\left[\varphi_1(t_0,t_1)\sharp\dots\sharp\varphi_{j-1}(t_{j-2},t_j) \sharp\right.\underbrace{(\varphi_{j+1}(t_j,t_j)\sharp\varphi_j(t_j,t_{j+1}))}_{=\varphi_j(t_j,t_{j+1})}\sharp \\[1ex] \left.\sharp\varphi_{j+2}(t_{j+1},t_{j+2})\sharp\dots\varphi_{M+1}(t_M,t_{M+1})\right](x,\xi) \\[1ex] -\left[\varphi_1(t_0,t_1)\sharp\dots\sharp\varphi_{j-1}(t_{j-2},t_{j}) \sharp\right.\underbrace{(\varphi_j(t_j,t_{j+1})\sharp\varphi_{j+1}(t_{j+1},t_{j+1}))}_{=\varphi_j(t_j,t_{j+1})}\sharp \\[1ex] \left.\sharp\varphi_{j+2}(t_{j+1},t_{j+2})\sharp\dots\sharp\varphi_{M+1}(t_M,t_{M+1})\right](x,\xi) =0. \end{multline*} \par % Then, the method of characteristics, applied to the linear, non-homogeneous PDE \eqref{newPsiComp}, shows that we can write $\Psi_j$ in the form \begin{equation}\label{finalFormOfPsi_j} \Psi_j(\mathbf{t}_{M+1};x,\xi)=\int_{t_j}^{t_{j-1}}\widetilde{F}_j(\mathbf{t}_{M+1,j-1}(\tau); \theta(\tau;\tilde{\theta}(\tau;x,\xi),\xi),\xi)d\tau, \end{equation} where \begin{align*} \widetilde{F}_j(\mathbf{t}_{M+1};x,\xi)&=-F_j(\mathbf{t}_{M+1,j}(Z_j(\mathbf{t}_{M+1};x,\xi));x,\xi), \\[1ex] \theta(\tau;y,\xi)&=\phantom{-}\theta(\mathbf{t}_{M+1,j}(\tau);y,\xi), \\[1ex] \tilde{\theta}(\tau;x,\xi)&=\phantom{-}\tilde{\theta}(\mathbf{t}_{M+1,j-1}(\tau);x,\xi), \end{align*} for suitable vector-valued functions $\theta,\tilde{\theta}$. By arguments similar to those in Subsection \ref{subs:auxpde} (cf. \cite{Taniguchi02}), both $\theta$ and $\tilde{\theta}$ turn out to be elements of $C^\infty(\Delta(T^\prime);{S}^{1,0}\otimes\R^n)$, satisfying % \[ \jb{\theta(\mathbf{t}_{M+1,j}(\tau);y,\xi)}\sim\jb{y},\quad \jb{\tilde{\theta}(\mathbf{t}_{M+1,j-1}(\tau);x,\xi)}\sim\langle x\rangle, \] % with constants independent of $\mathbf{t}_{M+1}\in\Delta(T^\prime)$, $x,\xi\in\R^n$. Such result, together with the properties of $Z_j$ and another application of Lemma \ref{lem:invariantcomsymbol}, allows to conclude that $\Psi_j\in C^\infty(\Delta(T');{S}^{0,0})$, and it is identically zero when $d_j\equiv0$, as claimed. The proof is complete. % \end{proof} % \par \begin{remark}\label{rem:commiterint} % As we will see in the next Section \ref{sec:fundsol}, the fundamental solution of a system of the form \eqref{sysint} involves iterated integrals (on subintervals of $[0,T^\prime]$) of multi-products of SG FIOs with matrix-valued symbols of order $(0,0)$, and regular, parameter-dependent SG phase functions $\varphi_{N_j}$, $N_j=1,\dots,N$, $j=1,\dots, \nu$, $\nu\in\N$, obtained as solutions to eikonal equations, associated with an involutive family of Hamiltonians $\{\lambda_j\}_{j=1}^N$. In view of \cite{AsCor}, such multi-product is a SG FIO, with symbol of order $(0,0)$ and regular phase fuction given by the multi-product $\phi=\varphi_{N_1}\#\cdots\#\varphi_{N_\nu}$. Theorem \ref{cruthm}, together with some standard manipulation of multiple integrals and change of time-parameters of the form $t_j=Z_j(\mathbf{t}_{\nu+1,j}(\widetilde{t}_j);x,\xi)$, implies that it is possible to change the order of two (or more) factors in the multi-product $\varphi_{N_1}\#\cdots\#\varphi_{N_\nu}$ within the iterated integral. The result is still an iterated integral, on the same subintervals, of a SG FIOs with regular phase function $\widetilde{\phi}$, given by a differently sorted multi-product of the same phase functions $\varphi_{N_j}$, $N_j=1,\dots,N$, $j=1,\dots, \nu$, $\nu\in\N$, and symbol of order $(0,0)$. Indeed, the symbol is altered only by the change of the $t_j$ parameter and (products of) factors of the form $\partial_{t_j}Z_j\cdot\exp(-i\Psi_j)$, which are of order $(0,0)$ and elliptic, by Theorem \ref{cruthm} and Lemma \ref{lem:invariantcomsymbol}, again taking into account that $h\in{S}^{0,0}\Rightarrow \exp(h)\in{S}^{0,0}$ (see \cite{Abdeljawadthesis, Coriasco:998.1, Morimoto, Taniguchi02} for details). In particular, if the original symbol is elliptic, the same property holds true for the transformed symbol. % \end{remark} % \section{Cauchy problems for weakly SG-hyperbolic systems}\label{sec:fundsol} \setcounter{equation}{0} In the present section we deal with the Cauchy problem \begin{equation}\label{sys} \begin{cases} \mathbf{L} U(t,s) = F(t), & (t,s)\in\Delta_T, \\[1ex] U(s,s) = G, & s\in[0,T), \end{cases} \end{equation} on the simplex $\Delta_{T}:=\{(t,s)\vert\ 0\leq s\leq t\leq T\}$, where \begin{equation}\label{L} \mathbf{L}(t,D_t;x,D_x) = D_t + \Lambda(t;x,D_x) +R(t;x,D_x), \end{equation} $\Lambda$ is a ($N\times N$)-dimensional, diagonal operator matrix, whose entries $\lambda_j(t;x,D_x)$, $j=1,\dots, N$, are pseudo-differential operators with real-valued, parameter-dependent symbols $\lambda_j(t;x,\xi)\in C^\infty([0,T]; {S}^{1,1})$, $R$ is a parameter-dependent, ($N\times N$)-dimensional operator matrix of pseudo-differential operators with symbols in $C^\infty([0,T];{S}^{0,0})$, $F\in C^\infty([0,T],H^{r,\varrho}\otimes\R^N)$, $G\in H^{r,\varrho}\otimes\R^N$, $r,\varrho\in\R$. The system \eqref{L} is then of hyperbolic type, since the principal symbol part $ \mathrm{diag} (\lambda_j(t;x,\xi))_{j=1,\dots,N}$ of the coefficient matrix is diagonal and real-valued. Then, its fundamental solution $E(t,s)$ exists (see \cite{CO}), and can be obtained as an infinte sum of matrices of Fourier integral operatos (see \cite{Kumano-go:1,Taniguchi02} and Section 5 of \cite{AsCor} for the SG case). Here we are going to show that if \eqref{sys} is of involutive type, then its fundamental solution $E(t,s)$ can be reduced to a finite sum expression, modulo a smoothing remainder, in the same spirit of \cite{Kumano-go:1, Taniguchi02}, by applying the results from Section \ref{sec:sgphfcomlaw} above. The fundamental solution of \eqref{sys} is a family $\{E(t,s)\vert (t,s)\in \Delta_{T'}\}$ of operators satisfying \begin{equation}\label{tocheck} \begin{cases} \mathbf{L} E(t,s) = 0, & (t,s)\in \Delta_{T'}, \\[1ex] E(s,s)=I, & s\in[0,T'), \end{cases} \end{equation} for $0< T'\leq T$. For $T'$ small enough, see Section 5 of \cite{AsCor}, it is possible to express $\{E(t,s)\}$ in the form % \begin{equation}\label{eq:E} E(t,s) = I_\varphi(t,s) + \int_s^t I_\varphi(t,\theta)\sum_{\nu=1}^\infty W_\nu(\theta,s)\,d\theta, \end{equation} % where $I_\varphi(t,s)$ is the operator matrix defined by % \[ I_\varphi(t,s) = \begin{pmatrix} I_{\varphi_1}(t,s) & & 0 \\[1ex] & \ddots & \\[1ex] 0 & & I_{\varphi_N}(t,s)\end{pmatrix} \] % and $I_{\varphi_j}:=Op_{\varphi_j}(1)$, $1\leq j\leq N$. The phase functions $\varphi_j = \varphi_j(t,s;x,\xi),\ 1\leq j\leq N$, defined on $\Delta_{T'}\times\R^{2n}$, are solutions to the eikonal equations \eqref{eik} with $-\lambda_j$ in place of $a$. The sequence of ($N\times N$)-dimensional matrices of SG \text{Fourier integral operators}\ $\{W_\nu(t,s);(t,s)\in\Delta({T'})\}_{\nu\in\N}$ is defined recursively as \begin{equation}\label{eq:Wn+1} W_{\nu+1}(t,s;x,D_x) = \int_s^t W_1(t,\theta;x,D_x)W_\nu(\theta,s;x,D_x)d\theta, \end{equation} starting with $W_1$ defined as \begin{equation}\label{heart} \mathbf{L} I_{\varphi}(t,s) = i W_1(t,s). \end{equation} % We also set \begin{equation}\label{omega_j} w_{j}(t,s;x,\xi)=\sigma(W_{j}(t,s;x,D_x)),\quad j=1,\dots,\nu+1,\ldots . \end{equation} % the (matrix-valued) symbol of $W_{j}$, \par The following result about existence and uniqueness of a solution $U(t,s)$ to the Cauchy problem \eqref{sys} is a SG variant of the classical Duhamel formula, see \cite{AsCor,CO,Coriasco:998.2}. \par \begin{proposition}\label{wp} For $F\in C^\infty([0,T]; H^{r,\varrho}(\R^n)\otimes\R^N)$ and $G\in H^{r,\varrho}(\R^n)\otimes\R^N$, the solution $U(t,s)$ of the Cauchy problem \eqref{sys}, under the SG-hyperbolicity assumptions explained above, exists uniquely for $(t,s)\in\Delta_{T'}$, $T'\in(0,T]$ suitably small, it belongs to the class $\displaystyle\bigcap_{k\in\mathbb{Z}_+}C^k(\Delta_{T'}; H^{r-k,\varrho-k}(\R^n)\otimes\R^N)$, and is given by $$ U(t,s)=E(t,s)G+i\displaystyle\int_s^t E(t,\sigma)F(\sigma)d\sigma,\quad (t,s)\in\Delta_{T'}, s\in[0,T'). $$ \end{proposition} Notice that, since the phase functions $\varphi_j$ are solutions of eikonal equations \eqref{eik} associated with the Hamiltonians $-\lambda_j$, we have the relation \arraycolsep1.5pt\begin{eqnarray*} D_t {\operatorname{Op}} _{\varphi_j(t,s)}(1) + {\operatorname{Op}} (\lambda_j(t)) {\operatorname{Op}} _{\varphi_j(t,s)}(1) = {\operatorname{Op}} _{\varphi_j(t,s)}(b_{0,j}(t,s)),\quad b_{0,j}(t,s)\in {S}^{0,0}(\R^{2n}), \end{eqnarray*}\arraycolsep5pt % $j=1,\dots,N$. Then, % \arraycolsep1.5pt\begin{eqnarray}\label{eq:W_1} W_1(t,s):= -i\left( \begin{pmatrix} B_{0,1}(t,s) & & 0 \\[1ex] & \ddots & \\[1ex] 0 & & B_{0,N}(t,s) \end{pmatrix} + R(t) I_{\varphi}(t,s)\right), \end{eqnarray}\arraycolsep5pt with $B_{0,j}(t,s)= {\operatorname{Op}} _{\varphi_j(t,s)}(b_{0,j}(t,s))$ and $b_{0,j}(t,s)\in S^{0,0}$, $j=1,\dots, N$. % \par By \eqref{eq:W_1} and Theorem \ref{thm:compi}, one can rewrite equation \eqref{heart} as \begin{equation}\label{heartbis} LI_{\varphi}(t,s) = \sum_{j=1}^{N} \widetilde{W}_{\varphi_j}(t,s), \end{equation} % where $ \widetilde{W}_{\varphi_j}(t,s)$ are ($N\times N$)-dimensional matrices, with entries given by Fourier integral operators with parameter-dependent phase function $\varphi_j$ and symbol in $S^{0,0}$, $1\leq j\leq N$. Thus, if we set $M_\nu=\{\mu=(N_1,\dots,N_\nu):N_k=1,\dots,N, \ k=1,\dots,\nu\}$ for $\nu \geq 2$, the operator matrix $W_\nu(t,s)$ can be written in the form of iterated integrals, namely \begin{align} \int_s^t\int_s^{\theta_1}\ldots\int_s^{\theta_{\nu-2}} \sum\limits_{\mu\in M_\nu} W ^{(\mu)}(t,\theta_1,\dots,\theta_{\nu-1},s)d\theta_{\nu-1}\ldots d\theta_1, \end{align} where $$ W ^{(\mu)}(t,\theta_1,\dots,\theta_{\nu-1},s) = W_{\varphi_{N_1}}(t,\theta_1) W_{\varphi_{N_2}}(\theta_1,\theta_2) \dots W_{\varphi_{N_\nu}}(\theta_{\nu-1},s) $$ is the product of $\nu$ Fourier integral operators matrices with regular phase functions $\varphi_{N_j}$ and elliptic symbols $\sigma(W_{\varphi_{N_j}}(\theta_{j-1},\theta_{j})) =-i\sigma(\widetilde{W}_{\varphi_{N_j}}(\theta_{j-1},\theta_{j}))\in S^{0,0}$. By (2) in Proposition \ref{thm:main}, $W ^{(\mu)}(t,\theta_1,\dots,\theta_{\nu-1},s)$ is a matrix of Fourier integral operators with phase function $\phi ^{(\mu)}=\varphi_{N_1}\sharp\dots\sharp\varphi_{N_\nu}$ and parameter-dependent elliptic symbol $\omega ^{(\mu)}(t,\theta_1,\dots,\theta_{\nu-1},s)$ of order $(0,0)$. Consequently, we can write % \begin{equation}\label{eq:fundsol} \begin{aligned} E(t,s)&= I_\varphi(t,s) + \int_s^t I_\varphi(t,\theta)\bigg\{\sum_{j=1}^{N} W_{\varphi_j}(\theta,s) \\[1ex] &+\sum\limits_{\nu=2}^\infty \sum\limits_{\mu\in M_\nu}\int_s^\theta\int_s^{\theta_1} \!\!\!\!\ldots\int_s^{\theta_{\nu-2}} \!\!\!W ^{(\mu)}(\theta,\theta_1,\dots,\theta_{\nu-1},s)d\theta_{\nu-1}\ldots d\theta_1\bigg\}d\theta. \end{aligned} \end{equation} Given the commutative properties of the product of the Fourier integral operators appearing in the espression of $E(t,s)$ under Assumption \ref{assumpt:invo}, which follow from the results proved in Section \ref{sec:sgphfcomlaw}, our second main result, the next Theorem \ref{thm:mainthm}, can be proved by an argument analogous to the one illustrated in \cite{Taniguchi02} (the details of the proof in the SG symbol classes can be found in \cite{Abdeljawadthesis}). \begin{theorem}\label{thm:mainthm} Let \eqref{sys} be an involutive SG-hyperbolic system, that is, Assumption \ref{assumpt:invo} is fulfilled by the family $\{\lambda_j\}_{j=1}^N$. Then, the fundamental solution \eqref{eq:fundsol} can be reduced, modulo smoothing terms, to \begin{equation}\label{eq:reducedfundsol} \begin{aligned} E(t,s)=&I_\varphi(t,s)+\sum\limits_{j=1}^{N}W_{\varphi_j}^\nmid(t,s) \\[1ex] &+\sum\limits_{j=2}^N \sum\limits_{\mu\in M_j^\nmid}\int_s^t\int_s^{\theta_1}\ldots\int_s^{\theta_{j-2}} {W}^{(\mu^\nmid)}(t,\theta_1,\dots,\theta_{j-1},s)d\theta_{j-1}\ldots d\theta_1, \end{aligned} \end{equation} where the symbol of $W_{\varphi_j}^\nmid(t,s)$ is $\displaystyle{\int_s^t} w_{j}(\theta,s)\,d\theta$, with $w_j$ in \eqref{omega_j}, $\mu^\nmid=(N_1,\dots,N_j)\in M_j^\nmid:=\{N_1<\dots<N_j:\,N_k=1,\dots,N,\ k=1,\dots,j\}$, and ${W}^{(\mu^\nmid)}(t,\theta_1,\dots,\theta_{j-1},s)$ is a ($N\times N$)-dimensional matrix of SG \text{Fourier integral operators}\ with regular phase function ${\phi}^{(\mu^\nmid)}=\varphi_{N_1}\sharp\dots\sharp\varphi_{N_j}$ and matrix-valued, parameter-dependent symbol ${\omega}^{(\mu^\nmid)}(t,\theta_1,\dots,\theta_{j-1},s)\in {S}^{0,0}(\R^{2n})$. \end{theorem} \begin{remark}\label{rem:fundsol} % Theorem \ref{thm:mainthm} can clearly be applied to the case of a $N\times N$ system such that $\Lambda$ is diagonal and its symbol entries $\lambda_j$, $j=1,\dots, N$, coincide with the (repeated) elements of a family of real-valued, parameter-dependent symbols $\{\tau_j\}_{j=1}^m$, $1 < m < N$, satisfying Assumption \ref{assumpt:invo}. In such situation, working initially ``block by block'' of coinciding elements, and then performing the reduction of \eqref{eq:fundsol} to \eqref{eq:reducedfundsol}, through further applications of the commutative properties proved above, as well as of the associative properties of the multi-products mentioned in Proposition \ref{prop:tderivofmultiph}, we see that \eqref{eq:reducedfundsol} can be further reduced to \begin{equation}\label{eq:reducedfundsolbis} \begin{aligned} E(t,s)&=I_\varphi(t,s)+\sum\limits_{j=1}^{m}W_{\varphi_j}^\nmid(t,s) \\[1ex] &+\sum\limits_{j=2}^m \sum\limits_{\mu\in M_j^\nmid}\int_s^t\int_s^{\theta_1}\ldots\int_s^{\theta_{j-2}} {W}^{(\mu^\nmid)}(t,\theta_1,\dots,\theta_{j-1},s)d\theta_{j-1}\ldots d\theta_1, \end{aligned} \end{equation} % with $\mu^\nmid=(m_1,\dots,m_j)\in M_j^\nmid:=\{m_1<\dots<m_j:\,m_k=1,\dots,m,\ k=1,\dots,j\}$, and ($N\times N$)-dimensional matrices of \text{Fourier integral operators}\ with phase function ${\phi}^{(\mu^\nmid)}=\varphi_{m_1}\sharp\dots\sharp\varphi_{m_j}$ and matrix-valued, parameter-dependent symbol as above. % \end{remark} \section{Cauchy problems for weakly SG-hyperbolic linear operators} \label{sec:sginvcp}\setcounter{equation}{0} Here we employ the results from the previous section to the study of Cauchy problems associated with linear hyperbolic differential operators of SG type. After obtaining the fundamental solution, we study the propagation of singularities in the case of SG-classical coefficients. We recall here just a few basic definitions, see \cite{CO, Coriasco:998.1, Coriasco:998.2, Cothesis, CoPa} for more details. \begin{definition}\label{def:4.13} Let $m\in \N$, $T>0$, and $L$ be a differential operator of order $m$, that is \begin{equation}\label{eq:4.69} L(t,D_t;x,D_x)= D_{t}^m + \sum _{j=1}^{m}p_j(t;x,D_x)D_t^{m-j} = D_{t}^m + \sum _{j=1}^{m}\sum_{|\alpha|\le j}c_{j\alpha}(t;x)D_x^\alpha D_t^{m-j} \end{equation} with symbol $p_{j}(t;x,\xi)=\displaystyle\sum_{|\alpha|\le j} c_{j\alpha}(t;x)\xi^\alpha$ such that $p_j\in C^\infty([0,T];{S}^{j,j}(\R^{2n}))$, that is \[ |\partial_t^k\partial^\beta_x c_{j\alpha}(t;x)|\lesssim \langle x\rangle^{j-|\beta|}, \quad \beta\in\mathbb{Z}_+^{n}, j=1,\dots,m,\alpha\in\mathbb{Z}_+^n, |\alpha|\le j . \] We denote by \[ L(t,\tau;x,\xi) = \tau^m + \sum _{j=1}^{m}p_j(t;x,\xi)\tau ^{m-j} \] the symbol of the operator $L$, and by \[ L_m(t,\tau;x,\xi) = \tau^m + \sum _{j=1}^{m}q_j(t;x,\xi)\tau ^{m-j} \] its principal symbol, where, for $j=1,\dots,m$, $q_{j}(t;x,\xi)=\displaystyle \sum_{|\alpha|=j}\widetilde{c}_{j\alpha}(t;x) \xi^\alpha$ belongs to $C^\infty([0,T];{S}^{j,j}(\R^{2n}))$ and is such that \begin{equation*}\label{eq:4.72} (L-L_m)(t,\tau;x,\xi) = \sum_{j=1}^m r_{j}(t;x,\xi) \tau^{m-j}, \quad r_{j} \in C^\infty([0,T];{S}^{j-1,j-1}(\R^{2n})). \end{equation*} \end{definition} \begin{definition}\label{def:sghyp} An operator $L$ of the type introduced in Definition \ref{def:4.13} is called hyperbolic if \begin{equation}\label{roots} L_m(t,\tau;x,\xi)=\prod_{j=1}^m\left(\tau-\tau_j(t,x,\xi)\right), \end{equation} with real-valued, smooth roots $\tau_j$, $j=1,\dots,m$. The roots $\tau_j$ are usually called \textit{characteristics}. More precisely, $L$ is called: \begin{enumerate} \item {\em strictly SG-hyperbolic}, if $L_m$ satisfies \eqref{roots} with real-valued, distinct and separated roots $\tau_j$, $j=1,\dots,m$, in the sense that there exists a constant $C>0$ such that \begin{equation}\label{def:strict} |\tau_j(t;x,\xi)-\tau_k(t;x,\xi)|\geq C\langle x\rangle\langle \xi \rangle,\quad \forall j\neq k,\ (t;x,\xi)\in[0,T]\times\R^{2n}; \end{equation} \item {\em (weakly) SG-hyperbolic with (roots of) constant multiplicities}, if $L_m$ satisfies \eqref{roots} and the real-valued, characteristic roots can be divided into $\mu$ groups ($1\leq \mu\leq m$) of distinct and separated roots, in the sense that, possibly after a reordering of the $\tau_j$, $j=1,\dots, m$, there exist $l_1,\ldots l_\mu\in\N$ with $l_1+\ldots+l_\mu=m$ and $n$ sets \[ G_1=\{\tau_1=\cdots=\tau_{l_1}\}, G_2=\{\tau_{l_1+1}=\cdots=\tau_{l_1+l_2}\}, \ldots G_\mu=\{\tau_{m-l_\mu+1}=\cdots=\tau_{m}\},\] satisfying, for a constant $C>0$, \begin{equation}\label{def:constmult}\tau_j\in G_p,\tau_k\in G_q ,\ p\neq q,\ 1\leq p,q\leq \mu\Rightarrow |\tau_j(t,x,\xi)-\tau_k(t,x,\xi)|\geq C\langle x\rangle\langle \xi \rangle,\end{equation} for all $(t,x,\xi)\in[0,T]\times\R^{2n}$; notice that, in the case $n=1$, we have only one group of $m$ coinciding roots, that is, $L_m$ admits a single real root of multiplicity $m$, while for $n=m$ we recover the strictly hyperbolic case; the number $l=\max_{j=1,\dots,n}l_j$ is the \textit{maximum multiplicity of the roots of $L_m$}; \item {\em (weakly) SG-hyperbolic with involutive roots (or SG-involutive)}, if $L_m$ satisfies \eqref{roots} with real-valued characteristic roots such that the family $\{\tau_j\}_{j=1}^m$ satisfies Assumption \ref{assumpt:invo}. \end{enumerate} \end{definition} \par \begin{remark}\label{rem:4.14} Roots of constant multiplicities are involutive. The converse statement is not true in general, see e.g., \cite{AsCoSu:1,Morimoto}. \end{remark} \par \subsection{Fundamental solution for linear SG-involutive operators of order $m\in\N$}% We will focus here on SG-involutive operators, see the references quoted above for the known results about SG-hyperbolic operators with constant multiplicities. In particular, we assume that there is no couple of characteristic roots $\tau_j$, $\tau_k$, $j,k=1,\dots,m$, $k\not=j$, satisfying \eqref{def:constmult}. It is possible to translate the Cauchy problem \begin{equation} \label{eq:4.74} \left\{ \begin{array}{ll} L u(t,s) = f(t), & (t,s)\in \Delta_T, \\[1ex] D_{t}^k u(s,s) = g_{k}, & k=0,\dots,m-1, s\in[0,T), \end{array} \right. \end{equation} for a SG-involutive operator $L$ in the sense of Definition \ref{def:sghyp}, into a Cauchy problem for an involutive system \eqref{sys} with suitable initial conditions, under an appropriate factorization condition, see below. \par We write $\Theta _{j}= {\operatorname{Op}} (\tau_j)$, and also set, for convenience below, $\Gamma_{j} = D_{t} - \Theta _{j}$, $j=1,\dots,m$. Moreover, with the permutations $M_{k}$ of $k$ elements of the set $\{1, \dots, m\}$ from Section \ref{sec:fundsol}, and their sorted counterparts $ \perm^\nmid _k$, $1 \leq k\leq m$, we introduce the notation \[ M_{0} = \{ \emptyset \}, \quad M = \bigcup_{k=0}^{m-1} M_{k}, \quad \perm^\nmid = \bigcup_{k=1}^{m} \perm^\nmid _{k}. \] For $\alpha \in M_{k}$, $0 \leq k\leq m$, we define $\card{\alpha} = k$ and \[ \tilde{D} _{\emptyset} = I, \qquad \tilde{D}_{\alpha} = \tilde{D} _{\alpha_{1}}\dots \tilde{D} _{\alpha_{k}},\, \alpha =(\alpha _{1}, \dots, \alpha _{k} )\in M_k, k\ge1. \] \par The proof of the following Lemma \ref{lemma:4.16.2.1} can be found in \cite{Cothesis}. Analogous results are used in \cite{Kumano-go:1} and \cite{Morimoto}. \par \begin{lemma} \label{lemma:4.16.2.1} When $\{ \lambda _{j} \}$ is an involutive system, for all $\alpha \in M_{m}$ we have % \begin{equation} \label{eq:4.112.7} \Gamma _{\alpha} = \Gamma _{1} \dots \Gamma _{m} + \sum_{\beta \in M} {\operatorname{Op}} (q^{\alpha}_{\beta}(t)) \Gamma _{\beta}, \end{equation} % where $q^{\alpha}_{\beta} \in C^\infty([0,T];{S}^{0,0}(\R^{2n}))$. \end{lemma} \par A systemization and well-posedness (with loss of decay and regularity) theorem can be stated for the Cauchy problem \eqref{eq:4.74} under a suitable condition for the operator $L$. This result is due, in its original local form, to Morimoto \cite{Morimoto} and it has been extended to the SG case in \cite{Coriasco:998.2}, where the proof of the next result, based on Lemma \ref{lemma:4.16.2.1}, can be found. \par \begin{proposition}\label{thm:4.20} Assume the SG-hyperbolic operator $L$ to be of the form \begin{equation}\label{eq:4.117} L = \Gamma _{1} \cdots \Gamma _{m} + \sum_{\alpha \in \perm^\nmid } {\operatorname{Op}} (p_{\alpha}(t)) \Gamma _{\alpha} \; \mathrm{mod} \operatorname{Op} (C^\infty([0,T]; {S}^{-\infty,-\infty}(\R^{2n}))), \end{equation} with $p_{\alpha} \in C^\infty([0,T];{S}^{0,0}(\R^{2n}))$. Moreover, assume that the family of its characteristic roots $\{\tau_j\}_{j=1}^m$ sastisfies Assumption \ref{assumpt:invo}. Then, the Cauchy problem \eqref{eq:4.74} for $L$ is equivalent to a Cauchy problem for a suitable first order system \eqref{sys} with diagonal principal part, of the form \begin{equation} \label{eq:4.119} \left\{ \begin{array}{l} \displaystyle\left(D_t +K(t)\right) U(t,s) = F(t), \mbox{ } (t,s) \in \Delta_T, \\[1ex] U(s,s) = G, \hspace*{2.5cm} s\in[0,T), \end{array} \right. \end{equation} where $U$, $F$ and $G$ are $N$-dimensional vector, $K$ a ($N\times N$)-dimensional matrix, with $N$ given by \eqref{eq:4.120.1}. $U$ is defined in \eqref{eq:4.120}, \eqref{eq:4.121}, and \eqref{eq:4.122}. Namely, \begin{equation} \label{eq:4.120.1} N = \sum_{j=0}^{m-1} \frac{m!}{(m-j)!}, \end{equation} \begin{equation}\label{eq:4.120} U = \ ^t\left(u_{\emptyset}\equiv u, u_{(1)}, \dots, u_{(m)}, u_{(1,2)}, u_{(2,1)}, \dots, u_{\alpha}, \dots\right), \end{equation} with $\alpha \in M$, and \begin{itemize} \item[-] for $\alpha \in M_{k}$, $0\le k \le m - 2$ and $j=\max \{ 1, \dots, m \} \setminus \alpha$, we set % \begin{equation} \label{eq:4.121} \tilde{D}_{j} u_{\alpha} = u_{\alpha_{j}} \end{equation} % with $\alpha_{j}= (j, \alpha_{1}, \dots, \alpha_{k}) \in M_{k+1}$; \item[-] for $\alpha \in M_{m-1}$ and $j \notin \{ \alpha \}$, we set % \begin{equation} \label{eq:4.122} \tilde{D}_{j} u_{\alpha} = f - \sum_{\beta \in \perm^\nmid } {\operatorname{Op}} (p_{\beta}(t)) u_{\beta} + \sum_{\beta \in M} {\operatorname{Op}} (q^{\alpha_{j}}_{\beta}(t)) u_{\beta}, \end{equation} % with $\alpha_{j}= (j, \alpha_{1}, \dots, \alpha_{k}) \in M_{m}$ and the symbols $p_\beta$, $q^{\alpha}_\beta$ from \eqref{eq:4.112.7} and \eqref{eq:4.117}. \end{itemize} \end{proposition} \par \begin{remark}\label{rem:Levi} % We call the SG-hyperbolic operators $L$ satisfying the factorization condition \eqref{eq:4.117} ``operators of Levi type''. % \end{remark} \par \begin{remark}\label{rem:incond} Since, for $\alpha \in M_{k}$, $k\ge1$, we have \[ \tilde{D}_{\alpha} = D^k_{t} + \sum_{j=0}^{k-1} {\operatorname{Op}} (\Upsilon^j_{\alpha}(s)) D^j_{t}, \mbox{ } \Upsilon^j_{\alpha} \in C^\infty([0,T];{S}^{k-j,k-j}(\R^{2n})), \] the initial conditions $G$ for $U$ can be expressed as \begin{equation} \label{eq:4.124} \left\{ \begin{array}{l} G_{\emptyset}(s,s) = g_{0}, \\ G_{\alpha}(s,s) = g_{\card{\alpha}} + \ssum{j=0}{\card{\alpha}-1} {\operatorname{Op}} (\Upsilon^j_{\alpha}(s)) g_{j}, \quad \alpha\in M, \card{\alpha}>0. \end{array} \right. \end{equation} Notice that, in view of the continuity properties of the SG pseudo-differential operators and of the orders of the $\Upsilon^i_\alpha$, \eqref{eq:4.124} implies \begin{equation}\label{eq:Gaord} G_{\alpha}\in H^{m-1-\card{\alpha},\mu-1-\card{\alpha}}(\R^n), \quad \alpha\in M. \end{equation} \end{remark} \par The next Theorem \ref{thm:mainLastSecThm} is our third main result, namely, a well-posedness result, with decay and regularity loss, for SG-involutive operators of the form \eqref{eq:4.119}. It is a consequence of Proposition \ref{thm:4.20} in combination with the main results of Section \ref{sec:fundsol}. \par \begin{theorem}\label{thm:mainLastSecThm} Let the operator $L$ in \eqref{eq:4.74} be SG-involutive, of the form considered in Proposition \ref{thm:4.20}. Let $f\in C^\infty([0,T]; H^{r,\varrho}(\R^n))$ and $g_k\in H^{r+m-1-k,\varrho+m-1-k}(\R^n)$, $k=0,\dots,m-1$. Then, for a suitable $T^\prime\in(0,T]$, the Cauchy problem \eqref{eq:4.74} admits a unique solution $u(t,s)$, belonging to $\displaystyle\bigcap_{k\in\mathbb{Z}_+}C^k(\Delta_{T^\prime}; H^{r-k,\varrho-k}(\R^n))$, given, modulo elements in $C^\infty(\Delta_{T^\prime};{\mathcal S}(\R^n))$, by % \begin{equation}\label{eq:soleqordm} u(t,s) = \sum_{\alpha\in M}W_{\alpha}(t,s) G_{\alpha}+ \sum_{\alpha\in M_{m-1}}\int_s^t W_{\alpha}(t,\sigma) f(\sigma)\,d\sigma, \quad (t,s)\in\Delta_{T^\prime}, s\in[0,T^\prime), \end{equation} % for suitable linear combinations of parameter-dependent families of (iterated integrals of) regular SG Fourier integral operators $W_{\alpha}(t,s)$, $\alpha\in M$, $(t,s)\in\Delta_{T^\prime}$, with phase functions and matrix-valued symbols determined through the characteristic roots of $L$. \end{theorem} \begin{proof} % By the procedure explained in Proposition \ref{thm:4.20} and Remark \ref{rem:incond}, we can switch from the Cauchy problem \eqref{eq:4.74} to an equivalent Cauchy problem \eqref{eq:4.119}, with $u\equiv U_\emptyset$. The uniqueness of the solution is then a consequence of known results about symmetric SG-hyperbolic systems, see \cite{CO}, of which \eqref{eq:4.119} is a special case. The fundamental solution of \eqref{eq:4.119} is given by \eqref{eq:reducedfundsolbis}, in view of Theorem \ref{thm:mainthm} and Remark \ref{rem:fundsol}. It is a matrix-valued, parameter-dependent operator family $E(t,s)=(E_{\mu\mu^\prime})_{\mu,\mu^\prime\in M}(t,s)$, whose elements $E_{\mu\mu^\prime}(t,s)$, $\mu,\mu^\prime\in M$, are, modulo elements with kernels in $C^\infty(\Delta_{T^\prime};{\mathcal S})$, linear combinations of parameter-dependent families of (iterated integrals of) regular SG Fourier operators, with phase functions of the type % \begin{align*} \phi^{(\mu^\nmid)}&=\varphi_{m_1}, && \mu^\nmid=(m_1)\in M^\nmid_1, \\ \phi^{(\mu^\nmid)}&=\varphi_{m_1}\sharp\dots\sharp\varphi_{m_j}, &&\mu^\nmid=(m_1,\dots,m_j)\in M^\nmid_j,j\ge2, \end{align*} % $\varphi_k$ solution of the eikonal equation associated with the characteristic root $\tau_k$ of $L$, $k=1,\dots,m$, and parameter-dependent, matrix-valued symbols of the type % \[ {\omega}^{(\mu^\nmid)}(t,\theta_1,\dots,\theta_{j-1},s)\in {S}^{0,0}, \quad \mu \in M^\nmid_j, \] % $j=1,\dots,m$. Then, the component $U_\emptyset\equiv u$ of the solution $U$ of \eqref{sys} has the form \eqref{eq:soleqordm}, with $W_\alpha=E_{\varnothing\alpha}$, taking into account \eqref{eq:4.122} and \eqref{eq:4.124}. We observe that the $k$-th order $t$-derivatives of the operators $W_\alpha$, $\alpha\in M$, map continuously $H^{r,\varrho}$ to $H^{r-k,\varrho-k}$, $k\in\mathbb{Z}_+$, in view of Theorem \ref{thm:fiocont} and of the fact that, of course, % \[ \partial_t[ {\operatorname{Op}} _{\phi^{(\mu^\nmid)}(t,s)}(w^{(\mu^\nmid)}(t,s))] = {\operatorname{Op}} _{\phi^{(\mu^\nmid)}(t,s)}(i(\partial_t\phi^{(\mu^\nmid)})(t,s)\cdot w^{(\mu^\nmid)}(t,s) +\partial_t w^{(\mu^\nmid)}(t,s)), \] % obtaining a symbol of orders $1$-unit higher in both components at any $t$-derivative step. This fact, together with the hypothesis on $f$, implies that the second sum in \eqref{eq:soleqordm} belongs to $\displaystyle\bigcap_{k\in\mathbb{Z}_+}C^k(\Delta_{T^\prime}; H^{r-k,\varrho-k}(\R^n))$. The same is true for the elements of the first sum. In fact, recalling the embedding among the Sobolev-Kato spaces and \eqref{eq:Gaord}, since $\alpha\in M\Rightarrow 0\le\card{\alpha}\le m-1$, we find % \begin{align*} W_\alpha(t,s)G_\alpha \in& \bigcap_{k\in\mathbb{Z}_+} C^k(\Delta_{T^\prime}; H^{r+m-1-\card{\alpha}-k,\varrho+m-1-\card{\alpha}-k}) \\ \hookrightarrow &\bigcap_{k\in\mathbb{Z}_+} C^k(\Delta_{T^\prime}; H^{r-k,\varrho-k}), \quad \alpha\in M, \end{align*} % and this concludes the proof. % \end{proof} \subsection{Propagation of singularities for classical SG-involutive operators} Theorem \ref{thm:mainLastSecThm}, together with the propagation results proved in \cite{CJT4}, implies our fourth main result, Theorem \ref{thm:propwfs} below, about the global wave-front set of the solution of the Cauchy problem \eqref{eq:4.74}, in the case of a classical SG-involutive operator $L$ of Levi type. We first recall the necessary definitions, adapting some materials appeared in \cite{CJT2,CJT4,CoMa}. \begin{definition}\label{def:admspace} Let ${\mathcal B}$ be a topological vector space of distributions on $\R^n$ such that $$ {\mathcal S}(\R^n)\subseteq {\mathcal B} \subseteq {\mathcal S}^\prime(\R^n) $$ with continuous embeddings. Then ${\mathcal B}$ is called SG-admissible when $ {\operatorname{Op}} _t(a)$ maps ${\mathcal B}$ continuously into itself, for every $a\in {S} ^{0,0}(\R^{2n})$. If ${\mathcal B}$ and ${\mathcal C}$ are SG-admissible, then the pair $({\mathcal B} ,{\mathcal C} )$ is called SG-ordered (with respect to $(m,\mu)\in\R^2$), when the mappings $$ {\operatorname{Op}} _t(a)\, :\, {\mathcal B} \to {\mathcal C} \quad \text{and}\quad {\operatorname{Op}} _t(b)\, :\, {\mathcal C} \to {\mathcal B} $$ are continuous for every $a\in {S} ^{m,\mu}(\R^{2n})$ and $b\in {S} ^{-m,-\mu}(\R^{2n})$. \end{definition} \begin{remark} ${\mathcal S}(\R^n)$, $H^{r,\varrho}(\R^n)$, $r,\varrho\in\R$, and ${\mathcal S}^\prime(\R^n)$ are SG-admissible. $({\mathcal S}(\R^n),{\mathcal S}(\R^n))$, $(H^{r,\varrho}(\R^n),H^{r-m,\varrho-\mu}(\R^n))$, $r,\varrho\in\R$, $({\mathcal S}^\prime(\R^n),{\mathcal S}^\prime(\R^n))$ are SG-ordered (with respect to any $(m,\mu)\in\R^2$). The same holds true for (suitable couples of) modulation spaces, see \cite{CJT2}. \end{remark} \begin{definition}\label{admspacesdef} Let $\varphi \in {\cal{P}}^\epsilon $ be a regular phase function, ${\mathcal B}$, ${\mathcal B}_1$, ${\mathcal B}_2$, ${\mathcal C}$, ${\mathcal C}_1$, ${\mathcal C}_2$, be SG-admissible and $\Omega \subseteq \R^n$ be open. Then the pair $({\mathcal B} ,{\mathcal C} )$ is called weakly-I SG-ordered (with respect to $(m,\mu,\varphi,\Omega )$), when the mapping $$ \operatorname{Op}_\varphi(a)\, :\, {\mathcal B} \to {\mathcal C} $$ is continuous for every $a\in {S} ^{m,\mu}(\R^{2n})$ with support such that the projection on the $\xi$-axis does not intersect $\R^n\setminus \Omega$. Similarly, the pair $({\mathcal B} ,{\mathcal C} )$ is called weakly-II SG-ordered (with respect to $(m,\mu,\varphi,\Omega )$), when the mapping $$ \operatorname{Op}_\varphi^*(b)\, :\, {\mathcal C} \to {\mathcal B} $$ is continuous for every $b\in {S} ^{m,\mu}(\R^{2n})$ with support such that the projection on the $x$-axis does not intersect $\R^n\setminus \Omega$. Furthermore, $({\mathcal B}_1, {\mathcal C}_1, {\mathcal B}_2, {\mathcal C}_2)$ are called SG-ordered (with respect to $m_1,\mu_1, m_2,\mu_2$, $\varphi$, and $\Omega$), when $({\mathcal B}_1,{\mathcal C}_1)$ is a weakly-I SG-ordered pair with respect to $(m_1,\mu_1,\varphi ,\Omega )$, and $({\mathcal B}_2,{\mathcal C}_2)$ is a weakly-II SG-ordered pair with respect to $(m_2,\mu_2,\varphi ,\Omega )$. \end{definition} \begin{remark} $({\mathcal S}(\R^n),{\mathcal S}(\R^n))$, $(H^{r,\varrho}(\R^n),H^{r-m,\varrho-\mu}(\R^n))$, $r,\varrho\in\R$, $({\mathcal S}^\prime(\R^n),{\mathcal S}^\prime(\R^n))$ are weakly-I and weakly-II SG-ordered pairs (with respect to any $(m,\mu)\in\R^2$, $\varphi\in {\cal{P}}^\epsilon $, and $\Omega=\emptyset$). The situation is more delicate in the case of modulation spaces, even just on Sobolev-Kato spaces modeled on $L^p(\R^n)$, $p\in[1,\infty)$, $p\not=2$, see \cite{CJT2} and the references quoted therein. \end{remark} Now we recall the definition given in \cite{CJT2} of global wave-front sets for temperate distributions with respect to Banach or Fr\'echet spaces and state some of their properties. First of all, we recall the definitions of set of characteristic points that we use in this setting. We need to deal with the situations where \eqref{eq:ellcond} holds only in certain (conic-shaped) subset of $\R^n \times \R^n$. Here we let $\Omega _m$, $m=1,2,3$, be the sets \begin{equation}\label{omegasets} \begin{aligned} \Omega _1= \R^n\times (&\R^n\setminus \{0\}),\qquad \Omega _2 = (\R^n \setminus \{0\})\times \R^n, \\[1ex] \Omega _3 &= (\R^n \setminus \{0\})\times (\R^n \setminus \{0\}), \end{aligned} \end{equation} \par \begin{definition}\label{defchar} Let $\Omega _k$, $k=1,2,3$ be as in \eqref{omegasets}, and let $a\in {S} ^{m,\mu}(\R^{2n})$. \par \begin{enumerate} \item $a$ is called \emph{locally} or \emph{type-$1$ invertible} with respect to $m,\mu$ at the point $(x_0,\xi_0)\in \Omega _1$, if there exist a neighbourhood $X$ of $x_0$, an open conical neighbourhood $\Gamma$ of $\xi _0$ and a positive constant $R$ such that \eqref{eq:ellcond} holds for $x\in X$, $\xi\in \Gamma$ and $|\xi|\ge R$. \par \item $a$ is called \emph{Fourier-locally} or \emph {type-$2$ invertible} with respect to $m,\mu$ at the point $(x_0,\xi_0)\in \Omega _2$, if there exist an open conical neighbourhood $\Gamma$ of $x_0$, a neighbourhood $X$ of $\xi _0$ and a positive constant $R$ such that \eqref{eq:ellcond} holds for $x\in \Gamma$, $|x|\ge R$ and $\xi\in X$. \par \item $a$ is called \emph{oscillating} or \emph{type-$3$ invertible} with respect to $m,\mu$ at the point $(x_0,\xi_0)\in \Omega _3$, if there exist open conical neighbourhoods $\Gamma _1$ of $x_0$ and $\Gamma _2$ of $\xi _0$, and a positive constant $R$ such that \eqref{eq:ellcond} holds for $x\in \Gamma _1$, $|x|\ge R$, $\xi \in \Gamma _2$ and $|\xi |\ge R$. \end{enumerate} \par If $k\in \{ 1,2,3\}$ and $a$ is \emph{not} type-$k$ invertible with respect to $m,\mu$ at $(x_0,\xi_0)\in \Omega _k$, then $(x_0,\xi_0)$ is called \emph{type-$k$ characteristic} for $a$ with respect to $m,\mu$. The set of type-$k$ characteristic points for $a$ with respect to $m,\mu$ is denoted by $\operatorname{Char} _{m,\mu}^k(a)$. \par The \emph{(global) set of characteristic points} (the characteristic set), for a symbol $a\in {S}^{m,\mu}(\R^{2n})$ with respect to $m,\mu$ is defined as $$ \operatorname{Char} (a)=\operatorname{Char} _{m,\mu}(a)=\operatorname{Char} ^1_{m,\mu}(a)\bigcup\operatorname{Char} ^2 _{m,\mu}(a)\bigcup\operatorname{Char} ^3_{m,\mu}(a). $$ \end{definition} \par In the next Definition \ref{cuttdef} we introduce different classes of cutoff functions (see also Definition 1.9 in \cite{CJT1}). \par \begin{definition}\label{cuttdef} Let $X\subseteq \R^n$ be open, $\Gamma \subseteq \R^n \setminus \{0\}$ be an open cone, $x_0\in X$ and $\xi _0\in \Gamma$. \begin{enumerate} \item A smooth function $\varphi$ on $\R^n$ is called a cutoff (function) with respect to $x_0$ and $X$, if $0\le \varphi \le 1$, $\varphi \in C_0^\infty (X)$ and $\varphi =1$ in an open neighbourhood of $x_0$. The set of cutoffs with respect to $x_0$ and $X$ is denoted by $\mathscr C_{x_0}(X)$ or $\mathscr C_{x_0}$. \par \item A smooth function $\psi$ on $\R^n$ is called a directional cutoff (function) with respect to $\xi_0$ and $\Gamma$, if there is a constant $R>0$ and open conical neighbourhood $\Gamma _1\subseteq \Gamma$ of $\xi _0$ such that the following is true: \begin{itemize} \item $0\le \psi \le 1$ and $ {\operatorname{supp}} \psi \subseteq \Gamma$; \par \item $\psi (t\xi )=\psi (\xi )$ when $t\ge 1$ and $|\xi |\ge R$; \par \item $\psi (\xi )=1$ when $\xi \in \Gamma _1$ and $|\xi |\ge R$. \end{itemize} \par \noindent The set of directional cutoffs with respect to $\xi _0$ and $\Gamma$ is denoted by $\mathscr C^{\operatorname{dir}} _{\xi _0}(\Gamma )$ or $\mathscr C^{\operatorname{dir}} _{\xi _0}$. \end{enumerate} \end{definition} \par \begin{remark}\label{psiinvremark} Let $X\subseteq \R^n$ be open and $\Gamma ,\Gamma _1,\Gamma _2 \subseteq \R^n\back 0$ be open cones. Then the following is true. \begin{enumerate} \item if $x_0\in X$, $\xi _0\in \Gamma$, $\varphi \in \mathscr C _{x _0}(X)$ and $\psi \in \mathscr C ^{{\operatorname{dir}}} _{\xi _0}(\Gamma )$, then $c_1=\varphi \otimes \psi$ belongs to ${S}^{0,0}(\R^{2n})$, and is type-$1$ invertible at $(x_0,\xi _0)$; \par \item if $x_0\in \Gamma$, $\xi _0\in X$, $\psi \in \mathscr C ^{{\operatorname{dir}}} _{x_0}(\Gamma )$ and $\varphi \in \mathscr C _{\xi _0}(X)$, then $c_2=\varphi \otimes \psi$ belongs to ${S}^{0,0}(\R^{2n})$, and is type-$2$ invertible at $(x_0,\xi _0)$; \par \item if $x_0\in \Gamma _1$, $\xi _0\in \Gamma _2$, $\psi _1\in \mathscr C ^{{\operatorname{dir}}} _{x_0}(\Gamma _1)$ and $\psi _2\in \mathscr C ^{{\operatorname{dir}}} _{\xi _0}(\Gamma _2)$, then $c_3=\psi _1 \otimes \psi _2$ belongs to ${S}^{0,0}(\R^{2n})$, and is type-$3$ invertible at $(x_0,\xi _0)$. \end{enumerate} \end{remark} \par The next Proposition \ref{charequiv} shows that $\operatorname{Op} _t(a)$ for $t\in \mathbb R$ satisfies convenient invertibility properties of the form \begin{equation}\label{locinvop} \operatorname{Op} _t(a)\operatorname{Op} _t(b) = \operatorname{Op} _t(c) + \operatorname{Op} _t(h), \end{equation} outside the set of characteristic points for a symbol $a$. Here $\operatorname{Op} _t(b)$, $\operatorname{Op} _t(c)$ and $\operatorname{Op} _t(h)$ have the roles of ``local inverse'', ``local identity'' and smoothing operators respectively. From these statements it also follows that our set of characteristic points in Definition \ref{defchar} are related to those in \cite{CoMa,Ho1}. \par \begin{proposition}\label{charequiv} Let $k\in \{1,2,3\}$, $m,\mu\in\R$, and let $a\in {S} ^{m,\mu}(\R^{2n})$. Also let $\Omega _k$ be as in \eqref{omegasets}, $(x_0,\xi _0)\in \Omega _k$, when $k$ is equal to $1$, $2$ and $3$, respectively. Then the following conditions are equivalent, $k=1,2,3$: \medspace \begin{enumerate} \item $(x_0,\xi _0)\notin \operatorname{Char} _{m,\mu}^k (a)$; \par \item there is an element $c\in {S}^{0,0}(\R^{2n})$ which is type-$k$ invertible at $(x_0,\xi _0)$, and an element $b\in {S}^{-m,-\mu}(\R^{2n})$ such that $ab=c$; \par \item \eqref{locinvop} holds for some $c\in {S}^{0,0}(\R^{2n})$ which is type-$k$ invertible at $(x_0,\xi _0)$, and some elements $h\in {S}^{-1,-1}(\R^{2n})$ and $b\in {S}^{-m,-\mu}(\R^{2n})$; \par \item \eqref{locinvop} holds for some $c_k\in {S}^{0,0}(\R^{2n})$ in Remark \ref{psiinvremark} which is type-$k$ invertible at $(x_0,\xi _0)$, and some elements $h$ and $b\in {S}^{-m,-\mu}(\R^{2n})$, where $h\in {\mathcal S}$ when $k\in \{ 1,3\}$ and $h\in {S} ^{-\infty ,0}(\R^{2n})$ when $k=2$. \par Furthermore, if $t=0$, then the supports of $b$ and $h$ can be chosen to be contained in $X\times \R^n$ when $k=1$, in $\Gamma \times \R^n$ when $k=2$, and in $\Gamma _1\times \R^n$ when $k=3$. \end{enumerate} \end{proposition} \par We can now introduce the complements of the wave-front sets. More precisely, let $\Omega _k$, $k\in \{ 1,2,3\}$, be given by \eqref{omegasets}, ${\mathcal B}$ be a Banach or Fr{\'e}chet space such that ${\mathcal S}(\R^n)\subseteq {\mathcal B} \subseteq {\mathcal S}^\prime(\R^n)$, and let $f\in {\mathcal S}^\prime(\R^n)$. Then the point $(x_0,\xi _0)\in \Omega _k$ is called \emph{type-$k$} regular for $f$ with respect to ${\mathcal B}$, if \begin{equation}\label{ImageOpcm} {\operatorname{Op}} (c_k)f\in {\mathcal B} , \end{equation} for some $c_k$ in Remark \ref{psiinvremark}, $k=1,2,3$. The set of all type-$k$ regular points for $f$ with respect to ${\mathcal B}$, is denoted by $\Theta ^k_{{\mathcal B}}(f)$. \par \begin{definition}\label{def:wfsMB} Let $k\in \{ 1,2,3\}$, $\Omega _k$ be as in \eqref{omegasets}, and let ${\mathcal B}$ be a Banach or Fr\'echet space such that ${\mathcal S}(\R^n)\subseteq {\mathcal B} \subset {\mathcal S}^\prime(\R^n)$. \begin{enumerate} \item The \emph{type-$k$ wave-front set} of $f\in {\mathcal S}^\prime(\R^n)$ with respect to ${\mathcal B}$ is the complement of $\Theta ^k_{{\mathcal B}}(f)$ in $\Omega _k$, and is denoted by $\operatorname{WF} ^k_{{\mathcal B}}(f)$; \par \item The \emph{global wave-front set} $\operatorname{WF}_{{\mathcal B}}(f)\subseteq (\R^n\times\R^n)\back 0$ is the set \begin{equation*} \operatorname{WF}_{{\mathcal B}}(f) \equiv \operatorname{WF} ^1_{\mathcal B} (f) \bigcup \operatorname{WF} ^2_{\mathcal B} (f) \bigcup \operatorname{WF} ^3_{\mathcal B} (f). \end{equation*} \end{enumerate} \end{definition} \par The sets $\operatorname{WF} ^1_{{\mathcal B}}(f)$, $\operatorname{WF} ^2_{{\mathcal B}}(f)$ and $\operatorname{WF} ^3_{{\mathcal B}}(f)$ in Definition \ref{def:wfsMB}, are also called the \emph{local}, \emph{Fourier-local} and \emph{oscillating} wave-front set of $f$ with respect to ${\mathcal B}$. \par \begin{remark} In the special case when ${\mathcal B}=H^{r,\varrho}(\R^n)$, $r,\varrho\in\R$, we write $\operatorname{WF} ^k_{r,\varrho}(f)$, $k=1,2,3$. In this situation, $\operatorname{WF}_{r,\varrho}(f) \equiv \operatorname{WF} ^1_{r,\varrho} (f) \bigcup \operatorname{WF} ^2_{r,\varrho} (f) \bigcup \operatorname{WF} ^3_{r,\varrho} (f)$ coincides with the scattering wave front set of $f\in{\mathcal S}^\prime(\R^n)$ introduced by Melrose \cite{Me}. In the case when ${\mathcal B}={\mathcal S}(\R^n)$, $\operatorname{WF}_{{\mathcal B}}(f)$ coincides with the ${\mathcal S}$-wave-front set considered in \cite{CoMa} (see also \cite{Schulz}). \end{remark} \par \begin{remark}\label{rem:wfcon} Let $\Omega _m$, $m=1,2,3$ be the same as in \eqref{omegasets}. % \begin{enumerate} % \item If $\Omega \subseteq \Omega _1$, and $(x_0,\xi_0)\in \Omega \ \Longleftrightarrow \ (x_0,\sigma \xi_0)\in \Omega$ for $\sigma \ge 1$, then $\Omega$ is called \emph{$1$-conical}; % \item If $\Omega \subseteq \Omega _2$, and $(x_0,\xi_0)\in \Omega \ \Longleftrightarrow \ (sx_0,\xi_0)\in\Omega$ for $s \ge 1$, then $\Omega$ is called \emph{$2$-conical}; % \item If $\Omega \subseteq \Omega _3$, and $(x_0,\xi_0) \in \Omega\ \Longleftrightarrow \ (sx_0,\sigma\xi_0)\in \Omega$ for $s,\sigma \ge 1$, then $\Omega$ is called \emph{$3$-conical}. % \end{enumerate} By \eqref{ImageOpcm} and the paragraph before Definition \ref{def:wfsMB}, it follows that if $m=1,2,3$, then $\Theta^m_{\mathcal B} (f)$ is $m$-conical. The same holds for $\operatorname{WF}^m_{\mathcal B}(f)$, $m=1,2,3$, by Definition \ref{def:wfsMB}, noticing that, for any $x_0\in \R^{r} \setminus \{0\}$, any open cone $\Gamma\ni x_0$, and any $s>0$, $\mathscr C ^{{\operatorname{dir}}} _{x_0}(\Gamma ) =\mathscr C ^{{\operatorname{dir}}} _{s x_0}(\Gamma )$. For any $R>0$ and $m\in \{1,2,3\}$, we set \begin{gather*} \Omega_{1,R} \equiv \sets {(x,\xi )\in \Omega _1}{|\xi |\geq R}, \quad \Omega_{2,R} \equiv \sets {(x,\xi )\in \Omega _2}{|x |\geq R}, \\[1ex] \Omega_{3,R} \equiv \sets {(x,\xi )\in \Omega _3}{|x|, |\xi |\geq R} \end{gather*} Evidently, $\Omega_m^R$ is $m$-conical for every $m\in \{ 1,2,3\}$. % \end{remark} \par From now on we assume that ${\mathcal B}$ in Definition \ref{def:wfsMB} is SG-admissible, and recall that Sobolev-Kato spaces and, more generally, modulation spaces, and $\mathscr S(\R^d)$ are SG-admissible, see \cite{CJT2, CJT4}. \par The next result describes the relation between ``regularity with respect to ${\mathcal B}$\,'' of temperate distributions and global wave-front sets. \par \begin{proposition}\label{mainthm4} Let ${\mathcal B}$ be SG-admissible, and let $f\in{\mathcal S}^\prime (\R^ d)$. Then \begin{equation*} f\in{\mathcal B} \quad \Longleftrightarrow \quad \operatorname{WF}_{{\mathcal B}}(f)=\emptyset. \end{equation*} \end{proposition} \par Theorem \ref{thm:propwfs} extends the analogous result in \cite{CJT4} to the more general case of a classical, SG-hyperbolic involutive operator $L$ of Levi type, and the one in \cite{Taniguchi02} to the global wave-front sets introduced above. It is a consequence of Theorem \ref{thm:mainLastSecThm} and of Theorem 5.17 in \cite{CJT4}. \begin{theorem}\label{thm:propwfs} % Let $L$ in \eqref{eq:4.74} be a classical, SG-hyperbolic, involutive operator of Levi type, that is, of the type considered in Proposition \ref{thm:4.20} with SG-classical coefficients, of the form \eqref{eq:4.117}. Let $g_k\in{\mathcal S}^\prime(\R^n)$, $k=0,\dots,m-1$, and assume that the $m$-tuple $({\mathcal B}_0,\dots,{\mathcal B}_{m-1})$ consists of SG-admissible spaces. Also assume that the SG-admissible space ${\mathcal C}$ is such that $({\mathcal B}_k,{\mathcal C})$, $k=0,\dots,m-1$, are weakly-I SG-ordered pairs with respect to % \[ k-j, k-j, k=0,\dots,m-1, j=0,\dots, k, \phi^{(\alpha)}, \alpha\in M, \text{ and } \emptyset. \] % Then, for the solution $u(t,s)$ of the Cauchy problem \eqref{eq:4.74} with $f\equiv0$, $(t,s)\in\Delta_{T^\prime}$, $s\in[0,T^\prime)$, we find % \begin{equation}\label{eq:wfprop} \operatorname{WF}^k_{\mathcal C}( u(t,s) ) \subseteq \bigcup_{j=1}^m \bigcup_{\alpha\in M^\nmid_j} \bigcup_{\substack{\mathbf{t}_j\in\Delta_j(T^\prime) \\ t_0=t,t_j=s}} \bigcup_{\ell=0}^{m-1} (\Phi_{\alpha}(\mathbf{t}_j)(\operatorname{WF}^k_{{\mathcal B}_\ell} (g_\ell)))^{\mathrm{con}_k} , \quad k=1,2,3, \end{equation} % where $V^{\mathrm{con}_k}$ for $V\subseteq \Omega_k$, is the smallest $k$-conical subset of $\Omega_k$ which includes $V$, $k\in\{1,2,3\}$ and $\Phi_\alpha(\mathbf{t}_j)$ is the canonical transformation of $T^\star \R^{n}$ into itself generated by the parameter-dependent SG-classical phase functions $\phi^{(\alpha)}(\mathbf{t}_j)\in {\cal{P}}^\epsilon $, $\alpha\in M_j^\nmid$, $\mathbf{t}_j\in\Delta_j(T^\prime)$, $t_0=t$, $t_j=s$, $j=1,\dots,m$, appearing in \eqref{eq:soleqordm}. % \end{theorem} \begin{proof} We prove \eqref{eq:wfprop} only for the case $k=3$, since the arguments for the cases $k=1$ and $k=2$ are analogous. Let \[ (x_0,\xi_0)\in \bigcap_{j=1}^m \bigcap_{\alpha\in M^\nmid_j} \bigcap_{\substack{\mathbf{t}_j\in\Delta_j(T^\prime) \\ t_0=t,t_j=s}} \bigcap_{\ell=0}^{m-1} [\Omega_3\setminus(\Phi_{\alpha}(\mathbf{t}_j)(\operatorname{WF}^3_{{\mathcal B}_\ell} (g_\ell)))^{\mathrm{con}_3}]. \] Then, by \eqref{eq:soleqordm} and \cite[Theorem 5.17]{CJT4}, in view of \eqref{eq:4.124} and the hypotheses on ${\mathcal C}$, ${\mathcal B}_k$, $k=0,\dots,m-1$, there exists $c_3\in {S}^{0,0}$, type-$3$ invertible at $(x_0,\xi_0)$, such that $ {\operatorname{Op}} (c_3)(W_\alpha(t,s)G_\alpha)\in{\mathcal C}$, $\alpha\in M^\nmid_j$, $j=1,\dots,m$. In fact, for the terms such that $\alpha\in M^\nmid_1$, this follows by a direct application of Theorem 5.17 in \cite{CJT4}. For the terms with $\alpha\in M^\nmid_j$, $j=2,\dots, m$, we first observe that we can bring $ {\operatorname{Op}} (c_3)$ within the iterated integrals analogous to those appearing in \eqref{eq:reducedfundsol}. Then, in view of the hypothesis on $(x_0,\xi_0)$, again by Theorem 5.17 in \cite{CJT4} and the properties of the operators $W_\alpha$ in \eqref{eq:soleqordm}, the integrand belongs to ${\mathcal C}$ for any $\mathbf{t}_j$, $j=2,\dots,m$, in the integration domain. In view of the smooth dependence on $\mathbf{t}_j$, the iterated integral belongs to ${\mathcal C}$ as well. Since the right-hand side of \eqref{eq:soleqordm} is a finite sum, modulo a term in ${\mathcal S}$, we conclude that $ {\operatorname{Op}} (c_3)(u(t,s))\in {\mathcal C}$, which implies $(x_0,\xi_0)\notin\operatorname{WF}^3_{\mathcal C}( u(t,s) )$ and proves the claim. \end{proof} \begin{remark} % \begin{enumerate} % \item We recall that the canonical transformation generated by an arbitrary regular phase function $\varphi\in {\cal{P}}^\epsilon $ is defined by the relations % \[ (x,\xi )=\Phi_\varphi (y,\eta ) \quad \Longleftrightarrow \quad \begin{cases} y = \varphi^\prime_{\xi} (x,\eta) = \varphi^\prime_{\eta} (x,\eta), \\[1ex] \xi = \varphi^\prime_x (x, \eta). \end{cases} \] % \item \label{point:wfsbichar} Assume that the hypotheses of Theorem \ref{thm:propwfs} hold true. Then $\operatorname{WF}^k_{\mathcal C}(u(t,s))$, $(t,s)\in \Delta_{T^\prime}$, $k=1,2,3$, consists of unions of arcs of bicharacteristics, generated by the phase functions appearing in \eqref{eq:soleqordm} and emanating from points belonging to $\operatorname{WF}^m_{{\mathcal B}_k} (g_k)$, $k=0,\dots,m-1$, cf. \cite{CoMa, Morimoto, Taniguchi02}. % \item The hypotheses on the spaces ${\mathcal B}_k$, $k=0,\dots,m-1$, ${\mathcal C}$, are authomatically fulfilled for ${\mathcal B}_k=H^{r+m-1-k,\varrho+m-1-k}(\R^n)$, ${\mathcal C}=H^{r,\varrho}(\R^n)$, $r,\varrho\in\R$, $k=0,\dots,m-1$. That is, the results in Theorem \ref{thm:propwfs} and in point \ref{point:wfsbichar} above hold true, in particular, for the $\operatorname{WF}^k_{r,\varrho}(u(t,s))$ wave-front sets, $r,\varrho\in\R$, $k=1,2,3$. % \item A result similar to Theorem \ref{thm:propwfs} holds true for the solution $U(t,s)$ of the system \eqref{sys} when $F\equiv0$ and $\Lambda$ and $R$ are matrices of SG-classical operators. % \end{enumerate} \end{remark} \section{Stochastic Cauchy problems for weakly SG-hyperbolic linear operators}\label{sec:sghypspde}% \setcounter{equation}{0} This section is devoted to the proof of the existence of random-field solutions of a stochastic PDE of the form \begin{equation}\label{eq:SPDE} L(t,D_t;x,D_x)u(t,x) = \gamma(t,x) + \sigma(t,x)\dot{\Xi}(t,x), \end{equation} associated with the initial conditions $D_t^ku(0,x)=g_k(x)$, $k=0,\dots,m-1$, for a SG-involutive operator $L$, where $\gamma$ and $\sigma$ are suitable real-valued functions, $\dot\Xi$ is a random noise, described below, and $u$ is an unknown stochastic process called \emph{solution} of the SPDE. Since the sample paths of the solution $u$ are, in general, not in the domain of the operator $L$, in view of the singularity of the random noise, we rewrite \eqref{eq:SPDE} in its corresponding integral (i.e., \textit{weak}) form and look for \emph{mild solutions of \eqref{eq:SPDE}}, that is, stochastic processes $u(t,x)$ satisfying \begin{equation}\label{eq:mildsolutionSPDE} u(t,x) = v_0(t,x)+\int_0^t\int_{\R^n} \Lambda(t,s,x,y)\gamma(s,y)dyds +\int_0^t\int_{\R^n} \Lambda(t,s,x,y)\sigma(s,y)\dot\Xi(s,y)dyds, \end{equation} where $\Lambda$ is a suitable distribution, associated with the fundamental solution of the operator $L$, and in \eqref{eq:mildsolutionSPDE} we adopted the usual abuse of notation involving \textit{distributional integrals}. Based on the results of the previous Section \ref{sec:sginvcp}, and on the analysis in \cite{AsCoSu:1}, we can show that \eqref{eq:mildsolutionSPDE} has a meaning, and we call it the solution of \eqref{eq:SPDE} with the associated initial conditions. \subsection{Stochastic integration with respect to a martingale measure.}\label{subs:stochastics} We recall here the definition of stochastic integral with respect to a martingale measure, using material coming from \cite{AsCoSu:1,AsCoSu:2}, to which we refer the reader for further details. Let us consider a distribution-valued Gaussian process $\{\Xi(\phi);\; \phi\in\mathcal{C}_0^\infty(\mathbb{R}_+\times\R^n)\}$ on a complete probability space $(\Omega, \scrF, \P)$, with mean zero and covariance functional given by \begin{equation} \E[\Xi(\phi)\Xi(\psi)] = \int_0^\infty\int_{\R^n} \big(\phi(t)\ast\tilde{\psi}(t)\big)(x)\,\Gamma(dx) dt, \label{eq:correlation} \end{equation} where $\widetilde{\psi}(t,x) := \psi(t,-x)$, $\ast$ is the convolution operator and $\Gamma$ is a nonnegative, nonnegative definite, tempered measure on $\R^n$. Then \cite[Chapter VII, Th\'{e}or\`{e}me XVIII]{schwartz} implies that there exists a nonnegative tempered measure $\mu$ on $\R^n$ such that $\caF\mu = \widehat{\mu}=\Gamma$. By Parseval's identity, the right-hand side of \eqref{eq:correlation} can be rewritten as \begin{equation*} \E[\Xi(\phi)\Xi(\psi)] = \int_0^{\infty}\int_{\R^n}[\caF\phi(t)](\xi)\ \cdot \overline{[\caF\psi(t)](\xi)}\,\mu(d\xi) dt. \end{equation*} The tempered measure $\Gamma$ is usually called \emph{correlation measure}. The tempered measure $\mu$ such that $\Gamma=\widehat\mu$ is usually called \emph{spectral measure}. In this section we consider the SPDE \eqref{eq:SPDE} and its mild solution \eqref{eq:mildsolutionSPDE}: this is the way in which we understand \eqref{eq:SPDE}; we provide conditions to show that each term on the right-hand side of \eqref{eq:mildsolutionSPDE} is meaningful. \\ In fact, we call \textit{(mild) random-field solution to \eqref{eq:SPDE}} an $L^2(\Omega)$-family of random variables $u(t,x)$, $(t,x)\in[0,T]\times\R^n$, jointly measurable, satisfying the stochastic integral equation \eqref{eq:mildsolutionSPDE}. \\ To give a precise meaning to the stochastic integral in \eqref{eq:mildsolutionSPDE} we define \arraycolsep1.5pt\begin{eqnarray}\label{intm} \displaystyle\int_0^t\int_{\R^n}\Lambda(t,s,x,y)\sigma(s,y)\dot\Xi(s,y)dsdy := \int_0^t\int_\R^n \Lambda(t,s,x,y)\sigma(s,y)M(ds,dy), \end{eqnarray}\arraycolsep5pt where, on the right-hand side, we have a stochastic integral with respect to the martingale measure $M$ related to $\Xi$. As explained in \cite{dalangfrangos}, by approximating indicator functions with $C^\infty_0$-functions, the process $\Xi$ can indeed be extended to a worthy martingale measure $M=(M_t(A);\; t\in\R_+, A\in\scrB_b(\R^n))$, where $\scrB_b(\R^n)$ denotes the bounded Borel subsets of $\R^n$. The natural filtration generated by this martingale measure will be denoted in the sequel by $(\scrF_t)_{t\geq 0}$. The stochastic integral with respect to the martingale measure $M$ of stochastic processes $f$ and $g$, indexed by $(t,x)\in[0,T]\times\R^n$ and satisfying suitable conditions, is constructed by steps (see \cite{conusdalang,dalang,walsh}), starting from the class $\caE$ of simple processes, and making use of the pre-inner product (defined for suitable $f,g$) \begin{equation}\label{eq:norm02} \begin{aligned} \langle f,g\rangle_{0}&= \E\bigg[\int_0^T\int_\R^n \big(f(s)\ast\tilde{g}(s)\big)(x)\,\Gamma(dx) ds \bigg] \\ & = \E\bigg[\int_0^T\int_\R^n [\caF f(s)](\xi)\cdot\overline{[\caF g(s)](\xi)}\,\mu(d\xi) ds\bigg], \end{aligned} \end{equation} with corresponding semi-norm $\|\cdot\|_0$, as follows. \begin{enumerate} \item For a {\it simple process} \[ g(t,x;\omega) = \sum_{j=1}^m 1_{(a_j,b_j]}(t)1_{A_j}(x)X_j(\omega)\in\mathcal E, \] (with $m\in\N$, $0\leq a_j < b_j\leq T$, $A_j\in\scrB_b(\R^n)$, $X_j$ bounded, and $\scrF_{A_j}$-measurable random variable for all $1\leq j\leq n$) the stochastic integral with respect to $M$ is given by \[ (g\cdot M)_t := \sum_{j=1}^m \big(M_{t\wedge b_j}(A_j) - M_{t\wedge a_j}(A_j)\big)X_j, \] where $x\wedge y := \min\{x,y\}$. One can show, by applying the definition, that the fundamental isometry \begin{equation}\label{eq:isometry} \E\big[(g\cdot M)_t^2\big] = \|g\|^2_0 \end{equation} holds true for all $g\in\caE$. \item Since the pre-inner product \eqref{eq:norm02} is well-defined on elements of $\mathcal E$, if now we define $\caP_0$ as the completion of $\caE$ with respect to $\langle\cdot,\cdot\rangle_0$, then, for all the elements $g$ of the Hilbert space $\caP_0$, we can construct the stochastic integral with respect to $M$ as an $L^2(\Omega)$-limit of simple processes via the isometry \eqref{eq:isometry}. So, $\caP_0$ turns out to be the space of all integrable processes (with respect to $M$). Moreover, by Lemma 2.2 in \cite{sanzsuess1} we know that $\caP_0=L^2_{p}([0,T]\times\Omega,\ensuremath{\mathcal{H}})$, where here $L^2_p(\ldots)$ stands for the predictable stochastic processes in $L^2(\ldots)$ and $\ensuremath{\mathcal{H}}$ is the Hilbert space which is obtained by completing the Schwartz functions with respect to the inner product $\langle\cdot,\cdot\rangle_0$. Thus, $\caP_0$ consists of predictable processes which may contain tempered distributions in the $x$-argument (whose Fourier transforms are functions, $\P$-almost surely). \end{enumerate} Now, to give a meaning to the integral \eqref{intm}, we need to impose conditions on the distribution $\Lambda$ and on the coefficient $\sigma$ such that $\Lambda\sigma\in \mathcal P_0$. In \cite{alessiandre}, sufficient conditions for the existence of the integral on the right-hand side of \eqref{intm} have been given, in the case that $\sigma$ depends on the spatial argument $y$, assuming that the spatial Fourier transform of the function $\sigma$ is a complex-valued measure with finite total variation. Namely, we assume that, for all $s\in[0,T]$, \[ |\caF\sigma(\cdot,s)| = |\caF\sigma(\cdot,s)|(\R^n) = \sup_\pi \sum_{A\in\pi} |\caF\sigma(\cdot,s)|(A) < \infty, \] where $\pi$ is any partition on $\R^n$ into measurable sets $A$, and the supremum is taken over all such partitions. Let, in the sequel, $\nu_s:=\caF\sigma(\cdot,s)$, and let $|\nu_s|_\mathrm{tv}$ denote its total variation. We summarize all these conditions in the following theorem; for its proof, see \cite[Theorem 2.6]{alessiandre}. \begin{theorem}\label{thm:existenceanduniquenessconstant} Let $\Delta_T$ be the simplex given by $0\leq t\leq T$ and $0\le s \le t$. Let, for $(t,s,x)\in\Delta_T\times\R^n$, $\Lambda(t,s,x)$ be a deterministic function with values in $ {\mathcal{S}} '(\R^n)_\infty$, the space of rapidly decreasing temperate distributions, and let $\sigma$ be a function in $L^2([0,T],C_b(\R^n))$. Assume that: \begin{enumerate} \item[(A1)]\label{ass:A1} the function $(t,s,x,\xi)\mapsto[\caF\Lambda(t,s,x)](\xi)$ is measurable, the function $s\mapsto\caF\sigma(s)=\nu_s \in L^2([0,T],\mathcal M_b(\R^n))$, and, moreover, \begin{equation}\label{eq:condition1} \int_0^T\bigg(\sup_{\eta\in\R^n} \int_{\R^n} |[\caF\Lambda(t,s,x)](\xi+\eta)|^2 \mu(d\xi)\bigg) |\nu_s|^2_\mathrm{tv} \, ds < \infty; \end{equation} \item[(A2)]\label{ass:A2} $\Lambda$ and $\sigma$ are as in (A1) and \begin{align*} \lim_{h\downarrow0} \int_0^T & \bigg(\sup_{\eta\in\R^n} \int_{\R^n} \sup_{r\in(s,s+h)} |[\caF(\Lambda(t,s,x)-\Lambda(t,r,x))](\xi+\eta)|^2 \mu(d\xi)\bigg) |\nu_s|^2_\mathrm{tv}\,ds = 0. \end{align*} \end{enumerate} Then $\Lambda\sigma\in\caP_0$, so the stochastic integral on the right-hand side of \eqref{intm} is well-defined and \begin{align*} \E\big[((\Lambda(t,\cdot,x,\ast)\sigma(\cdot,\ast))\cdot M)_t^2\big] \leq \int_0^t\bigg(\sup_{\eta\in\R^n} \int_{\R^n} |[\caF\Lambda(t,s,x)](\xi+\eta)|^2 \mu(d\xi)\bigg) |\nu_s|^2_\mathrm{tv}\,ds. \end{align*} \end{theorem} The reason for the assumption that $\Lambda(t)\in {\mathcal{S}} '(\R^n)_\infty$ is that, in this case, the Fourier transform in the second spatial argument is a smooth function of slow growth and the convolution of such a distribution with any other distribution in $\mathcal S'(\R^n)$ is well-defined, see \cite[Chapter VII, Section 5]{schwartz}. A necessary and sufficient condition for $T\in {\mathcal{S}} '(\R^n)_\infty$ is that each regularization of $T$ with a $C^\infty_0$-function is a Schwartz function. This is true in our application, due to next Proposition \ref{prop:ftofschwartzkernel} and the fact that the Fourier transform is a bijection on the Schwartz functions. \begin{proposition}\label{prop:ftofschwartzkernel} Let $A= {\operatorname{Op}} _\varphi(a)$ be a SG Fourier integral operator, with symbol $a\in S^{m,\mu}(\R^{2n})$, $(m,\mu)\in\R^2$, and phase function $\varphi$. Let $K_A$ denote its Schwartz kernel. Then, the Fourier transform with respect to the second argument of $K_A$, $\caF_{\cdot\mapsto\eta}K_{A}(x,\cdot)$, is given by % \begin{equation}\label{eq:ftofschwartzkernel} \caF_{\cdot\mapsto\eta}K_{A}(x,\cdot) = {\mathrm{e}} ^{i \varphi(x,-\eta)}a(x,-\eta). \end{equation} % \end{proposition} In the next subsection we will apply Theorem \ref{thm:existenceanduniquenessconstant}, Proposition \ref{prop:ftofschwartzkernel} and the theory developed in the previous sections to prove the existence of random-field solutions for stochastic PDEs associated with a SG-involutive operator. \subsection{Random field solutions of stochastic linear SG-involutive equations}\label{subs:inv} We conclude the paper with our fifth main result. \begin{theorem}\label{thm:linearin} Let us consider the Cauchy problem \begin{equation}\label{eq:cpstoch} \left\{ \begin{array}{ll} L(t,D_t;x,\partial_x)u(t,x) = \gamma(t,x) + \sigma(t,x)\dot{\Xi}(t,x), & t\in[0,T], x\in\R^n, \\[1ex] D_{t}^k u(0,x) = g_{k}(x), & k=0,\dots,m-1, x\in\R^n, \end{array} \right. \end{equation} for a SPDE associated with a SG-involutive operator $L$ of the type \eqref{eq:4.69}, $m\in\N$, of Levi type, that is, satisfying \eqref{eq:4.117}. Assume also, for the initial conditions, that $g_k\in H^{z+m-k-1, \zeta+m-k-1}(\R^n)$, $0\leq k\leq m-1$, with $z\in\R$ and $\zeta>n/2$. Furthermore, assume the Gaussian noise $\dot{\Xi}$ to be of the type described in Subsection \ref{subs:stochastics}, with the associated spectral measure such that \begin{equation}\label{eq:measinv} \int_{\R^n} \mu(d\xi) < \infty. \end{equation} Finally, assume that $\gamma\in C([0,T]; H^{z,\zeta}(\R^n))$, $\sigma \in C([0,T], H^{0,\zeta}(\R^n))$, and $s\mapsto\mathcal F\sigma(s)=\nu_s \in L^2([0,T],\mathcal M_b(\R^n))$. Then, for some $T^\prime\in(0,T]$, there exists a random-field solution $u$ of \eqref{eq:cpstoch}. Moreover, $\displaystyle\E[u]\in C([0,T^\prime], H^{z,\zeta}(\R^n))$. \end{theorem} \begin{proof} The operator $L$ is of the form considered in Section \ref{sec:sginvcp}. In particular, by Theorem \ref{thm:mainLastSecThm} we know that the solution to \eqref{eq:cpstoch} is formally given by \eqref{eq:soleqordm}, that is \begin{equation}\label{theu} u(t) = \sum_{\alpha\in M}E_{\varnothing \alpha}(t,0) G_{\alpha}+ \sum_{\alpha\in M_{m-1}}\int_0^t E_{\varnothing \alpha}(t,s) f(s)\,ds, \quad t\in[0,T^\prime], \end{equation} where $M=\bigcup_{k=0}^m M_k$ with $M_k$ the permutations of $k$ elements of the set $\{1,\ldots,m\}$, and where $E_{\varnothing \alpha}(t,s)$, $\alpha\in M$, $(t,s)\in\Delta_{T^\prime}$ are (modulo elements with kernels in $C^\infty(\Delta_{T'};\mathcal S)$) suitable linear combinations of parameter-dependent families of iterated integrals of regular SG Fourier integral operators, with phase functions given by (sorted) sharp products of solutions to the eikonal equations associated with the characteristic roots of $L$, and matrix-valued symbols in $S^{(0,0)}$ determined again through the characteristic roots of $L$. Let us insert now $f(t,x)=\gamma(t,x) + \sigma(t,x)\dot{\Xi}(t,x)$ in \eqref{theu}, so that, formally, \begin{equation}\label{eq:sollin1} \begin{aligned} u(t,x)&=v_0(t,x)+\int_0^t\int_{\R^d}\Lambda(t,s,x,y)\gamma(s,y)\,dyds +\int_0^t\int_{\R^d}\Lambda(t,s,x,y)\sigma(s,y)\dot{\Xi}(s,y)\,dyds \\ &=v_0(t,x)+v_1(t,x)+v_2(t,x), \end{aligned} \end{equation} In \eqref{eq:sollin1}, $\Lambda(t,s,x,y)$ is the kernel of the finite sum $\sum_{\alpha\in M_{m-1}}E_{\varnothing \alpha}(t,s)$, that is, $\Lambda$ is (modulo an element of $C^\infty(\Delta_{T'};\mathcal S)$) a linear combination of (iterated integrals of) kernels of parameter-dependent FIOs with symbols of order $(0,0)$. We have already observed (see the last lines of the proof of Theorem \ref{thm:mainLastSecThm}) that $\displaystyle v_0\in C([0,T'],H^{z,\zeta})$. Since, by assumption, $\zeta>\frac{n}{2}$, $v_0$ produces a function which is continuous and $L^2$ with respect to $x\in\R^n$ and $t\in[0,T']$, respectively. We have that $$\forall (t,x)\in[0,T]\times\R^n,\ v_0(t,x)\ is\ finite.$$ Since all the operators $E_{\varnothing\alpha}(t,s)$ in \eqref{theu} are linear and continuous from $H^{z,\zeta}$ to itself, and $\gamma\in C([0,T'],H^{z,\zeta})$, the first integral in \eqref{eq:sollin1} certainly makes sense, and also $\displaystyle v_1\in C([0,T'],H^{z,\zeta})$. This gives: $$\forall (t,x)\in[0,T]\times\R^n,\ v_1(t,x)\ is\ finite.$$ Let us now focus on the term $v_2$. We can rewrite it as \arraycolsep1.5pt\begin{eqnarray}\label{eq:v2} v_2(t,x)=\int_0^t\int_{\R^d}\Lambda(t,s,x,y)\sigma(s,y) M(ds, dy), \end{eqnarray}\arraycolsep5pt where $M$ is the martingale measure associated with the stochastic noise $\Xi$, as defined in Section \ref{subs:stochastics}. By Proposition \ref{prop:ftofschwartzkernel} we then find \begin{equation}\label{yeah} |[\caF_{y\mapsto\eta}\Lambda(t,s,x,\cdot)](\eta)|^2 \leq C_{t,s} \langle x\rangle^{0}\langle\eta\rangle^{0}=C_{t,s}, \end{equation} where $C_{t,s}$ can be chosen to be continuous in $s$ and $t$, since $\Lambda$ differs by an element of $\mathcal C^\infty(\triangle_{T'},\mathcal S)$ from the kernel of a linear combination of (iterated integrals of) kernels of (parameter-dependent) SG FIOs with symbol in $\mathcal C(\triangle_{T'},S^{0,0})$. Using \eqref{yeah}, we get that condition (A1) in Theorem \ref{ass:A1} is satisfied if \[ \int_0^t \left(\sup_{\xi\in\R^n}\int_{\R^n} |[\caF_{y\mapsto\eta}\Lambda(t,s,x,\cdot)](\eta+\xi)|^2\mu(d\eta)\right)|\nu_s|^2_\mathrm{tv}\,ds \lesssim \int_0^t |\nu_s|^2_\mathrm{tv}\, ds \int_{\R^n}\mu(d\eta)<\infty. \] In view of the assumptions on $\sigma$, we conclude that assumption $(A1)$ holds true as long as \eqref{eq:measinv} does. To check the continuity condition $(A2)$ in Theorem \ref{ass:A1}, since $\Lambda$ is regular with respect to $s$ and $t$, it suffices to show that \begin{equation}\label{ourstar} \sup_{r\in(s,s+h)} |\caF(\Lambda(t,s,x)-\Lambda(t,r,x))(\xi+\eta)|^2\leq C_{t,s,h}^2, \end{equation} with $C_{t,s,h}\to 0$ as $h\to 0$ and $C_{t,s,h}\leq C_{T'}$ for every $h\in [0,t-s],$ $(t,s)\in\Delta_{T'}$. Indeed, if \eqref{ourstar} holds, then (A2) holds via Lebesgue's Dominated Convergence Theorem, in view of assumption \eqref{eq:measinv}, the fact that $|\nu_s|^2_\mathrm{tv}\in L^1[0,T]$, and $C_{t,s,h}\leq C_{T_0}$. Then, it only remains to show that \eqref{ourstar} holds true. But this follows from the fact that the function $s\mapsto \caF\Lambda(t,s,\cdot)(\ast)$ is, by \eqref{yeah}, uniformly continuous on $[0,t]$ with values in the Fr\'echet space $S^{0,0}(\R^{2n})$, endowed with the norm $$||a-b||=\displaystyle\sum_{\ell=0}^\infty\frac1{2^\ell}\frac{\|a-b\|_\ell^{0,0}}{1+\|a-b\|_\ell^{0,0}},$$ so its modulus of continuity, \begin{align*} \omega_{t,s}(h)=\sup_{r\in(s,s+h)} ||(\caF\Lambda(t,s,\cdot)(\ast)-\caF\Lambda(t,r,\cdot)(\ast))|| \end{align*} tends to $0$ as $h\to 0$. For more details see \cite{AsCoSu:1}. \indent The argument above shows that we can apply Theorem \ref{ass:A1} to get that $v_2$ is well-defined as a stochastic integral with respect to the martingale measure canonically associated with $\Xi$. Summing up, the random-field solution $u(t,x)$ in \eqref{eq:sollin1} makes sense: its deterministic part is well-defined for every $(t,x)\in[0,T]$ and its stochastic part makes sense as a stochastic integral with respect to a martingale measure.% The regularity claim $\displaystyle\E[u]\in C([0,T'],H^{z,\zeta}(\R^d))$ follows from the regularity properties of the $E_{\varnothing\alpha}$, of $\gamma$ and of the Cauchy data, taking expectation on both sides of \eqref{eq:sollin1}, and recalling the fact that the expected value $\E[v_2]$ of the stochastic integral is zero, being $\Xi$ a Gaussian process with mean zero. It follows that the regularity of $\E[u]$ is the same as the one of the solution of the associated deterministic Cauchy problem. \end{proof} \begin{remark} % One could say that the random-field solution $u$ of \eqref{eq:cpstoch} found in Theorem \ref{thm:linearin} ``is unique'' in the following sense. First, when $\sigma\equiv 0$, it reduces to the unique solution of the deterministic Cauchy problem \eqref{eq:4.74}, with $f\equiv \gamma$ and $s=0$. Moreover, by linearity, if $u_1$ and $u_2$ are two solutions of the linear Cauchy problem \eqref{eq:cpstoch}, $u=u_1-u_2$ satisfies the deterministic equation $Lu=0$ with trivial initial conditions, and such Cauchy problem admits in $\mathcal S'$ only the trivial solution. The latter follows immediately by the $\mathcal S'$ well-posedness (with loss of smoothness and decay) of the Cauchy problem for the homogeneous deterministic linear equation $Lu=0$ proved in Theorem \ref{thm:mainLastSecThm}. % \end{remark} \bibliographystyle{abbrv}

proofpile-arXiv_069-10929

\section{Introduction} We consider the incompressible 3D Navier--Stokes equation in\\$(0,\,T)\times\Omega$, under Lions boundary conditions,% \begin{subequations}\label{sys-u-flat} \begin{align} \partial_t u+\langle u\cdot\nabla\rangle u-\nu\Delta u+\nabla p+h&=0, &\mathop{\rm div}\nolimits u &=0,\label{sys-u-flat-eq}\\ \left.\begin{pmatrix} u\cdot\mathbf n\\ \mathop{\rm curl}\nolimits u-(\mathbf n \cdot \mathop{\rm curl}\nolimits u)\mathbf n\end{pmatrix}\right|_{\partial\Omega}&=\begin{pmatrix}0\\0\end{pmatrix}, &u(0,\,x)&=u_0(x), \end{align} \end{subequations} where~$\Omega\subset\R^3$ is an arbitrary three-dimensional cylinder \[ \Omega = (0,L_1) \times (0,L_2) \times \left( \frac{2L_3}{2 \pi} \mathbb{S}^1 \right), \] whose boundary is denoted by $\partial{\Omega} \coloneqq \Big(\big(\{0,L_1\} \times (0,L_2)\big))\cup\big((0,L_1) \times \{0,L_2\}\big)\Big) \times \left(\frac{{2L_3}}{2 \pi} \mathbb{S}^1 \right)$. As usual $u=(u_1,u_2,u_3)$ and~$p$, defined for $( t,x_1,x_2,x_3)\in I\times\Omega$, are respectively the unknown velocity field and pressure of the fluid, $\nu>0$ is the viscosity, the operators~$\nabla$ and~$\Delta$ are respectively the well known gradient and Laplacian in the space variables $(x_1,x_2,x_3)$, $\langle u\cdot\nabla\rangle v$ stands for $(u\cdot\nabla v_1,u\cdot\nabla v_2,u\cdot\nabla v_3)$, $\mathop{\rm div}\nolimits u\coloneqq \sum_{i=1}^3 \partial_{x_i}u_i$, $\mathop{\rm curl}\nolimits u \coloneqq \left(\partial_{x_2} u_3 - \partial_{x_3}u_2,~\partial_{x_3} u_1 -\partial_{x_1} u_3, \partial_{x_1} u_2 - \partial_{x_2}u_1 \right)^\top $, the vector~$\mathbf n$ stands for the outward unit normal vector to~$\partial\Omega$, and $h$ is a fixed function. Notice that this is equivalent to take appropriate mixed Lions--periodic boundary conditions in the infinite channel~$\mathrm R_{\mathrm C} = (0,L_1) \times (0,L_2) \times \R$: \begin{subequations} \label{eq-BoundCond} \begin{align} \begin{pmatrix}u\cdot\mathbf n\\ \mathop{\rm curl}\nolimits u-(\mathbf n\cdot\mathop{\rm curl}\nolimits u)\mathbf n \end{pmatrix}&=0, \quad\mbox{on}\quad\Bigl(\bigl(\{0,L_1\} \times (0,L_2)\bigr))\cup\bigl((0,L_1) \times \{0,L_2\}\bigr)\Bigr) \times\R,\\ u(x_1,x_2,x_3)&=u(x_1,x_2,x_3+2L_3),\quad (x_1,x_2,x_3)\in\mathrm R_{\mathrm C}. \end{align} \end{subequations} The problem can be described as the model where the fluid is contained in a long (infinite) 3D channel with Lions boundary conditions on the bottom and the top of the channel, and with the periodicity assumption on the long (infinite) direction. Lions boundary condition is a particular case of Navier boundary conditions. For works and motivations concerning Lions and Navier boundary conditions (in both 2D and 3D cases) we refer to~\cite{XiaoXin07,XiaoXin13,PhanRod17,PhanRod18JDCS, ChemetovCiprianoGavrilyuk10,Kelliher06,IlyinTiti06} and references therein. \medskip We set the spaces \begin{align*} H &\coloneqq\{u\in L^2(\Omega,\,\R^3)\mid\mathop{\rm div}\nolimits u=0\mbox{ and }(u\cdot\mathbf n)\left.\kern-2\nulldelimiterspace\right|_{\partial\Omega}=0\}, \\ V&\coloneqq\{u\in H^1(\Omega,\,\R^3)\mid u\in H\}. \end{align*} We consider $H$, endowed with the norm inherited from $L^2(\Omega,\,\R^3)$, as a pivot space, that is, $H=H'$. For $u,v,w\in V$, we define \begin{align} A&\colon V\to V',&\langle A u,v\rangle_{V',V}&\coloneqq \nu(\mathop{\rm curl}\nolimits u,\mathop{\rm curl}\nolimits v)_{L^2(\Omega,\,\R^3)},\label{LStokes}\\ B&\colon V\times V\to V',&\langle B(u,v),w\rangle_{V',V}&\coloneqq -\int_\Omega(\langle u\cdot\nabla\rangle w)\cdot v\,\mathrm d\Omega.\label{Bop} \end{align} The domain of operator $A$ is denoted as \begin{align*} \D(A)\coloneqq\{u\in H^2(\Omega,\,\R^3)\mid u\in H, \quad\mathop{\rm curl}\nolimits u-(\mathbf n\cdot\mathop{\rm curl}\nolimits u)\mathbf n\left.\kern-2\nulldelimiterspace\right|_{\partial\Omega}=0\}. \end{align*} We will refer to~$A$ as the Stokes operator, under Lions boundary conditions. Further, we have the continuous, dense, and compact inclusions~$\D(A)\xhookrightarrow{\rm d,c}V\xhookrightarrow{\rm d,c}H$. \begin{remark} The notation $S\xhookrightarrow{} R$ above means that the inclusion $S \subseteq R$ is continuous. The letter ``$\rm d$'' (respectively ``$\rm c$'') means that, in addition, the inclusion is also dense (respectively compact). \end{remark} \begin{remark} Under the definition of $A$ in \eqref{LStokes}, it turns out that~$\D(A)=\{u\in H\mid Au\in H\}$ is the domain of~$A$. Indeed, by using the formula \begin{align*} \nabla(\mathop{\rm div}\nolimits u) - \mathop{\rm curl}\nolimits(\mathop{\rm curl}\nolimits u) = \Delta u, \end{align*} we can verify that if $u \in \D(A)$, $\mathop{\rm div}\nolimits (\Delta u) = 0$ and $(\Delta u \cdot \mathbf n) = 0$ on $\partial \Omega$. \end{remark} The eigenvalues of $A$, repeated accordingly with their multiplicity, form an increasing sequence $(\underline\lambda_k)_{k\in\N_0}$, \[ 0<\underline\lambda_1\le\underline\lambda_2\le\underline\lambda_3\le\underline\lambda_4\le\dots, \] with $\underline\lambda_k$ going to $+\infty$ with $k$. \medskip We can rewrite system~\eqref{sys-u-flat} as an evolutionary system \begin{equation*} \dot u+A u+B(u,u)+h=\eta,\quad u(0)=u_0, \end{equation*} in the subspace~$H$ of divergence free vector fields which are tangent to the boundary. We may suppose that~$h$ and~$\eta$ take their values in~$H$ (otherwise we just take their orthogonal projections onto~$H$). Denoting by~$\Pi$ the orthogonal projection in $L^2(\Omega,\,\R^3)$ onto $H$, for~$u,v\in\D(A)$ we may write~$Au\coloneqq\Pi(\nu\Delta u)$, and $B(u,\,v)\coloneqq\Pi(\langle u\cdot\nabla\rangle v)$. \subsection{Saturating sets and approximate controllability}\label{sS:saturH} In the pioneering work~\cite{AgraSary05}, the authors introduced a method which led to the controllability of finite-dimensional Galerkin approximations of the~$\mathrm{2D}$ and~$\mathrm{3D}$ Navier--Stokes system, and to the approximate controllability of the~$\mathrm{2D}$ Navier--Stokes system, by means of \\ low modes/degenerate forcing. Hereafter~$U\subseteq H$ will stand for a linear subspace of~$H$, and we denote \[ {\mathcal B}(a,b)\coloneqq B(a,b)+B(b,a), \qquad \text{for} \quad (a,~b) \in U \times U. \] \begin{definition}\label{D:FF-l} Let~${\mathcal C}=\{W_k\mid k\in\{1,\,2,\,\dots,\,M\}\}$ and let~$E$ be a finite-dimensional space so that~${\mathcal C}\subset E\subset U$. The finite-dimensional subspace ${\mathcal F}_{\tt L}(E)\subset U$ is given by \[ {\mathcal F}_{\tt L}(E)\coloneqq E+\mathop{\rm span}\nolimits\{{\mathcal B}(a,b)\mid a\in{\mathcal C},\,b\in E,\,\mbox{ and } (B(a,a),B(b,b))\in H\times H\}\textstyle\bigcap U. \] \end{definition} \begin{definition}\label{D:satur-l} A given finite subset~${\mathcal C}=\{W_k\mid k\in\{1,\,2,\,\dots,\,M\}\} \subset U$ is said $({\tt L},U)$-saturating if for the following sequence of subspaces ~${\mathcal G}^j\subset U$, defined recursively by \begin{align*} {\mathcal G}^0&\coloneqq\mathop{\rm span}\nolimits{\mathcal C},\qquad {\mathcal G}^{j+1}\coloneqq{\mathcal F}_{\tt L}({\mathcal G}^{j}), \end{align*} we have that the union~$\mathop{\bigcup}\limits_{j\in\N}{\mathcal G}^j$ is dense in~$H$. \end{definition} In~\cite[Section~4]{AgraSary06} the authors presented an explicit saturating set for the~$\mathrm{2D}$ Navier--Stokes system. We would like to refer also to the works~\cite{Romito04,HairerMatt06,EMat01}, where the notion of saturating set was used to derive ergodicity for the Navier--Stokes system under degenerate stochastic forcing (compare the sequence of subsets ${\mathcal Z}_n$ in~\cite[section~4]{HairerMatt06} with the sequence of subsets ${\mathcal K}^n$ in~\cite[section~8]{AgraSary05}). In the pioneering work~\cite{AgraSary05} the set~$U$ in~\eqref{D:satur-l} is taken to be~$\D(A)$, the same is done in~\cite{AgraSary06,Rod-Sev05,Rod06,Shirikyan06}. Later, in~\cite{Rod-Thesis08,Rod-wmctf07,PhanRod-ecc15}, $U$ is taken as~$V$ in order to deal either with Navier-type boundary conditions or with internal controls supported in a small subset. In~\cite{Shirikyan06}, the method introduced in~\cite{AgraSary05} was developed in the case where the well-posedness of the Cauchy problem is not known. Though the author focuses on no-slip boundary conditions, i.e. $u\left.\kern-2\nulldelimiterspace\right|_{\partial\Omega}=0$, the results also hold for other boundary conditions. The author considered the case of periodic boundary conditions, and presented an explicit saturating set~${\mathcal C}$ (for the case of~$(1,1,1)$-periodic vectors). This saturating set consisted of ~$64$ eigenfunctions of the Stokes operator (i.e., the Laplacian). For a general period~$q=(q_1,q_2,q_3)\in\R_0^3$ the existence of a saturating set was also proven in~\cite[Section 2.3, Theorem~2.5]{Shirikyan06} even though the form of the saturating set was less explicit. In \cite{PhanRod18JDCS}, the approximate controllability also follows from the existence of a~$({\tt L},\D(A))$-saturating set. For any given length triplet~$L=(L_1,L_2,L_3)$ of a {\rm 3D} rectangle, we presented an explicit $({\tt L},\D(A))$-saturating set~${\mathcal C}$ for the~$\mathrm{3D}$ rectangle~$\Omega=(0,L_1)\times(0,L_2)\times(0,L_3)$ (which will be recalled below). The elements of~${\mathcal C}_{\rm R}$ are~$81$ eigenfunctions of the Stokes operator under Lions boundary conditions. In various works of this topic, to tackle different types of boundary conditions as well as domains, some different definitions of saturating set has been proposed. Here we follow the definition of saturating set as in the previous work (see \cite{PhanRod18JDCS}) because it leads to some advantages in computations. For further results concerning the controllability and approximate controllability of Navier--Stokes (and also other) systems by a control with low finite-dimensional range (independent of the viscosity coefficient) in several domains (including the $\mathrm{2D}$ Sphere and Hemisphere) we refer the readers to~\cite{Nersisyan10,Nersisyan11,Nersesyan10,Shirikyan08,Shirikyan_Evian07,Shirikyan07,Sarychev12,AgraSary06,AgraSary08,AgrKukSarShir07}. We also mention Problem~VII raised by A.~Agrachev in~\cite{Agrachev13_arx} where the author inquired about the achievable controllability properties for controls taking values in a saturating set whose elements are localized/supported in a small subset~$\omega\subset\Omega$. The existence of such saturating sets is an open question (except for~$\mathrm{1D}$ Burgers in ~\cite{PhanRod-ecc15}). The controllability properties implied by such saturating set is an open question. There are some negative results, as for example in the case we consider the~$\mathrm{1D}$ Burgers equations in~$\Omega=(0,1)$ and take controls in~$L^2(\omega,\R)$, $w\subset\Omega$, the approximate controllability fails to hold. Instead, to drive the system from one state~$u_0=u(0)$ at time~$t=0$ to another one~$u_T=u(T)$ at time~$t=T$, we may need $T$ to be big enough. Though we do not consider localized controls here, we refer the reader to the related results in~\cite{FerGue07,Shirikyan17} and references therein. \subsection{The main contribution} We will present an explicit saturating set in the case of three-dimensional cylinder domain. The saturating set consists of finite number of eigenfunctions of Stokes operator (see Theorem \ref{T:satur3Dcyl} hereafter). The saturating set has 355 elements (or a simpler version with 260 elements in corollary \ref{cor-sat-cyl}). In some particular cases, it may exist other saturating sets with less elements. However, we want to emphasize that our goal is not to find a saturating set with minimal number of elements. In all cases, the existence of a $({\tt L},\D(A))$-saturating set must be independent of the viscosity coefficient $\nu$. In particular, the linear space ${\mathcal G}^1$, where the control $\eta$ takes its value, does not change with $\nu$. To construct a saturating set, we firstly introduce a system of eigenfunctions in Section \ref{subsec:eigfun}. In this type of domain, we have two types of eigenfunctions $ Y^{j(k),k}$ and $ Z^{j(k),k}$. The form of eigenfunctions $ Y^{j(k),k}$ are analogous to the ones in 3D Rectangles. Nevertheless the apperance of another type of eigenfunctions $Z^{j(k),k}$ yields to some difficulties. The construction of all eigenfunctions $ Z^{j(k),k}$ is based on the expression of $\left( Z^{j(k),k} \cdot \nabla \right) Y^{j(m),m} + \left( Y^{j(m),m} \cdot \nabla \right) Z^{j(k),k}$ (see Section \ref{subsec_YkZm}). To construct the eigenfunctions $ Z^{j(k),k}$, Lemma \ref{lemma-lin-indp_Z} (a similar version of Lemma 3.1 in \cite{PhanRod18JDCS}) is not enough to prove the linear independence. Therefore another Lemma \ref{lemma-lin-indp2} will be introduced and used mostly in the proof. The Lemma \ref{lemma-lin-indp2} is a fruitful tool to prove the linear independence property in most of the cases (for example in Step \ref{Step352}, equation \eqref{eq-lem-fail}) where Lemma \ref{lemma-lin-indp_Z} fails to prove the property. Another remark comes from the difference of boundary condition in the third direction from other directions. We notice that the procedure can be applied analogously in the first two directions because we consider Lions boundary conditions in the first two directions (see the proof in Section \ref{subsec-proof.genR12}). However, the third direction must be addressed separately in Section \ref{subsec-proof.genR3} because we consider the periodicity assumption in the third direction. In conclusion we believe that the proof in the case of ${\rm 3D}$ Cylinder is inspired from the case of ${\rm 3D}$ Rectangle but it cannot follow line by line. \bigskip The rest of the paper is organized as follows. In Section \ref{sec-pre}, we recall some results of the approximate controllability for 3D Navier-Stokes equations under Lions boundary conditions. An explicit saturating set in the case of three-dimensional Rectangle will be revisited in Section~\ref{sec: Recall3DRect}. In Section~\ref{sec: Sat3DCyl}, we construct a $({\tt L},\D(A))$-saturating set in the case of three-dimensional cylinders. The proof of the main Theorem \ref{T:satur3Dcyl} needs more complicated computations. The core ideas of the proof will be presented immediately. More detailed computations will be found in Section \ref{sec: ProofSat3DCyl}. \section{Preliminaries} \label{sec-pre} \subsection{Approximate controllability} Hereafter $u_0\in V$, ~$h\in L^2_{\rm loc}(\R_0,H)$, and~$E\subset\D(A)$ is a finite-dimensional subspace. Let us consider the system \begin{equation}\label{sys_u_E} \dot u+A u+B(u,u)+h=\eta,\quad u(0)=u_0, \end{equation} where the control~$\eta$ takes its values in~$E$. For simplicity we will denote \[ I_T\coloneqq(0,T),\quad\mbox{and}\quad \overline{I_T}\coloneqq [0,T],\qquad T>0. \] \begin{definition} Let $T$ be a positive constant. System~\eqref{sys_u_E} is said to be $E$-approximately controllable in time~$T$ if for any~$\varepsilon > 0$ and any pair~$(u_0,\hat u)\in V\times \D(A)$, there exists a control function $\eta\in L^\infty(I_T, E)$ and a corresponding solution\\ $u\in C(\overline{I_T},V)\bigcap L^2(I_T, \D(A))$, such that $\norm{u(T)-\hat u}{V} <\varepsilon$. \end{definition} We recall the Main Theorem in \cite{PhanRod18JDCS} which shows the approximate controllability of 3D Navier-Stokes system from the existence of a $({\tt L},\D(A))$-saturating set. \begin{theorem} Let $(u_0,\hat u)\in V\times V$, ~$\varepsilon>0$, and~$T > 0$. If~${\mathcal C}$ is a $({\tt L},\D(A))$-saturating set, then we can find a control~$\eta\in L^\infty((0,T),{\mathcal G}^1)$ so that the solution of system~\eqref{sys_u_E} satisfies $\norm{u(T)-\hat u}{V}<\varepsilon$. \end{theorem} \begin{remark} In \cite{Shirikyan06}, the author introduced a definition of $(\rm B, D(A))$ saturating set and proved that the existence of a $(\rm B, D(A))$ saturating set implies the approximate controllability of the 3D Navier-Stokes systems, at time $T>0$. In \cite{PhanRod18JDCS} and this work, we are using another definition of saturating set, so-called $(\rm L, D(A))$ saturating set. The main advantage to use this definition is in the computation below. \end{remark} \subsection{An explicit saturating set in 3D Rectangles} \label{sec: Recall3DRect} In this section, we recall a $({\tt L},\D(A))$-saturating set containing a finite number of suitable eigenfunctions of the Stokes operator~$A$ in the $\mathrm{3D}$ rectangle \[ R=(0,L_1) \times (0,L_2) \times(0,L_3) \] under Lions boundary conditions (see~\eqref{LStokes}), where $L_1$, $L_2$, and~$L_3$ are positive numbers. For a given $k \in \N^3$, let $\#_0(k)$ stand for the number of vanishing components of $k$. The index $j(k) \in \N$ is such that $1 \le j(k) \le 2 - \#_0(k)$. For example, if $k = (2,1,0)$, $\#_0(k) = 1$ and $j(k) \in \{ 1 \}$; or if $k = (2,1,2)$, $\#_0(k) = 0$ and $j(k) \in \{ 1,2 \}$. \begin{definition} For $x, y \in \R^3\setminus\{(0,0,0)\}$, the scalar product $(\cdot, \cdot)_{[L]}$ is defined as \begin{align} (x,\,y)_{[L]} &\coloneqq \frac{ x_1 y_1}{L_1} + \frac{ x_2 y_2}{L_2} + \frac{x_3 y_3}{L_3}. \end{align} For $k \in \N^3$, we denote the orthogonal space $\{k\}^{\perp_{[L]}}_0$ as follows \begin{align} \{k\}^{\perp_{[L]}}_0&\coloneqq\{z\in\R^3\setminus\{(0,0,0)\}\mid \left( z,\, k \right)_{[L]} =0, \mbox{ and }z_i=0\mbox{ if }k_i=0\}. \end{align} \end{definition} A complete system of eigenfunctions $\left\{ Y_{k} \right\} $ is given by \begin{subequations}\label{FamilyEig3DRect} \begin{equation} \label{FamilyEig3DRect.Y} Y^{j\left(k\right),k} \coloneqq \begin{pmatrix} w_1^{{j\left(k\right),k}} \sin \left( \frac{k_1 \pi x_1}{L_1} \right) \cos \left( \frac{k_2 \pi x_2}{L_2} \right) \cos \left( \frac{ k_3 \pi x_3}{L_3} \right)\\ w_2^{{j\left(k\right),k}} \cos \left( \frac{k_1 \pi x_1}{L_1} \right) \sin \left( \frac{k_2 \pi x_2}{L_2} \right) \cos \left( \frac{k_3 \pi x_3}{L_3} \right)\\ w_3^{{j\left(k\right),k}} \cos \left( \frac{k_1 \pi x_1}{L_1} \right) \cos \left( \frac{k_2 \pi x_2}{L_2} \right) \sin \left( \frac{k_3 \pi x_3}{L_3} \right) \end{pmatrix},\quad \#_0(k)\le1, \end{equation} with \begin{equation} \{ w ^{j(k),k}\mid j(k)\in\{1,2-\#_0(k)\}\}\subset\{k\}^{\perp_{[L]}}_0 \end{equation} a linearly independent and orthogonal family. \end{subequations} Notice that $2 - \#_0(k)$ is the dimension of the subspace $\{ k \}^{\perp_{[L]}}_0$ and that the orthogonality of the family~$\{ w ^{j(k),k}\mid j(k)\in\{1,2-\#_0(k)\}\}$ implies that the family in~\eqref{FamilyEig3DRect.Y} is also orthogonal. We recall the result in \cite[Theorem 3.1]{PhanRod18JDCS} about saturating set in 3D rectangles \begin{theorem}\label{T:satur3Drect} The set~$~{\mathcal C}_{\rm Y} \coloneqq\left\{ Y^{j(n),n} \left|\begin{array}{l}n\in\N^3,\quad 0 \le n_i \le 3,\\ \#_0(n) \le 1,\quad j(n) \in \{ 1, 2-\#_0(n) \}\end{array} \right. \right\}$ is $({\tt L},\D(A))$-saturating. \end{theorem} The theorem plays a remarkable role to prove the saturating set in 3D-cylinder case in Section \ref{sec: Sat3DCyl}. Particularly, the inclusion \eqref{Sat_Inc_CY} will be obtained directly from this theorem. \section{A saturating set in the 3D-cylinder case} \label{sec: Sat3DCyl} \subsection{A system of eigenfunctions} \label{subsec:eigfun} The system of eigenfunctions in the 3D-cylinder case consists two types of eigenfunctions $Y^{j\left(k\right),k}$ and $ Z^{j\left(k\right),k}$. The first type of eigenfunctions was defined before in \eqref{FamilyEig3DRect.Y}. Another type of eigenfunctions $Z^{j\left(k\right),k}$ is defined as follows \begin{align} \label{FamilyEig3DCylT2} Z^{j\left(k\right),k} &= \begin{pmatrix} w_1^{{j\left(k\right),k}} \sin \left( \frac{k_1 \pi x_1}{L_1} \right) \cos \left( \frac{k_2 \pi x_2}{L_2} \right) \sin \left( \frac{k_3 \pi x_3}{L_3} \right)\\ w_2^{{j\left(k\right),k}} \cos \left( \frac{k_1 \pi x_1}{L_1} \right) \sin \left( \frac{k_2 \pi x_2}{L_2} \right) \sin \left( \frac{k_3 \pi x_3}{L_3} \right)\\ -w_3^{{j\left(k\right),k}} \cos \left( \frac{k_1 \pi x_1}{L_1} \right) \cos \left( \frac{k_2 \pi x_2}{L_2} \right) \cos \left( \frac{k_3 \pi x_3}{L_3} \right) \end{pmatrix}, \end{align} either with $\#_0(k)\le1$, or with $\#_0(k)=2$ and $k_3=0$, or with $\#_0(k)=3$. Furthermore we assume that the vectors $w ^{j(k),k}$ are chosen satisfying \begin{itemize} \item if~$\#_0(k)= 3$, $w ^{j(k),k} = (0,0,-1)$. Then $Z^{(0,0,0)} = (0,0,1)^\top$. \item if~$\#_0(k)= 2$ and~$k_3=0$, $w ^{j(k),k}=w ^{1,k}=(0,0,w ^{1,k}_3)$, with~$w ^{1,k}_3\ne0$, \item if~$\#_0(k)\le 1$, then $ w ^{j(k),k} \in \{k\}^{\perp_{[L]}}_0. $ \end{itemize} \begin{remark} Even though the domain~$\Omega$ is different, namely~$x\in R=(0,L_1) \times (0,L_2) \times(0,L_3)$ and $x\in\Omega \sim(0,L_1) \times (0,L_2) \times(0,2L_3)$, the eigenfunctions $Y^{j\left(k\right),k}$ are the same. \end{remark} These functions of the forms \eqref{FamilyEig3DRect.Y} and \eqref{FamilyEig3DCylT2} are eigenfunctions of the shifted Stokes operator~$A$ in~$\Omega$ under the boundary conditions \eqref{eq-BoundCond}. Indeed, it is clear that they are eigenfunctions of the usual Laplacian operator. So it remains to check that they are divergence-free and satisfy the boundary conditions \eqref{eq-BoundCond}. The divergence free condition follows from the choices of the vectors~$w^{j(k),k}$. It is also clear that $u\cdot\mathbf n$ vanishes at the boundary~$\partial\Omega$. Finally we can see that the $\mathop{\rm curl}\nolimits$ is normal to the boundary, from the expressions \begin{align*} \mathop{\rm curl}\nolimits Y ^{j(k),k} &= -\pi\begin{pmatrix}\left(\textstyle\frac{k_2}{L_2} w ^{j(k),k}_3-\textstyle\frac{k_3}{L_3} w ^{j(k),k}_2\right) \cos(\textstyle\frac{k_1\pi x_1}{L_1})\sin(\textstyle\frac{k_2\pi x_2}{L_2})\sin(\textstyle\frac{k_3\pi x_3}{L_3})\\ \left(\textstyle\frac{k_3}{L_3} w ^{j(k),k}_1-\textstyle\frac{k_1}{L_1} w ^{j(k),k}_3\right) \sin(\textstyle\frac{k_1\pi x_1}{L_1})\cos(\textstyle\frac{k_2\pi x_2}{L_2})\sin(\textstyle\frac{k_3\pi x_3}{L_3})\\ \left(\textstyle\frac{k_1}{L_1} w ^{j(k),k}_2-\textstyle\frac{k_2}{L_2} w ^{j(k),k}_1\right) \sin(\textstyle\frac{k_1\pi x_1}{L_1})\sin(\textstyle\frac{k_2\pi x_2}{L_2})\cos(\textstyle\frac{k_3\pi x_3}{L_3}) \end{pmatrix},\\ \mathop{\rm curl}\nolimits Z^{j(k),k} &= \pi \begin{pmatrix} \left( \frac{k_2}{L_2} w_3^{{j\left(k\right),k}} - \frac{k_3}{L_3} w_2^{{j\left(k\right),k}} \right) \cos \left( \frac{k_1 \pi x_1}{L_1} \right) \sin \left( \frac{k_2 \pi x_2}{L_2} \right) \cos \left( \frac{k_3 \pi x_3}{L_3} \right) \\ \left( \frac{k_3}{L_3} w_1^{{j\left(k\right),k}} - \frac{k_1}{L_1} w_3^{{j\left(k\right),k}} \right) \sin \left( \frac{k_1 \pi x_1}{L_1} \right) \cos \left( \frac{k_2 \pi x_2}{L_2} \right) \cos \left( \frac{k_3 \pi x_3}{L_3} \right) \\ \left( \frac{k_2}{L_2} w_1^{{j\left(k\right),k}} - \frac{k_1}{L_1} w_2^{{j\left(k\right),k}} \right) \sin \left( \frac{k_1 \pi x_1}{L_1} \right) \sin \left( \frac{k_2 \pi x_2}{L_2} \right) \sin \left( \frac{k_3 \pi x_3}{L_3} \right) \end{pmatrix}, \end{align*} which we can derive by direct computations. For example, at the lateral boundary~$x_1=L_1$, that is, for $x\in \{L_1\} \times (0,L_2)\times\frac{2L_3}{2 \pi} \mathbb{S}^1$, we have $\mathbf n=(1,0,0)$ and \begin{align*} \mathop{\rm curl}\nolimits Y ^{j(k),k} &= -\pi\begin{pmatrix}\left(\textstyle\frac{k_2}{L_2} w ^{j(k),k}_3-\textstyle\frac{k_3}{L_3} w ^{j(k),k}_2\right) \sin(\textstyle\frac{k_2\pi x_2}{L_2})\sin(\textstyle\frac{k_3\pi x_3}{L_3})\\ 0\\ 0 \end{pmatrix},\\ \mathop{\rm curl}\nolimits Z^{j(k),k} &= \pi \begin{pmatrix} \left( \frac{k_2}{L_2} w_3^{{j\left(k\right),k}} - \frac{k_3}{L_3} w_2^{{j\left(k\right),k}} \right) \sin \left( \frac{k_2 \pi x_2}{L_2} \right) \cos \left( \frac{k_3 \pi x_3}{L_3} \right) \\ 0 \\ 0 \end{pmatrix}, \end{align*} which show that~$\mathop{\rm curl}\nolimits Y ^{j(k),k}$ and~$\mathop{\rm curl}\nolimits Z^{j(k),k}$ have the same direction as the normal vector~$\mathbf n$. \begin{lemma} The system of eigenfunctions \begin{align*} &\{ Y^{j(k), k},~ Z^{j(k),k} \mid k\in \N^3\mbox{ and }\#_0(k) \le 1 \}\\ & {\textstyle\bigcup} \{Z^{1,k} \mid k \in \N^3,~\#_0(k)=2,\mbox{ and }k_3=0\} ~ {\textstyle\bigcup} \left\{Z^{(0,0,0)} =(0,0,1)^\top \right \} \end{align*} is complete. \end{lemma} \begin{proof} Recalling that, for~$r>0$, \[ \{\sin(\textstyle\frac{k\pi x_1}{r})\mid k\in\N_0\}\quad\mbox{and}\quad\{\cos(\textstyle\frac{k\pi x_1}{r})\mid k\in\N\} \] are two complete systems in~$L^2((0,r),\R)$. And \[ \{\sin(\textstyle\frac{k\pi x_1}{r})\mid k\in\N_0\}\bigcup\{\cos(\textstyle\frac{k\pi x_1}{r})\mid k\in\N\} \] is a complete system in~$L^2((0,2r),\R)$. Then the proof can be done by following the arguments in~\cite[Section 6.6]{PhanRod17}. We skip the details. \end{proof} Now we can present the saturating set. \begin{theorem}\label{T:satur3Dcyl} The set of eigenfuntions~ \begin{align*} {\mathcal C} \coloneqq &\left\{ Y^{j(n),n} \mid~n\in\N^3,~\#_0(n)\le1,~n_i\le 4,~j(n) \in \{ 1, 2-\#_0(n) \}\right\} \\ & {\textstyle\bigcup} \left\{ Z^{j(n),n} \mid~n\in\N^3,~\#_0(n)\le1,~n_i\le {4},~j(n) \in \{ 1, 2-\#_0(n) \}\right\} \\ & {\textstyle\bigcup} \left\{ Z^{j(n),n} \mid~n \in \N^3,~\#_0(n)=2,~n_3=0 \right\} ~ {\textstyle\bigcup} \left\{Z^{(0,0,0)} =(0,0,1)^\top \right \} \end{align*} is $({\tt L},\D(A))$-saturating. \end{theorem} \begin{proof} Firstly, let us denote $\N_4 \coloneqq \{q \in \N \mid q \ge 4 \}$. We recall the index subsets ${\mathcal S}^q_{\mathrm R},\,{\mathcal R}^q_m,\,{\mathcal L}^q_{m_1,m_2}$ defined in the proof of rectangle case (see \cite[Section 3.4]{PhanRod18JDCS}) for $q \in \N_4 \coloneqq \{q \in \N \mid q \ge 4 \}$. \begin{equation}\label{setsSC_R} \begin{split} {\mathcal S}^q_{\rm R} &\coloneqq \left\{ n\in\N^3 \mid 0 \le n_i \le q,~ \#_0(n) \le 1 \right\},\\ {\mathcal C}^q_{\rm R} &\coloneqq \left\{ Y^{j(n),n} \mid~n \in {\mathcal S}^q_{\rm R},~ j(n) \in \{ 1, 2-\#_0(n) \} \right\},\\ {\mathcal R}^q_m &\coloneqq \left\{ n \in {\mathcal S}^q_{\rm R} \mid n_m = q,~0 \le n_i \le q-1~\mbox{for }i \neq m \right\},\\ {\mathcal L}^q_{m_1,m_2} & \coloneqq \left\{ n \in {\mathcal S}^q_{\rm R} \mid ~n_{m_1} = q= n_{m_2},~m_1\ne m_2,~0 \le n_i \le q-1,~i \notin \{m_1,m_2\} \right\}. \end{split} \end{equation} Next, we define some new sets \begin{equation}\label{setsSC_C} \begin{array}{rcl} {\mathcal S}^q_{\rm C} &\coloneqq& {\mathcal S}^q_{\rm R} \cup \{ (n_1,0,0), (0,n_2,0) \mid 0 < n_1 \le q,~0 < n_2 \le q \},\\ {\mathcal C}^q_{\rm C} &\coloneqq& \left\{ Y^{j(n),n},\, Z^{j(n),n} \mid~n \in {\mathcal S}^q_{\rm R},~ j(n) \in \{ 1, 2-\#_0(n) \} \right\}\\ & &\bigcup \left\{ Z^{j(n),n} \mid~n \in {\mathcal S}^q_{\rm C} \setminus {\mathcal S}^q_{\rm R},~ j(n) =1 \right\} \bigcup \left\{ Z^{(0,0,0)}\right\}. \end{array}\hspace*{-1em} \end{equation} We can see that Theorem~\ref{T:satur3Dcyl} is a corollary of the following inclusions \begin{equation}\label{Sat_Inc_C} {\mathcal C}^q_{\rm C}\subseteq {\mathcal G}^{q-1},\qquad \mbox{for all}\quad q\in\N_4. \end{equation} Let us decompose~${\mathcal C}^q_{\rm C}={\mathcal C}^q_{Y}\cup{\mathcal C}^q_{Z}$ with \begin{subequations}\label{splitCYZ} \begin{align} {\mathcal C}^q_{Y}&\coloneqq\left\{ Y^{j(n),n} \mid~n \in {\mathcal S}^q_{\rm R},~ j(n) \in \{ 1, 2-\#_0(n) \} \right\}\\ {\mathcal C}^q_{Z}&\coloneqq \left\{ Z^{j(n),n} \mid~n \in {\mathcal S}^q_{\rm R},~ j(n) \in \{ 1, 2-\#_0(n) \} \right\} \nonumber \\ &\qquad {\textstyle\bigcup} \left\{ Z^{1,n} \mid ~n \in {\mathcal S}^q_{\rm C}\setminus {\mathcal S}^q_{\rm R} \right\} {\textstyle\bigcup} \left\{ Z^{(0,0,0)}\right\}. \end{align} \end{subequations} Inspiring from the proof of Theorem \ref{T:satur3Drect}, we get \begin{equation}\label{Sat_Inc_CY} {\mathcal C}^q_{Y}\subseteq {\mathcal G}^{q-1},\qquad \mbox{for all}\quad q\in\N_4. \end{equation} So it remains to prove that \begin{equation}\label{Sat_Inc_CZ} {\mathcal C}^q_{Z}\subseteq {\mathcal G}^{q-1},\qquad \mbox{for all}\quad q\in\N_4, \end{equation} The inclusion \eqref{Sat_Inc_CZ} is proved by induction argument. \\ \medskip \textbf{Base step} By definition we have that ${\mathcal C}={\mathcal C}^4_{\rm C}\supset{\mathcal C}^4_{Z}$ and~$\mathop{\rm span}\nolimits{\mathcal C}={\mathcal G}^0$. Therefore \begin{equation} \mbox{Inclusion~\eqref{Sat_Inc_CZ} holds for }q=4. \label{BaseIndStep_C} \end{equation} \textbf{Induction Step} The induction hypothesis is \addtocounter{equation}{1} \begin{equation}\label{IH.C} \mbox{ ${\mathcal C}^4_{Z}\subseteq {\mathcal G}^{0}$ and the inclusion ${\mathcal C}^q_{Z}\subseteq {\mathcal G}^{q-1}$ holds true for a given $q\in\N_4$.} \end{equation} We want to prove that ${\mathcal C}^{q+1}_{Z}\subseteq {\mathcal G}^{q}$. Notice that \begin{equation} \label{subset-sq-3Dcyl} \begin{split} {\mathcal S}^{q+1}_{\rm C} = &{\mathcal S}^{q}_{\rm C} \bigcup \left( {\mathcal R}^{q+1}_1\cup{\mathcal R}^{q+1}_2\cup{\mathcal R}^{q+1}_3 \right) \bigcup \{ (q+1, 0, 0),\, (0,q+1, 0) \}\\ &\bigcup \left( {\mathcal L}^{q+1}_{1,2} \cup {\mathcal L}^{q+1}_{2,3} \cup {\mathcal L}^{q+1}_{3,1} \right) \bigcup \{ (q+1, q+1, q+1) \}. \end{split} \end{equation} The decomposition \eqref{subset-sq-3Dcyl} is sketched in Figure \ref{fig_induction}. Based on this decomposition \eqref{subset-sq-3Dcyl},we introduce five Lemmas below. \begin{lemma} \label{lem-genR3} $Z^{j(n),n} \in {\mathcal G}^q$ for all $ n \in {\mathcal R}^{q+1}_3$. \end{lemma} \begin{lemma} \label{lem-genR12} $Z^{j(n),n} \in {\mathcal G}^q$ for all $ n \in\left( {\mathcal R}^{q+1}_1\cup{\mathcal R}^{q+1}_2 \right)$. \end{lemma} \begin{lemma} \label{lem-gen00} $Z^{j(n),n} \in {\mathcal G}^q$ for all $ n \in \{ (q+1, 0, 0),\, (0,q+1, 0) \}$. \end{lemma} \begin{lemma} \label{lem-genL} $Z^{j(n),n} \in {\mathcal G}^q$ for all $ n \in \left( {\mathcal L}^{q+1}_{1,2} \cup {\mathcal L}^{q+1}_{2,3} \cup {\mathcal L}^{q+1}_{3,1} \right)$. \end{lemma} \begin{lemma} \label{lem-genq+1} $Z^{\{1,2\},(q+1,q+1,q+1)} \subset {\mathcal G}^q$. \end{lemma} \begin{figure}[ht] \centering \includegraphics[width=\textwidth]{Induction} \caption{Induction Step based on the decomposition \eqref{subset-sq-3Dcyl}.} \label{fig_induction} \end{figure} Under the composition \eqref{subset-sq-3Dcyl}, we will sequentially construct eigenfunctions whose indices belong to each parts of the decomposition. The first part containing three rectangles ${\mathcal R}^{q+1}_1,~{\mathcal R}^{q+1}_2,~{\mathcal R}^{q+1}_3$ will be considered in Lemmas \ref{lem-genR3} and \ref{lem-genR12}. We need to consider the third direction separately due to the difference of boundary condition in this direction. The Lemma \ref{lem-gen00} deals with two special eigenfunctions which did not appear in the proof of Theorem 3.1 in \cite{PhanRod18JDCS} (the indices are two green points in Figure \ref{fig_induction}). The Lemma \ref{lem-genL} tackles these eigenfunctions whose indices belong to three blue lines ${\mathcal L}^{q+1}_{1,2},~{\mathcal L}^{q+1}_{2,3},~{\mathcal L}^{q+1}_{3,1}$ in Figure \ref{fig_induction}. The last Lemma \ref{lem-genq+1} will take on two eigenfunctions whose index is $(q+1, q+1, q+1)$ (the orange point in Figure \ref{fig_induction}). To prove the Lemmas \ref{lem-genR3}, \ref{lem-genR12}, and \ref{lem-genL}, we will process them by induction. The particular proofs of all Lemmas will be presented in following Section \ref{sec: ProofSat3DCyl}. Following all results of Lemmas \ref{lem-genR3}--\ref{lem-genq+1}, we conclude that ${\mathcal C}^{q+1}_{Z\rm C}\subseteq {\mathcal G}^{q}$. By induction hypothesis and \eqref{BaseIndStep_C}, we can conclude that the inclusion \eqref{Sat_Inc_CZ} hold true, which implies the statement of Theorem \ref{T:satur3Dcyl}. \end{proof} \medskip To prove Lemmas \ref{lem-genR3}--\ref{lem-genq+1}, we will next introduce some fruitful tools. \subsection{The expression for $(Y^k \cdot \nabla) Z^m + (Z^m \cdot \nabla) Y^k $.} \label{subsec_YkZm} Here we will present the expression for the coordinates of $\left( Y^{j(k),k} \cdot \nabla \right) Z^{j(m),m} + \left( Z^{j(m),m} \cdot \nabla \right) Y^{j(k),k}$ for given eigenfunctions in~\eqref{FamilyEig3DRect.Y} and \eqref{FamilyEig3DCylT2}. In order to shorten the following expressions and simplify the writing, we will write \begin{align}Y^{k}=Y^{j\left(k\right),k},\quad Y^{m}=Y^{j\left(m\right),m},\quad w ^{k}=w ^{j(k),k},\quad \mbox{and}\quad w ^{m}=w ^{j(m),m} \end{align} by neglecting the indices $j(k),j(m)$. We will also denote \[ \textstyle {\rm C}_i(k_i)\coloneqq \cos \left( \frac{k_i \pi x_i}{L_i} \right)\quad\mbox{and}\quad {\rm S}_i(k_i)\coloneqq \sin \left( \frac{k_i \pi x_i}{L_i} \right),\qquad \text{for~~} i\in \{1,2,3\}. \] Proceeding as in the case of the rectangle (see \cite[Section 3.1]{PhanRod18JDCS}), we can obtain \begin{align*} \left( Y^k \cdot \nabla \right) Z^m &= \left( \begin{matrix} Y^k \cdot w^m_1 \left( \ \begin{matrix} \frac{m_1\pi}{L_1} {\rm C}_1(m_1) {\rm C}_2(m_2) {\rm S}_3(m_3) \\ -\frac{m_2\pi}{L_2} {\rm S}_1(m_1) {\rm S}_2(m_2) {\rm S}_3(m_3) \\ \frac{m_3\pi}{L_3} {\rm S}_1(m_1) {\rm C}_2(m_2) {\rm C}_3(m_3) \end{matrix} \right) \\ Y^k \cdot w^m_2 \left( \ \begin{matrix} -\frac{m_1\pi}{L_1} {\rm S}_1(m_1) {\rm S}_2(m_2) {\rm S}_3(m_3) \\ \frac{m_2\pi}{L_2} {\rm C}_1(m_1) {\rm C}_2(m_2) {\rm S}_3(m_3) \\ \frac{m_3\pi}{L_3} {\rm C}_1(m_1) {\rm S}_2(m_2) {\rm C}_3(m_3) \end{matrix} \right)\\ Y^k \cdot w^m_3 \left( \ \begin{matrix} \frac{m_1\pi}{L_1} {\rm S}_1(m_1) {\rm C}_2(m_2) {\rm C}_3(m_3) \\ \frac{m_2\pi}{L_2} {\rm C}_1(m_1) {\rm S}_2(m_2) {\rm C}_3(m_3) \\ \frac{m_3\pi}{L_3}{\rm C}_1(m_1) {\rm C}_2(m_2) {\rm S}_3(m_3) \end{matrix} \right) \end{matrix} \right), \\ \left( Z^m \cdot \nabla \right) Y^k &= \left( \begin{matrix} Z^m \cdot w^k_1 \left( \ \begin{matrix} \frac{k_1\pi}{L_1} {\rm C}_1(k_1) {\rm C}_2(k_2) {\rm C}_3(k_3) \\ -\frac{k_2\pi}{L_2} {\rm S}_1(k_1) {\rm S}_2(k_2) {\rm C}_3(k_3) \\ -\frac{k_3\pi}{L_3} {\rm S}_1(k_1) {\rm C}_2(k_2) {\rm S}_3(k_3) \end{matrix} \right) \\ Z^m \cdot w^k_2 \left( \ \begin{matrix} -\frac{k_1\pi}{L_1} {\rm S}_1(k_1) {\rm S}_2(k_2) {\rm C}_3(k_3) \\ \frac{k_2\pi}{L_2} {\rm C}_1(k_1) {\rm C}_2(k_2) {\rm C}_3(k_3) \\ -\frac{k_3\pi}{L_3} {\rm C}_1(k_1) {\rm S}_2(k_2) {\rm S}_3(k_3) \end{matrix} \right)\\ Z^m \cdot w^k_3 \left( \ \begin{matrix} -\frac{k_1\pi}{L_1} {\rm S}_1(k_1) {\rm C}_2(k_2) {\rm S}_3(k_3) \\ -\frac{k_2\pi}{L_2} {\rm C}_1(k_1) {\rm S}_2(k_2) {\rm S}_3(k_3) \\ \frac{k_3\pi}{L_3} {\rm C}_1(k_1) {\rm C}_2(k_2) {\rm C}_3(k_3) \end{matrix} \right) \end{matrix} \right), \end{align*} We next denote coefficient $\beta^{\star_1\star_2\star_3}_{w^{k},m}$ as follows \begin{align} \label{BetaStar} \beta^{\star_1\star_2\star_3}_{w^{k},m} \coloneqq \frac{\pi}{8} \textstyle\left( \star_1~\frac{w_1^{k} m_1}{L_1}~\star_2~\frac{w_2^{k} m_2}{L_2}~\star_3~\frac{w_3^{k}m_3 }{L_3}\right), \qquad\mbox{for}\quad {(\star_1, \star_2, \star_3) \in \{+,- \}^3.} \end{align} For example, we have \begin{align*} \beta^{+++}_{w^{k},m} \coloneqq \frac{\pi}{8} \textstyle\left( +\frac{w_1^{k} m_1}{L_1} + \frac{w_2^{k} m_2}{L_2} + \frac{w_3^{k}m_3 }{L_3}\right), \quad \beta^{-+-}_{w^{k},m} \coloneqq \frac{\pi}{8} \textstyle\left( -\frac{w_1^{k} m_1}{L_1} + \frac{w_2^{k} m_2}{L_2} - \frac{w_3^{k}m_3 }{L_3}\right). \end{align*} By straightforward computation, we can find the expression for the coordinates of $\left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k$ as follows \begin{subequations}\label{CoorYZ_Cyl} \begin{gather} \begin{split} \label{1stCoorMix} &\left( \left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k \right)_1\\ &\hspace{4em}= \left(w_1^{m}\beta^{+++}_{w^k,m} + w_1^{k} \beta^{+++}_{w^m,k} \right) {\rm S}_1 (k_1 + m_1) {\rm C}_2(k_2 + m_2) {\rm S}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( -w_1^{m}\beta^{+++}_{w^k,m} + w_1^{k} \beta^{+++}_{w^m,k}\right) {\rm S}_1 (k_1 - m_1) {\rm C}_2(k_2 - m_2) {\rm S}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( - w_1^{m}\beta^{++-}_{w^k,m} - w_1^{k} \beta^{++-}_{w^m,k} \right) {\rm S}_1 (k_1 + m_1) {\rm C}_2(k_2 + m_2) {\rm S}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( w_1^{m}\beta^{++-}_{w^k,m} - w_1^{k} \beta^{++-}_{w^m,k} \right) {\rm S}_1 (k_1 - m_1) {\rm C}_2(k_2 - m_2) {\rm S}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( w_1^{m}\beta^{+-+}_{w^k,m} + w_1^{k} \beta^{+-+}_{w^m,k} \right) {\rm S}_1 (k_1 + m_1) {\rm C}_2(k_2 - m_2) {\rm S}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( -w_1^{m}\beta^{+-+}_{w^k,m} + w_1^{k} \beta^{+-+}_{w^m,k} \right) {\rm S}_1 (k_1 - m_1) {\rm C}_2(k_2 + m_2) {\rm S}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( -w_1^{m}\beta^{+--}_{w^k,m} - w_1^{k} \beta^{+--}_{w^m,k} \right) {\rm S}_1 (k_1 + m_1) {\rm C}_2(k_2 - m_2) {\rm S}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( w_1^{m} \beta^{+--}_{w^k,m} - w_1^{k} \beta^{+--}_{w^m,k}\right) {\rm S}_1 (k_1 - m_1) {\rm C}_2(k_2 + m_2) {\rm S}_3 (k_3 + m_3), \end{split} \end{gather} \begin{gather} \begin{split} \label{2ndCoorMix} &\left( \left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k \right)_2\\ &\hspace{4em}= \left(w_2^{m}\beta^{+++}_{w^k,m} + w_2^{k} \beta^{+++}_{w^m,k} \right) {\rm C}_1 (k_1 + m_1) {\rm S}_2(k_2 + m_2) {\rm S}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( -w_2^{m}\beta^{+++}_{w^k,m} + w_2^{k} \beta^{+++}_{w^m,k}\right) {\rm C}_1 (k_1 - m_1) {\rm S}_2(k_2 - m_2) {\rm S}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( - w_2^{m}\beta^{++-}_{w^k,m} - w_2^{k} \beta^{++-}_{w^m,k} \right) {\rm C}_1 (k_1 + m_1) {\rm S}_2(k_2 + m_2) {\rm S}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( w_2^{m}\beta^{++-}_{w^k,m} - w_2^{k} \beta^{++-}_{w^m,k} \right) {\rm C}_1 (k_1 - m_1) {\rm S}_2(k_2 - m_2) {\rm S}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( -w_2^{m}\beta^{+-+}_{w^k,m} + w_2^{k} \beta^{+-+}_{w^m,k} \right) {\rm C}_1 (k_1 + m_1) {\rm S}_2(k_2 - m_2) {\rm S}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( w_2^{m}\beta^{+-+}_{w^k,m} + w_2^{k} \beta^{+-+}_{w^m,k} \right) {\rm C}_1 (k_1 - m_1) {\rm S}_2(k_2 + m_2) {\rm S}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( w_2^{m}\beta^{+--}_{w^k,m} - w_2^{k} \beta^{+--}_{w^m,k} \right) {\rm C}_1 (k_1 + m_1) {\rm S}_2(k_2 - m_2) {\rm S}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( -w_2^{m} \beta^{+--}_{w^k,m} - w_2^{k} \beta^{+--}_{w^m,k}\right) {\rm C}_1 (k_1 - m_1) {\rm S}_2(k_2 + m_2) {\rm S}_3 (k_3 + m_3), \end{split} \end{gather} \begin{gather} \begin{split} \label{3rdCoorMix} &\left( \left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k \right)_3\\ &\hspace{4em}= \left(-w_3^{m}\beta^{+++}_{w^k,m} - w_3^{k} \beta^{+++}_{w^m,k} \right) {\rm C}_1 (k_1 + m_1) {\rm C}_2(k_2 + m_2) {\rm C}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( w_3^{m}\beta^{+++}_{w^k,m} - w_3^{k} \beta^{+++}_{w^m,k}\right) {\rm C}_1 (k_1 - m_1) {\rm C}_2(k_2 - m_2) {\rm C}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( - w_3^{m}\beta^{++-}_{w^k,m} + w_3^{k} \beta^{++-}_{w^m,k} \right) {\rm C}_1 (k_1 + m_1) {\rm C}_2(k_2 + m_2) {\rm C}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( w_3^{m}\beta^{++-}_{w^k,m} + w_3^{k} \beta^{++-}_{w^m,k} \right) {\rm C}_1 (k_1 - m_1) {\rm C}_2(k_2 - m_2) {\rm C}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( -w_3^{m}\beta^{+-+}_{w^k,m} - w_3^{k} \beta^{+-+}_{w^m,k} \right) {\rm C}_1 (k_1 + m_1) {\rm C}_2(k_2 - m_2) {\rm C}_3 (k_3 + m_3) \\ &\hspace{5em}+ \left( w_3^{m}\beta^{+-+}_{w^k,m} - w_3^{k} \beta^{+-+}_{w^m,k} \right) {\rm C}_1 (k_1 - m_1) {\rm C}_2(k_2 + m_2) {\rm C}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( -w_3^{m}\beta^{+--}_{w^k,m} + w_3^{k} \beta^{+--}_{w^m,k} \right) {\rm C}_1 (k_1 + m_1) {\rm C}_2(k_2 - m_2) {\rm C}_3 (k_3 - m_3) \\ &\hspace{5em}+ \left( w_3^{m} \beta^{+--}_{w^k,m} + w_3^{k} \beta^{+--}_{w^m,k}\right) {\rm C}_1 (k_1 - m_1) {\rm C}_2(k_2 + m_2) {\rm C}_3 (k_3 + m_3). \end{split} \end{gather} \end{subequations} \begin{remark} We do not present the coordinates of the vector \\$\left( Z^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Z^k$ because we will not need them in the construction of $Z^{j(n),n}$. The vectors generate functions of the type $Y^{j(n),n}$. \end{remark} \subsection{Two fruitful lemmas} Next we introduce two fruitful lemmas which play the main role in the proof below. These lemmas help us to avoid complicated computations to obtain an explicit form of the operator ${\mathcal B}(a,b)$ as some works before in 2D cases (see \cite{Rod-Sev05,PhanRod-ecc15,Rod06,Rod-Thesis08}). \begin{lemma} \label{lemma-lin-indp_Z} Let us be given $\alpha, \gamma, \in \R^3$ and $k \in \N_0^3$. Then \[\mathop{\rm span}\nolimits\{ \Pi {\mathcal Z}_\alpha^k, \Pi {\mathcal Z}_\gamma^k\} =\mathop{\rm span}\nolimits Z^{\{1,2\},k}\] if, and only if, the family~$\{\alpha,\gamma,k\}$ is linearly independent. \end{lemma} In Lemma~\ref{lemma-lin-indp_Z}, we denote \begin{align*} {\mathcal Z}^n\coloneqq\begin{pmatrix} z_1 \sin \left( \frac{k_1 \pi x_1}{L_1} \right) \cos \left( \frac{k_2 \pi x_2}{L_2} \right) \sin \left( \frac{k_3 \pi x_3}{L_3} \right)\\ z_2 \cos \left( \frac{k_1 \pi x_1}{L_1} \right) \sin \left( \frac{k_2 \pi x_2}{L_2} \right) \sin \left( \frac{k_3 \pi x_3}{L_3} \right)\\ z_3 \cos \left( \frac{k_1 \pi x_1}{L_1} \right) \cos \left( \frac{k_2 \pi x_2}{L_2} \right) \cos \left( \frac{k_3 \pi x_3}{L_3} \right), \end{pmatrix} \end{align*} for given~$z\in\R^3$ and $n\in\N^3$. The proof is analogous to the Lemma 3.1 in \cite{PhanRod18JDCS}. We will also need the following relaxed version. \begin{lemma} \label{lemma-lin-indp2} Let us be given $\alpha, \gamma, \delta \in \R^3$ and $k \in \N_0^3$. Then \[\mathop{\rm span}\nolimits\{ \Pi {\mathcal Z}_\alpha^k, \Pi {\mathcal Z}_\gamma^k, \Pi {\mathcal Z}_\delta^k\} =\mathop{\rm span}\nolimits Z^{\{1,2\},k}\] if, and only if, at least one of the families~$\{\alpha,\gamma,k\}$, $\{\alpha,\delta,k\}$, and~$\{\gamma,\delta,k\}$ is linearly independent. \end{lemma} \begin{proof} Notice that the inclusion~$\mathop{\rm span}\nolimits\{ \Pi {\mathcal Z}_\alpha^k, \Pi {\mathcal Z}_\gamma^k, \Pi {\mathcal Z}_\delta^k\} \subseteq\mathop{\rm span}\nolimits Z^{\{1,2\},k}$ holds true for all~$\alpha, \gamma, \delta \in \R^3$. Then, the reverse inclusion holds if, and only if, we can set two vectors in~$\{ \Pi {\mathcal Z}_\alpha^k, \Pi {\mathcal Z}_\gamma^k, \Pi {\mathcal Z}_\delta^k\}$ which are linearly independent. The Lemma follows straightforwardly from Lemma~\ref{lemma-lin-indp_Z}. \end{proof} \begin{remark} Two lemmas above will help us to prove the linear independent property. However mostly in the proof (for example in Step \ref{Step352}, equation \eqref{eq-lem-fail}), we can not prove the linear independence by using only Lemma \ref{lemma-lin-indp_Z} (as the procedure that we used before in the case of 3D Rectangle in \cite{PhanRod18JDCS}). Therefore we must introduce the Lemma \ref{lemma-lin-indp2} and apply it for these situations. The main difference between two Lemmas is that we need three choices to generate the vectors $Z^k$ in Lemma \ref{lemma-lin-indp2} comparing with two choices in Lemma \ref{lemma-lin-indp_Z}. \end{remark} \subsection{Proof of all Lemmas \ref{lem-genR3}--\ref{lem-genq+1}} \label{sec: ProofSat3DCyl} We introduce notations used frequently in proofs below. For $z_1, z_2, z_3 \in \R^3$, the matrix $\left( z_1 \mid z_2 \mid z_3 \right)$ is a square matrix whose each column is the vector $z_i$ for $i \in \{1,2,3\}$; and $\det\left( z_1 \mid z_2 \mid z_3 \right)$ denotes its determinant. \bigskip \subsubsection{Proof of Lemma \ref{lem-genR3}} \label{subsec-proof.genR3} In this proof, we will construct all eigenfunctions $Z^{j(n),n}$ with $n \in {\mathcal R}^{q+1}_3$. It is equivalent that the third coordinates of indices $n$ are always $p+1$. We divide this proof into three steps \begin{enumerate \renewcommand{\theenumi}{{\sf3.3.\arabic{enumi}}} \renewcommand{\labelenumi}{} \item $\bullet$~Step~\theenumi:\label{Step351}{ Generating $Z^{1,n}$ with $n \in \{ (0,l, q+1),~(l,0, q+1) \mid 1 \le l \leq q \}$.} \item $\bullet$~Step~\theenumi:\label{Step352}{ Generating $Z^{\{1,2\},n}$ with $n \in \{ (1,l, q+1),~(l,1, q+1) \mid 1 \le l \le q \}$.} \item $\bullet$~Step~\theenumi:\label{Step353}{ Generating all $Z^{\{1,2\},n}$ with $n \in \{ (n_1,n_2, q+1) \mid 2 \le n_1 \le q, 2 \le n_2 \le q \}$ by induction.} \end{enumerate} \bigskip \textbf{Step \ref{Step351}:} Generating $Z^{1,n}$ with $n \in \{ (0,l, q+1),~(l,0, q+1) \mid 1 \le l \leq q \}$. In this step, we will generate eigenfunctions with indices $n$ whose first and second coordinates belongs two orthogonal sides of the square $[0,q] \times [0,q]$ in Figure \ref{fig_step1}. Notice that the index $(0,0,q+1)$ does not correspond with any eigenfunction. \begin{figure}[ht] \centering \includegraphics[width=0.5\textwidth]{Step1} \caption{Step \ref{Step351}.} \label{fig_step1} \end{figure} {\em The case $n=(0,l,q+1)$.} We may follow the result in 2D Cylinder addressed in~\cite{PhanRod-ecc15}. Indeed from~\eqref{CoorYZ_Cyl} we find that for~$k$ and~$m$ such that~$k_1=m_1=0$, \[ Y^k=\begin{pmatrix} 0\\\widehat Y^k \end{pmatrix}\qquad\mbox{and}\qquad Z^k=\begin{pmatrix} 0\\\widehat Z^k \end{pmatrix} \] where for suitable constants~$C_1$ and~$C_2$ \[ \widehat Y^k=C_1\begin{pmatrix} \frac{-k_3\pi}{L_3}\sin(\frac{k_2\pi x_2}{L_2})\cos(\frac{k_3\pi x_3}{L_3})\\ \frac{k_2\pi}{L_2}\cos(\frac{k_2\pi x_2}{L_2})\sin(\frac{k_3\pi x_3}{L_3}) \end{pmatrix}~~\mbox{and}~~\widehat Z^k=C_1\begin{pmatrix} \frac{k_3\pi}{L_3}\sin(\frac{k_2\pi x_2}{L_2})\sin(\frac{k_3\pi x_3}{L_3})\\ \frac{k_2\pi}{L_2}\cos(\frac{k_2\pi x_2}{L_2})\cos(\frac{k_3\pi x_3}{L_3}) \end{pmatrix}, \] where the functions~$\widehat Y^k$ and~$\widehat Z^k$ are eigenfuntions of the Stokes operator in~${\rm C}_2=(0,L_2)\times \frac{L_3}{\pi}\mathbb{S}^1$ under Lions boundary conditions. Using an argument that is similar to the one used to derive~$\mathop{\rm span}\nolimits\{Y^n\mid n\in{\mathcal S}^{q+1}_{\rm R}, \#_0(n)=1\}\subset {\mathcal G}^{q}$ as in \cite[Section 3.4]{PhanRod18JDCS}, we can derive that $$ {\mathcal B}(Y^k)Z^k=\begin{pmatrix} 0\\\Pi_2\left( \left( \widehat Y^k \cdot \nabla_2 \right) \widehat Z^m + \left( \widehat Z^m \cdot \nabla_2 \right) \widehat Y^k \right) \end{pmatrix} $$ with~$\Pi_2$ being the orthogonal projection onto $H_2=\{u\in L^2({\rm C}_2,T{\rm C}_2)\mid u\cdot\mathbf n=0, \mathop{\rm div}\nolimits_2 u=0\}$ and~$\nabla_2$ and~$\mathop{\rm div}\nolimits_2$ being the gradient and divergence operators in~${\rm C}_2\sim(0,L_2)\times(0,2L_3)$. Therefore from~\cite[proof of Theorem~4.1]{PhanRod-ecc15} we know that \begin{subequations}\label{step351.res} \begin{align} &\{Z^{1,n}\mid n=(0,l,q+1), 0<l\le q\}\subset{\mathcal G}^{q}. \intertext{ and a similar argument gives us} &\{Z^{1,n}\mid n=(l,0,q+1), 0<l\le q\}\subset{\mathcal G}^{q}. \end{align} \end{subequations} \bigskip \textbf{Step \ref{Step352}:} Generating $Z^{\{1,2\},n}$ with $n \in \{ (1,l, q+1),~(l,1, q+1) \mid 1 \le l \le q \}$. After Step \ref{Step351}, we obtain eigenfunctions with indices $n$ whose first and second coordinates belongs two orthogonal sides of the square. In this step, we will continue with two new lines plotted in Figure \ref{fig_step2}. \begin{figure}[ht] \centering \includegraphics[width=0.5\textwidth]{Step2} \caption{Step \ref{Step352}.} \label{fig_step2} \end{figure} {\em The case $n = (1,1,q+1)$}. Firstly, we choose \begin{align*} k &=(1,0,q),& m &=(0,1,1),\\ w^k &= (L_1q,0,-L_3),& w^m &= (0,L_2,-L_3). \end{align*} From \eqref{CoorYZ_Cyl}, this choice gives us \begin{align*} \left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k = {\mathcal Z}^{(1,1,q+1)}_{z_{\alpha^1}} + {\mathcal Z}^{(1,1,q-1)}_{z_{\alpha^2}}, \end{align*} for suitable $z_{\alpha^1}, z_{\alpha^2} \in \R^3$. By the induction hypothesis~\eqref{Sat_Inc_CZ}, we have $Z^{\{1,2 \},(1,1,q-1)} \in {\mathcal G}^{q-1}\subset {\mathcal G}^{q} $. It is equivalent that ${\mathcal Z}^{(1,1,q-1)}_{z_{\alpha^2}} \in {\mathcal G}^{q}$. Next, from \begin{align*} \beta_{w^k,m}^{\star_1\star_2+} = -\frac{\pi}{8},\quad\beta_{w^k,m}^{\star_1\star_2-} = \frac{\pi}{8},\quad\beta_{w^m,k}^{\star_1\star_2+} = -\frac{\pi}{8}q, \quad\mbox{and}\quad\beta_{w^m,k}^{\star_1\star_2-} = \frac{\pi}{8}q, \end{align*} with $\{\star_1,\star_2\}\subseteq \{+,-\}$, following the expressions in~\eqref{CoorYZ_Cyl}, we find \begin{align*} z_{\alpha^1} &= \begin{pmatrix} 0 + {L_1}q\left( \beta_{w^m,k}^{+++} - \beta_{w^m,k}^{++-} + \beta_{w^m,k}^{+-+} - \beta_{w^m,k}^{+--} \right) \\ {L_2} \left( \beta_{w^k,m}^{+++} + \beta_{w^k,m}^{++-} \mathop{\rm sign}\nolimits(0-1) - \beta_{w^k,m}^{+-+} \mathop{\rm sign}\nolimits(0-1) -\beta_{w^k,m}^{+--} \right) + 0 \\ -{L_3} \left(- \beta_{w^k,m}^{+++} - \beta_{w^k,m}^{+-+} + \beta_{w^k,m}^{+--} + \beta_{w^k,m}^{++-} \right) - {L_3} \left( -\beta_{w^m,k}^{+++} - \beta_{w^m,k}^{+-+} + \beta_{w^m,k}^{+--} + \beta_{w^m,k}^{++-} \right) \end{pmatrix} \\&= -\frac{\pi}{2} \begin{pmatrix} L_1q^2 \\ { L_2} \\ L_3(q+1) \end{pmatrix}, \end{align*} and we can conclude that $\Pi {\mathcal Z}^{(1,1,q+1)}_{z_{\alpha^1}} \in {\mathcal G}^{q}$. Secondly, we choose \begin{align*} k &=(0,1,q),& m &=(1,0,1),\\ w^k &= (0,L_2q, -L_3),& w^m &= (L_1,0,-L_3) \end{align*} which again gives us~$\left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k = {\mathcal Z}^{(1,1,q+1)}_{z_{\gamma^1}} + {\mathcal Z}^{(1,1,q-1)}_{z_{\gamma^2}}$ with \begin{align*} z_{\gamma^1} = -\frac{\pi}{2}\begin{pmatrix} L_1 \\ L_2q^2 \\ L_3 (q+1) \\ \end{pmatrix}. \end{align*} Again, we get that $\Pi{\mathcal Z}^{(1,1,q+1)}_{z_{\gamma^1}} \in {\mathcal G}^q$. We observe that \begin{align*}\det\left(n\mid z_{\alpha^1}\mid z_{\gamma^1}\right) &= \frac{\pi^2}{4} (q+1) \det \begin{pmatrix} 1 & L_1 q^2 & L_1\\ 1 & L_2 & L_2q^2\\ 1 & L_3 & L_3 \end{pmatrix} \\ &= \frac{\pi^2}{4} (q+1)^2 (q-1) \left( L_1L_2 q^2 + L_1L_2 - L_2L_3 - L_1L_3 \right), \end{align*} and that, since $q\ge4$, \begin{equation} \label{eq-lem-fail} \det\left(n\mid z_{\alpha^1}\mid z_{\gamma^1}\right) = 0 \quad \Longleftrightarrow\quad q^2 = \frac{L_2L_3+L_1L_3-L_1L_2}{L_1L_2}. \end{equation} Thus from Lemma~\ref{lemma-lin-indp_Z} we conclude that~$ \Pi {\mathcal Z}^{(1,1,q+1)}_{z_{\alpha^1}}$ and $\Pi {\mathcal Z}^{(1,1,q+1)}_{z_{\gamma^1}}$ are not necessarily linearly independent. So, next we want to use Lemma~\ref{lemma-lin-indp2}. For that, we choose the third quadruple \begin{align*} k &=(1,0,q-1),& m &=(0,1,2),\\ w^k &= (L_1(q-1),0,-L_3),& w^m &= (0,2L_2,-L_3), \end{align*} which gives us \begin{align*} \left( Y^k \cdot \nabla \right) Z^m + \left( Y^m \cdot \nabla \right) Z^k = {\mathcal Z}^{(1,1,q+1)}_{z_{\delta^1}} + {\mathcal Z}^{(1,1,q-3)}_{z_{\delta^2}}, \end{align*} for suitable $z_{\delta^1}, z_{\delta^2} \in \R^3$. Since by~\eqref{Sat_Inc_CZ} $Z^{\{1,2\},(1,1,q-3)} \in {\mathcal G}^{q-1} \subset {\mathcal G}^{q}$, we can conclude that $\Pi {\mathcal Z}^{(1,1,q+1)}_{z_{\delta^1}} \in {\mathcal G}^{q}$. We can also find \begin{align*} &~z_{\delta^1}= -\frac{\pi}{2} \begin{pmatrix} L_1(q-1)^2 \\ 4 L_2 \\ L_3(q+1) \end{pmatrix}. \end{align*} Now, we compute \begin{align*} &\frac{4}{\pi^2(q+1)}\det\left(n \mid z_{\alpha^1}\mid z_{\delta^1} \right) = \det \begin{pmatrix} 1 & L_1 q^2 & L_1 (q-1)^2\\ 1 & L_2 & 4L_2\\ 1 & L_3 & L_3 \end{pmatrix} \\ &\qquad= 3L_1L_2 q^2 + 2 (L_1L_2 - L_1L_3) q + L_1L_3 - L_1L_2 - 3L_2L_3 \\ &\frac{4}{\pi^2(q+1)}\det \left(n \mid z_{\gamma^1} \mid z_{\delta^1} \right) = \det \begin{pmatrix} 1 & L_1 & L_1 (q-1)^2\\ 1 & L_2 q^2 & 4L_2\\ 1 & L_3 & L_3 \end{pmatrix} \\ &\qquad= -L_1L_2 q^4 + 2L_1L_2 q^3 + (L_2L_3+ L_1L_3 - L_1L_2) q^2 -2L_1L_3q + 4L_1L_2 - 4L_2L_3. \end{align*} From which we conclude that $$\det\left(n\mid z_{\alpha^1}\mid z_{\gamma^1}\right) = \det\left(n \mid z_{\alpha^1}\mid z_{\delta^1} \right)= \det \left(n \mid z_{\gamma^1} \mid z_{\delta^1} \right) = 0$$ if, and only if, \begin{subequations} \begin{align} L_1L_2 q^2 + L_1L_2 - L_2L_3 - L_1L_3 &= 0 \label{step12-eq1} \\ L_2 ( L_3- L_1) (q-2) &= 0 \label{step12-eq2}\\ L_1 (L_2-L_3) (q-2) &= 0. \label{step12-eq3} \end{align} \end{subequations} Since $q \ge 4$ and $L_1,\,L_2,\,L_3 >0$, from \ref{step12-eq2} and \ref{step12-eq3}, it arrives that $L_1 = L_2 = L_3$. In this case, from \ref{step12-eq1} we arrive to the contradiction $q = 1$. Hence, at least one of the families~$\{n,z_{\alpha^1},z_{\gamma^1}\}$, $\{n,z_{\alpha^1},z_{\delta^1}\}$, or~$\{n,z_{\gamma^1},z_{\delta^1}\}$ is linearly independent. From Lemma~\ref{lemma-lin-indp2} we can conclude that \begin{align} \label{res-step21-3Dcyl} Z^{\{1,2\},(1,1,q+1)} \in {\mathcal G}^{q}. \end{align} \medskip {\em The case $n = (1, l , q +1 ) $ with $2 \le l \le q$.}\\ We will prove by induction. Assume that \begin{equation}\label{IH-C1l} Z^{j(k),k} \subset {\mathcal G}^{q} \quad \text{with} \quad k = (1,l-2,q+1). \end{equation} We will prove that $Z^{\{1,2 \},(1,l, q+1)}$. Firstly, we choose \begin{align*} k &=(1,l-1,q),& m &=(0,1,1),\\ w^k &= (0,L_2q,L_3(1-l)),& w^m &= (0,L_2,-L_3), \end{align*} This choice gives us \begin{align*} \left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k = {\mathcal Z}^{(1,l,q+1)}_{z_{\alpha^1}} + {\mathcal Z}^{(1,l-2,q+1)}_{z_{\alpha^2}} + {\mathcal Z}^{(1,l,q-1)}_{z_{\alpha^3}} + {\mathcal Z}^{(1,l-2,q-1)}_{z_{\alpha^4}}. \end{align*} From~\eqref{IH.C} and~\eqref{IH-C1l}, we have that ${\mathcal Z}^{(1,l-2,q+1)}_{z_{\alpha^2}}$, ${\mathcal Z}^{(1,l,q-1)}_{z_{\alpha^3}}$, and ${\mathcal Z}^{(1,l-2,q-1)}_{z_{\alpha^4}}$ belong in ${\mathcal G}^{q}$. Therefore, we can conclude that $\Pi{\mathcal Z}^{(1,l,q+1)}_{z_{\alpha^1}} \in {\mathcal G}^{q}$.\\ Next, from \begin{align*} \beta_{w^k,m}^{+++} =\beta_{w^k,m}^{-++} = \frac{\pi (q-l +1)}{8}\quad\mbox{and}\quad\beta_{w^m,k}^{+++} =\beta_{w^m,k}^{-++} = \frac{\pi(l-q-1)}{8}, \end{align*} we obtain \begin{align*} z_{\alpha^1} &= \begin{pmatrix} 0 \\ L_2 \left( \beta_{w^k,m}^{+++} + \beta_{w^k,m}^{-++} \right) + L_2q \left( \beta_{w^m,k}^{+++} + \beta_{w^m,k}^{-++} \right) \\ -L_3\left( -\beta_{w^k,m}^{+++} - \beta_{w^k,m}^{-++} \right) + L_3(1-l) \left( -\beta_{w^m,k}^{+++} - \beta_{w^m,k}^{-++} \right) \end{pmatrix}\\ &= -\frac{\pi}{4} \begin{pmatrix} 0 \\ L_2(q-l+1)(q-1)\\ L_3 (q-l+1)(l-2) \end{pmatrix}. \end{align*} Secondly we choose \begin{align*} k &=(1,l-1,q),& m &=(1,1,1),\\ w^k &= (L_1q,0,-L_3),& w^m &= (0,L_2,-L_3), \end{align*} which allow us to obtain $\Pi{\mathcal Z}^{(1,l,q+1)}_{z_{\gamma^1}} \in {\mathcal G}^{q}$, and from \[ \beta_{w^k,m}^{+++} = \beta_{w^k,m}^{-++} = -\frac{\pi}{8}\quad\mbox{and}\quad\beta_{w^m,k}^{+++} = \beta_{w^m,k}^{-++} = \frac{\pi}{8} (l-q-1), \] we can find \begin{align*} z_{\gamma^1}= -\frac{\pi}{4} \begin{pmatrix} L_1 q(q-l+1) \\ L_2 \\ L_3(q-l+2) \end{pmatrix}. \end{align*} Thirdly, we choose \begin{align*} k &=(1,l-1,q),& m &=(1,1,1),\\ w^k &= (0,L_2q,L_3(1-l)),& w^m &= (L_1,-L_2,0), \end{align*} which gives us that $\Pi{\mathcal Z}^{(1,l,q+1)}_{z_{\delta^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\delta^1}= -\frac{\pi}{4} \begin{pmatrix} - L_1 (q-l+1) \\ L_2 (ql-l+1) \\ L_3(l-1)^2 \end{pmatrix}. \end{align*} Next we compute \begin{align*} \det \left(n\mid z_{\alpha^1}\mid z_{\gamma^1}\right) &= -\frac{\pi^2}{16} q (q-l+1)^2 \left( L_1L_2 q^2 - L_1L_2 - L_2L_3 - L_1L_3 l(l-2)\right), \\ \det \left(n \mid z_{\alpha^1}\mid z_{\delta^1}\right) &=\frac{\pi^2}{16} (q-l+1)^2 \left( L_1L_2 q^2 + L_2L_3 - L_1L_2-L_1L_3 l(l-2)\right) , \end{align*} and observe that $\det \left(n\mid z_{\alpha^1}\mid z_{\gamma^1}\right)=\det \left(n \mid z_{\alpha^1}\mid z_{\delta^1}\right)=0$ if, and only if, \begin{align*} L_1L_2 q^2 - L_1L_2 - L_2L_3 - L_1L_3 l(l-2) = L_1L_2 q^2 + L_2L_3 - L_1L_2-L_1L_3 l(l-2) = 0, \end{align*} because $2 \le l \le q$, which implies $2 L_2L_3 = 0$. This contradicts the fact that $L_1,\,L_2,\,L_3 >0$. Therefore one of the families~$\{n,z_{\alpha^1},z_{\gamma^1}\}$ or~$\{n,z_{\alpha^1},z_{\delta^1}\}$ is linearly independent and, by Lemma~\ref{lemma-lin-indp2}, it follows that $Z^{\{1,2\},(1,l,q+1)} \in {\mathcal G}^{q}$. By induction, using~\eqref{step351.res},~\eqref{res-step21-3Dcyl}, and the induction hypothesis~\eqref{IH-C1l} it follows that \begin{subequations}\label{step352.res} \begin{align} Z^{(1,l,q+1)} \in {\mathcal G}^{q}, \qquad\mbox{for all}\quad 0\le l\le q\\ \intertext{and proceeding similarly we can also derive that} Z^{(l,1,q+1)} \in {\mathcal G}^{q},\qquad\mbox{for all}\quad 0\le l\le q. \end{align} \end{subequations} \bigskip \textbf{Step \ref{Step353}:} Generating the family $Z^{\{1,2\},n}$ with $n = (n_1,n_2, q+1)$ where $2 \le n_1 \le q $ and $2 \le n_2 \le q$. In the final step, we will generate all remaining eigenfunctions with indices $n$ whose the first and second coordinates belongs to the square. The process is based on the induction \eqref{IH-Cn1n2}. \begin{figure}[ht] \centering \includegraphics[width=.5\textwidth]{Step3} \caption{Step \ref{Step353}.} \label{fig_step3} \end{figure} Firstly, we introduce an induction hypothesis. Assume that \addtocounter{equation}{1} \begin{align}\label{IH-Cn1n2} &Z^{j(\kappa),\kappa} \in {\mathcal G}^{q},\\ &\mbox{for}\quad \kappa\in\{(n_1-2,n_2-2,q+1),(n_1-2,n_2,q+1),(n_1,n_2-2,q+1)\}.\notag \end{align} We will prove that $Z^{\{1,2\},(n_1,n_2,q+1)} \in {\mathcal G}^q$. By choosing \begin{align*} k &=(n_1-1,n_2-1,q),& m &=(1,1,1),\\ w^k &= (0,L_2q,L_3(1-n_2)),& w^m &= (0,L_2,-L_3), \end{align*} we obtain \begin{align*} \left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k = {\mathcal Z}^{(n_1,n_2,q+1)}_{z_{\alpha^1}} + {\sum_{i=2}^8{{\mathcal Z}^{\kappa^i}_{z_{\alpha^i}}},} \end{align*} with $\kappa^i \in \{(n_1-2,n_2-2,q-1), (n_1,n_2-2,q-1), (n_1-2,n_2,q-1),(n_1,n_2,q-1) (n_1-2,n_2-2,q+1), (n_1,n_2-2,q+1), (n_1-2,n_2,q+1) \}$. Using the inductive hypothesis~\eqref{IH.C}, we find that $ Z^{j(\kappa), \kappa} \in {\mathcal G}^{q}, $ for $\kappa \in \{ (n_1-2,n_2-2,q-1), (n_1,n_2-2,q-1), (n_1-2,n_2,q-1),(n_1,n_2,q-1) \}$. From the inductive hypothesis~\eqref{IH-Cn1n2} we also have $ Z^{j(\kappa), \kappa} \in {\mathcal G}^{q}, $ for $\kappa \in \{ (n_1-2,n_2-2,q+1), (n_1,n_2-2,q+1), (n_1-2,n_2,q+1) \}$. Thus, we can conclude that $\Pi {\mathcal Z}^{(n_1,n_2,q+1)}_{z_{\alpha^1}} \in {\mathcal G}^{q}$. Next, from \[ \beta_{w^k,m}^{+++} = \frac{\pi}{8} (q-n_2+1)\quad\mbox{and}\quad\beta_{w^m,k}^{+++} = \frac{\pi}{8} (n_2-q-1), \] we obtain \begin{align*} z_{\alpha^1} = -\frac{\pi}{8} \begin{pmatrix} 0 \\ L_2(q-n_2+1)(q-1) \\ L_3(q-n_2+1)(n_2-2) \end{pmatrix}. \end{align*} A second choice is \begin{align*} k &=(n_1-1,n_2-1,q),& m &=(1,1,1),\\ w^k &= (L_1q,0,L_3(1-n_1)),& w^m &= (0,L_2,-L_3), \end{align*} which gives us $\Pi {\mathcal Z}^{(n_1,n_2,q+1)}_{z_{\gamma^1}} \in {\mathcal G}^{q}$. From \[ \beta_{w^k,m}^{+++} = \frac{1}{8} \left( q-n_1 +1 \right),\quad\mbox{and}\quad\beta_{w^m,k}^{+++} = \frac{1}{8} (n_2 - q-1), \] we obtain \begin{align*} z_{\gamma^1}= \frac{\pi}{8} \begin{pmatrix} -L_1q(q-n_2+1) \\ L_2(q-n_1+1) \\ L_3(1-n_1)(q-n_2+1) + L_3(q-n_1+1) \end{pmatrix}. \end{align*} Another choice is \begin{align*} k &=(n_1,n_2-1,q),& m &=(0,1,1),\\ w^k &= (L_1(n_2-1),-L_2n_1,0),& w^m &= (0,L_2,-L_3), \end{align*} which gives us $\Pi {\mathcal Z}^{(n_1,n_2,q+1)}_{z_{\delta^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\delta^1}= \frac{\pi}{8} \begin{pmatrix} -L_1(q-n_2+1)(n_2-1) \\ L_2n_1 (q-n_2) \\ -L_3n_1 \end{pmatrix}. \end{align*} Next, from $2 \le n_1,\,n_2 \le q$ and \begin{align*} &-\frac{64}{\pi^2(q-n_2+1)}\det \left(n\mid z_{\alpha^1}\mid z_{\gamma^1}\right)\\ &\qquad= \det \begin{pmatrix} n_1 & 0 & -L_1(q-n_2+1)(n_2-1)\\ n_2 & L_2(q-1)&L_2(q-n_1+1) \\ q+1 & L_3(n_2-2) &L_3(1-n_1)(q-n_2+1) + L_3(q-n_1+1) \end{pmatrix} \\ &\qquad= q(q-n_2+1) \left( L_1L_2 q^2 - L_1L_2 - L_2L_3n_1 (n_1 - 2) - L_1L_3n_2 (n_2 - 2) \right), \\ &-\frac{64}{\pi^2(q-n_2+1)}\det \left(n\mid z_{\alpha^1}\mid z_{\delta^1}\right)= \det \begin{pmatrix} n_1 & 0 & -L_1q(q-n_2+1)\\ n_2 & L_2(q-1)&L_2n_1 (q-n_2) \\ q+1 & L_3(n_2-2) & -L_3n_1 \end{pmatrix} \\ &\qquad= (n_2 -1) (q-n_2 +1) \left( L_1L_2 q^2 - L_1L_2 - L_2L_3n_1^2 - L_1L_3n_2 (n_2 - 2) \right), \end{align*} we have that $\det \left(n\mid z_{\alpha^1}\mid z_{\gamma^1}\right)=\det \left(n\mid z_{\alpha^1}\mid z_{\delta^1}\right)=0 $ if only if $2L_2L_3 n_1 = 0$. This contradicts the fact that~$L_2$, $L_3$, and~$n_1$ are positive. Thus, one of the families~$\{n,z_{\alpha^1},z_{\gamma^1}\}$ or~$\{n,z_{\alpha^1},z_{\delta^1}\}$ is linearly independent. By Lemma~\ref{lemma-lin-indp2} it follows that $Z^{\{1,2\},(n_1,n_2,q+1)}\in{\mathcal G}^{q}$. Using~\eqref{IH.C},~\eqref{step351.res},~\eqref{step352.res}, and the induction hypothesis~\eqref{IH-Cn1n2}, we conclude that $Z^{j(n),n} $ with $n = (n_1,n_2,q+1)$. Finally, we obtain \begin{align} \label{lem35.res} Z^{j(n),n} \in {\mathcal G}^{q} \quad \text{for all~} n \in {\mathcal R}^{q+1}_3. \end{align} \qed \subsubsection{Proof of Lemma \ref{lem-genR12}} \label{subsec-proof.genR12} Notice that the cases~$n \in {{\mathcal R}^{q+1}_1} $ and~$n \in {{\mathcal R}^{q+1}_2}$ are analogous. On the other hand, since we consider the periodicity assumption in the third direction and Lions boundary conditions in the first two directions, these cases must be addressed separately from the case~$n \in {{\mathcal R}^{q+1}_3}$ treated in section~\ref{subsec-proof.genR3}. Let us take~$n \in {{\mathcal R}^{q+1}_1}$. Again we divide this proof into three main steps \begin{enumerate \renewcommand{\theenumi}{{\sf3.4.\arabic{enumi}}} \renewcommand{\labelenumi}{} \item $\bullet$~Step~\theenumi:\label{Step361} Generating $Z^{1,n}$ with $n \in \{ (q+1,l,0), (q+1,0,l) \mid 1 \le l\le q \}$. \item $\bullet$~Step~\theenumi:\label{Step362} Generating $Z^{\{1,2\},n}$ with $n \in \{(q+1,l,1), (q+1,l,1) \mid 1 \le l \le q \}$. \item $\bullet$~Step~\theenumi:\label{Step363} Generating $Z^{\{1,2\},n}$ with $n \in \{ (q+1,n_1,n_2) \mid 2 \le n_1,n_2 \le q \}$. \end{enumerate} \bigskip \textbf{Step \ref{Step361}:} Generating $Z^{1,n}$ with $n \in \{ (q+1,l,0), (q+1,0,l) \mid 1 \le l\le q \}$. {\em The case $n = (q+1,l,0)$}. We choose \begin{align*} k &=(q,l,0),& m &=(1,0,0),\\ w^k &= (L_1 l,-L_2 q, 0),& w^m &= (0,0,1), \end{align*} which gives us $\Pi {\mathcal Z}^{(q+1,l,0)}_{z_{\gamma^1}} \in {\mathcal G}^{q}$. From $ \beta_{w^k,m}^{+\star_1\star_2} = \frac{\pi}{8}l,\quad\beta_{w^m,k} = 0, $ with $\{\star_1,\star_2\}\subseteq \{+,-\}$, we get \begin{align*} z_{\gamma^1} = \frac{\pi}{2} \begin{pmatrix} 0 \\ 0 \\ l \end{pmatrix}. \end{align*} Observe that ${\mathcal Z}^{1,(q+1,l,0)}_{z_{\gamma^1}}=\frac{\pi}{2} l\begin{pmatrix} 0\\0\\\cos(\frac{(q+1)\pi x_1}{L_1})\cos(\frac{l \pi x_2}{L_2}) \end{pmatrix}$, it means that for a suitable constant $\zeta \neq 0$, $Z^{1,(q+1,l,0)} = \zeta {\mathcal Z}^{1,(q+1,l,0)}_{z_{\gamma^1}} $. Hence, we conclude that \begin{equation}\label{step361.res1} Z^{1,(q+1,l,0)}\in{\mathcal G}^{q},\quad\mbox{for all}\quad 1\le l\le q. \end{equation} To generate $Z^{1,n}$ with $n = (q+1,0,l)$, we can use the result for the 2D cylinder in~\cite{PhanRod-ecc15}. Notice that, for some constant~$\zeta\ne0$, \[ Z^{1,(q+1,0,l)} = \zeta\begin{pmatrix} \frac{l \pi }{L_3}\sin \left( \frac{(q+1) \pi x_1}{L_1} \right) \sin \left( \frac{l \pi x_3}{L_3} \right)\\ 0\\ \frac{(q+1) \pi }{L_1} \cos \left( \frac{(q+1) \pi x_1}{L_1} \right) \cos \left( \frac{l \pi x_3}{L_3} \right), \end{pmatrix} \] which is an eigenfunction of the Stokes operator in the 2D cylinder~$(0,L_1)\times\frac{L_2}{\pi}\mathbb{S}^1$, under Lions boundary conditions. It follows, from~\cite[Theorem~4.1]{PhanRod-ecc15}, that \begin{equation}\label{step361.res2} Z^{1,(q+1,0,l)}\in{\mathcal G}^{q},\quad\mbox{for all}\quad 1\le l\le q. \end{equation} \bigskip \textbf{Step \ref{Step362}:} Generating $Z^{\{1,2\},n}$ with $n \in \{(q+1,l,1), (q+1,l,1) \mid 1 \le l \le q \}$. {\em The case $n = (q+1,1,1)$ }. We firstly choose \begin{align*} k &=(q,0,1),& m &=(1,1,0),\\ w^k &= (L_1,0,-L_3q),& w^m &= (L_1,-L_2,0), \end{align*} Then, by changing the roles of~$k$ and~$m$ in from~\eqref{CoorYZ_Cyl}, we obtain \begin{align*} \left( Z^k \cdot \nabla \right) Y^m + \left( Y^m \cdot \nabla \right) Z^k = {\mathcal Z}^{(q+1,1,1)}_{z_{\alpha^1}} + {\mathcal Z}^{(q-1,1,1)}_{z_{\alpha^2}}, \end{align*} for suitable $z_{\alpha^1}, z_{\alpha^2} \in \R^3$. By~\eqref{IH.C}, we have $Z^{\{1,2 \},(q-1,1,1)} \in {\mathcal G}^{q}$. Therefore we derive that~$\Pi{\mathcal Z}^{(q+1,1,1)}_{z_{\alpha^1}}\in{\mathcal G}^{q}$, and we can also find \begin{align*} z_{\alpha^1}= \frac{\pi}{2} \begin{pmatrix} L_1 (q+1)\\ - L_2\\ L_3q^2 \end{pmatrix}. \end{align*} Secondly, we compute~$\left( Z^m \cdot \nabla \right) Y^k + \left( Y^k \cdot \nabla \right) Z^m$ with the choice \begin{align*} k &=(q,1,1),& m &=(1,0,0),\\ w^k &= (0,L_2,-L_3),& w^m &= (0,0,L_3), \end{align*} Analogously, we obtain that $\Pi {\mathcal Z}^{(q+1,1,1)}_{z^{\gamma^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\gamma^1}= \frac{\pi}{2} \begin{pmatrix} 0 \\ L_2 \\ L_3 \end{pmatrix}. \end{align*} Thirdly, we compute~$\left( Z^m \cdot \nabla \right) Y^k + \left( Y^k \cdot \nabla \right) Z^m$ with \begin{align*} k &=(q,1,1),& m &=(1,0,0),\\ w^k &= (L_1,0,-L_3q),& w^m &= (0,0,L_3), \end{align*} which gives us $\Pi {\mathcal Z}^{(q+1,1,1)}_{z^{\delta^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\delta^1}= \frac{\pi}{2} \begin{pmatrix} L_1 \\ 0 \\ L_3(q-1) \end{pmatrix}. \end{align*} Now, observe that $\det\left( n \mid z_{\alpha^1}~\mid z_{\gamma^1} \right) = \det \left(n \mid z_{\gamma^1} \mid z_{\delta^1} \right)=0$ if and only if \begin{align*} L_2L_3 q^2 +L_1L_3+ L_2L_3-L_1L_2 = L_2L_3 q^2 -L_2L_3+L_1L_3-L_1L_2 = 0, \end{align*} which implies the contradiction $2L_2L_3 = 0$, because $L_2,L_3 >0$. Therefore, by Lemma~\ref{lemma-lin-indp2}, \begin{equation} \label{step362.res1} Z^{\{1,2\},(q+1,1,1)} \in {\mathcal G}^{q}. \end{equation} \medskip {\em The case $n=( q +1,1,l)$}. Let us introduce the induction hypothesis \addtocounter{equation}{1} \begin{equation}\label{IH-Cq1l} Z^{j(k),k} \in {\mathcal G}^{q} ,\quad\mbox{if}\quad k=(q+1,1,l-2),\quad\mbox{for given }2\le l\le q. \end{equation} We prove that $ Z^{j(k),k} \in {\mathcal G}^{q}$ with $k=(q+1,1,l)$. \medskip To generate $n = (q+1,1,l)$, firstly we compute $\left( Z^m \cdot \nabla \right) Y^k + \left( Y^k \cdot \nabla \right) Z^m$ with the choice \begin{align*} k &=(q,1,l),& m &=(1,0,0),\\ w^k &= (0,L_2l,-L_3),& w^m &= (0,0,L_3), \end{align*} which allow us to conclude that $\Pi {\mathcal Z}^{(q+1,1,l)}_{z^{\alpha^1}} \in {\mathcal G}^{q}$ with \begin{align*} z_{\alpha^1}= \frac{\pi}{2} \begin{pmatrix} 0 \\ L_2 l^2 \\ L_3 l \end{pmatrix}. \end{align*} Secondly, we compute $\left( Z^m \cdot \nabla \right) Y^k + \left( Y^k \cdot \nabla \right) Z^m$ with \begin{align*} k &=(q,1,l),& m &=(1,0,0),\\ w^k &= (L_1l,0,-L_3q),& w^m &= (0,0,L_3), \end{align*} which gives us $\Pi {\mathcal Z}^{(q+1,1,l)}_{z^{\gamma^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\gamma^1}= \frac{\pi}{2} \begin{pmatrix} L_1 l^2 \\ 0 \\ L_3 l (q-1) \end{pmatrix}. \end{align*} Thirdly we compute $\left( Z^k \cdot \nabla \right) Y^m + \left( Y^m \cdot \nabla \right) Z^k$ with \begin{align*} k &=(q,0,l),& m &=(1,1,0),\\ w^k &= (L_1l,0,-L_3q),& w^m &= (L_1,-L_2,0), \end{align*} which gives us $\Pi {\mathcal Z}^{(q+1,1,l)}_{z^{\delta^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\delta^1}= \frac{\pi}{2} \begin{pmatrix} L_1 l (q+1) \\ - L_2l \\ L_3 q^2 \end{pmatrix}. \end{align*} Next we observe that $\det\left( n \mid z_{\alpha^1}~\mid z_{\gamma^1} \right) = \det\left( n \mid z_{\alpha^1}~\mid z_{\delta^1} \right) =0 $ if, and only if, \begin{align*} l^3 (L_2L_3 q^2 - L_2L_3 + L_1L_3 - L_1L_2 l^2) = l^2 (q+1) (L_2L_3 q ^2 + L_2L_3 + L_1L_3 - L_1L_2l) = 0 \end{align*} which leads to the contradiction $0=2L_1L_2l(l-1) + 2 L_2L_3 \ge 2L_2(L_1 + L_3) > 0$, since $2 \le l \le q$. Then from Lemma~\ref{lemma-lin-indp2} we conclude that $ {\mathcal Z}^{(q+1,1,l)}_{z^{\delta^1}} \in {\mathcal G}^{q}. $ By induction, using~\eqref{step361.res1}, \eqref{step362.res1}, and the induction hypothesis~\eqref{IH-Cq1l}, it follows that \begin{equation}\label{step362.res2} {\mathcal Z}^{(q+1,1,l)}_{z^{\delta^1}} \in {\mathcal G}^{q},\quad\mbox{for all}\quad 1\le l\le q. \end{equation} \medskip {\em The case $n=( q +1,l,1)$.} Let us introduce the induction hypothesis \addtocounter{equation}{1} \begin{equation}\label{IH-Cql1} Z^{j(k),k} \in {\mathcal G}^{q} ,\quad\mbox{with}\quad k=(q+1,1,l-2),\quad l\ge2. \end{equation} We prove that $ Z^{j(k),k} \in {\mathcal G}^{q}$ with $ k=(q+1,1,l)$. To generate $n = (q+1,l,1)$, firstly we choose \begin{align*} k &=(q,l,1),& m &=(1,0,0),\\ w^k &= (0,L_2,-L_3 l),& w^m &= (0,0,L_3), \end{align*} which allow us to conclude that $\Pi {\mathcal Z}^{(q+1,1,l)}_{z^{\alpha^1}} \in {\mathcal G}^{q}$ with \begin{align*} z_{\alpha^1}= \frac{\pi}{2} \begin{pmatrix} 0 \\ L_2 \\ L_3 l \end{pmatrix}. \end{align*} Secondly, we compute $\left( Z^m \cdot \nabla \right) Y^k + \left( Y^k \cdot \nabla \right) Z^m$ with \begin{align*} k &=(q,l,1),& m &=(1,0,0),\\ w^k &= (L_1,0,-L_3q),& w^m &= (0,0,L_3), \end{align*} which gives us $\Pi {\mathcal Z}^{(q+1,1,l)}_{z^{\gamma^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\gamma^1}= \frac{\pi}{2} \begin{pmatrix} L_1 \\ 0 \\ L_3 l (q-1) \end{pmatrix}. \end{align*} Thirdly we compute $\left( Z^k \cdot \nabla \right) Y^m + \left( Y^m \cdot \nabla \right) Z^k$ with \begin{align*} k &=(q,l,0),& m &=(1,0,1),\\ w^k &= (L_1 l,-L_2 q, 0),& w^m &= (L_1,0, -L_3), \end{align*} which gives us $\Pi {\mathcal Z}^{(q+1,l,1)}_{z^{\delta^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\delta^1}= \frac{\pi}{2} \begin{pmatrix} L_1 l (q+1) \\ - L_2 q^2 \\ L_3 l \end{pmatrix}. \end{align*} Next we observe that $\det\left( n \mid z_{\alpha^1} \mid z_{\gamma^1} \right) = \det\left( n \mid z_{\alpha^1} \mid z_{\delta^1} \right) =0 $ if, and only if, \begin{align*} L_2 L_3 l q (q-1) + L_1 (L_3 l^2 - L_2) = L_2 L_3 q (q^2 + 1) + L_1 (L_3 l^2 - L_2) (q+1) = 0 \end{align*} which leads to the contradiction $0= (l-1) (q^2 -1) - 2 \ge q^2-3 > 0$, since $2 \le l \le q$. Then from Lemma~\ref{lemma-lin-indp2} we conclude that $ {\mathcal Z}^{(q+1,l,1)}_{z^{\delta^1}} \in {\mathcal G}^{q}. $ By induction, using~\eqref{step361.res2}, \eqref{step362.res1}, and the induction hypothesis~\eqref{IH-Cql1}, it follows that \begin{align} \label{step362.res3} Z^{\{1,2 \}(q+1,l,1)} \in {\mathcal G}^{q} \quad \mbox{for all~} 1 \le l \le q. \end{align} \medskip \textbf{Step \ref{Step363}:} Generating $Z^{\{1,2\},n}$ with $n \in \{ (q+1,n_1,n_2) \mid 2 \le n_1,n_2 \le q \}$. \medskip Let us introduce the inductive hypothesis \begin{align}\label{IH-Cylqn1n2} \begin{split} &Z^{j(\kappa),\kappa} \in {\mathcal G}^{q},\\ &\mbox{for}\quad \kappa\in \{ (q+1,n_1-2,n_2-2), (q+1,n_1,n_2-2), (q+1,n_1-2,n_2) \}. \end{split} \end{align} We will prove that $Z^{j(\kappa),\kappa} \in {\mathcal G}^{q}$ with $\kappa = (q+1,n_1,n_2)$. \medskip To generate $n = (q+1, n_1,n_2)$, firstly we compute~$\left( Z^m \cdot \nabla \right) Y^k + \left( Y^k \cdot \nabla \right) Z^m$ with \begin{align*} k &=(q,n_1,n_2),& m &=(1,0,0),\\ w^k &= (0,L_2n_2,-L_3n_1),& w^m &= (0,0,L_3), \end{align*} which leads to $\Pi {\mathcal Z}^{(q+1,n_1,n_2)}_{z_{\alpha^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\alpha^1}= \frac{\pi}{2} \begin{pmatrix} 0 \\ L_2 n_2^2 \\ L_3 n_1 n_2 \end{pmatrix}. \end{align*} Secondly we compute~$\left( Z^m \cdot \nabla \right) Y^k + \left( Y^k \cdot \nabla \right) Z^m$ with \begin{align*} k &=(q,n_1,n_2),& m &=(1,0,0),\\ w^k &= (L_1 n_2,0,-L_3q),& w^m &= (0,0,L_3), \end{align*} which leads us to $\Pi {\mathcal Z}^{(q+1,n_1,n_2)}_{z_{\gamma^1}} \in {\mathcal G}^{q}$ where \begin{align*} z_{\gamma^1}= \frac{\pi}{2} \begin{pmatrix} L_1 n_2^2 \\ 0 \\ L_3 n_2 (q-1) \end{pmatrix}. \end{align*} Thirdly we compute~$\left( Z^k \cdot \nabla \right) Y^m + \left( Y^m \cdot \nabla \right) Z^k$ with \begin{align*} k &=(q,n_1-1,n_2),& m &=(1,1,0),\\ w^k &= (L_1(n_1-1),-L_2q,0),& w^m &= (L_1,-L_2,0), \end{align*} which gives us $\Pi {\mathcal Z}^{(q+1,n_1,n_2)}_{z_{\delta^1}} \in {\mathcal G}^{q}$ with \begin{align*} z_{\delta^1}= \frac{\pi}{4} \begin{pmatrix} L_1 (q-n_1+1)(n_1-2) \\ L_2 (q-n_1+1)(2-q) \\ 0 \end{pmatrix}. \end{align*} Now $\det\left( n \mid z_{\alpha^1} \mid z_{\gamma^1} \right) = \det\left( n \mid z_{\alpha^1} \mid z_{\delta^1} \right) =0 $ if, and only if, \begin{subequations} \begin{align} n_2^3 (L_2L_3 q^2 - L_2L_3 + L_1L_3 n_1^2 - L_1L_2 n_2^2) &= 0 \label{step23-eq1} \\ (q-n_1 + 1) n_2 ( L_2L_3 n_1 (q+1) (q-2) + (n_1 -2) (L_1L_3 n_1^2 - L_1L_2 n_2^2) )&= 0 \label{step23-eq2} \end{align} \end{subequations} Since $0<2 \le n_2,n_1\le q$, from \ref{step23-eq1}, we have $L_1L_3 n_1^2 - L_1L_2 n_2^2 = L_2L_3 (1-q^2)$. Then, after substitution into \ref{step23-eq2} and since~$n_2(q-n_1 + 1)\ge n_2>0$, we arrive to the contradiction~$0=L_2L_3 (q+1) (2q -n_1 -2) >L_2L_3 (q+1) (q-2)>0$, because $q\ge4$ and~$L_2,L_3>0$. Therefore by Lemma~\ref{lemma-lin-indp2} it follows that~$Z^{(q+1,n_1,n_2)}_{z_{\delta^1}} \in {\mathcal G}^{q}$. By induction, using~\eqref{step361.res1}, \eqref{step361.res2}, \eqref{step362.res2}, \eqref{step362.res3}, and the induction hypothesis~\eqref{IH-Cylqn1n2}, we obtain \begin{subequations}\label{lem36.res} \begin{align} Z^{j(n),n} \in {\mathcal G}^{q} \quad \text{for all~} n \in {\mathcal R}^{q+1}_1, \intertext{and a similar argument gives us} Z^{j(n),n} \in {\mathcal G}^{q} \quad \text{for all~} n \in {\mathcal R}^{q+1}_2. \end{align} \end{subequations} \qed \subsubsection{Proof of Lemma \ref{lem-gen00}} \label{subsec-proof.gen00} To generate $n = (q+1,0,0)$, we choose \begin{align*} k &=(q, 0,1),& m &=(1,0,1),\\ w^k &= (L_1, 0, -L_3q),& w^m &= (L_1,0,-L_3), \end{align*} which gives us \begin{align*} \left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k = {\mathcal Z}^{(q+1,0,0)}_{z_{\alpha^1}} + {\mathcal Z}^{(q+1,0,2)}_{z_{\alpha^2}} + {\mathcal Z}^{(q-1,0,0)}_{z_{\alpha^3}} + {\mathcal Z}^{(q-1,0,2)}_{z_{\alpha^3}}. \end{align*} From~\eqref{IH.C} and~\eqref{lem36.res}, we can conclude that $\Pi {\mathcal Z}^{(q+1,0,0)}_{z_{\alpha^1}} \in {\mathcal G}^{q}$ where \begin{align*} z_{\alpha^1}= -\frac{\pi }{4} \begin{pmatrix} 0 \\ 0 \\ L_3 (q-1)^2 \\ \end{pmatrix}. \end{align*} Now, since $L_3 (q-1)^2\ne0$ we have that $Z^{1,(q+1,0,0)} =\zeta {\mathcal Z}^{(q+1,0,0)}_{z_{\alpha^1}}$. Therefore \begin{subequations}\label{lem37.res} \begin{align} &Z^{1,(q+1,0,0)} \in {\mathcal G}^{q}, \intertext{and a similar argument gives us} &Z^{1,(0,q+1,0)} \in {\mathcal G}^{q}. \end{align} \end{subequations} \qed \subsubsection{Proof of Lemma \ref{lem-genL}} \label{subsec-proof.genL} Due to two different types of boundary conditions, we divide the proof into two steps \begin{enumerate \renewcommand{\theenumi}{{\sf3.8.\arabic{enumi}}} \renewcommand{\labelenumi}{} \item $\bullet$~Step~\theenumi:\label{Step381} Generating $Z^{j(n),n}$ with $n \in {\mathcal L}^{q+1}_{2,3}\bigcup {\mathcal L}^{q+1}_{3,1}$. \item $\bullet$~Step~\theenumi:\label{Step382} Generating $Z^{j(n),n}$ with $n \in {\mathcal L}^{q+1}_{1,2}$. \end{enumerate} \medskip \textbf{Step \ref{Step381}:} Generating $Z^{j(n),n}$ with $n \in {\mathcal L}^{q+1}_{2,3}\bigcup {\mathcal L}^{q+1}_{3,1}$. To generate~$n = (l,\,q+1,\,q+1)$, with~$1 \le l \le q$. We start computing~$\left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k$ with \begin{align*} k &=(l, q-1,q),& m &=(0,2,1),\\ w^k &= (0, L_2q, L_3(1-q)),& w^m &= (0,L_2,-2L_3), \end{align*} and obtain that $\Pi {\mathcal Z}^{(l,q+1,q+1)}_{z_{\alpha^1}} \in {\mathcal G}^{q}$ with \begin{align*} z_{\alpha^1} = -\frac{\pi}{4} \begin{pmatrix} 0 \\ L_2 (q^2-1) \\ L_3(q+1) (q-3) \end{pmatrix}. \end{align*} Next we compute~$\left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k$ \begin{align*} k &=(l, q,q),& m &=(0,1,1),\\ w^k &= (L_1q, -L_2l, 0),& w^m &= (0,L_2,-L_3), \end{align*} and obtain $\Pi{\mathcal Z}^{(l,q+1,q+1)}_{z_{\gamma^1}}\in {\mathcal G}^{q}$ with \begin{align*} z_{\gamma^1} = -\frac{\pi}{4} \begin{pmatrix} 0 \\ L_2l \\ L_3l \end{pmatrix}. \end{align*} Since \begin{align*} \frac{16}{\pi^2} \det \left(n \mid z_{\alpha^1} \mid z_{\gamma^1} \right)= \det \begin{pmatrix} l & 0 & 0 \\ q+1 & L_2(q^2-1) & L_2l \\ q+1 & L_3(q+1)(q-3) & L_3l \end{pmatrix} = 2 L_2L_3 l^2 (q+1) > 0, \end{align*} from Lemma~\ref{lemma-lin-indp_Z} we obtain that \begin{subequations}\label{step381.res1} \begin{align} Z^{\{1,2\}, (l,q+1,q+1)}\in {\mathcal G}^{q},\quad\mbox{for}\quad1 \le l \le q, \intertext{and a similar argument gives us} Z^{\{1,2\}, (q+1,l,q+1)}\in {\mathcal G}^{q},\quad\mbox{for}\quad1 \le l \le q, \end{align} \end{subequations} {\em The case~$n= (0,q+1,q+1)$}. We can use again~\cite[Theorem~4.1]{PhanRod-ecc15} to conclude that \begin{subequations}\label{step381.res2} \begin{align} &Z^{1,(0,q+1,q+1)} \in {\mathcal G}^{q}, \intertext{and a similarly} &Z^{1,(q+1,0,q+1)} \in {\mathcal G}^{q}. \end{align} \end{subequations} \medskip \textbf{Step \ref{Step382}:} Generating $Z^{j(n),n}$ with $n \in {\mathcal L}^{q+1}_{1,2}$. To generate $n = (q+1, q+1, l)$ with $1 \le l \le q$, we first compute~$\left( Y^m \cdot \nabla \right) Z^k + \left( Z^k \cdot \nabla \right) Y^m$ with \begin{align*} k &=(q, q,l),& m &=(1,1,0),\\ w^k &= (L_1l, 0, -L_3q),& w^m &= (L_1,-L_2,0), \end{align*} which gives us~$\Pi {\mathcal Z}^{(q+1,q+1,l)}_{z_{\alpha^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\alpha^1} = \frac{\pi}{4} \begin{pmatrix} L_1l \\ - L_2l \\ 0 \end{pmatrix}. \end{align*} Next, we choose compute~$\left( Y^m \cdot \nabla \right) Z^k + \left( Z^k \cdot \nabla \right) Y^m$ with \begin{align*} k &=(q, q-1,l),& m &=(1,2,0),\\ w^k &= (L_1(q-1), -L_2q, 0),& w^m &= (2L_1,-L_2,0), \end{align*} which gives us~$\Pi {\mathcal Z}^{(q+1,q+1,l)}_{z_{\gamma^1}} \in {\mathcal G}^{q}$, with \begin{align*} z_{\gamma^1} = \frac{\pi}{4} \begin{pmatrix} L_1(q+1)(q-3) \\ -L_2 (q^2-1) \\ 0 \end{pmatrix}. \end{align*} From \begin{align*} \frac{16}{\pi^2} \det \left(n \mid z_{\alpha^1}\mid z_{\gamma^1} \right)= \det \begin{pmatrix} q+1 &L_1l & L_1(q+1)(q-3) \\ q+1 & -L_2l & -L_2 (q^2-1) \\ l & 0 & 0 \end{pmatrix} = \frac{32}{\pi^2}L_1L_2 l^2 (q+1) > 0, \end{align*} and Lemma~\ref{lemma-lin-indp_Z} we obtain that \begin{equation}\label{step382.res1} Z^{\{1,2\}, (q+1,q+1,l)}\in {\mathcal G}^{q}, \quad\mbox{for}\quad 1 \le l \le q. \end{equation} To generate $n = (q+1,q+1,0)$ we choose \begin{align*} k &=(q, q-1,1),& m &=(1,2,1),\\ w^k &= (L_1, 0, -L_3q),& w^m &= (L_1,0,-L_3), \end{align*} which gives us \[ \left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k={\mathcal Z}^{(q+1,q+1,0)}_{z_{\alpha^1}}+{\mathcal Z}^{(q+1,q+1,2)}_{z_{\alpha^2}}+\sum_{i=3}^8{\mathcal Z}^{k^i}_{z_{\alpha^i}} \] with~$k^i\in\{(q+1,q-3,0),(q+1,q-3,2),(q-1,q+1,0),(q-1,q+1,2),(q-1,q-3,0),(q-1,q-3,2)\}$. Recalling that~$q\ge4$, from~\eqref{IH.C} and~\eqref{lem36.res} it follows that $\sum_{i=3}^8\Pi{\mathcal Z}^{k^i}_{z_{\alpha^i}}\in {\mathcal G}^{q}$ and, from~\eqref{step382.res1} we have that~$\Pi{\mathcal Z}^{(q+1,q+1,2)}_{z_{\alpha^2}}\in{\mathcal G}^{q}$. Therefore we obtain $\Pi {\mathcal Z}^{(q+1,q+1,0)}_{z_{\alpha^1}} \in {\mathcal G}^{q}$, and we can find \begin{align*} z_{\alpha^1} = \frac{\pi}{8} \begin{pmatrix} -L_1(q+1) \\ 0 \\ (q+1)(-L_3q+L_3) \end{pmatrix}. \end{align*} Observe that~${\mathcal Z}^{1,(q+1,q+1,0)}_{z_{\alpha^1}}=-\frac{\pi}{8}(q^2-1)L_3\begin{pmatrix} 0\\0\\\cos(\frac{(q+1)\pi x_1}{L_1})\cos(\frac{(q+1)\pi x_2}{L_2}) \end{pmatrix}$, that is, for a suitable constant~$\zeta\ne0$, we have $Z^{1,(q+1,q+1,0)}=\zeta{\mathcal Z}^{1,(q+1,q+1,0)}_{z_{\alpha^1}}$. In particular~$\Pi{\mathcal Z}^{1,(q+1,q+1,0)}_{z_{\alpha^1}}={\mathcal Z}^{1,(q+1,q+1,0)}_{z_{\alpha^1}}$ and it follows \begin{align} \label{step382.res2} Z^{1,(q+1,q+1,0)} \in {\mathcal G}^{q}. \end{align} To sum up, from~\eqref{step381.res1}, \eqref{step381.res2}, \eqref{step382.res1}, and~\eqref{step382.res2}, it follows that \begin{align}\label{lem38.res} Z^{j(n),n} \in {\mathcal G}^{q},\quad\mbox{for all}\quad n\in{\mathcal L}^{q+1}_{1,2}{\textstyle\bigcup}{\mathcal L}^{q+1}_{2,3}{\textstyle\bigcup}{\mathcal L}^{q+1}_{3,1}. \end{align} \qed \subsubsection{Proof of Lemma \ref{lem-genq+1}}\label{subsec-proof.genq+1} Firstly, we compute~$\left( Y^k \cdot \nabla \right) Z^m + \left( Z^m \cdot \nabla \right) Y^k$ \begin{align*} k &=(q,q-1,q),& m &=(1,2,1),\\ w^k &= (L_1,0,-L_3),& w^m &= (2L_3,-L_2,0), \end{align*} which gives us~$\Pi {\mathcal Z}^{(q+1,q+1,q+1)}_{z_{\alpha^1}}\in {\mathcal G}^{q}$ where \begin{align*} z_{\alpha^1} = \frac{\pi}{8} \begin{pmatrix} 2 L_1 (q+1)\\ - L_2(q+1) \\ 0 \end{pmatrix}. \end{align*} Secondly, we compute~$\left( Y^m \cdot \nabla \right) Z^k + \left( Z^k \cdot \nabla \right) Y^m$ \begin{align*} k &=(q,q-1,q),& m &=(1,2,1),\\ w^k &= (L_1,0,-L_3),& w^m &= (2L_3,-L_2,0), \end{align*} which gives us~$\Pi {\mathcal Z}^{(q+1,q+1,q+1)}_{z_{\gamma^1}}\in {\mathcal G}^{q}$, with \begin{align*} z_{\gamma^1} = \frac{\pi}{8} \begin{pmatrix} - L_1 (q+1)(q-3)\\ L_2(q^2-1) \\ 0 \end{pmatrix}. \end{align*} Since \begin{align*} \frac{64}{\pi^2(q+1)^3} \det \left(n \mid z_{\alpha^1}\mid z_{\gamma^1} \right) = \det \begin{pmatrix} 1 & 2L_1 & -L_1 (q-3) \\ 1 & -L_2 & L_2 (q-1) \\ 1 & 0 & 0 \end{pmatrix} = L_1L_2(q+1) > 0, \end{align*} by Lemma~\ref{lemma-lin-indp_Z}, we find \begin{align} \label{lem39.res} Z^{\{1,2\}, (q+1,q+1,q+1)}\in {\mathcal G}^{q}. \end{align} \qed \section{Final remarks} Following the approximate controllability by degenerate low modes forcing proven in \cite{PhanRod18JDCS,Shirikyan06}, we present an explicit ${(\rm L, D}(A))$-saturating set in a general 3D Cylinder. This case is as an extended result in the work of 2D Cylinder (see \cite{PhanRod-ecc15}). However we just get the control $\eta \in L^\infty((0,T), {\mathcal G}^1)$ instead of $L^\infty((0,T), {\mathcal G}^0)$ in 2D Cylinder case. The reason is that we do not have the equality $B(Y^k, Y^k) = 0$ for all $k$ as in 2D Cylinder case (see more details in \cite[Theorem 3.2]{PhanRod18JDCS}). We underline that the presented saturating set is (by definition) independent of the viscosity coefficient $\nu$. That is, approximate controllability holds by means of controls taking values in ${\mathcal G}^1 = {\rm span} {\mathcal C} + {\rm span} {\mathcal B}({\mathcal C}, {\rm span} {\mathcal C}) = {\rm span}\left({\mathcal C} \cup {\mathcal B}({\mathcal C}, {\mathcal C}) \right)$, for any $\nu > 0$. Note that a ${(\rm L, D}(A))$-saturating set with less elements does exist. One of them will be introduced in next corollary. \begin{cor} \label{cor-sat-cyl} The set of eigenfuntions~ \begin{align*} \widetilde{{\mathcal C}} \coloneqq &\left\{ Y^{j(n),n} \mid~n\in\N^3,~\#_0(n)\le1,~n_i\le 3,~j(n) \in \{ 1, 2-\#_0(n) \}\right\} \\ & {\textstyle\bigcup} \left\{ Z^{j(n),n} \mid~n\in\N^3,~\#_0(n)\le1,~n_i\le {4},~j(n) \in \{ 1, 2-\#_0(n) \}\right\} \\ & {\textstyle\bigcup} \left\{ Z^{j(n),n} \mid~n \in \N^3,~\#_0(n)=2,~n_3=0 \right\} ~ {\textstyle\bigcup} \left\{Z^{(0,0,0)} \right \} \end{align*} is saturating. \end{cor} \begin{proof} Denote $\widetilde{{\mathcal C}} = {\mathcal C}_Y \cup \widetilde{{\mathcal C}}_Z$ where \begin{align*} {\mathcal C}_Y &\coloneqq \left\{ Y^{j(n),n} \mid~n\in\N^3,~\#_0(n)\le1,~n_i\le 3,~j(n) \in \{ 1, 2-\#_0(n) \}\right\}, \\ \widetilde{{\mathcal C}}_Z &\coloneqq \left\{ Z^{j(n),n} \mid~n\in\N^3,~\#_0(n)\le1,~n_i\le {4},~j(n) \in \{ 1, 2-\#_0(n) \}\right\} \\ &\quad {\textstyle\bigcup} \left\{ Z^{j(n),n} \mid~n \in \N^3,~\#_0(n)=2,~n_3=0 \right\} ~ {\textstyle\bigcup} \left\{Z^{(0,0,0)} \right \}. \end{align*} Notice that ${\mathcal C}_Y$ is the same set as in Theorem \ref{T:satur3Drect}. We recall the definition of ${\mathcal C}^q_Y$ and ${\mathcal C}^q_Z$ as in \eqref{splitCYZ}. Using Theorem \ref{T:satur3Drect}, we get that ${\mathcal C}^q_Y \subseteq {\mathcal G}^{q-2} \subseteq {\mathcal G}^{q-1}$ for all $q \in \N_4$. Repeating the arguments in the proof of Theorem \ref{T:satur3Dcyl}, we get that ${\mathcal C}^q_Z \subseteq {\mathcal G}^{q-1}$ for all $q \in \N_4$. In conclusion it yields that $\widetilde{{\mathcal C}}$ is also a saturating set with less elements than ${\mathcal C}$ in Theorem \ref{T:satur3Dcyl}. \end{proof} As mentioned in the beginning of this work, it is not our goal to find a saturating set with minimal number of elements.

proofpile-arXiv_069-10935

\section{Proof sketches}\label{AppendixProofSketches} \subsection{Correctness of the adapted protocol}\label{ConceptProtocolAdaptationsCorrectness} \subsubsection{Bottom quartile}\label{ConceptProtocolAdaptationsCorrectnessBottomQuartile} Assume the original protocol as presented in~\cite{KER10} is correct. Consequently, the rank set computed in step 3 is a set, not a multiset, containing the $n$ consecutive integers $1,...,n$~\cite{KER10}. The bottom quartile is the element at position (rank) $\lceil \frac{n}{4}\rceil$ of the sorted list of inputs. Step 6B yields \begin{equation}\label{eq:ConceptProofCorrectness6B2} E^{bq}_i = E\left(x_{\phi(i)} + r_{6B_i}\right) \end{equation} for the one player $P_i$ with rank $rank_{\phi\left(i\right)} = \lceil \frac{n}{4}\rceil$ and \begin{equation}\label{eq:ConceptProofCorrectness6B1} E^{bq}_i = E\left(r_{6B_i}\right) \end{equation} for each of the remaining $n-1$ players $P_i$ with rank $rank_{\phi\left(i\right)} \neq \lceil \frac{n}{4}\rceil$. In step 12B, each player's $E^{bq}_i$ is rerandomised by multiplying it with the encrypted identity element $0$. This does not affect the corresponding plaintext, which is \begin{equation}\label{eq:ConceptProofCorrectness12B1} \begin{aligned} D\left(E^{bq\;\prime}_i\right) &= D\left(E^{bq}_i \cdot E\left(0\right)\right)\\&= D\left(E\left(x_{\phi(i)} + r_{6B_i} + 0\right)\right)\\&= D\left(E\left(x_{\phi(i)} + r_{6B_i}\right)\right)\\&= D\left(E^{bq}_i\right)\\&= x_{\phi(i)} + r_{6B_i} \end{aligned} \end{equation} for the one player $P_i$ with rank $rank_{\phi\left(i\right)} = \lceil \frac{n}{4}\rceil$ and \begin{equation}\label{eq:ConceptProofCorrectness12B2} \begin{aligned} D\left(E^{bq\;\prime}_i\right) &= D\left(E^{bq}_i \cdot E\left(0\right)\right)\\&= D\left(E\left(r_{6B_i} + 0\right)\right)\\&= D\left(E\left(r_{6B_i}\right)\right)\\&= D\left(E^{bq}_i\right)\\&= r_{6B_i} \end{aligned} \end{equation} for each of the remaining $n-1$ players $P_i$ with rank $rank_{\phi\left(i\right)} \neq \lceil \frac{n}{4}\rceil$ (see Equation~(\ref{eq:PreliminariesHomomorphicEncryptionRerandomisationAddition})). In step 17B, the service provider $P_S$ computes the product of the $n$ rerandomised ciphertexts $E^{bq\;\prime}_i$ each multiplied by the corresponding encrypted, negated random value $r_{6B_i}$. According to Equation~(\ref{eq:PreliminariesHomomorphicEncryptionAdditionMult}), this yields \begin{equation}\label{eq:ConceptProofCorrectness17B1} \begin{aligned} \prod_{i=1}^n &E_i^{bq\;\prime} \cdot E\left(-r_{6B_i}\right)\\&= E\left(r_{6B_1} + \cdots + r_{6B_n} + x_{\phi(i)} - r_{6B_1} - \cdots - r_{6B_n}\right)\\&= E\left(x_{\phi(i)}\right) = E\left(bq\right). \end{aligned} \end{equation} Furthermore, this product of ciphertexts is multiplied by an encrypted random value $r_{10B}$ resulting in the encrypted, blinded statistical measure \begin{equation}\label{eq:ConceptProofCorrectness17B2} \begin{aligned} E\left(bq\right) \cdot E\left(r_{10B}\right) = E\left(bq + r_{10B}\right). \end{aligned} \end{equation} Decrypting this ciphertext in step 25B yields the decrypted, blinded statistical measure \begin{equation}\label{eq:ConceptProofCorrectness25B} \begin{aligned} D\left(E\left(bq + r_{10B}\right)\right) = bq + r_{10B}. \end{aligned} \end{equation} The subtraction of $r_{10B}$ by $P_S$ in step 30B results in the statistical measure \begin{equation}\label{eq:ConceptProofCorrectness30B} \begin{aligned} bq + r_{10B} - r_{10B} = bq. \end{aligned} \end{equation} Assume that all the players $P_i$ and the service provider $P_S$ use the same MAC function $MAC(\cdot)$ and the same cryptographic hash function $h(\cdot)$. Furthermore, assume that the symmetric key of the MAC function is known to every player $P_i$ but not to $P_S$. Equality of the two hashes computed in 34B given the MAC tags of steps 26B proves integrity of the statistical measure bottom quartile~\cite{KER10}. \\ This completes the proof of correctness of the steps related to the bottom quartile computation.~$\hfill\square$ \\ \subsubsection{Top quartile}\label{ConceptProtocolAdaptationsCorrectnessTopQuartile} The correctness proof of the top quartile computation is very similar to the one of the bottom quartile computation. The only difference apart from step labels and variable names is the selection criterion of the OT step 6C, i.e. $\lfloor \frac{3\cdot n}{4} +1 \rfloor$, which is the position of the top quartile element. \subsection{Privacy of the adapted protocol}\label{ConceptProtocolAdaptationsPrivacy} For our protocol to be secure in the semi-honest model, it has to be provided that anything an adversary can learn during a protocol execution can as well be learned only from the input and the output of the protocol~\cite{KER10}. To prove that, one needs to show that the view $V$ of an adversary can be simulated only based on the input and output~\cite{KER10}. The protocol privately computes the respective quartile if a simulator $S$ is able to generate a view $V^\prime$ that is computationally indistinguishable from a participant’s view $V$~\cite{KER10}. This simulator creates the protocol input and the coin tosses himself. The former can be done by taking the original input while the latter is simulated by using the same pseudorandom generator (PRG) that is used for generating the random numbers in the protocol. Therefore, only the messages $m_i$ that the participant receives are relevant~\cite{KER10}. For these, the simulator has to generate a message $m_i^\prime$ for each message $m_i$ of the view $V$ such that both are computationally indistinguishable. \subsubsection{The players' view}\label{ConceptProtocolAdaptationsPrivacyPlayersView} Similar to the proof of the original protocol in~\cite{KER10}, the OT protocol is substituted by its ideal functionality for simplification purposes. Since the players have the secret decryption key, in case the decrypted messages can be simulated by $S_{P_i}$ computationally indistinguishable, the same applies to the encrypted messages. This is true since encryption can be regarded as a deterministic mapping of probability distributions~\cite{KER10}. During a protocol execution, player $P_i$ receives the following messages. The arrow ``$\rightarrow$'' shows the corresponding plaintext that $P_i$ can compute himself. \begin{itemize} \addtolength{\itemindent}{0.25cm} \item[6B.] $E^{bq}_i = \begin{cases} E\left(x_{\phi(i)} + r_{6B_i}\right) \rightarrow x_{\phi(i)} + r_{6B_i}\\ E\left(r_{6B_i}\right) \rightarrow r_{6B_i} \end{cases}$ \item[6C.] $E^{tq}_i = \begin{cases} E\left(x_{\phi(i)} + r_{6C_i}\right) \rightarrow x_{\phi(i)} + r_{6C_i}\\ E\left(r_{6C_i}\right) \rightarrow r_{6C_i} \end{cases}$ \item[17B.] $E\left(bq+r_{10B}\right) \rightarrow bq+r_{10B}$ \item[17C.] $E\left(tq+r_{10C}\right) \rightarrow tq+r_{10C}$ \item[30B.] $bq$ \item[30C.] $tq$ \item[34B.] $\begin{aligned}[t]h(&MAC(bq+r_{10B}||1,K_{MAC}),...,\\&MAC(bq+r_{10B}||n,K_{MAC}))\end{aligned}$ \item[34C.] $\begin{aligned}[t]h(&MAC(tq+r_{10C}||1,K_{MAC}),...,\\&MAC(tq+r_{10C}||n,K_{MAC}))\end{aligned}$ \end{itemize} \subsubsection{The service provider's view}\label{ConceptProtocolAdaptationsPrivacyServiceProvidersView} Different from the players, the service provider $P_S$ does not have an input. His output are the statistical measures $bq$ and $tq$. Since he does not have access to the secret decryption key, he cannot decrypt encrypted messages~\cite{KER10}. The messages he receives are as follows. \begin{itemize} \addtolength{\itemindent}{0.25cm} \item[12B.] $E^{bq\;\prime}_i = E^{bq}_i \cdot E\left(0\right)$ \item[12C.] $E^{tq\;\prime}_i = E^{tq}_i \cdot E\left(0\right)$ \item[25B.] $bq+r_{10B}$ \item[25C.] $tq+r_{10C}$ \item[26B.] $MAC\left(bq+r_{10B}||i,K_{MAC}\right)$ \item[26C.] $MAC\left(tq+r_{10C}||i,K_{MAC}\right)$ \end{itemize} \subsubsection{The service provider's and the players' simulators}\label{ConceptProtocolAdaptationsPrivacySimulators} Given that $dom(\cdot)$ denotes the domain of a function, the simulator $S_{P_i}$ for a player $P_i$ generates the following simulated messages. The names of the random values are not related to those of the random values used in the protocol description. \begin{itemize} \addtolength{\itemindent}{0.25cm} \item[6B.] A random value $r_1$, uniformly chosen from $dom(D(\cdot))$ \item[6C.] A random value $r_2$, uniformly chosen from $dom(D(\cdot))$ \item[17B.] A random value $r_3$, uniformly chosen from $dom(D(\cdot))$ \item[17C.] A random value $r_4$, uniformly chosen from $dom(D(\cdot))$ \item[30B.] $bq$ \item[30C.] $tq$ \item[34B.] If the validation bit $v_{34B_i}$ equals $1$, the simulator's value is\\ $\begin{aligned}h(&MAC(bq+r_3||1,K_{MAC}),...,\\&MAC(bq+r_3||n,K_{MAC}))\end{aligned}$\\otherwise it is a random value $r_5$, uniformly chosen from $dom(h(\cdot))$ \item[34C.] If the validation bit $v_{34C_i}$ equals $1$, the simulator's value is\\ $\begin{aligned}h(&MAC(tq+r_4||1,K_{MAC}),...,\\&MAC(tq+r_4||n,K_{MAC}))\end{aligned}$\\ otherwise it is a random value $r_6$, uniformly chosen from $dom(h(\cdot))$ \end{itemize} The corresponding simulator $S_{P_S}$ for the service provider $P_S$ generates the following simulated messages. \begin{itemize} \addtolength{\itemindent}{0.25cm} \item[12B.] A random value $r_7$, uniformly chosen from $dom(E(\cdot))$ \item[12C.] A random value $r_8$, uniformly chosen from $dom(E(\cdot))$ \item[25B.] $bq+r_9$ where $r_9$ is an internal coin toss \item[25C.] $tq+r_{10}$ where $r_{10}$ is an internal coin toss \item[26B.] A random value $r_{11}$, uniformly chosen from $dom(MAC(\cdot))$ \item[26C.] A random value $r_{12}$, uniformly chosen from $dom(MAC(\cdot))$ \end{itemize} \setcounter{footnote}{0} \renewcommand*{\thefootnote}{\alph{footnote}} \begin{table*}[!t] \renewcommand{\arraystretch}{1.3} \caption{Computational and Communication Complexity of the Enhanced Protocol per Participant} \centering \begin{tabular}{l|c|c|c|c|c|c|c} \hline \textbf{Step} & \textbf{$E(\cdot)$} & \textbf{$D(\cdot)$} & \textbf{Exp} & \textbf{Mult} & \textbf{Add} & \textbf{Inv} & \textbf{Values to send}\\\hline\hline \textbf{1} & $1$ & & & & & & $1$\\\hline \textbf{2} & $1$ & & & $n$ & & & $n$\\\hline \textbf{3} & $n^2$ & & $n^2$ & $2\!\cdot\!n^2$ & & $n^2$ & $n^2$\\\hline \textbf{4-6C ($P_i$)} & $1$ & & & & $2$ & & $1$\\\hline \textbf{4-6C ($P_S$)} & $n$ & & & $n$ & & & $n$\\\hline \textbf{7} & & $1$ & & & & & $1$\\\hline \textbf{8} & & & \footnoteref{ftn:AppendixComplexityDependsMAC} & \footnoteref{ftn:AppendixComplexityDependsMAC} & \footnoteref{ftn:AppendixComplexityDependsMAC} & & $1$\\\hline \textbf{9} & & & & & $1$ & & $n$\\\hline \textbf{10-12C} & $1$ & & & $1$ & & & $1$\\\hline \textbf{13} & $1$ & & & $2$ & $1$ & & $1$\\\hline \textbf{14} & $1$ & & & $n$ & & & $n$\\\hline \textbf{15-17C} & $n+1$ & & & $2\!\cdot\!n$ & & & $n$\\\hline \textbf{18} & & & \footnoteref{ftn:AppendixComplexityDependsHash} & \footnoteref{ftn:AppendixComplexityDependsHash} & \footnoteref{ftn:AppendixComplexityDependsHash} & & $n$\\\hline \textbf{19, 21, 23, 25, 25B, 25C} & & $1$ & & & & & $1$\\\hline \textbf{20, 22, 24, 26, 26B, 26C} & & & \footnoteref{ftn:AppendixComplexityDependsMAC} & \footnoteref{ftn:AppendixComplexityDependsMAC} & \footnoteref{ftn:AppendixComplexityDependsMAC} & & $1$\\\hline \textbf{27-30C} & & & & & $1$ & & $n$\\\hline \textbf{31-34C} & & & \footnoteref{ftn:AppendixComplexityDependsHash} & \footnoteref{ftn:AppendixComplexityDependsHash} & \footnoteref{ftn:AppendixComplexityDependsHash} & & $n$\\\hline\hline \textbf{Total $\left(P_i\right)$} & \textbf{$8$} & \textbf{$7$} & \footnoteref{ftn:AppendixComplexityDependsMAC} & \textbf{$7$}\footnoteref{ftn:AppendixComplexityDependsMAC} & \textbf{$11$}\footnoteref{ftn:AppendixComplexityDependsMAC} & & \textbf{$26$}\\\hline \textbf{Total $\left(P_S\right)$} & \textbf{$n^2\!+\!10\!\cdot\!n\!+\!7$} & \textbf{$0$} & \textbf{$n^2$}\footnoteref{ftn:AppendixComplexityDependsHash} & \textbf{$2\!\cdot\!n^2\!+\!17\!\cdot\!n$}\footnoteref{ftn:AppendixComplexityDependsHash} & \textbf{$7$}\footnoteref{ftn:AppendixComplexityDependsHash} & \textbf{$n^2$} & \textbf{$n^2\!+\!26\!\cdot\!n$}\\\hline \end{tabular} \vspace{0.2in} \label{tab:AppendixComplexityComputational} \end{table*} \subsubsection{Comparison}\label{ConceptProtocolAdaptationsPrivacyComparison} Privacy is proven if the simulator generates an output that is computationally indistinguishable from a participant's view. To prove computational indistinguishability, it is sufficient to show that two functions are identically distributed~\cite{KER10}. In case of steps 6B, 6C, 17B, and 17C, the secret values are blinded by a random number that is added to it. Consequently, these sums are identically distributed to uniformly chosen random numbers~\cite{KER10}. Therefore, they are computationally indistinguishable from the uniformly chosen random numbers provided by the simulator $S_{P_i}$~\cite{KER10}. The messages of steps 12B and 12C are rerandomised encrypted values that cannot be decrypted by the receiver. According to the proof of semantic security of Paillier's cryptosystem, the ciphertexts are computationally indistinguishable from uniformly chosen random numbers~\cite{KER10}. Therefore, they are also computationally indistinguishable from the corresponding output provided by the simulator $S_{P_S}$~\cite{KER10}. In steps 25B and 25C, the messages are blinded statistical measures, that is, sums of a statistical measure and a uniformly chosen random value. The statistical measure is part of the receiver's, i.e. the service provider's, output. Therefore, the simulator can copy it from the output~\cite{KER10}. The random variable was previously generated by $P_S$ himself and is hence known to him~\cite{KER10}. Given the same PRG, the simulator can generate a random value which is identically distributed~\cite{KER10}. Therefore, the messages and $S_{P_S}$'s output are identically distributed and hence computationally indistinguishable~\cite{KER10}. Assuming that the service provider is not capable of MAC forgery and does not know the secret MAC key $K_{MAC}$, MAC tags are computationally indistinguishable from random numbers for him~\cite{KER10}. Therefore, the messages of steps 26B and 26C are computationally indistinguishable from the uniformly chosen random numbers provided by the simulator $S_{P_S}$. The simulation of steps 30B and 30C only requires copying $P_i$'s outputs, i.e. the statistical measures bottom quartile and top quartile, that are known to the simulator. The messages of player $P_i$ and the output of the simulator $S_{P_i}$ are therefore computationally indistinguishable~\cite{KER10}. In steps 34B and 34C, the hashes sent from the service provider can either match the ones computed by player $P_i$ or not. If they do not match, the received hash appears as a uniformly chosen random number to $P_i$ as he cannot compute a pre-image of the hash~\cite{KER10}. If the hashes match, the simulator's output is the actual hash~\cite{KER10}. In both cases, the simulator's outputs and the player's view are identically distributed and therefore computationally indistinguishable~\cite{KER10}. \\ This completes the proof of privacy of the adapted protocol.~$\hfill\square$ \subsection{Computational and communication complexity of the adapted protocol}\label{AppendixComplexity} For each step, the number of required encryptions, decryptions, exponentiations, multiplications, and additions as well as the number of modular inversions is given in Table~\ref{tab:AppendixComplexityComputational}. Together they form the computational complexity. The communication complexity is given in the rightmost column. Consecutive steps with the same complexity, such as steps 4 to 6C, are shown in a single row of the same table. \pagebreak The lower-case letters in this table do not represent variables but instead correspond to the footnotes~\footnote{Depends on the MAC function being used\label{ftn:AppendixComplexityDependsMAC}}~and~\footnote{Depends on the hash function being used\label{ftn:AppendixComplexityDependsHash}}. \section{Related Work}\label{Approaches} We assess the suitability of several existing approaches and protocols for privacy-preserving benchmarking regarding the requirements described above. It covers the generic approaches of garbled circuits, secret sharing, and homomorphic encryption as well as custom protocols. The former can be used for computing arbitrary functions while the custom protocols only enable computation of predefined functions. \subsection{Garbled circuits}\label{ApproachesBMR90} The protocol presented by Beaver, Micali, and Rogaway in~\cite{BMR90} implements the garbled circuits approach for multiple players. The players create a Boolean circuit that implements $f(x)$. Even though one might consider garbled circuits to be impractical~\cite{HEK11}, it was shown that they can compete with custom protocols regarding efficiency~\cite{HEK12}. These findings were driven by improvements of the garbled circuits approach, such as free XOR and garbled row reduction (see~\cite{HEK11,KAS08,PSS09}). For specific tasks, e.g. calculation of the mean, one only needs to create a garbled circuit that implements the corresponding operations instead of developing an entire protocol. The protocol of~\cite{BMR90} consists of two phases, one for jointly generating a common garbled circuit $C$ together with the garbled input and a second for publishing and evaluating this circuit. Since the protocol implements the generic approach of garbled circuits, it can be used for calculating any computable function~\cite{BMR90}. However, up-front effort is required to create the necessary circuits. The most important drawback of this approach is its non-centralised communication model. It requires pairwise communication between the players, which precludes inherent anonymity among them. Therefore, the approach presented in~\cite{BMR90} does not suit the requirements properly. \subsection{Linear secret sharing and homomorphic encryption}\label{ApproachesLinearSecretSharingHomomorphicEncryption} This approach is a combination of linear secret sharing and homomorphic encryption. The idea of secret sharing is to split the secret values into shares and spread these shares among the players involved in the scheme~\cite{CDN15}. The function $f(x)$ for these inputs can then be computed given the shares~\cite{CDN15}. In this approach, the players $P_i$ share their secret input values $x_i$ among the set of $n$ players using a linear secret sharing scheme like Shamir's scheme (see~\cite{SHA79}). Each player $P_j$ holds one share $[\![ x_i ]\!]_j$ for each of the $n$ secret-shared values $[\![ x_i ]\!]$. Addition of shares is a linear operation and can be performed locally while multiplication requires a subprotocol that introduces overhead~\cite{HIR01}. Such a subprotocol for two players is presented by Atallah et al. in~\cite{ABL04}. The complexity of this subprotocol is exponential in $n$. In its original form, the approach has a non-centralised communication model requiring pairwise communication between the players. Therefore, the anonymity requirement is not met. The most important drawback of this approach is the complexity of the multiplication subprotocol making it impractical for large $n$, i.e. large peer groups. Consequently, the approach described in this Section does not suit the requirements properly. \subsection{Fully homomorphic encryption}\label{ApproachesFullyHomomorphicEncryption} Given a FHE scheme with an encryption function $E(\cdot)$ and the encrypted secret values $E(x_1),...,E(x_n)$, one can compute the encrypted result $E(y)=E(f(x_1,...,x_n))$ for any efficiently computable function $f(x)$ without needing to decrypt~\cite{GEN09b}. Given that, one can build a secure MPC system where one player conducts the entire computation of $E(f(x))$ locally without learning anything about $x_1,...,x_n$ or $y$. In~\cite{GEN09b}, Gentry presents the first ever FHE scheme. This seminal work started a new research field and led to many significant improvements. However, even enhanced approaches to FHE schemes, such as the one presented by Brakerski, Gentry, and Vaikuntanathan in~\cite{BGV14}, have complexity at least polynomial in the size of the respective circuit with large constants. Therefore, this approach can reasonably be assumed to not suit the performance requirements properly. \subsection[Custom protocol]{Custom benchmarking protocol}\label{ApproachesKER10} In~\cite{KER10}, Kerschbaum presents a secure MPC protocol designed for privacy-preserving benchmarking platforms that has constant cost\footnote{Constant cost here means constant round complexity and constant, i.e. linear in the size of the security parameter $\kappa$, communication complexity. Both are independent of the peer group size~\cite{KER08}.}, provides anonymity among the players, and is centralised. It requires a single server; the service provider $P_S$ of the benchmarking platform who acts as a processor. The protocol enables the benchmarking platform to compute the statistical measures mean, variance, median, maximum, and best-in-class~\cite{KER10}. The protocol is based on an additively partially homomorphic encryption scheme. The protocol natively implements the service provider model and therefore can ensure anonymity among the players as well as cloud suitability. Input privacy is ensured in the semi-honest model as well as in the constrained malicious model~\cite{KER10}. Output integrity is ensured via message authentication codes (MACs). With a fixed set of four rounds, its round complexity is constant. However, the protocol only offers a subset of the required statistical measures. The protocol's computational and communication complexity both are quadratic in $n$. This may prove critical for large peer groups. \subsection[Custom aggregation protocol]{Custom aggregation protocol}\label{ApproachesBIK16} In~\cite{BIK16}, Bonawitz et al. present a protocol for secure aggregation. This protocol assumes two kinds of participants: one server $P_S$, acting as a processor, and a set of $n$ players $P_i$, each providing a secret input $x_i$. Players only communicate with the server $P_S$, who acts as a mediator between players. Only $P_S$ learns the output $y=\sum_{i=1}^n x_i$, i.e. the sum of the secret values. The players do not learn anything new~\cite{BIK16}. Executed in a central server scenario, the protocol enables\linebreak anonymity among the players as well as cloud suitability. Rank-based statistical measures such as quartiles cannot directly be computed using this protocol. Even though its round complexity is constant, the protocol has computational and communication complexity that is quadratic in the number of players $n$, which may prove critical for large $n$~\cite{BIK16}. \subsection{Summary and selection of a suitable approach}\label{ApproachesSummary} The three generic approaches garbled circuits, secret sharing, and (fully) homomorphic encryption do not meet the requirements mostly due to their complexity. On top of that, their implementation in a centralised communication model would require additional effort and further increase the complexity. Otherwise, they would not meet the required level of anonymity. The less complex protocol presented in~\cite{BIK16} is not suitable due to the lack of rank-based statistical measures. The most suitable approach is the privacy-preserving benchmarking protocol presented in~\cite{KER10}. Even though its computational and communication complexity of $\mathcal{O}(n^2)$ may prove critical for large peer groups, we build upon this protocol. \section{Conclusion and Future Work}\label{Conclusion} This paper provides an overview of benchmarking based on secure multi-party computation. It elaborates requirements for a secure benchmarking system in the context of product cost optimisation. Based on these requirements, generic approaches to secure KPI benchmarking were discussed, the existing related work was reviewed, and fitting approaches were described. The most suitable approach was selected, extended, implemented in a prototype, and extensively evaluated. The resulting protocol meets all of the requirements for a privacy-preserving benchmarking system in the given context of product cost optimisation. However, its computational and communication complexity that is quadratic in the number of players $n$, i.e. $\mathcal{O}(n^2)$, shows potential for further optimisation. In our future work, we will focus on reducing the computational complexity by simplifying the rank computation. In the presented protocol, this sorting step compares each player's value to any other player's value, causing a quadratic number of comparisons. Fortunately, the lower bound for comparisons in sorting is $\mathcal{O}(n \cdot \log n)$~\cite{CLRS09}. Such a sorting mechanism with less comparisons would likely need to be more interactive, causing a higher communication complexity. Furthermore, it would require an additional step for hiding the order of the inputs. Otherwise, the service provider would learn the players' ranks in the sorted list of inputs. \section{Design}\label{Design} The privacy-preserving benchmarking protocol of~\cite{KER10} (see Section~\ref{ApproachesKER10}) only offers a subset of the required statistical measures, i.e. mean, variance, median, maximum, and best-in-class. To provide the additional statistical measures bottom quartile $bq$ and top quartile $tq$, the protocol had to be enhanced. The full adapted protocol is given in this Section. Prior to the protocol execution, each player $P_i$ learns the following two keys, e.g. with the help of a certificate authority as described in~\cite{KER10}. \begin{itemize} \item $K_{DEC}$: The secret decryption key of the PHE scheme. \item $K_{MAC}$: The symmetric key of the MAC. \end{itemize} Every participant, including the service provider $P_S$, also learns the public encryption key $K_{ENC}$ corresponding to $K_{DEC}$. The players use the same secret key for decryption. They directly communicate only with the service provider, via pairwise channels that are secured based on standard methods for protecting transmission over insecure networks~\cite{KER10}. \subsection{Adapted protocol}\label{DesignAdaptedProtocol} Both the original protocol and our enhanced version are combinations of the techniques summation, rank computation, selection, and decryption~\cite{KER10}, which will be introduced first. Summation of encrypted values is conducted by multiplying the ciphertexts (see Equation~(\ref{eq:PreliminariesHomomorphicEncryptionAdditionMult}))~\cite{KER10}. For $n$ values $x_i$, the encrypted sum is \begin{equation}\label{eq:ApproachesKER10Summation} E\left(sum\right) = E\left(\sum^n_{i=1} x_i\right) = \prod^n_{i=1}E\left(x_i\right). \end{equation} Summation is required for calculation of the mean $mean$ (steps 1 and 2) and of the variance $var$ (steps 13 and 14). The sum is blinded by adding a random value~\cite{KER10}. Since the players know the size $n$ of the peer group, each player can compute the mean himself by dividing the sum by $n$~\cite{KER10}. Rank computation yields the rank of a value $x_i$ in a list which is sorted in ascending order~\cite{KER10}. To achieve that, the value $x_i$ is compared to each value $x_j$. For that comparison, the indices of the secret values are permuted by the permutations $\phi$ and $\phi^\prime$~\cite{KER10}. The assigned element of $i$ is denoted by $\phi(i)$ while the one of $j$ has index $\phi^\prime(j)$. The difference between $x_{\phi(i)}$ and $x_{\phi^\prime(j)}$ is blinded by two random values $1 \leq r_{2_j}$ and $0 \leq r_{3_j} \ll r_{2_j}$~\cite{KER10}. These are chosen individually for each $j$. The blinded difference \begin{equation}\label{eq:ApproachesKER10RankDifference} c_{\phi(i)_{\phi^\prime(j)}} = r_{2_j} \cdot \left(x_{\phi(i)} - x_{\phi^\prime(j)}\right) + r_{3_j} \end{equation} is stored in the vector $\vec {c_{\phi(i)}}$. Counting the non-negative elements $pos(\vec {c_{\phi(i)}})$ of that vector yields the number of input values that are smaller than $x_{\phi(i)}$~\cite{KER10}. Given that, its rank is \begin{equation}\label{eq:ApproachesKER10Rank} rank_{\phi\left(i\right)} = pos\left(\vec {c_{\phi\left(i\right)}}\right) + 1. \end{equation} Now, due to the permutations, each player $P_i$ holds the rank of the value $x_{\phi(i)}$ of some player $P_{\phi(i)}$~\cite{KER10}. Rank computation is done once in the protocol (step 3). It is required for calculation of the the median $med$, the best-in-class $bic$, the maximum $max$, the bottom quartile $bq$, and the top quartile $tq$. Selection is the act of computing the ciphertext of a secret value with specific, i.e. selected, rank~\cite{KER10}. First, $P_S$ chooses a random value $r_i$ individually for each player $P_i$ and computes the ciphertexts $E(x_{\phi(i)}+r_i)$ and $E(r_i)$~\cite{KER10}. That is, the value of $P_i$'s assigned rank blinded by $r_i$ and a $0$ blinded by $r_i$. By using a 1-out-of-2 OT protocol (see Section~\ref{PreliminariesPrimitivesOT}), a player $P_i$ receives $E(x_{\phi(i)}+r_i)$, i.e. the blinded secret value, only if his assigned rank is the one selected~\cite{KER10}. The other players receive the blinded $0$. As these OT steps are identical, we use a template to increase readability of the protocol. Let \begin{equation} \begin{split} OT_{\circ}(m,r_i,p,i) &=\ P_S \xrightarrow{\mathrm{OT}} P_i:\\E^{m}_i&= \begin{cases} E\left(x_{\phi(i)} + r_{i}\right) & \text{if}\;\;\; rank_{\phi\left(i\right)} \circ p\\ E\left(r_{i}\right) & \text{otherwise} \end{cases} \end{split} \end{equation} be the OT step, indexed with a binary comparison operator $\circ$ that takes a statistical measure descriptor $m$, a blinding parameter $r_i$, an array position in the sorted list $p$, and an index $i$. After the OT step, each player rerandomises the value he received by multiplying it by an encrypted $0$ (see Equation~(\ref {eq:PreliminariesHomomorphicEncryptionRerandomisationAddition})) and sends the product to the service provider~\cite{KER10}. The service provider then multiplies the encrypted values he received, removes the random values $r_i$, and gets the ciphertext of $x_{\phi(i)}$~\cite{KER10}. Selection is required for computing the median, best-in-class, maximum, bottom quartile, and the top quartile. It occurs in steps 4 to 6C (OT), steps 10 to 12C (returning the selected values), and steps 15 to 17C (computing the results). Decryption by the service provider is required since he is supposed to learn the result first~\cite{KER10}. Thus, he is able to round the result before sending it to the players~\cite{KER10}. To decrypt the result $v$, $P_S$ first blinds the result with a random value $r$ and sends the ciphertext $E(v+r)$ to the players~\cite{KER10}. Each player $P_i$ decrypts the blinded result and sends the plaintext $v+r$ together with the corresponding MAC tag \begin{equation}\label{eq:ApproachesKER10DecryptionMAC} \theta_i = MAC\left(v+r||i, K_{MAC}\right) \end{equation} back to $P_S$~\cite{KER10}. The service provider gets $v$ by subtracting the random value $r$. To prove that he sent the same encrypted, blinded result to every player, $P_S$ computes the hash \begin{equation}\label{eq:ApproachesKER10DecryptionHash} \begin{split} h(&\theta_1=MAC(v+r||1, K_{MAC}),\ldots,\\&\theta_n=MAC(v+r||n, K_{MAC})) \end{split} \end{equation} of the MAC tags $\theta_i$ he received by using a cryptographic hash function~\cite{KER10}. Together with the result $v$, $P_S$ sends this hash to the players. Each $P_i$ then computes the MAC tags and the hash and compares the hash to the one received from the service provider and obtains a validation bit $v_{s_i}$~\cite{KER10}. This bit, where $s$ indicates the protocol step, is $1$ in case of successful hash validation and $0$ otherwise. It states whether the service provider has sent the same statistical measure to each $P_i$~\cite{KER10}. Decryption is required for each of the statistical measures mean, variance, median, best-in-class, maximum, bottom quartile, and top quartile~\cite{KER10}. It occurs in steps 2 and 14 to 17C (sending encrypted results), steps 7, 8, and 19 to 26C (returning decrypted, blinded results), steps 9 and 27 to 30C (sending decrypted results), and steps 18 and 31 to 34C (sending the hashed MAC tags). Based on these preliminaries, our full enhanced protocol is given below in Table~\ref{tab:DesignAdaptedProtocolTable} together with descriptions of the steps we added for the bottom quartile and top quartile computation. These steps are marked with the letter ``B'' and ``C'' in the step label, respectively. \begin{table*}[t!] \renewcommand{\arraystretch}{1.3} \centering \caption{Enhanced Benchmarking Protocol with Step Labels and Computations} \subfloat{% \begin{tabular}[t]{p{0.7cm}|p{6.35cm}} \hline \textbf{Step} & \textbf{Computation}\\\hline\hline 1 & \underline{$P_i \rightarrow P_S$:} $E(x_i)$ \\ \hline 2 & \underline{$P_S \rightarrow P_i$:} $E(sum+r_1) = E(\sum_{i=1}^n x_i) \cdot E(r_1)$ \\ 3 & $\begin{aligned}[t]\!E(\vec {c_{\phi(i)}}) = &(\ldots, E(c_{\phi(i)_{\phi^\prime(j)}})\\&= E(r_{2_j} \cdot (x_{\phi(i)} - x_{\phi^\prime(j)}) + r_{3_j}),\ldots)\end{aligned}$ \\ 4 & $\mathrm{OT}_{=}(med,r_4,\lceil \frac{n}{2}\rceil,i)$ \\ 5 & $\mathrm{OT}_{\geq}(bic,r_5,\lfloor \frac{3\cdot n}{4}+1\rfloor,i)$ \\ 6 & $\mathrm{OT}_{=}(max,r_6,n,i)$ \\ 6B & $\mathrm{OT}_{=}(bq,r_{6B},\lceil \frac{n}{4}\rceil,i)$ \\ 6C & $\mathrm{OT}_{=}(tq,r_{6C},\lfloor \frac{3\cdot n}{4}+1\rfloor,i)$ \\ 7 & \underline{$P_i \rightarrow P_S$:} $sum+r_1 = D(E(sum+r_1))$ \\ 8 & $MAC(sum+r_1||i,K_{MAC})$ \\ 9 & \underline{$P_S \rightarrow P_i$:} $sum = sum+r_1-r_1$ \\ 10 & \underline{$P_i \rightarrow P_S$:} $E^{med\;\prime}_i = E^{med}_i \cdot E(0)$ \\ 11 & $E^{bic\;\prime}_i = E^{bic}_i \cdot E(0)$ \\ 12 & $E^{max\;\prime}_i = E^{max}_i \cdot E(0)$ \\ 12B & $E^{bq\;\prime}_i = E^{bq}_i \cdot E(0)$ \\ 12C & $E^{tq\;\prime}_i = E^{tq}_i \cdot E(0)$ \\ 13 & $E((x_i-mean)^2) = E((x_i-\frac{sum}{n})^2)$ \\ \hline 14 & \underline{$P_S \rightarrow P_i$:} $\begin{aligned}[t]E&(var+r_7)\\&=E(\textstyle\sum_{i=1}^n(x_i-mean)^2) \cdot E(r_7)\\&= (\textstyle\prod_{i=1}^n E((x_i-mean)^2)) \cdot E(r_7)\end{aligned}$ \\ 15 & $E(med+r_8) = (\prod_{i=1}^n E_i^{med\;\prime} \cdot E(-r_{4_i})) \cdot E(r_8)$ \\ 16 & $E(bic+r_9) = (\prod_{i=1}^n E_i^{bic\;\prime} \cdot E(-r_{5_i})) \cdot E(r_9)$ \\ 17 & $E(max+r_{10}) = (\prod_{i=1}^n E_i^{max\;\prime} \cdot E(-r_{6_i})) \cdot E(r_{10})$ \\ 17B & $E(bq+r_{10B}) = (\prod_{i=1}^n E_i^{bq\;\prime} \cdot E(-r_{6B_i})) \cdot E(r_{10B})$ \\ 17C & $E(tq+r_{10C}) = (\prod_{i=1}^n E_i^{tq\;\prime} \cdot E(-r_{6C_i})) \cdot E(r_{10C})$ \\ 18 &$\begin{aligned}[t]h(&MAC(sum+r_1||1,K_{MAC}),\ldots,\\&MAC(sum+r_1||n,K_{MAC}))\end{aligned}$ \\ 19 & \underline{$P_i \rightarrow P_S$:} $var+r_7 = D(E(var+r_7))$ \\ \end{tabular}} \hfil \subfloat{% \begin{tabular}[t]{p{0.7cm}|p{6.35cm}} \hline \textbf{Step} & \textbf{Computation}\\\hline\hline 20 & $MAC(var+r_7||i,K_{MAC})$ \\ 21 & $med+r_8 = D(E(med+r_8))$ \\ 22 & $MAC(med+r_8||i,K_{MAC})$ \\ 23 & $bic+r_9 = D(E(bic+r_9))$ \\ 24 & $MAC(bic+r_9||i,K_{MAC})$ \\ 25 & $max+r_{10} = D(E(max+r_{10}))$ \\ 25B & $bq+r_{10B} = D(E(bq+r_{10B}))$ \\ 25C & $tq+r_{10C} = D(E(tq+r_{10C}))$ \\ 26 & $MAC(max+r_{10}||i,K_{MAC})$ \\ 26B & $MAC(bq+r_{10B}||i,K_{MAC})$ \\ 26C & $MAC(tq+r_{10C}||i,K_{MAC})$ \\ 27 & \underline{$P_S \rightarrow P_i$:} $var = var+r_7-r_7$ \\ 28 & $med = med+r_8-r_8$ \\ 29 & $bic = bic+r_9-r_9$ \\ 30 & $max = max+r_{10}-r_{10}$ \\ 30B & $bq = bq+r_{10B}-r_{10B}$ \\ 30C & $tq = tq+r_{10C}-r_{10C}$ \\ \hline 31 & \underline{$P_S \rightarrow P_i$:} $\begin{aligned}[t]h(&MAC(var+r_7||1,K_{MAC}),\ldots,\\&MAC(var+r_7||n,K_{MAC}))\end{aligned}$ \\ 32 & $\begin{aligned}[t]h(&MAC(med+r_8||1,K_{MAC}),\ldots,\\&MAC(med+r_8||n,K_{MAC}))\end{aligned}$ \\ 33 & $\begin{aligned}[t]h(&MAC(bic+r_9||1,K_{MAC}),\ldots,\\&MAC(bic+r_9||n,K_{MAC}))\end{aligned}$ \\ 34 & $\begin{aligned}[t]h(&MAC(max+r_{10}||1,K_{MAC}),\ldots,\\&MAC(max+r_{10}||n,K_{MAC}))\end{aligned}$ \\ 34B & $\begin{aligned}[t]h(&MAC(bq+r_{10B}||1,K_{MAC}),\ldots,\\&MAC(bq+r_{10B}||n,K_{MAC}))\end{aligned}$ \\ 34C & $\begin{aligned}[t]h(&MAC(tq+r_{10C}||1,K_{MAC}),\ldots,\\&MAC(tq+r_{10C}||n,K_{MAC}))\end{aligned}$ \\ \hline \end{tabular}} \label{tab:DesignAdaptedProtocolTable} \end{table*} \paragraph{Round 1 (step 1)}\label{ConceptProtocolRound1} Each player $P_i$ sends his encrypted input to the service provider $P_S$. \paragraph{Round 2 (steps 2-13)}\label{ConceptProtocolRound2} The service provider computes the encrypted, blinded sum of the input values and sends it to the players $P_i$. Furthermore, $P_S$ conducts a rank computation after which each player has the rank of some player $P_j$'s input value. Given that rank, each $P_i$ receives either an encrypted, blinded statistical measure or an encrypted random value via OT depending on whether his assigned rank fits the respective selection criterion. This is repeated for each of the statistical measures median, best-in-class, maximum, bottom quartile, and top quartile. Afterwards, the players decrypt the blinded sum they received and send it back to $P_S$ together with a MAC tag of the blinded sum. Furthermore, $P_S$ sends the sum to each $P_i$. Then, each player rerandomises his OT step outputs and sends them back to the service provider. Then each player computes the squared difference between his input and the mean and sends the encrypted result to $P_S$ as the basis for the variance computation. Steps 6B and 6C are OT steps that are part of the selection of the bottom quartile and top quartile values of the sorted list of inputs. In step 6B, the selection criterion of the OT protocol $rank_{\phi\left(i\right)} = \lceil \frac{n}{4}\rceil$ is the index of the sorted list's bottom quartile element. For step 6C, the top quartile index $\lfloor \frac{3\cdot n}{4} +1 \rfloor$ is used. Steps 12B and 12C rerandomise the ciphertext received from the service provider in steps 6B and 6C. \paragraph{Round 3 (steps 14-30C)}\label{ConceptProtocolRound3} The service provider computes the encrypted, blinded statistical measures variance, median, best-in-class, maximum, bottom quartile, and top quartile by multiplying the values received in round 2. He sends them to the players together with the hashed MAC tags of the blinded sum. The latter is then used by the players to validate whether each player previously received the same blinded sum. Similar to round 2, each player then decrypts the blinded statistical measures and sends them to $P_S$ together with the respective MAC tags. In the last steps of round 3, $P_S$ sends the unblinded statistical measures to the players. Steps 17B and 17C return the encrypted, rank-based statistical measures to the players. They are computed by multiplying the $n$ encrypted, rerandomised OT messages that $P_S$ received in steps 12B and 12C. Additionally, the statistical measures are blinded by multiplying the product by an encrypted random value $r_{10B}$ and $r_{10C}$, respectively. In steps 25B and 25C, each player decrypts the blinded statistical measures he received in steps 17B and 17C and sends the result to $P_S$. In steps 26B and 26C, each player sends the MAC tags of the blinded bottom quartile and top quartile, respectively. The blinded statistical measures are concatenated with player $P_i$'s index $i$ before computing the corresponding MAC tag. This MAC computation requires the symmetric MAC key $K_{MAC}$. Steps 30B and 30C are used by $P_S$ to send the output, i.e. the decrypted, unblinded statistical measures, to the players. \paragraph{Round 4 (steps 31-34C)}\label{ConceptProtocolRound4} The service provider sends to each player the hashed MAC tags of the blinded statistical measures variance, median, best-in-class, maximum, bottom quartile, and top quartile. These are used by the players for validation of output integrity. Steps 34B and 34C are used by the service provider to distribute the hashed MAC tags of the statistical measures. Step 34B is the hash of the $n$ bottom quartile MAC tags that the players sent to $P_S$ in step 26B. Step 34C in turn is the hash of the $n$ top quartile MAC tags of step 26C. \subsection{Implementation}\label{DesignImplementation} Our prototype consists of two main parts: the secure benchmarking client and the secure benchmarking service. Both are written in Java and have their own PostgreSQL database to enable persistent data storage. The service is a Java servlet, running on Cloud Foundry (see~\cite{CFF17}), while the client is a Java console application. Additionally, we developed a C\# front end add-in for the product costing suite. It utilises the client implementation to execute the benchmarking protocol for actual product costing key figures. Client and service communicate via HTTPS sending JSON strings. An overview of this secure benchmarking system is depicted in Fig.~\ref{fig:DesignPrototypeBlockDiagram}. \begin{figure*}[t!] \centering \includegraphics[scale=0.7]{block_diagram_prototype_anonymous_large.eps} \caption{Block Diagram of the Prototype's Architecture} \label{fig:DesignPrototypeBlockDiagram} \end{figure*} \section{Discussion}\label{Discussion} A variety of attacks that are possible for TTP-based benchmarking analyses can also be applied to our system. This includes inference attacks, where an adversary-controlled player $P_i$ repeatedly runs benchmarks for the same KPI and peer group, each time with a different input $x_i$, in order to gain additional knowledge of other players' secret inputs. Such attacks can be precluded via organisational measures like retention periods. In the described product cost optimisation scenario, benchmarks are not supposed to be executed on demand but rather once every one or two quarters on pre-defined dates. Given the long period of time between two benchmarks and assuming that at least some players' KPIs changed during that period, the value of such an attack would be rather limited. Attacks where a player inputs an incorrect KPI to temper with the results are much harder to circumvent as they require a mechanism for analysing the semantics of the inputs. However, as this is immanent in benchmarking in general, such behaviour was beyond the scope of our considerations. \section{Evaluation}\label{Evaluation} This Section provides an evaluation of the adapted approach implemented in the prototype. The evaluation refers to the requirements that are described in Section~\ref{Introduction}. \subsection{Statistical measures}\label{EvaluationStatisticalMeasures} The selected protocol of~\cite{KER10} computes the statistical measures mean, variance, median, maximum, and best-in-class while the adapted protocol additionally computes the bottom quartile and top quartile. The correctness of the adapted protocol has been proven (see Appendix~\ref{ConceptProtocolAdaptationsCorrectness}). Therefore, the adapted protocol fulfils the requirements for necessary and optional statistical measures in the context of product costing. \subsection{Communication}\label{EvaluationCommunication} As described in Section~\ref{ApproachesKER10}, the protocol of~\cite{KER10} is based on the service provider model. The same applies to the adapted protocol since our adaptations presented in Section~\ref{DesignAdaptedProtocol} only add further computation steps but do not affect the communication. The service provider's implementation of the protocol runs as a Cloud Foundry service (see Section~\ref{DesignImplementation}) proving the protocol's cloud suitability. No pairwise communication between players takes place. Messages sent from $P_i$ to $P_S$ and vice versa do not contain any private data of some player $P_{j\neq i}$ that allows for an identification of $P_j$. This enables anonymity among the players in case of a non-colluding \(P_S\). \subsection{Performance}\label{EvaluationPerformance} As stated in Section~\ref{ApproachesSummary}, the quadratic complexity of the selected approach may prove critical for large $n$. Therefore, we conducted an extensive evaluation of the prototype's performance. We examined the computational and communication complexity of our protocol both theoretically and empirically. \subsubsection{Theoretical performance of the adapted protocol}\label{EvaluationPerformanceTheoretical} The most expensive part of the original benchmarking protocol is the rank computation in step 3. Its computational and communication complexity is quadratic in the number of players $n$, i.e. $\mathcal{O}(n^2)$~\cite{KER10}. This also holds for the rank computation of the adapted protocol since step 3 is not affected by the adaptations. Since the newly added steps each have complexity that is below $\mathcal{O}(n^2)$, the total computational and communication complexity of the protocol is $\mathcal{O}(n^2)$. The computational and communication complexity of each step of the adapted protocol is given in Appendix~\ref{AppendixComplexity}. \subsubsection{Practical performance of the prototype}\label{EvaluationPerformancePractical} To compare the prototype's performance to the theoretical complexity of the adapted protocol, a practical evaluation is performed in this Section. For this evaluation, a number of benchmarks were executed while the net execution time was measured. The net execution time is the time required for one entire protocol execution minus the time during which an inactive participant delays the execution, causing the others to wait. These benchmarks were conducted in different scenarios to determine the influence of the parameters peer group size $n$, network latency, bandwidth, and asymmetric (Paillier) key length. Furthermore, two worst case scenarios were considered combining several of these parameters. For the empirical analysis, the Java servlet was uploaded to a Cloud Foundry trial instance. The $n$ clients were run in the form of the Java console application on three local client notebooks. Preliminary analyses indicated hardware limitations on the part of the service provider and the client machines. Hence, the performance evaluation was conducted for peer groups of $n\leq 60$ players and extrapolated for larger $n$. These $n$ clients were equally distributed among the three PCs. In each of the corresponding tests, the net execution time $t$ was measured for a default Paillier key length of $768$ bits, a default bandwidth of circa $17$ Mbit/s, and a network latency to the service provider of about $6$ milliseconds. \paragraph{Peer group size}\label{EvaluationPerformancePracticalEvaluationPeergroupSize} Fig.~\ref{fig:EvaluationPeerGroupSize} depicts the extrapolated net execution time for peer group sizes of $n\leq300$ players given the measured net execution time for $n\leq60$ players. With circa $1.200$ seconds, the net execution time was below the required maximum of $24$ hours. For the remaining parameters network latency, bandwidth, and key length, the default values were used. This scenario serves as the baseline scenario for the remaining analyses of the performance evaluation. \begin{figure*}[t!] \centering \centerline{ \subfloat[Different Peer Group Sizes, Extrapolated]{\includegraphics[width=2.24in]{peer_group_size_extrapolated.eps}% \label{fig:EvaluationPeerGroupSize}} \hfil \subfloat[Worst Case 1, Extrapolated]{\includegraphics[width=2.24in]{worst_case_1_extrapolated.eps}% \label{fig:EvaluationWorstCase1Estimated}} \hfil \subfloat[Worst Case 2, Extrapolated]{\includegraphics[width=2.24in]{worst_case_2_extrapolated.eps}% \label{fig:EvaluationWorstCase2Estimated}}} \caption{Net Execution Times of the Enhanced Benchmarking Protocol} \end{figure*} \paragraph{Network latency}\label{EvaluationPerformancePracticalEvaluationNetworkLatency} To examine the effect of the network latency, the net execution time was measured for the default network latency of $6$ milliseconds as well as for this default value plus an offset. The network latency offset had to be suitable for simulating global communication. Based on the network latencies for different continents as well as for intercontinental communication given in~\cite{VER17}, an average offset of $150$ milliseconds was chosen. Compared to the baseline, the offset only significantly affects the net execution time for smaller peer groups of $n\leq 40$ players. With increasing peer group size $n$, the effect of the network latency decreases. \paragraph{Bandwidth}\label{EvaluationPerformancePracticalEvaluationBandwidth} Customers can reasonably be assumed to have access to a broadband network, which is defined as a network with a transmission capacity of at least $2$ Mbit/s~\cite{ITU11}. Even for such a low bandwidth, the difference between the net execution times was small. Increasing the bandwidth by a factor of $8.5$ barely affected the net execution time. \paragraph{Key length}\label{EvaluationPerformancePracticalEvaluationKeyLength} For investigating the effect of the key length on the net execution time, the key lengths $768$, $1024$, $1536$, and $2048$ bits were taken into consideration. These investigations showed two facts. On the one hand, the net execution time increased disproportionately with growing key length. This is due to the complexity of the modular exponentiation of Paillier's cryptosystem, which is cubic in the key length~\cite{KER10}. On the other hand, for keys of $2048$ bits, the net execution time for $n=300$ was about $25.000$ seconds, i.e. approximately $7$ hours. \paragraph{Worst case}\label{EvaluationPerformancePracticalEvaluationWorstCase} For the worst case analysis, two scenarios are taken into consideration. This analysis refers to the worst case from a net execution time perspective, not from a security perspective. The first scenario investigates the combined effect of the peer group size, network latency, and bandwidth while the second worst case scenario also takes the key length into consideration. The extrapolated net execution time for the first worst case scenario is depicted in Fig.~\ref{fig:EvaluationWorstCase1Estimated}. For growing peer group size, the fitted curves diverge. The net execution time for $n=300$ was below the required maximum of $24$ hours. The fitted curves for both cases grow faster than they diverge. Hence, the influence of the computations on the net execution time can be assumed to be higher than the impact of the communication. The extrapolated net execution time of the second worst case scenario is depicted in Fig.~\ref{fig:EvaluationWorstCase2Estimated}. Due to the large difference in the key length, which was $768$ bits in the baseline case and $2048$ bits in the worst case, the fitted curves grow very differently. However, both grow quadratically. The net execution time of the second scenario was circa $7$ hours. Hence, the prototype's performance meets the requirements even for larger keys of $2048$ bits, poor internet connection, and peer groups of $n\leq300$ players. However, with its quadratic complexity, the net execution time of this system would soon exceed the desired maximum of $24$ hours for larger peer groups. This is mostly due to the extensive rank computation. \subsection{Security}\label{EvaluationSecurity} We proved the adapted protocol to be secure against computationally bounded semi-honest adversaries (see Appendix~\ref{ConceptProtocolAdaptationsPrivacy}). This privacy proof shows that up to $n-1$ players can be corrupted without them learning anything about the non-corrupted players' inputs that cannot be inferred from the corrupted players' inputs and the output. If the service provider himself is corrupted, privacy of the inputs ensures that he learns nothing about the players’ secret values as long as none of the players is corrupted at the same time~\cite{KER10}. The possibility of a player eavesdropping on the communication between another player and the service provider is precluded as they communicate over pairwise secure channels. Integrity of the output is enabled via cryptographic hashes and MACs. Further security related properties, such as secure data storage, access control, availability of the service, and robustness regarding players dropping out of a protocol execution, are beyond the scope of this paper. \subsection{Summary of the evaluation}\label{EvaluationSummary} The statistical measures provided by the implemented protocol are the mean, variance, median, maximum, best-in-class, bottom quartile, and the top quartile, and the best-in-class. The prototype implements the service provider model, enabling anonymity among the players as well as cloud suitability. Our enhanced secure benchmarking protocol has been proven to ensure confidentiality of the players' secret inputs against computationally bounded semi-honest adversaries. Output integrity is also provided. The computational and communication complexity both are $\mathcal{O}(n^2)$. The net execution time for $300$ players is circa $7$ hours. Consequently, the prototype implementing the adapted protocol meets each of the requirements defined in Section~\ref{Introduction}. It even provides additional statistical measures. However, our extensive evaluation showed the bottleneck of our protocol, which is the quadratic rank computation. \section{Introduction}\label{Introduction} Benchmarking is the comparison of key performance indicators (KPIs) of a company's peer group~\cite{KER08}. KPIs are statistical quantities that can be used for evaluating the performance of a company~\cite{KER08}. A peer group is a set of companies that take advantage of comparing KPIs~\cite{KER08}. Usually, the companies of a peer group are competitors of the same industry, which implies a demand for privacy of the KPIs~\cite{KER08}. We define a privacy-preserving benchmarking analysis as the process of comparing KPIs as secure inputs across different companies~\cite{CDN15}. Every company learns how it performs compared to the other companies being involved but not their private KPIs~\cite{CDN15}. One approach to privacy-preserving benchmarking is using a trusted third party (TTP) that conducts the calculation of a function $f(x)$ without revealing any private data. However, with mutually distrusting companies, finding a TTP might not be trivial~\cite{GMW87}. An approach that does not require trust can be found in secure multi-party computation (MPC). In MPC, participants providing an input $x_i$ are called players, those who compute $f(x)$ are called processors~\cite{HIR01}. A participant can be both player and processor at the same time. An MPC is secure in the sense that the participants only learn the outputs, their own input, and what they can infer from that~\cite{CDN15}. In this paper, we investigate approaches to adding privacy-\linebreak preserving benchmarking to a product costing suite of the software company SAP. Such product costing software enables a company to calculate the costs of its products. This includes quotation costing as well as cost estimation from the project acquisition to the design and production to the disposal. To increase profit, a company, e.g. of the automotive industry, might want to reduce production costs. To do so, it needs to determine those areas with the best ratio between cost savings and optimisation effort, requiring knowledge of the company's performance. To make well-informed optimisation decisions, optimisation can be facilitated by comparing the company's production KPIs, e.g. assembly time of a car engine, to those of other companies of the industry, i.e. via benchmarking analyses. If according to the benchmarking results a company finds itself among the best performing, it might be rather expensive to further improve the compared KPI, e.g. assembly time. A performance below average might imply higher potential for cost savings. However, as companies might be reluctant to provide their confidential production KPIs, benchmarking needs to be conducted in a manner that ensures privacy of the KPIs and still provides the desired statistical measures. Such privacy-preserving benchmarking analyses could be repeated on a regular basis, e.g. once every quarter, to investigate performance development over time relatively to the industry. The requirements for our system were defined given the corporate context. They are based on the requirements of an existing, TTP-based benchmarking suite of SAP, serving as a lower bound for functionality and performance. In this TTP-based system, benchmarks are conducted individually for each KPI at least once every six to seven months on pre-defined dates rather than on demand. It provides the statistical measures mean, bottom quartile, top quartile, and best-in-class. The latter is the mean of the top quarter of the sorted list of inputs. As in the existing benchmarking system and similar to Atallah et al. in~\cite{ABL04}, we assume companies comparing their product costing KPIs to be interested in correct results and therefore to behave honestly. This can reasonably be assumed as companies need such benchmarking results for well-informed, far-reaching business decisions. These results are only correct if every participant follows the protocol. Hence, we require our system to provide input privacy against semi-honest adversaries. To guarantee that each company gets the same, correct results, output integrity is required. Similar to the TTP-based system, our system should provide anonymity among the players in the sense that participants must not be referred to with any persistent identifier during benchmarking~\cite{KER10}. This might be of interest in case of computing benchmarks only for a subset of a peer group. Further aspects like availability were beyond the scope of our considerations. Following the company's cloud strategy, our system should run as a cloud service. This provides scalability and availability and can be assumed to enable protocol execution even for large peer groups~\cite{AFG10}. In the TTP-based system, such large peer groups contain up to $300$ players, which is sufficient for the product costing context. As suggested in Kerschbaum's guideline for non-functional requirements of a benchmarking platform~\cite{KER10}, a benchmark should at most take $24$ hours. Our main contributions are the construction and implementation of a privacy-preserving benchmarking system in the context of product cost optimisation. Furthermore, we conducted security and complexity analyses and an extensive performance evaluation of this protocol in the given context. Focusing on the technical feasibility of privacy-preserving benchmarking systems, we first performed a state-of-the-art analysis for possible approaches (Section~\ref{Approaches}). We then selected one approach, adapted it to better suit the product costing context, designed a prototype (Section~\ref{Design}), and evaluated it extensively regarding the given requirements (Section~\ref{Evaluation}). \section{Preliminaries}\label{Preliminaries} \subsection{Oblivious transfer}\label{PreliminariesPrimitivesOT} In an oblivious transfer (OT) protocol, a player $P_1$ has $k$ secret messages $m_1,...,m_k$ with $k \geq 2$. A player $P_2$ wants to select and receive message $m_i$ without $P_1$ learning the value $i$~\cite{KER10}. Furthermore, $P_1$ does not want $P_2$ to learn any message apart from $m_i$~\cite{KER10}. We denote such a protocol by $P_1 \xrightarrow{\mathrm{OT}} P_2$. \subsection{Homomorphic encryption}\label{PreliminariesHomomorphicEncryption} Assume a cryptosystem with a (randomised) encryption function $E(\cdot)$ and a decryption function $D(\cdot)$~\cite{KAL08}. Homomorphic cryptosystems enable computations for secret values $x_1,...,x_n$ based on their ciphertexts $E(x_i),...,E(x_n)$ without needing the decryption key~\cite{KER08}. Applying an operation to such ciphertexts yields the ciphertext of the result of a corresponding homomorphic operation applied to the plaintexts~\cite{KER08}. Partially homomorphic encryption (PHE) schemes enable one operation on the plaintext, e.g. addition or multiplication~\cite{KER10}. Fully homomorphic encryption (FHE) schemes allow for the computation of arbitrary functions, e.g. by providing both addition and multiplication~\cite{GEN09b}. For example, Paillier's asymmetric, additively (partially) homomorphic encryption scheme has the properties given in Equations~(\ref{eq:PreliminariesHomomorphicEncryptionAdditionMult}) and~(\ref{eq:PreliminariesHomomorphicEncryptionAdditionExp})~\cite{KER08}. \begin{equation}\label{eq:PreliminariesHomomorphicEncryptionAdditionMult} D\left(E\left(x_1\right) \cdot E\left(x_2\right)\right) = x_1 + x_2 \end{equation} \begin{equation}\label{eq:PreliminariesHomomorphicEncryptionAdditionExp} D\left(E\left(x_1\right)^{x_2}\right) = x_1 \cdot x_2 \end{equation} Homomorphic semantically secure cryptosystems provide rerandomisation of ciphertexts as follows~\cite{KER10}. \begin{equation}\label{eq:PreliminariesHomomorphicEncryptionRerandomisationAddition} E(x_i + 0) = E(x_i) \cdot E(0) = E^\prime(x_i) \end{equation} With high probability, $E(x_i) \neq E^\prime(x_i)$ is provided such that $E(x_i)$ and $E^\prime(x_i)$ are computationally indistinguishable~\cite{KER10}.

proofpile-arXiv_069-10946

\section{Introduction} Quantum entanglement is an important theme of current research in theoretical and applied sciences because at the level of foundational theory it is the ``uniquely distinct feature of quantum" \cite{Schrodinger,Schrodinger2} and, as the primary resource of quantum information processing, it acts as the fountain spring for quantum computing, communication, control, cryptography and metrology which usher in the so-called second revolution of quantum sciences and engineering we are witnessing now. Unfortunately, when considered under realistic physical conditions, a system unavoidably interacts with its many environments, and model studies (see, e.g., \cite{YuEberly,AEPW02,FicTan06,ASH06,ASH09,Goan,AnZhang,Mani,Mani2,LCH08,LinHu09,entrev-horodecki,Ludwig,entg-book1,entg-book2,Kanu,Wilson,HH15AOP,HH15PRD,HH15PLB,HH15JHEP} and references therein) have found that entanglement in a realistic system can easily be degraded by various noises in the environments. It is well-known that thermal noise is the nemesis of many quantum features, quantum coherence and entanglement are no exception. Thus it came somewhat as a surprise when Galve et al \cite{galve-prl} announced their interesting finding that entanglement can be kept at high temperature (`hot entanglement'~\cite{Vedral}) if one considers a parametrically driven quantum system with time-dependent coupling between the two parties, each interacting with its own bath. This is the theme we shall focus on here and probe deeper into. A related noteworthy paper by Estrada and Pach\'on \cite{EstPac} explores nonMarkovian effects on hot entanglement in the same setup as Galve et al by introducing a finite cutoff frequency in the baths. They show that non-Markovian dynamics, when compared to the Markovian case, allow for the survival of stationary entanglement at higher temperatures, with larger coupling strength to the baths and at smaller driving rates. However, neither of these two important works considered the work cost of the driving protocol with which the entanglement is sustained. Let us begin our story with some background on hot entanglement. \subsection{Background on `hot entanglement'} The observation of quantum entanglement in hot and/or large scale systems has been of interest with different motivations. The relevance of entanglement in the context of quantum computation and information (see Refs.~\cite{densecoding,entg-dist1} as earlier examples of works relating entanglement distillation to quantum dense coding, communication, Ref.~\cite{entg-dist2} for the role of entanglement in quantum error correction, Ref.~\cite{qcomp-exp-spup1} necessity of entanglement for exponential speedup in quantum computation with pure states) and the challenges to sustain entanglement over times much larger than the decoherence timescale need to be mentioned as one of the major motivations behind many of the works on hot entanglement (see Ref.~\cite{qcomp-hotentg1} for hot environment mediated entanglement between solid state spin qubits, Ref.~\cite{qcomp-hotentg2} for entanglement between two-level atoms in an optical cavity and Ref.~\cite{qcomp-negtemp-entg} for entanglement with negative temperature Markovian baths). Other important directions in the research on hot entanglement include the relation between the heat current and entanglement between two qubits coupled to Markovian reservoirs at different temperatures~\cite{qcomp-entg-flow}, non-linear Hamiltonians, bath spectrum filtering and coupling via an auxiliary system~\cite{qcomp-entg-aux}. One other important field in which the entanglement between macroscopic objects at hot temperatures is relevant is the relatively new field of quantum biology where the quantum nature of molecules with hundreds of atoms is still under debate (see Ref.~\cite{qbio-entg-dyn} for an example of the interplay between the classical and quantum aspects of biomolecules in the context of driving induced entanglement in hot environments) and this ongoing debate is promising to open new horizons in molecular biology such as, but not limited to, enhanced biological measurements with quantum spectroscopy using entangled photons~\cite{qbio-entg-meas}, the effects of entanglement in photosynthetic light harvesting complexes on the efficiency of excitation transfer (see Ref.~\cite{qbio-entg-lhc} and references therein), chemical compass in birds~\cite{qbio-entg-comp1,qbio-entg-comp2}, electron tunneling assisted by phonon degrees of freedom of an odorant as an accurate model of olfaction~\cite{qbio-entg-olf1,qbio-entg-olf2} and possible effects on mutation rates~\cite{qbio-entg-mut}. Some other contexts in which entanglement is relevant are touched upon in Refs.~\cite{entrev-horodecki,entg-book1,entg-book2} and references therein. \subsection{This work: Key issues} This paper is a sequel of a 2015 paper by two of us \cite{HH15JHEP} on the possibility of sustaining entanglement at high temperatures between two coupled harmonic oscillators each interacting with its own bath at different temperatures, and related to a recent paper \cite{HAH22} where we studied the effect of nonMarkovian baths on hot entanglement. Both studies assume constant inter-oscillator coupling. The conclusion drawn in \cite{HH15JHEP} is, in most realistic circumstances, for bosonic systems with \textit{constant} bilinear coupling interacting with Ohmic (scalar field) baths, `hot entanglement' is unlikely. Similar qualitative conclusions were found in \cite{HAH22} for the same system with nonMarkovian baths. (In a later work \cite{AHH-nM} we shall address the nonMarkovian bath issues in comparison with Estrada and Pach\'on \cite{EstPac}). Here, we allow the inter-oscillator coupling to change in time (driven parametric coupling) in the same set up as Galve et al, where the baths are Ohmic with infinite\footnote{{Here the Caldeira-Leggett approximation have been implemented because the high-temperature bath is considered.}} cutoff frequencies. At high temperature, the dynamics of the reduced system is Markovian and the equations are a lot easier to solve than the nonMarkovian cases. Our analysis of the system's dynamics suggests that the parameters Galve et al used to report on hot entanglement fall in the dynamically unstable regimes. This is probably known to them, but they didn't probe into the special circumstances which allow for entanglement to be sustained and face the unwelcome consequences of dynamically unstable driven systems. As we shall see there are many subtle issues, perhaps known, not widely, but not addressed earlier (hot entanglement functions only in the unstable regimes) or hitherto unknown (instability is a necessary but not sufficient condition, {unreliability} of the Caldeira-Leggett approximation) and some concerning factors (costly power intake) which makes actual implementation of hot entanglement doubtful. \subsection{Major findings: Unstable dynamics, squeezed systems and energy budget} Three key issues we wish to address here are: Q1) In What parameter regimes are hot entanglement functional? Does high temperature entanglement need be operated in a dynamically unstable regime? Q2) Would squeezed systems without parametric drive allow for hot entanglement? Q3) If yes is the answer to Q1, would the energy budget needed for the parametric drive to sustain entanglement at high temperature be cost-efficient? The answers to these questions are, put succinctly, 1) Yes, 2) No, 3) Not really. 1) \textit{Unstable dynamics}: Our analysis suggests that the parameters Galve et al use to report on hot entanglement fall in the dynamically unstable regimes. Whether the system can maintain a steady state needs be addressed. Similar concerns as ours have been expressed earlier, e.g., {Figueiredo Roque and Roversi} \cite{FigFov13} show that quantum correlations and entanglement in a system of two harmonic oscillators with time-dependent coupling and in contact with a {\it common} heat bath (note: this is different from our set-up) can survive even at very high temperatures. They show that a`remarkable' relation exists between entanglement and the instability of the system, and that quantum correlations are much more sensitive to the parameters of the oscillators than the temperature of the bath.Chakraborty and Sarma \cite{ChaSar18} study the entanglement dynamics of two coupled mechanical oscillators within a modulated optomechanical system and identified the critical mechanical coupling strength for transition from stationary to dynamical entanglement, which is extremely robust against the oscillator temperature. They show that this entanglement dynamics is strongly related to the stability of the normal modes. 2) \textit{Squeezing}. {Unstable quantum (open) dynamics intrinsic in the system or due to external drive is expected to effectively induce large squeezing~\cite{GP85, GS86, HH22, GV20}. We are interested in whether the squeezing of a quantum system is sufficient to produce and sustain entanglement. A lesser known aspect in this context is that runaway dynamics also incites large thermal fluctuations. Survivability of quantum entanglement then can be understood as a tug of war between squeezing and thermal fluctuations. To show that squeezing is not sufficient condition we use an {amplifying harmonic (anti-damping)} oscillator interacting with a single bath to demonstrate that effective squeezing generated by dynamical instability is not strong enough to sustain entanglement. Galve et al's result indicate that parametric drive is needed to boost squeezing and at the same time tame the stimulated thermal fluctuations. Thus instability does not guarantee entanglement: the unstable motion may induce a much larger effective thermal fluctuations than the effective squeezing can suppress. In contrast, for stable dynamics due to constant inter-oscillator coupling, as studied in \cite{HAH22}, squeezing in the initial state does not help late-time sustainability of entanglement because its benefit is severely weakened by the relaxation process.} 3) \textit{Power intake}. If it is confirmed that hot entanglement can exist only in the unstable regime then the natural question to ask is, lacking a NESS, how long can one operate in such a regime and reap the benefits of sustained entanglement at high temperatures. For this inquiry we shall carry out an energy budget analysis to assess the power the external drive needs to deliver to the system to this end. It turns out to be unattainable, not entirely surprisingly for unstable systems. This paper is organized as follows. In Sec.~\ref{S:ebtgdfg} we present the essential theoretical elements to treat quantum entanglement of the above-described parametrically-driven system. In Sec.~\ref{numer-ohmic} we compare the time evolution of hot entanglement when the system undergoes unstable and stable motion. In particular we focus on its sustainability at late times. In Sec.~\ref{S:eoujhbdf} we use the model of an amplifying harmonic oscillator as a counter example to illustrate the point that instability does not guarantee hot entanglement although it seems necessary. We introduce terms familiar in quantum optics -- effective squeezing and effective temperature -- to describe entanglement. Finally in Sec.~\ref{S:ebgdfhg}, we examine the energy flow between the external agent, the reduced system and the thermal bath. We arrive at the conclusion that hot entanglement under the present model and set-up \cite{galve-prl} does not seem tenable, by virtue of the fact that it needs to operate in the regime of unstable dynamics wherein a tremendous amount of energy consumption from the external agent is required. Appendix~\ref{S:peotr} collects concise information about two-mode squeezing, needed for Sec.~\ref{S:eoujhbdf}. In Appendix~\ref{S:gbdghf}, we take a closer look at the Caldeira-Leggett approximation, applied to a high-temperature unstable system, and point out the ambiguities that are not present in stable dynamics. \section{Parametrically coupled oscillators}\label{S:ebtgdfg} \subsection{Theoretical framework} It is physically transparent and mathematically simple to use the Langevin equations to describe the dynamics of the oscillator in this NESS setting~\cite{HH15AOP}. Since they are linear equations, we can readily write down its formal solutions from which we may construct the covariance matrix. This matrix is of central importance for the Gaussian system because it encompasses the essential features of the full dynamics~\cite{Peres, Horod, PPTSimon, Adesso, Adesso05, Illuminati04}. In particular, the quantum entanglement measure used here, logarithmic negativity, can be expressed in terms of the covariance matrix elements. Suppose both oscillators have the same mass $m$, and physical frequency $\omega$ but we allow different oscillator-bath coupling strengths $e_{i}$ with $i=1$, 2. Let the displacements of the oscillators be denoted by $\chi_{i}$ with $i=1$, 2, and are grouped into a column vector $\bm{\chi}=(\chi_{1}\;\chi_{2})^{T}$. The Langevin equations for the current setup take on a compact form \begin{align} \ddot{\bm{\chi}}(s) + 2\bm{\gamma} \cdot \dot{\bm{\chi}}(s) + \bm{\Omega}^2_{\textsc{r}}(s) \cdot \bm{\chi}(s) = \frac{1}{m}\, \bm{\xi}(s) \label{lange-sf}\,, \end{align} where $s$ is the time variable and the matrices $\bm{\gamma}$, $\bm{\Omega}^2_{\textsc{r}}(s)$ are given by \begin{align} \bm{\gamma} &= \begin{pmatrix} \gamma_{1} &0 \\ 0 &\gamma_{2} \end{pmatrix}\,,&\bm{\Omega}^2_{\textsc{r}}(s) &=\begin{pmatrix} \omega^2 &\sigma(s) \\ \sigma(s) &\omega^2 \end{pmatrix}\, . \end{align} where $\gamma_{i}=e^2_i/(8\pi m)$ denotes the damping constant for oscillator $i$, and $\sigma(s)$ the time-dependent, inter-oscillator coupling strength. On the righthand side \eqref{lange-sf}, $\bm{\xi}(s)$ is a Gaussian noise from the private baths, with the properties \begin{align} \langle \bm{\xi}(s) \rangle &= 0, &\langle \bm{\xi}(s) \bm{\xi}^{T}(s') \rangle &= \begin{pmatrix} \nu^{(1)}(s,s') &0 \\ 0 & \nu^{(2)}(s,s') \end{pmatrix} \label{xi-covar}\,. \end{align} It reflects the quantum fluctuations of the thermal baths the oscillators are separately attached to. In particular, the kernel function $\nu^{(i)}(s,s')=e_{i}^{2}G_{H,0}^{(i)}(s,s')$ contains the noise kernel $G_{H,0}^{(i)}(s,s')$ of the private bath $i$, and the coupling strength $e_{i}$. Thus the Langevin equations \eqref{lange-sf} describe the \textit{damped} dynamics of two parametrically coupled harmonic oscillators, driven by the quantum thermal noises from the their own baths. Here, although the equation contains a damping term, the evolution of such a system is not guaranteed to be stable. It could possess runaway solutions, depending on the choice of parametric driving. Following Ref.~\cite{HH15AOP} with generalization to the time-dependent frequency $\bm{\Omega}_{\textsc{r}}(s)$, the solution of the Langevin equation can be conveniently expressed in terms of the fundamental solution matrices $\bm{D}_1 (s,s')$ and $\bm{D}_2 (s,s')$ with the following conditions \begin{align} \bm{D}_1 (s,s) &=1\,, &\dot{\bm{D}}_1 (s,s)&=0\,, &\bm{D}_2 (s,s) &= 0\,, &\dot{\bm{D}}_2 (s,s) &= 1\,,\label{d2-init} \end{align} and $\bm{D}_{i}(s,s')=0$ with $i=1$, 2 if $s<s'$. The matrices $\bm{D}_{i}$ have two time arguments due to the fact that the coefficients in the Langevin equation are time dependent, meaning that the response to the impulse depends on when the response is measured as well as when the impulse is applied. Thus it is not time translationally invariant, i.e., $\bm{D}_i(s,s')\neq\bm{D}_i(s-s')$. However, when the parametric driving is periodic, i.e., the matrix $\bm{\Omega}^2_{\textsc{r}}(s)$ being periodic, and the motion is stable, the fundamental solution matrices at late times have a nice property~\cite{paz-cool1} \begin{align}\label{E:dkjgbfkse} \bm{D}_{i}(s,s') &= \bm{D}_{i} (s - n \tau_d, s' - n \tau_d)\,, &i&=1,2\,, \end{align} where $\tau_d$ is the driving period and $n$ is an integer. Having defined the fundamental solution matrices, we find the displacement $\bm{\chi}$ and the corresponding canonical momentum $\bm{p}= m \dot{\bm{\chi}}(s) $ are given by \begin{align} \bm{\chi}(s) &= \bm{D}_1 (s,0) \cdot \bm{\chi}(0) + \frac{1}{m}\,\bm{D}_2 (s,0) \cdot \bm{p}(0) + \frac{1}{m} \int_0^s ds'\; \bm{D}_2 (s,s') \cdot \bm{\xi}(s')\,, \label{chi} \\ \bm{p}(s) &= m\dot{\bm{D}}_1 (s,0) \cdot \bm{\chi}(0) +\dot{\bm{D}}_2 (s,0) \cdot \bm{p}(0) + \int_0^s ds' \;\dot{\bm{D}}_2 (s,s') \cdot \bm{\xi}(s') \label{p} \end{align} where an overdot represents taking the derivative with respect to the first time arguments of the matrices $\bm{D}_i$. The quantum Langevin equation (\ref{lange-sf}) of the coupled harmonic oscillators allows us to readily write down the evolution equations for the first moments of the canonical variables of the reduced system~\cite{qle-ford} \begin{align} \frac{d \langle \chi_{i} \rangle}{dt} &= \frac{1}{m} \langle p_{i} \rangle \label{q-1st-mom-eqn} \\ \frac{d \langle p_{i} \rangle}{dt} &= -2\gamma_{i} \langle p_{i}\rangle - m\omega^2 \langle \chi_{i} \rangle - m\sigma(t) \langle \chi_{j} \rangle \label{p-1st-mom-eqn} \end{align} where the index $j=3-i$ with $i=1$, 2. This is a form of the Ehrenfest theorem. Notice that the noise term in the Langevin equation (\ref{lange-sf}) does not appear in Eqs.~\eqref{q-1st-mom-eqn} and \eqref{p-1st-mom-eqn} since the noise has zero mean, as assumed in \eqref{xi-covar}. For simplicity, we will assume that the means of the canonical variables are zero for the initial Gaussian state of the oscillators. Then, the equations of motion of the second moments take simpler forms~\cite{qle-ford} \begin{align} \frac{d}{dt}\langle\chi_{i}^2\rangle&= \frac{1}{m} \langle \{ \chi_{i},\,p_{i} \} \rangle\,, \\ \frac{d}{dt}\langle \{\chi_{i},\, \chi_{j}\bigr\} \rangle&= \frac{1}{m} \langle\{ p_{i},\, \chi_{j}\}\rangle+ \frac{1}{m} \langle\{\chi_{i}, p_{j}\}\rangle\,, \\% \langle p^{(i)} \chi^{(\bar{i})} + \chi^{(i)} p^{(\bar{i})} + p^{(\bar{i})} \chi^{(i)} + \chi^{(\bar{i})} p^{(i)}\rangle \\ \frac{d}{dt}\langle\{ \chi_{i},\, p_{i}\}\rangle& =\frac{2}{m}\langle p_{i}^2 \rangle + \langle \{\xi_{i}-2\gamma_{i} p_{i} -m\omega^2 \chi_{i} -m\sigma(t)\chi_{j},\, \chi_{i}\rangle\,, \\ \frac{d }{dt}\langle\{ \chi_{i},\, p_{j}\}\rangle & = \frac{1}{m}\langle \{p_{i},\,p_{j}\}\rangle +\langle \{\xi_{j}-2\gamma_{j} p_{j} -m\omega^2 \chi_{j} -m\sigma(t)\chi_{i},\, \chi_{i}\} \rangle \\ \frac{d}{dt}\langle \{p_{i},\, p_{j}\}\rangle& =\langle \{\xi_{i}-2\gamma_{i} p_{i} -m\omega^2 \chi_{i} -m\sigma(t)\chi_{j},\, p_{j}\} \rangle\notag \\ &\qquad\qquad\qquad\qquad\qquad+\langle \{\xi_{j}-2\gamma_{j} p_{j} -m\omega^2 \chi_{j} -m\sigma(t)\chi_{i},\, p_{i}\} \rangle\,,\\ \frac{d}{dt}\langle p_{i}^2\rangle&= \langle \{\xi_{i}-2\gamma_{i} p_{i} -m\omega^2 \chi_{i} -m\sigma(t)\chi_{j},\, p_{i}\}\rangle\,, \label{E:ughdf} \end{align} with $j=3-i$ and $i=1$, 2. These constitute a set of coupled differential equations for the second moments, i.e, covariance matrix elements, of the canonical variables for the coupled oscillator system, which are formally defined as~\cite{PPTSimon,Illuminati04,HH15JHEP} \begin{equation} \bm{\sigma} =\frac{1}{2}\langle \{\bm{Z},\bm{Z}^{T} \} \rangle \,, \end{equation} by the phase space variale $\bm{Z}=(\chi_{1}\;\chi_{2}\;p_{1}\;p_{2})^{T}$. Finding the numerical solutions of the covariance matrix elements is both mathematically and computationally challenging in the parametrically driven open-system setting on account that both $\chi_{i}$ and $p_{i}$ are formally given by \eqref{chi} and \eqref{p}. However, at the high bath temperature regime, great simplification is possible by resorting to the Caldeira-Leggett limit~\cite{hm94} of the kernel function $\nu^{(i)}(s,s')$ \begin{equation} \nu^{(i)}(s,s') = \frac{4m\gamma_{i}}{\beta_{i}^{\textsc{b}}}\,\delta(s-s')\,, \end{equation} where $\beta_{i}^{\textsc{b}}$ is the initial inverse temperature of the $i$th private bath\footnote{There are a few subtleties when implementing the limit in the unstable system. {For example, other than the result given by the Caldeira-Leggett approximation, the $\langle\xi_ip_i\rangle$ in \eqref{E:ughdf} has a component which oscillates with exponentially increasing amplitude.} We will elaborate them in Appendix~\ref{S:gbdghf}.}. This expression implies that at high temperature, the thermal noise has a white spectrum to make the kernel function local in time, and that the coupling, in the form of $\gamma_{i}$, is sufficiently weak such that any contribution that depends on the frequency cutoff is ignorable. In this limit we arrive at a much simpler set of differential equations \begin{align} \frac{d}{dt}\bm{\sigma}_{\chi_i\chi_i}&= \frac{2}{m} \,\bm{\sigma}_{\chi_ip_i}\,, \label{covar-ohmic-1st} \\ \frac{d }{dt}\bm{\sigma}_{\chi_i\chi_j}&= \frac{1}{m} \,\bm{\sigma}_{p_i\chi_j}+ \frac{1}{m} \,\bm{\sigma}_{\chi_ip_j}\,, \\ \frac{d }{dt}\bm{\sigma}_{\chi_ip_i} &= \frac{1}{m}\bm{\sigma}_{p_ip_i} -2\gamma_{i}\bm{\sigma}_{\chi_ip_i} - m\omega^{2}\bm{\sigma}_{\chi_i\chi_i}-m\sigma(t)\bm{\sigma}_{\chi_i\chi_j}\,,\\ \frac{d }{dt}\bm{\sigma}_{\chi_ip_j}& = \frac{1}{m}\bm{\sigma}_{p_ip_j}- 2\gamma_{j}\bm{\sigma}_{\chi_ip_j}-m\omega^2\bm{\sigma}_{\chi_i\chi_j} -m\sigma(t)\bm{\sigma}_{\chi_i\chi_i}\,, \\ \frac{d }{dt}\bm{\sigma}_{p_ip_j}&= - 2(\gamma_{i} + \gamma_{j})\bm{\sigma}_{p_ip_j}- m\omega^2\bigl[\bm{\sigma}_{p_i\chi_j}+\bm{\sigma}_{\chi_ip_j}\bigr]-m \sigma(t)\bigl[\bm{\sigma}_{\chi_ip_i}+\bm{\sigma}_{\chi_jp_j}\bigr]\,, \\ \frac{d }{dt}\bm{\sigma}_{p_ip_i}&= \frac{4m\gamma_{i}}{\beta_{i}^{\textsc{b}}} - 4\gamma_{i} \bm{\sigma}_{p_ip_i} - 2m\omega^2\bm{\sigma}_{\chi_ip_i} -m\sigma(t)\bm{\sigma}_{p_i\chi_j}\,,\label{covar-ohmic-last} \end{align} with $j=3-i$ and $i=1$, 2. \begin{figure} \centering \scalebox{1.5}{\includegraphics[width=0.45\textwidth]{stability}} \caption{\label{fig:stability} Phase diagram for stability in terms of driving frequency $\omega_{d}=\frac{2\pi}{\tau_{d}}$ and coupling parameter $c_{1}$ when $\sigma(t)$ takes the form $\sigma(t)=c_{0}+c_{1}\,\cos\omega_{d}t$. Black regions in the diagram are unstable while the white regions are stable. The constant parameters are $m = 1$, $\omega = 1$, $\gamma_{1}=\gamma_{2}= 0.005$, and $c_0 = 0$.} \end{figure} Before proceeding to investigating parametrically driven open system dynamics, let us comment on the stability of the quantum Langevin equation \eqref{lange-sf}. Following Ref.~\cite{mathieu-stbl}, we will define a matrix $\bm{C}$ that describes the movement in the phase space over one driving period $\tau_d $ from the initial phase space position $\bm{Z}(0)$. We first rewrite Eqs.~\eqref{q-1st-mom-eqn} and \eqref{p-1st-mom-eqn} in terms of this phase space variable $\bm{Z}(t)$, \begin{align} \langle\dot{\bm{Z}}(t)\rangle&=\bm{W}(t)\cdot \langle\bm{Z}(t)\rangle\,,&\bm{W}(t)&= \begin{prmatrix} 0 &0 &m^{-1} &0 \\ 0 &0 &0 &m^{-1} \\ -m\omega^2 &-m\sigma(t) &-2\gamma_{1} &0 \\ -m\sigma(t) &-m\omega^2 &0 &+2\gamma_{2} \end{prmatrix}\,. \end{align} This implies that we may introduce the matrix $\bm{\mathfrak{D}}(t)$, in a similar fashion as the fundamental solutions $\bm{D}_{i}(t)$ in the configuration space, that maps the initial state $\bm{Z}(0)$ to the current state $\bm{Z}(t)$ by $\langle\bm{Z}(t)\rangle=\bm{\mathfrak{D}}(t)\cdot\langle\bm{Z}(0)\rangle$. The matrix $\bm{D}$ satisfies $\dot{\bm{\mathfrak{D}}}(t)=\bm{W}(t)\cdot\bm{\mathfrak{D}}(t)$, and is explicitly related to the fundamental solution matrices $\bm{D}_{i}$ by \begin{equation} \bm{\mathfrak{D}} = \begin{pmatrix} \bm{D}_1 &\bm{D}_2 \\ \dot{\bm{D}}_1 &\dot{\bm{D}}_2 \end{pmatrix}\,. \end{equation} Then, the matrix $\bm{C}$ is defined as $\bm{C} = \bm{\mathfrak{D}}(\tau_d)$ and the stability is defined by the requirement that all eigenvalues of $\bm{C}$ should have the modulus less than one~\cite{mathieu-stbl,mathieu-stbl2}. Fig.~\ref{fig:stability} shows a phase diagram of the parameter space $(c_{1},\omega_{d})$. The black shade represents the dynamically unstable regime, while the white area denotes the stable regime. \subsection{Entanglement dynamics}\label{entng-ohmic} We now study the evolution of the entanglement of the coupled oscillators, in contact with their own private high temperature baths. We use logarithmic negativity as the entanglement measure~\cite{part-tr,Plenio}. This is achieved by numerically solving the coupled evolution equations of the covariance matrix elements, \eqref{covar-ohmic-1st}--\eqref{covar-ohmic-last}. The logarithmic negativity is defined as $E_{\mathcal{N}} = \max\{0,-\ln(2\lambda_{<}^{\textsc{pt}})\}$, where $\lambda_{<}^{\textsc{pt}}$ is the smaller symplectic eigenvalue of the partially transposed covariance matrix $\bm{\sigma}^{\textsc{pt}}$. When $E_{\mathcal{N}}>0$, the oscillators are entangled. The symplectic eigenvalues of $\bm{\sigma}^{\textsc{pt}}$ can be calculated most easily from the eigenvalues of the matrix $i\,\Sigma\cdot\bm{\sigma}^{\textsc{pt}}$ with \begin{equation} \Sigma = \begin{pmatrix} 0 &0 &+1 &0\\ 0 &0 &0 &+1 \\ -1 &0 &0 &0\\ 0 &-1 &0 &0 \end{pmatrix}\,, \end{equation} for our choice of phase space vector $\bm{Z}$. Thus the symplectic eigenvalues come in pairs, $(\pm\lambda_{>}^{\textsc{pt}},\pm\lambda_{<}^{\textsc{pt}})$, so we set $\lambda_{>}^{\textsc{pt}} > \lambda_{<}^{\textsc{pt}} > 0$ for definiteness. \section{Unstable vs stable dynamics: Instability a necessary condition}\label{numer-ohmic} \subsection{Unstable dynamics} We consider a bath that has an Ohmic spectral density function with infinite cutoff frequency, thus a Markovian bath. We choose the same set of parameters as used in Fig.~2(b) of Ref.~\cite{galve-prl} to first reproduce how the logarithmic negativity changes with time\footnote{Note we use natural logarithm in the definition of logarithmic negativity whereas Ref.~\cite{galve-prl} defines it with base 2. In addition, our definition of the decay coefficients $\gamma$ differs from that in Ref.~\cite{galve-prl} by a factor of 2.} in Fig.~\ref{fig:logneg-galve}. Our primary purpose here is to make explicit the relation between hot entanglement and unstable open system dynamics. \begin{figure}[h!] \centering \scalebox{1.5}{\includegraphics[width=0.45\textwidth]{logneg_galve}} \caption{\label{fig:logneg-galve} Time evolution of the logarithmic negativity when we choose the parameters by $m = 1$, $ \omega = 1$, $\gamma_{1}=\gamma_{2}= 0.0025$, $\beta_{1}^{\textsc{b}}=\beta_{2}^{\textsc{b}}= 0.2$, $\sigma(t) = c_0 + c_1\cos(\omega_d t)$ with $c_0 = 0$, $c_1 = 0.5$, $\omega_d = 1.996$. The parameter $\beta_i$ in the plot legend denotes the initial inverse temperature of the oscillators.} \end{figure} In Fig.~\ref{fig:logneg-galve} we assume that both oscillators are initially prepared in thermal states of the same temperature $\beta_i^{-1}$. This initial oscillator temperature is high with respect to the oscillator's natural frequency $\omega$, so the thermal fluctuations completely destroy the initial entanglement between the oscillators. This can be seen from the plot that $E_{\mathcal{N}}$ is essentially zero at the beginning. We also let the bath temperature $(\beta_{i}^{\textsc{b}}){^{-1}}$ set in the high temperature regime. In the conventional setting considered in~\cite{HH15PRD}, at late times the entanglement will not survive when $\beta^{\textsc{b}}_{i}<\mathcal{O}(\omega^{-1})$. However, as have shown by Ref.~\cite{galve-prl} and seen in Fig.~\ref{fig:logneg-galve}, the entanglement is still sustained at late times in this high temperature setting for the parametrically coupled oscillators. Fig.~\ref{fig:logneg-galve} furthermore confirms the conclusion of Ref.~\cite{galve-prl} that logarithmic negativity at late times does not depend on the initial state of the oscillators. This kind of phenomenon usually occurs when the dynamics has a steady state. There, the decaying behavior of the fundamental solution in \eqref{chi} and \eqref{p} renders any dependence on the initial state negligibly small at late times. Thus it may be surprising to see this phenomenon even when the dynamics is unstable. The instability of the system can be inferred from the observation that the parameters $\omega_{d}$, $c_{0}$, and $c_{1}$ used in Fig.~\ref{fig:logneg-galve} fall inside the dark shade in Fig.~\ref{fig:stability}. When the system is unstable, the fundamental solutions in \eqref{chi} and \eqref{p} will grow indefinitely, so it is natural to expect that the contributions from the initial state to become more and more significant. But here lies the subtlety: for example, the covariance matrix elements of the system can be divided into an active component and a passive component. The former depends on the initial conditions, and describes the system's intrinsic behavior even when the interaction with the environment is turned off. The latter is generated by the bath after the evolution starts, and is independent of the initial condition. Thus, if the passive component of {the quantities of interest} dominates over the active component, then at late times this quantity, though still growing with time, can barely depend on its initial condition. Now let us inspect a little more the symplectic eigenvalue $\lambda_{<}^{\textsc{pt}}$ of the partially transposed covariance matrix. It can be expressed by two symplectic invariants $\Delta(\bm{\sigma}^{\textsc{pt}})$ and $\det \bm{\sigma}$~\cite{PPTSimon,Illuminati04}, \begin{align} (\lambda_{<}^{\textsc{pt}})^{2} = \frac{1}{2}\left\{ \Delta(\bm{\sigma}^{\textsc{pt}}) - \sqrt{\Delta^2(\bm{\sigma}^{\textsc{pt}}) - 4\det \bm{\sigma}} \right\}\,, \label{sympeigen-exp} \end{align} with $\Delta(\bm{\sigma}^{\textsc{pt}}) = I_{1}+I_{2}-2I_{3}$, and \begin{align} I_1 &=\bm{\sigma}_{\chi_1\chi_1}\bm{\sigma}_{p_1p_1} -\frac{1}{4}\bm{\sigma}_{\chi_1p_1}^2\,, &I_2 &=\bm{\sigma}_{\chi_2\chi_2}\bm{\sigma}_{p_2p_2} -\frac{1}{4}\bm{\sigma}_{\chi_2p_2}^2\,,\notag \\ I_3 &=\bm{\sigma}_{\chi_1\chi_2}\bm{\sigma}_{p_1p_2}-\bm{\sigma}_{\chi_1p_2}\bm{\sigma}_{\chi_2p_1}\,. \label{sympeigen-exp-det} \end{align} \begin{figure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{galve-sympeigen-terms2} \caption{with private baths} \label{fig:galve-sympeigen-terms-wbath} \end{subfigure} \hfill \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{galve-sympeigen-wobath-terms2} \caption{without private baths} \label{fig:galve-sympeigen-terms-wobath} \end{subfigure} \caption{Evolution of $\Delta(\bm{\sigma}^{\textsc{pt}})$ and $\det\bm{\sigma}$ of $(\lambda_{<}^{\textsc{pt}})^{2}$ given in Eq.~\eqref{sympeigen-exp}. The same parameters as in Fig.~\ref{fig:logneg-galve} are used except that $\gamma = 0.005$ in (a). In {contrast, in} (b), the coupled oscillators form a closed system without contact with the thermal baths.} \label{fig:galve-sympeigen-terms} \end{figure} In Fig.~\ref{fig:galve-sympeigen-terms}, we show the time evolution of $\Delta(\bm{\sigma}^{\textsc{pt}})$ and $\det\bm{\sigma}$ that constitute of $(\lambda_{<}^{\textsc{pt}})^{2}$ according to Eq.~\eqref{sympeigen-exp}. We observe that both $\Delta(\bm{\sigma}^{\textsc{pt}})$ and $\det \bm{\sigma} $ in Eq.~\eqref{sympeigen-exp} grow exponentially with time. Intriguingly the difference between $\ln\Delta(\bm{\sigma}^{\textsc{pt}})$ and $\ln\det \bm{\sigma}$ seems to remain constant with time when $t$ is sufficiently large. From this difference we can infer that \begin{align}\label{E:dfhgvd} \ln\Delta(\bm{\sigma}^{\textsc{pt}})&>\ln\det \bm{\sigma}\,,&&\Rightarrow&2\ln\Delta(\bm{\sigma}^{\textsc{pt}})&>\ln\Delta(\bm{\sigma}^{\textsc{pt}})>\ln\det\bm{\sigma}\,,\notag\\ &&&\Rightarrow&\Delta^{2}(\bm{\sigma}^{\textsc{pt}})&\gg\det\bm{\sigma}\,. \end{align} This is an important criterion. Since entanglement occurs when $(\lambda_{<}^{\textsc{pt}})^{2}<1/4$, it implies that $\Delta(\bm{\sigma}^{\textsc{pt}})$ is barely greater than $\sqrt{\Delta^{2}(\bm{\sigma}^{\textsc{pt}})-4\det \bm{\sigma}}$. This is possible only when $\Delta^{2}(\bm{\sigma}^{\textsc{pt}})\gg4\det \bm{\sigma}$. In this case the square of the symplectic eigenvalue $\lambda_{<}^{\textsc{pt}}$ can be approximately given by \begin{equation} (\lambda_{<}^{\textsc{pt}})^2\simeq \frac{\det\bm{\sigma}}{\Delta(\bm{\sigma}^{\textsc{pt}})}\,. \end{equation} Thus entanglement exists when $\det\bm{\sigma}<\Delta(\bm{\sigma}^{\textsc{pt}})/4$. Later we will give another example in which the dynamics is unstable but $\Delta^{2}(\bm{\sigma}^{\textsc{pt}})\gtrsim4\det \bm{\sigma}$. In that case entanglement is not realizable at high temperatures. The observation $\ln\Delta(\bm{\sigma}^{\textsc{pt}})-\ln\det \bm{\sigma}\simeq\text{const}>0$ implies that for sufficiently large time, $(\lambda_{<}^{\textsc{pt}})^{2}$ will be a constant smaller than unity. On the other hand, the fact that entanglement can be sustained when $(\lambda_{<}^{\textsc{pt}})^{2}<1/4$ in turn tells us that the difference between $\ln\Delta(\bm{\sigma}^{\textsc{pt}})$ and $\ln\det \bm{\sigma}$ needs to be greater than $\ln4\sim1.39$. Apparently this requirement is well satisfied in Fig.~\ref{fig:galve-sympeigen-terms-wbath}, not to mention Fig.~\ref{fig:galve-sympeigen-terms-wobath}. The small ripples in the curve of $\ln\Delta(\bm{\sigma}^{\textsc{pt}})$ does not affect the above arguments; they are manifested in the oscillatory behavior of $E_{\mathcal{N}}$ in Fig.~\ref{fig:logneg-galve}. Fig.~\ref{fig:galve-sympeigen-terms-wobath} shows that the determinant of the covariance matrix remains constant for the closed system case because the evolution is unitary when the bath is absent. It is fixed by our choice of the initial states of the oscillator. As $\Delta(\bm{\sigma}^{\textsc{pt}})$ keeps growing under the influence of parametric coupling, the symplectic eigenvalue $\lambda_{<}^{\textsc{pt}}$ in Eq. \eqref{sympeigen-exp} will approach zero, equivalent to an infinite logarithmic negativity. That is, entanglement is well preserved. Comparing Figs.~\ref{fig:galve-sympeigen-terms-wbath} with \ref{fig:galve-sympeigen-terms-wobath}, we may conclude that the growth of $\det \bm{\sigma}$ clearly results from the intervention of the baths. However, at this point it is not yet {fully} clear why 1) $\Delta(\bm{\sigma}^{\textsc{pt}})$ barely depends on the baths, 2) $\ln\Delta(\bm{\sigma}^{\textsc{pt}})-\ln\det \bm{\sigma}$ is a constant (apart from small ripples), and 3) the late-time entanglement is independent of the initial states when the oscillators are parametrically coupled. {We conjecture that 3) may result from the possibility that the contributions which depend on the initial condition play a subdominant role.} This scenario is particularly likely when the bath temperature is high {since the contributions from the bath can be amplified by the high temperature}. \begin{figure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{logneg_stable_wd} \caption{} \label{fig:stable-wd} \end{subfigure} \hfill \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{logneg_stable_str} \caption{} \label{fig:stable-str} \end{subfigure} \caption{Time evolution of the logarithmic negativity at high bath temperature in the stable regime for different (a) driving frequencies and (b) coupling strengths. The parameters shared in both cases are $m = 1$, $\omega = 1$, $\gamma_{1}=\gamma_{2}= 0.005$, $\beta^{\textsc{b}}_{1}=\beta^{\textsc{b}}_{2}= 0.2$, $\beta_i = 2\times 10^4$, $c_0 = 0$, but in (a) $c_1 = 0.5$ and in (b) $\omega_d = 1.525$.} \label{Fi:ebgtsd} \end{figure} \subsection{Stable dynamics} Next we turn to stable dynamics. In this case, the contributions from the initial conditions in \eqref{chi} and \eqref{p} will be exponentially small at late times due to the decaying behavior of $\bm{D}_{i}(s,0)$. However, the periodicity property of $\bm{D}_{i}(s,s')$ in \eqref{E:dkjgbfkse} implies that the contributions from the quantum fluctuations of the private baths tend to be periodic in time as well. Thus in this regime, the covariance matrix elements will evolve to have finite, but periodic behavior. This is in strong contrast to the situation that the system is not parametrically driven. In the latter case, the observables of the system tends to rest on constant values {or constant rates} at late times in the stationary state. Therefore, in the current case when the system is parametrically driven by a periodic external agent and has stable dynamics, we understand only in an average sense that the system reaches a \textit{stationary} state. That is, the system's observables will appear to be constants only after we average them over the driving period at late times. With this dynamic feature in mind, we now examine the entanglement in the stable regime. \begin{figure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{galve-sympeigen-stable-terms2} \caption{with the baths} \label{fig:galve-sympeigen-stable-terms-wbath} \end{subfigure} \hfill \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{fig5b} \caption{without the baths} \label{fig:galve-sympeigen-stable-terms-wobath} \end{subfigure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{logneg-fig5a} \caption{with the baths} \label{fig:logneg-fig5a} \end{subfigure} \hfill \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{fig5d} \caption{without the baths} \label{fig:galve-det-wobath-stable} \end{subfigure} \caption{First row:~evolution of $\Delta(\bm{\sigma}^{\textsc{pt}})(t)$ and $\det \bm{\sigma}(t)$ where the parameters $m=1$, $\omega=1$, $\gamma_{i} = 0.005$, $\beta_i^{\vphantom{\textsc{b}}}=2\times10^4$, $\beta_i^{\textsc{b}}=0.2$, $\omega_d = 1.5$, $c_0=0$ and $c_1=0.5$ are chosen for stable evolution of the system. In (a) the parametrically coupled oscillators are in contact with their private thermal bath, but in (b) the contact is removed. Second row:~evolution of the corresponding logarithmic negativity. In (c) the entanglement drops to zero rather quickly, compared with the $\gamma_i^{-1}$.} \label{fig:galve-sympeigen-stable-terms} \end{figure} Fig.~\ref{fig:stable-wd} shows the time evolution of logarithmic negativity at high bath temperature in the stable regime for different driving frequencies and Fig.~\ref{fig:stable-str} shows the time evolution of logarithmic negativity in the stable regime for different coupling strengths. We see that the entanglement measure drops to zero very rapidly, compared to the time scale $\gamma_{i}^{-1}\sim200$, which is the typical relaxation time when the system is not parametrically driven. Before the entanglement measure vanishes, it seems that we will have greater transient entanglement when the parameters $(c_{1},\omega_{d})$ are located closer to the boundary between the stable and the unstable regimes in Fig.~\ref{fig:stability}. We further note that after the sudden death~\cite{YuEberly} of quantum entanglement, it never revives at late times in the high bath temperature $(T^{(\textsc{b})}_{1,2}\sim5\omega)$ case. This is consistent with the findings in the non-parametric driving case in~\cite{HH15PLB,Anders,AndWin,HH15JHEP}. The evolution of $\Delta(\bm{\sigma}^{\textsc{pt}})$ and $\det \bm{\sigma} $ in the expression, Eq.~(\ref{sympeigen-exp}), of $\lambda_{<}^{2}$ is plotted in Fig.~\ref{fig:galve-sympeigen-stable-terms}. We observe that $\Delta(\bm{\sigma}^{\textsc{pt}})$ keeps oscillating around an average value for both open and closed system cases, {where in the open-system case, shown in Fig.~\ref{fig:logneg-fig5a}, the state of the coupled system rapidly becomes separable, not long after the evolution starts}. Thus, following the discussion around the criterion \eqref{E:dfhgvd}, we can draw the conclusion that {in the open system setting of Fig.~\ref{fig:galve-sympeigen-stable-terms}, since} $\Delta(\bm{\sigma}^{\textsc{pt}})\gg1$ and $\Delta^{2}(\bm{\sigma}^{\textsc{pt}})\gtrsim4 \det \bm{\sigma}$, the symplectic eigenvalues of the partially transposed covariance matrix satisfy $\lambda_{>}\gtrsim\lambda_{<}\gg1$ and entanglement { may not survive} at late times. For comparison with Fig.~\ref{Fi:ebgtsd}, we show the evolution of logarithmic negativity, when the motion of coupled oscillator is unstable, for different driving frequencies in Fig.~\ref{fig:unstable-wd} and different $c_{1}$ in Fig.~\ref{fig:unstable-str}. In addition, we also observe a similar phenomenon, seen in Fig.~\ref{Fi:ebgtsd}, that when the parameter pair $(c_1,\omega_{d})$ is more deeply located inside the dark regime, the sustained entanglement seems stronger because the corresponding negativity is greater~\cite{Eisert98,Virmani}. Thus it seems that dynamical instability is a necessary ingredient to sustain a non-zero logarithmic negativity, allowing quantum entanglement to last over long times. \begin{figure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{logneg_unstable_wd} \caption{} \label{fig:unstable-wd} \end{subfigure} \hfill \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{logneg_unstable_str} \caption{} \label{fig:unstable-str} \end{subfigure} \caption{Time evolution of the logarithmic negativity in the unstable regime for $m = 1$, $\omega = 1$, $\gamma_{1}=\gamma_{2}= 0.005$, $\beta^{\textsc{b}}_{1}=\beta^{\textsc{b}}_{2}= 0.2$, $\beta_i = 2\times 10^4$, and $c_0 = 0$. In (a) we choose $c_1 = 0.5$ and in (b) $\omega_d = 1.9525$.} \label{fig:galve-sympeigen-stable-terms2} \end{figure} \section{Instability is not a sufficient condition}\label{S:eoujhbdf} Here we give an example to show that dynamical instability, though necessary, is not sufficient to ensure high-temperature entanglement. Following that, we propose an alternative but equivalent description for entanglement, in terms of squeezing and thermal fluctuations. \subsection{Coupled amplifying harmonic oscillators} Let us consider the coupled amplifying harmonic oscillators, each interacting with its individual bath, following the equations of motion given by \begin{align} \ddot{\chi}_{1}(t)-2\mathsf{g}\,\dot{\chi}_{1}(t)+\omega^{2}\chi_{1}(t)+c_{0}\,\chi_{2}(t)&=\frac{e}{m}\,\xi_{1}(t)\,,\label{E:eotugbsd1}\\ \ddot{\chi}_{2}(t)-2\mathsf{g}\,\dot{\chi}_{2}(t)+\omega^{2}\chi_{1}(2)+c_{0}\,\chi_{1}(t)&=\frac{e}{m}\,\xi_{2}(t)\,.\label{E:eotugbsd2} \end{align} Since this phenomenological model is used to illustrate a point, we may leave out questions about the mechanism of amplification. Here, `amplifying' or `anti-damping' means the opposite of `damping': the system oscillator's dynamical equation has a damping term with a sign opposite to that in a normal oscillator. We assume that both oscillators in our system have the same mass $m$, physical oscillating frequency $\omega$, amplification constant $\mathsf{g}$, and the same coupling strength $e$ with their individual private bath. The strength, denoted by $c_{0}$, of inter-oscillator coupling is assumed to be time-independent. The noise force $\xi_{i}$ accounts for the quantum fluctuations of the private bath $i$, which initially is assumed to be in a thermal state of temperature $\beta^{-1}$. The corresponding dissipative backreaction of the private bath is assumed to be overwhelmed by the unspecified amplification mechanism, so that their overall effect is given by $-2\mathsf{g}\,\dot{\chi}_{i}(t)$ in the equations of motion. In principle $\mathsf{g}$ is not necessarily equal to $\gamma=e^{2}/(8\pi m)$, but we assume that it takes on this value for the moment. We still suppose the baths satisfy the standard fluctuation-dissipation relation associated with their initial state. This choice of parameters allows us to easily decouple the equations of motion into \begin{align} \ddot{\chi}_{\pm}(t)-2\mathsf{g}\,\dot{\chi}_{\pm}(t)+\omega^{2}_{\pm}\chi_{\pm}(t)&=\frac{e}{m}\,\xi_{\pm}(t)\,,\label{E:fgbkdgye} \end{align} where $\omega_{\pm}^{2}=\omega^{2}\pm\sigma$ by the normal modes \begin{equation} \chi_{\pm}(t)=\frac{1}{\sqrt{2}}\,\chi_{1}(t)\pm\frac{1}{\sqrt{2}}\,\chi_{2}(t)\,. \end{equation} Since the Langevin equations \eqref{E:fgbkdgye} are homogeneous in time, the fundamental solutions are given by \begin{align} d_{1}^{(\pm)}(t)&=e^{\mathsf{g}t}\Bigl[\cos\Omega_{\pm}t-\frac{\mathsf{g}}{\Omega_{\pm}}\,\sin\Omega_{\pm}t\Bigr]\,,&d_{2}^{(\pm)}(t)&=e^{\mathsf{g}t}\,\frac{1}{\Omega_{\pm}}\sin\Omega_{\pm}t\,, \end{align} with $\Omega_{\pm}^{2}=\omega_{\pm}^{2}-\mathsf{g}^{2}=\omega^{2}-\mathsf{g}^{2}\pm\sigma$. They grow exponentially due to the amplification effect. In this case $D_{2}^{(\pm)}(t,s)=d_{2}^{(\pm)}(t-s)$. The behavior of the covariance matrix elements may not be completely controlled by the bath at late time because the contributions from the initial conditions can be significant due to runaway dynamics. However, their relative dominance depends on the bath temperature. \begin{figure} \centering \scalebox{0.35}{\includegraphics{AO_ppMode}} \caption{The time evolution of the covariance matrix elements $\bm{\sigma}_{\chi_{+}\chi_{+}}(t)$, $\bm{\sigma}_{p_{+}p_{+}}(t)$ and $\bm{\sigma}_{\chi_{+}p_{+}}(t)$ for the coupled amplifying oscillators in the NESS configuration. We choose the parameters $m=1$, $\mathsf{g}=\gamma=0.02$, $\sigma=0.2$, and $\eta=2$. The temperature of two private bath is $\beta^{-1}=10$, which belongs to the conventional high temperature regime. All the parameters and the evolution time are normalized to $\omega$ or $\omega^{-1}$.}\label{Fi:AO_ppMode} \end{figure} Let us consider the time evolution of the covariant matrix elements. Suppose that the initial state of the amplifying oscillators is a two-mode squeezed vacuum state, so both oscillators are initially entangled, Thus in the normal modes, $\bm{Z}_{\pm}=(\chi_{+}, p_{+}, \chi_{-}, p_{-})^{T}$, the covariance matrix element has the form \begin{align*} \bm{\sigma}_{\pm}(0)&=\begin{pmatrix}\sigma_{\chi_{+}\chi_{+}}(0) &0 &0 &0 \\[4pt]0 &\sigma_{p_{+}p_{+}}(0) &0 &0\\[4pt]0 &0 &\sigma_{\chi_{-}\chi_{-}}(0) &0\\[4pt] 0 &0 &0 &\sigma_{p_{-}p_{-}}(0) \end{pmatrix}\,, \end{align*} with \begin{align*} \sigma_{\chi_{+}\chi_{+}}(0)&=\frac{1}{2m\omega}e^{-2\eta}\,,&\sigma_{p_{+}p_{+}}(0)&=\frac{m\omega}{2}e^{+2\eta}\,,\\ \sigma_{\chi_{-}\chi_{-}}(0)&=\frac{1}{2m\omega}e^{+2\eta}\,,&\sigma_{p_{+}p_{+}}(0)&=\frac{m\omega}{2}e^{-2\eta}\,. \end{align*} Here for example, $\sigma_{\chi_{+}\chi_{+}}$ represents \begin{equation} \sigma_{\chi_{+}\chi_{+}}=\frac{1}{2}\langle\{\chi_{+},\,\chi_{+}\}\rangle \end{equation} and $\eta$ is the {initial} squeeze parameter. Since both modes are decoupled, the covariance matrix is always in block form: \begin{align}\label{E:pworihfbd} \bm{\sigma}_{\pm}(t)&=\begin{pmatrix}\bm{A}_+(t) &\bm{0}\\\bm{0} &\bm{A}_-(t)\end{pmatrix}\,,&\bm{A}_{\pm}&=\begin{pmatrix} \bm{\sigma}_{\chi_{\pm}\chi_{\pm}} &\bm{\sigma}_{\chi_{\pm}p_{\pm}}\\\bm{\sigma}_{\chi_{\pm}p_{\pm}} &\bm{\sigma}_{p_{\pm}p_{\pm}} \end{pmatrix}\,, \end{align} with elements like $\bm{\sigma}_{\chi_{\pm}\chi_{\pm}}(t)$ given by \begin{align} \bm{\sigma}_{\chi_{\pm}\chi_{\pm}}(t)&=d_1^{(\pm)}(t)\,\bm{\sigma}_{\chi_{\pm}\chi_{\pm}}(0)+\frac{1}{m^2}\,d_2^{(\pm)}(t)\,\bm{\sigma}_{p_{\pm}p_{\pm}}(0)\notag\\ &\qquad\qquad\qquad\qquad+\frac{e^2}{m^2}\int_0^t\!ds\int_0^t\!ds\;d_2^{(\pm)}(t-s)d_2^{(\pm)}(t-s')\,G_{H,0}^{(++)}(s,s')\,, \end{align} where \begin{equation} G_{H,0}^{(ij)}(s,s')=\frac{1}{2}\,\langle\xi_i(s)\xi_j(s')\rangle \end{equation} and $i$, $j=+$, $-$. The other elements in $\bm{A}_\pm$ can be constructed similarly. \begin{figure} \centering \scalebox{0.5}{\includegraphics{AO_sympPT}} \caption{The time evolution of the smaller of the squared symplectic eigenvalues of the partially transposed covariance matrix in terms of the canonical variables. Its value quickly rises past $1/4$ after roughly one cycle of motion. Afterwards, the entanglement of the partite system vanishes, and the eigenvalue seem to increase indefinitely. We choose the parameters $m=1$, $\mathsf{g}=\gamma=0.02$, $\sigma=0.2$, and $\eta=2$. The temperature of two private bath is $\beta^{-1}=10$, which belongs to the conventional high temperature regime. All the parameters and the evolution time are normalized to $\omega$ or $\omega^{-1}$.}\label{Fi:AO_sympPT} \end{figure} Fig.~\ref{Fi:AO_ppMode} shows the generic behavior of the covariance matrix elements with time. They are oscillatory but with increasing amplitudes, indicating runaway dynamics. When we turn to the time evolution of $(\lambda_{<}^{\textsc{pt}})^{2}$ in Fig.~\ref{Fi:AO_sympPT}, a distinct difference shows. The eigenvalue rapidly grows out of bound. When we zoom into the early time regime, we observe that the initial entanglement, $(\lambda_{<}^{\textsc{pt}})^{2}<1/4$, is destroyed right after about one cycle of motion. Thus it is barely sustained, in strong contrast to the unstable parametric driven case in~\cite{galve-prl}. The results shown in Fig.~\ref{Fi:AO_sympPT} hint that \begin{align}\label{E:gfkdbsert} \lambda^{\textsc{pt}}_{>}&\gtrsim\lambda^{\textsc{pt}}_{<}\,,&&\text{but}&\lambda^{\textsc{pt}}_{>}\lambda^{\textsc{pt}}_{<}&\gg1\,. \end{align} According to \eqref{sympeigen-exp}, they in turns imply \begin{align} \Delta(\bm{\sigma}^{\textsc{pt}})&\gg1\,, &\det\bm{\sigma}&\gg1\,,&&\text{but} &\Delta^{2}(\bm{\sigma}^{\textsc{pt}})\gtrsim4\det\bm{\sigma}\,, \end{align} such that $\Delta(\bm{\sigma}^{\textsc{pt}})\gg\sqrt{\Delta^{2}(\bm{\sigma}^{\textsc{pt}})-4\det\bm{\sigma}}$, since \begin{align} \bigl(\lambda^{\textsc{pt}}_{>}\bigr)^{2}\bigl(\lambda^{\textsc{pt}}_{<}\bigr)^{2}=\det\bm{\sigma}\,. \end{align} Comparing with the unstable parametrically driven oscillator case, even this cursory analysis shows that in order to have sustained entanglement, we need to have a configuration that keeps $\det\bm{\sigma}$ in check such that $\Delta^{2}(\bm{\sigma}^{\textsc{pt}})\gg4\det\bm{\sigma}$. It is exactly the case found in \eqref{E:dfhgvd} for the unstable, parametrically driven case. This ensures the contribution from the thermal fluctuations is subdominant. It seems to tell that cross correlation between the oscillators is so strong as to overcome the fluctuation effect in each oscillator. Through this example we see that dynamical instability is a necessary but not sufficient condition for hot entanglement, it also illustrates the simple criterion in the link between dynamical instability and sustained entanglement. \subsection{Entanglement expressed in terms of quantum optics parameters} Here we propose an interpretation of entanglement in terms of squeezing and thermal fluctuations parameters. Following the discussion in Appendix~\ref{S:peotr}, we notice that the covariance matrix \eqref{E:bskads} of the normal modes, which correspond to two canonical pairs that define the two mode squeezed state, has the same structure as the covariance matrix \eqref{E:pworihfbd}. It prompts the possibility to express \eqref{E:pworihfbd} at any instant by an effective two-mode squeezed state. The state corresponding to \eqref{E:pworihfbd} in fact is a product of two single-mode squeezed state, each of which rotates at different angular velocities due to different normal-mode frequencies. In this sense the two-mode squeezed state, having fewer number of independent parameters than the product of two single-mode squeezed states do, is only an effective description, requiring the effective squeeze parameter and the effective temperature to be time-dependent. Notwithstanding, the clear advantage is to describe entanglement with familiar quantities in quantum optics. We can write the symplectic eigenvalues of the partially transposed covariance matrix for the canonical pairs of the coupled oscillator as \begin{equation}\label{E:ewtdgs} \lambda^{\textsc{pt}}_{\gtrless}=e^{\pm2\eta_{\textsc{eff}}}\,\bigl(\bar{n}_{\textsc{eff}}+\frac{1}{2}\bigr)\,, \end{equation} in analogy to \eqref{E:bskads} in terms of the effective squeeze parameter $\eta_{\textsc{eff}}$ and the effective inverse temperature $\beta_{\textsc{eff}}$, satisfying \begin{align} \bar{n}_{\textsc{eff}}+\frac{1}{2}=\frac{1}{2}\,\coth\frac{\beta_{\textsc{eff}}\omega}{2}&=\sqrt{\lambda_{>}^{\textsc{pt}}\lambda_{<}^{\textsc{pt}}}\,,&\eta_{\textsc{eff}}&=\frac{1}{4}\ln\frac{\lambda^{\textsc{pt}}_{>}}{\lambda^{\textsc{pt}}_{<}}\,. \end{align} \begin{figure} \centering \scalebox{0.45}{\includegraphics{AO_effective}} \caption{The time evolution of the effective squeeze parameter $\eta_{\textsc{eff}}(t)$ and the effective temperature $T_{\textsc{eff}}(t)=\beta_{\textsc{eff}}^{-1}(t)$. Actually, the squeeze parameter more or less decreases with time but the effective temperature grows rapidly. We choose the parameters $m=1$, $\mathsf{g}=\gamma=0.02$, $\sigma=0.2$, and $\eta=2$. The temperature of two private bath is $\beta^{-1}=10$, which belongs to the conventional high temperature regime. All the parameters and the evolution time are normalized to $\omega$ or $\omega^{-1}$.}\label{Fi:AO_effective} \end{figure} Here the effective temperature is determined up to a frequency factor, and we choose the physical frequency of the oscillator. If another time-independent frequency scale is chosen, it merely rescales the effective temperature but will not affect its generic feature. Both effective parameters are time-dependent, and thus are nonequilibrium in nature. They are shown in Fig.~\ref{Fi:AO_effective}. In sub-figure~\ref{Fi:AO_effective}-(a), we see the effective squeeze parameter immediately drops from its initial value, and oscillates with time, but overall it decrease with time. This could be a bad sign if we use the instability configuration to sustain entanglement. Fig.~\ref{Fi:AO_effective}-(b) shows the time evolution of the effective temperature of the system. It grows more and more rapidly with time, and zooms past the value of the initial bath temperature at roughly $t=6\omega^{-1}$. This implies that in this case instability actually induces larger and larger effective thermal fluctuations {such that they overwhelm} the effect of squeezing. This is quite different from the stable case where the contribution from thermal fluctuations approaches a constant value with time. The compound effect of decreasing squeeze parameter and increasing thermal fluctuations makes it impossible to sustain late-time quantum entanglement of such a bipartite system in the NESS configuration. Next we consider the energy budget in the unstable, parametrically driven case. \section{Energy Budget operating in the unstable, parametrically driven regimes}\label{S:ebgdfhg} Finally, in this section we scrutinize the scheme of hot entanglement proposed by Galve et al ~\cite{galve-prl} operating in the unstable regimes of parametric driven coupling from a realistic experimental feasibility perspective. We numerically study the energy flows to/from the two baths, between the coupled oscillators and calculate the work cost of driving to see if it is within reasonable experimental reach. \subsection{Hot entanglement can only be a transient effect from work cost considerations} We first define the internal energy $U$ of the system of coupled harmonic oscillators as the expectation value of the system Hamiltonian at a given time \begin{equation} U(t) =\sum_{i=1}^{2} \left( \frac{m\omega^2 \langle\chi_{i}^2\rangle}{2} + \frac{\langle p_{i}^2\rangle}{2m} \right) + m\sigma(t)\,\langle\chi_{1}\chi_{2} \rangle\, . \label{sysen} \end{equation} Taking the derivative of Eq.~\eqref{sysen} with respect to time and making use of Eqs.~\eqref{covar-ohmic-1st}--\eqref{covar-ohmic-last}, we obtain \begin{align}\label{E:bkeirs} \frac{dU}{dt} &=\sum_{i=1}^{2} \Bigl( \frac{m\omega^2}{2} \frac{d}{dt}\langle\chi_{i}^2\rangle + \frac{1}{2m}\frac{d}{dt}\langle p_{i}^2\rangle\Bigr) + \frac{m\sigma}{2} \frac{d}{dt}\langle\{\chi_{1},\chi_{2}\}\rangle + \frac{m}{2}\frac{d\sigma}{dt}\langle\{\chi_{1},\chi_{2}\}\rangle\,. \end{align} On the other hand, from the Langevin equation \eqref{lange-sf}, we have \begin{align} \sum_{i=1}^{2} \Bigl( \frac{m\omega^2}{2} \frac{d}{dt}\langle\chi_{i}^2\rangle + \frac{1}{2m}\frac{d}{dt}\langle p_{i}^2\rangle\Bigr) + \frac{m\sigma}{2} \frac{d}{dt}\langle\{\chi_{1},\chi_{2}\}\rangle=\sum_{i=1}^2\Bigl(\langle\xi_i\dot{\chi}_i\rangle-2m\gamma_i\langle\dot{\chi}_i^2\rangle\Bigr)\,. \end{align} Comparing with \eqref{E:bkeirs}, we find \begin{align} \frac{dU}{dt}&= \sum_{i=1}^{2} \Bigl( \frac{2\gamma_{i}}{\beta^{\textsc{b}}_{i}} - \frac{2\gamma_{i}}{m} \langle p_{i}^2 \rangle \Bigr) + \frac{m}{2}\frac{d\sigma}{dt}\langle\{\chi_{1},\chi_{2}\}\rangle+\cdots\,,\label{E:kfgjbdfg} \end{align} where we have implemented the Caldeira-Leggett approximation and assumed the driving protocol, $\sigma(t)=c_{0}+c_{1}\,\cos\omega_{d}t$. The first term on the righthand side of \eqref{E:kfgjbdfg} describes a constant energy flow from the private baths to their respective system oscillators in the form of a white thermal noise due to our high temperature Ohmic bath assumption. The second term results from the dissipative term in Eq.~\eqref{lange-sf}, effecting an energy flow from the system oscillators to their respective baths. The last term describes the work transfer from the external agent that performs the time-dependent driving on the system. Following the notation in Ref.~\cite{HH15AOP}, we rewrite the time derivative of the internal energy as \begin{equation}\label{E:kgdfge} \frac{dU}{dt} = \sum_{i=1}^{2} \left({P_{\xi}}_i + {P_{\gamma}}_i \right) + P_{\text{dr}}\,. \end{equation} Eq.~\eqref{E:kgdfge} expresses the standard energy conservation in thermodynamics: the work done by the external agent plus the heat from the thermal bath cause the internal energy of the system to change. With the same parameters used in Fig.~\ref{fig:logneg-galve}, we find in Figs.~\ref{fig:galve-logenergy} and \ref{fig:galve-logpower} that the internal energy $U$, dissipated power $\lvert P_{\gamma_{i}}\rvert$ and the driving power $P_{\text{dr}}$ {increase exponentially with time}. (The constant contributions $P_{\xi_{i}}$ from the bath noises are not plotted.) The plots show that after a sufficiently long time, the system, which is composed of two parametrically coupled oscillators in the two bath configuration, draws a huge amount of energy from the external agent, resulting in the internal energy and the dissipation power to both baths grow exponentially fast. In other words, because quantum entanglement between the oscillators can be kept only in the dynamically unstable regimes, the external agent must have an exponentially large energy reservoir to {supply} the system's consumption. From a realistic experimental viewpoint this can only be supported for a short period of time, but not sustainable. We conclude that hot entanglement in this setup can only be a transient effect. \begin{figure} \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{galve-logenergy} \caption{} \label{fig:galve-logenergy} \end{subfigure} \hfill \begin{subfigure}{0.49\textwidth} \includegraphics[width=\textwidth]{galve-logpower} \caption{} \label{fig:galve-logpower} \end{subfigure} \caption{(a) Time evolution of the internal energy. (b) Time evolution of the dissipation power $\lvert P_{\gamma_{i}}\rvert$ and driving powers $P_{\text{dr}}$. Note that the dissipation power is always negative, the driving power is positive except for a few data points in the plot. We attribute the existence of negative values of $P_{\text{dr}}$ to numerical errors since these negative values are negligibly small. We use the same parameters aa in Fig.~\ref{fig:logneg-galve}} \label{fig:galve-sympeigen-stable-terms3} \end{figure} \subsection{No fluctuation-dissipation relation at late times} Since the energy input from the external driving agent mainly goes to the internal energy of the system and flows to the surrounding bath via the damping channel, the power delivered by the noise force from the bath plays a negligible role, even when the bath temperature is high. From this viewpoint of energy flow, the classical description seems to suffice, as if the noise term in the equation of motion \eqref{lange-sf} is absent. However, this is not true, because of the presence of quantum entanglement. The huge order of magnitude discrepancy between the dissipation power and the noise power clearly indicates it is impossible to have any steady state at late times, and thus there cannot be any fluctuation-dissipation relation for such parametrically driven systems. \section{Conclusion} In this paper, we probe deeper into the conditions for sustaining the entanglement between two coupled harmonic oscillators (the system) each interacting with its own heat bath, while the system is subjected to external parametric drive, the same set up as in prior works for Markovian~\cite{galve-prl} and non-Markovian~\cite{EstPac} baths. Our results based on numerical solutions of the quantum Langevin equations show that entanglement can be sustained, but only in the dynamically unstable regimes, and, as such, the work cost for modulating the coupling strength increases exponentially over time to make the scheme untenable. To cover a fuller ground we also investigated the possibility of sustaining entanglement in the stable regime, maximizing the effect of weak driving on the system by choosing driving frequencies slightly detuned from one of the parametric resonance frequencies. We concluded that under stable dynamics the energy cost-sustained entanglement trade-off has no particular advantage. Varying the temperature ranges we also found, with no surprise, that entanglement becomes insignificant at late times at very low temperatures. Fully considered, we are able to conclude that sustaining entanglement at high temperatures with driving protocols operating the system in the dynamically unstable regime for a long time is practically impossible due to its exponentially rising energy cost whereas operating in the stable regime requiring only constant power consumption fails to achieve hot entanglement. It remains an open challenge to investigate hot entanglement under different conditions such as making the inter-oscillator coupling or external driving nonlinear, and to try out different set ups such as sharing a common bath, and important generic systems such as two-level systems. This is an important subject which, as far as we can see, is still in its infancy awaiting more thorough and systematic analysis. \\ \textbf{Acknowledgment} OA is supported by a research assistantship from the Maryland Center for Fundamental Physics. J.-T. Hsiang is supported by the Ministry of Science and Technology of Taiwan, R.O.C. under Grant No.~MOST 111-2811-M-008-022. BLH appreciates the hospitality of the National Center for Theoretical Sciences of Taiwan during his visit in the finishing stage of this paper. \newpage

proofpile-arXiv_069-10973

\section{Introduction} \label{sec:intro} Existing approaches for on-device training are neither efficient nor practical enough to satisfy the resource constraints of edge devices (Figure \ref{fig:method_matrix}). This is because these methods do not properly address a fundamental problem in on-device training, namely \textit{the computational and memory complexity of the back-propagation (BP) algorithm.} More precisely, although the architecture modification~\cite{cai2020tinytl} and layer freezing~\cite{lin2022device, lee2022surgical} can help skipping the BP for some layers, for other layers, the complexity remains high. Gradient quantization~\cite{chen2020statistical, banner2018scalable} can reduce the cost of arithmetic operations but cannot reduce the number of operations (\textit{e.g.}, multiplications); thus, the speedup in training remains limited. Moreover, gradient quantization is not supported by existing deep-learning frameworks (e.g., CUDNN~\cite{chetlur2014cudnn}, MKLDNN~\cite{onednn}, PyTorch~\cite{pytorch} and Tensorflow~\cite{tensorflow2015-whitepaper}). To enable on-device training, there are two important questions must be addressed: \begin{itemize} \item \textit{How can we reduce the computational complexity of back propagation through the convolution layers?} \item \textit{How can we reduce the data required by the gradient computation during back propagation?} \end{itemize} In this paper, we propose \textit{gradient filtering}, a new research direction, to address both questions. By addressing the first question, we reduce the computational complexity of training; by addressing the second question, we reduce the memory consumption. In general, the gradient propagation through a convolution layer involves multiplying the gradient of the output variable with a Jacobian matrix constructed with data from either the input feature map or the convolution kernel. We aim at simplifying this process with the new gradient filtering approach proposed in Section \ref{sec:method}. Intuitively, if the gradient map w.r.t. the output has the same value for all entries, then the computation-intensive matrix multiplication can be greatly simplified, and the data required to construct the Jacobian matrix can be significantly reduced. Thus, our gradient filtering can approximate the gradient w.r.t. the output by creating a new gradient map with a special (\textit{i.e.}, spatial) structure and fewer unique elements. By doing so, the gradient propagation through the convolution layers reduces to cheaper operations, while the data required (hence memory) for the forward propagation also lessens. Through this filtering process, we trade off the gradient precision against the computation complexity during BP. We note that gradient filtering does not necessarily lead to a worse precision, \textit{i.e.}, models sometimes perform better with filtered gradients when compared against models trained with vanilla BP. In summary, our contributions are as follows: \begin{itemize} \item We propose \textit{gradient filtering}, which reduces the computation and memory required for BP by more than two orders of magnitude compared to the exact gradient calculation. \item We provide a rigorous error analysis which shows that the errors introduced by the gradient filtering have only a limited influence on model accuracy. \item Our experiments with multiple DNN models and computer vision tasks show that we can train a neural network with significantly less computation and memory costs, with only a marginal accuracy loss compared to baseline methods. Side-by-side comparisons against other training acceleration techniques also suggest the effectiveness of our method. \item Our method is easy to deploy with highly optimized deep learning frameworks (\textit{e.g.}, MKLDNN~\cite{onednn} and CUDNN~\cite{chetlur2014cudnn}). Evaluations on resource-constrained edge (Raspberry Pi and Jetson Nano) and high-performance devices (CPU/GPU) show that our method is highly suitable for real life deployment. \end{itemize} The paper is organized as follows. Section \ref{sec:related_work} reviews relevant work. Section \ref{sec:method} presents our method in detail. Section \ref{sec:analysis} discusses error analysis, computation and memory consumption. Experimental results are presented in Section \ref{sec:exp}. Finally, Section \ref{sec:conclusion} summarizes our main contributions. \section{Related Work} \label{sec:related_work} \noindent\textbf{Architecture Modification:} Authors of \cite{cai2020tinytl} propose to attach small branches to the original neural network. During training, the attached branches and biases in the original model are updated. Though memory consumption is reduced, updating these branches still needs gradient propagation through the entire network; moreover, a large computational overhead for inference is introduced. \noindent\textbf{Layer Freezing:} Authors of \cite{lee2022surgical, lin2022device} propose to only train parts of the model. \cite{lee2022surgical} makes layer selection based on layer importance metrics, while \cite{lin2022device} uses evolutionary search. However, the layers selected by all these methods are typically computationally heavy layers (\textit{e.g.}, the last few layers in ResNet~\cite{he2016resnet}) which consume most of the resources. Thus, the speedup achieved by these approaches is limited. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{figs/matrix.pdf} \caption{Matrix of orthogonal directions for on-device training. ``Arch'' is short for ``architecture''. Our approach opens up a new direction of research for on-device training for EdgeAI.} \label{fig:method_matrix} \end{figure} \noindent\textbf{Gradient Quantization:} \cite{alistarh2016qsgd, bernstein2018signsgd} quantize gradient after back-propagation, which means these methods cannot accelerate the training on a single device. Work in \cite{chen2020statistical, banner2018scalable, sun2020ultra, hubara2017quantized, zhao2021distribution, hong2022efficient, wang2019e2} accelerates training by reducing the cost for every arithmetic operation. However, these methods do not reduce the number of operations, which is typically huge for SOTA CNNs, so their achievable speedup is limited. Also, all these methods are not supported by the popular deep learning frameworks~\cite{chetlur2014cudnn, onednn, tensorflow2015-whitepaper, pytorch}. In contrast to the prior work, our method opens up a new research direction. More precisely, we reduce the number of computations and memory consumption required for training a single layer via gradient filtering. Thus, our method can be combined with any of the methods mentioned above. For example, in Section \ref{sec:supp_gq} in the Supplementary, we illustrate how our method can work together with the gradient quantization methods to enable a higher speedup. \begin{figure*}[tp] \centering \includegraphics[width=\textwidth]{figs/method.pdf} \caption{(a) Computation procedures for vanilla training method (upper) and our method (lower). (b) Example of gradient propagation with gradient filtering. Numbers in this example are chosen randomly for illustration purposes. In this case, the patch size selected for the gradient filter is $2\times 2$. Thus, the $4\times 4$ gradient map $g_y$ is approximated by $\tilde g_y$, which has four $2\times 2$ patches with one unique value for each patch. Also, input feature map $x$ and mirrored convolution kernel $\theta'$ are spatial summed to $\tilde x$ and $\tilde \theta'$. Since $\tilde x$ has fewer unique values than $x$, memory consumption is reduced. Finally, with $\tilde g_y, \tilde x$ and $\tilde \theta$, we compute the gradient w.r.t. kernel and input feature map with much fewer operations than the standard back propagation method.} \label{fig:method} \end{figure*} \begin{table}[b] \centering \begin{tabular}{c|c} \toprule $C_x$ & Number of channels of $x$ \\ \hline $W_x, H_x$ & Width and height of $x$ \\ \hline $\theta$ & Convolution kernel \\ \hline $\theta'$ & Rotated $\theta$, \textit{i.e.}, $\theta'=\text{rot180}(\theta)$ \\ \hline $r$ & Patch size ($r \times r$ ) \\ \hline $g_x, g_y, g_\theta$ & Gradients w.r.t. $x, y, \theta$\\ \hline $\tilde g_y$ & Approximated gradient $g_y$ \\ \hline \multirow{2}{*}{$\tilde x, \tilde \theta'$} & Sum of $x$ and $\theta'$ over \\ ~ & spatial dimensions (height and width) \\ \hline \multirow{2}{*}{$x[n, c_i, h, w]$} & Element for feature map $x$ \\ ~ & at batch $n$, channel $c_i$, pixel $(h, w)$\\ \hline \multirow{3}{*}{$\theta[c_o, c_i, u, v]$} & Element for convolution kernel $\theta$ \\ ~ & at output channel $c_o$, input channel $c_i$,\\ ~ & position $(u, v)$\\ \bottomrule \end{tabular} \caption{Table of symbols we use.} \label{tab:symbols} \end{table} \section{Proposed Method} \label{sec:method} In this section, we introduce our gradient filtering approach to accelerate BP. To this end, we target the most computation and memory heavy operation, \textit{i.e.}, convolution (Figure \ref{fig:method}(a)). Table \ref{tab:symbols} lists some symbols we use. \subsection{Problem Setup} The computations for both forward and backward paths are shown in Figure~\ref{fig:method}(a). For the standard (vanilla) approach (upper Figure~\ref{fig:method}(a)), starting with input $x$, the forward propagation convolves the input feature map $x$ with kernel $\theta$ and returns output $y$, which is further processed by the other layers in the neural network (dotted arrow) until the loss value $l$ is calculated. As shown in Figure \ref{fig:method}(a), the BP of the convolution layer starts with the gradient map w.r.t. output $y$ ($g_y$). The gradient w.r.t. input ($g_x$) is calculated by convolving $g_y$ with the \textit{rotated} convolution kernel $\theta'$, \textit{i.e.}, $g_x=g_y\circledast \text{rot180}(\theta)=g_y\circledast \theta'$. The gradient w.r.t. convolution kernel, namely $g_\theta$, is calculated with the Frobenius inner product \cite{horn2012matrix} between $x$ and $g_y$, \textit{i.e.}, $g_\theta=g_y\text{ \scriptsize\circled{F} }x$. The lower half of Figure~\ref{fig:method}(a) shows our method, where several changes are made: We introduce the gradient filter ``{\scriptsize\circled{A}}'' after $g_y$ to generate the approximate gradient for BP. Also, instead of using the accurate $x$ and $\theta'$ values for gradient computation, we sum over spatial dimensions (height and width dimensions), \textit{i.e.}, $\tilde{x}$ and $\tilde{\theta}'$, respectively. Finally, the convolution layer now multiplies the approximate gradient $\tilde g_y$ with spatial kernel $\tilde\theta'$ instead of convolving with it to calculate $\tilde g_x$. Figure~\ref{fig:method}(b) shows an example of gradient propagation with our gradient filter. \subsection{Preliminary Analysis} \label{sec:preliminary_analysis} Consider the vanilla BP for convolution in Figure \ref{fig:method}(a). Equation~\eqref{equ:vanilla_comp} shows the number of computations (\#FLOPs) required to calculate $g_x$ given $g_y$: \begin{equation} \text{\#FLOPs} = 2C_{x}C_{y} \cdot W_y H_y \cdot W_{\theta}H_{\theta} \label{equ:vanilla_comp} \end{equation} The computation requirements in Equation \eqref{equ:vanilla_comp} belong to three categories: number of channels, number of \textit{unique elements} per channel in the gradient map, and \textit{kernel size}. Our method focuses on the last two categories. \textbf{i. Unique elements:} $(W_y H_y)$ represents the number of unique elements per channel in the gradient w.r.t. output variable $y$ ($g_y$). Given the high-resolution images we use, this term is huge, so if we manage to reduce the number of unique elements in the spatial dimensions (height and width), the computations required are greatly reduced too. \textbf{ii. Kernel size:} $(W_{\theta}H_{\theta})$ represents the number of unique elements in the convolution kernel. If the gradient $g_y$ has some special structure, for example $g_y=1_{H_y\times W_y}\cdot v$ (\textit{i.e.}, every element in $g_y$ has the same value $v$), then the convolution can be simplified to $(\sum\theta')v1_{H_y\times W_y}$ (with boundary elements ignored). With such a special structure, only one multiplication and $(W_{\theta}H_{\theta}-1)$ additions are required. Moreover, $\sum\theta'$ is independent of data so the result can be shared across multiple images until $\theta$ gets updated. \subsection{Gradient Filtering} \label{sec:gf} To reduce the number of unique elements and create the special structure in the gradient map, we apply the gradient filter after the gradient w.r.t. output ($g_y$) is provided. During the backward propagation, the gradient filter {\scriptsize\circled{A}} \textit{approximates} the gradient $g_y$ by spatially cutting the gradient map into $r\times r$-pixel patches and then replacing all elements in each patch with their average value (Figure \ref{fig:method}(b)): \begin{equation} \tilde{g}_y[n, c_o, h, w] = \frac{1}{r^2}\sum_{i=\lfloor h/r\rfloor r}^{\lceil h/r\rceil r}\sum_{j=\lfloor w/r\rfloor r}^{\lceil w/r\rceil r} g_y[n, c_o, i, j] \end{equation} For instance in Figure \ref{fig:method}(b), we replace the 16 distinct values in the gradient map $g_y$ with 4 average values in $\tilde g_y$. So given a gradient map $g_y$ with $N$ images per batch, $C$ channels, and $H\times W$ pixels per channel, the gradient filter returns a structured approximation of the gradient map containing only $N\times C \times \lceil \frac{H}{r}\rceil \times \lceil\frac{W}{r}\rceil$ blocks, with \textit{one unique value per patch}. We use this matrix of unique values to represent the approximate gradient map $\tilde g_y$, as shown in Figure \ref{fig:method}(b). \subsection{Back Propagation with Gradient Filtering} We describe now the computation procedure used after applying the gradient filter. Detailed derivations are provided in Supplementary Section \ref{sec:supp_gf}. \noindent\textbf{Gradient w.r.t. input:} The gradient w.r.t. input is calculated by convolving $\theta'$ with $g_y$ (Figure \ref{fig:method}(a)). With the approximate gradient $\tilde g_y$, this convolution simplifies to: \begin{equation} \tilde g_{x}[n, c_i, h, w] = \sum_{c_o}\tilde g_{y}[n, c_o, h, w] \odot \tilde\theta'[c_o, c_i] \label{equ:grad_x} \end{equation} where $\tilde\theta'[c_o, c_i]=\sum_{u, v}\theta'[c_o, c_i, u, v]$ is the spatial sum of convolution kernel $\theta$, as shown in Figure~\ref{fig:method}(b). \noindent\textbf{Gradient w.r.t. kernel:} The gradient w.r.t. the kernel is calculated by taking the Frobenius inner product between $x$ and $g_y$, \textit{i.e.}, $g_\theta[c_o, c_i, u, v] = x \text{ \scriptsize\circled{F} } g_y$, namely: \begin{equation} g_\theta[c_o, c_i, u, v] = \sum_{n, i, j} x[n, c_i, i+u, j+v] g_y[n, c_o, i, j] \end{equation} With the approximate gradient $\tilde g_y$, the operation can be simplified to: \begin{equation} \tilde g_{\theta}[c_o, c_i, u, v] = \sum_{n,i,j}\tilde x[n, c_i, i, j] \tilde g_{y}[n, c_o, i, j] \label{equ:grad_w} \end{equation} with $\tilde x[n, c_i, i, j] = \sum_{h=\lfloor i/r\rfloor r}^{\lceil i/r\rceil r}\sum_{w=\lfloor j/r\rfloor r}^{\lceil j/r\rceil r} x[n, c_i, h, w]$. As shown in Figure \ref{fig:method}(b), $\tilde x[n, c_i, i, j]$ is the spatial sum of $x$ elements in the same patch containing pixel $(i, j)$. \section{Analyses of Proposed Approach} \label{sec:analysis} In this section, we analyze our method from three perspectives: gradient filtering approximation error, computation reduction, and memory cost reduction. \subsection{Error Analysis of Gradient Filtering} \label{sec:err_analysis} We prove that the approximation error introduced by our gradient filtering is bounded during the gradient propagation. Without losing generality, we consider that all variables have only one channel, \textit{i.e.}, $C_{x_0}=C_{x_1}=1$. \noindent\textbf{Proposition 1:} \textit{For any input-output channel pair $(c_o, c_i)$ in the convolution kernel $\theta$, assuming the DC component has the largest energy value compared to all components in the spectrum\footnote{As a reminder, the energy of a signal is the sum of energy of the DC component and the energy of its AC components.}, then the signal-to-noise-ratio (SNR) of $\tilde g_x$ is greater than SNR of $\tilde g_y$.} \noindent\textbf{Proof:} We use $G_x, G_y$ and $\Theta$ to denote the gradients $g_x, g_y$ and the convolution kernel $\theta$ in the \textit{frequency domain}; $G_x[u,v]$ is the spectrum value at frequency $(u,v)$ and $\delta$ is the 2D discrete Dirichlet function. To simplify the discussion, we consider only one patch of size $r \times r$. The gradient returned by the gradient filtering can be written as: \begin{equation} \tilde g_y = \frac{1}{r^2} 1_{r\times r} \circledast g_y \label{equ:gf_spatial} \end{equation} where $\circledast$ denotes convolution. By applying the discrete Fourier transformation, Equation \eqref{equ:gf_spatial} can be rewritten in the frequency domain as: \begin{equation} \tilde G_y[u,v] = \frac{1}{r^2}\delta[u, v]G_y[u, v] \label{equ:gf_freq} \end{equation} $\tilde g_y$ is the approximation of $g_y$ (\textit{i.e.}, the ground truth for $\tilde g_y$ is $g_y$), and the SNR of $\tilde g_y$ equals to: \begin{equation} \begin{aligned} \text{SNR}_{\tilde g_y} &= \frac{\sum_{(u,v)}G_y^2[u,v]}{\sum_{(u,v)} (G_y[u,v]-\frac{1}{r^2}\delta[u,v]G_y[u,v])^2} \\ &= (1 - \frac{2r^2-1}{r^4}\frac{G_y^2[0,0]}{\sum_{(u,v)}G_y^2[u,v]})^{-1} \\ \end{aligned} \label{equ:snr_g3} \end{equation} For the convolution layer, the gradient w.r.t. the approximate variable $\tilde x$ in the frequency domain is\footnote{Because $g_y$ is convolved with the \textbf{rotated} kernel $\theta'$, in the frequency domain, we use $\Theta[-u,-v]$ instead of $\Theta[u,v]$.}: \begin{equation} \begin{aligned} \tilde G_{x}[u,v] &= \Theta[-u,-v]\tilde G_{y}[u,v]\\ &=\frac{1}{r^2}\Theta[-u,-v]\delta[u,v]G_y[u,v] \end{aligned} \end{equation} and its ground truth is: \begin{equation} G_{x}[u,v] = \Theta[-u,-v]G_y[u,v] \end{equation} Similar to Equation \eqref{equ:snr_g3}, the SNR of $g_{\tilde x}$ is: \begin{equation} \begin{aligned} \text{SNR}_{\tilde g_x} = (1 - \frac{2r^2-1}{r^4}\frac{(\Theta[0,0]G_y[0,0])^2}{\sum_{(u,v)}{(\Theta[u,v]G_y[u,v])^2}})^{-1} \end{aligned} \label{equ:snr_approx_gx} \end{equation} Equation~\eqref{equ:snr_approx_gx} can be rewritten as: \begin{equation}\label{equ:raw_prop} \begin{aligned} \frac{r^4(1-\text{SNR}^{-1}_{\tilde g_x})}{2r^2-1} &=\frac{(\Theta[0,0]G_y[0,0])^2}{\sum_{(u,v)} (\Theta[-u,-v]G_y[u,v])^2}\\ &= \frac{G_y^2[0,0]}{\sum_{(u,v)} (\frac{\Theta[-u,-v]}{\Theta[0, 0]}G_y[u,v])^2} \end{aligned} \end{equation} Furthermore, the main assumption (\textit{i.e.}, the DC component dominates the frequency spectrum of $\Theta$) can be written as: \begin{equation}\label{equ:dcac_ratio} \Theta^2[0,0]/\text{max}_{(u,v)\ne(0,0)} \Theta^2[u,v] \ge 1 \end{equation} that is, $\forall (u, v), \frac{\Theta^2[-u,-v]}{\Theta^2[0,0]} \le 1$; thus, by combining Equation~\eqref{equ:raw_prop} and Equation~\eqref{equ:dcac_ratio}, we have: \begin{equation} \begin{aligned} \frac{G_y^2[0,0]}{\sum_{(u,v)} (\frac{\Theta[-u,-v]}{\Theta[0, 0]}G_y[u,v])^2} &\ge \frac{G_y^2[0,0]}{\sum_{(u,v)} (G_y[u,v])^2} \\ \Leftrightarrow \frac{r^4(1-\text{SNR}^{-1}_{\tilde g_x})}{2r^2-1} &\ge \frac{r^4(1-\text{SNR}^{-1}_{\tilde g_y})}{2r^2-1} \end{aligned} \end{equation} which means that: $\text{SNR}_{\tilde g_x} \ge \text{SNR}_{\tilde g_y}$. This completes our proof for error analysis. $\blacksquare$ In conclusion, as the gradient propagates through the network, the noise introduced by our gradient filter becomes weaker compared to the real gradient signal. This property ensures that the error in gradient has only a limited influence on the quality of BP. We validate Proposition 1 later in the experimental section. \subsection{Computation and Overhead Analysis} \label{sec:comp_analysis} \begin{figure} \centering \includegraphics[width=0.85\linewidth]{figs/comp_approx.pdf} \caption{Computation analysis for a specific convolution layer\protect\footnotemark. Minimum achievable computation is given in Equation \eqref{equ:lb_comp}. By reducing the number of unique elements, computations required by our approach drop to about $1/r^2$ compared with the standard BP method. By combining it with structured gradient map, computations required by our approach drop further, getting very close to the theoretical limit.} \label{fig:comp_approx} \end{figure} \footnotetext{The layer is from U-Net~\cite{ronneberger2015unet}. The size of the input is assumed to be $120\times 160$ pixels with 192 channels; the output has the same resolution, but with only 64 channels. The kernel size of the convolution layer is $3\times3$. Analysis for ResNet is included in the supplementary material.} In this section, we analyse the computation required to compute $g_x$, the gradient w.r.t. input $x$. Figure \ref{fig:comp_approx} compares the computation required to propagate the gradient through this convolution layer under different patch sizes $r\times r$. A patch size $1\times 1$ means the vanilla BP algorithm which we use as the baseline. As discussed in the preliminary analysis section (Section \ref{sec:preliminary_analysis}), two terms contribute to the computation savings: fewer unique elements in the gradient map and the structured gradient map. \noindent\textbf{Fewer unique elements:} In vanilla BP, there are $H_y W_y$ unique elements in the gradient map. After applying gradient filtering with a patch size $r\times r$, the number of unique elements reduces to only $\lceil \frac{H_y}{r} \rceil \lceil\frac{W_y}{r}\rceil$. This reduction contributes the most to the savings in computation (orange line in Figure \ref{fig:comp_approx}). \begin{table*}[htbp] \centering \resizebox{\textwidth}{!}{% \begin{tabular}{cc|ccc|cc|ccc} \toprule \textbf{MobileNetV2}\cite{sandler2018mobilenetv2} & \textbf{\#Layers} & \textbf{Accuracy} & \textbf{FLOPs} & \textbf{Mem} & \textbf{ResNet-18}\cite{he2016resnet} & \textbf{\#Layers} & \textbf{Accuracy} & \textbf{FLOPs} & \textbf{Mem} \\ \hline No Finetuning & 0 & 4.2 & 0 & 0 & No Finetuning & 0 & 4.7 & 0 & 0 \\ \hline \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 75.1 & 1.13G & 24.33MB & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 73.1 & 5.42G & 8.33MB \\ \cdashline{2-5}\cdashline{7-10} & 2 & 63.1 & 113.68M & 245.00KB & & 2 & 70.4 & 489.20M & 196.00KB \\ & 4 & 62.2 & 160.00M & 459.38KB & & 4 & 72.3 & 1.14G & 490.00KB \\ \hline TinyTL~\cite{cai2020tinytl} & N/A & 60.2 & 663.51M & 683.00KB & TinyTL~\cite{cai2020tinytl} & N/A & 69.2 & 3.88G & 1.76MB \\ \hline \multirow{2}{*}{\textbf{Ours}} & 2 & 63.1 & 39.27M & 80.00KB & \multirow{2}{*}{\textbf{Ours}} & 2 & 68.6 & 28.32M & 64.00KB \\ & 4 & 63.4 & 53.96M & 150.00KB & & 4 & 68.5 & 61.53M & 112.00KB \\ \hline \textbf{MCUNet}\cite{lin2020mcunet} & \textbf{\#Layers} & \textbf{Accuracy} & \textbf{FLOPs} & \textbf{Mem} & \textbf{ResNet-34}\cite{he2016resnet} & \textbf{\#Layers} & \textbf{Accuracy} & \textbf{FLOPs} & \textbf{Mem} \\ \hline No Finetune & 0 & 4.1 & 0 & 0 & No Finetune & 0 & & 0 & 0 \\ \hline \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 68.5 & 231.67M & 9.17MB & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 70.8 & 11.17G & 13.11MB \\ \cdashline{2-5}\cdashline{7-10} & 2 & 62.1 & 18.80M & 220.50KB & & 2 & 69.6 & 489.20M & 196.00KB \\ & 4 & 64.9 & 33.71M & 312.38KB & & 4 & 72.3 & 1.21G & 392.00KB \\ \hline TinyTL~\cite{cai2020tinytl} & N/A & 53.1 & 148.01M & 571.5KB & TinyTL~\cite{cai2020tinytl} & N/A & 72.9 & 8.03G & 2.95MB \\ \hline \multirow{2}{*}{\textbf{Ours}} & 2 & 61.8 & 6.34M & 72.00KB & \multirow{2}{*}{\textbf{Ours}} & 2 & 68.6 & 28.32M & 64.00KB \\ & 4 & 64.4 & 11.01M & 102.00KB & & 4 & 70.6 & 64.07M & 128.00KB \\ \bottomrule \end{tabular}% } \caption{Experimental results for ImageNet classification with four neural networks (MobileNet-V2, ResNet18/34, MCUNet). ``\#Layers'' is short for ``the number of \textit{active} convolutional layers''. For example, \#Layers equals to 2 means that only the last two convolutional layers are trained. For memory consumption, we only consider the memory for input feature $x$. Strategy ``No Finetuning'' shows the accuracy on new datasets without finetuning the pretrained model. Since TinyTL~\cite{cai2020tinytl} changes the architecture, ``\#Layers'' is not applicable (N/A).} \label{tab:imagenet} \end{table*} \begin{table*}[] \centering \resizebox{\textwidth}{!}{% \begin{tabular}{cc|ccc|cc|ccc|cc|ccc} \toprule \textbf{PSPNet\cite{zhao2017pyramid}} & \textbf{\#Layers} & \textbf{GFLOPs} & \textbf{mIoU} & \textbf{mAcc} & \textbf{PSPNet-M\cite{zhao2017pyramid}} & \textbf{\#Layers} & \textbf{GFLOPs} & \textbf{mIoU} & \textbf{mAcc} & \textbf{FCN}\cite{long2015fully} & \textbf{\#Layers} & \textbf{GFLOPs} & \textbf{mIoU} & \textbf{mAcc} \\ \hline Calibration & 0 & 0 & 12.86 & 19.74 & Calibration & 0 & 0 & 14.20 & 20.46 & Calibration & 0 & 0 & 10.95 & 15.69 \\ \hline \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 166.5 & 55.01 & 68.02 & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 42.4 & 48.48 & 61.48 & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 170.3 & 45.22 & 58.80 \\ \cdashline{2-5}\cdashline{7-10}\cdashline{12-15} & 5 & 15.0 & 39.54 & 51.86 & & 5 & 12.22 & 36.35 & 47.09 & & 5 & 59.5 & 27.41 & 37.90 \\ & 10 & 110.6 & 53.15 & 67.10 & & 10 & 22.46 & 46.01 & 58.70 & & 10 & 100.9 & 43.87 & 57.58 \\ \hline \multirow{2}{*}{\textbf{Ours}} & 5 & 0.14 & 39.34 & 51.86 & \multirow{2}{*}{\textbf{Ours}} & 5 & 0.11 & 36.14 & 46.86 & \multirow{2}{*}{\textbf{Ours}} & 5 & 0.58 & 27.42 & 37.88 \\ & 10 & 0.79 & 50.88 & 64.73 & & 10 & 0.76 & 44.90 & 57.50 & & 10 & 0.96 & 36.30 & 48.82 \\ \hline \textbf{DLV3}\cite{chen2017rethinking} & \textbf{\#Layers} & \textbf{GFLOPs} & \textbf{mIoU} & \textbf{mAcc} & \textbf{DLV3-M}\cite{chen2017rethinking} & \textbf{\#Layers} & \textbf{GFLOPs} & \textbf{mIoU} & \textbf{mAcc} & \textbf{UPerNet}\cite{xiao2018unified} & \textbf{\#Layers} & \textbf{GFLOPs} & \textbf{mIoU} & \textbf{mAcc} \\ \hline Calibration & 0 & 0 & 13.95 & 20.62 & Calibration & 0 & 0 & 21.96 & 36.15 & Calibration & 0 & 0 & 14.71 & 21.82 \\ \hline \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 151.2 & 58.32 & 71.72 & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 54.4 & 55.66 & 68.95 & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Vanilla\\ BP\end{tabular}} & All & 541.0 & 64.88 & 77.13 \\ \cdashline{2-5}\cdashline{7-10}\cdashline{12-15} & 5 & 18.0 & 40.85 & 53.16 & & 5 & 14.8 & 38.21 & 49.35 & & 5 & 503.9 & 47.93 & 61.67 \\ & 10 & 102.0 & 54.65 & 68.64 & & 10 & 33.1 & 47.95 & 61.49 & & 10 & 507.6 & 48.83 & 63.02 \\ \hline \multirow{2}{*}{\textbf{Ours}} & 5 & 0.31 & 33.09 & 44.33 & \multirow{2}{*}{\textbf{Ours}} & 5 & 0.26 & 35.47 & 46.35 & \multirow{2}{*}{\textbf{Ours}} & 5 & 1.97 & 47.04 & 60.44 \\ & 10 & 2.96 & 47.11 & 60.28 & & 10 & 1.40 & 45.53 & 58.99 & & 10 & 2.22 & 48.00 & 62.07 \\ \bottomrule \end{tabular}% } \caption{Experimental results for semantic segmentation task on augmented Pascal VOC12 dataset~\cite{chen2017rethinking}. Model name with postfix ``M'' means the model uses MobileNetV2 as backbone, otherwise ResNet18 is used. ``\#Layers'' is short for ``the number of \textit{active} convolutional layers'' that are trained. All models are pretrained on Cityscapes dataset~\cite{Cordts2016Cityscapes}. Strategy ``Calibration'' shows the accuracy when only the classifier and normalization statistics are updated to adapt different numbers of classes between augmented Pascal VOC12 and Cityscapes.} \label{tab:seg} \end{table*} \noindent\textbf{Structured Gradient Map:} By creating the structured gradient map, the convolution over the gradient map $\tilde g_y$ is simplified to the element-wise multiplication and channel-wise addition. Computation is thus reduced to $(H_{\theta}W_{\theta})^{-1}$ of its original value. For instance, the example convolution layer in Figure \ref{fig:comp_approx} uses a $3\times3$ convolution kernel so around $89\%$ computations are removed. The blue line in Figure \ref{fig:comp_approx} shows the \#FLOPs after combining both methods. Greater reduction is expected when applying our method with larger convolution kernels. For instance, FastDepth~\cite{wofk2019fastdepth} uses $5\times5$ convolution kernel so as much as $96\%$ reduction in computation can be achieved, in principle. \noindent\textbf{Minimum Achievable Computation:} With the two reductions mentioned above, the computation required to propagate the gradient through the convolution layer is: \begin{equation} \text{\#FLOPs}(r) = \lceil \frac{H_{y}}{r} \rceil \lceil\frac{W_{y}}{r}\rceil C_{x}(2C_{y}-1) + o(H_{y}W_{y}) \label{equ:min_comp} \end{equation} where $o(H_{y}W_{y})$ is a constant term which is independent of $r$ and negligible compared to $H_y W_y$. When the patch is as large as the feature map, our method reaches the minimum achievable computation (blue dashed line in Figure \ref{fig:comp_approx}): \begin{equation} \text{min}_r~\text{\#FLOPs}(r)=2C_{x}C_{y} - C_{x} + o(H_{y}W_{y}) \label{equ:lb_comp} \end{equation} In this case, each channel of the gradient map is represented with a single value, so the computation is controlled by the number of input and output channels. \noindent\textbf{Overhead:} The overhead of our approach comes from approximating the feature map $x$, gradient $g_y$, and kernel $\theta$. As the lower part of Figure \ref{fig:method}(a) shows, the approximation for $x$ is considered as part of forward propagation, while the other two as back propagation. Indeed, with the patch size $r$, the ratio of forward propagation overhead is about $1/(2C_o W_\theta H_\theta)$, while the ratio of backward propagation overhead is about $(r^2-1)/(2C_x)$. Given the large number of channels and spatial dimensions in typical neural networks, both overhead values take less than 1\% computation in the U-Net example above. \subsection{Memory Analysis} As Figure \ref{fig:method}(a) shows, the standard back propagation for a convolution layer relies on the input feature map $x$, which needs to be stored in memory during forward propagation. Since every convolution layer requiring gradient for its kernel needs to save a copy of feature map $x$, the memory consumption for storing $x$ is huge. With our method, we simplify the feature map $x$ to approximated $\tilde x$, which has only $\lceil\frac{H_x}{r}\rceil\lceil\frac{W_x}{r}\rceil$ unique elements for every channel. Thus, by saving only these unique values, our method achieves around $(1 - \frac{1}{r^2})$ memory savings, overall. \section{Experiments} \label{sec:exp} In this section, we first present our experimental results on ImageNet classification~\cite{imagenet_cvpr09} and semantic segmentation. Then, we study the impact of different hyper-parameter selections. Furthermore, we present the evaluation result running our method on real hardware. Lastly, we empirically validate the assumption in Section \ref{sec:err_analysis}. \subsection{Experimental Setup} \label{sec:setup} \noindent\textbf{Classification:} Following \cite{mcmahan2017communication}, we split every dataset into two highly non-i.i.d. partitions with the same size. Then, we pretrain our models on the first partition with a vanilla training strategy, and finetune the model on the other partition with different configurations for the training strategy (\textit{i.e.}, with/without gradient filtering, hyper-parameters, number of convolution layers to be trained). More details (\textit{e.g.}, hyper-parameters) are in the Supplementary. \noindent\textbf{Segmentation:} Models are pretrained on Cityscapes~\cite{Cordts2016Cityscapes} by MMSegmentation~\cite{mmseg2020}. Then, we calibrate and finetune these models with different training strategies on the augmented Pascal-VOC12 dataset following \cite{chen2017rethinking}, which is the combination of Pascal-VOC12~\cite{voc12} and SBD~\cite{sbd}. More details are included in the supplementary material. \noindent\textbf{On-device Performance Evaluation:} For CPU performance evaluation, we implement our method with MKLDNN~\cite{onednn} (a.k.a. OneDNN) v2.6.0 and compare it with the convolution BP method provided by MKLDNN. We test on three CPUs, namely Intel 11900KF, Quad-core Cortex-A72 (Jetson Nano) and Quad-core Cortex-A53 (Raspberry Pi 3b). For GPU performance evaluation, we implement our method on CUDNN v8.2 \cite{chetlur2014cudnn} and compare with the BP method provided by CUDNN. We test on two GPUs, RTX 3090Ti and the edge GPU on Jetson Nano. Since both MKLDNN and CUDNN only support float32 BP, we test float32 BP only. Additionally, for the experiments on Jetson Nano, we record the energy consumption for CPU and GPU with the embedded power meter. More details (\textit{e.g.}, frequency) are included in the supplementary material. \subsection{ImageNet Classification} Table \ref{tab:imagenet} shows our evaluation results on the ImageNet classification task. As shown, our method significantly reduces the FLOPs and memory required for BP, with very little accuracy loss. For example, for ResNet34, our method achieves 18.9$\times$ speedup with 1.7\% accuracy loss when training four layers; for MobileNetV2, we get a 1.2\% better accuracy with 3.0$\times$ speedup and 3.1$\times$ memory savings. These results illustrate the effectiveness of our method. On most networks, TinyTL has a lower accuracy while consuming more resources compared to the baselines methods. \subsection{Semantic Segmentation} \label{sec:exp_seg} Table~\ref{tab:seg} shows our evaluation results on the augmented Pascal-VOC12 dataset. On a wide range of networks, our method constantly achieves significant speedup with marginal accuracy loss. For the large network UPerNet, our method achieves 229$\times$ speedup with only 1\% mIoU loss. For the small network PSPNet, our method speedups training by 140$\times$ with only 2.27\% mIoU loss. This shows the effectiveness of our method on a dense prediction task. \subsection{Hyper-Parameter Selection} \label{sec:exp_hyper} Figure \ref{fig:hyper_parameter} shows our experimental results for ResNets under different hyper-parameter selection, \textit{i.e.} number of convolution layers and patch size of gradient filter $r\times r$. Of note, the y-axis (MFLOPs) in Figure \ref{fig:hyper_parameter} is log scale. More results are included in Supplementary Section~\ref{sec:supp_hyper}. We highlight three qualitative findings in Figure \ref{fig:hyper_parameter}: \begin{figure}[htbp] \centering \includegraphics[width=\linewidth]{figs/acc_flops.pdf} \caption{Computation (\#MFLOPs, log scale) and model accuracy [\%] under different hyper-parameter selection. ``Baseline'' means vanilla BP; ``Ours-R2/4'' uses gradient filtering with patch size $2\times 2$/$4\times 4$ during BP.} \label{fig:hyper_parameter} \end{figure} \begin{enumerate} \item[\textbf{a.}] For a similar accuracy, our method greatly reduces the number of operations (1 to 2 orders of magnitude), while for a similar number of computations, our method achieves a higher accuracy (2\% to 5\% better). \end{enumerate} This finding proves the effectiveness of our method. \begin{enumerate} \item[\textbf{b.}] Given the number of convolution layers to be trained, the more accurate method returns a better accuracy. Baseline (\textit{i.e.}, standard BP) uses the most accurate gradient, Ours-R4 (BP with gradient filter with patch size $4\times4$) uses the least accurate gradient; thus, Baseline $>$ Ours-R2 $>$ Ours-R4. \end{enumerate} This finding is intuitive since the more accurate method should introduce smaller noise to the BP, \textit{e.g.}, the gradient filtering with patch size $2\times 2$ (Ours-R2) introduces less noise than with patch size $4\times 4$ (Ours-R4). In Figure~\ref{fig:snr}, we evaluate the relationship between accuracy and noise level introduced by gradient filtering. With a higher SNR (\textit{i.e.}, a lower noise level), a better accuracy is achieved. \begin{figure}[h] \centering \includegraphics[width=0.9\linewidth]{figs/acc_snr.pdf} \caption{Relationship between accuracy and noise level introduced by the gradient filtering. As shown, accuracy increases as the SNR increases, \textit{i.e.}, noise level decreases.} \label{fig:snr} \end{figure} \begin{enumerate} \item[\textbf{c.}] Given the number of computations, the less accurate method returns the better accuracy by training more layers, \textit{i.e.}, Ours-R4 $>$ Ours-R2 $>$ baseline. \end{enumerate} This finding suggests that for neural network training with relatively low computational resources, training more layers with less accurate gradients is preferable than training fewer layers with more accurate gradients. \begin{figure*}[htbp] \centering \includegraphics[width=\linewidth]{figs/speedup_mem.pdf} \caption{Speedup and normalized memory consumption results on multiple CPUs and GPUs under different test cases (\textit{i.e.} different input sizes, numbers of channels, etc.) Detailed configuration of these test cases are included in the supplementary material. ``R2'', ``R4'' mean using gradient filtering with $2\times 2$ and $4\times 4$ patch sizes, respectively. Our method achieves significant speedup with low memory consumption compared to all baseline methods. For example, on Jetson CPU with patch size $4\times 4$ (``Jetson-R4'' in left top figure), our method achieves 114$\times$ speedup with only 33\% memory consumption for most test cases.} \label{fig:speedup_mem} \end{figure*} \subsection{On-device Performance Evaluation} \label{sec:exp_ondevice} Figure \ref{fig:speedup_mem} and Table \ref{tab:energy} show our evaluation results on real devices. More results are included in the Supplementary Section~\ref{sec:supp_ondevice}. As Figure~\ref{fig:speedup_mem} shows, on CPU, most convolution layers achieve speedups over 20$\times$ with less than 50\% memory consumption for gradient filtering with patch sizes $2\times 2$; for gradient filtering with patch size $4\times 4$, the speedups are much higher, namely over 60$\times$. On GPU, the speedup is a little bit lower, but still over 10$\times$ and 25$\times$, respectively. Furthermore, as Table \ref{tab:energy} shows, our method saves over 95\% energy for both CPU and GPU scenarios, which largely resolves one of the most important constraints on edge devices. All these experiments on real devices show that our method is practical for the real deployment of both high-performance and IoT applications. \begin{table}[] \centering \begin{tabular}{cc|c} \toprule Device & Patch Size & Normalized Energy Cost {[}STD{]} \\ \hline \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Edge\\ CPU\end{tabular}} & $2\times 2$ & 4.13\% {[}0.61\%{]} \\ & $4\times 4$ & 1.15\% {[}0.18\%{]} \\ \hline \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Edge\\ GPU\end{tabular}} & $2\times 2$ & 3.80\% {[}0.73\%{]} \\ & $4\times 4$ & 1.22\% {[}1.10\%{]} \\ \bottomrule \end{tabular} \caption{Normalized energy consumption for BP with gradient filtering for different patch sizes. Results are normalized w.r.t. the energy cost of standard BP methods. For instance, for edge CPU with a $4\times 4$ patch, only 1.15\% of energy in standard BP is used. Standard deviations are shown within brackets.} \label{tab:energy} \end{table} \begin{table}[htbp] \centering \resizebox{\linewidth}{!}{ \begin{tabular}[t]{lc|lc} \toprule Model & Ratio & Model & Ratio \\ \midrule (Wide)ResNet18-152 & 1.462 & VGG(bn)11-19 & 1.497 \\ DenseNet121-201 & 2.278 & EfficientNet b0-b7 & 1.240 \\ \bottomrule \end{tabular}} \caption{Evaluation of energy ratio defined in Equation~\eqref{equ:dcac_ratio} on models published on Torchvision. The ratio greater than 1 empirically verifies our assumption.} \label{tab:dcac} \end{table} \subsection{Main Assumption Verification} We now empirically verify the assumption that the DC component dominates the frequency spectrum of the convolution kernel (Section~\ref{sec:err_analysis}). To this end, we collect the energy ratio shown in Equation \eqref{equ:dcac_ratio} from trained models published in Torchvision~\cite{marcel2010torchvision}. As Table \ref{tab:dcac} shows, for the convolution kernels in all these networks, we get a ratio greater than one, which means that the energy of DC components is larger than energy of all AC components. Thus, our assumption in Section~\ref{sec:err_analysis} empirically holds true in practice. \section{Conclusions} \label{sec:conclusion} In this paper, we have addressed the on-device model training for resource-constrained edge devices. To this end, a new gradient filtering method has been proposed to systematically reduce the computation and memory consumption for the back-propagation algorithm, which is the key bottleneck for efficient model training. In Section \ref{sec:method}, a new gradient filtering approach has been proposed to reduce the computation required for propagating gradients through the convolutional layers. The gradient filtering creates an approximate gradient feature map with fewer unique elements and a special structure; this reduces the computation by more than two orders of magnitude. Furthermore, we proved that the error introduced during back-propagation by our gradient filter is bounded so the influence of gradient approximation is limited. Extensive experiments in Section \ref{sec:exp} have demonstrated the efficiency and wide applicability of our method. Indeed, models can be finetuned with orders of magnitudes fewer computations, while having only a marginal accuracy loss compared to popular baseline methods. \vspace{20pt} {\small \bibliographystyle{ieee_fullname}

proofpile-arXiv_069-11077

\section{Introduction} \IEEEPARstart{E}{ver} since discovered in 1940s, nuclear magnetic resonance (NMR) has manifested itself as one of the most significant tools to derive detailed information on the atomic scale about material properties. Nowadays, the trend of performing NMR experiments in high magnetic field is actively driven not only by higher sensitivity and resolution, but also by the desirable exploration of field-induced physics in modern material science\cite{ref1,ref2,ref3,ref4,ref5,ref6,ref7,ref8,ref9,ref10,ref11,ref12,ref13,ref14,ref15,ref16,ref17,ref18}. For instance, researches on half-integer quadrupolar nuclei such as $^{27}$Al and $^{17}$O notably benefit from the high magnetic field, because the quadrupolar interaction induced shift and broadening of the central transition are inversely proportional to the square of the magnetic field strength. The reduction in second-order quadrupolar broadening brings resolution enhancement and additional information, which provides new opportunities for measurements of nuclei not probed before\cite{ref1,ref2,ref3,ref4,ref5,ref8,ref10}. On the other side, since the high magnetic field often causes electronic and structural transitions of materials, the interests in field-induced physics like high-$T_{\rm c}$ cuprate superconductors have been increasing in the community of modern condensed matter physics. The NMR spectroscopy can provide more fundamental insights in the properties of materials, hence people have been making efforts to carry out NMR experiments in higher magnetic field\cite{ref6,ref7,ref9,ref11,ref12,ref13,ref14,ref15,ref16,ref17,ref18}. At present, the high magnetic field greater than 30 T can be generated by continuous-operation (also called as “steady-state” or “DC”) or pulsed magnets. The DC magnets with operation time of more than seconds include a variety of water-cooled resistive magnets (up to 41.5 T\cite{ref19}), superconducting magnets (up to 32.35 T\cite{ref20}) and hybrid magnets (up to 45.5 T\cite{ref21}). The continuous magnetic field is applicable to perform NMR experiments, because there is enough time to correct the field both in stability and homogeneity, as well as to execute various sequences for measurements and achieve signal averaging. However, large construction and operation costs and limited field strength of superconducting conductors hinder the development of NMR experiments to higher DC magnetic field. Currently, only pulsed resistive magnets are practically available to produce the high magnetic field exceeded 45 T and even up to 100 T for timescales typically in the range of 1-100 ms\cite{ref22,ref23}. Although some NMR experiments were successfully optimized and achieved in the short and time-varying pulsed field\cite{ref6,ref7,ref8,ref9,ref10,ref11,ref12,ref13,ref14,ref15,ref16}, it is still desirable to perform NMR measurements in a stable magnetic field\cite{ref17,ref18,ref24}. Firstly, a large real-time bandwidth up to tens of MHz or higher is required to excite and detect NMR signals due to the low repeatability and time-varying of the pulse field, which brings great difficulties to hardware design and signal search\cite{ref25}. Besides, the instability of the pulsed magnetic field causes the violent fluctuation of the resonance frequency and leads to the distortion of the NMR spectra. Despite the application of the complex signal post correction algorithms, the signal quality is still limited\cite{ref13,ref15}. Last but not least, some highly accurate parameter measurements such as the relaxation time $T_1$ and $T_2$ demand that the time dependence of the magnetic field should be reduced to zero during the long enough sequence durations\cite{ref24}. Hence, it is still challenging to carry out NMR measurements in the pulsed magnetic field. Fortunately, lots of efforts have been being made to create a quasi-stable stage (usually called “flat-top”) near the peak of the pulsed magnetic field\cite{ref26,ref27,ref28,ref29,ref30,ref31,ref32}, opening a new avenue for the pulsed magnetic field NMR measurements\cite{ref18}. In the early stage, Weickert $et$ $al.$\cite{ref26} tried to use a long-pulse magnet powered by high-power capacitor bank to creat an approximate 44 T flat-top for NMR experiments. However, the unregulated long pulsed magnetic field is inefficient in improving the flat-top stability and has a long repetition time more than 8 hours. Recently, Ihara and Kohama $et$ $al.$\cite{ref17,ref24} systematically peformed NMR measurements in a dynamically controlled flat-top field, showing great advantages of the flat-top pulsed magnetic field (FTPMF). Nevertheless, their measurements are limited in the low field of 13 T, and a higher FTPMF is desired in the community. In this work, we present a practically available FTPMF for NMR measurements up to 40 T at Wuhan National High Magnetic Field Center (WHMFC). In Sec. II, we briefly introduce basic concepts of NMR measurements. Furthermore, experimental setups are presented in Sec. III including a two-stage corrected flat-top pulsed magnetic field, a modular GHz NMR spectrometer and a sample probe. In Sec. IV, NMR measurements of $^{93}$Nb carried out in the DC field and FTPMF are described and discussed. Finally, the paper is concluded in Sec. V. \section{Basic Concepts of NMR} To understand basic concepts of NMR measurements, a brief description is presented here and shown in Fig. 1. The interaction between an external magnetic field $\boldsymbol{B}_0$ and nuclear spin $I$ ($I>0$) is described by the Zeeman effect, in which the spectral line splits into multiple discrete levels. For the simplest case of $I=\frac{1}{2}$, the Zeeman energy levels are given by \begin{equation} E_{\mathrm{Zee}}=\pm \frac{1}{2}h\gamma B_0 \end{equation} where $\gamma$ is gyromagnetic ratio (in MHz/T), and $h$ is Planck's constant. The difference of energy levels can be written as \begin{equation} \varDelta E=hf_{\mathrm{L}}=h\gamma B_0 \end{equation} where the $f_{\mathrm{L}}$ is called the Larmor frequency and $f_{\mathrm{L}}=\gamma B_0$. If a radio-frequency (RF) field $\boldsymbol{B}_1\left( t \right) =B_1\cos \left( 2\pi f_{\mathrm{L}}t \right)$ perpendicular to the $\boldsymbol{B}_0$ excites the nucleus, transitions between these energy levels can occur and NMR phenomenon can be observed. Commonly, a ${\pi /2}$ RF pulse is used to irradiate the NMR free induction decay (FID) signals which have the form of \begin{equation} s_{\mathrm{FID}}\left( t \right) \sim \cos \left( 2\pi f_{\mathrm{L}}t \right) \cdot e^{-\frac{t}{T_{2}^{*}}} \end{equation} where $T_{2}^{*}$ is the decay time. In the corresponding Fourier transform spectrum, the standard NMR spectrum is of the following form \begin{equation} S\left( f \right) \sim \frac{1/T_{2}^{*}}{\left( 1/T_{2}^{*} \right) ^2+\left( 2\pi f-2\pi f_L \right) ^2} \end{equation} which is called the absorption Lorentzian. It is obvious that the peak of the spectrum is at the Larmor frequency $f_{\mathrm{L}}$. When the external magnetic field $B_0$ changes with the time during the period of the FID, the $f_{\mathrm{L}}$ also changes proportionally and this leads to the distortion of the NMR spectrum (an example is shown in Fig. 1(d)). Hence, high stability of $B_0$ with time is one of the key prerequisites for NMR measurements. Another essential parameter in the NMR spectrum is the full width at half maxima (FWHM) parameterized by $1/\left( \pi T_{2}^{*} \right)$, which represents the spectral resolution. In fact, the FWHM includes the intrinsic homogeneous broadening quantified by the transverse relaxation time $T_2$ and the inhomogeneous broadening due to the spatial inhomogeneity of the $B_0$. Consequently, $B_0$ with less spatial inhomogeneity $\varDelta B_0\left( r \right)$ is desired for higher spectral resolution. \begin{figure}[!t] \centering \includegraphics[width=3.3in]{F1.jpg} \caption{Basic principles of NMR measurements. (a) The Zeeman energy levels of a nucleus with spin 1/2 in the presence of an external magnetic field $B_0$. NMR transition occurs at the Larmor frequency $f_{\mathrm{L}}=\gamma B_0$. (b) FID signals irradiated by a ${\pi /2}$ RF pulse. (c) A standard absorption Lorentzian NMR spectrum. the FWHM of $1/\left( \pi T_{2}^{*} \right)$ includes the intrinsic homogeneous broadening quantified by the transverse relaxation time $T_2$ and the inhomogeneous broadening due to the spatial inhomogeneity $\varDelta B_0\left( r \right)$ of the $B_0$. (d) The distortion of the NMR spectrum due to the temporal instability of the $B_0$. The $\varDelta B_0\left( t \right)$ reflects the degree of the instability.} \end{figure} \section{Experimental Setups} \subsection{Flat-top Pulsed Magnetic Field} For decades, the FTPMF technique has been pursuing to improve the stability and duration of the flat-top stage while maintaining the high field strength\cite{ref26,ref27,ref28,ref29,ref30,ref31,ref32}. The economical and reliable high-voltage capacitor bank is the most commonly used main power source for generating the high pulsed magnetic field, but it is difficult to regulate the discharge current for controlling the magnetic field. Despite all this, some solutions have been proposed, such as sequentially discharging multiple capacitors\cite{ref27,ref28,ref32}, or using a small compensation coil built into the bore of the main magnet to finely regulate the magnetic field\cite{ref29}. The former method can achieve a long flat-top duration of more than 10 ms but has high instability beyond 10$^3$ ppm, while the latter stands the opposite. Hence, we combined the advantages of the two schemes, and the flat-top realization is divided into two processes as shown in Fig. 2. \begin{figure}[!t] \centering \includegraphics[width=3.4in]{F2.jpg} \caption{Scheme of the two-stage corrected flat-top pulsed magnetic field for NMR measurements. (a) Diagram of connection of main components in the two-stage corrected FTPMF system. (b) Illustration of magnetic field waveforms generated by the two-stage corrected FTPMF system.} \end{figure} In terms of concrete implementation, the main magnetic field was produced by a standard 60 T pulsed magnet with the height of 150 mm and bore of 20 mm, which has an inductance of 3.5 mH and a resistance of 30 m$\Omega$ at 77 K (the magnet was immersed in the liquid nitrogen for cooling). The main magnet was connected in series with the primary winding of a pulse transformer (refer to Ref. \cite{ref28} for specific parameters) and powered by 8 parallel capacitor banks (3.84 mF, 1.2 MJ). The secondary winding of the pulse transformer was powered by 6 parallel capacitor banks (3.84 mF, 1.2 MJ) and formed the auxiliary circuit. Before the current of the main circuit reached its peak, the auxiliary circuit was discharged to achieve the first correction. According to the different optimized discharge voltages of both circuits and trigger time of the auxiliary circuit, the flat-top pulsed magnetic field up to 45 T after the first correction were produced, whose stability was within 0.5 T, and duration was within 12–15 ms depending on the field strength. The compensation coil used for the second correction consists of a main regulating coil with the height of 36 mm at the center of the main magnet and two oppositely wound decoupling coils with the height of 18 mm at the ends of the main regulating coil. Due to the strict radial space limitation in the main magnet bore of 20 mm, the copper compensation coil was double wound around a 17 mm Helium cryostat tail with only 0.4 mm wire diameter and reinforced by a Zylon/epoxy composite layer of 0.5 mm thickness. The compensation coil was powered by twelve 12-V lead-acid batteries, and its current can be linearly regulated by insulated gate bipolar transistor (IGBT) modules with a homemade driver to generate an adjustable magnetic field from 0 to 1 T beyond 10 ms. magnetic field signals at the sample area were sensed by a pick-up coil and the pick-up voltage was acquired at a sample rate of 100 kS/s by a 16-Bit analog-to-digital converter equipped with a field programmable gate array (FPGA) module. Digital proportional integral derivative (PID) control was applied to achieve the fine compensation of the magnetic field for the second correction. Finally, the magnetic field fluctuation after the two-stage correction was limited less than 10 mT, which is shown in Fig. 3. Due to the establishment time of the PID control and the decrease of the regulation capability of the compensation coil caused by the main magnet heating, the flat-top time after the second correction was reduced by about 4 ms compared with that of the first-order correction. It is noteworthy that both centers of the main magnet and compensation coil should be aligned accurately to provide a high magnetic field homogeneity area for NMR experiments. The axial locating processes were realized by using the pick-up coil and a lock-in amplifier with an error less than 2 mm. Considering the aging of the main magnet, we would perform NMR experiments in the FTPMF up to 40 T for safety. \begin{figure}[!t] \centering \includegraphics[width=3.3in]{F3.jpg} \caption{Magnetic field profiles up to 45 T generated by the two-stage corrected flat-top system and measured by the pick-up coil. The insert shows that the magnetic field fluctuation of the flat-top was limited less than 10 mT (integral drift had been corrected).} \end{figure} \subsection{Modular GHz NMR Spectrometer} A flexible modular GHz NMR spectrometer was developed, in which a series of stand-alone instruments were combined for use. The advantages of the modular spectrometer are convenient for testing and maintenance, as well as further upgrading and expansion. According to the properties of the FTPMF, several requirements of the NMR spectrometer are as follows: (1) Since the pulsed field is transient and its duration is typically less than 100 ms, the spectrometer should be controlled by a sophisticated timing system and has excellent transient responses. (2) The spectrometer should support high radio frequency up to several GHz, due to the fact that the resonance frequency is proportional to the field strength. For example, 2.6 GHz is needed for $^1$H at 60 T. (3) Low noise is required because the electromagnetic environment of the pulsed field is far worse than that of the DC field, and the number of times of the signal averaging is limited in the FTPMF. A diagram of the self-built modular NMR spectrometer is presented in Fig. 4. The construction of the spectrometer was based on a National Instruments (NI) PXI system including a chassis and an embedded computer, which was controlled via a self-written LabVIEW interface. Currently, the PXI system hosted three main modules, consisting of a RF generator NI PXIe-5651, a vector signal analyzer NI PXI-5661 and a pulse programmer NI PXI-6542. The RF generator can produce RF signals with frequency ranges from 500 kHz to 3.3 GHz. Correspondingly, the vector signal analyzer can support intermediate frequency (IF) down-conversion of NMR signals up to 2.7 GHz with a 20 MHz real-time bandwidth, and the NMR signals are finally recorded by a 100 MS/s and 14-Bit oscilloscope with using quadrature digital down-conversion. The 100 MHz digital pulse programmer was used to receive an external trigger when the flat-top field arrived, and then trigger each independent module in the spectrometer. The all components were synchronously clocked by a 10 MHz reference clock from the RF generator. \begin{figure}[thbp] \centering \includegraphics[width=3.3in]{F4.jpg} \caption{Diagram of the modular NMR spectrometer.} \end{figure} In the preceding stage circuit for excitation, the continuous RF signals were modulated into desired NMR sequences by a pulse gate (Analog Devices, HMC427ALP3E) with switching time smaller than 10 ns and isolation up to 40 dB. Then, the power of the modulated signals was amplified by a Tomco 100 W RF amplifier with band 5-650 MHz. The duplexer is a key three ports component for switching between the excitation of the high-power RF signals and the reception of the weak NMR signals. A 1 GHz bandwidth homemade active duplexer based on PIN diodes and various impedance lines was developed with 0.56 dB insert loss, 500 W power capacity, 37 dB isolation and 1 $\rm {\mu}$s switching time. To protect the receiving part from the crosstalk of the high-power excitation signals, a 70 dB isolation switch (Mini-Circuits, ZASWA-2-50DRA+) was applied with switching time of 20 ns. A lower noise amplifier N141-306CB up to 1 GHz with gain greater than 30 dB and noise figure smaller than 0.8 dB was used to amplifier the weak NMR signals. \subsection{Sample Probe} Compared with the conventional NMR probe for the DC field, the probe for the pulsed field is challenging to design because of the strict space limitation in the bore of the pulsed magnet, as well as serious electromagnetic interference and mechanical vibration environment. In practice, the bore space of the magnet available for the probe design is only 10 mm, but the probe needs to have multiple functional components, for example, including a sample chamber with enough space, a well tuned and matched resonance circuit, a temperature sensor and a heater, as well as a magnetic field pick-up coil. Closed loops of the conducting materials leading to eddy current have to be avoid, and all components should be well fixed. \begin{figure}[thpb] \centering \includegraphics[width=3.3 in]{F5.jpg} \caption{(a) Schematic illustration of the sample probe designed for the pulsed field. The partial photo of the sample probe and connection of the resonance circuit are also shown. (b) Photo of the experimental station layout.} \end{figure} Fig. 5(a) shows the schematic illustration of the sample probe designed for the pulsed field. The probe was located in a cryostat that can use liquid helium or nitrogen to perform low-temperature experiments. The volume of the sample chamber was about several tens of cubic millimeters and it was surrounded by a micro RF coil to excite and detect NMR signals. A tuning capacitor and a matching coil were connected to the resonant circuit which had been preliminarily calculated and tested at room temperature. Although there was deviation about several MHz from the expected resonant frequency in the working condition, precise tuning was not very necessary in the pulsed field experiments. The return loss of the resonance circuit at the resonance frequency was adjusted to be less than -10 dB, meaning that more than 90\% of the power was absorbed by the RF coil. The magnetic field pick-up coil was wounded on the outer sleeve of the probe and at the same axial height as the center of the sample to ensure that the magnetic field at the sample were accurate collected. The heater and temperature sensor were located near the resonance circuit to control the local temperature around the sample. Most components were arranged vertically to improve the space utilization. Based on the establishment of the above experimental setups, a photo of the experimental station layout is presented in Fig. 5(b). $^{93}$Nb powder (99.999\% purity) uniformly mixed by resin was selected as the testing sample. $^{93}$Nb is ideal for NMR measurements under the pulsed field, because it has the natural abundance of 100\%, a large gyromagnetic ratio $\gamma$=10.405 MHz/T, a short longitudinal relaxation time $T_1$ of 4.7 ms at 77 K, and a high relative sensitivity of 48{\%} compared with $^1$H. The mixed sample was cut into a cylinder and placed in the RF coil wound by copper wire (about 2 mm in diameter, 4 mm in length, 0.3 mm wire diameter with 6 turns). Considering the Knight shift ($K$), the resonance frequency of $^{93}$Nb is $\gamma \left( 1~+~ ^{93}K \right) B_0$, where $^{93}K= 0.87\%$. \section{Results and Discussion} \subsection{NMR Measurements in the DC Field and FTPMF} Since the pick-up coil was not well calibrated for measuring the pulsed magnetic field with an estimated error lager than 1{\%} and the repetition time for the pulsed magnet cooling was usually longer than 30 minutes, a pre-optimization process in the DC field was necessary to achieve the maximum sensitivity of NMR signals in a single measurement as possible. This process was carried out with our self-built spectrometer and probe in a superconducting magnet (Oxford) with high stability (better than 10 ppm/h) and homogeneity (better than 10 ppm/cm$^3$). The FID measurements of $^{93}$Nb in the DC field were performed at 14.55 T and 77 K, and the corresponding resonance frequency is 152.7 MHz. In order to irradiate the FID signals with both high sensitivity and bandwidth, the power and duration of the $\pi/2$ RF pulse were optimized. Finally, the 50 W and 1.5 $\rm \mu$s RF pulse was implemented with the maximum signal-to-noise ratio (SNR) of 20 dB and bandwidth of 600 kHz. The FID signals after quadrature down-conversion are shown in Fig. 6(a), which were measured at the given RF of 152.6 MHz for the clarity of the oscillation. Ignoring the switching dead time of 1 $\rm \mu$s, the decay time $T_{2}^{*} $ of the FID signals about 8 $\rm \mu$s was observed, which can be approximately regarded as the $T_2$ due to the high stability and homogeneity of the DC field. For quickly locating the NMR signals and calibrating the pick-up coil in the FTPMF, a field-sweep mode with slope-top pulse was performed. According to the bandwidth of excitation and longitudinal relaxation time $T_1$ of $^{93}$Nb, the sweep slope rate was set to 10 mT/ms and the interval of excitation was 1 ms. After several sweep tests, the NMR measurements can be normally carried out in the FTPMF. The FID signals at the field of 23.24 T and excited RF of 244 MHz after quadrature down-conversion are shown in Fig. 6(b). It is obvious that the decay time of the FID signals reduces from 8 $\rm \mu$s to 3 $\rm \mu$s in the FTPMF compared to the DC field, which was mainly caused by the inhomogeneity of the FTPMF and will be discussed later. Although the field strength of the FTPMF is 1.6 times higher than that of the DC field, the sensitivity of FID signals do not significantly increase due to the short polarization time of nuclei, inhomogeneity of the magnetic field and so on. These $^{93}$Nb FID results can be further confirmed by a spin-echo (SE) technique. A standard SE measurement ($\pi /2-\tau -\pi$, 1.5-20-3 $\rm \mu$s in this case) was performed in the FTPMF, the result of which is shown in Fig. 7. It is clear that the SE signals refocus after waiting for 20 $\rm \mu$s, as expected. \begin{figure}[htbp] \centering \includegraphics[width=3.3in]{F6.jpg} \caption{FID signals after quadrature down-conversion: (a) at the DC field of 14.55 T and excited RF of 152.6 MHz; (b) at the FTPMF of 23.24 T and excited RF of 244 MHz. (c) at the FTPMF of 39.54 T and excited RF of 415 MHz. All measurements were carried out at temperature of 77 K.} \end{figure} \begin{figure}[htbp] \centering \includegraphics[width=3.3in]{F7.jpg} \caption{ NMR RF pulse sequence followed by the spin-echo signals at the FTPMF of 23.26 T and excited RF of 244 MHz. The given SE sequence is 1.5-20-3.0 $\mu$s. } \end{figure} The FID signals in the FTPMF of 39.54 T and excited RF of 415 MHz were measured and are presented in Fig. 6(c). It is noteworthy that the given $\pi/2$ RF pulse was reoptimized with the duration of 9 $\rm \mu$s to achieve the highest possible signal sensitivity. The results show that the signal sensitivity in the high field of 39.54 T increases about 1.6 times in proportion to the magnetic field strength, compared with that in the low field of 23.24 T. However, the decay time of FID further decreases to 2 $\rm \mu$s due to the degradation of the magnetic field homogeneity in the absolute value. \subsection{Stability of the FTPMF} The NMR technique itself is also an accurate method for measuring the magnetic field, hence the FID signals were excited several times to check the stability of the FTPMF during the period of the flat-top. Fig. 8(a) shows the nine excitation points with an interval of 1 ms during the FTPMF at 39.55 T measured by the pick-up coil. The corresponding NMR spectra at the excited RF of 415 MHz are presented in Fig. 8(b). The central frequency offset of the nine spectra was limited within 100 kHz, meaning the field fluctuation was less than 10 mT. It well corresponds to the result measured by the pick-up coil. Overall, the current stability of the FTPMF in the timescale about 10 ms would potentially find some applications for NMR experiments in condensed matter physics\cite{ref33}. A higher resolution ADC for the acquisition of the magnetic field and smaller feedback control cycle for the second correction system are desired to further improve the stability of the FTPMF. \begin{figure}[htbp] \centering \includegraphics[width=3.3in]{F8.jpg} \caption{Stability of the FTPMF checked by the NMR spectra. (a) Nine excitation points of the FID signals with an interval of 1 ms during the FTPMF at 39.55 T measured by the pick-up coil. (b) The corresponding NMR spectra at the excited RF of 415 MHz. The central frequency offset of the nine spectra was limited within 100 kHz, which corresponds well to the magnetic field fluctuation of 10 mT measured by the pick-up coil.} \end{figure} \subsection{Homogeneity of the FTPMF} In the section II, we revealed that the decay time $T_{2}^{*}$ of the FID signals decreases with the increase of the magnetic field spatial inhomogeneity. Besides, the inhomogeneity can also lead to the broadening of the NMR spectra, which is harmful to the high resolution NMR measurements. Compared with the commercial superconducting magnet with sophisticated shimming techniques, the pulsed magnet has great magnetic field inhomogeneity up to 1000 ppm or higher in 1 cm$^3$ space of the magnet center. Ignoring the winding error and magnet deformation, the axial distribution of the magnetic field generated by the solenoid-type magnet is approximately a quadratic function of $B_0\left( z \right) =B_0-az^2$ , where the coefficient $a>0$. Hence, small volume and highly accurate axial positioning of the sample in the pulsed magnet are important to weaken the influence of the magnetic field inhomogeneity. The NMR spectra measured in the DC field of 14.55 T and measured in the FTPMF of 23.25 T are compared in Fig. 9. The axial position of the sample in the FTPMF was well optimized with an accuracy less than 1 mm (labelled at 0 mm). Because of the high homogeneity better than 10 ppm/cm$^3$ in the DC field, the FWHM of 300 ppm can be approximately attributed to the intrinsic homogeneous broadening. Although the strength of the FTPMF is higher than that of the DC field, the FWHM increases to 550 ppm due to the magnetic field inhomogeneity. Because the magnetic field inhomogeneity was caused by both the compensation coil and main magnet, the original pulsed magnetic field at the peak with change rate less than 1 mT/10 $\rm \mu$s was applied to check the inhomogeneity induced by the main magnet. The results show that the main magnet is the main source of the magnetic field inhomogeneity and the compensation coil has little impact. The spectrum of the sample positioned at 3 mm away the optimized position was also measured and is shown in Fig. 9. It is obvious that the FWHM increases to 1440 ppm due the heavier magnetic field inhomogeneity. The NMR spectrum in the FTPMF of 39.55 T is also presented in Fig. 9. Despite greater absolute broadening of the spectrum in 39.55 T compared with that in the low field of 23.25 T, the relative broadening almost remains unchanged due to the increasing of the resonance frequency from 244 MHz up to 415 MHz. On balance, the inhomogeneity effect of the pulsed magnet can be reduced to the level of 10$^2$ ppm with a sample volume of 10 mm$^3$ by the location optimization of the sample. A main magnet after specially designed with higher magnetic field homogeneity is needed to further promote the resolution of NMR measurements\cite{ref34}. \begin{figure}[!t] \centering \includegraphics[width=3.3in]{F9.jpg} \caption{The NMR spectra of $^{93}$Nb under different conditions. Relative FWHM is marked for comparison.} \end{figure} \section{Conclusion} In this paper, we report the nuclear magnetic resonance measurements in the high flat-top pulsed magnetic field up 40 T. The scheme of FTPMF with two-stage correction was proposed, whose magnetic field fluctuation was limited within 10 mT and duration was beyond 9 ms. Besides, the NMR spectrometer and probe suitable for the pulsed field condition were also developed. The NMR measurements of $^{93}$Nb were preoptimized in the DC field and finally carried out in the FTMPF. The results show that the stability and homogeneity of the FTPMF can reach an order of 10$^2$ ppm / 10 ms and 10$^2$ ppm / 10 mm$^3$ respectively, which is sufficient for exploring microscopic structures of some condensed matters in the high magnetic field. Crucial avenues of research in the future are the improvements of the FTPMF in many aspects including field strength, duration, stability, homogeneity and so on. \bibliographystyle{IEEEtran}

proofpile-arXiv_069-11080

\section{Introduction} Littlewood's identity reads as \begin{equation} \label{littlewood} \sum_{\lambda} s_{\lambda}(X_1,\ldots,X_n) = \prod_{i=1}^{n} \frac{1}{1-X_i} \prod_{1 \le i < j \le n} \frac{1}{1-X_i X_j}, \end{equation} where $s_{\lambda}(X_1,\ldots,X_n)$ denotes the Schur polynomial associated with the partition $\lambda$ and the sum is over all partitions $\lambda$. In fact, the identity was already known to Schur, see \cite[p.\ 163]{schur1} or \cite[p.\ 456]{schur2}, and written down by Littlewood in \cite[p.\ 238]{Littlewood}. This identity has a beautiful combinatorial proof that is based on the Robinson-Schensted-Knuth correspondence and exploits its symmetry, see Appendix~\ref{combclassical}, and e.g., \cite{Sta99} for details. In recent papers \cite{Fis19b,Fis19a,fourfold}, where ``alternating sign matrix objects'' (namely, \emph{alternating sign triangles} and \emph{alternating sign trapezoids}) have been connected to certain ``plane partition objects'' (namely, \emph{totally symmetric self-complementary plane partitions} and \emph{column strict shifted plane partitions of fixed class}, which generalize the better known \emph{descending plane partitions}), a very similar identity played the crucial role to establish this still mysterious \cite{cube} connection. All these proofs are not of a combinatorial nature and involve rather complicated calculations, and so the study of the combinatorics of our Littlewood-type identity is very likely lead to a better understanding of the combinatorics of this relation. In order to formulate the identity, we rewrite \eqref{littlewood} using the bialternant formula for the Schur polynomial \cite[7.15.1]{Sta99} $$ s_{(\lambda_1,\ldots,\lambda_n)}(X_1,\ldots,X_n) = \frac{\det_{1 \le i,j \le n} \left( X_i^{\lambda_j+n-j} \right)} {\prod_{1 \le i < j \le n} (X_i-X_j)} = \frac{ \mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{i=1}^n X_i^{\lambda_i+n-i} \right] } {\prod_{1 \le i < j \le n} (X_i-X_j)}, $$ allowing zeros at the end section of $(\lambda_1,\ldots,\lambda_n)$, with $$\mathbf{ASym}_{X_1,\ldots,X_n} f(X_1,\ldots,X_n) = \sum_{\sigma \in {\mathcal S}_n} \sgn \sigma \cdot f(X_{\sigma(1)},\ldots,X_{\sigma(n)})$$ as follows. $$ \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \sum_{0 \le k_1 < k_2 < \ldots < k_n} X_1^{k_1} X_2^{k_2} \cdots X_n^{k_n} \right]}{\prod_{1 \le i < j \le n} (X_j-X_i)} = \prod_{i=1}^{n} \frac{1}{1-X_i} \prod_{1 \le i < j \le n} \frac{1}{1-X_i X_j} $$ We have used the following identity in \cite{Fis19b,Fis19a}. There it is proved by induction with respect to $n$. \begin{multline} \label{littlewoodASM1} \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} (1+X_j + X_i X_j) \sum_{0 \le k_1 < k_2 < \ldots < k_n} X_1^{k_1} X_2^{k_2} \cdots X_n^{k_n} \right]}{\prod_{1 \le i < j \le n} (X_j-X_i)} \\ = \prod_{i=1}^{n} \frac{1}{1-X_i} \prod_{1 \le i < j \le n} \frac{1+X_i + X_j}{1-X_i X_j} \end{multline} In \cite{fourfold}, an additional parameter has been introduced, which has to be set to $1$ to obtain \eqref{littlewoodASM1}. \begin{multline} \label{Hans} \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[\prod_{1 \le i < j \le n} (Q+(Q-1) X_i + X_j + X_i X_j) \sum_{0 \le k_1 < k_2 < \ldots < k_n} \prod_{i=1}^n \left( \frac{X_i(1+X_i)}{Q+X_i} \right)^{k_i} \right]}{\prod_{1 \le i < j \le n} (X_j-X_i)} \\ = \prod_{i=1}^n \frac{Q+X_i}{Q-X_i^2} \frac{ \prod_{1 \le i < j \le n} (Q(1+X_i)(1+X_j)- X_i X_j )}{\prod\limits_{1 \le i < j \le n} (Q-X_i X_j)}. \end{multline} Among other things, we will see in this paper that we can also introduce another parameter in \eqref{littlewoodASM1} as follows: \begin{multline} \label{littlewoodASM2} \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} (1+ w X_i + X_j + X_i X_j) \sum_{0 \le k_1 < k_2 < \ldots < k_n} X_1^{k_1} X_2^{k_2} \cdots X_n^{k_n} \right]}{\prod_{1 \le i < j \le n} (X_j-X_i)} \\ = \prod_{i=1}^{n} \frac{1}{1-X_i} \prod_{1 \le i < j \le n} \frac{1+X_i + X_j + w X_i X_j}{1-X_i X_j}, \end{multline} in fact, there is even the following common generalization of \eqref{Hans} and \eqref{littlewoodASM2}. \begin{multline} \label{mostgeneralunbounded} \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} (Q+(Q+r) X_i + X_j + X_i X_j) \sum_{0 \le k_1 < k_2 < \ldots < k_n} \prod_{i=1}^n \left( \frac{X_i(1+X_i)}{Q+X_i} \right)^{k_i} \right]} {\prod_{1 \le i < j \le n} (X_j-X_i)} \\ = \prod_{i=1}^n \frac{Q+X_i}{Q-X_i^2} \frac{ \prod_{1 \le i < j \le n} Q(1+X_i)(1+X_j)+ r X_i X_j}{\prod\limits_{1 \le i < j \le n} (Q-X_i X_j)} \end{multline} The main purpose of this paper is to derive bounded versions of these identities and to provide combinatorial interpretations of the identities that would allow us to approach them with a combinatorial proof, possibly by a variant of the Robinson-Schensted-Knuth correspondence that mimics the proof for the classical Littlewood identity. By bounded version we mean that the sums $\sum_{0 \le k_1 < k_2 < \ldots < k_n}$ are restricted to, say, $\sum_{0 \le k_1 < k_2 < \ldots < k_n \le m}$. Macdonald \cite{macdonald} has provided such a bounded version of the classical identity \eqref{littlewood}, namely \begin{multline} \label{littlewoodbounded} \sum_{\lambda \subseteq (m^n)} s_{\lambda}(X_1,\ldots,X_n) = \sum_{0 \le k_1 \le k_2 \le \ldots \le k_n \le m} s_{(k_n,k_{n-1},\ldots,k_1)}(X_1,\ldots,X_n) \\ = \frac{ \det_{1 \le i, j \le n} \left( X_i^{j-1} - X_i^{m+2n-j} \right) }{\prod_{i=1}^n (1-X_i) \prod_{1 \le i < j \le n} (X_j-X_i)(1-X_i X_j)}, \end{multline} which he used to prove MacMahon's conjecture. Very recent work on bounded Littlewood identities can be found in \cite{RW21}. More specifically, we will prove the following. \begin{theorem} \label{boundedrQ} For $n \ge 1$, we have \begin{multline} \label{boundedrQid} \frac{1}{\prod\limits_{1 \le i < j \le n} (X_j-X_i)} \mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} (Q+(Q+r) X_i+X_j + X_i X_j) \right. \\ \times \left. \sum_{0 \le k_1 < k_2 < \ldots < k_n \le m} \left( \frac{X_1(1+X_1)}{Q+X_1} \right)^{k_1} \left( \frac{X_2(1+X_2)}{Q+X_2} \right)^{k_2} \cdots \left( \frac{X_n(1+X_n)}{Q+X_n} \right)^{k_n} \right]  \\ = { \frac{\det_{1 \le i, j \le n} \left( a_{j,m,n}(Q,r;X_i) \right)}{\prod\limits_{1 \le i \le j \le n} (Q-X_i X_j) \prod\limits_{1 \le i < j \le n} (X_j-X_i)}} \end{multline} with \begin{multline*} a_{j,m,n}(Q,r;X) = (1+Q X^{-1}) X^{j} (1+X)^{j-1} (Q+ r X + Q X)^{n-j} \\ - X^{2 n} Q^{-n} \left( \frac{(1+X) X}{Q+X} \right)^{m} (1+X) \left( Q X^{-1} \right)^{j} (1+Q X^{-1})^{j-1} (Q+r Q X^{-1} + Q^2 X^{-1})^{n-j}. \end{multline*}  \end{theorem} Setting $Q=1$ and $r=w-1$, we obtain, after simplifying the right-hand side, the following corollary. \begin{cor} For $n \ge 1$, we have \begin{multline} \label{rincluded} \frac{1}{\prod\limits_{1 \le i < j \le n} (X_j-X_i)} \mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} (1+w X_i+X_j + X_i X_j) \sum_{0 \le k_1 < k_2 < \ldots < k_n \le m} X_1^{k_1} X_2^{k_2} \cdots X_n^{k_n} \right]  \\ = \frac{\det_{1 \le i, j \le n} \left( X_i^{j-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{m+2n-j} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right)}{\prod\limits_{i=1}^n (1-X_i) \prod\limits_{1 \le i < j \le n} (1-X_i X_j)(X_j-X_i)}. \end{multline} \end{cor} In the second part of the paper, we will then provide combinatorial interpretations for both sides of the identity in the corollary. \subsection*{Outline} In Section~\ref{proof}, we give a proof of \eqref{boundedrQ}. In Appendix~\ref{combclassical}, we discuss a point of view on the combinatorics of the classical Littlewood identity \eqref{littlewood} and its bounded version \eqref{littlewoodbounded} that is beneficial for possible combinatorial proofs of the Littlewood-type identities that we establish in this paper. Recall that this is of interest because such identities have been used several times \cite{Fis19a,Fis19b,fourfold} to establish connections between alternating sign matrix objects and plane partition objects. To approach this, we offer combinatorial interpretations of the left-hand sides of \eqref{littlewoodASM2} and \eqref{rincluded} in Section~\ref{LHS} and in Appendix~\ref{furtherLHS}. Then, in Section~\ref{RHS}, we offer a combinatorial interpretation of the right-hand sides of \eqref{littlewoodASM2} and \eqref{rincluded}. These interpretations are nicest in the cases $w=0,1$. In Section~\ref{outlook}, we offer an outlook on related work on the cases $w=0,-1$, which will appear in a forthcoming paper with Florian Schreier-Aigner. \section{Proof of Theorem~\ref{boundedrQ}} \label{proof} Bressoud's elementary proof \cite{BressoudElement} of \eqref{littlewoodbounded} turned out to be useful to obtain the following (still elementary, but admittedly very complicated) proof of Theorem~\ref{boundedrQ} provided here. Conceptually the proof is not difficult: We use induction with respect to $n$ and show that both sides satisfy the same recursion. \subsection{The case $m \to \infty$.} We start by proving the $m \to \infty$ case of Theorem~\ref{boundedrQ}. This is equivalent to proving that \begin{multline} \label{infinite} \mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} (Q+(Q+r) X_i + X_j + X_i X_j) \prod_{i=1}^n \left( \frac{X_i(1+X_i)}{Q+X_i} \right)^{i-1} \prod_{i=1}^n \frac{1}{1- \prod_{j=i}^n \frac{X_j(1+X_j)}{Q+X_j}} \right] \\ = \prod_{i=1}^n \frac{Q+X_i}{Q-X_i^2} \frac{ \prod_{1 \le i < j \le n} (X_j - X_i)(Q(1+X_i)(1+X_j)+ r X_i X_j )}{\prod\limits_{1 \le i < j \le n} (Q-X_i X_j)}, \end{multline} which is just \eqref{mostgeneralunbounded} multiplied on both sides with $\prod_{1 \le i < j \le n} (X_j-X_j)$. To see this, we rewrite the left-hand side of \eqref{boundedrQid} by using the summation formula for the geometric series $n$ times. As $m \to \infty$, $a_{j,m,n}(Q,r;X_i)$ simplifies to $$ (1+Q X^{-1}) X^{j} (1+X)^{j-1} (Q+ r X + Q X)^{n-j} $$ in a formal power series sense, and $$ \det_{1 \le i,j \le n} \left( (1+Q X^{-1}) X^{j} (1+X)^{j-1} (Q+ r X + Q X)^{n-j} \right) $$ can be computed using the Vandermonde determinant evaluation, we are led to the right-hand side of \eqref{infinite} eventually. We denote by $L_n(X_1,\ldots,X_n)$ the left-hand side of \eqref{infinite} and observe that the following recursion is satisfied. \begin{multline} \label{rec} L_n(X_1,\ldots,X_n) = \sum_{k=1}^n (-1)^{k-1} \frac{1}{1-\prod_{i=1}^n \frac{X_i(1+X_i)}{Q+X_i} } L_{n-1}(X_1,\ldots,\widehat{X_k},\ldots,X_n) \\ \times \prod_{1 \le j \le n, \atop j \not= k} \frac{X_j(1+X_j)(Q+(Q+r) X_k + X_j + X_k X_j)}{Q+X_j}, \end{multline} where $\widehat{X_k}$ means that we omit $X_k$. Indeed, suppose more generally that $$ P(X_1,\ldots,X_n) = \mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} s(X_i,X_j) \prod_{i=1}^n t(X_i)^{i-1} \prod_{i=1}^n \frac{1}{1- \prod_{j=i}^n u(X_j)} \right], $$ then \begin{equation} \label{rec1} \begin{aligned} P(X_1,\ldots,X_n) &= \sum_{k=1}^n \sum_{\sigma \in {\mathcal S}_n: \atop \sigma(1)=k} \sgn \sigma \frac{1}{1- \prod_{j=1}^n u(X_j)} \prod_{1 \le j \le n, \atop j \not= k} s(X_k,X_j) t(X_j) \\ &\qquad \times \sigma \left[ \prod_{2 \le i < j \le n} s(X_i,X_j) \prod_{i=2}^n t(X_i)^{i-2} \prod_{i=2}^n \frac{1}{1- \prod_{j=i}^n u(X_j)} \right] \\ &= \sum_{k=1}^n (-1)^{k-1} \frac{1}{1- \prod_{j=1}^n u(X_j)} P(X_1,\ldots,\widehat{X_k},\ldots,X_n) \\ &\qquad \times \prod_{1 \le j \le n, \atop j \not= k} s(X_k,X_j) t(X_j). \end{aligned} \end{equation} The last equality follows from the fact that the sign of $\sigma$ is the product of $ (-1)^{k-1}$ and the sign of the restriction of $\sigma$ to $\{2,3,\ldots,n\}$, assuming $\sigma(1)=k$ and ``identifying'' the preimage $\{2,3,\ldots,n\}$ as well as the image $\{1,\ldots,n\} \setminus \{k\}$ with $\{1,2,\ldots,n-1\}$ in the natural way. We show \eqref{infinite} by induction with respect to $n$. The case $n=1$ is easy to check. It suffices to show that the right-hand side of \eqref{infinite} satisfies the recursion \eqref{rec}, i.e., \begin{multline*} \prod_{i=1}^n \frac{Q+X_i}{Q-X_i^2} \frac{ \prod_{1 \le i < j \le n} (X_j - X_i)(Q(1+X_i)(1+X_j)+ r X_i X_j )}{\prod\limits_{1 \le i < j \le n} (Q-X_i X_j)} \\ = \sum_{k=1}^n (-1)^{k-1} \frac{1}{1-\prod_{i=1}^n \frac{X_i(1+X_i)}{Q+X_i} } \prod_{1 \le j \le n, \atop j \not= k} \frac{X_j(1+X_j)(Q+(Q+r) X_k + X_j + X_k X_j)}{Q-X_j^2} \\ \times \frac{ \prod_{1 \le i < j \le n, i,j \not=k} (X_j - X_i)(Q(1+X_i)(1+X_j)+ r X_i X_j )}{\prod\limits_{1 \le i < j \le n, i,j \not=k} (Q-X_i X_j)}. \end{multline*} We multiply by $\left(1 - \prod_{i=1}^n \frac{X_i(1+X_i)}{Q+X_i} \right) \prod_{1 \le i \le j \le n} (Q-X_i X_j)$ and obtain \begin{equation} \label{leftright} \begin{aligned} & \left(\prod_{i=1}^{n} (Q+X_i) -\prod_{i=1}^n X_i(1+X_i) \right) \prod_{1 \le i < j \le n} (X_j - X_i)(Q(1+X_i)(1+X_j)+ r X_i X_j ) \\ &\quad = \prod_{1 \le i < j \le n} (X_j - X_i)(Q(1+X_i)(1+X_j)+ r X_i X_j ) \\ & \qquad \times \sum_{k=1}^n (Q-X_k^{2}) \prod_{1 \le j \le n \atop j \not= k} \frac{X_j (X_j+1)(Q+(Q+r) X_k + X_j + X_k X_j)(Q-X_j X_k)}{(X_j-X_k)(Q(1+X_j)(1+X_k)+ r X_j X_k)}. \end{aligned} \end{equation} For each $s \in \{1,2,\ldots,n\}$, both sides are polynomials in $X_s$ of degree no greater than $2n$. It is not hard to see that both sides vanish for $X_s = X_t$ and $X_s =-\frac{Q(1+X_t)}{Q+Q X_t + r X_t}$ for any $t \in \{1,2,\ldots,n\} \setminus \{s\}$. Moreover, it is also not hard to see that the evaluations also agree for $X_s=0,-1$, which gives a total of $2n$ evaluations for each $X_s$. It follows that the difference of the left-hand side and the right-hand side is up to a constant in $\mathbb{Q}(Q,r)$ equal to \begin{equation} \label{difference} \prod_{i=1}^n X_i(1+X_i) \prod_{1 \le i < j \le n} (X_j - X_i)(Q(1+X_i)(1+X_j)+ r X_i X_j). \end{equation} To show that this constant is indeed zero, we consider the following specialization $$(X_1,X_2,X_3,X_4,\ldots) = \left(X_1,\frac{Q}{X_1},X_3,\frac{Q}{X_3},\ldots \right).$$ Note first that \eqref{difference} does not vanish at this specialization, and, therefore, it suffices to show that the left-hand side and the right-hand side of \eqref{leftright} agree on this specialization. If $n$ is even, this is particularly easy to see, because both sides vanish (on the right-hand side all summands vanish, which is due to the factor $Q-X_j X_k$). If $n$ is odd, then only the last summand on the right-hand side remains and it is not hard to see that it is equal to the left-hand side. \subsection{The general case} We rewrite the identity from Theorem~\ref{boundedrQ} that we need to prove as follows. \begin{multline} \label{rewrite} \det_{1 \le i, j \le n} \left( a_{j,m,n}(Q,r;X_i) \right) \\ \ = \mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{i=1}^n (Q-X_i^2) \prod_{1 \le i < j \le n} (Q+(Q+r) X_i+X_j + X_i X_j)(Q-X_i X_j) \right. \\ \times \left. \sum_{0 \le k_1 < k_2 < \ldots < k_n \le m} \left( \frac{X_1(1+X_1)}{Q+X_1} \right)^{k_1} \left( \frac{X_2(1+X_2)}{Q+X_2} \right)^{k_2} \cdots \left( \frac{X_n(1+X_n)}{Q+X_n} \right)^{k_n} \right]  \end{multline} We set \begin{multline*} F(m;X_1,\ldots,X_n)= \prod_{i=1}^n (Q-X_i^2) \prod_{1 \le i < j \le n} (Q+(Q+r) X_i+X_j + X_i X_j)(Q-X_i X_j) \\ \times \sum_{0 \le k_1 < k_2 < \ldots < k_n \le m} \left( \frac{X_1(1+X_1)}{Q+X_1} \right)^{k_1} \left( \frac{X_2(1+X_2)}{Q+X_2} \right)^{k_2} \cdots \left( \frac{X_n(1+X_n)}{Q+X_n} \right)^{k_n} \end{multline*} and observe that \begin{multline*} F(m;X_1,\ldots,X_n) = (Q-X_1^2) \prod_{j=2}^n (Q+(Q+r) X_1+X_j + X_1 X_j)(Q-X_1 X_j) \\ \times \sum_{l=0}^m \frac{Q+X_1}{X_1(1+X_1)} \left( \prod_{i=1}^n \frac{X_i(1+X_i)}{Q+X_i} \right)^{l+1} F(m-1-l;X_2,\ldots,X_n). \end{multline*} We set $$ A(m;X_1,\ldots,X_n) = \mathbf{ASym}_{X_1,\ldots,X_n} F(m;X_1,\ldots,X_n), $$ and observe that \begin{align*} A(m;X_1,\ldots,X_n) &= \sum_{k=1}^n \sum_{l=0}^m (-1)^{k+1} (Q-X_k^2) \frac{Q+X_k}{X_k(1+X_k)} \left( \prod_{i=1}^n \frac{X_i(1+X_i)}{Q+X_i} \right)^{l+1} \\  &\quad \times A(m-l-1;X_1,\ldots,\widehat{X_k},\ldots,X_n) \\ &\quad \times \prod_{1 \le i \le n, i \not= k} (Q+(Q+r) X_k+X_i + X_i X_k)(Q-X_i X_k), \end{align*} by the same argument that has led to \eqref{rec1}. By the induction hypothesis, we have $$ A(m-l-1;X_1,\ldots,\widehat{X_k},\ldots,X_n) = \det_{1 \le i \le n, i \not= k \atop 1 \le j \le n-1} \left( a_{j,m-l-1,n-1}(Q,r;X_i) \right). $$ Therefore, the right-hand side of \eqref{rewrite} is \begin{multline} \label{expand} \sum_{k=1}^n (-1)^{k+1} (Q-X_k^2) \prod_{1 \le i \le n, i \not= k} (Q+(Q+r) X_k+X_i + X_i X_k)(Q-X_i X_k) \\ \times \sum_{l=0}^m \left( \frac{X_k(1+X_k)}{Q+X_k} \right)^{l} \det_{1 \le i \le n, i \not= k \atop 1 \le j \le n-1} \left( \left( \frac{X_i(1+X_i)}{Q+X_i} \right)^{l+1} a_{j,m-l-1,n-1}(Q,r;X_i) \right) \end{multline} and we need to show that it is equal to $\det_{1 \le i, j \le n} \left( a_{j,m,n}(Q,r;X_i) \right)$. Noting that \begin{equation} \begin{aligned} & \left( \frac{X(1+X)}{Q+X} \right)^{l+1} a_{j,m-l-1,n-1}(Q,r;X) \\ &\quad = (1+Q X^{-1}) X^{j+l+1} (1+X)^{j+l} (Q+ r X + Q X)^{n-1-j} (Q+X)^{-l-1} \\ &\qquad - X^{2 n-2} Q^{-n+1} \left( \frac{(1+X) X}{Q+X} \right)^{m} (1+X) \left( Q X^{-1} \right)^{j} (1+Q X^{-1})^{j-1} (Q+r Q X^{-1} + Q^2 X^{-1})^{n-1-j} \\ &\quad = X^{j+l} (1+X)^{j+l} (Q+X)^{-l} (Q+ r X + Q X)^{n-1-j} \\ &\qquad - X^{-j+m+n} (1+X)^{m+1} (Q+X)^{j-m-1} (X+r+Q)^{n-1-j}, \end{aligned}  \end{equation} we can write the determinant in \eqref{expand} as \begin{multline*} \sum_{\sigma, S} (-1)^{I(\sigma)+|S|} \prod_{i \in S} X_i^{-\sigma(i)+m+n} (1+X_i)^{m+1} (Q+X_i)^{\sigma(i)-m-1} (X_i+r+Q)^{n-1-\sigma(i)} \\ \times \prod_{i \in \overline{S}} X_i^{\sigma(i)+l} (1+X_i)^{\sigma(i)+l} (Q+X_i)^{-l} (Q+ r X_i + Q X_i)^{n-1-\sigma(i)}, \end{multline*} where the sum is over all bijections $\sigma: \{1,2,\ldots,n \} \setminus \{k\} \to \{1,2,\ldots,n-1\}$, all subsets $S$ of $ \{1,2,\ldots,n \} \setminus \{k\}$ and $I(\sigma)$ is the number of all inversions, i.e., pairs $i,j \in \{1,2,\ldots,n \} \setminus \{k\} $ with $i<j$ and $\sigma(i)>\sigma(j)$. Moreover, $\overline{S}$ denotes the complement of $S$ in $\{1,2,\ldots,n \} \setminus \{k\}$. Comparing with \eqref{expand}, we multiply by $\left( \frac{X_k(1+X_k)}{Q+X_k} \right)^{l}$ and take the sum over $l$. \begin{multline*} \sum_{\sigma, S} (-1)^{I(\sigma)+|S|} \prod_{i \in S} X_i^{-\sigma(i)+m+n} (1+X_i)^{m+1} (Q+X_i)^{\sigma(i)-m-1} (X_i+r+Q)^{n-1-\sigma(i)} \\ \times \prod_{i \in \overline{S}} X_i^{\sigma(i)} (1+X_i)^{\sigma(i)} (Q+ r X_i + Q X_i)^{n-1-\sigma(i)} \sum_{l=0}^{m} \left( \frac{X_k (1+X_k)}{Q+X_k} \right)^l \prod_{i \in \overline{S}} \left( \frac{X_i (1+X_i)}{Q+X_i} \right)^l. \\ \end{multline*} We evaluate the sum and rearrange some terms. \begin{multline*} \sum_{S} (-1)^{|S|} \frac{1-\left( \frac{X_k (1+X_k)}{Q+X_k} \right)^{m+1} \prod_{i \in \overline{S}} \left( \frac{X_i (1+X_i)}{Q+X_i} \right)^{m+1}} {1-\frac{X_k (1+X_k)}{Q+X_k} \prod_{i \in \overline{S}} \frac{X_i (1+X_i)}{Q+X_i}} \\ \times \prod_{i \in S} X_i^{m+n-1} (1+X_i)^{m+1} (Q+X_i)^{-m} (X_i+r+Q)^{n-2} \prod_{i \in \overline{S}} X_i (1+X_i) (Q+r X_i + Q X_i)^{n-2} \\ \times \sum_{\sigma} (-1)^{I(\sigma)}\prod_{i \in S} (Q X_i^{-1})^{\sigma(i)-1} (1+Q X_i^{-1})^{\sigma(i)-1} (Q+ r Q X_i^{-1} + Q^2 X_i^{-1})^{-\sigma(i)+1} \\ \times \prod_{i \in \overline{S}} X_i^{\sigma(i)-1} (1+X_i)^{\sigma(i)-1} (Q+ r X_i + Q X_i)^{-\sigma(i)+1} \end{multline*} The inner sum is a Vandermonde determinant, which we evaluate. We obtain \begin{multline*} \sum_{S} (-1)^{|S|} \frac{1-\left( \frac{X_k (1+X_k)}{Q+X_k} \right)^{m+1} \prod_{i \in \overline{S}} \left( \frac{X_i (1+X_i)}{Q+X_i} \right)^{m+1}} {1-\frac{X_k (1+X_k)}{Q+X_k} \prod_{i \in \overline{S}} \frac{X_i (1+X_i)}{Q+X_i}} \\ \times \prod_{i \in S} X_i^{m+n-1} (1+X_i)^{m+1} (Q+X_i)^{-m} (X_i+r+Q)^{n-2} \prod_{i \in \overline{S}} X_i (1+X_i) (Q+r X_i + Q X_i)^{n-2} \\ \times \prod_{1 \le i < j \le n, i,j \not=k} \left(\frac{Y_j (1+Y_j)}{Q+r Y_j + Q Y_j}-\frac{Y_i (1+Y_i)}{Q+r Y_i + Q Y_i} \right), \end{multline*} with $Y_i=X_i$ if $i \in \overline{S}$ and $Y_i=Q X_i^{-1}$ if $i \in S$. From \eqref{expand}, we add the sum over all $k$ and finally have the full right-hand side of \eqref{rewrite}. We exchange the sum over $k$ and $S$: now we sum over all proper subsets $S \subseteq [n]$ and all $k$ not in $S$. If we write $i \notin S$, then we mean $i \in \{1,2,\ldots,n\} \setminus S$. \begin{equation} \begin{aligned} \label{zwischen} &\sum_{S} (-1)^{|S|} \frac{1- \prod_{i \notin S} \left( \frac{X_i (1+X_i)}{Q+X_i} \right)^{m+1}} {1- \prod_{i \notin S} \frac{X_i (1+X_i)}{Q+X_i}} \\ &\quad \times \prod_{i \in S} X_i^{m+n-1} (1+X_i)^{m+1} (Q+X_i)^{-m} (X_i+r+Q)^{n-2} \prod_{i \notin S} X_i (1+X_i) (Q+r X_i + Q X_i)^{n-2} \\ &\quad \times \sum_{k \notin S} (-1)^{k+1} (Q-X_k^2) X_k^{-1} (1+X_k)^{-1} (Q+r X_k + Q X_k)^{-n+2} \\ &\qquad \times \prod_{1 \le i \le n, i \not= k} (Q+(Q+r) X_k+X_i + X_i X_k)(Q-X_i X_k) \\ &\qquad \times \prod_{1 \le i < j \le n, i,j \not=k} \left( \frac{Y_j (1+Y_j)}{Q+r Y_j + Q Y_j}- \frac{Y_i (1+Y_i)}{Q+r Y_i + Q Y_i} \right) \end{aligned} \end{equation} We rewrite $$ \begin{aligned} &(-1)^{k-1} \prod_{1 \le i \le n, i \not= k} (Q+(Q+r) X_k+X_i + X_i X_k)(Q-X_i X_k) \\ &\quad = \prod_{1 \le i \le k-1 \atop i \notin S} (Q+(Q+r) X_k+X_i + X_i X_k)(X_i X_k - Q) \\ &\qquad \times \prod_{k+1 \le i \le n \atop i \notin S} (Q+(Q+r) X_k+X_i + X_i X_k)(Q-X_i X_k) \\ &\qquad \times \prod_{i \in S} X_i (X_i + r + Q)(Q+r X_k + Q X_k) \\ &\qquad \times \prod_{1 \le i \le k-1 \atop i \in S} \left( \frac{X_k (1+X_k)}{Q+r X_k + Q X_k} - \frac{Q X_i^{-1} (1+Q X_i^{-1})}{Q+r Q X_i^{-1} Q^2 X_i^{-1}} \right) \\ &\qquad \times \prod_{k+1 \le i \le n \atop i \in S} \left( \frac{Q X_i^{-1} (1+Q X_i^{-1})}{Q+r Q X_i^{-1} Q^2 X_i^{-1}} - \frac{X_k (1+X_k)}{Q+r X_k + Q X_k} \right). \end{aligned} $$ We use this to rewrite \eqref{zwischen} as follows. \begin{align*} &\sum_{S} (-1)^{|S|} \frac{1- \prod_{i \notin S} \left( \frac{X_i (1+X_i)}{Q+X_i} \right)^{m+1}} {1- \prod_{i \notin S} \frac{X_i (1+X_i)}{Q+X_i}} \prod_{i} X_i \\ &\quad \times \prod_{i \in S} X_i^{m+n-1} (1+X_i)^{m+1} (Q+X_i)^{-m} (X_i+r+Q)^{n-1} \prod_{i \notin S} (1+X_i) (Q+r X_i + Q X_i)^{n-1} \\ &\quad \times \sum_{k \notin S} (Q-X_k^2) X_k^{-1} (1+X_k)^{-1} \\ &\qquad \times \prod_{1 \le i \le k-1 \atop i \notin S} \frac{(Q+(Q+r) X_k+X_i + X_i X_k)(X_i X_k - Q)}{(Q+r X_k + Q X_k)(Q+r X_i + Q X_i)} \\ &\qquad \times \prod_{k+1 \le i \le n \atop i \notin S} \frac{(Q+(Q+r) X_k+X_i + X_i X_k)(Q-X_i X_k)}{(Q+r X_k + Q X_k)(Q+r X_i + Q X_i)} \\ &\qquad \times \prod_{1 \le i \le k-1 \atop i \in S} \left( \frac{X_k (1+X_k)}{Q+r X_k + Q X_k} - \frac{Q X_i^{-1} (1+Q X_i^{-1})}{Q+r Q X_i^{-1} Q^2 X_i^{-1}} \right) \\ & \qquad \times \prod_{k+1 \le i \le n \atop i \in S} \left( \frac{Q X_i^{-1} (1+Q X_i^{-1})}{Q+r Q X_i^{-1} Q^2 X_i^{-1}} - \frac{X_k (1+X_k)}{Q+r X_k + Q X_k} \right) \\ &\qquad \times \prod_{1 \le i < j \le n, i,j \not=k} \left( \frac{Y_j (1+Y_j)}{Q+r Y_j + Q Y_j}-\frac{Y_i (1+Y_i)}{Q+r Y_i + Q Y_i} \right) \end{align*} This is further equal to \begin{equation} \label{next} \begin{aligned} &\sum_{S} (-1)^{|S|} \frac{1- \prod_{i \notin S} \left( \frac{X_i (1+X_i)}{Q+X_i} \right)^{m+1}} {1- \prod_{i \notin S} \frac{X_i (1+X_i)}{Q+X_i}} \prod_{i} X_i \\ &\quad \times \prod_{i \in S} X_i^{m+n-1} (1+X_i)^{m+1} (Q+X_i)^{-m} (X_i+r+Q)^{n-1} \\ &\quad \times \prod_{i \notin S} (1+X_i) (Q+r X_i + Q X_i)^{n-1} \\ &\quad \times \prod_{1 \le i < j \le n, \{i,j\} \cap S \not= \emptyset} \left( \frac{Y_j (1+Y_j)}{Q+r Y_j + Q Y_j}-\frac{Y_i (1+Y_i)}{Q+r Y_i + Q Y_i} \right) \\ &\quad \times \sum_{k \notin S} (Q-X_k^2) X_k^{-1} (1+X_k)^{-1} \\ &\qquad \times \prod_{1 \le i \le k-1 \atop i \notin S} \frac{(Q+(Q+r) X_k+X_i + X_i X_k)(X_i X_k - Q)}{(Q+r X_k + Q X_k)(Q+r X_i + Q X_i)} \\ &\qquad \times \prod_{k+1 \le i \le n \atop i \notin S} \frac{(Q+(Q+r) X_k+X_i + X_i X_k)(Q-X_i X_k)}{(Q+r X_k + Q X_k)(Q+r X_i + Q X_i)} \\ &\qquad \times \prod_{1 \le i < j \le n, i,j \notin S \cup \{k\}} \left( \frac{X_j (1+X_j)}{Q+r X_j + Q X_j}-\frac{X_i (1+X_i)}{Q+r X_i + Q X_i} \right). \end{aligned} \end{equation} We divide \eqref{leftright} by $\prod_{i=1}^n (Q+r X_i + Q X_i)^{n-1} X_i (1+X_i)$, and, after some further modifications, we obtain $$ \begin{aligned} & \left(\prod_{i=1}^{n} \frac{Q+X_i}{X_i (1+X_i)} - 1 \right) \prod_{1 \le i < j \le n} \left( \frac{X_j (1+X_j)}{Q+r X_j + Q X_j}-\frac{X_i (1+X_i)}{Q+r X_i + Q X_i} \right) \\ &= \sum_{k=1}^n (Q-X_k^{2}) X_k^{-1} (1+X_k)^{-1} \\ &\quad \times \prod_{j=1}^{k-1} \frac{(Q+(Q+r) X_k + X_j + X_k X_j)(X_j X_k-Q)}{(Q+r X_j + Q X_j)(Q+r X_k + Q X_k)} \\ &\quad \times \prod_{j=k+1}^{n} \frac{(Q+(Q+r) X_k + X_j + X_k X_j)(Q-X_j X_k)}{(Q+r X_j + Q X_j)(Q+r X_k + Q X_k)} \\ &\quad \times \prod_{1 \le i < j \le n, i,j \not=k} \left( \frac{X_j (1+X_j)}{Q+r X_j + Q X_j}-\frac{X_i (1+X_i)}{Q+r X_i + Q X_i} \right). \end{aligned} $$ We can use this to replace the sum over all $k \in S$ in \eqref{next} by something simpler. \begin{align*} &\sum_{S} (-1)^{|S|} \frac{1- \prod_{i \notin S} \left( \frac{X_i (1+X_i)}{Q+X_i} \right)^{m+1}} {1- \prod_{i \notin S} \frac{X_i (1+X_i)}{Q+X_i}} \prod_{i} X_i \\ &\quad \times \prod_{i \in S} X_i^{m+n-1} (1+X_i)^{m+1} (Q+X_i)^{-m} (X_i+r+Q)^{n-1} \prod_{i \notin S} (1+X_i) (Q+r X_i + Q X_i)^{n-1} \\ &\quad \times \prod_{1 \le i < j \le n, \{i,j\} \cap S \not= \emptyset} \left( \frac{Y_j (1+Y_j)}{Q+r Y_j + Q Y_j}-\frac{Y_i (1+Y_i)}{Q+r Y_i + Q Y_i} \right) \left(\prod_{i \notin S} \frac{Q+X_i}{X_i (1+X_i)} - 1 \right) \\ &\quad \times \prod_{1 \le i < j \le n, i,j \notin S} \left( \frac{X_j (1+X_j)}{Q+r X_j + Q X_j}-\frac{X_i (1+X_i)}{Q+r X_i + Q X_i} \right) \end{align*} This can be further simplified as follows, \begin{multline*} \sum_{S} (-1)^{|S|} \left( 1- \prod_{i \notin S} \left( \frac{X_i (1+X_i)}{Q+X_i} \right)^{m+1} \right) \\ \times \prod_{i \in S} X_i^{m+n} (1+X_i)^{m+1} (Q+X_i)^{-m} (X_i+r+Q)^{n-1} \prod_{i \notin S} (Q+X_i) (Q+r X_i + Q X_i)^{n-1} \\ \times \prod_{1 \le i < j \le n} \left( \frac{Y_j (1+Y_j)}{Q+r Y_j + Q Y_j}-\frac{Y_i (1+Y_i)}{Q+r Y_i + Q Y_i} \right), \end{multline*} recalling that $Y_i=X_i$ if $i \notin S$ and $Y_i = Q X_i^{-1}$ if $i \in S$. We write this as \begin{multline} \label{twosum} \sum_{S,\sigma} (-1)^{|S|+I(\sigma)} \prod_{i \in S} X_i^{m+n-\sigma(i)+1} (1+X_i)^{m+1} (Q+X_i)^{-m+\sigma(i)-1} (X_i+r+Q)^{n-\sigma(i)} \\ \times \prod_{i \notin S} X_i^{\sigma(i)-1} (1+X_i)^{\sigma(i)-1} (Q+X_i) (Q+r X_i + Q X_i)^{n-\sigma(i)} \\ - \sum_{S,\sigma} (-1)^{|S|+I(\sigma)} \prod_{i \in S} X_i^{m+n-\sigma(i)+1} (1+X_i)^{m+1} (Q+X_i)^{-m+\sigma(i)-1} (X_i+r+Q)^{n-\sigma(i)} \\ \times \prod_{i \notin S} X_i^{m+\sigma(i)} (1+X_i)^{m+\sigma(i)} (Q+X_i)^{-m} (Q+r X_i + Q X_i)^{n-\sigma(i)}. \end{multline} Recall that the sums are over all proper subsets $S$, but since the sums are equal for $S=\{1,2,\ldots,n\}$ we can also sum over all subsets $S$. Now the second sum is equal to \begin{multline*} \prod_{i=1}^{n} X_i^{m+1} (1+X_i)^{m+1} (Q+X_i)^{-m} \\ \times \det_{1 \le i, j \le n} \left( X_i^{j-1} (1+X_i)^{j-1} (Q+ r X_i + Q X_i)^{n-j} - X_i^{n-j} (Q+X_i)^{j-1} (X_i + r + Q)^{n-j} \right). \end{multline*} The determinant can be seen to vanish as follows: First observe that it is a polynomial in $X_1,\ldots,X_n$ of degree no greater than $2n-2$ in each $X_i$. For $1 \le i < j \le n$, the $i$-th row and the $j$-th row of the underlying matrix are collinear when setting $X_i=X_j$ or $X_i=Q X_j^{-1}$. Moreover, the $i$-th row vanishes when setting $X_i^2=Q$. It follows that $\prod_{i=1}^{n} (X_i^2-Q) \prod_{1 \le i < j \le n} (X_j-X_j)(1-Q X_i X_j)$ is a divisor of the determinant, but since it is of degree $2n$ in each $X_i$, the determinant vanishes. The second sum in \eqref{twosum} remains and it can easily be seen to be equal to $\det_{1 \le i,j \le n} \left( a_{j,m,n}(Q,r;X_i) \right)$. This concludes the proof of Theorem~\ref{boundedrQ}. \section{Combinatorial interpretations of the left-hand sides} \label{LHS} \subsection{Arrowed Gelfand-Tsetlin patterns} \label{agtp} To continue the analogy with the ordinary Little\-wood identity \eqref{littlewood} and Macdonald's bounded version \eqref {littlewoodbounded} of it, both sides of the identities \eqref{littlewoodASM2} and \eqref{rincluded} will be interpreted combinatorially. For the left-hand side, this was accomplished in another recent paper \cite{nASMDPP}, and we will describe the result and adjust to our context next. In order to motivate the definition for the combinatorial objects, recall the combinatorial interpretation of the left-hand sides of \eqref{littlewood} and \eqref{littlewoodbounded} in terms of Gelfand-Tsetlin patterns, which is described in Appendix~\ref{gtpattern}. We need to extend the discussion from there in so far that there is also a sensible extension of the definition of Gelfand-Tsetlin patterns to arbitrary integers sequences $(\lambda_1,\ldots,\lambda_n)$. The notion of signed intervals is crucial for this: $$ \si{a}{b} = \begin{cases} [a,b], & a \le b \\ \emptyset, & b=a-1 \\ [b+1,a-1], & b < a-1 \end{cases} $$ If we are in the last case, then the interval is said to be \emph{negative}. The condition that defines Gelfand-Tsetlin pattern can also be written as $a_{i,j} \in [a_{i+1,j},a_{i+1,j+1}]$. If the bottom row is weakly increasing, we can replace this condition also by $a_{i,j} \in \si{a_{i+1,j}}{a_{i+1,j+1}}$ (since we then have $a_{i+1,j} \le a_{i+1,j+1}$ as can be seen inductively with respect to $n$). We use this now as the definition for arbitrary bottom rows: A (generalized) Gelfand-Tsetlin pattern is a triangular array $A=(a_{i,j})_{1 \le j \le i \le n}$ of integers with $a_{i,j} \in \si{a_{i+1,j}}{a_{i+1,j+1}}$ for all $i,j$. Then the sign of a Gelfand-Tsetlin pattern $A$ is $$ (-1)^{\# \text{ of negative intervals $\si{a_{i+1,j}}{a_{i+1,j+1}}$}}=: \sgn A. $$ Then \begin{equation} \label{schurextension} s_{(\lambda_1,\ldots,\lambda_n)}(X_1,\ldots,X_n) = \sum_{A=\left( a_{i,j} \right)_{1 \le j \le i \le n}} \sgn A \prod_{i=1}^n X_i^{\sum_{j=1}^i a_{i,j} - \sum_{j=1}^{i-1} a_{i-1,j}}, \end{equation} where the sum is over all Gelfand-Tsetlin patterns $A=(a_{i,j})_{1 \le j \le i \le n}$ with bottom row $(\lambda_n,\lambda_{n-1},\ldots,\lambda_1)$ and $$s_{(\lambda_1,\ldots,\lambda_n)}(X_1,\ldots,X_n) = \frac{\det_{1 \le i,j \le n} \left( X_i^{\lambda_j+n-j} \right)}{\prod_{1 \le i < j \le n} (X_i-X_j)}.$$ This result is a special case of Theorem~\ref{robbins} below that will also cover the combinatorial interpretation of the left-hand side of \eqref{littlewoodASM2} and \eqref{rincluded}. However, this special case appeared essentially also earlier in \cite{Fis05} (with some details missing). \begin{definition} \label{def:AGTP} An \emph{arrowed Gelfand-Tsetlin pattern} (AGTP)\footnote{They appeared first in \cite{nASMDPP} as extended arrowed monotone triangles.} is a triangular array of the following form $$ \begin{array}{ccccccccccccccccc} & & & & & & & & a_{1,1} & & & & & & & & \\ & & & & & & & a_{2,1} & & a_{2,2} & & & & & & & \\ & & & & & & \dots & & \dots & & \dots & & & & & & \\ & & & & & a_{n-2,1} & & \dots & & \dots & & a_{n-2,n-2} & & & & & \\ & & & & a_{n-1,1} & & a_{n-1,2} & & \dots & & \dots & & a_{n-1,n-1} & & & & \\ & & & a_{n,1} & & a_{n,2} & & a_{n,3} & & \dots & & \dots & & a_{n,n} & & & \end{array}, $$ where each entry $a_{i,j}$ is an integer decorated with an element from $\{\nwarrow, \nearrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow,\emptyset \}$ and the following is satisfied for each entry $a$ not in the bottom row: Suppose $b$ is the $\swarrow$-neighbor of $a$ and $c$ is the $\searrow$-neighbor of $a$, respectively, i.e., $$ \begin{array}{ccc} &a& \\ b&&c \end{array}. $$ Depending on the decoration of $b, c$, denoted by $\operatorname{decor}(b)$ and $\operatorname{decor} ( c )$, respectively, we need to consider four cases: \begin{itemize} \item $(\operatorname{decor}(b),\operatorname{decor}( c )) \in \{\nwarrow,\emptyset\} \times \{\nearrow, \emptyset\}$: $a \in \si{b}{c}$ \item $(\operatorname{decor}(b),\operatorname{decor}( c )) \in \{\nwarrow,\emptyset\} \times \{\nwarrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow\}$: $a \in \si{b}{c-1}$ \item $(\operatorname{decor}(b),\operatorname{decor}( c )) \in \{\nearrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow \} \times \{\nearrow,\emptyset\}$: $a \in \si{b+1}{c}$ \item $(\operatorname{decor}(b),\operatorname{decor}( c)) \in \{\nearrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow \} \times \{\nwarrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow\}$: $a \in \si{b+1}{c-1}$ \end{itemize} \end{definition} An example is provided next. We write $^\nwarrow e, e^\nearrow, ^\nwarrow e^\nearrow, e$ if the entry $e$ is decorated with $\nwarrow, \nearrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow,\emptyset$, respectively. $$ \begin{array}{ccccccccccccccccc} & & & & & & & & ^\nwarrow2 & & & & & & & & \\ & & & & & & & 2 & & ^\nwarrow3^\nearrow & & & & & & & \\ & & & & & & ^\nwarrow2 & & 2^\nearrow & & 3^\nearrow & & & & & & \\ & & & & & 3 & & ^\nwarrow2 & & ^\nwarrow3^\nearrow & & ^\nwarrow3^\nearrow & & & & & \\ & & & & 2^\nearrow & & 4 & & ^\nwarrow2^\nearrow & & 3^\nearrow & & 2 & & & & \\ & & & ^\nwarrow6 & & ^\nwarrow2^\nearrow & & 5 & & 1^\nearrow & & ^\nwarrow4 & & ^\nwarrow2^\nearrow & & & \end{array} $$ We define the sign of an AGTP $A=(a_{i,j})_{1 \le j \le i \le n}$ as follows: Each negative interval $\si{a_{i+1,j}(+1)}{a_{i+1,j+1}(-1)}$ with $i \ge 1$ and $j \le i$ contributes a multiplicative $-1$, choosing $a_{i+1,j}+1$ iff $\operatorname{decor}(a_{i+1,j}) \in \{\nearrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow \}$ and $a_{i+1,j}$ otherwise, and choosing $a_{i+1,j+1} - 1$ iff $\operatorname{decor}(a_{i+1,j+1}) \in \{ \nwarrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow \}$ and $a_{i+1,j+1}$ otherwise. There are no negative intervals in rows $1,2,3$, two in rows $4,5$ and three in row $6$, so that the sign of the pattern is $-1$. We associate the following weight to a given arrowed Gelfand-Tsetlin pattern $A=(a_{i,j})_{1 \le j \le i \le n}$: $$ {\operatorname{W}}(A) = \sgn(A) t^{\# \emptyset} u^{\# \nearrow} v^{\# \nwarrow} w^{\# \nwarrow \!\!\!\!\!\;\!\! \nearrow} \prod_{i=1}^{n} X_i^{\sum_{j=1}^i a_{i,j} - \sum_{j=1}^{i-1} a_{i-1,j} + \# \nearrow \text{in row $i$ } - \# \nwarrow \text{in row $i$ }} $$ The weight of our example is $$ - t^5 u^5 v^5 w^6 X_1 X_2^3 X_3^3 X_4^3 X_5^4 X_6^6. $$ For this paper only arrowed Gelfand-Tsetlin patterns with weakly increasing bottom row are relevant and in this case the description of the objects can be simplified considerably as follows. \begin{prop} An arrowed Gelfand-Tsetlin pattern with weakly increasing bottom row is an ordinary Gelfand-Tsetlin pattern (i.e., with weakly increasing rows), where each entry is decorated with an element from $\{\nwarrow, \nearrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow,\emptyset \}$ such that the following is satisfied. \begin{itemize} \item Suppose an entry $a$ is equal to its $\nearrow$-neighbour and $a$ is decorated with either $\nearrow$ or $\nwarrow \!\!\!\!\!\;\!\! \nearrow$ (i.e., an arrow is pointing from $a$ to its $\nearrow$-neighbour), then the entry right of $a$ in the same row is also equal to $a$ and decorated with $\nwarrow$ or $\nwarrow \!\!\!\!\!\;\!\! \nearrow$. \item Suppose an entry $a$ is equal to its $\nwarrow$-neighbour and $a$ is decorated with either $\nwarrow$ or $\nwarrow \!\!\!\!\!\;\!\! \nearrow$ (i.e., an arrow is pointing from $a$ to its $\nwarrow$-neighbour), then the entry left of $a$ in the same row is also equal to $a$ and decorated with $\nearrow$ or $\nwarrow \!\!\!\!\!\;\!\! \nearrow$. \end{itemize} The sign is $-1$ to the number of entries $a$ that are equal to their $\swarrow$-neighbor $b$ as well as to to their $\searrow$-neighbor $c$, and $b$ is decorated with $\nearrow$ or $\nwarrow \!\!\!\!\!\;\!\! \nearrow$ and $c$ is decorated with $\nwarrow$ and $\nwarrow \!\!\!\!\!\;\!\! \nearrow$. \end{prop} \begin{proof} Suppose $(a_{i,j})_{1 \le j \le i \le n}$ is an AGTP. If $a_{i+1,j} < a_{i+1,j+1}$ for particular $i,j$, then $a_{i+1,j} \le a_{i,j} \le a_{i+1,j+1}$. The first inequality has to be strict if the decoration of $a_{i+1,j}$ contains an arrow pointing towards $a_{i,j}$ (i.e., $\operatorname{decor}(a_{i+1,j}) \in \{\nearrow,\nwarrow \!\!\!\!\!\;\!\! \nearrow\}$), while the second inequality has to be strict if $a_{i+1,j+1}$ contains an arrow pointing towards $a_{i,j}$ (i.e., $\operatorname{decor}(a_{i+1,j+1}) \in \{\nwarrow,\nwarrow \!\!\!\!\!\;\!\! \nearrow\}$). On the other hand, if $a_{i+1,j} = a_{i+1,j+1}$ for particular $i,j$, then $a_{i+1,j}=a_{i,j}=a_{i+1,j+1}$. In this case $$ (\operatorname{decor}(a_{i+1,j}),\operatorname{decor}(a_{i+1,j+1})) \in \{\emptyset,\nwarrow\} \times \{\emptyset,\nearrow\} $$ or \begin{equation} \label{sign} (\operatorname{decor}(a_{i+1,j}),\operatorname{decor}(a_{i+1,j+1})) \in \{\nearrow,\nwarrow \!\!\!\!\!\;\!\! \nearrow\} \times \{\nwarrow,\nwarrow \!\!\!\!\!\;\!\! \nearrow\}, \end{equation} where in the second case there is a contribution of $-1$ to the sign of the object. These observations imply that, if the bottom row is weakly increasing, then the underlying undecorated triangular array is an ordinary Gelfand-Tsetlin pattern and that the properties on the decoration stated in the proposition are satisfied. The only instance when we have a contribution to the sign is in the case of \eqref{sign}. Conversely, a decoration of a given Gelfand-Tsetlin pattern that follows the rule as given in the statement of the proposition is eligible for an arrowed Gelfand-Tsetlin pattern according to Definition~\ref{def:AGTP}. \end{proof} \begin{remark} In the case that the bottom row of an arrowed Gelfand-Tsetlin pattern is strictly increasing and we forbid the decoration $\emptyset$, we have that all rows are strictly increasing and we obtain a monotone triangle. Recall that monotone triangles are defined as Gelfand-Tsetlin patterns with strictly increasing rows; their significance comes from the fact that monotone triangles with bottom row $1,2,\ldots,n$ are in easy bijective correspondence with $n \times n$ alternating sign matrices, see, e.g., \cite{Bre99}. In such a case, there is no instance where we gain a $-1$ that contributes to the sign. These objects were used in \cite{nASMDPP} to study alternating sign matrices. Among other things, the generating function of these decorated monotone triangles can be interpreted as a generating function of (undecorated) monotone triangles, thus of alternating sign matrices. \end{remark} The following explicit formula for the generating function of arrowed Gelfand-Tsetlin patterns with fixed bottom row $k_1,k_2,\ldots,k_n$ is proved in \cite{nASMDPP}. \begin{theorem} \label{robbins} The generating function of arrowed Gelfand-Tsetlin patterns with bottom row $k_1,\ldots,k_n$ is $$ \prod_{i=1}^{n} (t + u X_i + v X_i^{-1} + w) \prod_{1 \le i < j \le n} \left( t + u {\operatorname{E}}_{k_i} + v {\operatorname{E}}_{k_j}^{-1} + w {\operatorname{E}}_{k_i} {\operatorname{E}}_{k_j}^{-1} \right) s_{(k_n,k_{n-1},\ldots,k_1)}(X_1,\ldots,X_n), $$ where ${\operatorname{E}}_x$ denotes the shift operator, defined as ${\operatorname{E}}_x p(x) = p(x+1)$. \end{theorem} The formula has to be applied as follows: First interpret $k_1,\ldots,k_n$ as variables and apply the operator $\prod_{1 \le i < j \le n} \left( t + u {\operatorname{E}}_{k_i} + v {\operatorname{E}}_{k_j}^{-1} + w {\operatorname{E}}_{k_i} {\operatorname{E}}_{k_j}^{-1} \right)$ to $s_{(k_n,k_{n-1},\ldots,k_1)}(X_1,\ldots,X_n)$. This will result in a linear combination of expressions of the form $s_{(k_n+i_n,k_{n-1}+i_{n-1},\ldots,k_1+i_1)}(X_1,\ldots,X_n)$ for some (varying) integers $i_j$. The $k_j$ are only specialized to the actual integers after that. Note that we do not necessarily have $k_n+i_n \ge k_{n-1}+i_{n-1} \ge \ldots \ge k_1+i_1$ even if $k_n \ge k_{n-1} \ge \ldots \ge k_1$, so that the extension of the Schur polynomial in \eqref{schurextension} is necessary. \medskip \begin{example} We illustrate the theorem on the example $(k_1,k_2,k_3)=(1,2,3)$. We list the $8$ Gelfand-Tsetlin pattern with bottom row $1,2,3$ and indicate the possible decorations (one will be listed twice with a disjoint set of decorations), where $L=\{\emptyset, \nwarrow~\}$, $R=\{\emptyset, \nearrow\}$ and $LR= \{\emptyset,\nwarrow,\nearrow,\nwarrow \!\!\!\!\!\;\!\! \nearrow\}$, and on the right we indicate the generating function restricted to the particular underlying Gelfand-Tsetlin patterns with the indicated decorations, where we use $$ L(X)=t+ v X^{-1}, R(X)=t+u X \quad \text{and} \quad LR(X) = t + u X + v X^{-1} + w. $$ \begin{tabular}{cl} $ \begin{array}{ccccc} && {LR \atop 1} && \\ & {L \atop 1} && {LR \atop 2} & \\ {L \atop 1} && {L \atop 2} && {LR \atop 3} \end{array} $ & $X_1 X_2^2 X_3^3 LR(X_1) L(X_2) LR(X_2) L(X_3)^2 LR(X_3)$ \vspace{3mm} \\ $ \begin{array}{ccccc} && {LR \atop 2} && \\ & {LR \atop 1} && {R \atop 2} & \\ {L \atop 1} && {L \atop 2} && {LR \atop 3} \end{array} $ & $X_1^2 X_2 X_3^3 LR(X_1) LR(X_2) R(X_2) L(X_3)^2 LR(X_3)$ \vspace{3mm} \\ $ \begin{array}{ccccc} && {LR \atop 1} && \\ & {L \atop 1} && {LR \atop 3} & \\ {L \atop 1} && {LR \atop 2} && {R \atop 3} \end{array} $ & $X_1 X_2^3 X_3^2 LR(X_1) L(X_2) LR(X_2) L(X_3) LR(X_3) R(X_3)$ \vspace{3mm} \\ $ \begin{array}{ccccc} && {LR \atop 2} && \\ & {LR \atop 1} && {LR \atop 3} & \\ {L \atop 1} && {LR \atop 2} && {R \atop 3} \end{array} $ & $X_1^2 X_2^2 X_3^2 LR(X_1) LR(X_2)^2 L(X_3) LR(X_3) R(X_3)$ \vspace{3mm} \\ $ \begin{array}{ccccc} && {LR \atop 3} && \\ & {LR \atop 1} && {R \atop 3} & \\ {L \atop 1} && {LR \atop 2} && {R \atop 3} \end{array} $ & $X_1^3 X_2 X_3^2 LR(X_1) LR(X_2) R(X_2) L(X_3) LR(X_3) R(X_3)$ \vspace{3mm} \\ $ \begin{array}{ccccc} && {LR \atop 2} && \\ & {L \atop 2} && {LR \atop 3} & \\ {LR \atop 1} && {R \atop 2} && {R \atop 3} \end{array} $ & $X_1^2 X_2^3 X_3 LR(X_1) L(X_2) LR(X_2) LR(X_3) R(X_3)^2$ \vspace{3mm} \\ $ \begin{array}{ccccc} && {LR \atop 3} && \\ & {LR \atop 2} && {R \atop 3} & \\ {LR \atop 1} && {R \atop 2} && {R \atop 3} \end{array} $ & $X_1^3 X_2^2 X_3 LR(X_1) LR(X_2) R(X_2) LR(X_3) R(X_3)^2$ \vspace{3mm} \\ $ \begin{array}{ccccc} && {LR \atop 2} && \\ & {L \atop 2} && {R \atop 2} & \\ {LR \atop 1} && {\emptyset \atop 2} && {LR \atop 3} \end{array} $ & $X_1^2 X_2^2 X_3^2 LR(X_1) L(X_2) R(X_2) t LR(X_3)^2 $ \vspace{3mm} \\ $ \begin{array}{ccccc} && {LR \atop 2} && \\ & {\{ \nearrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow \} \atop 2} && { \{ \nwarrow, \nwarrow \!\!\!\!\!\;\!\! \nearrow \} \atop 2} & \\ {LR \atop 1} && {\emptyset \atop 2} && {LR \atop 3} \end{array} $ & $-X_1^2 X_2^2 X_3^2 LR(X_1) (w+uX_2)(w+v X_2^{-1}) t LR(X_3)^2 $ \vspace{3mm} \\ \end{tabular} \end{example} It is convenient for us to rewrite the formula from Theorem~\ref{robbins} as follows. \begin{cor} \label{bialternant} The generating function of arrowed Gelfand-Tsetlin patterns with bottom row $k_1,\ldots,k_n$ is \begin{equation} \label{asym} \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i \le j \le n} \left(v + w X_i + t X_j + u X_i X_j \right) \prod_{i=1}^{n} X_i^{k_i-1} \right] }{\prod_{1 \le i < j \le n} (X_j - X_i)}. \end{equation} \end{cor} \begin{proof} Observe that \begin{align*} & \prod_{i=1}^{n} (t + u X_i + v X_i^{-1}+w) \prod_{1 \le i < j \le n} \left(t+ u {\operatorname{E}}_{k_i} + v {\operatorname{E}}_{k_j}^{-1} + w {\operatorname{E}}_{k_i} {\operatorname{E}}_{k_j}^{-1} \right) s_{(k_n,k_{n-1},\ldots,k_1)}(X_1,\ldots,X_n) \notag \\ \quad &= \prod_{i=1}^{n} (u X_i + v X_i^{-1}+w) \prod_{1 \le i < j \le n} \left(t + u {\operatorname{E}}_{k_i} + v {\operatorname{E}}_{k_j}^{-1} + w {\operatorname{E}}_{k_i} {\operatorname{E}}_{k_j}^{-1} \right) \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{i=1}^{n} X_i^{k_i+i-1} \right] }{\prod_{1 \le i < j \le n} (X_j - X_i)} \notag \\ \quad &= \prod_{i=1}^{n} (t + u X_i + v X_i^{-1}+w) \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} \left(t + u {\operatorname{E}}_{k_i} + v {\operatorname{E}}_{k_j}^{-1} + w {\operatorname{E}}_{k_i} {\operatorname{E}}_{k_j}^{-1} \right) \prod_{i=1}^{n} X_i^{k_i+i-1} \right] }{\prod_{1 \le i < j \le n} (X_j - X_i)} \notag \\ \quad &= \prod_{i=1}^{n} (t + u X_i + v X_i^{-1}+w) \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i < j \le n} \left(t + u X_i + v X_j^{-1} + w X_i X_j^{-1} \right) \prod_{i=1}^{n} X_i^{k_i+i-1} \right] }{\prod_{1 \le i < j \le n} (X_j - X_i)} \notag \\ \quad &= \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i \le j \le n} \left(v + w X_i + t X_j + u X_i X_j \right) \prod_{i=1}^{n} X_i^{k_i-1} \right] }{\prod_{1 \le i < j \le n} (X_j - X_i)} \end{align*} and the assertion follows. \end{proof} \begin{remark} Suppose $(k_1-1,k_2-1,\ldots,k_n-1)$ is a partition (allowing zero parts) then, when setting $u=v=0$, $w=1$ and replacing $t$ by $-t$ \eqref{asym}, we obtain the Hall-Littlewood polynomials \cite{macdonald} up to a factor that is a rational function in $t$. \end{remark} \subsection{Generating function with respect to a Schur polynomial weight} We are now ready to obtain our first interpretation. Multiplying \eqref{littlewoodASM2} and \eqref{rincluded} with $\prod_{i=1}^{n} (X_i^{-1} + 1+w + X_i)$ gives \begin{multline} \label{littlewoodASM3} \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i \le j \le n} (1+ w X_i + X_j + X_i X_j) \sum_{0 \le k_1 < k_2 < \ldots < k_n} X_1^{k_1-1} X_2^{k_2-1} \cdots X_n^{k_n-1} \right]}{\prod_{1 \le i < j \le n} (X_j-X_i)} \\ = \prod_{i=1}^{n} (X_i^{-1} + 1+w + X_i) \prod_{i=1}^{n} \frac{1}{1-X_i} \prod_{1 \le i < j \le n} \frac{1+X_i + X_j + w X_i X_j}{1-X_i X_j}, \end{multline} and \begin{multline} \label{rincluded1} \frac{\mathbf{ASym}_{X_1,\ldots,X_n} \left[ \prod_{1 \le i \le j \le n} (1+w X_i+X_j + X_i X_j) \sum_{0 \le k_1 < k_2 < \ldots < k_n \le m} X_1^{k_1-1} X_2^{k_2-1} \cdots X_n^{k_n-1} \right]}{\prod_{1 \le i < j \le n} (X_j-X_i)}  \\ = \prod_{i=1}^{n} (X_i^{-1} + 1+w + X_i) \\ \times \frac{\det_{1 \le i, j \le n} \left( X_i^{j-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{m+2n-j} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right)}{\prod\limits_{i=1}^n (1-X_i) \prod\limits_{1 \le i < j \le n} (1-X_i X_j)(X_j-X_i)}, \end{multline} respectively, and we can now interpret the left-hand sides as the generating function of arrowed Gelfand-Tsetlin patterns with non-negative strictly increasing bottom row, where we need to specialize $t=u=v=1$ in the weight and in the second case the entries in the bottom row are less than or equal to $m$. \begin{remark} \begin{enumerate} \item For $\mathbf{X}=(X_1,\ldots,X_n)$, let $\mathcal{AGTP}(t,u,v,w;\mathbf{k};\mathbf{X})$ denote the generating function of arrowed Gelfand-Tsetlin patterns with bottom row $\mathbf{k}=(k_1,\ldots,k_n)$. Then, using \eqref{asym}, it follows by changing $(X_1,\ldots,X_n)$ to $(X_n,X_{n-1},\ldots,X_1)$ that $$ \mathcal{AGTP}(t,u,v,w;\mathbf{k};\mathbf{X}) = (-1)^{\binom{n}{2}} \mathcal{AGTP}(w,u,v,t;\overline{\mathbf{k}};\mathbf{X}), $$ where $\overline{\mathbf{k}}=(k_n,\ldots,k_1)$. Therefore, the left-hand sides are up to the sign $(-1)^{\binom{n}{2}}$ also the generating function of AGTPs with strictly \emph{decreasing} bottom row of non-negative integers, where we need to set $u=v=w=1$ and replace $t$ by $w$ in the weight, and, in the case of \eqref{rincluded}, the entries in the bottom row are less than or equal to $m$. \item For the case $t=0$, there is worked out a possibility in \cite{nASMDPP} to get around the multiplication with the extra factor $\prod_{i=1}^{n} (X_i^{-1} + 1+w + X_i)$ by working with ``down arrows'' as decorations. In our application, this can be used in combination with our second combinatorial interpretation concerning AGTPs with strictly decreasing bottom row to give combinatorial interpretations of the left-hand sides of \eqref{littlewoodASM2} and \eqref{rincluded} in the special case $w=0$. It is an open problem to explore whether the down-arrowed array can be extended to general $t$. \end{enumerate} \end{remark} In Appendix~\ref{furtherLHS}, we develop some other (maybe less interesting) combinatorial interpretations of the left-hand sides, which we include for the sake of completeness. \section{Combinatorial interpretations of the right-hand sides of \eqref{littlewoodASM3} and \eqref{rincluded1}} \label{RHS} \subsection{Right-hand side of \eqref{littlewoodASM3}} For the right-hand side of \eqref{littlewoodASM3}, which is \begin{equation} \label{RHSsimple} \prod_{i=1}^{n} \frac{X_i^{-1}+1+w+X_i}{1-X_i} \prod_{1 \le i < j \le n} \frac{1+X_i + X_j + w X_i X_j}{1-X_i X_j}, \end{equation} it is straightforward to give a combinatorial interpretation as a generating function. Recall that, in the ordinary case \eqref{littlewood}, the right-hand side $\prod_{i=1}^{n} \frac{1}{1-X_i} \prod_{1 \le i < j \le n} \frac{1}{1-X_i X_j}$ is interpreted as two-line arrays with entries in $\{1,2,\ldots,n\}$, ordered lexicographically, with the top element of each column being greater than or equal to its bottom element. The exponent of $X_i$ in the weight is computed by subtracting from the total number of $i$'s in the two-line array the number of columns with $i$ as top and bottom element. To extend this to an interpretation of \eqref{RHSsimple}, we have one additional column $\binom{j}{i}$ for all pairs $i \le j$, which are either overlined, underlined, both or neither. An overlined column $\binom{j}{i}$ with $i<j$, contributes an additional multiplicative $X_j$ to the weight, while an underlined column with $i$ as bottom element contributes an additional $X_i$, and if a column is overlined and underlined then such a column contributes, in addition to $X_i X_j$, $w$. Moreover, an overlined column $\binom{i}{i}$ contributes an additional $X_i$ to the weight and if it is underlined then it contributes $X_i^{-1}$ to the weight, and, again, if the column is overlined and underlined, then it contributes also $w$. In both cases, if the column is neither underlined nor overlined, it contributes nothing in addition. \subsection{Right-hand side of \eqref{rincluded1}} The following theorem provides an interpretation of the right-hand side of \eqref{rincluded1} as a weighted count of (partly non-intersecting) lattice paths. This right-hand side differs from the right-hand side of \eqref{rincluded} by a simple multiplicative factor. We work as long as possible with general $w$, however, it will turn out that we need to specialize to $w=0,1$ at some point to obtain a nicer interpretation. We present two different proofs to obtain the result, where the second one is only sketched. Figure~\ref{examplepaths} seeks to illustrate the theorem in the case that $m$ is odd. \begin{theorem} \label{RHScomplicated} (1) Assume that $m=2l+1$. Then the right-hand side of \eqref{rincluded1} has the following interpretation as weighted count of families of $n$ lattice paths. \begin{itemize} \item The $i$-th lattice path starts in one point in the set $A_i=\{(-3i+1,-i+1),(-i+1,-3i+1)\}$, $i=1,2,\ldots,n$, and the end points of the paths are $E_j=(n-j+l+1,j-l-2)$, $j=1,2,\ldots,n$. \item Below and on the line $x+y=0$, the step set is $\{(1,1),(-1,1)\}$ for steps that start in $(-3i+1,-i+1)$ and it is $\{(1,1),(1,-1)\}$ for steps that start in $(-i+1,-3i+1)$. Steps of type $(-1,1)$ and $(1,-1)$ with distance $0,2,4,\ldots$ from $x+y=0$ are equipped with the weights $X_1,X_2,X_3,\ldots$, respectively, while such steps with distance $1,3,5,\ldots$ are equipped with the weights $X_1^{-1},X_2^{-1},X_3^{-1},\ldots$, respectively. \item Above the line $x+y=0$, the step set is $\{(1,0),(0,1)\}$. Above the line $x+y=j-1$, horizontal steps of the path that ends in $E_j$ are equipped with the weight $w$. \item The paths can be assumed to be non-intersecting below the line $x+y=0$. In case $w=1$, we can also assume them to be non-intersecting above the line $x+y=0$. In case $w=0$, $E_j$ can be replaced by $E'_j=(n-j+l+1,2j-n-l-2)$, $j=1,2,\ldots,n$, and then we can also assume the paths to be non-intersecting above the line $x+y=0$. \item The sign of family of paths is the sign of the permutation $\sigma$ with the property that the $i$-th path connects $A_i$ to $E_{\sigma(i)}$ with an extra contribution of $-1$ if we choose $(-i+1,-3i+1)$ from $A_i$. Moreover, we have an overall factor of $$ (-1)^{\binom{n+1}{2}} \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w + X_i)(1+X_i). $$ \item In case $w=0,1$, when restricting to non-intersecting paths, let $1 \le i_1 < i_2,\ldots < i_m < n$ be the indices for which we chose $(-3i+1,-i+1)$ from $A_i$. Then the sign can assumed to be $(-1)^{i_1+\ldots+i_m}$ and the overall factor is $$ \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w + X_i)(1+X_i). $$ \end{itemize} (2) Assume that $m=2l$. Then, to obtain an interpretation for the right-hand side of \eqref{rincluded1}, we only need to replace $E_j$ by a set of two possible endpoints $E_j=\{(n-j+l+1,j-l-2),(n-j+l,j-l-1)\}$. The overall factor is $$ (-1)^{\binom{n+1}{2}} \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w + X_i) $$ in the case when we do not specialize $w$. The endpoints are replaced by $E'_j=\{(n-j+l+1,2j-n-l-2),(n-j+l,2j-n-l-1)\}$ if $w=0$. In case $w=0,1$ if we restrict to non-intersecting paths and the sign is taken care of as above, then the overall factor is $$ \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w + X_i). $$ \end{theorem} We discuss the weight and the sign on the example in Figure~\ref{examplepaths}. The weights that come from the individual paths are $$ X_1^{-1} \cdot X_1^{-1} \cdot X_2 \cdot X_1 X_3 \cdot X_2 X_3^{-1} \cdot X_5^{-2}, $$ where the factors are arranged in a manner that the $i$-th factor is the weight of the path that starts in the set $A_i$. To compute the sign, observe that $\sigma = (6 \, 5 \, 4 \, 3 \, 2 \, 1)$ in one-line notation so that $\sgn \sigma = -1$ and that we choose the second starting point in $A_i$ except for $i=1$, so that the total sign is $(-1)\cdot(-1)^5=1$. \begin{figure}[htb] \centering \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-17.75,0)--(8.75,0) node[right]{$x$}; \draw[->,thick] (0,-17.75)--(0,2.75) node[above]{$y$}; \draw[dotted,thick] (-2.75,2.75)--(8.75,-8.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (-1.75,2.75)--(8.75,-7.75) node[right]{\tiny $x+y=1$}; \draw[dotted,thick] (-0.75,2.75)--(8.75,-6.75) node[right]{\tiny $x+y=2$}; \draw[dotted,thick] (0.25,2.75)--(8.75,-5.75) node[right]{\tiny $x+y=3$}; \draw[dotted,thick] (1.25,2.75)--(8.75,-4.75) node[right]{\tiny $x+y=4$}; \draw[dotted,thick] (2.25,2.75)--(8.75,-3.75) node[right]{\tiny $x+y=5$}; \draw [help lines,step=1cm,dashed] (-17.75,-17.75) grid (8.75,2.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (-8,-2) circle (5pt) node[right]{$A_3$}; \fill (-11,-3) circle (5pt) node[right]{$A_4$}; \fill (-14,-4) circle (5pt) node[right]{$A_5$}; \fill (-17,-5) circle (5pt) node[right]{$A_6$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (-2,-8) circle (5pt) node[left]{$A_3$}; \fill (-3,-11) circle (5pt) node[left]{$A_4$}; \fill (-4,-14) circle (5pt) node[left]{$A_5$}; \fill (-5,-17) circle (5pt) node[left]{$A_6$}; \fill (8,-3) circle (5pt) node[right]{$E_1$}; \fill (7,-2) circle (5pt) node[right]{$E_2$}; \fill (6,-1) circle (5pt) node[right]{$E_3$}; \fill (5,0) circle (5pt) node[right]{$E_4$}; \fill (4,1) circle (5pt) node[right]{$E_5$}; \fill (3,2) circle (5pt) node[right]{$E_6$}; \path[decoration=arrows, decorate] (-2,0) --++ (-1,1) --++ (1,1) --++ (1,0) --++ (1,0) --++ (1,0) --++ (1,0) --++ (1,0); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1) --++ (1,1) --++ (1,-1) --++ (1,1) --++ (0,1) --++ (0,1) --++ (1,0) --++ (0,1) --++ (0,1); \path[decoration=arrows, decorate] (-2,-8) --++ (1,1) --++ (1,1) --++ (1,1) --++ (1,-1) --++ (1,1) --++ (1,1) --++ (0,1) --++ (0,1) --++ (1,0)--++(0,1)--++(0,1); \path[decoration=arrows, decorate] (-3,-11) --++ (1,1) --++ (1,1) --++ (1,1) --++ (1,-1) --++ (1,1) --++ (1,1) --++ (1,1) --++ (1,1)--++ (1,-1) --++ (0,1) --++ (0,1)--++(0,1)--++(0,1) --++(0,1); \path[decoration=arrows, decorate] (-4,-14) --++ (1,1) --++ (1,1) --++ (1,1) --++ (1,1) --++ (1,-1) --++ (1,1) --++ (1,1) --++ (1,1)--++ (1,-1) --++ (1,1) --++ (1,1)--++ (0,1) --++ (0,1)--++(0,1) --++(0,1)--++(0,1); \path[decoration=arrows, decorate] (-5,-17) --++ (1,1) --++ (1,1) --++ (1,-1) --++ (1,-1) --++ (1,1) --++ (1,1) --++ (1,1) --++ (1,1)--++ (1,1) --++ (1,1) --++ (1,1)--++ (1,1) --++ (1,1) --++ (0,1) --++ (0,1)--++(0,1) --++(0,1)--++(0,1); \end{tikzpicture} \caption{\label{examplepaths} An example of families of lattice paths in Theorem~\ref{RHScomplicated}.} \end{figure} In the case that $m$ is odd, we always need to choose the second lattice point in $A_i$ if $l \ge n-2$ because then all $E_i$ have a non-positive $y$-coordinate and this implies that they cannot be reached by any of the first lattice points in $A_i$ since any lattice path starting from the first lattice point in $A_i$ intersects the line $x+y=0$ in a lattice point with positive $y$-coordinate. This implies that, in the non-intersecting case, the sign is always $1$. In the case that $m$ is even, the condition is $l \ge n-1$. In theses cases and when we have in addition $w=0$, we can translate the lattice paths easily into pairs of plane partitions. The case $m=2l+1$ is illustrated in Figure~\ref{odd1}, while the case $m=2l$ is illustrated in Figure~\ref{even}. A similar result can in principal be derived for the case $w=1$, but we omit this here. \begin{figure} \begin{tikzpicture}[scale=.40,baseline=(current bounding box.center)] \draw[->,thick] (-7.75,0)--(20.75,0) node[right]{\tiny$x$}; \draw[->,thick] (0,-27.75)--(0,2.75) node[above]{\tiny$y$}; \draw [help lines,step=1cm,dashed] (-7.75,-27.75) grid (20.75,2.75); \fill[red] (0,0) circle (5pt); \draw[dotted,thick] (-2.75,2.75)--(20.75,-20.75) node[below]{\footnotesize $x+y=0$}; \fill (0,-2) circle (5pt) node[left]{\tiny$A_1$}; \fill (-1,-5) circle (5pt) node[left]{\tiny$A_2$}; \fill (-2,-8) circle (5pt) node[left]{\tiny$A_3$}; \fill (-3,-11) circle (5pt) node[left]{\tiny$A_4$}; \fill (-4,-14) circle (5pt) node[left]{\tiny$A_5$}; \fill (-5,-17) circle (5pt) node[left]{\tiny$A_6$}; \fill (-6,-20) circle (5pt) node[left]{\tiny$A_7$}; \fill (19,-19) circle (5pt) node[right]{\tiny$E_1$}; \fill (18,-17) circle (5pt) node[right]{\tiny$E_2$}; \fill (17,-15) circle (5pt) node[right]{\tiny$E_3$}; \fill (16,-13) circle (5pt) node[right]{\tiny$E_4$}; \fill (15,-11) circle (5pt) node[right]{\tiny$E_5$}; \fill (14,-9) circle (5pt) node[right]{\tiny$E_6$}; \fill (13,-7) circle (5pt) node[right]{\tiny$E_7$}; \path[decoration=arrows, decorate] (0,-2)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)--++(1,1) --++ (1,0)node[above left]{\tiny$1$}--++ (1,0)node[above left]{\tiny$2$} --++(0,1)--++ (1,0) node[above left]{\tiny$4$}--++(1,0) node[above left]{\tiny$5$} --++(1,0) node[above left]{\tiny$6$}; \path[decoration=arrows, decorate] (-1,-5)node[below=0.1cm]{\tiny$4$} --++ (1,-1)node[below=0.1cm]{\tiny$4$} --++ (1,-1)node[below=0.1cm]{\tiny$4$}--++ (1,-1)node[below=0.1cm]{\tiny$4$} --++ (1,-1) --++ (1,1)node[below=0.1cm]{\tiny$3$} --++ (1,-1)node[below=0.1cm]{\tiny$3$} --++ (1,-1)node[below=0.1cm]{\tiny$3$} --++(1,-1) --++ (1,1)node[below=0.1cm]{\tiny$2$}--++(1,-1)node[below=0.1cm]{\tiny$2$}--++(1,-1)--++(1,1)--++(0,1) --++(1,0)node[above left]{\tiny$2$}--++(1,0)node[above left]{\tiny$3$}--++(0,1)--++(1,0)node[above left]{\tiny$5$}; 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\path[decoration=arrows, decorate] (-4,-14)node[below=0.1cm]{\tiny$10$} --++(1,-1)node[below=0.1cm]{\tiny$10$}--++(1,-1)node[below=0.1cm] {\tiny$10$}--++(1,-1)node[below=0.1cm]{\tiny$10$}--++(1,-1)node[below=0.1cm]{\tiny$10$}--++ (1,-1)--++(1,1)--++(1,1)node[below=0.1cm]{\tiny$8$}--++(1,-1)node[below=0.1cm]{\tiny$8$}--++(1,-1)--++(1,1)node[below=0.1cm]{\tiny$7$}--++(1,-1)node[below=0.1cm]{\tiny$7$}--++(1,-1)--++(1,1)--++(1,1)node[below=0.1cm]{\tiny$5$} --++(1,-1)node[below=0.1cm]{\tiny$5$}--++(1,-1)node[below=0.1cm]{\tiny$5$}--++(1,-1)--++(1,1)--++(1,1)--++(1,1)--++(1,1)--++(0,1)--++(0,1); \path[decoration=arrows, decorate] (-5,-17)--++(1,1)node[below=0.1cm]{\tiny$11$}--++(1,-1)node[below=0.1cm]{\tiny$11$}--++ (1,-1)node[below=0.1cm]{\tiny$11$}--++(1,-1)node[below=0.1cm]{\tiny$11$}--++ (1,-1)node[below=0.1cm]{\tiny$11$}--++(1,-1)node[below=0.1cm]{\tiny$11$}--++(1,-1)--++(1,1)node[below=0.1cm]{\tiny$10$}--++(1,-1)node[below=0.1cm]{\tiny$10$}--++(1,-1)--++(1,1)--++(1,1)node[below=0.1cm]{\tiny$8$}--++(1,-1)node[below=0.1cm]{\tiny$8$}--++ (1,-1)node[below=0.1cm]{\tiny$8$}--++(1,-1)--++(1,1)node[below=0.1cm]{\tiny$7$}--++(1,-1)--++(1,1)--++(1,1)--++(1,1)--++(1,1)--++(1,1)--++(1,1) --++(0,1); \path[decoration=arrows, decorate] (-6,-20)--++(1,1)--++(1,1)node[below=0.1cm]{\tiny$12$}--++(1,-1)node[below=0.1cm]{\tiny$12$}--++(1,-1)node[below=0.1cm]{\tiny$12$}--++(1,-1)node[below=0.1cm]{\tiny$12$} --++(1,-1)node[below=0.1cm]{\tiny$12$}--++(1,-1)node[below=0.1cm]{\tiny$12$}--++(1,-1)node[below=0.1cm]{\tiny$12$}--++(1,-1)--++(1,1)node[below=0.1cm]{\tiny$11$}--++(1,-1)--++(1,1)--++(1,1)node[below=0.1cm]{\tiny$9$}--++(1,-1)node[below=0.1cm]{\tiny$9$}--++ (1,-1)node[below=0.1cm]{\tiny$9$}--++(1,-1)node[below=0.1cm]{\tiny$9$}--++(1,-1)--++(1,1)--++(1,1)--++(1,1)--++(1,1)--++(1,1)--++(1,1)--++(1,1)--++(1,1); \end{tikzpicture} \tiny $$ \Downarrow $$ $$ \begin{ytableau} 12 & 12 & 12 & 12 & 12 & 12 & 12 & 11 & 9 & 9 & 9 & 9 \\ 11 & 11 & 11 & 11 & 11 & 11 & 10 & 10 & 8 & 8 & 8 & 7 \\ 10 & 10 & 10 & 10 & 10 & 8 & 8 & 7 & 7 & 5 & 5 & 5 \\ 8 & 8 & 8 & 6 & 6 & 6 & 6 & 6 & 6 & 4 & 4 & 2 \\ 6 & 6 & 6 & 5 & 5 & 5 & 4 & 4 & 4 & 3 \\ 4 & 4 & 4 & 4 & 3 & 3 & 3 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{ytableau} \qquad \begin{ytableau} 6 & 5 & 4 & 2 & 1 \\ 5 & 3 & 2 \\ 3 & 1 \end{ytableau} $$ \caption{\label{odd1} Illustration of Corollary~\ref{signless} (1) for $n=7$ and $l=12$.} \end{figure} \begin{figure} \begin{tikzpicture}[scale=.40,baseline=(current bounding box.center)] \draw[->,thick] (-7.75,0)--(20.75,0) node[right]{\tiny$x$}; \draw[->,thick] (0,-27.75)--(0,2.75) node[above]{\tiny$y$}; \draw [help lines,step=1cm,dashed] (-7.75,-27.75) grid (20.75,2.75); \fill[red] (0,0) circle (5pt); \draw[dotted,thick] (-2.75,2.75)--(20.75,-20.75) node[below]{\footnotesize $x+y=0$}; \fill (0,-2) circle (5pt) node[left]{\tiny$A_1$}; \fill (-1,-5) circle (5pt) node[left]{\tiny$A_2$}; \fill (-2,-8) circle (5pt) node[left]{\tiny$A_3$}; \fill (-3,-11) circle (5pt) node[left]{\tiny$A_4$}; \fill (-4,-14) circle (5pt) node[left]{\tiny$A_5$}; \fill (-5,-17) circle (5pt) node[left]{\tiny$A_6$}; \fill (-6,-20) circle (5pt) node[left]{\tiny$A_7$}; \fill (20,-20) circle (5pt) node[right]{\tiny$E_1$}; \fill (19,-18) circle (5pt) node[right]{\tiny$E_2$}; \fill (18,-16) circle (5pt) node[right]{\tiny$E_3$}; \fill (16,-13) circle (5pt) node[right]{\tiny$E_4$}; \fill (15,-11) circle (5pt) node[right]{\tiny$E_5$}; \fill (14,-9) circle (5pt) node[right]{\tiny$E_6$}; \fill (13,-7) circle (5pt) node[right]{\tiny$E_7$}; \path[decoration=arrows, decorate] (0,-2)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)node[below=0.1cm]{\tiny$2$} --++ (1,-1)--++(1,1) --++ (1,0)node[above left]{\tiny$1$}--++ (1,0)node[above left]{\tiny$2$} --++(0,1)--++ (1,0) node[above left]{\tiny$4$}--++(1,0) node[above left]{\tiny$5$} --++(1,0) node[above left]{\tiny$6$}; \path[decoration=arrows, decorate] (-1,-5)node[below=0.1cm]{\tiny$4$} --++ (1,-1)node[below=0.1cm]{\tiny$4$} --++ (1,-1)node[below=0.1cm]{\tiny$4$}--++ (1,-1)node[below=0.1cm]{\tiny$4$} --++ (1,-1) --++ (1,1)node[below=0.1cm]{\tiny$3$} --++ (1,-1)node[below=0.1cm]{\tiny$3$} --++ (1,-1)node[below=0.1cm]{\tiny$3$} --++(1,-1) --++ (1,1)node[below=0.1cm]{\tiny$2$}--++(1,-1)node[below=0.1cm]{\tiny$2$}--++(1,-1)--++(1,1)--++(0,1) --++(1,0)node[above left]{\tiny$2$}--++(1,0)node[above left]{\tiny$3$}--++(0,1)--++(1,0)node[above left]{\tiny$5$}; \path[decoration=arrows, decorate] (-2,-8)node[below=0.1cm]{\tiny$6$} --++ (1,-1)node[below=0.1cm]{\tiny$6$} --++ (1,-1)node[below=0.1cm] {\tiny$6$}--++ (1,-1)--++ (1,1)node[below=0.1cm]{\tiny$5$} --++ (1,-1)node[below=0.1cm]{\tiny$5$} --++ (1,-1)node[below=0.1cm]{\tiny$5$} --++ (1,-1)--++(1,1)node[below=0.1cm]{\tiny$4$} --++ (1,-1)node[below=0.1cm]{\tiny$4$}--++(1,-1)node[below=0.1cm]{\tiny$4$}--++(1,-1)--++(1,1)node[below=0.1cm]{\tiny$3$}--++ (1,-1)--++(1,1)--++(1,1)--++(1,0)node[above left]{\tiny$1$} --++(0,1)--++(1,0)node[above left]{\tiny$3$} --++(0,1); 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\end{tikzpicture} \tiny $$ \Downarrow $$ $$ \begin{ytableau} 12 & 12 & 12 & 12 & 12 & 12 & 12 & 11 & 9 & 9 & 9 & 9 \\ 11 & 11 & 11 & 11 & 11 & 11 & 10 & 10 & 8 & 8 & 8 & 7 \\ 10 & 10 & 10 & 10 & 10 & 8 & 8 & 7 & 7 & 5 & 5 & 5 \\ 8 & 8 & 8 & 6 & 6 & 6 & 6 & 6 & 6 & 4 & 4 & 2 \\ 6 & 6 & 6 & 5 & 5 & 5 & 4 & 4 & 4 & 3 \\ 4 & 4 & 4 & 4 & 3 & 3 & 3 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{ytableau} \qquad \begin{ytableau} \none & 6 & 5 & 4 & 2 & 1 \\ \none & 5 & 3 & 2 \\ \none &3 & 1 \\ 2 \\ 1 \end{ytableau} $$ \caption{\label{even} Illustration of Corollary~\ref{signless} (1) for $n=7$ and $l=13$.} \end{figure} \begin{cor} \label{signless} Let $w=0$. (1) Assume that $m=2l+1$. In case $l \ge n-2$, the right hand side of \eqref{rincluded1} is the generating function of plane partitions $(P,Q)$ of shapes $\lambda, \mu$, respectively, where $\mu$ is the complement of $\lambda$ in the $n \times l$-rectangle, $P$ is a column-strict plane partition such that the entries in the $i$-th row are bounded by $2n+2-2i$, and $Q$ is a row-strict plane partition of positive integers such that the entries in the $i$-th row are bounded by $n-i$. The weight is $$ \prod_{i=1}^n X_i^l (X_i^{-1} +1 + X_i)(1+X_i) X_i^{\# \text{ of } 2i-1 \text{ in } P} X_i^{- \, \# \text{ of } 2i \text{ in } P}. $$ (2) Assume that $m=2l$. In case $l \ge n-1$, the right-hand side of \eqref{rincluded1} is the generating function of plane partitions $(P,Q)$ of (straight) shape $\lambda$ and skew shape $\mu$, respectively, such that $\mu$ is the complement of $\lambda$ in the $n \times (l-1)$-rectangle after possibly deleting the first column of $\mu$, $P$ is a column strict plane partition such that the entries in the $i$-th row are bounded by $2n+2-2i$ and $Q$ is a row-strict plane partition such that the entries in the $i$-th row are bounded by $n-i$. The weight is $$ \prod_{i=1}^n X_i^l (X_i^{-1} +1 + X_i) X_i^{\# \text{ of } 2i-1 \text{ in } P} X_i^{- \, \# \text{ of } 2i \text{ in } P}. $$ \end{cor} \begin{proof} We consider the case $m$ is odd. Assume that $1 \le k_1 < k_2 < \ldots < k_n$ are chosen such that $(k_i,-k_i)$ is the last point in the intersection of the line $x+y=0$ with the path that connects $A_i$ to $E'_{n+1-i}$ when traversing the path from $A_i$ to $E'_{n+1-i}$. Note that the portion of the path from $A_i$ to $(k_i,-k_i)$ has $k_i-i$ steps of type $(1,-1)$ and $2i-1$ steps of type $(1,1)$. These portions correspond to the plane partition $P$ is follows: The $i$-th path corresponds to the $(n+1-i)$-th row where the $(1,-1)$-steps correspond to the parts, where we fill the cells in the Ferrers diagram from left to right when traversing the path from $A_i$ to $(k_i,-k_i)$, and a $(1,-1)$-step at distance $d$ from $x+y=0$ gives the entry $d+1$. It follows that the length of row $i$ is $k_{n+1-i}-n-1+i$ and that the entries in row $i$ are bounded by $2n+2-2i$. Now the portion of the path from $(k_i,-k_i)$ to $E'_{n+1-i}$ corresponds to the $i$-th row of the plane partition $Q$. More precisely, the horizontal steps correspond to the parts, where we fill the cells in the Ferrers diagram from right to left when traversing the path from $(k_i,-k_i)$ to $E'_{n+1-i}$, where the $j$-th step gives the entry $j$. Note that there are $i-k_i+l$ steps of type $(1,0)$ in this portion, while there are $n-i$ steps in total, so that the length of the $i$-th row is $i-k_i+l$ and the entries in row $i$ are bounded by $n-i$. The case $m$ is even is very similar and it is therefore omitted here. \end{proof} \begin{remark} (1) The plane partitions $P$ in the corollary are in easy bijection with symplectic tableaux as defined in \cite[Section~4]{KoiTer90}. Also the weight is up to an overall multiplicative factor essentially just the weight that is used for symplectic tableaux. As a consequence, the corollary can be interpreted as to provide the expansion of the generating function of arrowed Gelfand-Tsetlin into symplectic characters. This is in the vein of main results in \cite{TSPP} and in \cite[Remark 2.6]{nVASMDPP}. (2) In the case $m$ is odd, the plane partitions $Q$ are in easy bijective correspondence with $2n \times 2n \times 2n$ totally symmetric self-complementary plane partitions. The bijection is provided in \cite[Remark 2.6]{nVASMDPP}. In the case $m$ is even, we place the part $n+1-i$ into the cell in the $i$-th row of the inner shape and that way we obtain plane partitions that are in easy bijective correspondence with $(2n+2) \times (2n+2) \times (2n+2)$ totally symmetric self-complementary plane partitions. \end{remark} \subsection{The cases $n=2$ and $m=2,3$} In this section, we give a list of all objects for the left-hand side and right-hand side of \eqref{rincluded1} in the case $n=2$ and $m=2,3$. We start with the case that $m=3$, since this is easier on the right-hand side. Note that $m=3$ implies $l=1$. The arrowed monotone triangles are as follows, using the notation from Section~\ref{LHS}. \begin{multline*} \begin{array}{ccc} & {LR \atop 0} & \\ {L \atop 0} && {LR \atop 1} \end{array}, \begin{array}{ccc} & {LR \atop 1} & \\ {LR \atop 0} && {R \atop 1} \end{array}, \begin{array}{ccc} & {LR \atop 0} & \\ {L \atop 0} && {LR \atop 2} \end{array}, \begin{array}{ccc} & {LR \atop 1} & \\ {LR \atop 0} && {LR \atop 2} \end{array}, \begin{array}{ccc} & {LR \atop 2} & \\ {LR \atop 0} && {R \atop 2} \end{array}, \begin{array}{ccc} & {LR \atop 0} & \\ {L \atop 0} && {LR \atop 3} \end{array},\\ \begin{array}{ccc} & {LR \atop 1} & \\ {LR \atop 0} && {LR \atop 3} \end{array}, \begin{array}{ccc} & {LR \atop 2} & \\ {LR \atop 0} && {LR \atop 3} \end{array}, \begin{array}{ccc} & {LR \atop 3} & \\ {LR \atop 0} && {R \atop 3} \end{array}, \begin{array}{ccc} & {LR \atop 1} & \\ {L \atop 1} && {LR \atop 2} \end{array}, \begin{array}{ccc} & {LR \atop 2} & \\ {LR \atop 1} && {R \atop 2} \end{array}, \begin{array}{ccc} & {LR \atop 1} & \\ {L \atop 1} && {LR \atop 3} \end{array}, \\ \begin{array}{ccc} & {LR \atop 2} & \\ {LR \atop 1} && {LR \atop 3} \end{array}, \begin{array}{ccc} & {LR \atop 3} & \\ {L \atop 1} && {LR \atop 3} \end{array}, \begin{array}{ccc} & {LR \atop 2} & \\ {L \atop 2} && {LR \atop 3} \end{array}, \begin{array}{ccc} & {LR \atop 3} & \\ {LR \atop 2} && {R \atop 3} \end{array} \end{multline*} The weights are \begin{multline} \label{first} X_2(1+X_2^{-1}), X_1(1+X_2), X_2^2(1+X_2^{-1}), X_1 X_2(X_2^{-1}+1+w+X_2), X_1^2(1+X_2), X_2^3(1+X_2^{-1}),\\ X_1 X_2^2(X_2^{-1}+1+w+X_2), X_1^2 X_2(X_2^{-1}+1+w+X_2), X_1^3(1+X_2), X_1 X_2^2(1+X_2^{-1}), X_1^2 X_2(1+X_2),\\ X_1 X_2^3 (1+X_2^{-1}), X_1^2 X_2^2 (X_2^{-1} + 1 + w + X_2), X_1^3 X_2 (1+X_2), X_1^2 X_2^3(1+X_2^{-1}), X_1^3 X_2^2(1+X_2), \end{multline} up to the overall factor $LR(X_1) LR(X_2)$, setting $t=u=v=1$. The corresponding paths from Theorem~\ref{RHScomplicated} are as follows. \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (2,-1) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1) --++ (1,0); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1) --++ (1,1) --++ (1,1) --++(1,0); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; 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\fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (2,-1) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1) --++ (1,0); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,-1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (2,-1) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1) --++ (1,0); \path[decoration=arrows, decorate] (-1,-5) --++ (1,-1)--++(1,1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (2,-1) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++(1,-1) --++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1) --++ (1,1) --++(1,-1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (2,-1) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++(1,-1) --++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,-1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (2,-1) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++(1,-1) --++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,-1)--++(1,1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (2,-1) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,-1)--++(1,1) --++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,-1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (2,-1) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,-1)--++(1,1) --++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,-1)--++(1,1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} The weights are \begin{equation} \label{first}  -w,-X_1,-X_1^{-1},-X_2,-X_2^{-1},-X_1^{-1} -X_2, -X_1^{-1} X_2^{-1},-1,-X_1 X_2, -X_1 X_2^{-1}, \end{equation}  up to the overall factor $$- X_1 X_2 (1+X_1)(1+X_2)(X_1^{-1} + 1 + w + X_1)(X_2^{-1} +1 +w + X_2)= -X_1 X_2 (1+X_1)(1+X_2)LR(X_1) LR(X_2), $$ and, as can easily be seen, the sum of weights agrees with those for the arrowed Gelfand-Tsetlin patterns. Now we consider the case $m=2$. We have $l=1$. The arrowed monotone triangles are as follows. \begin{multline*} \begin{array}{ccc} & {LR \atop 0} & \\ {L \atop 0} && {LR \atop 1} \end{array}, \begin{array}{ccc} & {LR \atop 1} & \\ {LR \atop 0} && {R \atop 1} \end{array}, \begin{array}{ccc} & {LR \atop 0} & \\ {L \atop 0} && {LR \atop 2} \end{array}, \begin{array}{ccc} & {LR \atop 1} & \\ {LR \atop 0} && {LR \atop 2} \end{array}, \begin{array}{ccc} & {LR \atop 2} & \\ {LR \atop 0} && {R \atop 2} \end{array}, \begin{array}{ccc} & {LR \atop 1} & \\ {L \atop 1} && {LR \atop 2} \end{array}, \\ \begin{array}{ccc} & {LR \atop 2} & \\ {LR \atop 1} && {R \atop 2} \end{array}\end{multline*} The weights are \begin{multline*} X_2(1+X_2^{-1}), X_1(1+X_2), X_2^2(1+X_2^{-1}), X_1 X_2(X_2^{-1}+1+w+X_2), X_1^2(1+X_2), X_1 X_2^2(1+X_2^{-1}), \\ X_1^2 X_2(1+X_2), \end{multline*} up to the overall factor $LR(X_1) LR(X_2)$, setting $t=u=v=1$. As for the lattice paths, the situation is very similar to the case $m=3,l=1$, only $E_1=(3,-2)$ is replaced by the set $E_1=\{(2,-1),(3,-2)\}$ and $E_2=(2,-1)$ is replaced by the set $E_2=\{(1,0),(2,-1)\}$. It follows that all the families of paths from the case $m=3$ and $l=1$ also appear here. In addition, we have the following families of lattice paths. \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (2,-1) circle (5pt) node[right]{$E_1$}; \fill (1,0) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (1,0) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,1)--++(1,1)--++(1,0); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (1,0) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,1)--++(1,1)--++(1,-1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (1,0) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,1)--++(1,-1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (1,0) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,-1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (3,-2) circle (5pt) node[right]{$E_1$}; \fill (1,0) circle (5pt) node[right]{$E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++ (0,1); \path[decoration=arrows, decorate] (-1,-5) --++ (1,-1)--++(1,1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} \begin{tikzpicture}[scale=.45,baseline=(current bounding box.center)] \draw[->,thick] (-5.75,0)--(3.75,0) node[right]{$x$}; \draw[->,thick] (0,-6.75)--(0,0.75) node[above]{$y$}; \draw[dotted,thick] (-0.75,0.75)--(3.75,-3.75) node[right]{\tiny $x+y=0$}; \draw[dotted,thick] (0.25,0.75)--(3.75,-2.75) node[right]{\tiny $x+y=1$}; \draw [help lines,step=1cm,dashed] (-5.75,-6.75) grid (3.75,0.75); \fill[red] (0,0) circle (5pt); \fill (-2,0) circle (5pt) node[right]{$A_1$}; \fill (-5,-1) circle (5pt) node[right]{$A_2$}; \fill (0,-2) circle (5pt) node[left]{$A_1$}; \fill (-1,-5) circle (5pt) node[left]{$A_2$}; \fill (2,-1) circle (5pt) node[right]{$E_1=E_2$}; \path[decoration=arrows, decorate] (0,-2) --++ (1,1)--++ (1,0); \path[decoration=arrows, decorate] (-1,-5) --++ (1,1)--++(1,1)--++(1,1)--++(0,1); \end{tikzpicture} Thus, in addition to the weights in \eqref{first}, these families of lattice paths give $$ -1,-w,-X_1,-X_1^{-1},-X_2,-X_2^{-1},-1,w, $$ up to the overall factor $$- X_1 X_2 (X_1^{-1} + 1 + w + X_1)(X_2^{-1} +1 +w + X_2)= -X_1 X_2 LR(X_1) LR(X_2), $$ where the last two weights come from the last picture, first by interpreting the endpoint of the path that starts in $A_1$ as element of $E_2$ and second as element of $E_1$. \subsection{First proof of Theorem~\ref{RHScomplicated}} The approach of the first proof of Theorem~\ref{RHScomplicated} is closely related to the approach we used in the proof of Theorem~2.2 in \cite{nVASMDPP}. We consider the following bases for Laurent polynomials in $X$ that are invariant under the transformation $X \to X^{-1}$: let $$ q_i(X)=\frac{X^i-X^{-i}}{X-X^{-1}} \qquad \text{and} \qquad b_i(X)=(X+X^{-1})^i, $$ then $(q_i(X))_{i \ge 0}$ and $(b_i(X))_{i \ge 0}$ are two such bases. It is not hard to verify that \begin{equation} \label{transform} q_m(X)= \sum_{r=0}^{(m-1)/2} (-1)^r \binom{m-r-1}{r} b_{m-1-2r}(X). \end{equation} In order to derive a combinatorial interpretation of the right-hand side of \eqref{rincluded1}, consider \begin{equation} \label{det} \det_{1 \le i, j \le n} \left( X_i^{j-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{m+2n-j} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right). \end{equation} \emph{We start by considering the case that $m$ is odd:} We set $m=2l+1$, and pull out $\prod_{i=1}^n X_i^{l+n}$. \begin{equation} \label{odd} \prod_{i=1}^n X_i^{l+n} \det_{1 \le i, j \le n} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right) \end{equation} The entry in the $i$-th row and $j$-th column of the matrix underlying the determinant is obtained from \begin{multline*} \frac{X^{j-l-n-1} (1+X)^{j-1} (1+w X)^{n-j} - X^{-j+l+n+1} (1+X^{-1})^{j-1} (1+w X^{-1})^{n-j}}{X-X^{-1}} \\ = \sum_{p,q \ge 0} \binom{j-1}{p} \binom{n-j}{q} w^q \frac{X^{j-l-n+p+q-1}-X^{-j+l+n-p-q+1}}{X-X^{-1}} \end{multline*} by multiplying with $X-X^{-1}$ and then setting $X=X_i$. Note that this expression is invariant under replacing $X$ by $X^{-1}$. From \eqref{transform}, it follows that this is further equal to \begin{multline*} \sum_{p,q,r \ge 0 \atop |j-l-n+p+q-1|-1-2r \ge 0} \sgn(j-l-n+p+q-1) (-1)^r w^q \binom{j-1}{p} \binom{n-j}{q} \\ \times \binom{|j-l-n+p+q-1|-r-1}{r} b_{|j-l-n+p+q-1|-1-2r}(X). \end{multline*} We apply the following lemma. A proof can be found in \cite[Lemma 7.2]{nASMDPP}. Note that the lemma also involves complete homogeneous symmetric polynomials $h_k$ with negative $k$ as defined in \cite[Section~5]{nASMDPP}. Concretely, we define $h_k(X_1,\ldots,X_n)=0$ for $k=-1,-2,\ldots,-n+1$ and \begin{equation} \label{reci} h_k(X_1,\ldots,X_n) = (-1)^{n+1} X_1^{-1} \ldots X_n^{-1} h_{-k-n}(X_1^{-1},\ldots,X_n^{-1}) \end{equation} for $k \le -n$. Note that a consequence of this definition is that the latter relation is true for any $k$. \begin{lemma} \label{limit} Let $f_j(Y)$ be formal Laurent series for $1 \le j \le n$, and define $$ f_j[Y_1,\ldots,Y_i]=\sum_{k \in \mathbb{Z}} \langle Y^{k} \rangle f_j(Y) \cdot h_{k-i+1}(Y_1,\ldots,Y_i), $$ where $\langle Y^{k} \rangle f_j(Y)$ denotes the coefficient of $Y^{k}$ in $f_j(Y)$ and $h_{k-i+1}$ denotes the complete homogeneous symmetric polynomial of degree~$k-i+1$. Then $$ \frac{\det_{1 \le i, j \le n} \left( f_j(Y_i) \right) }{\prod_{1 \le i < j \le n} (Y_j - Y_i)} = \det_{1 \le i, j \le n} \left( f_j[Y_1,\ldots,Y_i] \right). $$ \end{lemma} Noting that a Laurent polynomial in $X$ that is invariant under the replacement $X \to X^{-1}$ can be written as a polynomial in $X+X^{-1}$, we use the lemma to basically rewrite \eqref{odd} as follows. \begin{multline*}  \frac{ \det_{1 \le i, j \le n} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right)} {\prod_{1 \le i < j \le n} (X_j + X_j^{-1} - X_i - X_i^{-1})} \\ = \prod_{i=1}^n (X_i - X_i^{-1}) \det_{1 \le i, j \le n} \left( \sum_{p,q,r \ge 0 \atop |j-l-n+p+q-1|-i-2r \ge 0} \sgn(j-l-n+p+q-1) (-1)^r w^q \binom{j-1}{p} \binom{n-j}{q} \right. \\ \left. \phantom{\sum_{p,q,r \ge 0}} \times \binom{|j-l-n+p+q-1|-r-1}{r} h_{|j-l-n+p+q-1|-i-2r}(X_1+X_1^{-1},\ldots,X_i+X_i^{-1}) \right) \end{multline*} Now, as $X_j + X_j^{-1} - X_i - X_i^{-1}=(X_i-X_j)(1-X_i X_j)X_i^{-1} X_j^{-1}$, in order to find a combinatorial interpretation for the right-hand side of \eqref{rincluded1}, we need to find a combinatorial interpretation of \begin{multline*} (-1)^{\binom{n}{2}} \prod_{i=1}^{n} X_i^{l+1} (X_i^{-1}+1+w + X_i) (X_i - X_i^{-1})(1-X_i)^{-1} \\ \times \det_{1 \le i, j \le n} \left( \sum_{p,q,r \ge 0 \atop |j-l-n+p+q-1|-i-2r \ge 0} \sgn(j-l-n+p+q-1) (-1)^r w^q \binom{j-1}{p} \binom{n-j}{q} \right. \\ \left. \phantom{\sum_{p,q,r \ge 0}} \times \binom{|j-l-n+p+q-1|-r-1}{r} h_{|j-l-n+p+q-1|-i-2r}(X_1+X_1^{-1},\ldots,X_i+X_i^{-1}) \right)\\ =(-1)^{\binom{n+1}{2}} \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w + X_i)(1+X_i) \\ \times \det_{1 \le i, j \le n} \left( \sum_{p,q,r \ge 0 \atop |j-l-n+p+q-1|-i-2r \ge 0} \sgn(j-l-n+p+q-1) (-1)^r w^q \binom{j-1}{p} \binom{n-j}{q} \right. \\ \left. \phantom{\sum_{p,q,r \ge 0}} \times \binom{|j-l-n+p+q-1|-r-1}{r} h_{|j-l-n+p+q-1|-i-2r}(X_1+X_1^{-1},\ldots,X_i+X_i^{-1}) \right). \end{multline*} For this purpose, we find a combinatorial interpretation of the entry of the underlying matrix, i.e., \begin{multline*} \sum_{p,q,r \ge 0 \atop |j-l-n+p+q-1|-i-2r \ge 0} \sgn(j-l-n+p+q-1) (-1)^r w^q \binom{j-1}{p} \binom{n-j}{q} \binom{|j-l-n+p+q-1|-r-1}{r} \\ \times h_{|j-l-n+p+q-1|-i-2r}(X_1+X_1^{-1},\ldots,X_i+X_i^{-1}) \end{multline*} in terms of a lattice paths generating function. We simplify the expression using the transformation $q \to n-j-q$. \begin{multline} \label{triple} \sum_{p,q,r \ge 0 \atop |p-q-l-1|-i-2r \ge 0} \sgn(p-q-l-1) (-1)^r w^{n-j-q} \binom{j-1}{p} \binom{n-j}{q} \binom{|p-q-l-1|-r-1}{r} \\ \times h_{|p-q-l-1|-i-2r}(X_1+X_1^{-1},\ldots,X_i+X_i^{-1}) \end{multline} We simplify the expression further using the following lemma. A combinatorial proof of it using a sign-reversing involution is provided in \cite[Lemma 7.7]{nVASMDPP}. \begin{lemma} \label{h} Let $a,i$ be positive integers with $i \le a$. Then $$ \sum_{r=0}^{(a-i)/2} (-1)^r \binom{a-r-1}{r} h_{a-i-2r}(X_1+X_1^{-1},\ldots,X_i+X_i^{-1}) = h_{a-i}(X_1,X_1^{-1},\ldots,X_i,X_i^{-1}). $$ \end{lemma} Therefore, the sum in \eqref{triple} is equal to \begin{equation} \label{path} \sum_{p,q} \sgn(p-q-l-1) w^{n-j-q} \binom{j-1}{p} \binom{n-j}{q} \\ h_{|p-q-l-1|-i}(X_1,X_1^{-1},\ldots,X_i,X_i^{-1}). \end{equation} We claim the following: If $p-q-l-1 \ge 0$, then \eqref{path} is the generating function of lattice paths from $(-3i+1,-i+1)$ to $(n-j+l+1,j-l-2)$ such that the following is satisfied. \begin{itemize} \item Below and on the line $x+y=0$, the step set is $\{(1,1),(-1,1)\}$. Steps of type $(-1,1)$ with distances $0,2,4,\ldots$ from $x+y=n$ are equipped with the weights $X_1,X_2,X_3,\ldots$, respectively, while steps of type $(-1,1)$ with distances $1,3,5,\ldots$ are equipped with the weights $X_1^{-1},X_2^{-1},X_3^{-1},\ldots$, respectively. \item Above the line $x+y=0$, the step set is $\{(1,0),(0,1)\}$. Above the line $x+y=j-1$, horizontal steps are equipped with the weight $w$. \end{itemize} Namely, if we assume that there are $q$ steps of type $(0,1)$ above the line $x+y=j-1$, and, therefore, $n-j-q$ steps of type $(1,0)$, then the path intersects the line $x+y=j-1$ in the lattice point $(l+1+q,j-l-2-q)$, assuming that the endpoint of the path is $(n-j+l+1,j-l-2)$, and there are $\binom{n-j}{q}$ of such paths each of them contributing $w^{n-j-q}$ to the weight. Note that this weight depends on $j$ if $w \not= 0,1$, and this causes complications when applying the Lindstr\"om-Gessel-Viennot lemma. If we further assume that there are $p$ steps of type $(1,0)$ below the line $x+y=j-1$, and, therefore, $j-p$ steps of type $(0,1)$, then the last lattice point of such a path on the line $x+y=0$ when traversing the path from $(-3i+1,-i+1)$ to $(n-j+l+1,j-l-2)$ is $(-p+q+l+1,p-q-l-1)$. Note that by the assumption $p-q-l-1 \ge 0$, the lattice point $(-p+q+l+1,p-q-l-1)$ is in the second quadrant, i.e., $\{(x,y)|x \le 0, y \ge 0 \}$. Finally, lattice paths from $(-3i+1,-i+1)$ to $(-p+q+l+1,p-q-l-1)$ with step set $\{(1,1),(-1,1)\}$ have $p-q-l-1-i$ steps of type $(-1,1)$ and $2i-1$ steps of type $(1,1)$. The generating function of such paths is clearly $h_{p-q-l-1-i}(X_1,X_1^{-1},\ldots,X_i,X_i^{-1})=h_{|p-q-l-1|-i}(X_1,X_1^{-1},\ldots,X_i,X_i^{-1})$. The situation is very similar if $p-q-l-1 \le 0$, except that we need to replace the starting point $(-3i+1,-i+1)$ by $(-i+1,-3i+1)$ and the step set is $\{(1,1),(1,-1)\}$ below the line $x+y=0$. Again we can assume that $(-p+q+l+1,p-q-l-1)$ is the last lattice point on the line $x+y=0$ when traversing the path from $(-i+1,-3i+1)$ to $(-p+q+l+1,p-q-l-1)$. In this case, $(-p+q+l+1,p-q-l-1)$ lies in the fourth quadrant $\{(x,y)|x \le 0, y \le 0 \}$. We have $-p+q+l+1-i=|p-q-l-1|-i$ steps of type $(1,-1)$ and $2i-1$ steps of type $(1,1)$, thus the generating function in this segment is also $h_{|p-q-l-1|-i}(X_1,X_1^{-1},\ldots,X_i,X_i^{-1})$. Consequently, we can conclude that the right-hand side of \eqref{rincluded1} has the following combinatorial interpretation: We consider families of $n$ lattice paths from $A_i=\{(-3i+1,-i+1),(-i+1,-3i+1)\}$, $i=1,2,\ldots,n$, to $E_j=(n-j+l+1,j-l-2)$, $j=1,2,\ldots,n$, with steps sets and weights as described above. By the Lindstr\"om-Gessel-Viennot lemma \cite{Lin73,GesVie85,GesVie89}, the paths can be assumed to be non-intersecting on and below the line $x+y=0$. In case $w=0,1$, we can also assume them to be non-intersecting. This is clear for $w=1$. In case $w=0$, we can assume that there are no steps of type $(1,0)$ above the line $x+y=j-1$, and, therefore, we can also have $(n-j+l+1,2j-n-l-2)$ on the line $x+y=j-1$ as endpoint since above the line all the $n-j$ steps have to be of type $(0,1)$. Whenever we choose $(-i+1,-3i+1)$, this contributes $-1$ to the weight. In the non-intersecting setting, suppose we choose $(-i+1,-3i+1)$ from $A_i$ for $1 \le i_1 < \ldots < i_m \le n$, then the sign of the permutation $\sigma$ such that $A_i$ is connected to $E_{\sigma(i)}$ via the paths is $(-1)^{i_1+i_2+\ldots+i_m-m}$. This gives a total sign of $(-1)^{i_1+i_2+\ldots+i_m}$. Recall also that we have an additional overall weight of $$ (-1)^{\binom{n+1}{2}} \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w + X_i)(1+X_i). $$ Combining the sign from above with $(-1)^{\binom{n+1}{2}}=(-1)^{1+2+\ldots+n}$, the sign can also be computed as follows: suppose we choose $(-3i+1,-i+1)$ from $A_i$ precisely for $i_1,\ldots,i_m$, then the sign is $(-1)^{i_1+i_2+\ldots+i_m}$ and in this setting the overall weight is $$ \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w + X_i)(1+X_i). $$ This concludes the proof of the first part of Theorem~\ref{RHScomplicated}. \bigskip \emph{Now we consider the case that $m$ is even:} We set $m=2l$ in \eqref{det}, and pull out $\prod_{i=1}^n X_i^{l+n}$. $$ \prod_{i=1}^n X_i^{l+n} \det_{1 \le i, j \le n} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{-j+l+n} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right). $$ The entry in the $i$-th row of the $j$-th column of the matrix underlying the determinant is obtained from \begin{multline*} \frac{X^{j-l-n-1} (1+X)^{j-1} (1+w X)^{n-j} - X^{-j+l+n} (1+X^{-1})^{j-1} (1+w X^{-1})^{n-j}}{1-X^{-1}} \\ = \sum_{p,q \ge 0} \binom{j-1}{p} \binom{n-j}{q} w^q \frac{X^{j-l-n+p+q-1}-X^{-j+l+n-p-q}}{1-X^{-1}} \end{multline*} when multiplying with $1-X^{-1}$ and then setting $X=X_i$. Now note that $$ \frac{X^m-X^{-m-1}}{1-X^{-1}} = q_{m+1}(X) + q_{m}(X) $$ for any integer $m$, so that we obtain $$ \sum_{p,q \ge 0} \binom{j-1}{p} \binom{n-j}{q} w^q \left(q_{j-l-n+p+q}(X) + q_{j-l-n+p+q-1}(X) \right). $$ It follows from \eqref{transform} that this is \begin{multline*} \sum_{p,q,r \ge 0 \atop |j-l-n+p+q|-1-2r \ge 0} \sgn(j-l-n+p+q) (-1)^r w^q \binom{j-1}{p} \\  \times \binom{n-j}{q} \binom{|j-l-n+p+q|-r-1}{r} b_{|j-l-n+p+q|-1-2r}(X) \\ + \sum_{p,q,r \ge 0 \atop |j-l-n+p+q-1|-1-2r \ge 0} \sgn(j-l-n+p+q-1) (-1)^r w^q \binom{j-1}{p} \binom{n-j}{q} \\ \times \binom{|j-l-n+p+q-1|-r-1}{r} b_{|j-l-n+p+q-1|-1-2r}(X). \end{multline*} Also here we simplify the expression using the replacement $q \to n-j-q$. \begin{multline*} \sum_{p,q,r \ge 0 \atop |-l+p-q|-1-2r \ge 0} \sgn(-l+p-q) (-1)^r w^{n-j-q} \binom{j-1}{p} \binom{n-j}{q} \binom{|-l+p-q|-r-1}{r} b_{|-l+p-q|-1-2r}(X) \\ + \sum_{p,q,r \ge 0 \atop |-l+p-q-1|-1-2r \ge 0} \sgn(-l+p-q-1) (-1)^r w^{n-j-q} \binom{j-1}{p} \binom{n-j}{q} \binom{|-l+p-q-1|-r-1}{r} b_{|-l+p-q-1|-1-2r}(X) \end{multline*} This implies the following. \begin{multline*}  \frac{ \det_{1 \le i, j \le n} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{-j+l+n} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right)} {\prod_{1 \le i < j \le n} (X_j + X_j^{-1} - X_i - X_i^{-1})} \\ = \prod_{i=1}^n (1-X_i^{-1}) \det_{1 \le i, j \le n} \left( a_{i,j} \right), \end{multline*} with \begin{multline*} a_{i,j} = \sum_{p,q,r \ge 0 \atop |p-q-l|-1-2r \ge 0} \sgn(p-q-l) (-1)^r w^{n-j-q} \binom{j-1}{p} \binom{n-j}{q} \\ \times \binom{|p-q-l|-r-1}{r} h_{|p-q-l|-i-2r}(X_1+X_1^{-1},\ldots,X_i+X_i^{-1}) \ \\ + \sum_{p,q,r \ge 0 \atop |p-q-l-1|-1-2r \ge 0} \sgn(p-q-l-1) (-1)^r w^{n-j-q} \binom{j-1}{p} \binom{n-j}{q} \\ \times \binom{|p-q-l-1|-r-1}{r} h_{|p-q-l-1|-i-2r}(X_1+X_1^{-1},\ldots,X_i+X_i^{-1}). \end{multline*} Using Lemma~\ref{h}, we see that this is equal to \begin{multline} \label{path2} b_{i,j} = \sum_{p,q \ge 0} \sgn(p-q-l) w^{n-j-q} \binom{j-1}{p} \binom{n-j}{q} h_{|p-q-l|-i}(X_1,X_1^{-1},\ldots,X_i,X_i^{-1}) \ \\ + \sum_{p,q \ge 0} \sgn(p-q-l-1) w^{n-j-q} \binom{j-1}{p} \binom{n-j}{q} h_{|p-q-l-1|-i}(X_1,X_1^{-1},\ldots,X_i,X_i^{-1}). \end{multline} Here we need to find a combinatorial interpretation of $$ (-1)^{\binom{n+1}{2}} \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w + X_i) \det_{1 \le i, j \le n} \left( b_{i,j} \right). $$ The only modification compared to the odd case is that the endpoints have to be replaced by the following set of two endpoints $E_j=\{(n-j+l+1,j-l-2),(n-j+l,j-l-1)\}$ and that the overall factor is $$ \prod_{i=1}^{n} X_i^{l} (X_i^{-1}+1+w+ X_i) , $$ given that the sign is taken care of as above. This concludes the proof of Theorem~\ref{RHScomplicated}. \subsection{Right-hand side of \eqref{rincluded1}, second proof} In this section, we sketch a second proof of Theorem~\ref{RHScomplicated}. It is closely related to the proof of Theorem~2.4 in \cite{nVASMDPP}. We only study the case $m=2l+1$. Again, we need to consider \begin{equation} \label{odd2nd} \det_{1 \le i, j \le n} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right). \end{equation} We have the following lemma. \begin{lemma} \label{22} For $n \ge 1$ and $l \in \mathbb{Z}$, the following identity holds. \begin{multline*} \frac{1}{\prod_{1 \le i < j \le n} (X_j-X_i)(X_j^{-1} - X_i^{-1}) \prod_{i,j=1}^n (X_j^{-1} - X_i)} \\ \times \det_{1 \le i, j \le n} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} - X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right) \\ \times \det_{1 \le i, j \le n} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} + X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right) \\ = \frac{(-1)^n}{2} \det_{1 \le i, j \le n} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k-i+1}-h_{k+i-1-2n}) \right) \\ \times \det_{1 \le i,j \le n} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k+i-1-n}+ h_{k-i+1-n}) \right) \end{multline*} \end{lemma} \begin{proof} We use $$ \det(A-B) \det(A+B) = \det \left( \begin{array}{c|c} A-B & B \\ \hline 0 & A+B \end{array} \right) = \det \left( \begin{array}{c|c} A-B & B \\ \hline B-A & A \end{array} \right) = \det \left( \begin{array}{c|c} A & B \\ \hline B & A \end{array} \right) $$ to see that the product of determinants on the left-hand side in the assertion of the lemma is equal to $$ \det \left( \begin{array}{c|c} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} \right)_{1 \le i, j \le n} & \left( X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right)_{1 \le i,j \le n} \\ \hline \left( X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right)_{1 \le i,j \le n} & \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} \right)_{1 \le i,j \le n} \end{array} \right). $$ Setting $X_{n+i} = X_i^{-1}$ for $i=1,2,\ldots,n$, we can also write this is as $$ \det \left( \begin{array}{c|c} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} \right)_{1 \le i \le 2n \atop 1 \le j \le n} & \left( X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right)_{1 \le i \le 2n \atop 1 \le j \le n} \end{array} \right). $$ We apply Lemma~\ref{limit} to $$ \frac{\det \left( \begin{array}{c|c} \left( X_i^{j-l-n-1} (1+X_i)^{j-1} (1+ w X_i)^{n-j} \right)_{1 \le i \le 2n \atop 1 \le j \le n} & \left( X_i^{-j+l+n+1} (1+X_i^{-1})^{j-1} (1+w X_i^{-1})^{n-j} \right)_{1 \le i \le 2n \atop 1 \le j \le n} \end{array} \right)}{\prod_{1 \le i < j \le 2n} (X_j-X_i)} $$ and obtain \begin{multline*} \det \left( \left. \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q h_{k-i+1}(X_1,\ldots,X_i) \right)_{1 \le i \le 2n \atop 1 \le j \le n} \right| \right. \\ \left. \left( \sum_{k,q} \binom{j-1}{-j-k+l+n-q+1} \binom{n-j}{q} w^q h_{k-i+1}(X_1,\ldots,X_i) \right)_{1 \le i \le 2n \atop 1 \le j \le n} \right). \end{multline*} We multiply from the left with the following matrix $$ (h_{j-i}(X_j,X_{j+1},\ldots,X_{2n}))_{1 \le i,j \le 2n} $$ with determinant $1$. For this purpose, note that $$ \sum_{l=1}^{2n} h_{l-i}(X_l,X_{l+1},\ldots,X_{2n}) h_{k-l+1}(X_1,\ldots,X_l)=h_{k-i+1}(X_1,\ldots,X_{2n}), $$ and, therefore, the multiplication results in \begin{multline*} \det \left( \left. \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q h_{k-i+1}(X_1,\ldots,X_{2n}) \right)_{1 \le i \le 2n \atop 1 \le j \le n} \right| \right. \\ \left. \left( \sum_{k,q} \binom{j-1}{-j-k+l+n-q+1} \binom{n-j}{q} w^q h_{k-i+1}(X_1,\ldots,X_{2n}) \right)_{1 \le i \le 2n \atop 1 \le j \le n} \right). \end{multline*} We set $X_{i+n}=X_i^{-1}$ for $i=1,2,\ldots,n$ (so that the arguments of all complete symmetric functions are $(X_1,\ldots,X_n,X_1^{-1},\ldots,X_n^{-1})$) and omit the $X_i$'s now. Also note that, under this specialization, the denominator $\prod_{1 \le i < j \le 2n} (X_j-X_i)$ specializes to the denominator on the left-hand side in the assertion of the lemma. $$ \det \left( \begin{array}{c|c} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q h_{k-i+1} \right)_{1 \le i \le 2n \atop 1 \le j \le n} & \left( \sum_{k,q} \binom{j-1}{-j-k+l+n-q+1} \binom{n-j}{q} w^q h_{k-i+1} \right)_{1 \le i \le 2n \atop 1 \le j \le n} \end{array} \right) $$ With this specialization, we have $h_k=-h_{-k-2n}$ using \eqref{reci}. Therefore, the above is \begin{multline*} (-1)^n \det \left( \begin{array}{c|c} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q h_{k-i+1} \right)_{1 \le i \le 2n \atop 1 \le j \le n} & \left( \sum_{k,q} \binom{j-1}{-j-k+l+n-q+1} \binom{n-j}{q} w^q h_{-k+i-1-2n} \right)_{1 \le i \le 2n \atop 1 \le j \le n} \end{array} \right) \\ = (-1)^n \det \left( \begin{array}{c|c} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q h_{k-i+1} \right)_{1 \le i \le 2n \atop 1 \le j \le n} & \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q h_{k+i-1-2n} \right)_{1 \le i \le 2n \atop 1 \le j \le n} \end{array} \right). \end{multline*} Now, for $j=1,2,\ldots,n$, we subtract the $(j+n)$-th column from the $j$-th column. \begin{multline*} (-1)^n \det \left( \left. \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k-i+1}-h_{k+i-1-2n}) \right)_{1 \le i \le 2n \atop 1 \le j \le n} \right| \right. \\ \left. \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q h_{k+i-1-2n} \right)_{1 \le i \le 2n \atop 1 \le j \le n} \right) \end{multline*} For $i=n+2,n+3,\ldots,2n$, we add the $(2n+2-i)$-th row to the $i$-th row. This gives a zero block for $\{(i,j) | n+1 \le i \le 2n, 1 \le j \le n\}$, since $$ \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k-i+1}-h_{k+i-1-2n}+h_{k-(2n+2-i)+1}-h_{k+(2n+2-i)-1-2n})=0. $$ The lower right block is \begin{multline*} \det_{n+1 \le i \le 2n \atop 1 \le j \le n} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k+i-1-2n}+ [i\not=n+1]h_{k+2n+2-i-1-2n}) \right) \\ = \det_{n+1 \le i \le 2n \atop 1 \le j \le n} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k+i-1-2n}+ [i\not=n+1]h_{k-i+1}) \right) \\ = \frac{1}{2} \det_{1 \le i,j \le n} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k+i-1-n}+ h_{k-i+1-n}) \right). \end{multline*} This concludes the proof of the lemma. \end{proof} The identity in the lemma involves, up to factors, a product of two determinants on the left-hand side and also a product of two determinants on the right-hand side. This suggests that each of the determinants on the left-hand side equals up to factors a determinant on the right-hand side. This is indeed the case. More specifically, one can show that \eqref{odd2nd} is up to factors equal to $$ (-1)^n \prod_{i=1}^n (1+X_i) X_i^l \det_{1 \le i,j \le n} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k+i-1-n}+ h_{k-i+1-n}) \right). $$ This can be shown by induction with respect to $n$ as suggested in \cite[Remark~7.4]{nVASMDPP}. Thus Lemma~\ref{22} would actually not have been necessary, however, it explains how the expression was obtained much better than the proof by induction. Using Lemma~7.5 from \cite{nVASMDPP}, we can conclude further that the expression is equal to \begin{multline*} (-1)^n \prod_{i=1}^n (1+X_i) X_i^l \\ \times \det_{1 \le i,j \le n} \left( \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q h_{k+i-1-n}(X_1,X_1^{-1},\ldots,X_{n-i+1},X_{n-i+1}^{-1}) \right). \end{multline*} Also this formula can be proven directly by induction with respect to $n$. Our next goal would be to give a combinatorial interpretation of $$ \sum_{k,q} \binom{j-1}{-j+k+l+n-q+1} \binom{n-j}{q} w^q (h_{k+i-1-n}(X_1,X_1^{-1},\ldots,X_{n-i+1},X_{n-i+1}^{-1}). $$ We replace $i$ by $n-i+1$ and $q$ by $n-j-q$, and then we get rid of $k$ by setting $p=k+l+q+1$. $$ \sum_{p,q} \binom{j-1}{p} \binom{n-j}{q} w^{n-j-q} h_{p-q-l-1-i}(X_1,X_1^{-1},\ldots,X_{i},X_{i}^{-1}). $$ This is equal to \eqref{path} when taking into account the definition of complete symmetric functions $h_k$ for negative $k$'s as given in \eqref{reci}. \section{Explicit product formulas in case $(X_1,\ldots,X_n)=(1,\dots,1)$ and $w=0,-1$} \label{outlook} When evaluating the specializations of the LHS or RHS of \eqref{rincluded} at $(X_1,\ldots,X_n)=(1,\ldots,1)$ in the cases $w=0,-1$ for small values of $n$, one observes that the numbers involve only small prime factors, and, therefore, it is likely that they are expressible by product formulas. (A similar observation is true for the case $w=1$, but there the explanation is simple, since $\prod_{1 \le i < j \le n} (1+ w X_i+X_j + X_i X_j)$ on the left-hand side of \eqref{rincluded} is symmetric then.) For the LHS and the case $m=n-1$, these are unpublished conjectures of Florian Schreier-Aigner from 2018. For instance, in the case $w=-1$ and $m=n-1$, we obtain the following numbers $$ 1,4,60,3328,678912,\ldots = 2^{n(n-1)/2} \prod_{j=0}^{n-1} \frac{(4j+2)!}{(n+2j+1)!} $$ that have also appeared in recent work of Di Francesco \cite{twenty}. Related conjectures for arbitrary $m$ have now been proven and will appear in forthcoming work with Florian Schreier-Aigner. The approach we have been successful with involves the transformation of the bialternant formula on the RHS of \eqref{rincluded} into a Jacobi-Trudi type determinant. Then we can easily set $X_i=1$ and we were able to guess the LU-decompositions of the relevant matrices. Proving these guesses involves the evaluation of certain triple sums, which is possible using Sister Celine's algorithm \cite{celine} and the fabulous Mathematica packages provided by RISC \cite{risc}. We state the results next. First we deal with the case $w=0$ . \begin{theorem} The specialization of the generating function of arrowed Gelfand-Tsetlin patterns with $n$ rows and strictly increasing non-negative bottom row where the entries are bounded by $m$ at $(X_1,\ldots,X_n)=(1,\ldots,1)$, $u=v=t=1$ and $w=0$ is equal to $$ 3^{\binom{n+1}{2}} \prod_{i=1}^n \frac{(2n+m+2-3i)_i}{(i)_i}. $$ \end{theorem} Now we turn to the case $w=-1$. \begin{theorem} The specialization of the generating function of arrowed Gelfand-Tsetlin patterns with $n$ rows and strictly increasing non-negative bottom row where the entries are bounded by $m$ at $(X_1,\ldots,X_n)=(1,\ldots,1)$, $u=v=t=1$ and $w=-1$ is equal to $$ 2^n \prod_{i=1}^n \frac{(m-n+3i+1)_{i-1} (m-n+i+1)_i}{\left(\frac{m-n+i+2}{2} \right)_{i-1} (i)_i}. $$ \end{theorem} Other future work concerns extending results that have been obtained using \eqref{littlewoodASM1} by replacing this identity by Theorem~\ref{boundedrQ} or specializations thereof.

proofpile-arXiv_069-11098

\section{Introduction} An unsolved problem in active galactic nuclei (AGN) feedback on clusters is how to account for the the morphology and stability of buoyant bubbles and their interactions with the ambient intracluster medium (ICM) \citep{mn07}. \citet{fstacji06} showed that in the Perseus cluster, such bubbles can stay intact far from cluster centers where they were inflated by AGN jets. However, studies of kinetic-energy dominated jets in the purely hydrodynamic limit did not explain the observed long term persistence of the buoyant bubbles, which are prone to Rayleigh-Taylor (RTI) and Kelvin-Holmhotz (KHI) instabilities and fragment entirely within $100\;\unit{Myr}$, contrary to observations [but see \citet{rmfs05,ps06,ga07}]. Appreciable magnetic energy has been observed in both cluster and radio lobe plasmas \citep{oek00,kdl01,chh05}. The magnetic fields could play a vital role in the dynamics of the rising bubble, as shown by a series of 2D magnetohydrodynamic (MHD) studies for bubbles of several $10^8$~years \citep{bk01,rdrr04,jd05}. \citet{sg08a} showed that uniform strong magnetic fields do not suppress RTI completely, but sheared fields at the bubble interface quench the instability. This implies that not only the field strength but also the configuration matters for bubble stability. \citet{rebhp07} did a comprehensive study of the influence of different magnetic field configurations upon the stability of a rising bubble. They found that the internal bubble helical magnetic field moderately stabilizes the bubble, however the bubble has an initial plasma parameter $\beta\equiv\left<2nT/B^2\right>\gg1$, still in thermal energy-dominated regime. Recently, \citet{llf06} proposed a different bubble magnetic field configuration for jet/lobes. Here we adopt the field configuration of \citet{llf06} and focus on the late stage of the evolution of magnetically dominated bubbles in 3D\@. We show that this spheromak-like magnetic field configuration strongly stabilizes the instabilities and prevents the bubble from breaking up, possibly forming the intact but detached ``ghost cavity" observed in systems such as Perseus-A\@. This {\it Letter} is organized as follows. In Sec.~\ref{setup}, we describe the problem setup. The simulation results and discussions are given in Sec.~\ref{results}. \section{Problem Set Up}\label{setup} The background ICM is assumed to be hydrostatic and isothermal. The density and pressure profile is given as $\rho=p=[1+(R/R_c)^2]^{-\kappa}$, where the parameters $\kappa$ and $R_c$ are taken to be $1.0$ and $4.0$ respectively. The fixed yet distributed gravitational field $-\nabla\psi(R)$ dominated by the dark matter is assumed \citep{nll06}. We simulate only the phase after the AGN has inflated a low-density bubble which is in pressure equilibrium with the ICM\@ initially. The static magnetized bubble initially lies at $(x_b,y_b,z_b)=(0,0,5)$ with density $\rho_b=0.1$, spherical radius $r_d=2$ for the density profile and $r_b=1$ for the magnetic configuration. The specific heat $\gamma$ inside and outside the bubble is taken to be $5/3$. Physical quantities are normalized by the characteristic system length scale $R_0=25\;\unit{kpc}$, density $\rho_0=1.67\times10^{-26}\;\unit{g\;cm^{-3}}$, and velocity $V_{0}=6.2\times10^{7}\;\unit{cm\;s^{-1}}$. The initial sound speed, $C_s|_{t=0}=\gamma^{1/2}\sim1.29$, is constant throughout the computational domain. Other quantities are normalized as: time $t=1$ gives $R_0/V_0=38.6\;\unit{Myr}$, magnetic field $B=1$ gives $(2\rho_0V_{0}^2)^{1/2}=1.13\times10^{-5}\;\unit{G}$ and energy $E=1$ gives $\rho_0 V_{0}^2R_0^3=2.71\times10^{58}\;\unit{ergs}$. The magnetic field setup of the bubble follows \citet{llf06} with the ratio of toroidal to poloidal bubble field $\alpha$ taken to be $\sqrt{10}$, which corresponds to a minimum initial Lorentz force. The latter is reasonable since we want to mimic the situation that the magnetic bubble has substantially relaxed and detached from the jet tip. The initial total magnetic energy is around $2.0\times10^{59}\;\unit{ergs}$. The computational domain is taken to be $|x|\le16$, $|y|\le16$, and $0\le z\le32$ corresponding to a $(800\;\unit{kpc})^3$ box in the actual length scales. The numerical resolution used here is $400^3$, where the grid points are assigned uniformly in every direction. A cell $\delta x$ corresponds to $2.0\;\unit{kpc}$. We use ``outflow" boundary conditions at every boundary except the boundary at $z=0$, where we use ``reflecting" boundary conditions, which guarantee that the energy flux through this boundary is zero. \section{Results \& Discussion}\label{results} Energy and density evolution reveal the different stages of the bubble. Figure~\ref{energy}(\emph{top}) presents the time evolution of various energies at the early stage, in which the gravitational energy $E_g=\int \rho\psi dV$ and the internal energy $E_T=\int p/(\gamma-1)dV$, where $dV$ is the infinitesimal volume and the integral is over the entire computation domain, have been reduced by their initial values $E_{g,0}$ and $E_{T,0}$, respectively. The total energy is defined as $E_{\rm total}=E_m+E_k+E_T+E_g$, where the kinetic energy is defined as $E_k=\int 1/2\rho v^2 dV$ and the magnetic energy is defined as $E_m=\int B^2/2 dV$. At the early stage ($t\lesssim5$), the kinetic energy increases since the bubble accelerates upward due to buoyancy, and the magnetic energy decreases due to work done in expansion from the weak initial Lorentz force and conversion to shock and wave energy \citep{nll06}. The gravitational energy increases because there is a net outward mass flow in the axial direction. The passage of the shock wave heats and compresses the ICM and alters its pressure gradient. The total energy $E_{\rm total}$ is almost constant before $t\sim10$, at which time the shock and wavefront reach the boundary. Conservation of the total energy for $t\lesssim10$ is not strictly satisfied due to numerical diffusion. The initial bubble expansion ($t\le20$) is approximately adiabatic, which gives $E=Vp\propto V^{1-\gamma}$, where $V$ is the bubble volume. Therefore $E_2/E_1=(V_1/V_2)^{2/3}$, where the subscript $1$ and $2$ indicate the initial and final state, respectively. At $t=20$, the radius of the bubble has increased from $2$ to $\sim5$, which gives $E_2/E_1\sim16\%$, roughly matching the amount of magnetic energy in the bubble [$20\%$ from Fig.~\ref{energy}(\emph{top})]. For later times [Fig.~\ref{energy}(\emph{bottom})], fitting with $E_m(t)/E_{m,0}=\exp(-t/\tau_{\rm dis})$, we have: (1)~a fast dissipation stage ($t\lesssim10$) when magnetic energy dissipation time $\tau_{\rm dis}\sim11$; (2)~a slow dissipation stage ($t\gtrsim10$) when $\tau_{\rm dis}\sim114$. Both times are much smaller than the numerical dissipation time $\tau_{\rm res}$ at the corresponding stages (see discussions at the end of this section). The kinetic energy is oscillating while it is decaying slowly, which can be understood as follows. Gravity pulls the bubble down to the denser ICM, compressing the bubble and causing an increase in magnetic energy (since the magnetic flux is nearly conserved). This results in the magnetic energy oscillating in phase with the kinetic energy with period $\sim90$ ($3.5\times10^9\;\unit{yr}$). At $t=200$ after several periods of decaying oscillations, about 3.5\% of the initial magnetic energy remains. The magnetic field suppresses instabilities and therefore the bubble remains intact longer. Figure~\ref{overview} presents the typical density distribution (logarithmic scale) in 2D $x$--$z$ slices at $y=0$ at different times ($t=0,7.5,20,50,100,125$). The white solid contour lines indicate contours of constant magnetic field strength $|B|$. Consistent with Fig.~\ref{energy}(\emph{bottom}), after $t\gtrsim50$, the bubble undergoes a slowly decaying oscillation between $z\sim19$ and $z\sim24$. We have also performed simulations of an unmagnetized bubble and found that it disintegrates after $t=20$, whereas the magnetic bubble still clearly differentiates itself from the ambient medium at $t=200$ (Fig.~\ref{overview}). The formation of an ``umbrella" or a thin protective magnetic layer on the bubble working surface suppresses instabilities \citep{rebhp07}. The stronger magnetic field is also found to move the bubble faster and push the bubble farther away from its initial position. The position of the bubble top as a function of time is shown in Fig.~\ref{pos}. This position is calculated by plotting the axial profile of the density along the line of $(x,y)=(0,0)$ and finding the location of the first density jump from the top of the domain. Comparing the unmagnetized run (dash line) with the magnetized run (both without initial momentum/rotation), we can see that the rising speed increases from $~0.36$ to $0.6$. Note that the rough estimate of the terminal speed $v_t$ based on the initial gravitational acceleration $g$ and size $r_d$, both position-dependent, is $v_t\approx4/3\sqrt{2gr_d}\sim2$, assuming force balance between buoyancy and viscous drag force at the final stage \citep{mn07}. We investigate the effect of the magnetic field on bubble stability in more detail. Following \citet{nll07}, we first study KHI of the bubble. At both sides of the bubble, the magnetic field is almost uniform, not twisted like at the top of the bubble. From linear analysis, the instability criterion for nonaxisymmetric KHI surface modes is \citep{hr02} : $\Delta V > V_{As}=[(\rho_b+\rho_e)/(4\pi \rho_b\rho_e)(B_b^2+B_e^2)]^{1/2}$, where $\Delta V\equiv|V_b-V_e|$ is the velocity shear and $V_{As}$ is the surface Alfv\'en speed. The subscripts $b$ and $e$ indicate the bubble and external medium, respectively. The nonaxisymmetric body modes would be important if \citep{hr99}: (1) $V_b>V_f$, in which $V_f$ is the fast magnetosonic speed; or (2) $C_sV_A/(C_s^2+V_A^2)^{1/2}<V_b<V_s$, in which $V_A$ is the Alfv\'en speed and $V_s$ is the slow magnetosonic speed. The definitions of $V_{f,s}$ are: $V_{f,s}=\big\{1/2\{C_s^2+V_A^2\pm[(C_s^2+V_A^2)^2-4C_s^2V_A^2\cos^2\theta]^{1/2}\}\big\}^{1/2}$. Since we focus on the $x$-direction only, $V_A^2\cos^2\theta$ are taken to be $B_x^2/\rho$. Figure~\ref{khi} (\emph{top}) displays the transverse distribution of the bulk flow speed $V=(v_x^2+v_y^2+v_z^2)^{1/2}$, the density $\rho$, and the magnetic field strength $B$ at $t=7.5$ at $z=8.0$. A distinct velocity shear is identified at $x\sim3.5$. Across this shear, $\Delta V\sim0.75$ and $V_{As}\sim0.85$. The inequality $\Delta V<V_{As}$ holds. This means that the KHI surface modes are completely suppressed. Figure~\ref{khi} (\emph{bottom}) displays the transverse distribution of $V$, $V_f$, and $V_s$ at $t=7.5$ at $z=8.0$ . We see that the bulk flow $V$ lies between $V_f$ and $V_s$. The inequality $V_s<V<V_f$ is satisfied in the body of the buoyant bubble. This rules out the KHI body modes as well. RTI could also be suppressed by the magnetic field. For the idealized case of two conducting fluids separated by a contact discontinuity with a uniform magnetic field parallel to the interface undergoing constant acceleration $g$, \citet{chan61} demonstrated that RTI on a scale $L$ parallel to the field requires $B<B_c\equiv[Lg(\rho_h-\rho_l)]^{1/2}$, in which $\rho_h$ and $\rho_l$ are the densities in the heavy and light fluids respectively. Modes perpendicular to the field are unaffected. At the top of the bubble ($z\sim11.6$) (Fig.~\ref{rti}), $\rho_h$ and $\rho_l$ are found to be $0.15$ and $0.06$, respectively, and the gravitational acceleration $g$ is calculated to be $0.95$. If $L$ is chosen to be the computation domain size, the maximum possible mode wavelength in the simulations, then the critical magnetic field strength is $B_c\sim1.6$, which is larger than the magnetic field ($\sim0.8$) at that location. This means that only part of the parallel modes are suppressed regardless of the perpendicular modes. This is not consistent with the simulation results. However, as pointed out in \citet{sg08a}, the twisting nature of the bubble field at the top of the bubble introduces a current sheet at the surface, and the changes in the direction of the field at the interface must be on very small scales to inhibit the interchange modes. A more detailed study of this effect is beyond the scope of this paper and will be the subject of future study. Our simulation also provides one possible explanation of the morphology and origin of the large scale magnetic field and the generation of ``ghost cavities" observed in many clusters. Like \citet{rdrr04}, a magnetized high-density tail remains as the bubble rises (Fig.~\ref{overview}). The tails have an elongated morphology (Fig.~\ref{overview}), resembling H$\alpha$ filaments, which is found to indicate the history of the rising bubble. Interestingly, the magnetic field also helps stabilize this ``filament" as it is still visible at $t=200$. \citet{nb04} has argued that thermal conduction has to be strongly suppressed in the ICM; otherwise, such cold filaments would be rapidly evaporated. As in \citet{rebhp07}, our results provide a possibility that thermal conduction may be locally weaker in the bubble wake, thus preventing or slowing down filament evaporation. The magnetic field is distributed between $z\sim2.5$ and $z\sim25$ at $t=200$ with peak value around $0.156$ ($\sim1.8\;\unit{\mu G}$) or mean value around $0.078$ ($\sim0.88\;\unit{\mu G}$), close to the estimates of wider cluster fields \citep{ct02,tfa02}. This large-scale ($\gtrsim500\;\unit{kpc}$) magnetic field structure also mimics the morphology of the second class (``Phoenix") of ``radio relics" elongated from the cluster center to the periphery observed in A 115 \citep{gfg01}. The simulation results mean that the magnetic bubble rising from $\sim125\;\unit{kpc}$ with an initial magnetic energy of $2\times10^{59}\;\unit{ergs}$, will spread magnetic fields between $62.5\;\unit{kpc}\lesssim z\lesssim600\;\unit{kpc}$ and keep a magnetized bubble between $475\;\unit{kpc}$ and $600\;\unit{kpc}$ for at least $\sim8\times10^{9}\;\unit{yr}$. Therefore one could reproduce the intact but detached ``ghost cavity" observed in some systems such as Perseus-A. Real radio lobes possess complex, jet-driven flows, different from the static bubble that we use for our initial conditions. We study the effects of additional variations in the initial bubble by having: (1)~an initial uniform injection speed $v_{\rm inj}$ or (2)~an initial uniform rotation $\omega$. \citet{ntll08} reported that the radio lobe gains a momentum of $v_{\rm inj}\sim0.91C_{s0}=1.174$, and the rotation speed $\omega$ at the edge of the bubble reaches $\sim1.5C_{s0}/r_d=0.9675$ after jet-driven inflation and interaction with the ambient ICM, although neither are not uniform inside the bubble. Here we take these two values as our initial values for $v_{\rm inj}$ and $\omega$ and idealize both to be uniform throughout the bubble. Fig.~\ref{pos} presents the time evolution of the position of the bubble top with initial uniform injection velocity $v_{\rm inj}=1.174$ (long dash) and initial uniform rotation speed $\omega=0.9675$ (dash dot). It is seen that these two variables only have some influence on the evolution of the bubble during the early stage. The bubble with an initial rotation has a reduced buoyancy resulting from rotation-driven instabilities that in turn lead to a reduced pressure differential between the bubble and the ICM. The evolution, and structure of the bubble in the later times are, however, essentially the same. The effects of numerical diffusion are estimated as follows. If we fit the net toroidal magnetic flux $\psi_t=\int B_y dS$ (only positive $B_y$ is selected) as $\psi_t(t)/\psi_t(t=0)\equiv\exp(-t/\tau_{\rm res})$, the ``resistive" dissipation time due to numerical diffusion is $\tau_{\rm res}\sim 38$ for $t\le10$, $\tau_{\rm res}=138$ for $10\le t\le50$, and $\tau_{\rm res}=507$ for $50\le t \le 100$. Therefore the numerical diffusion is not important on the time scales of interest $(t\lesssim200)$. Increasing numerical resolution can further reduce the numerical diffusion. The oldest bubbles known so far are a few hundred $\unit{Myr}$ old \citep{mn07}, which are much younger than the long-sustained ($\sim$ a Hubble time) magnetized bubbles reported in this \emph{Letter}. The simulations show that at later times $t\gtrsim50$, the density and X-ray luminosity contrast between the bubble and ICM plasma at the periphery of the cluster is small, which makes the direct X-ray observation of the bubble faraway from the parent galaxy very difficult. But the radio bubble at the borders of the clusters may still be detectable through: (1) Faraday rotation measurement of the outskirts of the cluster if an outside synchrotron source exists; (2) or radio observation from re-energization of fossil radio bubble plasma by shocks or mergers. It is highly possible for the bubbles to pile up in the outer atmospheres of the clusters in deep Chandra observation, if those observations are technically viable. It is conceivable that part of the initial magnetic energy of the bubble accelerates cosmic rays. Throughout the lifetime of a cluster, there could be multiple AGNs injecting jets/lobes into the ICM\@. Both the cosmic rays and the remaining magnetic fields (distributed over large scales) could provide the energy sources for phenomena such as radio relics and radio haloes that have been observed for a number of clusters \citep{fgsbr08}.

proofpile-arXiv_069-11170

\section{Introduction} The origin of the non-vanishing but extremely small positive cosmological constant $\Lambda\simeq (10^{-3}\,{\rm eV})^4$, as deduced from precision measurements of the cosmic microwave background, is still elusive. Similarly, the origin and stability of the weak scale are still not clearly understood, although we hope to find evidence for an answer at the LHC. Various possibilities to explain the weak scale as well as the gauge hierarchy problem have been proposed, the most prominent surely being supersymmetry which is supported by gauge coupling unification at the scale $M_X=1.2\cdot 10^{16}\,{\rm GeV}$ within the MSSM. String theory motivates also more exotic ideas descending from the presence of extra space-dimensions such as Large Extra Dimensions \cite{ArkaniHamed:1998rs,Antoniadis:1998ig} (see also \cite{Antoniadis:1990ew}) or the Randall-Sundrum \cite{Randall:1999ee} scenario. For string theory to make contact with low energy phenomenology, the problem of moduli stabilisation has to be addressed. In fact, during the last years considerable progress has been made. In particular, KKLT \cite{Kachru:2003aw} proposed a scenario which led to meta-stable de Sitter vacua with all moduli stabilised. In this scenario, the complex structure moduli are fixed at tree-level by fluxes while the K\"ahler moduli are stabilised via non-perturbative contributions to the superpotential \begin{equation} \label{kklt} W=W_0+A\, e^{-\alpha\, T} \;. \end{equation} As a generalisation of this, including also next to leading order corrections to the K\"ahler potential, the volume of the compactification manifold can be stabilised at exponentially large values \cite{Balasubramanian:2005zx}. These large volume minima are quite generic \cite{Cicoli:2008va} and have been called LARGE Volume Scenarios (LVS). Their phenomenological features were studied in very much detail for the string scale in the intermediate regime $M_{\rm s}\simeq 10^{11}\,{\rm GeV}$ \cite{Conlon:2005ki,Conlon:2007xv} leading to intermediate scales for the neutrino and axion sector of the MSSM. In this paper, we further extend the KKLT respectively LARGE Volume Scenario by including the recently proposed poly-instanton corrections \cite{Blumenhagen:2008ji} to the superpotential. Let us emphasise that such corrections are of genuine stringy origin and are not expected to be understood from pure field theory. With these contributions it is quite simple to combine a race-track scenario with the LVS allowing for exponential control over effective parameters $W^{\rm eff}_0$ and $A^{\rm eff}$ in a superpotential such as \eqref{kklt}. In particular, the stabilised race-track modulus controls the size of $W^{\rm eff}_0$ and $A^{\rm eff}$ allowing for exponential small values without fine-tuning.\footnote{Moduli stabilisation via race-track type superpotentials in the context of M-theory has been discussed in \cite{Acharya:2007rc}.} One particular scenario we are aiming for is a conventional supersymmetric Grand Unified Theory (GUT) with MSSM gauge coupling unification at $M_X$. Computing the resulting soft masses for MSSM branes supported on small cycles of the Calabi-Yau manifold suggests two different setups. One is very similar to the LVS featuring a gravitino mass around the TeV scale with a supergravity mediated $\ln(M_{\rm Pl}/M_{3/2})$ suppression of the soft-terms. The second scenario utilises the observation that the race-track modulus is fixed in an almost supersymmetric minimum. The soft masses on $D7$-branes wrapping a four-cycle corresponding to this modulus are automatically hierarchically suppressed. In particular, the gravitino and soft scalar masses are obtained in an intermediate regime while the gaugino masses are found at the weak scale, dominantly generated by anomaly mediation. The outline of this paper is as follows. In section \ref{sec_mod_stab} we define and analyse our scenario which is a combination of a supersymmetric flux minimum and a race-track superpotential subject to string instanton corrections. In section \ref{sec_gut}, we present supersymmetric GUT scenarios with $M_{\rm s}=M_X$ where realistic gauge coupling unification arises dynamically, and in section \ref{sec_cc} we comment on the cosmological constant problem. Finally, in section \ref{sec_concl} we summarise our findings. \section{Moduli Stabilisation} \label{sec_mod_stab} We consider type IIB orientifolds of Calabi-Yau manifolds with $O7$- and $O3$-planes which are the currently best understood framework for moduli stabilisation. In order to cancel the $C_8$ and induced $C_4$ tadpoles, we introduce $D7$- and $D3$-branes as well as the combination $G_3=F_3+S\, H_3$ of R-R and NS-NS fluxes. \bigskip The $G_3$-flux gives rise to a tree-level superpotential of the form \cite{Gukov:1999ya} \begin{equation} \label{w_GVW} W_{\rm{flux}} = \int_{\mathcal{X}} G_3\wedge \Omega_3 \;, \end{equation} which in general allows to stabilise all complex structure moduli $U_i$ together with the axio-dilaton $S$ by the supersymmetry conditions $D_{U_i}W_{\rm{flux}}=D_S W_{\rm{flux}}=0$. In contrast to KKLT and LVS, we require in addition that \eq{ \label{flux_zero} W_{\rm flux}\,\bigr\rvert_{\rm min.}=0 \;. } This is an over-constrained system, but it was shown in \cite{DeWolfe:2004ns} that such solutions are not highly suppressed. In fact, it was provided evidence that the number of minima \eqref{flux_zero} compared to all flux minima scales as $\#(W_{\rm flux}\rvert_{\rm min.}=0)/\#({\rm tot})\sim 1/L^{\mathcal{D}/2}$ with $L$ the upper limit on the flux quanta and $\mathcal{D}$ an integer. Finally, we note that $D_{U_i}W$ and $D_S W$ appear at order $\mathcal{V}^{-2}$ in the scalar potential while the K\"ahler moduli, as we will see below, appear at orders $\mathcal{V}^{-1}$ and $\mathcal{V}^{-3}$. In the limit of large $\mathcal{V}$ we are interested in, we can therefore stabilise $U_i$ and $S$ independently of the K\"ahler moduli. \bigskip Let us now study in more detail the K\"ahler moduli $T_a=\tau_a+i\,\rho_a$ with $\tau_a$ the four-cycle volumes of the compact space and $\rho_a$ the axions originating from $C_4$. We compute the F-term potential using the standard K\"ahler potential including $\alpha'$-corrections \cite{Becker:2002nn} \begin{equation} \label{kahl-pot} \mathcal{K} = -2\ln \Bigl( \mathcal{V} + {\textstyle \frac{{\hat\xi}}{2} } \Bigr) - \ln \Bigl( S+\overline{S} \Bigr) + \mathcal{K}_{\rm CS} \end{equation} where $\hat\xi=\xi/g_s^{3/2}$ and $g_s$ is the string coupling. The resulting inverse K\"ahler metric for the K\"ahler moduli reads \begin{equation} G^{a \overline b} =-2\,\Bigl( \mathcal{V} + {\textstyle \frac{{\hat\xi}}{2} } \Bigr) \biggl( \frac{\partial^2 \mathcal{V}} {\partial\tau_a\,\partial\tau_b} \biggr)^{-1} +\tau_a\, \tau_b\, \frac{4\,\mathcal{V}-\hat\xi}{\mathcal{V}-\hat\xi }\; , \end{equation} where the inverse denotes the usual matrix inverse and we choose the volume $\mathcal{V}$ of the internal space to be of {\em swiss-cheese} form with three K\"ahler moduli \begin{equation} \label{vol1} \mathcal{V} = \bigl( \eta_{\rm b} \tau_{\rm b}\bigr)^{3/2} - \bigl(\eta_1 \tau_1 \bigr)^{3/2} - \bigl(\eta_2 \tau_2 \bigr)^{3/2} \; \end{equation} Here, $\tau_{\rm b}$ controls the overall volume $\mathcal{V}$ and $\tau_{(1,2)}$ are small holes in this geometry. The constants $\eta_{\rm b},\eta_1,\eta_2$ are determined by a specific choice of a compactification manifold. Furthermore, we assume gaugino condensation on two stacks of $D7$-branes wrapping the four-cycle $\Gamma_1$ corresponding to the K\"ahler modulus $T_1$.\footnote{On the type I side, such a setup can be realised for instance by discrete Wilson lines as it has been shown in \cite{Camara:2007dy}.} This leads to a race-track superpotential containing exponentials of the two gauge kinetic functions. In further developing the D-brane instanton calculus pioneered in \cite{Billo:2002hm,Blumenhagen:2006xt,Ibanez:2006da,Florea:2006si,Akerblom:2006hx,Bianchi:2007fx,Cvetic:2007ku,Argurio:2007vqa,Bianchi:2007wy,Ibanez:2007rs}, it has been argued in \cite{Akerblom:2007uc,Blumenhagen:2008ji} that also the gauge kinetic function receives non-perturbative corrections from euclidean $D3$-brane instantons. For such an instanton to contribute, it must have a zero-mode structure specified by $h_{2,0}(\Gamma_{E3})=1$ and $h_{1,0}(\Gamma_{E3})=0$ where $\Gamma_{E3}$ denotes the cycle wrapped by the instanton. Let us assume that such corrections can indeed arise from an instanton wrapping the four-cycle $\Gamma_2$ with K\"ahler modulus $T_2$. Note that, because of its zero-mode structure, this instanton will not contribute as a single instanton and so the superpotential at leading order reads (see also \cite{Grimm:2007xm}) \begin{align} \label{instanton} W_{\rm np} &= \hspace{6.5pt} \mathcal{A}\, e^{-a\,\left( T_1 + \mathcal{C}_1\, e^{-2\pi T_2}\right)} -\mathcal{B}\, e^{-b\,\left( T_1 + \mathcal{C}_2\, e^{-2\pi T_2}\right)} \\[3pt] \label{superpot_eff} &= \Bigl[ \mathcal{A}\, e^{-a\, T_1} - \mathcal{B}\, e^{-b\, T_1} \Bigr] \;-\; \Bigl[ \mathcal{A}\,\mathcal{C}_1\,a\, e^{-a\, T_1} - \mathcal{B}\,\mathcal{C}_2\,b\, e^{-b\, T_1} \Bigr]\, e^{-2\pi T_2} \;+ \ldots \end{align} with all K\"ahler moduli in Einstein frame. The constants $a=\frac{2\pi}{N}$ and $b=\frac{2\pi}{M}$ are determined by the rank of the two gauge groups $U(N)$ respectively $U(M)$ and $\mathcal{A},\mathcal{B},\mathcal{C}_{(1,2)}$ are constant after the complex structure moduli have been stabilised at tree-level via \eqref{w_GVW}. Next, we are going to analytically estimate the large volume minimum of this scenario. However, we would like to emphasise that the specific models of the following sections have been analysed numerically for the K\"ahler potential \eqref{kahl-pot} and superpotential \eqref{instanton} without any approximations. We start from the scalar F-term potential $V_F=e^{\mathcal{K}}\left( \lvert D W_{\rm np} \rvert^2-3\,\lvert W_{\rm np}\rvert^2\right)$ with $U_i$ and $S$ stabilised. Since we are interested in a minimum at large $\mathcal{V}$, we expand this expression in powers of $1/\mathcal{V}$ and keep only the leading order term in $T_1$ \begin{equation} \label{rt_pot} V_F \sim \frac{ \sqrt{\tau_1}\:\bigl\vert \partial_{T_1} W_{\rm np} \bigr\vert^2}{ \mathcal{V}} + \mathcal{O}\Bigl(\mathcal{V}^{-2},e^{-4\pi\tau_2}\Bigr) \;. \end{equation} The minimum of \eqref{rt_pot} is determined by $\partial_{T_1} W_{\rm np}=\partial_{\overline T_1} \overline W_{\rm np}=0$ stabilising $T_1$ at \begin{equation} \label{fixed_T1} \tau^*_1 \simeq \frac{1}{a-b}\: \ln\,\biggl( \frac{\mathcal{A}\,a}{\mathcal{B}\, b} \biggr) \;,\qquad \rho^*_1=0\;, \end{equation} if $\mathcal{A}>\mathcal{B}$ are real and $a>b$. Using this solution, we identify the first term in \eqref{superpot_eff} as $W_0^{\rm eff}$ and in the second term we identify $A^{\rm eff}$ such that \eqref{instanton} becomes \begin{equation} W^{\rm eff} = W_0^{\rm eff} - A^{\rm eff}\: e^{-2\pi T_2} + \ldots\;, \end{equation} which is of a form similar to \eqref{kklt}. Since both $W_0^{\rm eff}$ and $A^{\rm eff}$ scale as $\exp\,(-a\tau_1^*)$, it is possible to obtain exponentially small values for them without fine-tuning. We proceed and study the resulting effective potential for $\mathcal{V}$ and $T_2$ with $T_1$ stabilised at values \eqref{fixed_T1}. As for the LARGE Volume Scenario, we take the limit $\mathcal{V}\gg1$ and keep only the leading term in $\mathcal{V}$ at each order of $\exp\,(-2\pi \tau_2)$. The resulting potential then reads (in Einstein frame) \begin{align} \label{effspot} V_F \sim& \hspace{10pt} \frac{8}{3}\, \Biggl( \frac{\sqrt{\tau^*_1}}{\eta_1}\: \Bigl\lvert \gamma\, W_0^{\rm eff} \Bigr\rvert^2 + (2\pi)^2\,\frac{\sqrt{\tau_2}}{\eta_1}\: \Bigl\lvert A^{\rm eff}\Bigr\rvert^2 \Biggr)\, \frac{e^{-4\pi \tau_2}}{ \mathcal V} \nonumber \\ &-4\,\Bigl\lvert W_0^{\rm eff} \Bigr\rvert\, \Bigl\lvert \tau_1^*\, \gamma\, W_0^{\rm eff} + 2\pi \tau_2\, A^{\rm eff} \Bigr\rvert \: \frac{ e^{-2\pi \tau_2}}{\mathcal{V}^2} \\ & +\frac{3\,\hat\xi}{4}\, \Bigl\lvert W^{\rm eff}_0\Bigr\rvert^2\: \frac{1}{\mathcal{V}^3}\; , \nonumber \end{align} with $\gamma=(a\,\mathcal{C}_1-b\,\mathcal{C}_2)\left(\frac{1}{a}-\frac{1}{b}\right)^{-1}$ and the axion $\rho_2$ stabilised such that the second term in \eqref{effspot} is minimised. The analytical analysis of this potential reveals that $\mathcal{V}$ is fixed at \begin{equation} \label{minV} \mathcal{V}^* = P\Bigl(A^{\rm eff},W_0^{\rm eff},\tau_1^*,\tau_2^*, \ldots\Bigr) \; e^{2\pi \tau_2^*} \end{equation} where $P$ is a rather complicated algebraic function of various quantities and $\tau^*_2$ is determined by an implicit equation. However, using the scaling $2\pi\tau_2\sim\ln\mathcal{V}$ obtained from \eqref{minV}, we observe that those terms in \eqref{effspot} involving $\tau_1^*$ scale as $\mathcal{V}^{-3}$ and can therefore be absorbed into $\hat\xi$. The resulting potential is of a form as in the LVS and so we expect to find a non-supersymmetric AdS minimum for large values of $\mathcal{V}$. \bigskip In order to obtain a positive cosmological constant, eventually the AdS minimum has to be uplifted. This is achieved for instance by anti $D3$-branes at the bottom of a Klebanov-Strassler warped throat \cite{Klebanov:2000hb}. The corresponding uplift potential has the form \begin{equation} \label{uplift} V_{\rm up} \sim \frac{a^4}{\mathcal{V}^2}\: M_{\rm Pl}^4 \qquad\mbox{and}\qquad a = e^{-\:\frac{2\pi K}{3 g_s M}} \end{equation} is the warp factor at the bottom of the throat with $M,K$ the $A$- respectively $B$-cycle flux-quanta. \section{Supersymmetric GUT Scenarios} \label{sec_gut} In the previous section, we have described a scenario featuring exponential control over $W^{\rm eff}_0$ and $A^{\rm eff}$ in an effective superpotential. As we will show in the following, by tuning the initial parameters (mostly at the order of 10\%), we are able to dynamically fix the moduli such that $\mathcal{V}\sim 10^5$ and $\lvert W_0^{\rm eff}\rvert\sim 10^{-10}$. Expressing the string scale $M_{\rm s}=(\alpha')^{-1/2}$ and the gravitino mass $M_{3/2}$ in terms of $W_0^{\rm eff.}$ and $\mathcal{V}$ (in Einstein frame) \begin{equation} \label{masses_01} M_{\rm s}= \frac{\sqrt{\pi}\, g_s^{1/4}}{\sqrt{\mathcal{V}}} \: M_{\rm Pl} \;,\hspace{40pt} M_{3/2}\sim \frac{\lvert W^{\rm eff}_0\rvert} {\mathcal{V}}\: M_{\rm Pl}\, , \end{equation} we see that for these values we obtain $M_{\rm s}\simeq M_X\simeq 1.2\cdot 10^{16}\,{\rm GeV}$ and $M_{3/2}$ in the TeV range. Note that for the usual LVS, a high degree of tuning is needed to have $M_{\rm s}\simeq M_X$ while keeping the susy breaking scale in the TeV regime. \bigskip Let us now investigate where in such a scenario the MSSM respectively the Grand Unified Theory might be localised. \begin{itemize} \item A first guess is to wrap the MSSM $D7$-branes on the four-cycle $\Gamma_{\rm b}$ with K\"ahler modulus $T_{\rm b}$ leading to a gauge coupling $\alpha^{-1}\simeq \tau_{\rm b}\sim \mathcal{V}^{2/3}$ which is however too small. \item A second possibility is to place the branes on $\Gamma_2$ with K\"ahler modulus $T_2$ giving $\alpha^{-1}\simeq\tau_2 \simeq \frac{1}{2\pi} \ln\mathcal{V}\simeq 1.8$ which differs from the GUT gauge coupling $\alpha^{-1}_X=25$ by an order of magnitude. \item A third possibility is to wrap the $D7$-branes along $\Gamma_1$ with gauge coupling $\alpha^{-1}\simeq \tau_1\sim -\ln\,\lvert W_0^{\rm eff} \rvert\sim 21$ which is in the right ballpark. \end{itemize} Note that we are certainly aware of the problem raised in \cite{Blumenhagen:2007sm} about stabilising K\"ahler moduli via instantons if the MSSM or a GUT is realised by D-branes. By considering a manifold with a further small K\"ahler modulus $T_4$ and wrapping the branes along $\Gamma_1+\Gamma_4$ with chiral intersection on $\Gamma_4$, we can avoid this issue. The additional modulus then has to be stabilised by D-terms or, as suggested in \cite{Cicoli:2008va}, by string loop-effects \cite{Berg:2007wt}. \bigskip Having identified a natural choice for the MSSM branes in our setup, let us now take a different point of view and fix $\mathcal{V}/\sqrt{g_s}\simeq1.26\cdot10^5$ together with $\tau_1\simeq\alpha_{X}^{-1}\simeq25$. By scanning the parameters $a=\frac{2\pi}{N},b=\frac{2\pi}{M},\mathcal{A},\mathcal{B}$ in a natural range, $M\in [2,12]$, $0<N<M$ and $\ln\mathcal{A}\in[\ln0.1,\ln10]$, $\ln\mathcal{B}\in[\ln\frac{\mathcal{A}}{10},\ln\mathcal{A})$ in equidistant steps, we find the distribution of resulting values for $M_{3/2}$ shown in figure \ref{fighisto}. This illustrates that in our setup with input $M_{\rm s}=M_X$ and $\alpha^{-1}=25$, a gravitino mass in the TeV region is obtained rather naturally. \begin{figure}[t] \begin{center} \includegraphics[width=0.5\textwidth]{fig_statistics} \begin{picture}(0,0) \put(-225,82){$N$} \put(5,10){$\log_{10} \Bigl( \frac{M_{3/2}}{\rm TeV} \Bigr)$} \end{picture} \end{center} \vspace{-10pt} \caption{{\small Distribution of $M_{3/2}$ with $\tau_1=25\pm 0.25$ and $\mathcal{V}/\sqrt{g_s}\simeq1.26\cdot10^5$ for a scan over natural values of $a,b,\mathcal{A},\mathcal{B}$. The statistical mean of this distribution is $\left\langle \log_{10}\left( M_{3/2}/{\rm TeV} \right)\right\rangle =1.79$ and the standard deviation is $\sigma = 0.98$.}\label{fighisto}} \end{figure} \bigskip We close the general analysis of our GUT setup with a summary of formulas for various mass scales: \begin{itemize} \item Suppressing factors of $g_s$ and $2\pi$, the masses of the closed sector moduli fields scale in the following way \cite{Conlon:2005ki} \begin{equation} \label{masses_02} \arraycolsep1.5pt \begin{array}{lcllcl} M_{U} &\sim& \displaystyle \frac{M_{\rm Pl}}{\mathcal{V} }\;, \hspace{70pt} & M_{T_1} &\sim&\displaystyle \lvert W^{\rm eff}_0\rvert \, M_{\rm Pl}\;, \\[10pt] M_{T_2} &\sim&\displaystyle \frac{\lvert W^{\rm eff}_0\rvert} {\mathcal{V}}\: M_{\rm Pl}\;, & M_{T_{\rm b}} &\sim& \displaystyle \frac{\lvert W^{\rm eff}_0\rvert} {\mathcal{V}^{3/2}}\: M_{\rm Pl}\; , \end{array} \end{equation} and in the regime $W^{\rm eff}_0<{\cal V}^{-1}$ they are ordered from heavy to light. \item For gravity mediated supersymmetry breaking, the gaugino masses are calculated as\footnote{In the following, we can safely ignore the effect of the uplifting \cite{Choi:2005ge} which gives only subdominant contributions to the soft terms.} \eq{ \label{gaugino_grav} m^{\rm gravity}_{1/2} = \frac{1}{2\,\mbox{Re}\,(f_a)} \: F^{I} \partial_I\, \mbox{Re}\,(f_a) \;. } \item For anomaly mediated supersymmetry breaking, the following formula has to be evaluated \cite{Randall:1998uk,Bagger:1999rd} \eq{ \label{gaugino_anom} m^{\rm anomaly}_{1/2,\, a} & = -\:\frac{\alpha_a}{4\pi}\: \biggl( \bigl( 3 T_G-T_R\bigr)\:M_{3/2} + \bigl( T_G-T_R\bigr)\: K_I F^I \\ & \hspace{158pt} + \frac{2\, T_R}{d_R}\: \partial_I\bigl( \ln \det K\bigr\rvert_R'' \bigr) F^I \biggr) \;, } where a sum over all representations $R$ is understood. Furthermore, $T_G$ is the Dynkin index of the adjoint representation, $T_R$ is the Dynkin index of the representation $R$ and $d_R$ denotes its dimension. Finally, $K\rvert_R''$ is the K\"ahler metric restricted to the representation $R$ which for a {\em swiss-cheese} manifold takes the form \cite{Conlon:2006tj,Conlon:2006wz} \eq{ \bigl(K\bigr\rvert_R''\bigr)_{i\overline{j}} = \frac{h_{i\overline j}\bigl(\tau_{\rm m},U\bigr)}{\tau_{\rm b}} \sim \frac{\tau_{\rm m}^{\lambda}}{\tau_{\rm b}} \:X_{i\overline j}\,\bigl(U\bigr) \;+\; \ldots \;. } Here, $\tau_{\rm m}$ denotes the volume of the small cycle the matter is localised on, in the present case $\tau_1$, and $\lambda$ can take values between $0$ and $1$. However, later we will use the value $\lambda=1/3$ of the minimal {\em swiss-cheese} setup \cite{Conlon:2006wz}. Finally, $X_{i\overline j}\,(U)$ is a matrix depending in the complex structure moduli. \item The scalar masses obtained for gravity mediation of supersymmetry breaking read \eq{ \bigl( m_{0}^{\rm gravity} \bigr)^2 = M_{3/2}^2 + V_0 - F^I F^{\overline J}\partial_I\partial_J \ln \tilde{K}_{\alpha} \;, } where the potential in the minimum $V_0$ is set to zero and the K\"ahler potential for the matter fields is \cite{Conlon:2006tj,Conlon:2006wz} \eq{ \tilde{K}_{\alpha} = \frac{k_{\alpha}\bigl(\tau_{\rm m},U\bigr)} {\tau_{\rm b}} \sim \frac{\tau_{\rm m}^{\lambda}}{\tau_{\rm b}}\: k_{\alpha}^{(0)}\bigl(U\bigr) \;+\; \ldots \;. } \item Generically, the two-loop generated scalar masses for an anom\-a\-ly mediated scenario are always smaller than the supergravity mediated masses. Therefore, we do not display the relevant formulas here. \end{itemize} \subsection{Model 1: A Starter} \label{sec_gut_1} In the following subsections, we investigate the scalar F-term potential resulting from the K\"ahler potential \eqref{kahl-pot} and the superpotential \eqref{instanton} numerically, that is we are searching for minima at large volume with $\ln \mathcal{V}\sim2\,\pi \tau_2$. \bigskip Taking the observations from the beginning of this section into account, we first assume that the MSSM is localised on $D7$-branes wrapping the cycle $\Gamma_1$ associated to the race-track modulus $T_1$ with size $\tau_1\simeq 25$. A particular set of parameters realising this setup without fine-tuning is the following \begin{equation} \label{parameters_naive} \arraycolsep1.5pt \begin{array}{@{}lcllcllcllcllcllcl@{}} \mathcal{A} &=& 1.6\,, \;\;& \mathcal{B} &=& 0.2\,, \;\;& \mathcal{C}_1&=& 1\,, \;\;& \mathcal{C}_2&=& 3\,, \;\;& a &=& \frac{2\pi}{8}\,, \;\;& b &=& \frac{2\pi}{9}\,, \\[4pt] g_s &=& \frac{2}{5}\,, \;\;& \eta_{\rm b} &=& 1\,, \;\;& \eta_1 &=& \frac{1}{53}\,, \;\;& \eta_2 &=& \frac{1}{6}\,, \;\;& \chi &=& \multicolumn{4}{l}{-136\,. } \end{array} \end{equation} We computed the scalar potential using the K\"ahler potential \eqref{kahl-pot} and the superpotential \eqref{instanton} without approximations and employed {\sf Mathematica} to determine the minimum with high precision. The resulting values are \eq{ \mathcal{V}^* = 78559\;,\quad T_1^* = 25.18\;,\quad T_2^* = 2.88\;,\quad V_F^* = -1.5\cdot 10^{-36}\, M_{\rm Pl}^4\;, } which is indeed the AdS large volume minimum argued for in the last section. Three plots showing the potential in the vicinity of the minimum can be found in figures \ref{figgut1}. \begin{figure}[h] \centering \renewcommand{\subfigcapskip}{-0pt} \subfigure[$V_F(\mathcal{V},\tau_1)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_1_V_tau1} } \hfill \subfigure[$V_F(\mathcal{V},\tau_2)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_1_V_tau2} } \hfill \subfigure[$V_F(\tau_1,\tau_2)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_1_tau1_tau2} } \vspace*{-7.5pt} \caption{\small F-term potential of the GUT model 1 in the vicinity of the minimum. \label{figgut1}} \end{figure} \noindent The various masses are computed using the formulas from above. Let us however emphasise that for the gaugino and scalar masses it is crucial to work with precise values for $T_1^*$, $T_2^*$ and $\mathcal{V}^*$ in order to numerically obtain certain cancellations. \begin{equation*} \arraycolsep1.5pt \renewcommand{\arraystretch}{1.25} \begin{array}{@{\hspace{9pt}}lclll@{\hspace{9pt}}||@{\hspace{9pt}} lclll@{\hspace{9pt}}||@{\hspace{9pt}} lclll@{\hspace{9pt}}} \multicolumn{5}{@{\hspace{9pt}}c@{\hspace{9pt}}||@{\hspace{9pt}}} {\mbox{Fundamental Masses}} & \multicolumn{5}{c@{\hspace{9pt}}||@{\hspace{9pt}}} {\mbox{Moduli Masses}} & \multicolumn{5}{c@{\hspace{9pt}}}{\mbox{Soft Masses}} \\ \hline\hline M_{\rm s} &=& 1.2\,\cdot&10^{16} &{\rm GeV} & M_{U} &=& 3.1\,\cdot&10^{13} &{\rm GeV} & m_{1/2}^{\rm gravity} &=& 1.1\,\cdot&10^{-5} &{\rm GeV} \\ M_{3/2} &=& 1.6\,\cdot&10^4 &{\rm GeV} & M_{T_1} &=& 1.2\,\cdot&10^{9} &{\rm GeV} & m_{1/2,\, \scriptscriptstyle SU(3)}^{\rm anomaly} &=& 1.2\,\cdot&10^{-3} &{\rm GeV} \\ &&&&& M_{T_2} &=& 1.6\,\cdot&10^{4} &{\rm GeV} & m_{0}^{\rm gravity} &=& 64& &{\rm GeV} \\ &&&&& M_{T_{\rm b}} &=& 56 & &{\rm GeV} \end{array} \end{equation*} \pagebreak[1] \noindent Let us comment on these scales: \vspace*{-1.75pt} \begin{itemize} \setlength{\itemsep}{0pt} \item By construction, the string scale $M_{\rm s}$ coincides with the GUT scale and the gravitino mass $M_{3/2}$ is in the TeV regime. \item The closed sector moduli masses are ordered from heavy to light and take acceptable values except for $T_{\rm b}$ which is too small. For $M_s=M_X$, the lower bound for moduli masses not to be in conflict with cosmology is around $1\,{\rm TeV}$. Therefore, in this model we face the Cosmological Moduli Problem (CMP) \cite{Banks:1993en,deCarlos:1993jw}. \item In addition, the gaugino as well as the scalar masses are far too small. The main reason is that we realised the MSSM on the cycle $\Gamma_1$ which is related to the race-track modulus $T_1$. For this modulus we observe numerical cancellations giving $F^1\simeq 2\cdot 10^{-22}\, M_{\rm Pl}\sim 10^{-8}\, M_{3/2}$, i.e. supersymmetry breaking on the race-track cycle is eight orders of magnitude smaller than naively expected. The explanation is that the exact race-track minimum, in leading order given by the globally supersymmetric minimum \eqref{fixed_T1}, is almost supersymmetric. For the gravity mediated gaugino masses we therefore obtain a strong suppression \eq{ \label{soft_scaling_1} m_{1/2}^{\rm gravity}=\frac{1}{2\,\tau_1}\: F^1\sim \frac{1}{2\cdot 25} \:2\cdot 10^{-22} \:M_{\rm Pl} \sim 10^{-5}\, {\rm GeV} \;. } \item For the anomaly mediated gaugino masses $m_{1/2}^{\rm anomaly}$, we find the expected cancellation of $M_{3/2}$ at leading order (see \cite{Conlon:2006wz} for a detailed derivation), however the sub-leading order in $\mathcal{V}$ dominates over $F^{1}$ \eq{ \label{soft_scaling_2} m_{1/2,\, \scriptscriptstyle SU(3)}^{\rm anomaly} & \sim \frac{\alpha_a}{4\pi}\:\biggl( 3\,T_G\,M_{3/2}\, \Bigl( 1-1+ \mathcal{O}\bigl(\mathcal{V}^{-1}\bigr) \Bigr) +2\,\lambda\, T_R\: \frac{F^1}{2\,\tau_1} \biggr) \\ &\sim \frac{1}{300}\:\biggl(3\cdot3\cdot 10^4\,{\rm GeV}\cdot10^{-5} + 2\, \lambda\cdot6\cdot \frac{2\cdot 10^{-22}\,M_{\rm Pl}}{2\cdot 25} \biggr) \\[3pt] & \sim 10^{-3}\,{\rm GeV} \;. } This value is of course still too small but note it is larger than the gravity mediated term. The gaugino masses are thus dominantly generated via anomaly mediation. \item A similar mechanism is at work for the scalar masses where $M_{3/2}^2$ is cancelled at leading order and subleading corrections in $\mathcal{V}$ give the main contribution (see again \cite{Conlon:2006wz} for a detailed derivation of this expression) \eq{ \label{soft_scaling_3} \bigl( m_{0}^{\rm gravity}\bigr)^2 &\sim M_{3/2}^2\,\Bigl( 1-1 + \mathcal{O}\bigl( \mathcal{V}^{-1}\bigr) \Bigr) +\lambda\,\biggl( \frac{F^1}{2\,\tau_1} \biggr)^2 \\ &\sim \bigl( 10^4\, {\rm GeV} \bigr)^2\cdot 10^{-5} +\lambda\, \biggl( \frac{2\cdot 10^{-22}\, M_{\rm Pl}}{2\cdot 25} \biggr)^2 \\[3pt] &\sim \bigl( 10^{3/2}\,{\rm GeV} \bigr)^2 \;. } \end{itemize} In conclusion, although we were able to easily arrange for $M_{\rm s}\simeq M_X$, $\alpha^{-1}\simeq\alpha^{-1}_X\simeq25$ and a gravitino mass in the TeV range, the soft terms are much too small. In order to obtain realistic soft masses, two options seem viable. Either we take the present setup and scale the mass parameters by a factor of $10^6$, or we wrap the MSSM branes not only along $\Gamma_1$ but on $\Gamma_1+\Gamma_2$ giving also a contribution from $F^2$ to the soft terms. In the following two subsections, we discuss these two possibilities in more detail. \subsection[Model 2: A Mixed Anomaly-Supergravity Mediated Model]{Model 2: A Mixed Anomaly-Gravity Mediated Model} Recall that the minimum for the race-track modulus $T_1$ is approximately supersymmetric. To construct a model with gaugino masses in the TeV range, we use the same set of parameters \eqref{parameters_naive} of the previous setup but scale $\mathcal{A}$ and $\mathcal{B}$ to somewhat more unrealistic values \eq{ \label{model2_scaling} \mathcal{A}=1.6\;\times\;8\cdot10^5\;,\qquad \mathcal{B}=0.2\;\times\;8\cdot10^5\;. } The minimum of the resulting F-term potential is not changed except for the value of $V_F$ in the minimum \eq{ \mathcal{V}^* = 78559\;,\quad T_1^* = 25.18\;,\quad T_2^* = 2.88\;,\quad V_F^* = -9.5\cdot 10^{-25}\, M_{\rm Pl}^4\;. } Three plots showing the potential in the vicinity of the minimum can be found in figures \ref{figgut2} on page \pageref{figgut2} and the mass scales for the new setup are the following: \begin{equation*} \arraycolsep1.5pt \renewcommand{\arraystretch}{1.25} \begin{array}{@{\hspace{9pt}}lclll@{\hspace{9pt}}||@{\hspace{9pt}} lclll@{\hspace{9pt}}||@{\hspace{9pt}} lclll@{\hspace{9pt}}} \multicolumn{5}{@{\hspace{9pt}}c@{\hspace{9pt}}||@{\hspace{9pt}}} {\mbox{Fundamental Masses}} & \multicolumn{5}{c@{\hspace{9pt}}||@{\hspace{9pt}}} {\mbox{Moduli Masses}} & \multicolumn{5}{c@{\hspace{9pt}}}{\mbox{Soft Masses}} \\ \hline\hline M_{\rm s} &=& 1.2\,\cdot&10^{16} &{\rm GeV} & M_{U} &=& 3.1\,\cdot&10^{13} &{\rm GeV} & m_{1/2}^{\rm gravity} &=& 8.6 & &{\rm GeV} \\ M_{3/2} &=& 1.3\,\cdot&10^{10} &{\rm GeV} & M_{T_1} &=& 9.9\,\cdot&10^{14} &{\rm GeV} & m_{1/2,\, \scriptscriptstyle SU(3)}^{\rm anomaly} &=& 962 & &{\rm GeV} \\ &&&&& M_{T_2} &=& 1.3\,\cdot&10^{10} &{\rm GeV} & m_{0}^{\rm gravity} &=& 5.1\,\cdot&10^7 &{\rm GeV} \\ &&&&& M_{T_{\rm b}} &=& 4.5\,\cdot&10^7 &{\rm GeV} \end{array} \end{equation*} \pagebreak[2] \noindent We again comment on these scales: \begin{itemize} \item Since the value of the volume modulus is not changed compared to the previous setup, we similarly obtain $M_{\rm s}\simeq M_X$. However, the gravitino mass is in an intermediate regime due to the change in $W_0^{\rm eff}$. \item With the gravitino masses in the intermediate regime, the Cosmological Moduli Problem has been evaded, as $T_{\rm b}$ is much heavier then the TeV scale. We also see that $M_{T_1}>M_U$, but since we have a well-defined expansion in $1/\mathcal{V}$ we can safely stabilise $U$ and then study the stabilisation of $T_1$. \item Concerning the gravity mediated gaugino mass, the scaling \eqref{model2_scaling} results in a value of $W_0^{\rm eff}$ which is $8\cdot 10^5$ times larger than in the previous setup leading to $F^1\sim 10^{-16}\, M_{\rm Pl}$. The gravity mediated gaugino mass $m_{1/2}^{\rm gravity}$ is still too small but the anomaly mediated masses are now in the TeV regime. \item As expected from the first example, the gravity mediated scalar masses are at $m^{\rm gravity}_0\sim 10^7\,{\rm GeV}$ and therefore much heavier than the gaugino masses. However, they come in complete $SU(5)$ multiplets and therefore do not spoil gauge coupling unification. \item With this hierarchy between the gaugino masses and the scalar masses, we have a dynamical realisation of the split supersymmetry scenario \cite{ArkaniHamed:2004fb}. The Higgs sector masses, i.e. the canonically normalised $\hat\mu$-term and the soft term $\mu B$, are expected to be of the same order of magnitude as the scalar masses, simply for the reason that here both $F^{\rm b}\sim M_{3/2}$ and $F^1$ contribute. Therefore, in order to keep these at the weak scale, a fine-tuning of the supersymmetric $\mu$-term is necessary. \end{itemize} To summarise, tuning the initial parameters such that $\alpha^{-1}\simeq\alpha_X^{-1}\simeq 25$ and $M_s=M_X$, and localising the MSSM solely on the race-track cycle $\Gamma_1$, we find a high suppression of the gravity mediated gaugino masses due to the quasi-supersymmetry of the race-track minimum. Fixing the gaugino masses at the TeV scale leads to an intermediate supersymmetry breaking scenario with gravitino masses and scalar masses in the intermediate regime. Due to the large value of $M_{3/2}$, this evades the CMP and gives a stringy realisation of the split supersymmetry scenario proposed in \cite{ArkaniHamed:2004fb}.\footnote{For a local realisation of split supersymmetry see \cite{Antoniadis:2004dt}, and a realisation by mixed anomaly-D-term mediation has been reported in \cite{Dudas:2005vv}.} \subsection{Model 3: An LVS like Supergravity Mediated Model} \label{sec_gut_3} We now consider the second possibility from the end of section \ref{sec_gut_1} which indeed realises our initial goal, namely to naturally find large volume minima with the string scale at the GUT scale, gauge coupling unification and soft masses in the TeV regime. This provides a concrete moduli stabilisation scenario for which the analysis of \cite{Aparicio:2008wh} is applicable. There, the computation of soft masses and running to the weak scale has been studied quite systematically for moduli dominated supersymmetry breaking in F-theory respectively Type IIB orientifold compactifications realising the MSSM. \bigskip We modify our original setup such that the soft terms are dominantly generated via $T_2$ similarly to the original LVS. To do so, we place the MSSM $D7$-branes on the combination of cycles $\Gamma_1+\Gamma_2$. A concrete set of parameters realising this setup without fine-tuning is \begin{equation} \arraycolsep1.5pt \begin{array}{@{}lcllcllcllcllcllcl@{}} \mathcal{A} &=& 1.5\,, \;\;& \mathcal{B} &=& 0.25\,, \;\;& \mathcal{C}_1&=& 1\,, \;\;& \mathcal{C}_2&=& 3\,, \;\;& a &=& \frac{2\pi}{8}\,, \;\;& b &=& \frac{2\pi}{9}\,, \\[4pt] g_s &=& \frac{2}{5}\,, \;\;& \eta_{\rm b} &=& 1\,, \;\;& \eta_1 &=& \frac{1}{40}\,, \;\;& \eta_2 &=& \frac{1}{6}\,, \;\;& \chi &=& \multicolumn{4}{l}{-153\,. } \end{array} \end{equation} The minimum of the scalar F-term potential is again determined with the help of {\sf Mathematica} giving \eq{ \mathcal{V}^* = 92158\;,\quad T_1^* = 21.58\;,\quad T_2^* = 2.91\;,\quad V_F^*= -1.4\cdot 10^{-34}\, M_{\rm Pl}^4\;, } so that the gauge coupling of the D7-branes $\alpha^{-1}=\tau_1+\tau_2=24.8$ is again the unified gauge coupling at the GUT scale. Three plots showing the potential in the vicinity of the minimum can be found in figures \ref{figgut3} on page \pageref{figgut3}. The mass scales in this setup are calculated as follows: \begin{equation*} \arraycolsep1.5pt \renewcommand{\arraystretch}{1.25} \begin{array}{@{\hspace{9pt}}lclll@{\hspace{9pt}}||@{\hspace{9pt}} lclll@{\hspace{9pt}}||@{\hspace{9pt}} lclll@{\hspace{9pt}}} \multicolumn{5}{@{\hspace{9pt}}c@{\hspace{9pt}}||@{\hspace{9pt}}} {\mbox{Fundamental Masses}} & \multicolumn{5}{c@{\hspace{9pt}}||@{\hspace{9pt}}} {\mbox{Moduli Masses}} & \multicolumn{5}{c@{\hspace{9pt}}}{\mbox{Soft Masses}} \\ \hline\hline M_{\rm s} &=& 1.1\,\cdot&10^{16} &{\rm GeV} & M_{U} &=& 2.6\,\cdot&10^{13} &{\rm GeV} & m_{1/2}^{\rm gravity} &=& 819 & &{\rm GeV} \\ M_{3/2} &=& 168 & &{\rm TeV} & M_{T_1} &=& 1.5\,\cdot&10^{10} &{\rm GeV} & m_{1/2,\,\scriptscriptstyle SU(3)}^{\rm anomaly} &=& 10 & &{\rm GeV} \\ &&&&& M_{T_2} &=& 168 & &{\rm TeV} & m_{0}^{\rm gravity} &=& 817 & \, &{\rm GeV} \\ &&&&& M_{T_{\rm b}} &=& 553 & &{\rm GeV} \end{array} \end{equation*} \pagebreak[1] \noindent Let us again comment on the various scales arising in this large volume minimum: \begin{itemize} \item We arranged the parameters for the present setup such that $M_{\rm s}\simeq M_X$ together with the gravitino mass in the TeV range. \item The closed sector moduli masses take (almost) acceptable values with \linebreak[4]$M_{T_{\rm b}}$ close to the regime where the Cosmological Moduli Problem may be avoided. (Note that we were not careful with factors of $2\pi$.) \item The soft masses are dominantly generated via the supersymmetry breaking of $T_2$ specified by $F^2\sim10^{-14} M_{\rm Pl}$. Since $T_2$ is stabilised as in the original LARGE Volume Scenario, similar mechanisms generating the soft masses are at work \cite{Conlon:2006wz}. In particular, using the fact that $F^1\sim 10^{-20}M_{\rm Pl}\ll F^2$, the common term determining the gaugino as well as the scalar masses can be expressed as \eq{ \frac{F^1+F^2}{2\,\bigl( \tau_1+\tau_2\bigr)} \sim \frac{F^2}{2\,\bigl( \tau_1+\tau_2\bigr)} \sim \frac{\tau_2}{\tau_1+\tau_2}\: \frac{M_{3/2}}{\ln\left( M_{\rm Pl}/M_{3/2}\right)} \;. } Here we used that $F^2\simeq 2 \tau_2\,M_{3/2}/\ln\left( M_{\rm Pl}/M_{3/2}\right)$ which was obtained in \cite{Conlon:2006us}. For the gravity mediated gaugino mass we thus find \eq{ m_{1/2}^{\rm gravity}=\frac{F^1+F^2}{2\,\bigl(\tau_1+\tau_2\bigr)}\: \sim \frac{3}{25} \:\frac{1.7\cdot 10^{5}\,{\rm GeV}} {\ln\bigl( 10^{18}\cdot 10^{-5}\bigr)} \sim 700\, {\rm GeV} \;. } \item For the anomaly mediated gaugino mass we use equation \eqref{soft_scaling_2} to obtain \begin{align} \nonumber m_{1/2,\, \scriptscriptstyle SU(3)}^{\rm anomaly} & \sim \frac{\alpha_a}{4\pi}\:\biggl( 3\,T_G\,M_{3/2}\, \Bigl( 1-1+ \mathcal{O}\bigl(\mathcal{V}^{-1}\bigr) \Bigr) +2\,\lambda\, T_R\: \frac{F^1+F^2}{2\,\bigl(\tau_1+\tau_2\bigr)} \biggr) \\ &\sim \frac{1}{300}\:\biggl(3\cdot3\cdot 10^5\,{\rm GeV}\cdot10^{-6} + 2\, \lambda\cdot6\cdot 700\, {\rm GeV} \biggr) \\[3pt] \nonumber & \sim 10\,{\rm GeV} \;, \end{align} with $\lambda=1/3$ \cite{Conlon:2006wz}. Note that again the contribution of $M_{3/2}$ is cancelled at leading order but the subleading correction $M_{3/2}/\mathcal{V}$ is suppressed compared to $F^2$. Furthermore, in this supersymmetry breaking scheme, the anomaly mediated gaugino masses are significantly smaller than the gravity mediated ones. This is contrast to the so-called mirage mediation scheme arising for instance in the original KKLT scenario \cite{Endo:2005uy,Choi:2005uz,Choi:2007ka}, where both are of the same order of magnitude. \item Finally, we consider the gravity mediated scalar masses. Referring to formula \eqref{soft_scaling_3}, we obtain \eq{ \bigl( m_{0}^{\rm gravity}\bigr)^2 &\sim M_{3/2}^2\,\Bigl( 1-1 + \mathcal{O}\bigl( \mathcal{V}^{-1}\bigr) \Bigr) +\lambda\,\biggl( \frac{F^1+F^2}{2\,\bigl(\tau_1+\tau_2\bigr)} \biggr)^2 \\[3pt] &\sim \bigl( 10^5\, {\rm GeV} \bigr)^2\cdot 10^{-6} +\lambda\, \bigl( 700\,{\rm GeV} \bigr)^2 \\[4pt] &\sim \bigl( 10^{2}\,{\rm GeV} \bigr)^2 \;. } Note that for the scalar masses, the contribution from $M_{3/2}$ is cancelled at leading order but now the subleading corrections scale as $M_{3/2}/\sqrt{\mathcal{V}}$. Therefore, by accident, the two terms in the equation above are of the same order which is in contrast to the relation $m_{0}^{\rm gravity}= m_{1/2}^{\rm gravity}/\sqrt{3}$ obtained in the LVS \cite{Conlon:2006wz}. \item Assuming that the supersymmetric $\mu$-term vanishes, the canonical normalised Higgs parameters are also in the TeV regime. This would solve the $\mu$-problem via the Giudice-Masiero mechanism \cite{Giudice:1988yz}. \end{itemize} In summary, by wrapping the MSSM supporting $D7$-branes along $\Gamma_1+\Gamma_2$, we obtain a LARGE Volume Scenario with supergravity mediated soft masses in the TeV region and $M_{\rm s}=M_X$. This is different compared to the original LVS where the string scale is usually at an intermediate scale. Along the lines of \cite{Conlon:2007xv,Aparicio:2008wh}, the next step is to calculate the soft-terms at the weak scale to see whether distinctive patterns for the supersymmetric phenomenology to be tested at the LHC can be obtained. \section{Comment on the Cosmological Constant} \label{sec_cc} For the GUT setup discussed in the previous section, the tree-level cosmological constant is $V_F^*=-1.4\cdot 10^{-34}M_{\rm Pl}^4$ and therefore a high degree of fine-tuning in the uplift potential \eqref{uplift} is needed to obtain $\Lambda \simeq +10^{-120}\,M_{\rm Pl}^4$. After such a fine-tuning has been achieved, a LARGE volume scenario with $M_{\rm s}\simeq M_X$ contains a natural candidate serving as a quintessence field \cite{Choi:1999xn,Svrcek:2006hf,Quevedo:talksp08}. Indeed, taking into account also non-perturbative corrections corresponding to the large four-cycle $\Gamma_{\rm b}$, the scalar potential depending on the (canonically normalised) axion $\sigma_{\rm b}$ takes the form \eq{ \label{quintessence} \frac{V_Q}{M_{\rm Pl}^4}\simeq \left( \frac{M_{\rm Pl}\, M_{3/2}}{M_X^2}\right)\, \:e^{-\frac{2\pi}{L} \tau_{\rm b}}\: \biggl( 1-\cos\biggl( \frac{2\pi}{L}\, \sigma_{\rm b} \biggr)\biggr) \;. } For the minimum $\mathcal{V}^* \simeq \tau_{\rm b}^{2/3}\simeq 92158$ of our model from section \ref{sec_gut_3} and $L$ of the order $L=40\ldots50$, the prefactor in \eqref{quintessence} is of the right order of magnitude. \bigskip Although the GUT models from the previous section may contain a quint\-essence field, let us now take a different point of view. Since in our scenario we have exponential control over $W_0^{\rm eff}$ and $A^{\rm eff}$, one might ask whether it is possible to dynamically find a minimum $ V_F^* \simeq - \lvert W^{\rm eff}_0\rvert^2\, \mathcal{V}^{-3} \simeq - 10^{-120}M_{\rm Pl}^4$ realised without fine-tuning. Ignoring the weak scale for a moment, for the string scale we have $M_{\rm s}\sim\mathcal{V}^{-1/2}M_{\rm Pl}>{\rm TeV}$ which implies that $\mathcal{V}<10^{30}$. To keep the tuning of $a,b,\mathcal{A},\mathcal{B}$ moderate, we identify $\lvert W_0^{\rm eff}\rvert\sim10^{-15}$ and thus $\mathcal{V}\sim 10^{30}$ as a natural choice to realise $ V_F^* \simeq - 10^{-120}M_{\rm Pl}^4$. A set of parameters dynamically leading to such values is for instance \begin{equation*} \arraycolsep1.5pt \begin{array}{@{}lcllcllcllcllcllcl@{}} \mathcal{A} &=& 1\,, \;\;& \mathcal{B} &=& 0.1\,, \;\;& \mathcal{C}_1&=& 1\,, \;\;& \mathcal{C}_2&=& 3\,, \;\;& a &=& \frac{2\pi}{13}\,, \;\;& b &=& \frac{2\pi}{14}\,, \\[4pt] g_s &=& \frac{1}{5}\,, \;\;& \eta_{\rm b} &=& 1\,, \;\;& \eta_1 &=& \frac{1}{30}\,, \;\;& \eta_2 &=& \frac{1}{6}\,, \;\;& \chi &=& \multicolumn{4}{l}{-452\,, } \end{array} \end{equation*} and the plots showing the potential in the vicinity of the minimum can be found in figures \ref{figcc} on page \pageref{figcc}. The numerical values specifying the minimum are \eq{ \mathcal{V}^* = 6.4\cdot10^{28}\;,\quad T_1^* = 68.84\;,\quad T_2^* = 11.53\;,\quad V_F^*= -7.8\cdot 10^{-121}M_{\rm Pl}^4\;, } and because the AdS minimum is at $V_F^*\sim-10^{-120}M_{\rm Pl}^4$, the warp factor $a=10^{-15}$ in the uplift potential \eqref{uplift} does not involve any fine-tuning of the flux parameters $K$ and $M$. We conclude that there exist vacua of the scalar potential \eqref{effspot} whose tree-level cosmological constant has the right order of magnitude. However, this clearly does not solve the cosmological constant problem, as we have not yet identified the Standard Model and the origin of the weak scale. Once we try to introduce the MSSM into this set-up, we are confronted with the usual problems. Let us briefly explain three possibilities: \begin{itemize} \setlength{\itemsep}{0pt} \item Localising the MSSM on $D7$-branes wrapping the cycles $\Gamma_{(1,2)}$ leads to soft masses below the gravitino mass scale $M_{3/2}\simeq 10^{-18}\,{\rm eV}$ which itself is ridiculously small. \item Since $M_{\rm s}\sim {\rm TeV}$, we could break supersymmetry at the string scale and place a non-supersymmetric anti $D3$-brane configuration realising the \linebreak[4] MSSM at the bottom of a throat, i.e. on the TeV brane in the RS scenario. The uncancelled NS-NS tadpole of the non-supersymmetric brane configuration would be the red-shifted uplift term. However, all mass scales in the throat are red-shifted as well \cite{Randall:1999ee} so that the stringy excitations such as squarks have masses $M_{\tilde Q}\simeq a\, M_{\rm s}\simeq \Lambda^{1/4} = 10^{-3}\, {\rm eV}$. \item A third option is to place an explicitly supersymmetry breaking D-brane configuration in the bulk, i.e. on the Planck brane in the RS scenario. Then the superpartners have string scale masses in the TeV region, but we get an additional positive contribution $V\sim \mathcal{O}(M_{\rm s}^4)$ to the scalar potential. \end{itemize} In conclusion, even though we have exponential control over the effective parameters $W_0^{\rm eff}$ and $A^{\rm eff}$, the cosmological constant problem is not even touched. It can be phrased as the problem of hiding the TeV scale supersymmetry breaking of the Standard Model such that it does not induce a large contribution to the tree-level value $\Lambda_0$. \section{Summary and Conclusions} \label{sec_concl} In this paper, we have studied string theory compactifications of type IIB orientifolds where the complex structure moduli are assumend to be stabilised by fluxes such that $W_{\rm flux}\rvert_{\rm min.}=0$. In addition, we considered a compactification manifold of {\em swiss-cheese} type where gaugino condensation on two stacks of $D7$-branes leads to a race-track superpotential. Taking into account instanton corrections to the gauge kinetic function manifesting themselves as poly-instanton corrections to the race-track superpotential, we constructed a scenario featuring exponential control over the parameters $W_0^{\rm eff}$ and $A^{\rm eff}$ in an effective superpotential of the form $W^{\rm eff}=W_0^{\rm eff}-A^{\rm eff} \exp(-a\, T)$. Such superpotentials are used for moduli stabilisation in KKLT and LARGE Volume Scenarios. However, in our setup we can arrange for exponential small values of these parameters without fine-tuning. \bigskip Using this setup, we were able to find minima of the resulting scalar potential realising supersymmetric GUT scenarios featuring $M_{\rm s}=M_X\simeq 1.2\cdot 10^{16}\,{\rm GeV}$, $\alpha^{-1}\simeq\alpha^{-1}_X\simeq25$ and a gravitino mass in the TeV region without fine-tuning. However, despite the phenomenological interesting value of $M_{3/2}$, the soft terms in our first setup are strongly suppressed. The reason is that the race-track modulus is stabilised in a nearly supersymmetric minimum giving F-terms which are much smaller than expected. We proposed two resolutions of this issue and constructed the corresponding setups: \begin{itemize} \item First, we scaled our parameters such that effectively $W_0^{\rm eff}$ is scaled by a factor of $10^6$ leading to a larger gravitino mass but also to larger soft masses. The gaugino masses are dominantly generated by anomaly-mediation while the scalar masses are generated by gravity mediation leading to a stringy realisation of split supersymmetry if the Higgs and Higgsino masses are tuned to small values. \item The second possibility we considered was to generate the soft terms not by the F-term of the race-track modulus but also by the F-term of the small LVS K\"ahler modulus. The gaugino as well as the scalar masses are then generated by gravity mediation similarly to the original LARGE Volume Scenario. The lightest modulus was on the edge of posing problems with cosmology (CMP) and the $\mu$-problem could be solved by the Giudice-Masiero mechanism. \end{itemize} Clearly, a more detailed analysis of the phenomenological implications of these setups along the lines of \cite{Conlon:2007xv,Aparicio:2008wh} would be very interesting. Moreover, it remains to be seen whether one can indeed construct global string or F-theory compactifications realising all the features we assumed for this scenario. Not the least challenge is to concretely evade the problem of intrinsic tension between moduli stabilisation via instantons and a chiral MSSM sector raised in \cite{Blumenhagen:2007sm}. \bigskip Finally, in the last section we mentioned that for our GUT models there is, similarly to the LARGE Volume Scenarios, a natural candidate for a quintessence field. Taking a different point of view, we were also able to construct a setup with tree-level cosmological constant at the order of $\Lambda_0\sim -10^{-120} M^4_{\rm Pl}$ which can be uplifted to a positive value without fine-tuning. However, although we obtained an encouraging value for $\Lambda_0$, introducing the Standard Model in this setup will spoil this feature. \subsection*{Acknowledgements} We gratefully acknowledge discussions with N. Aker\-blom, E. Dudas, T. Grimm, D. L\"ust, M. Schmidt-Sommerfeld and T. Weigand. We also thank S. P. de Alwis for pointing out a possible unclarity in the first version of this paper. \clearpage \begin{figure}[p] \centering \renewcommand{\subfigcapskip}{-0pt} \subfigure[$V_F(\mathcal{V},\tau_1)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_2_V_tau1} } \hfill \subfigure[$V_F(\mathcal{V},\tau_2)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_2_V_tau2} } \hfill \subfigure[$V_F(\tau_1,\tau_2)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_2_tau1_tau2} } \vspace*{-5pt} \caption{\small F-term potential of the GUT model 2 in the vicinity of the minimum. \label{figgut2}} \vspace{-11pt} \end{figure} \begin{figure}[p] \centering \renewcommand{\subfigcapskip}{-0pt} \subfigure[$V_F(\mathcal{V},\tau_1)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_3_V_tau1} } \hfill \subfigure[$V_F(\mathcal{V},\tau_2)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_3_V_tau2} } \hfill \subfigure[$V_F(\tau_1,\tau_2)$]{ \includegraphics[width=0.3\textwidth]{fig_gut_3_tau1_tau2} } \vspace*{-5pt} \caption{\small F-term potential of the GUT model 3 in the vicinity of the minimum. \label{figgut3}} \vspace{-11pt} \end{figure} \begin{figure}[p] \centering \renewcommand{\subfigcapskip}{-0pt} \subfigure[$V_F(\mathcal{V},\tau_1)$]{ \includegraphics[width=0.3\textwidth]{fig_cc_V_tau1} } \hfill \subfigure[$V_F(\mathcal{V},\tau_2)$]{ \includegraphics[width=0.3\textwidth]{fig_cc_V_tau2} } \hfill \subfigure[$V_F(\tau_1,\tau_2)$]{ \includegraphics[width=0.3\textwidth]{fig_cc_tau1_tau2} } \vspace{-5pt} \caption{\small F-term potential of the $\Lambda$ model in the vicinity of the minimum.\label{figcc}} \vspace{-11pt} \end{figure} \clearpage \nocite{*}

proofpile-arXiv_069-11227

\section{Introduction}\label{S:intro} The method of approximate approximations is mainly directed to the numerical solution of partial integro-differential equations. The method provides simple formulas for quasi-interpolants, which approximate functions up to a prescribed precision very accurately, but in general the approximants do not converge. The lack of convergence, which is not perceptible in numerical computations, is offset by a greater flexibility in the choice of approximating functions. So it is possible to construct multivariate approximation formulas, which are easy to implement and have additionally the property that pseudodifferential operations can be effectively performed. This allows to create effective numerical algorithms for solving boundary value problems for differential and integral equations. Approximate quasi-interpolants on a uniform grid are of the form \begin{equation}\label{general0} Mu(x)=\mathcal D^{-n/2}\sum_{j\in\mathbb{Z}^{n}}\mathcal H\left(\frac{x-h j}{h\sqrt{\mathcal D}} \right)u(hj) \end{equation} with positive parameters, ``small'' $h$ and ``large'' $\mathcal D$, and the generating function $\mathcal H$ is sufficiently smooth and of rapid decay. Their properties have been studied in a series of papers \cite{Ma2,MS1,MS2,Ma1} (cf. also the monograph \cite{MSbook}). It has been shown that the quasi-interpolant approximates smooth functions with \begin{equation}\label{generalest} |Mu(x)-u(x)|\leq \varepsilon \sum_{|\beta|=0}^{N-1} (h \sqrt{\mathcal D})^{|\beta|}|\partial^{\beta} u(x)|+ c (h \sqrt{\mathcal D})^{N} \Vert \nabla_{N} u \Vert_{L_{\infty}} \, , \end{equation} as long as $\mathcal H$ is subject to the moment condition \begin{equation}\label{momcon} \int \limits_{R^n} x^\alpha \mathcal H(x) \, dx = \delta_{|\alpha|0}\, , \quad 0 \le |\alpha| < N \, . \end{equation} In \eqref {generalest} the constant $c$ depends only on $\mathcal H$ and $\varepsilon$ can be made arbitrarily small if the parameter $\mathcal D$ is sufficiently large, so that one can fix $\mathcal D$ such that in numerical computations $Mu$ approximates with the order $O(h^N)$. The construction of simple generating functions satisfying \eqref{momcon} for arbitrary $N$ has been addressed in \cite{MS3}, whereas \cite{LMS,MS4} extend the results to the case of nonuniform grids. In this paper we study more general quasi-interpolation formulas with values of a function $u$ and of some of its derivatives prescribed at the points of a uniform grid. More precisely, we consider approximants of the form \begin{equation}\label{generalqi} Mu(x)=\mathcal D^{-n/2}\sum_{j\in\mathbb{Z}^{n}}\mathcal H\left(\frac{x-h j}{h\sqrt{\mathcal D}} \right)\mathcal Q((-h\sqrt{\mathcal D})\partial)u(hj)\, , \end{equation} where $\mathcal Q(t),\, t\in \mathbb{R}^{n}$, is a polynomial with $\deg \mathcal Q<N$. We establish estimates of the type \eqref{generalest} if the generating function $\mathcal H$ and the coefficients of $\mathcal Q$ are connected by suitable conditions (see \eqref{3.10}). It is shown in particular, that for an arbitrary polynomial $\mathcal Q$ there exists $\mathcal H$ such that the approximate Hermite quasi-interpolant \eqref{generalqi} satisfies the estimate \eqref{generalest}. On the other hand, the same is true if $\mathcal H$ is the Gaussian or a related function and $\mathcal Q$ is chosen suitably. As a byproduct we obtain very simple approximants which provide high order approximations to solutions of the Laplace equation or other elliptic differential equations. Estimate \eqref{generalest} is proved in Section \ref{S:DUE} for \eqref{generalqi} with $\mathcal H$ and $\mathcal Q$ connected by conditions \eqref{3.10}. In Section \ref{S:QUA} we describe a simple method for the construction of $\mathcal H$ satisfying the conditions \eqref{3.10} for a given polynomial $\mathcal Q$. This result is applied in Section \ref{example} when we consider some examples of Hermite quasi-interpolants. In the particular case $\mathcal H(x)=\pi^{-n/2}(\det B)^{-1/2}{\rm e}^{-<B^{-1}x,x>}$, with $B=\{b_{ij}\}$ symmetric and positive definite real matrix, we obtain the following quasi-interpolant of order $\mathcal O((h\sqrt{\cD})^{2M})$ \[M u(x)=\frac{(\det B)^{-1/2}}{(\pi \mathcal D)^{n/2}} \sum_{m\in\mathbb{Z}^{n}}\sum_{s=0}^{M-1} \frac{(-h^2 \mathcal D )^{s}}{s!4^{s}} \, \mathcal B ^{s}u(hm)\,{\rm e}^{\, -\langle B^{-1}(x-hm),\,x-hm\rangle/(h^2\mathcal D)} \] where $\mathcal B$ is the second order partial differential operator $\mathcal B u=\sum_{i,j=1}^{n}b_{ij}\partial_{i}\partial_{j}u$. In Section \ref{appl} we use $M u$ for the approximation of solutions of the equation $\mathcal B u=0$. If $u$ satisfies the equation in $\mathbb{R}^{n}$ then $M u$ has the simple form of the quasi-interpolant of order $\mathcal O((h\sqrt{\cD})^{2})$ but gives an approximation of exponential order plus a small saturation error. The same approximation property holds for functions which satisfy the equation $\mathcal B u=0$ in a domain $\Omega\subset \mathbb{R}^{n}$. If the function $u$ is extended by zero outside $\Omega$ then $M u$ simplifies to \[ M u(x)=\frac{(\det B)^{-1/2}}{(\pi \mathcal D)^{n/2}} \sum_{hm\in\Omega}\, u(hm)\,{\rm e}^{\, -\langle B^{-1}(x-hm),\,x-hm\rangle/(h^2\mathcal D)} . \] We obtain that, for any $\varepsilon>0$, $M\,u$ approximates pointwise $u$ in a subdomain $\Omega'\subsetneq \Omega$ with $$|M u(x)-u(x)|\leq \varepsilon \sum_{|\beta|=0}^{2M-1} (h \sqrt{\mathcal D})^{|\beta|}|\partial^{\beta} u(x)|+ C_B (h \sqrt{\mathcal D})^{2M} c_{u},$$ if we choose $\mathcal D$ and $h$ appropriately; the constant $C_B$ is independent of $u,h,\mathcal D, M$ and $c_{u}$ depends on $u$. In Section \ref{deriv} we show that the Hermite quasi-interpolant \eqref{generalqi} gives the simultaneous approximation of the derivatives of $u$. If $\partial^{\beta}\mathcal H$ exists and is of rapid decay, for any function $u\in W_{\infty}^{L}(\mathbb{R}^{n})$ with $L\geq N+|\beta|$, the difference $\partial^{\beta}Mu(x)-\partial^{\beta}u(x)$ can be estimated by \[ |\partial^{\beta}Mu(x)-\partial^{\beta}u(x)|\leq \varepsilon\,\sum_{|\gamma|=0}^{L-1}(h\sqrt{\cD})^{|\gamma|-|\beta|}|\partial^{\gamma}u(x)|+ c_{1}\,(h\sqrt{\cD})^{N}\Vert \nabla_{N+|\beta|} u \Vert_{L_{\infty}}, \] with $c_{1}$ independent of $u$, $h$ and $\mathcal D$. \section{Quasi-interpolants with derivatives}\label{S:DUE} In this section we study the approximation of a smooth function $u(x)$, $x\in \mathbb{R}^{n}$, by the Hermite quasi-interpolation operator \begin{equation}\label{3.0} M u(x)=\mathcal D^{-n/2} \sum_{m\in \mathbb{Z}^{n}}\mathcal H\left( \frac{x-h m}{h \sqrt{\mathcal D}}\right) \mathcal Q\left(-h \sqrt{\mathcal D}\, {\partial }\right) u(hm) \end{equation} where $\mathcal Q(t)$, $t\in \mathbb{R}^{n}$, is a polynomial of degree at most $N-1$ \begin{equation}\label{polinomio} \mathcal Q(t)=\sum_{|\gamma|=0}^{N-1}a_{\gamma} t^{\gamma},\> t \in \mathbb{R}^{n}, \quad \mbox{with }a_{\gamma}\in \mathbb{R} \; \mbox{ and } a_{0}=1 \, , \end{equation} $\partial=\partial_1 \ldots\partial_n$, and $\mathcal H$ is a sufficiently smooth, rapidly decaying function. Then \begin{equation} M u(x)=\mathcal D^{-n/2} \sum_{m\in \mathbb{Z}^{n}}\left(\sum_{|\gamma|=0}^{N-1}(-h \sqrt{\mathcal D})^{|\gamma|} a_{\gamma} \partial^{\gamma} u(hm) \right) \mathcal H\left( \frac{x-h m}{h \sqrt{\mathcal D}} \right). \label{3.1} \end{equation} Our aim is to give conditions on the generating function $\mathcal H$ such that (\ref{3.1}) is an approximation formula of order $\mathcal O((h \sqrt{\mathcal D})^{N})$ plus terms, which can be made sufficiently small. Suppose that $u\in C^{N}(\mathbb{R}^{n})\cap L^\infty(\mathbb{R}^n)$. Taking the Taylor expansion of $\partial^{\gamma} u$ at each node $h m$ leads to \begin{equation} \label{3.2} \begin{split} \partial^{\gamma}u(h m)= &\sum_{|\alpha|=0}^{N-1-|\gamma|} \frac{(h m-x)^{\alpha}}{\alpha!} \, \partial^{\alpha+\gamma} u(x) \\ &+\sum_{|\alpha|=N-|\gamma|} \frac{(h m-x)^{\alpha}}{\alpha!} \, U_{\alpha+\gamma}(x,h m),\quad 0\leq |\gamma|<N, \end{split} \end{equation} with \begin{equation}\label{ualfa} U_{\alpha}(x,y)=N \int \limits_{0}^{1}s^{N-1}\partial^{\alpha} u(s x+(1-s)y)\, ds\,, \; |\alpha|=N. \end{equation} We write $Mu$ in the form \begin{equation*} M u(x)= \sum_{|\beta|=0}^{N-1} (- h \sqrt{\mathcal D})^{|\beta|}\partial^{\beta}u(x) \sum_{\alpha\leq \beta} \frac{ a_{\beta-\alpha}}{\alpha!} \sigma_{\alpha}(\frac{x}{h},\mathcal D,\mathcal H)+\mathcal R_{h,N}(x) \end{equation*} with the periodic functions \begin{equation*} \sigma_{\alpha}(\xi,\mathcal D,\mathcal H)=\mathcal D^{-n/2} \sum_{m \in \mathbb{Z}^{n}} \left( \frac{\xi- m}{\sqrt{\mathcal D}}\right)^{\alpha} \mathcal H\left( \frac{\xi- m}{\sqrt{\mathcal D}}\right),\quad 0\leq |\alpha|<N \, , \end{equation*} and the remainder term \begin{equation*} \begin{split} &\mathcal R_{h,N}(x)= \\ &\; (-h \sqrt{\mathcal D})^{N} \mathcal D^{-n/2}\sum_{|\beta|=N} \sum_{0<\alpha\leq\beta} \frac{a_{\beta-\alpha}}{\alpha!} \!\! \sum_{m\in \mathbb{Z}^{n}} U_{\beta}(x,h m) \left( \frac{x-h m}{h \sqrt{\mathcal D}}\right)^{\alpha}\!\!\! \mathcal H\left( \frac{x-h m}{h \sqrt{\mathcal D}}\right). \end{split} \end{equation*} Therefore by using the definition of $\sigma_{\alpha}$, \begin{equation}\label{3.6} \begin{split} M u(x)-u(x)=&\,u(x) \left( \sigma_{0}\big(\frac{x}{h},\mathcal D,\mathcal H\big)-1\right) \\ &+\sum_{|\beta|=1}^{N-1} (-h \sqrt{\mathcal D})^{|\beta|} \partial^{\beta}u(x) \sum_{\alpha\leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \sigma_{\alpha}\big(\frac{x}{h},\mathcal D,\mathcal H\big)+\mathcal R_{h,N}(x). \end{split} \end{equation} Let us introduce the functions \[ \mathcal E_{\alpha}(\xi,\mathcal D,\mathcal H)=\sigma_{\alpha}(\xi,\mathcal D,\mathcal H)- \int \limits_{\mathbb{R}^{n}} x^{\alpha} \mathcal H(x)\, dx \, ,\qquad 0\leq|\alpha|\leq N-1 . \] Then we have \begin{equation}\label{3.8} \sigma_{0}(\xi,\mathcal D,\mathcal H)-1=\mathcal E_{0}(\xi,\mathcal D,\mathcal H)+ \Big(\int \limits_{\mathbb{R}^{n}} \mathcal H(x) dx-1\Big) \end{equation} and for $|\beta|=1,\ldots,N-1$ \begin{equation}\label{3.9} \sum_{\alpha\leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \sigma_{\alpha}(\xi,\mathcal D,\mathcal H) = \sum_{\alpha\leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \mathcal E_{\alpha}(\xi,\mathcal D,\mathcal H) + \sum_{\alpha \leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \int \limits_{\mathbb{R}^{n}}x^{\alpha} \mathcal H(x) dx. \end{equation} \begin{theorem}\label{UNO} Suppose that $\mathcal H$ is differentiable up to the order of the smallest integer $n_{0}>n/2$, satisfies the decay condition: there exist $K>N+n$ and $C_{\beta}>0$ such that \begin{equation}\label{decaybis} |\partial ^{\beta} \mathcal H(x)|\leq C_{\beta}\, (1+|x|)^{-K},\quad 0\leq|\beta|\leq n_{0} \end{equation} and the conditions \begin{equation} \label{3.10} \begin{cases} \displaystyle{\qquad \int \limits_{\mathbb{R}^{n}} \mathcal H(x) dx }& \hskip-5pt = 1 \\[5mm] \displaystyle{\; \sum_{\alpha \leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \int \limits_{\mathbb{R}^{n}}x^{\alpha} \mathcal H(x) dx} & \hskip-5pt = 0 \,,\quad |\beta|=1,\ldots,N-1 . \end{cases} \end{equation} Then for any $\varepsilon>0$ there exists $\mathcal D>0$ such that, for all $u \in W^{N}_{\infty}(\mathbb{R}^{n})\cap C^{N}(\mathbb{R}^{n})$ the approximation error of the quasi-interpolant \eqref{3.1} can be pointwise estimated by \begin{equation}\label{3.14} |M u(x) -u(x)|\leq \varepsilon \sum_{|\beta|=0}^{N-1} (h \sqrt{\mathcal D})^{|\beta|}|\partial^{\beta} u(x)|+ c (h \sqrt{\mathcal D})^{N} \Vert \nabla_{N} u \Vert_{L_{\infty}} \end{equation} with the constant $c$ not depending on $h$, $u$ and $\mathcal D$. \end{theorem} {\bf Proof.} Under conditions (\ref{3.10}) the relations \eqref{3.8} and \eqref{3.9} simplify to \begin{equation*} \begin{array}{rl} \sigma_{0}(\xi,\mathcal D,\mathcal H)-1 & = \mathcal E_{0}(\xi,\mathcal D,\mathcal H) , \\ \\ \displaystyle{ \sum_{\alpha\leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \sigma_{\alpha}(\xi,\mathcal D,\mathcal H)} & = \displaystyle{\sum_{\alpha\leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \mathcal E_{\alpha}(\xi,\mathcal D,\mathcal H)}. \end{array} \end{equation*} Moreover, the assumptions (\ref{decaybis}) on $\mathcal H$ ensure that (see \cite{MS4}) \[ \{\partial^{\alpha}\mathcal F \mathcal H(\sqrt{\mathcal D} \nu)\}\in l_{1}(\mathbb{Z}^{n}),\qquad 0\leq |\alpha|<K-n, \] \begin{equation*} | \mathcal E_{\alpha}(\xi,\mathcal D,\mathcal H)|=\Big|\sigma_{\alpha}(\xi,\mathcal D,\mathcal H)- \int_{\mathbb{R}^{n}} x^{\alpha} \mathcal H(x)\, dx\Big|\leq (2\pi)^{-|\alpha|} \hskip-3pt \sum_{\nu\in\mathbb{Z}^{n}\setminus 0}\big|\partial^{\alpha}\mathcal F \mathcal H(\sqrt{\mathcal D} \nu)\big| , \end{equation*} and \begin{equation*} \sum_{\nu\in\mathbb{Z}^{n}\setminus 0}|\partial^{\alpha}\mathcal F \mathcal H(\sqrt{\mathcal D} \nu)| \longrightarrow 0 \;\mbox{ as }\; \mathcal D \to \infty . \end{equation*} Hence for any $\varepsilon>0$ there exists $\mathcal D$ such that the following estimates are valid \begin{equation} |\sigma_{0}(\xi,\mathcal D,\mathcal H)-1|=|\mathcal E_{0}(\xi,\mathcal D,\mathcal H)|< \varepsilon, \label{3.11} \end{equation} \begin{equation} \Big| \sum_{\alpha\leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \sigma_{\alpha}(\xi,\mathcal D,\mathcal H)\Big| \leq \sum_{\alpha\leq \beta} \frac{|a_{\beta-\alpha}|}{\alpha!} |\mathcal E_{\alpha}(\xi,\mathcal D,\mathcal H)| <\varepsilon,\quad |\beta|=1,\ldots,N-1. \label{3.12} \end{equation} The next step is to estimate the remainder term $\mathcal R_{h,N}$. Since \begin{equation*} |U_{\alpha}(x,y)|=N\left|\int \limits_{0}^{1}s^{N-1}\partial^{\alpha}u(s x+(1-s)y)ds\right|\leq ||\partial^{\alpha}u||_{L_{\infty}} \end{equation*} we obtain \begin{equation*} |\mathcal R_{h,N}(x)| \leq (h \sqrt{\mathcal D})^{N} \sum_{|\beta|=N} \big\Vert \partial^{\beta} u\big\Vert_{L^{\infty}} \sum_{0< \alpha\leq \beta} \frac{|a_{\beta-\alpha}|}{\alpha!} \Vert \rho_{\alpha}(\cdot,\mathcal D,\mathcal H) \Vert_{L_{\infty}} \end{equation*} where \begin{equation*} \rho_{\alpha}(\xi,\mathcal D,\mathcal H)= \mathcal D^{-n/2}\sum_{m\in\mathbb{Z}^{n}}\Big| \Big(\frac{\xi-m}{\sqrt{\mathcal D}}\Big)^{\alpha} \mathcal H \Big(\frac{\xi-m}{\sqrt{\mathcal D}}\Big)\Big|. \end{equation*} In view of the decay condition there exist constants $c_{\alpha}$ such that, for $\mathcal D$ sufficiently large (\cite{MS4}) \[\Vert \rho_{\alpha}(\cdot,\mathcal D,\mathcal H) \Vert_{L_{\infty}}\leq c_{\alpha}, \quad 0\leq |\alpha|\leq N.\] Hence we obtain \begin{equation}\label{3.13} |\mathcal R_{h,N}(x)| \leq c\, (h \sqrt{\mathcal D})^{N} \sum_{|\beta|=N} \big\Vert \partial^{\beta} u\big\Vert_{L^{\infty}} \end{equation} with a constant $c$ not depending on $h$, $\mathcal D$ and $u$. By \eqref{3.6}, this leads together with (\ref{3.11}) and (\ref{3.12}) to the estimate \eqref{3.14}. \section{Construction of generating functions for arbitrary $N$}\label{S:QUA} Here we describe a simple method for the construction of generating functions $\mathcal H$ satisfying the conditions (\ref{3.10}) for arbitrary given $a_{\gamma}$ with $a_{0}=1$. Let us denote by $\mathcal A=\{\mathcal A_{\alpha\beta}\}$ the triangular matrix with the elements \[ \mathcal A _{\alpha\beta}= \begin{cases} a_{\beta-\alpha}\quad\> &\alpha\leq \beta,\cr 0 \quad &{\rm otherwise} \end{cases} \quad |\beta|,|\alpha|=0,\ldots,N-1. \] The dimension of $\mathcal A$ is $(N+n-1)!/((N-1)! \,n!)$. Since $\det\mathcal A=1$ there exists the inverse matrix $\mathcal A^{-1}=\{\mathcal A_{\alpha\beta}^{(-1)}\}$ and (\ref{3.10}) leads to the following conditions \begin{equation}\label{3.10bis} \int_{\mathbb{R}^{n}} x^{\alpha} \mathcal H(x)\, dx= \alpha ! \, \mathcal A_{0 \alpha}^{(-1)},\quad |\alpha|=0,\ldots,N-1. \end{equation} These conditions can be rewritten as \[ \partial^{\alpha} ( \mathcal F \mathcal H)(0)= \alpha !(-2\pi i)^{|\alpha|} \mathcal A^{(-1)}_{0\alpha} ,\quad |\alpha|=0,\ldots,N-1. \] Let us assume (see \cite{MS3}) \[ \mathcal H(x)=\mathcal P_{N}\Big(\frac{1}{2 \pi i} \frac{ \partial}{\partial x}\Big)\, \eta(x), \] where $\mathcal P_{N}(t)$ is a polynomial of degree less than or equal to $N-1$ and $\eta$ is a smooth function rapidly decaying as $|x|\to \infty$ with $\mathcal F\eta(0) \neq 0$. Conditions (\ref{3.10bis}) give \[ \partial^{\alpha}\big(\mathcal P_{N}(\lambda) \mathcal F \eta(\lambda)\big)(0)=(-2 \pi i)^{|\alpha|}\alpha!\,\mathcal A^{(-1)}_{0\alpha},\, \quad |\alpha|=0,\ldots,N-1. \] We choose $\mathcal P_{N}$ as the Taylor polynomial of order $(N-1)$ of the function \[ Q(\lambda)=\sum_{|\alpha|=0}^{N-1} \frac{\mathcal A_{0\alpha}^{(-1)}(-2\pi i\lambda)^{\alpha}}{\mathcal F\eta(\lambda)},\quad \lambda \in \mathbb{R}^{n}. \] Since \[ \partial^{\beta}Q(0)=\sum_{\alpha\leq \beta}\mathcal A_{0 \alpha}^{(-1)}(-2\pi i)^{|\alpha|} \frac{\beta!}{(\beta-\alpha)!} \partial^{\beta-\alpha} (\mathcal F\eta)^{-1}(0), \] where we use the notation \[ \partial^{\beta-\alpha}(\mathcal F\eta)^{-1}(0)=\partial^{\beta-\alpha}\Big(\frac{1}{\mathcal F\eta(\lambda)}\Big)(0), \] we obtain \[ \mathcal P_{N}(\lambda)=\sum_{|\beta|=0}^{N-1}\frac{\lambda^{\beta}}{\beta!} \sum_{\alpha\leq \beta}\mathcal A_{0 \alpha}^{(-1)}(-2\pi i)^{|\alpha|} \frac{\beta!}{(\beta-\alpha)!} \partial^{\beta-\alpha} (\mathcal F\eta)^{-1}(0) \, . \] Therefore the equations \begin{align*} \partial^{\alpha} \big(\mathcal P_{N}(\lambda) \mathcal F \eta(\lambda)\big)(0) &= \, \partial^{\alpha}(Q(\lambda)\mathcal F\eta(\lambda))(0)\\ &= \, \partial^{\alpha} \Big( \sum_{|\gamma|=0}^{N-1} {\mathcal A_{0\gamma}^{(-1)}(-2\pi i\lambda)^{\gamma}} \Big)(0)=(-2 \pi i)^{|\alpha|}\alpha! \, \mathcal A^{(-1)}_{0\alpha} \end{align*} are valid for all $\alpha:0\leq|\alpha|\leq N-1$. We have thus proved the following \begin{theorem}\label{4.1T} Suppose that $\eta\in C^{N-1}(\mathbb{R}^{n})$ satisfies \begin{align*} &|\eta(x)|\leq A\, (1+|x|)^{-K},\quad x \in \mathbb{R}^{n}, \quad K>N+n, \\ &\int \limits_{\mathbb{R}^{n}} |x|^{N-1}|\partial^{\alpha}\eta(x)|dx<\infty,\quad 0 \leq |\alpha|\leq N-1 , \end{align*} and $\mathcal F\eta(0)\neq 0$. Then the function $$\mathcal H(x)=\sum_{|\beta|=0}^{N-1}\sum_{\alpha \leq \beta} \mathcal A_{0 \alpha}^{(-1)}{ (-1)^{|\alpha|}} \frac{\partial^{\beta-\alpha} (\mathcal F\eta)^{-1}(0) }{ (\beta-\alpha)!(2 \pi i)^{|\beta-\alpha|}} \partial^{\beta}\eta(x) $$ satisfies the conditions \eqref{3.10}. \end{theorem} Suppose that $\eta(x)$ is radial, that is $\eta(x)=\psi(r), r=|x|$. Then $\partial ^{\alpha}\mathcal F \eta(0)=0$ for any $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ containing at least one odd $\alpha_{i}$ and we obtain the formula \[ \mathcal H(x)=\sum_{|\beta|=0}^{N-1}(-\partial)^{\beta}\eta(x)\,\sum_{2\gamma \leq \beta} \mathcal A_{0, \beta-2\gamma}^{(-1)} \frac{\partial^{2\gamma} (\mathcal F\eta)^{-1}(0) }{ (2\gamma)!(-4 \pi^{2})^{|\gamma|}} \, . \] Let us consider the special case $\eta(x)=\pi^{-n/2} {{\rm e}}^{-|x|^{2}}$ with $\mathcal F\eta(\lambda)={{\rm e}}^{-\pi^{2}|\lambda|^{2}}$. Denoting by $H_{\beta}$ the Hermite polynomial of $n$ variables defined by \[ H_{\beta}(t)={{\rm e}}^{|t|^{2}}(-{\partial} )^{\beta}{{\rm e}}^{-|t|^{2}} \] we derive $\partial^{\gamma}(\mathcal F \eta)^{-1}(0)=(-\pi i)^{|\gamma|}H_{\gamma}(0)$. Note that $H_{\gamma}(0)=0$ if $\gamma$ has odd components, otherwise $H_{2 \gamma}(0)=(-1)^{\gamma}{(2\gamma)!}/{\gamma!}$. Hence \begin{equation}\label{4.1} \mathcal H(x)=\pi^{-n/2}\sum_{|\beta|=0}^{N-1}H_{\beta}(x){{\rm e}}^{-|x|^{2}} \sum_{2\gamma\leq \beta}\frac{(-1)^{|\gamma|}}{\gamma! 4^{|\gamma|}} {\mathcal A_{0\, ,\beta-2\gamma}^{(-1)}}. \end{equation} Assuming $a_{\gamma}=\delta_{|\gamma| 0}$ and $N=2M$ we find out \begin{align*} \mathcal H(x)= & \, \pi^{-n/2}\sum_{j=0}^{M-1}\frac{(-1)^{j}}{4^{j}} \sum_{|\beta|=j}\frac{H_{2\beta}(x)}{\beta!} \, {\rm e}^{-|x|^{2}}\\ =& \, \pi^{-n/2}\sum_{j=0}^{M-1}\frac{(-1)^{j}}{4^{j}j!}\Delta^{j} {{\rm e}}^{-|x|^{2}}=\pi^{-n/2}L_{M-1}^{(n/2)}(|x|^{2}){{{\rm e}}^{-|x|^{2}}} \,, \end{align*} where $L_{k}^{(\gamma)}$ are the generalized Laguerre polynomials, which are defined by \begin{equation*} L_{k}^{(\gamma)}(y)=\frac{{\rm e}^{\,y} y^{-\gamma}}{k!} \, \Big( {\frac{d}{dy}}\Big)^{k} \! \left({\rm e}^{\,-y} y^{k+\gamma}\right), \quad \gamma > -{\rm 1} \, . \end{equation*} In this case we obtain the classical generating function $\eta(x)=L^{(n/2)}_{M-1}(|x|^{2}){{\rm e}}^{-|x|^{2}}$ (see \cite{MS3,Sc}). \section{Examples}\label{example} \begin{example} If $N=2$, then formula \eqref{4.1} gives \begin{equation} \mathcal H(x)=\pi^{-n/2}\Big(1-2 \sum_{|\beta|=1} a_{\beta}x^{\beta}\Big){{\rm e}}^{-|x|^{2}}. \end{equation} For the one-dimensional case the corresponding quasi-interpolant is \[ M_{a} u(x)=(\pi \mathcal D)^{-1/2}\sum_{m\in \mathbb{Z}} (u(hm)-h \sqrt{\mathcal D}\, a \,u'(h m))(1-2 a \,\frac{x-h m}{h \sqrt{\mathcal D}}\,){{\rm e}}^{-(x-hm)^{2}/(h^{2}\mathcal D)}\, . \] For $u(x)=\cos x$, $x \in \mathbb{R}$, the difference between $u(x)$ and $M_{a} u(x)$ is plotted in \ref{fig8} by taking $h=0.1$, $\mathcal D=2$ for different values of $a$. \begin{figure}\begin{center} \epsfig{file=Msec.eps,width=3.6in,height=1.8in} \caption{The graphs of $(M_{a} -I)\cos x $ when $a=0$ (solid line), $a=1/8$ (dotted line) and $a=1/4$ (dashed line).} \label{fig8} \end{center} \end{figure} \end{example} \begin{example} Now we are looking for quasi-interpolants of order $\mathcal O(h^{4})$. In the one-dimensional case \[ \mathcal A^{(-1)}_{01}=-a_{1},\> \mathcal A_{02}^{(-1)}=a_{1}^{2}-a_{2},\> \mathcal A^{(-1)}_{03}=-a_{1}^{3}+2 a_{1}a_{2}-a_{3} \] and formula (\ref{4.1}) gives \[\begin{split} \mathcal H(x)=& \pi^{-1/2}{\rm e}^{-x^{2}} [(3/2-\mathcal A_{02}^{(-1)})+(5\mathcal A_{01}^{-1}-12\mathcal A_{03}^{(-1)})x \\ &+ (4\mathcal A_{02}^{(-1)}-1)x^{2}+(8\mathcal A_{03}^{(-1)}-2\mathcal A_{01}^{(-1)})x^{3}]. \end{split}\] If $a_{1}=a_{2}=a_{3}=0$, then we get the classical generating function $\eta(x)=\pi^{-1/2}{\rm e}^{-x^{2}}(3/2-x^{2})$. Assuming $a_{3}=a_{2}=0,\, a_{1}=\pm 1/2$ we obtain \[ \mathcal H(x)=\pi^{-1/2}(1\pm x){{\rm e}}^{-x^{2}} \] and the quasi-interpolant of order $\mathcal O(h^{4})$: \[ M_{1}u(x)=(\pi \mathcal D)^{-1/2}\sum_{m\in \mathbb{Z}}(u(h m)\pm \frac{h \sqrt{\mathcal D}}{2} u'(hm))(1\pm \frac{x-h m}{h \sqrt{\mathcal D}})\, {{\rm e}}^{-(x-h m)^2/(h^2 \mathcal D)} \,. \] With the choice $a_{1}=a_{3}=0,\, a_{2}=-1/4$ we obtain $\mathcal H(x)=\pi^{-1/2}{{\rm e}}^{-x^{2}}$ and the quasi-interpolant of order $\mathcal O(h^{4})$: \[ M_{2}u(x)=(\pi \mathcal D)^{-1/2}\sum_{m\in \mathbb{Z}}(u(h m)- \frac{h^{2} {\mathcal D}}{4} u''(hm))\, {{\rm e}}^{-(x-h m)^2/(h^2 \mathcal D)} \, . \] In the Figures \ref{fig5}, \ref{fig6}, \ref{fig7} we show the error graphs for the approximation of $u=\cos(x)$ with the quasi-interpolants $Mu, M_{1}u, M_{2}u$, respectively. We have taken $\mathcal D=2$, $h=0.05$, $h=0.1$ and $h=0.2$. The case of smaller $h$ gives different pictures: it is clearly visible that the error oscillates very fast with $M u$ and $M_{1}u$. \begin{figure}\begin{center} \subfigure[]{ \epsfig{file=M0uno.eps,width=1.6in,height=1.8in}} \subfigure[]{ \epsfig{file=M0due.eps,width=1.6in,height=1.8in}} \subfigure[]{ \epsfig{file=M0unobis.eps,width=1.6in,height=1.8in}} \caption{The graphs of $(M-I)\cos(x)$ by assuming $h=0.2$ (a), $h=0.1$ (b) and $h=0.05$ (c).} \label{fig5} \end{center} \end{figure} \begin{figure}\begin{center} \subfigure[]{ \epsfig{file=M1uno.eps,width=1.6in,height=1.8in}} \subfigure[]{ \epsfig{file=M1due.eps,width=1.6in,height=1.8in}} \subfigure[]{ \epsfig{file=M1duebis.eps,width=1.6in,height=1.8in}} \caption{The graphs of $(M_{1}-I)\cos(x)$ by assuming $h=0.2$ (a), $h=0.1$ (b) and $h=0.05$ (c).} \label{fig6} \end{center} \end{figure} \begin{figure}\begin{center} \subfigure[]{ \epsfig{file=M2uno.eps,width=1.6in,height=1.8in}} \subfigure[]{ \epsfig{file=M2due.eps,width=1.6in,height=1.8in}} \subfigure[]{ \epsfig{file=M2duebis.eps,width=1.6in,height=1.8in}} \caption{The graphs of $(M_{2}-I)\cos(x)$ by assuming $h=0.2$ (a), $h=0.1$ (b) and $h=0.05$ (c).} \label{fig7} \end{center} \end{figure} If $n>1$, by taking $a_{2\beta}=a$ and $a_{\beta}=0$ otherwise, we obtain the radial generating function \begin{align*} \mathcal H_{a}(x)=& (1-\sum_{|\beta|=1} (a_{2\beta}+1/4)(4 x^{2\beta}-2)) {{\rm e}}^{-|x|^{2}}\pi^{-n/2} \\ =&(1+n(2 a+1/2)-(1+4a)|x|^{2} ){{\rm e}}^{-|x|^{2}}{\pi^{-n/2}} \end{align*} for the approximate approximation of order $\mathcal O(h^{4})$ \[ M_{a}u(x)=(\pi \mathcal D)^{-n/2}\sum_{m\in \mathbb{Z}^{n}}(u(h m)+h^{2}\, \mathcal D\, a \,\Delta u(hm))\mathcal H_{a}\big( \frac{x-hm}{h\sqrt{\mathcal D}}\big). \] \end{example} \begin{example}\label{4.3} Let $n>1$ and set in (\ref{3.1}) $N=2M$ and \begin{align} \label{coefLap} \begin{cases} \; a_{\gamma}\,=0 \, ,\quad {\rm if}\; \gamma \; \mbox{has odd components,}\\[3mm] \; \displaystyle{a_{2\gamma}=\frac{(-1)^{|\gamma|}}{\gamma! 4^{|\gamma|}}},\quad 0\leq |\gamma|\leq M-1. \end{cases} \end{align} Keeping in mind that \begin{equation}\label{lapl} \sum_{|\gamma|=s}\frac{s!}{\gamma!} \, \partial^{2\gamma}u(x)=\Delta^{s}u(x) \end{equation} we obtain \[ \mathcal Q\left((-h \sqrt{\mathcal D})\partial \right)u = \sum_{j=0}^{M-1}\frac{(-1)^{j}(h \sqrt{\mathcal D})^{2j}}{4^{j}}\sum_{|\gamma|=j}\frac{\partial^{2 \gamma}}{\gamma!}u=\sum_{j=0}^{M-1}\frac{(-1)^{j}(h^2 \mathcal D)^j}{j!4^{j}}\Delta^{j}u. \] The function $\mathcal H(x)=\pi^{-n/2}{{\rm e}}^{-|x|^{2}}$ satisfies conditions (\ref{3.10}). In fact \[ \frac{1}{\pi^{n/2}}\int \limits_{\mathbb{R}^{n}}x^{\alpha} {\rm e}^{-|x|^2} dx=\frac{(-\pi i)^{\alpha}}{(2 \pi)^{\alpha}}H_{\alpha}(0)= \begin{cases} \; 0, & \mbox{if } \alpha \mbox{ has odd components,} \\[2mm] \displaystyle \frac{(2 \gamma)!}{\gamma!2^{2|\gamma|}} , & \mbox{if } \alpha=2\gamma \, . \end{cases} \] Then the equations (\ref{3.10}) are valid for all $\beta: 0 \leq |\beta|\leq M-1$ because of \[ \sum_{\alpha\leq \beta}\frac{a_{2(\beta-\alpha)}}{\alpha!2^{2|\alpha|}}= \sum_{\alpha\leq \beta} \frac{(-1)^{|\beta-\alpha|}}{(\beta-\alpha)!4^{|\beta-\alpha|}} \frac{1}{\alpha!4^{|\alpha|}}=\delta_{|\beta| 0},\quad |\beta|=0,\ldots,M-1. \] Therefore, a general approximation of order $N=2M$ is given by \begin{equation}\label{hope} M^{(N)}u(x)=(\pi \mathcal D)^{-n/2} \sum_{m\in\mathbb{Z}^{n}}\sum_{s=0}^{M-1} (h \sqrt{\mathcal D})^{2s} \frac{(-1)^{s}}{s!4^{s}}\Delta^{s}u(hm)\, {{\rm e}}^{-|x-h m|^2/(h^2 \mathcal D)} \, . \end{equation} If $M=2$ then we find the ``fourth order formula'' \begin{equation*} M^{(4)}u(x)=(\pi \mathcal D)^{-n/2}\sum_{m\in \mathbb{Z}^{n}} \big[u(h m)- \frac{h^2 \mathcal D}{4}\Delta u(h m) \big] {{\rm e}}^{-|x-h m|^2/(h^2 \mathcal D)} \, . \end{equation*} For $M=3$ we obtain the ``sixth order formula'' \[ M^{(6)}u(x)=(\pi \mathcal D)^{-n/2}\!\!\sum_{m\in \mathbb{Z}^{n}}\!\!\! \big[u(h m)- \frac{h^2\mathcal D}{4}\Delta u(h m) +\frac{h^4 \mathcal D^{2}}{32}\Delta^{2}u(hm) \big] {{\rm e}}^{-|x-h m|^2/(h^2 \mathcal D)} . \] Note the additive structure of the formula (\ref{hope}) \[ M^{(N+2)}u(x)=M^{(N)}u(x)+ \frac{(h \sqrt{\mathcal D})^{N}(-1)^{M}}{(\pi \mathcal D)^{n/2}M! 4^{M}}\!\!\!\!\sum_{m\in \mathbb{Z}^{n}} \Delta^{M} u(h m) \, {{\rm e}}^{-|x-h m|^2/(h^2 \mathcal D)}. \] \end{example} \begin{example}\label{generalcase} We consider the second order partial differential operator \[ \mathcal B u=\sum_{i,k=1}^{n} b_{ik} \partial_{i}\partial_{k} u \, , \] where the matrix $B=\{b_{ik}\}\in\mathbb{R}^{n\times n} $ is symmetric and positive definite. Define the quasi-interpolant \begin{equation}\label{gen} M u(x)=\frac{(\det B)^{-1/2}}{(\pi \mathcal D)^{n/2}} \sum_{m\in\mathbb{Z}^{n}}\sum_{s=0}^{M-1} \frac{(-h^2 \mathcal D )^{s}}{s!4^{s}} \, \mathcal B ^{s}u(hm)\,{\rm e}^{\, -\langle B^{-1}(x-hm),x-hm\rangle/(h^2\mathcal D)} . \end{equation} Assume $C$ such that $B^{-1}=C^{T}C$. If we consider the linear transformation $\xi=C\,x$ and introduce $U(\xi)=u(x)$, in the new coordinates we have (\cite[p.42]{Pe}) \[ \mathcal B u(x)=\Delta U(\xi). \] Then \eqref{gen} will take the form \begin{align*} M u(x)={\mathcal M} U(\xi)= \frac{\det C}{(\pi \mathcal D)^{n/2}} \sum_{m\in\mathbb{Z}^{n}}\sum_{s=0}^{M-1} \frac{(-h^2 \mathcal D)^{s}}{s!4^{s}}\Delta^{s} U(h\,C\,m)\, {\rm e}^{\, -|\xi-h\,C\,m|^2/(h^2\mathcal D)}. \end{align*} Keeping in mind \eqref{lapl} and \eqref{3.2} we rewrite \[ {\mathcal M}U(\xi)=\sum_{|\beta|=0}^{2M-1}(-h \sqrt{\mathcal D})^{|\beta|}\partial^{\beta}U(\xi) \, \mathcal E_{\beta}\big(\frac{\xi}{h},\mathcal D\big)+R_{h}(\xi) \, \] where \begin{align*} \mathcal E_{\beta}(\xi,\mathcal D) &=\sum_{2\gamma\leq\beta}\frac{(-1)^{|\gamma|}}{\gamma!(\beta-2\gamma)!4^{|\gamma|}} \Sigma_{\beta-2\gamma}({\xi},\mathcal D) \intertext{with} \Sigma_{\alpha}(\xi,\mathcal D) &=\frac{\det C}{(\pi \mathcal D)^{n/2}}\sum_{m\in\mathbb{Z}^{n}}\left(\frac{\xi-C\,m}{\sqrt{\mathcal D}}\right)^{\alpha} \!\!{{\rm e}}^{-|\frac{\xi-C\,m}{ \sqrt{\mathcal D}}|^{2}} \, , \end{align*} and \begin{align*} &R_{h}(\xi)= \, \frac{(h\sqrt{\mathcal D})^{2M}\det C}{(\pi \mathcal D)^{n/2}} \\ &\times \sum_{|\beta|=2M} \sum_{2\gamma\leq\beta}\frac{(-1)^{|\gamma|}}{4^{|\gamma|}\gamma!(\beta-2\gamma)!} \sum_{m\in\mathbb{Z}^{n}}U_{\beta}(\xi,hCm) \left(\frac{\xi-hCm}{h\sqrt{\mathcal D}}\right)^{\beta-2\gamma} {{\rm e}}^{-|\frac{\xi-hCm}{ h\sqrt{\mathcal D}}|^{2}}. \end{align*} Now we use Poisson's summation formula on affine grids (see \cite[p.23]{MSbook}) \begin{equation}\label{poisson} \begin{split} \frac{\det C}{ \mathcal D^{n/2}}\sum_{m\in\mathbb{Z}^n}\hskip-3pt \Big(\frac{\xi-C\,m}{\sqrt{\mathcal D}}\Big)^{\delta} \eta\Big(\frac{\xi-C\,m}{\sqrt{\mathcal D}}\Big)&= \\ \Big(\frac{i}{2\pi}\Big)^{|\delta|}&\sum_{\nu\in\mathbb{Z}^{n}}\partial^{\delta}\mathcal F\eta (\sqrt{\mathcal D}C^{-T}\nu){\rm e}^{2\pi i \langle \xi,C^{-T}\nu\rangle }, \end{split} \end{equation} where we denote by $C^{-T}= (C^T)^{-1}$. In our case $\eta(x)=\pi^{-n/2} {\rm e}^{-|x|^2}$ and $\mathcal F\eta(\lambda)={\rm e}^{-\pi^{2}|\lambda|^{2}}$. Since $\partial^{\delta}\mathcal F\eta(\lambda)=(-\pi)^{\delta}H_{\delta}(\pi\,\lambda) {\rm e}^{-\pi^{2}|\lambda|^{2}} $ we obtain \[ \partial^{\delta}\mathcal F\eta(0)=\begin{cases} 0\, , & \mbox{if } \delta \mbox{ has odd components,} \\[3mm] \displaystyle \pi^{2\gamma}\frac{(-1)^{|\gamma|}(2\gamma)!}{\gamma!}\, , & \mbox{if } \delta=2\gamma \, . \end{cases} \] Formula \eqref{poisson} applied to $\Sigma_{\alpha}$ gives \[ \Sigma_{\alpha}(\xi,\mathcal D)= \begin{cases} \displaystyle\Big(\frac{i}{2\pi}\Big)^{|\alpha|}\sum_{\nu\neq 0}\partial^{\alpha}\mathcal F\eta (\sqrt{\mathcal D}C^{-T}\nu){\rm e}^{2\pi i \langle\xi,C^{-T}\nu\rangle } , \, \alpha \mbox{ has odd components,} \\[4mm] \displaystyle \frac{\alpha!}{2^{|\alpha|} \gamma!}+ \Big(\frac{i}{2\pi}\Big)^{|\alpha|}\displaystyle\sum_{\nu\neq 0}\partial^{\alpha}\mathcal F\eta (\sqrt{\mathcal D}C^{-T}\nu){\rm e}^{2\pi i \langle\xi,C^{-T}\nu\rangle }, \, \alpha=2\gamma . \end{cases} \] We deduce that \[ \begin{split} \mathcal E_{\beta}(\xi,\mathcal D)&=\delta_{|\beta|0} \\ &+\sum_{2\gamma\leq\beta} \frac{(-1)^{|\gamma|}}{\gamma!(\beta-2\gamma)!\, 4^{|\gamma|}} \Big(\frac{i}{2\pi}\Big)^{|\beta-2\gamma|}\sum_{\nu\neq 0}\partial^{\beta-2\gamma}\mathcal F\eta (\sqrt{\mathcal D}C^{-T}\nu){\rm e}^{2\pi i \langle\xi,C^{-T}\nu\rangle } \end{split} \] and \[ |\mathcal E_{\beta}(\xi,\mathcal D)-\delta_{|\beta|0}|\leq \sum_{2\gamma\leq\beta}\frac{\pi^{|2\gamma-\beta|}}{\gamma!(\beta-2\gamma)!\, 2^{|\beta|}} \sum_{\nu\neq 0} \big|\partial^{\beta-2\gamma}\mathcal F\eta (\sqrt{\mathcal D}C^{-T}\nu)\big|. \] By repeating the same arguments used in the proof of Theorem \ref{UNO} we derive that $|\mathcal E_{\beta}(\xi,\mathcal D)-\delta_{|\beta|0}|<\varepsilon_{1}$ for prescribed $\varepsilon_{1}>0$ and sufficiently large $\mathcal D$, and the remainder is bounded by \[ |R_{h}(\xi)|\leq c_{B} (h\sqrt{\mathcal D})^{2M}\sum_{|\beta|=2M}\|\partial^{\beta}U\|_{L^{\infty}} \] with the constant $c_{B}$ independent of $U,h,\mathcal D$. Hence \begin{align*} |M u(x) -u(x)|&=|{\mathcal M} U(\xi)-U(\xi)|\\ & \leq \varepsilon_{1}\sum_{|\beta|=0}^{2M-1}(h \sqrt{\mathcal D})^{|\beta|}|\partial^{\beta}U(\xi)| +c_{B} (h\sqrt{\mathcal D})^{2M}\sum_{|\beta|=2M}||\partial^{\beta}U||_{L^{\infty}}\\ & \leq \varepsilon\sum_{|\beta|=0}^{2M-1}(h \sqrt{\mathcal D})^{|\beta|}|\partial^{\beta}u(x)| +C_{B} (h\sqrt{\mathcal D})^{2M}\sum_{|\beta|=2M}||\partial^{\beta}u||_{L^{\infty}} \, , \end{align*} where $C_{B}$ depends only on the matrix $B$. Therefore the quasi-interpolant \eqref{gen} approximates $u$ with the order $\mathcal O((h\sqrt{\mathcal D})^{2M})$ up to the saturation error. \end{example} \section{An application of formula (\ref{hope})}\label{appl} Here we consider the approximation of harmonic functions. Suppose that $\Delta u=0$ in some domain $\Omega \subseteq \mathbb{R}^n$. Then for any $N=2M$ and $x \in \Omega$ the Hermite quasi-interpolant \eqref{hope} has the simple form \begin{align} \label{harmonic} M^{(N)}u(x)=M u(x)=(\pi \mathcal D)^{-n/2} \sum_{hm\in \Omega} u(hm)\, {{\rm e}}^{-|x-h m|^2/(h^2 \mathcal D)} \, , \end{align} i.e., it coincides with the well known quasi-interpolation formula of second order. However, Theorem \ref{UNO} indicates higher approximation rates. This will be studied here in more detail. First we consider the case $\Omega = \mathbb{R}^{n}$. Then \[ u(\xi)=\sum_{|\beta|=0}^{\infty}\frac{\partial^{\beta}u(x)}{\beta!}\,(\xi-x)^{\beta},\quad \xi\in\mathbb{R}^{n} \] and the series converges absolutely in $\mathbb{R}^{n}$. Moreover, $u$ has the analytic extension \begin{align} \label{anext} \tilde u (\zeta)=\sum_{|\beta|=0}^{\infty}\frac{\partial^{\beta}u(x)}{\beta!}\, (\zeta-x)^{\beta},\quad \zeta\in\C^{n} \, , \end{align} cf. e.g. \cite{Se, Fi}. Using formula \eqref{3.6} for the quasi-interpolant with the generating function \eqref{hope} we obtain \begin{align} \label{4.6} M u(x)-u(x) = \mathcal E_{h,2M}(x)+\mathcal R_{h,2M}(x) \, , \end{align} where \begin{align*} &\mathcal E_{h,2M}(x) = u(x) \Big((\pi \mathcal D)^{-n/2} \sum_{m\in \mathbb{Z}^n} {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)} -1\Big) \\ &\quad +(\pi \mathcal D)^{-n/2} \sum_{|\beta|=1}^{2M-1} (-h \sqrt{\mathcal D})^{|\beta|} \partial^{\beta}u(x) \sum_{m\in \mathbb{Z}^n} {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)} \sum_{\alpha\leq \beta} \frac{a_{\beta-\alpha}}{\alpha!} \Big( \frac{x- hm}{h\sqrt{\mathcal D}}\Big)^{\alpha} \end{align*} constitutes the saturation error and the remainder term has the form \begin{equation} \label{rem} \begin{split} &\mathcal R_{h,2M}(x)= (-h \sqrt{\mathcal D})^{N} (\pi \mathcal D)^{-n/2} \\ &\quad \times \sum_{|\beta|=2M} \sum_{m\in \mathbb{Z}^n} U_{\beta}(x,h m) {{\rm e}}^{-|x-hm|^2/(h^2 \mathcal D)} \sum_{0<\alpha\leq\beta} \frac{a_{\beta-\alpha}}{\alpha!} \Big( \frac{x-h m}{h \sqrt{\mathcal D}}\Big)^{\alpha}\,. \end{split} \end{equation} From \eqref{coefLap} we see that \begin{align*} \sum_{\alpha\leq\beta} \frac{a_{\beta-\alpha}}{\alpha!} \,x^{\alpha} = \sum_{2\gamma\leq\beta}\frac{a_{2 \gamma}}{(\beta-2\gamma)!} \,x^{\beta-2\gamma} = \sum_{2\gamma\leq\beta}\frac{(-1)^{|\gamma|}}{\gamma! \, (\beta-2\gamma)! \, 2^{2|\gamma|}} \, x^{\beta-2\gamma} \, , \end{align*} which by using the representation of Hermite polynomials \begin{align*} H_k(\tau)= \sum_{0\le 2j \le k} \frac{(-1)^j k!}{j! \, (k-2j)!}\, (2\tau)^{k-2j} \,, \end{align*} shows that \begin{align*} \sum_{\alpha\leq\beta} \frac{a_{\beta-\alpha}}{\alpha!} \, x^{\alpha} = \frac{1}{\beta! \, 2^{|\beta|}} H_\beta (x) \,. \end{align*} Hence we obtain \begin{align*} \mathcal E_{h,2M}(x) = \sum_{|\beta|=0}^{2M-1} \Big( \hskip-3pt- \frac{h \sqrt{\mathcal D}}{ 2}\Big)^{|\beta|} \frac{\partial^{\beta}u(x)}{\beta! }\, \sigma_\beta \Big(\frac{x}{h}, \mathcal D\Big) \end{align*} with the functions \begin{equation} \label{defsigma} \begin{split} \sigma_0 (x, \mathcal D) &=(\pi \mathcal D)^{-n/2} \sum_{m\in \mathbb{Z}^{n}} {{\rm e}}^{-|x-m|^2/\mathcal D} -1 \, ,\\ \sigma_\beta (x, \mathcal D) &= (\pi \mathcal D)^{-n/2} \sum_{m \in \mathbb{Z}^n} H_\beta \Big( \frac{x-m}{ \sqrt{\mathcal D}}\Big)\, {{\rm e}}^{-|x-m|^2/\mathcal D} \,,\; |\beta|=1,\ldots,2M-1 \, . \end{split} \end{equation} It follows from the definition of $H_\beta$ and from Poisson's summation formula that \[ \sigma_\beta (x, \mathcal D)= (-1)^{|\beta|} \mathcal D^{|\beta|/2}\partial^{\beta}\sigma_0 (x, \mathcal D)= (-2 \pi i )^{|\beta|} \mathcal D^{|\beta|/2} \!\!\!\sum_{m \in \mathbb{Z}^{n}\setminus\{0\}}\!\!\!\! m^\beta {\rm e}^{ \,- \pi^2 \mathcal D|m|^2} {\rm e}^{ \,2 \pi i \langle m,x\rangle}. \] Thus the saturation error can be expressed as \begin{align} \label{saturN} \begin{split} \mathcal E_{h,2M}(x) &= \sum_{|\beta|=0}^{2M-1} \frac{\partial^{\beta}u(x)}{\beta! }\Big(\frac{h \mathcal D}{2} \Big)^\beta \, \partial^{\beta}\,\sigma_0 \Big(\frac{x}{h}, \mathcal D\Big)\\ &= \sum_{m \in \mathbb{Z}^{n}\setminus\{0\}} \tilde u_{2M}(x+i \pi h \mathcal D m) {\rm e}^{ \,- \pi^2 \mathcal D|m|^2} {\rm e}^{ \,2 \pi i \langle m,x\rangle/h} \, , \end{split} \end{align} where \[ \tilde u_N(\zeta)=\sum_{|\beta|=0}^{N-1}\frac{\partial^{\beta}u(x)}{\beta!}\, (\zeta-x)^{\beta} \] is the Taylor polynomial of the analytic extension $\tilde u$. Note that \eqref{saturN} is valid for any $M$. \begin{theorem}\label{thmarm} Suppose that the harmonic in $\mathbb{R}^n$ function $u$ is such that the series \begin{align} \label{asssqu} \sum_{|\beta|=0}^{\infty}\frac{\partial^{\beta}u(x)}{\sqrt{\beta!}}\, y^{\beta} \end{align} converges absolutely for any $y \in \mathbb{R}^n$. If $\sqrt{\mathcal D}h <1$, then the quasi-interpolant \eqref{harmonic} approximates $u$ with \begin{align*} M u(x)-u(x) = \lim_{M\to \infty}\mathcal E_{h,2M}(x)= \sum_{m\in \mathbb{Z}^{n}\setminus\{0\}} \tilde u(x+ \pi i h \mathcal D m) {\rm e}^{ \,- \pi^2 \mathcal D|m|^2} {\rm e}^{ \, 2 \pi i \langle m,x\rangle/h} , \end{align*} where $\tilde u$ is the analytic extension of $u$ onto $\C^n$. \end{theorem} {\bf Proof.} We have to show that $|\mathcal R_{h,2M}(x)| \to 0$ as $M \to \infty$. To estimate \eqref{rem} we rewrite \begin{align*} \sum_{0\le 2j < k} \frac{(-1)^j }{j! \, (k-2j)! \, 2^{2j}}\, \tau^{k-2j} &= \frac{1}{2^k \, k! } \big( H_k(\tau) - H_k(0) \big)\\ &= \frac{1}{2^{k-1} \, (k-1)! } \int \limits_0^\tau H_{k-1}(t) \, dt \, , \end{align*} which implies for $|\beta|=2M$ \begin{align*} \sum_{0< \alpha\leq\beta} \frac{a_{\beta-\alpha}}{\alpha!} \Big( \frac{x}{\sqrt{\mathcal D}}\Big)^{\alpha} &= \frac{1}{2^{2M} }\prod_{\beta_j>0} \frac{1}{\beta_j!}\Big( H_{\beta_j} \Big( \frac{x_j}{\sqrt{\mathcal D}}\Big) -H_{\beta_j} (0)\Big)\\ &= \frac{1}{2^{2M} }\prod_{\beta_j>0} \frac{2}{(\beta_j-1)!}\int \limits_0^{x_j/\sqrt{\mathcal D}}H_{\beta_j-1}(t) \, dt \, . \end{align*} Consequently, the remainder $\mathcal R_{h,2M}$ takes the form \begin{align*} \Big(\frac{h \sqrt{\mathcal D}}{ 2}\Big)^{N}(\pi \mathcal D)^{-n/2} \sum_{|\beta|=2M} \sum_{m\in \mathbb{R}^n} {U_{\beta}(x,h m)}&\\ \times{{\rm e}}^{-|x-hm|^2/(h^2 \mathcal D)}& \prod_{\beta_j>0} \frac{2}{(\beta_j-1)!}\int \limits_0^{z_j}H_{\beta_j-1}(t) \, dt \, , \end{align*} where we use the notation $z_j=(x_j-h m_j)/(h\sqrt{\mathcal D})$. Then Cramer's inequality for Hermite polynomials \begin{align} \label{cramer} |H_k(x)| \le 2^{k/2} \sqrt{k!} \, {\rm e}^{\, x^2/2} \end{align} (see \cite{SZ, IJ}), leads to the estimate \begin{align} |\mathcal R_{h,2M}(x)|& \le \Big(\frac{h \sqrt{\mathcal D}}{ 2}\Big)^{N}(\pi \mathcal D)^{-n/2} \nonumber\\[-5pt] & \quad \times \sum_{|\beta|=2M} \sum_{m\in \mathbb{R}^n} |U_{\beta}(x,h m)|\, {{\rm e}}^{-|x-hm|^2/(h^2 \mathcal D)} \prod_{\beta_j>0} \frac{2^{(\beta_j+1)/2}}{ \sqrt{(\beta_j-1)!} } \int \limits_0^{|z_j|} {\rm e}^{\, t^2/2} \, dt \nonumber \\ &\le \Big(\frac{h \sqrt{\mathcal D}}{ 2}\Big)^{N}(\pi \mathcal D)^{-n/2} \sum_{|\beta|=2M} C_\beta \sum_{m\in \mathbb{R}^n}|U_{\beta}(x,h m)| \, S_\beta(x-h m) \, .\label{rm} \end{align} For the last inequality we use the notations \begin{align*} &S_\beta(x-h m)=\prod_{\beta_j=0}{{\rm e}}^{-(x_j-hm_j)^2/(\mathcal D h^2)} \prod_{\beta_j>0} \frac{|x_j -hm_j|}{h \sqrt {2\mathcal D}}{{\rm e}}^{\, -(x_j -hm_j)^2/(2h^2\mathcal D)} \, , \\ &C_\beta = \prod_{\beta_j>0} \frac{ 2^{\beta_j+1/2}}{ \sqrt{(\beta_j-1)!} } = 2^{2M}\prod_{\beta_j>0} \sqrt{\frac{2}{(\beta_j-1)!} } \end{align*} and the estimate \[ {{\rm e}}^{\, -z_j^2}\int \limits_0^{|z_j|} {\rm e}^{\, t^2/2} \, dt \le |z_j| {\rm e}^{\,-z_j^2/2} \, . \] By \eqref{ualfa} we obtain for harmonic $u$ \begin{align*} |U_{\beta}(x,h m)| &= \Big|\int \limits_{0}^{1}s^{N-1}\partial^{\beta} u(x+(1-s)(x-hm))\, ds \Big| \\ &\le \sum_{|\alpha|=0}^{\infty} \frac{\big|\partial^{\beta+\alpha}u(x)\big|}{\alpha!} \big|(x-hm)^{\alpha}\big| N \int \limits_{0}^{1}s^{N-1}(1-s)^{|\alpha|} ds \, , \end{align*} which shows that \begin{align*} \sum_{m\in \mathbb{R}^n}|U_{\beta}(x,h m)| \, S_\beta(x-h m) &\le \\ \sum_{|\alpha|=0}^{\infty}& \frac{|\alpha|! \, N! }{ (N+|\alpha|)!} \frac{\big|\partial^{\beta+\alpha}u(x)\big|}{ \alpha! } \sum_{m\in \mathbb{R}^n} S_\beta(x-h m) \big|(x-hm)^{\alpha}\big| . \end{align*} To get an upper bound of the last sum we write \begin{align*} &\sum_{m\in \mathbb{R}^n} S_\beta(x-m)\big|(x-m)^{\alpha}\big|\\ &=\prod_{\beta_j=0} \sum_{m_j\in \mathbb{R}} |x_j -m_j|^{\alpha_j}{{\rm e}}^{-(x_j -m_j)^2/\mathcal D} \prod_{\beta_j>0} \sum_{m_j\in \mathbb{R}}\frac{|x_j -m_j|^{\alpha_j+1}}{ \sqrt {2\mathcal D}} {{\rm e}}^{\, -(x_j -m_j)^2/(2\mathcal D)}\, , \end{align*} and note that for $\mathcal D > \mathcal D_0$ \begin{align*} & \sum_{m_j\in \mathbb{R}} |x_j -m_j|^{\alpha_j}{{\rm e}}^{-(x_j -m_j)^2/\mathcal D} \!\le c \mathcal D^{(\alpha_j+1)/2} \int \limits_0^\infty y^{\alpha_j} {\rm e}^{\,-y^2} dy = \frac{c\mathcal D^{(\alpha_j+1)/2}}{2}\Gamma\Big(\frac{\alpha_j+1}{2}\Big),\\ & \sum_{m_j\in \mathbb{R}} \frac{|x_j -m_j|^{\alpha_j+1}}{ \sqrt {2\mathcal D}} {{\rm e}}^{-(x_j -m_j)^2/(2\mathcal D)} \le \frac{c \, (2\mathcal D)^{(\alpha_j+1)/2}}{2} \Gamma\Big(\frac{\alpha_j+2}{2}\Big). \end{align*} Hence we obtain the estimate \begin{align*} \sum_{m\in \mathbb{R}^n} S_\beta(x-m)\big|(x-m)^{\alpha}\big| \le \frac{c \, \mathcal D^{(|\alpha|+n)/2}}{2^n} \prod_{\beta_j=0}\Gamma\Big(\frac{\alpha_j+1}{2}\Big) \prod_{\beta_j>0} 2 ^{(\alpha_j+1)/2} \, \Gamma\Big(\frac{\alpha_j+2}{2}\Big) \end{align*} with a constant $c$ independent of $\alpha$, $\beta$, and $\mathcal D$. Now we use that for $\alpha_j \ge 2$ \begin{align*} \Gamma\Big(\frac{\alpha_j+1}{2}\Big) \le \frac{\sqrt[4]{\pi}}{2^{\alpha_j/2}} \sqrt{\alpha_j !} \, , \end{align*} which leads to \begin{align*} \sum_{m\in \mathbb{R}^n} S_\beta(x-m)\big|(x-m)^{\alpha}\big| \le c_1 \, \mathcal D^{(|\alpha|+n)/2} \prod_{\beta_j=0}\frac{ \sqrt{\alpha_j !}}{2^{\alpha_j/2}} \prod_{\beta_j>0} \sqrt{ (\alpha_j+1) !} \end{align*} with another constant $c_1$. Thus we derive from \eqref{rm} \begin{align*} &|\mathcal R_{h,2M}(x)| \\ &\le C (h \sqrt{\mathcal D})^{N}\!\!\! \sum_{|\beta|=2M} \sum_{|\alpha|=0}^{\infty} \frac{|\alpha|! \, |\beta|! \,\mathcal D^{(|\alpha|+n)/2}}{ |\beta+\alpha|!} \frac{\big|\partial^{\beta+\alpha}u(x)\big|}{ \alpha!} \prod_{\beta_j=0}\frac{ \sqrt{\alpha_j !}}{2^{\alpha_j/2}} \prod_{\beta_j>0} \frac{ \sqrt{2 (\alpha_j+1) !}}{\sqrt{(\beta_j-1)!}}\\ &= C (h \sqrt{\mathcal D})^{N}\!\!\! \sum_{|\beta|=2M} \sum_{|\alpha|=0}^{\infty} \frac{|\alpha|! \, |\beta|! \,\mathcal D^{(|\alpha|+n)/2}}{ |\beta+\alpha|!} \frac{\big|\partial^{\beta+\alpha}u(x)\big|}{ \sqrt{\alpha ! \, \beta!}} \!\!\!\prod_{\beta_j=0} 2^{-\alpha_j/2} \prod_{\beta_j>0} \sqrt{2 (\alpha_j+1)\beta_j} \end{align*} with a constant $C$ not depending on $M$, $\mathcal D$, $h$, and $u$. Since \[ \frac{|\alpha|! \, |\beta|! }{ |\beta+\alpha|!} \sqrt{\frac{(\alpha+ \beta)!}{\alpha! \, \beta!}} \le \frac{|\alpha|! \, |\beta|! }{ |\beta+\alpha|!} {\frac{(\alpha+ \beta)!}{\alpha! \, \beta!}} \le 1 \, , \] the remainder can be estimated by \begin{align*} &|\mathcal R_{h,2M}(x)| \le C (h \sqrt{\mathcal D})^{N} \sum_{|\beta|=2M} \sum_{|\alpha|=0}^{\infty} \frac{\big|\partial^{\beta+\alpha}u(x)\big|}{\sqrt{(\alpha+\beta)!}}\mathcal D^{(|\alpha|+n)/2} \prod_{\beta_j>0} \sqrt{2 (\alpha_j+1)\beta_j} \, . \end{align*} Thus, $|\mathcal R_{h,2M}(x)| \to 0$ for any fixed $\mathcal D$ if \eqref{asssqu} holds. \begin{remark} {\em The assertion of Theorem { \ref{thmarm}} is a concrete realization of a general approximation result for analytic functions. Let $u$ be an entire function in $\C^n$ of order less than 2. Theorem 7.1 in \cite{MS2} states that the semi-discrete convolution \begin{equation*} u_h(x) = \sum_{m \in {\mathbb{Z}}^n} u_{m} \, {\rm e}^{- {|x-hm|^2}/(h^2\mathcal D) } \end{equation*} with coefficients \begin{equation} \label{equ5e2} u_{m} := \int \limits_{\mathbb{R}^n} {\rm e}^{ - \pi^2 {\mathcal D}|y|^2} \, u(h m + i \pi {\mathcal D} h y) \, d y \end{equation} differs from $u$ by \[ u_h(x)-u(x)= \sum_{m \in \mathbb{Z}^{n}\setminus\{0\}} \tilde u(x+i \pi h \mathcal D m) {\rm e}^{ \,- \pi^2 \mathcal D|m|^2} {\rm e}^{ \,2 \pi i \langle m,x\rangle/h} \, , \] (cf. also \cite[Lemma~2.1]{MS5}). It can be easily seen from \eqref{anext} and \eqref{equ5e2} that the coefficients $u_{m}= (\pi \mathcal D)^{-n/2} u(hm)$ if the restriction of $u$ to $\mathbb{R}^n$ is harmonic. } \end{remark} We have applied the simple quasi-interpolant \eqref{harmonic} to the harmonic function $u(x_{1},x_{2})={\rm e}^{x_{1}}\cos x_{2}$ in $\mathbb{R}^{2}$ by assuming $\mathcal D=2$ (see Figure~\ref{harm1}), $\mathcal D=3$ (Figure~\ref{harm2}), $\mathcal D=4$ (Figure~\ref{harm3}), $h=2^{-3}$ and $2^{-7}$. The experiments confirm that the quasi-interpolation error $Mu-u$ has reached its saturation bound also for large $h$ because it does not decrease if $h$ becomes smaller. \begin{figure}\begin{center} \includegraphics[width=2.4in,height=1.6in]{d2harm1_8.eps} \includegraphics[width=2.4in,height=1.6in]{d2harm1_128.eps} \caption{ The graphs of $Mu-u$ with $u(x_{1},x_{2})={\rm e}^{x_{1}}\cos x_{2}$, $\mathcal D=2$, $h=2^{-3}$ (on the left) and $h=2^{-7}$ (on the right).} \label{harm1} \end{center} \end{figure} \begin{figure}\begin{center} \includegraphics[width=2.4in,height=1.6in]{d3harm1_8.eps} \includegraphics[width=2.4in,height=1.6in]{d3harm1_128.eps} \caption{The graphs of $Mu-u$ with $u(x_{1},x_{2})={\rm e}^{x_{1}}\cos x_{2}$, $\mathcal D=3$, $h=2^{-3}$ (on the left) and $h=2^{-7}$ (on the right).} \label{harm2} \end{center} \end{figure} \begin{figure}\begin{center} \includegraphics[width=2.4in,height=1.6in]{d4harm1_8.eps} \includegraphics[width=2.4in,height=1.6in]{d4harm1_128.eps} \caption{The graphs of $Mu-u$ with $u(x_{1},x_{2})={\rm e}^{x_{1}}\cos x_{2}$, $\mathcal D=4$, $h=2^{-3}$ (on the left) and $h=2^{-7}$ (on the right).} \label{harm3} \end{center} \end{figure} Let now $u$ be harmonic in some convex domain $\Omega \subset \mathbb{R}^n$ and we consider the approximant \begin{align} \label{harmonicdom} M u(x)=(\pi \mathcal D)^{-n/2} \sum_{hm\in \Omega} u(hm)\, {{\rm e}}^{-|x-h m|^2/(h^2 \mathcal D)} \, . \end{align} \begin{theorem} \label{harmdom} Suppose that the function $u$ is harmonic in a convex domain $\Omega \subset \mathbb{R}^n$ and satisfies for a given $N=2M$ \[ c_u = \sum_{|\beta|=2M} \|\partial^\beta u \|_{L_\infty(\Omega)} \prod_{\beta_j>0} \sqrt{\frac{2}{(\beta_j-1)!}} < \infty \, . \] Then for any $\varepsilon>0$ and subdomain $\Omega' \subsetneq \Omega$ there exists $\mathcal D>0$ and $h>0$ such that the quasi-interpolant \eqref{harmonicdom} provides for all $x \in \Omega'$ the estimate \begin{equation}\label{5.15} |u(x)- M u(x) |\leq C (h \sqrt{\mathcal D})^{N} c_u +\varepsilon \sum_{|\beta|=0}^{N-1} (h \sqrt{\mathcal D})^{|\beta|}|\partial^{\beta} u(x)| \, , \end{equation} where the constant $C$ depends only on the space dimension. \end{theorem} {\bf Proof.} Analogously to the case $\Omega = \mathbb{R}^n$ we obtain \[ |u(x)- M u(x) |\leq |\mathcal R_{h,2M}(x)| +|\mathcal E_{h,2M}(x)| \] with \begin{align*} |\mathcal E_{h,2M}(x)| &\le \, |u(x)| \Big|(\pi \mathcal D)^{-n/2} \sum_{hm\in \Omega} {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)} -1\Big| \\ +(\pi \mathcal D)^{-n/2}& \sum_{|\beta|=1}^{2M-1} \Big(\frac{h \sqrt{\mathcal D}}{ 2}\Big)^{|\beta|} \frac{|\partial^{\beta}u(x)|}{\beta! } \Big|\sum_{hm \in \Omega} H_\beta \Big( \frac{x-hm}{h \sqrt{\mathcal D}}\Big)\, {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)}\Big| . \end{align*} and \begin{align*} |\mathcal R_{h,2M}&(x)| \le \Big(\frac{h \sqrt{\mathcal D}}{ 2}\Big)^{N}(\pi \mathcal D)^{-n/2} \sum_{|\beta|=2M} C_\beta \sum_{hm\in \Omega}|U_{\beta}(x,h m)| \, S_\beta(x-h m) \, , \end{align*} see \eqref{rm}. Since \[ |U_{\beta}(x,h m)| \le \|\partial^\beta u \|_{L_\infty(\Omega)} \quad \mbox{and} \quad \sum_{hm\in \Omega} S_\beta(x-h m) \le c \,\mathcal D^{n/2} \] with a constant $c$ depending only on $n$, we get the inequality \[ |\mathcal R_{h,2M}(x)| \le C \, (h \sqrt{\mathcal D})^{N} c_u \, . \] To estimate $|\mathcal E_{h,2M}(x)|$ we use the functions $\sigma_\beta$ given by \eqref{defsigma} and write \begin{equation*} \begin{split} &(\pi \mathcal D)^{-n/2} \sum_{hm\in \Omega} {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)} -1 =\sigma_0 \Big(\frac{x}{h}, \mathcal D\Big)-(\pi \mathcal D)^{-n/2} \sum_{hm\notin \Omega} {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)} \, , \\ &(\pi \mathcal D)^{-n/2} \sum_{hm \in \Omega} H_\beta \Big( \frac{x-hm}{h \sqrt{\mathcal D}}\Big)\, {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)} \\ &\hskip3cm = \sigma_\beta \Big(\frac{x}{h}, \mathcal D\Big) - (\pi \mathcal D)^{-n/2} \sum_{hm \notin \Omega} H_\beta \Big( \frac{x-hm}{h \sqrt{\mathcal D}}\Big)\, {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)}\, . \end{split} \end{equation*} Furthermore, for $x \in \Omega$ we derive \[ \bigg| (\pi \mathcal D)^{-n/2} \sum_{hm \notin \Omega} H_\beta \Big( \frac{x-hm}{h \sqrt{\mathcal D}}\Big)\, {{\rm e}}^{-|x-hm|^2/(h^2\mathcal D)}\bigg| \le \delta_\beta(h^{-1} \dist(x,\partial \Omega), \mathcal D) \, , \] where $\delta_\beta(r, \mathcal D)$ , $r \ge 0$, denotes the rapidly decaying function \[ \delta_\beta(r, \mathcal D) = \sup_{x\in\mathbb{R}^{n}} \, (\pi \mathcal D)^{-n/2} \sum_{\substack{m \in \mathbb{Z}^n \\|x-m| > r}} \Big|H_\beta \Big( \frac{x-m}{ \sqrt{\mathcal D}}\Big)\Big|\, {{\rm e}}^{-|x-m|^2/\mathcal D} \, . \] Thus, for any domain $\Omega$, fixed parameter $h$ and multiindex $\beta$ we can find a subdomain $\Omega'_{\beta,h} \subsetneq \Omega$ such that \begin{align} \label{xxx} \sup_{x \in \Omega'_{\beta,h}} \delta_\beta(h^{-1} \dist(x,\partial \Omega), \mathcal D) \le \|\sigma_\beta (\cdot, \mathcal D)\|_{L_\infty}\, , \end{align} which gives \begin{align*} |\mathcal E_{h,2M}(x)| \le & \, 2 \sum_{|\beta|=0}^{2M-1} \Big(\frac{h \sqrt{\mathcal D}}{ 2}\Big)^{|\beta|} \frac{|\partial^{\beta}u(x)|}{\beta! }\|\sigma_\beta (\cdot, \mathcal D)\|_{L_\infty} \quad \mbox{for all} \quad x \in \bigcap_{|\beta|=0}^{2M-1} \Omega'_{\beta,h} \, . \end{align*} Now we have to choose $h$ such that $\Omega' \subset \bigcap \Omega'_{\beta,h}$, which is possible since $\Omega'_{\beta,h} \to \Omega$ as $h \to 0$. \begin{remark} {\em Obviously the assertion of Theorems {\em \ref{thmarm}} and {\em \ref{harmdom}} can be extended to the case that a solution $u$ of the second order equation \[ \sum_{i,k=1}^{n} b_{ik} \partial_{i}\partial_{k} u =0 \] in $\Omega$ is approximated by the quasi-interpolant \begin{align*} M u(x)=\frac{(\det B)^{-1/2}}{(\pi \mathcal D)^{n/2}} \sum_{hm \in \Omega}u(hm)\, {\rm e}^{\, -\langle B^{-1}(x-hm),x-hm\rangle/(h^2\mathcal D)} \end{align*} with the matrix $B=\{b_{ik}\}$.} \end{remark} \section{Approximation of derivatives}\label{deriv} Here we study the approximation of derivatives using Hermite quasi-interpolation operator \eqref{3.0}. We introduce the continuous convolution (see \cite{MSbook}) \begin{equation}\label{conv} C_{\delta}v(x)=\delta^{-n}\int_{\mathbb{R}^{n}}\mathcal H\left(\frac{x-y}{\delta}\right) \mathcal Q(-\delta \,\partial)v(y)dy \end{equation} where $\mathcal Q(t)$ is the polynomial in \eqref{polinomio}. \begin{theorem}\label{} Suppose that $\mathcal H$ satisfies the decay condition \eqref{decaybis} with $K>L+n$, $L\in \mathbb{N}$, $L\geq N$. For any $\varepsilon>0$ there exists $\mathcal D>0$ such that for any function $u\in W^{L}_{\infty}(\mathbb{R}^{n})$ \begin{equation} \label{MC} |M u(x)-C_{h\sqrt{\cD}} \,u(x)|\leq \varepsilon\,\sum_{|\gamma|=0}^{L-1}(h\sqrt{\cD})^{|\gamma|}|\partial^{\gamma}u(x)| +c_{1}\,(h\sqrt{\cD})^{L}\sum_{{|\gamma|=L}}||\partial^{\gamma}u||_{L_{\infty}} \, , \end{equation} where the constant $c_{1}$ does not depend on $u$, $h$ and $\sqrt{\cD}$. \end{theorem} {\bf Proof.} Suppose that the function $u\in W^{L}_{\infty}(\mathbb{R}^{n})$. The Taylor expansion \eqref{3.2} with $N$ replaced by $L$ gives the following form of the quasi-interpolant $Mu$ in \eqref{3.1} \begin{equation*}\begin{split} &M u(x)= \sum_{|\gamma|=0}^{N-1}a_{\gamma}\sum_{|\alpha|=0}^{L-1-|\gamma|} \frac{(- h \sqrt{\mathcal D})^{|\alpha+\gamma|}}{\alpha!}\partial^{\alpha+\gamma}u(x) \sigma_{\alpha}(\frac{x}{h},\mathcal D,\mathcal H) \\ &+ (-h \sqrt{\mathcal D})^{L} \sum_{|\gamma|=0}^{N-1}a_{\gamma}\!\!\!\sum_{|\alpha|=L-|\gamma|} \frac{\mathcal D^{-n/2}}{\alpha!} \!\!\sum_{m\in \mathbb{Z}^{n}} U_{\alpha+\gamma}(x,h m) \left( \frac{x-h m}{h \sqrt{\mathcal D}}\right)^{\alpha}\!\!\! \mathcal H\left( \frac{x-h m}{h \sqrt{\mathcal D}}\right). \end{split} \end{equation*} Similarly the Taylor expansion of $u$ around $y$ leads to \begin{equation}\label{conv2} \begin{split} C_{\delta}u(x)&= \sum_{|\gamma|=0}^{N-1}a_{\gamma}\sum_{|\alpha|=0}^{L-1-|\gamma|} (-\delta)^{|\alpha+\gamma|}\partial^{\alpha+\gamma}u(x) \frac{ 1}{\alpha!} \int_{\mathbb{R}^{n}}\tau^{\alpha}\mathcal H(\tau)d\tau\\ &+ (-\delta)^{L} \sum_{|\gamma|=0}^{N-1}a_{\gamma}\sum_{|\alpha|=L-|\gamma|} \frac{ 1}{\alpha!} \int_{\mathbb{R}^{n}}\tau^{\alpha}\mathcal H(\tau)U_{\alpha+\gamma}(x,x-\tau\delta)d\tau. \end{split} \end{equation} Setting $\delta=h \sqrt{\cD}$, we obtain \begin{equation*} \begin{split} & Mu(x)-C_{h \sqrt{\cD}}u(x)= \sum_{|\gamma|=0}^{N-1}a_{\gamma}\sum_{|\alpha|=0}^{L-1-|\gamma|} \frac{(- h \sqrt{\mathcal D})^{|\alpha+\gamma|}}{\alpha!} \partial^{\alpha+\gamma}u(x) \mathcal E_{\alpha}(\frac{x}{h},\mathcal D,\mathcal H)\\ & +(-h \sqrt{\mathcal D})^{L} \sum_{|\gamma|=0}^{N-1}a_{\gamma}\sum_{|\alpha|=L-|\gamma|} \frac{ 1}{\alpha!} [ \mathcal D^{-n/2}\!\!\!\sum_{m\in \mathbb{Z}^{n}} U_{\alpha+\gamma}(x,h m) \left( \frac{x-h m}{h \sqrt{\mathcal D}}\right)^{\alpha}\!\!\! \mathcal H\left( \frac{x-h m}{h \sqrt{\mathcal D}}\right)\\&-\int \limits_{\mathbb{R}^{n}} \tau^{\alpha}\mathcal H(\tau)U_{\alpha+\gamma}(x,x-\tau\delta)d\tau]. \end{split} \end{equation*} Then \begin{equation*}\begin{split} & |Mu(x) \, -C_{h \sqrt{\cD}}u(x)|\leq \sum_{|\gamma|=0}^{N-1}|a_{\gamma}|\sum_{|\alpha|=0}^{L-1-|\gamma|} \frac{( h \sqrt{\mathcal D})^{|\alpha+\gamma|}}{\alpha!} |\partial^{\alpha+\gamma}u(x)| \,\, |\mathcal E_{\alpha}(\frac{x}{h},\mathcal D,\mathcal H)| \\ &+(h \sqrt{\mathcal D})^{L} \sum_{|\gamma|=0}^{N-1}|a_{\gamma}|\sum_{|\alpha|=L-|\gamma|}\frac{ ||\partial^{\alpha+\gamma}u||_{L_{\infty}}}{\alpha!} \left(||\rho_{\alpha}(\cdot,\mathcal D,\mathcal H)||_{L_{\infty}}+ \int_{\mathbb{R}^{n}}|\tau^{\alpha}\mathcal H(\tau)|d\tau\right). \end{split} \end{equation*} Proceeding as in the proof of Theorem \ref{UNO} we deduce \eqref{MC}. \begin{theorem} \label{Cvv} If $\mathcal H$ satisfies the conditions \eqref{decaybis} with $K>N+n$ and \eqref{3.10}, then for any $v\in W^{N}_{\infty}(\mathbb{R}^{n})$ \begin{equation}\label{conv3} |C_{\delta}v(x)-v(x)|\leq c_{2} \delta^{N} \sum_{|\alpha|=N}\Vert \partial^{\alpha}v\Vert_{L_{\infty}}. \end{equation} \end{theorem} {\bf Proof.} The representation \eqref{conv2} with $L=N$ gives \begin{equation}\label{conv6} C_{\delta}v(x)= \sum_{|\alpha|=0}^{N-1}(-\delta)^{|\alpha|}\partial^{\alpha}v(x)\sum_{\gamma\leq \alpha}\frac{a_{\alpha-\gamma}}{\gamma!}\int_{\mathbb{R}^{n}}\tau^{\gamma}\mathcal H(\tau)d\tau+ R_{\delta,\mathcal H}(x) \end{equation} where the remainder $R_{\delta,\mathcal H}$ satisfies \begin{equation}\label{remconv} |R_{\delta,\mathcal H}(x)|\leq \delta^{N}\sum_{|\alpha|=N}\Vert \partial^{\alpha}v\Vert_{L_{\infty}} \sum_{\gamma\leq \alpha}\frac{|a_{\alpha-\gamma}|}{\gamma!} \int_{\mathbb{R}^{n}}|\tau^{\gamma}\mathcal H(\tau)|d\tau\leq c_{2} \delta^{N} \sum_{|\alpha|=N}\Vert \partial^{\alpha}v\Vert_{L_{\infty}}. \end{equation} If condition (\ref{3.10}) holds, in view of (\ref{conv6}), we obtain \eqref{conv3}. Theorem \ref{Cvv} leads immediately to the next corollary. \begin{corollary}\label{Cdudu} Suppose that $\mathcal H$ satisfies conditions \eqref{decaybis} with $K>N+n$ and \eqref{3.10}. Then for $u$ such that $\partial^{\beta}u\in W^{N}_{\infty}(\mathbb{R}^{n})$, \begin{equation}\label{conv7} |C_{\delta}\partial^{\beta}u(x)-\partial^{\beta}u(x)|\leq c_{2}\delta^{N}\sum_{|\alpha|=N}\Vert \partial^{\alpha+\beta}u\Vert_{L_{\infty}}. \end{equation} \end{corollary} If the derivative $\partial^{\beta}\mathcal H$ exists and satisfies the decay condition \eqref{decaybis} then the convolution satisfies \begin{eqnarray} \label{CuC} \partial^{\beta} C_{\delta}\,u(x)=(h \sqrt{\mathcal D})^{-|\beta|}C_{\delta,\partial^{\beta}\mathcal H}\,u(x)= C_{\delta}\partial^{\beta}u(x) \end{eqnarray} where \[ C_{\delta,\partial^{\beta}\mathcal H}u(x)=\delta^{-n}\int_{\mathbb{R}^{n}}\partial^{\beta}\mathcal H\left(\frac{x-y}{\delta}\right) \mathcal Q(-\delta \,\partial)u(y)dy \] and the quasi-interpolant $M u$ in (\ref{3.1}) satisfies the equation \begin{equation}\label{derivate}\partial^{\beta}M u=(h \sqrt{\mathcal D})^{-|\beta|}M_{\partial^{\beta}\mathcal H}u \end{equation} with \[ M_{\partial^{\beta}\mathcal H}u= \mathcal D^{-n/2} \sum_{m\in \mathbb{Z}^{n}}\partial^{\beta}\mathcal H\left( \frac{x-h m}{h \sqrt{\mathcal D}}\right) \mathcal Q\left(-h \sqrt{\mathcal D}\, {\partial }\right) u(hm). \] Hence keeping in mind \eqref{CuC} and \eqref{derivate} we write the difference \begin{align*} (h &\sqrt{\mathcal D})^{|\beta|}[\partial^{\beta}M u(x)-\partial^{\beta}u(x)]\\ &= (h \sqrt{\mathcal D})^{|\beta|}[\partial^{\beta}Mu(x)-\partial^{\beta}C_{\delta}u(x)]+ (h \sqrt{\mathcal D})^{|\beta|}[\partial^{\beta}C_{\delta}u(x)-\partial^{\beta}u(x)]\\ &= [M_{\partial^{\beta}\mathcal H}u(x)-C_{\delta,\partial^{\beta}\mathcal H}u(x)]+ (h \sqrt{\mathcal D})^{|\beta|}[C_{\delta}\partial^{\beta}u(x)-\partial^{\beta}u(x)]. \end{align*} From (\ref{conv7}) and (\ref{MC}) we obtain the following result. \begin{theorem} Suppose that $\mathcal H$ satisfies the conditions \eqref{decaybis} with $K>N+n$ and \eqref{3.10}. Moreover suppose that the derivative $\partial^{\beta} \mathcal H$ exists and satisfies the conditions \eqref{decaybis} with $K>L+n$. The for any $\varepsilon>0$ there exits $\mathcal D>0$ such that, for any $u\in W^{L}_{\infty}(\mathbb{R}^{n})$ with $L\geq N+|\beta|$, \begin{align*} |\partial^{\beta}M u(x)-\partial^{\beta}u(x)|\leq& \, \varepsilon\,\sum_{|\gamma|=0}^{L-1}(h\sqrt{\cD})^{|\gamma|-|\beta|}|\partial^{\gamma}u(x)|\\ +c_{1}\,&(h\sqrt{\cD})^{L-|\beta|}\sum_{{|\gamma|=L}}||\partial^{\gamma}u||_{L_{\infty}}+ c_{2}(h \sqrt{\cD})^{N}\sum_{|\alpha|=N}\Vert \partial^{\alpha+\beta}u\Vert_{L_{\infty}}. \end{align*} \end{theorem} We deduce that formula (\ref{3.0}) gives the simultaneous approximation of the derivatives $\partial^\beta u$ with the saturation term $\varepsilon h^{-|\beta|}$.

proofpile-arXiv_069-11382

\section{Introduction} Low momentum jets are estimated to produce 50\% of transverse energy in RHIC heavy ion collisions and 80\% at the LHC \cite{minijet}. Despite the large role these \emph{minijets} play, they have received little attention from the general community since the start of the RHIC experimental program. As minijet abundance increases at higher energies, the dynamics of minijet interactions are becoming essential to understanding heavy ion collisions. While low momentum jets are not individually resolvable, their combined effect generates an observable correlation. In a theoretical context minijets are typically defined within the range of applicability of pQCD by specifying a low hadron $p_t$ cutoff around 2 GeV/c, even though QCD interactions continue to lower $p_t$. We experimentally define minijets based on correlation structure rather than an {\it a priori} $p_t$ range. This requires a \emph{minimum-bias} two-particle correlation analysis where every possible pair of particles is considered instead of selecting a few trigger/associated pairs. Minijets are distinguished from other sources by decomposing the unique correlations. Previous analyses have used this technique to reveal large minijet contributions in transverse \cite{130xtxt} and axial ($\eta$, $\phi$) \cite{130CI} spaces at 130 GeV at four centralities. Here we report the detailed energy and centrality dependence of minijet angular correlations at RHIC. \section{Analysis} Charged particle tracks detected in the STAR TPC with $p_t > 0.15$ GeV/c, $|\eta| < 1$, and full $2\pi$ azimuth were analyzed from 1.2M minbias triggered 200 GeV Au+Au and 6.7M 62 GeV Au+Au events. Pair densities $\rho(\vec{p}_1, \vec{p}_2)$ were measured as number of pairs per unit area on relative angles ($\ensuremath{\eta_\Delta} \equiv \eta_1 - \eta_2$, $\ensuremath{\phi_\Delta} \equiv \phi_1 - \phi_2$) for all possible unique particle pairs. Particles within the same event form \emph{sibling} pair densities $\rho_{sib}$, while mixing particles from different events measures the uncorrelated \emph{reference} $\rho_{ref}$. These are formed into a normalized covariance to produce a correlation measure. The difference $\Delta \rho \equiv \rho_{sib} - \rho_{ref}$ measures the covariance in number of pairs between histogram bins, and the normalization is provided by bin-wise division of $\sqrt{\rho_{ref}}$. Thus we use the notation \ensuremath{\frac{\Delta\rho}{\sqrt{\rho_{\rm ref}}}}{} for a \emph{per-particle} correlation measure, shown in figure \ref{fig1} for selected centralities. \begin{figure}[htb] \begin{center} \includegraphics[width=\textwidth]{fig1.eps} \caption{Minimum-bias correlations for several centralities from peripheral (left) to central (right) in 200 GeV Au+Au collisions.} \label{fig1} \end{center} \end{figure} \section{Fit Results} Proton-proton collisions provide a reference for measuring the contributions to these structures. Analysis of minimum-bias correlations \cite{ppcorr} and single particle $p_t$ spectra \cite{ppspectra} show that p+p collisions are well described by a two-component soft and semi-hard scattering model, as commonly used in event generators such as \textsc{Pythia}. The soft component represents longitudinal fragmentation in unlike-sign pairs and produces a 1D gaussian correlation centered along \ensuremath{\eta_\Delta}=0. The semi-hard component contains a same-side peak, modeled as a 2D gaussian at the $\ensuremath{\eta_\Delta}=\ensuremath{\phi_\Delta}=0$ origin, and an away-side ridge centered at \ensuremath{\phi_\Delta}=$\pi$. For an inclusive $p_t$ range the away-side is completely represented by function $-\cos(\ensuremath{\phi_\Delta})$ that approximates a wide gaussian which narrows with increasing $p_t$ \cite{ppcorr}. The final component necessary to describe p+p data is a 2D exponential at the origin containing contributions from HBT in like-sign pairs and conversion $e^{\pm}$ in unlike-sign pairs. To ensure the simplest possible fit function for Au+Au collisions, we use these components from p+p collisions with only one additional $\cos(2\ensuremath{\phi_\Delta})$ quadrupole term to account for correlations conventionally attributed to elliptic flow \cite{FlowMethods}. The eleven parameter fit function used for the correlation structures in figure \ref{fig1} is then: \\ \begin{eqnarray} \label{eq:fit} \fl F = A_{\phi_\Delta}\cos(\phi_\Delta) + A_{2\phi_\Delta}\cos(2\, \phi_\Delta) + G_s(\ensuremath{\eta_\Delta}: A_0, \sigma_0) + G_h(\ensuremath{\eta_\Delta}, \ensuremath{\phi_\Delta}: A_1, \sigma_{\eta\Delta}, \sigma_{\phi\Delta}) \nonumber \\ + E(\ensuremath{\eta_\Delta}, \ensuremath{\phi_\Delta}: A_2, w_{\eta\Delta}, w_{\phi\Delta}) + A_3 \end{eqnarray} where $G_s$ and $G_h$ are the soft and hard Gaussian terms and $E$ is an exponential function with parameters listed after the colon. An example of this fit is shown in figure \ref{fig2}. \begin{figure}[htb] \begin{center} \includegraphics[width=\textwidth]{fig2.eps} \caption{An example of the fit function showing correlation data (first panel), model function (second panel), residual (third panel) defined as data minus model fit, and the same-side gaussian and exponential peaks (last panel).} \label{fig2} \end{center} \end{figure} Figure \ref{fig3} shows the measured fit parameters for the same-side peak amplitude, \ensuremath{\eta_\Delta}{} width, and volume ($=2\pi A_1 \sigma_{\eta\Delta} \sigma_{\phi\Delta}$). Fitting errors are shown and systematic error is estimated to be $\pm$9\% of the correlation amplitude and at most a few percent of the widths. The dashed lines show the binary scaling reference expected from independent nucleon-nucleon collisions. Using the Kharzeev and Nardi two-component model \cite{KN} and path length $\nu \equiv 2\langle N_{bin} \rangle / \langle N_{part} \rangle$, the minijet amplitude in Au+Au collisions is expected to scale as $A_1(\nu) = A_{1,pp} \: \nu / [1 + x(\nu - 1)]$ from the p+p value. Peripheral collisions follow the binary scaling reference closely, deviating only by small increases in \ensuremath{\eta_\Delta}{} and decreases in \ensuremath{\phi_\Delta}{} widths. The data show a sharp transition at approximately 55\% centrality for 200 GeV and 40\% for 62 GeV where the amplitude and \ensuremath{\eta_\Delta}{} widths increase dramatically while the \ensuremath{\phi_\Delta}{} widths continue to decrease slightly. Centrality in figure \ref{fig3} is represented by transverse particle density calculated as $\frac{3}{2} \frac{dN_{ch}}{d\eta} / \langle S \rangle$ with initial collision overlap area $\langle S \rangle$ from Monte Carlo Glauber. Transverse density brings the transition points for the two energies to coincidence, whereas conventional centrality measures displace the transition points and tend to compress the peripheral data. \begin{figure}[htb] \begin{center} \includegraphics[width=\textwidth]{fig3.eps} \caption{Same-side gaussian peak amplitude, \ensuremath{\eta_\Delta}{} width, and volume fits. Points show eleven centrality bins for each energy (84-93\%, 74-84\%, 65-74\%, 55-65\%, 46-55\%, 37-46\%, 28-37\%, 19-28\%, 9-19\%, 5-9\%, and 0-5\%) transformed to tranvserse density.} \label{fig3} \end{center} \end{figure} \section{Discussion} The correlation structures are modified at the transition, but are still likely to be associated with minijets for several reasons. First, these results, particularly when taken with a similar analysis of $p_t$ correlations \cite{ptcorr}, show that contributions from a new physical mechanism unrelated to minijets are unlikely. Any such hypothetical process must have \ensuremath{\phi_\Delta}{} widths and $p_t$ correlations that match seamlessly with minijets, which would be a remarkable coincidence. Second, the amplitude and \ensuremath{\eta_\Delta}{} width increases are consistent with further minijet interactions, which may be possible due to path-length considerations \cite{MinijetProduction}. Finally, it is possible that the new correlation structures are due to changes in minijet fragmentation. The trends in the data also suggest a lower $p_t$ manifestation of the ``ridge'' \cite{Ridge}, and these results may help to discriminate among the many competing models of ridge formation. The same-side peak volume gives the total number of correlated pairs, though finding the particle yield requires estimating the average number of correlated structures per event. Assuming each structure originates with a semi-hard parton and that semi-hard scattering follows binary scaling, we estimate that 30\% of all final-state hadrons in central 200 GeV Au+Au collisions are associated with this same-side correlation. As a source of correlated low momentum particles, minijets provide an extremely sensitive probe of the collision system. The binary scaling reference represents one extreme limit of a transparent medium, while the other extreme is a completely thermalized system opaque to minijets \cite{Therm}. These results call into question the existence of the latter system at RHIC energies. \Bibliography{10} \bibitem{minijet} Wang X-N and Gyulassy M 1991 {\it Phys. Rev. D} {\bf 44} 3501 \bibitem{130xtxt} Adams J \etal (STAR Collaboration) 2007 {\it J. Phys. G: Nucl. Part. Phys.} {\bf 34} 799 \bibitem{130CI} Adams J \etal (STAR Collaboration) 2006 {\it Phys. Rev. C} {\bf 73} 064907 \bibitem{ppcorr} Porter R J and Trainor T A 2005 {\it J. Phys.: Conf. Ser.} {\bf 27} 98 \bibitem{ppspectra} Adams J \etal (STAR Collaboration) 2006 {\it Phys. Rev D} {\bf 74} 032006 \bibitem{FlowMethods} Trainor T A and Kettler D T 2007 {\it Preprint} arXiv:0704.1674 [hep-ph] \bibitem{KN} Kharzeev D and Nardi M 2001 {\it Phys. Lett. B} {\bf 507} 121 \bibitem{ptcorr} Adams J \etal (STAR Collaboration) 2006 {\it J. Phys. G: Nucl. Part. Phys.} {\bf 32} L37 \bibitem{MinijetProduction} Kajantie K \etal 1987 {\it Phys. Rev. Lett.} {\bf 59} 2527 \bibitem{Ridge} Putschke J (STAR Collaboration) 2007 {\it Preprint} nucl-ex/0701074 \bibitem{Therm} Nayak G C \etal 2001 {\it Nucl. Phys. A} {\bf 687} 457; Shin G R and M\"{u}ller 2003 {\it J. Phys. G: Nucl. Part. Phys.} {\bf 29} 2485 \endbib \end{document}

proofpile-arXiv_069-11485

\section{Introduction} \label{sec:introduction} The \emph{Multidimensional Assignment Problem} (MAP) (abbreviated $s$-AP in the case of $s$ dimensions, also called \emph{(axial) Multi Index Assignment Problem}, MIAP,~\cite{Bandelt2004,Pardalos2000}) is a well-known optimization problem. It is an extension of the \emph{Assignment Problem} (AP), which is exactly the two dimensional case of MAP\@. While AP can be solved in the polynomial time~\cite{Kuhn1955}, $s$-AP for every $s \ge 3$ is NP-hard~\cite{Garey1979} and inapproximable~\cite{Burkard1996}\footnote{Burkard et al.\ show it for a special case of 3-AP and since 3-AP is a special case of $s$-AP the result can be extended to the general MAP}\@. The most studied case of MAP is the case of three dimensions~\cite{Aiex2005,Andrijich2001,Balas1991,Crama1992,Huang2006,Spieksma2000} though the problem has a host of applications for higher numbers of dimensions, e.g., in matching information from several sensors (data association problem), which arises in plane tracking~\cite{Murphey1998,Pardalos2000b}, computer vision~\cite{Veenman2003} and some others~\cite{Andrijich2001,Bandelt2004,Burkard1999}, in routing in meshes~\cite{Bandelt2004}, tracking elementary particles~\cite{Pusztaszeri1996}, solving systems of polynomial equations~\cite{Bekker2005}, image recognition~\cite{Grundel2004}, resource allocation~\cite{Grundel2004}, etc. For a fixed $s \ge 2$, the problem $s$-AP is stated as follows. Let $X_1 = X_2 = \ldots = X_s = \{ 1, 2, \ldots, n \}$. We will consider only vectors that belong to the Cartesian product $X = X_1 \times X_2 \times \ldots \times X_s$. Each vector $e \in X$ is assigned a non-negative weight $w(e)$. For a vector $e \in X$, the component $e_j$ denotes its $j$th coordinate, i.e., $e_j \in X_j$. A collection $A$ of $t \le n$ vectors $A^1, A^2, \ldots, A^t$ is a \emph{(feasible) partial assignment} if $A^i_j \neq A^k_j$ holds for each $i \neq k$ and $j \in \{ 1, 2, \ldots, s \}$. The \emph{weight} of a partial assignment $A$ is $w(A) = \sum_{i=1}^t w(A^i)$. An \emph{assignment} (or \emph{full assignment}) is a partial assignment with $n$ vectors. The objective of $s$-AP is to find an assignment of minimal weight. We also provide a \emph{permutation form} of the assignment which is sometimes more convenient. Let $\pi_1, \pi_2, \ldots, \pi_s$ be permutations of $X_1, X_2, \ldots, X_s$ respectively. Then $\pi_1 \pi_2 \ldots \pi_s$ is an assignment with the weight $\sum_{i=1}^n w(\pi_1(i) \pi_2(i) \ldots \pi_s(i))$. It is obvious that one permutation, say the first one, may be fixed without any loss of generality: $\pi_1 = 1_n$, where $1_n$ is the identity permutation of size $n$. Then the objective of the problem is as follows: $$ \min_{\pi_2, \ldots, \pi_s} \sum_{i=1}^n w(i \pi_1(i) \ldots \pi_s(i)) \ . $$ A graph formulation of the problem is as follows. Having an $s$-partite graph $G$ with parts $X_1$, $X_2$, \ldots, $X_s$, where $|X_i| = n$, find a set of $n$ disjoint cliques in $G$ of the minimal total weight if every clique $e$ in $G$ is assigned a weight $w(e)$. Finally, an integer programming formulation of the problem is as follows. $$ \min \sum_{i_1 \in X_1, \ldots, i_s \in X_s} w(i_1 \ldots i_s) \cdot x_{i_1 \ldots i_s} $$ subject to $$ \sum_{i_2 \in X_2, \ldots, i_s \in X_s} x_{i_1 \ldots i_s} = 1 \qquad {\forall i_1 \in X_1} \mbox{ ,} $$ $$ \ldots $$ $$ \sum_{i_1 \in X_1, \ldots, i_{s-1} \in X_{s-1}} x_{i_1 \ldots i_s} = 1 \qquad {\forall i_s \in X_s} \mbox{ ,} $$ where $x_{i_1 \ldots i_s} \in \{ 0, 1 \}$ for all $i_1$, \ldots, $i_s$ and $|X_1| = \ldots = |X_s| = n$. Sometimes the problem is formulated in a more general way if $|X_1| = n_1$, $|X_2| = n_2$, \ldots, $|X_s| = n_s$ and the requirement $n_1 = n_2 = \ldots = n_s$ is omitted. However this case can be easily transformed into the problem described above by complementing the weight matrix to an $n \times n \times \ldots \times n$ matrix with zeros, where $n = \max_i n_i$. The problem was studied by many researchers. Several special cases of the problem were intensively studied in the literature (see~\cite{Kuroki2007} and references there) and for few classes of instances polynomial time exact algorithms were found, see, e.g.,~\cite{Burkard1996a,Burkard1996,Isler2005}. In many cases MAP remains hard to solve~\cite{Kuroki2007,Spieksma1996}. For example, if there are three sets of points of size $n$ on a Euclidean plain and the objective is to find $n$ triples of points, one from each set, such that the total circumference or area of the corresponding triangles is minimal, the corresponding 3-AP is still NP-hard~\cite{Spieksma1996}. The asymptotic properties of some special instance families are studied in~\cite{Grundel2004}. As regards the solution methods, apart from exact and approximation algorithms~\cite{Balas1991,Crama1992,Kuroki2007,Pasiliao2005,Pierskalla1968}, several heuristics including construction heuristics~\cite{Balas1991,Gutin2008,GK_MAP_Construction,Oliveira2004}, greedy randomized adaptive search procedures~\cite{Aiex2005,Murphey1998,Oliveira2004,Robertson2001}, metaheuristics~\cite{Clemons2004,Huang2006} and parallel heuristics~\cite{Oliveira2004} are presented in the literature. Several local search procedures are proposed and discussed in~\cite{Aiex2005,Balas1991,Bandelt2004,Burkard1996,Clemons2004,Huang2006,Oliveira2004,Robertson2001}. The difference between the construction heuristics and local search is sometimes crucial. While a construction heuristic generates a solution from scratch and, thus, has some solution quality limitation, local search is intended to improve an existing solution and, thus, can be used after a construction heuristic or as a part of a more sophisticated heuristic, so called metaheuristic. The contribution of our paper is in collecting and generalizing all local search heuristics known from the literature, proposing some new ones and detailed theoretical and evaluating them both theoretically and experimentally. For the purpose of experimental evaluation we also thoroughly discuss, classify the existing instance families and propose some new ones. In this paper we consider only the general case of MAP and, thus, all the heuristics which rely on the special structures of the weight matrix are not included in the comparison. We also assume that the number of dimensions $s$ is a small fixed constant while the size $n$ can be arbitrary large. \section{Heuristics} \label{sec:heuristics} In this section we discuss some well known and some new MAP local search heuristics as well as their combinations. \subsection{Dimensionwise Variations Heuristics} \label{sec:dv} The heuristics of this group were first introduced by Bandelt et al.~\cite{Bandelt2004} for MAP with decomposable costs. However, having a very large neighborhood (see below), they are very efficient even in the general case. The fact that this approach was also used by Huang and Lim as a local search procedure for their memetic algorithm~\cite{Huang2006} confirms its efficiency. The idea of the dimensionwise variation heuristics is as follows. Consider the initial assignment $A$ in the permutation form $A = \pi_1 \pi_2 \ldots \pi_s$ (see Section~\ref{sec:introduction}). Let $p(A, \rho_1, \rho_2, \ldots, \rho_s)$ be an assignment obtained from $A$ by applying the permutations $\rho_1, \rho_2, \ldots, \rho_s$ to $\pi_1, \pi_2, \ldots, \pi_s$ respectively: \begin{equation} \label{eq:p} p(A, \rho_1, \rho_2, \ldots, \rho_s) = \rho_1(\pi_1) \rho_2(\pi_2) \ldots \rho_s(\pi_s) \ . \end{equation} Let $p_D(A, \rho)$ be an assignment $p(A, \rho_1, \rho_2, \ldots, \rho_s)$, where $\rho_j = \rho$ if $j \in D$ and $\rho_j = 1_n$ otherwise ($1_n$ is the identity permutation of size $n$): \begin{equation} \label{eq:p_D} p_D(A, \rho) = p\left(A, \left\{ \begin{array}{@{}l@{~}l@{}}\rho & \text{if } 1 \in D\\ 1_n & \text{otherwise}\end{array} \right., \left\{ \begin{array}{@{}l@{~}l@{}}\rho & \text{if } 2 \in D\\ 1_n & \text{otherwise}\end{array} \right., \ldots, \left\{ \begin{array}{@{}l@{~}l@{}}\rho & \text{if } s \in D\\ 1_n & \text{otherwise}\end{array} \right.\right) \ . \end{equation} On every iteration, the heuristic selects some nonempty set $D \subsetneq \{ 1, 2, \ldots, s \}$ of dimensions and searches for a permutation $\rho$ such that $w(p_D(A, \rho))$ is minimized. For every subset of dimensions $D$, there are $n!$ different permutations $\rho$ but the optimal one can be found in the polynomial time. Let $swap(u, v, D)$ be a vector which is equal to vector $u$ in all dimensions $j \in \{ 1, 2, \ldots, s \} \setminus D$ and equal to vector $v$ in all dimensions $j \in D$: \begin{equation} \label{eq:swap} swap(u, v, D)_j = \left\{ \begin{array}{l @{\quad} l} u_j & \text{if } j \notin D \\ v_j & \text{if } j \in D \\ \end{array} \right. \text{\qquad for } j = 1, 2, \ldots, s. \end{equation} Let matrix $[M_{i,j}]_{n \times n}$ be constructed as \begin{equation} \label{eq:M} M_{i,j} = w(swap(A^i, A^j, D)) \ . \end{equation} It is clear that the solution of the corresponding 2-AP is exactly the required permutation $\rho$. Indeed, assume there exists some permutation $\rho'$ such that $w(p_D(A, \rho')) < w(p_D(A, \rho))$. Observe that $p_D(A, \rho) = \{ swap(A^i, A^{\rho(i)}, D) :\ i \in \{ 1, 2, \ldots, n \} \}$. Then we have $$ \sum_{i = 1}^n w(swap(A^i, A^{\rho'(i)}, D)) < \sum_{i = 1}^n w(swap(A^i, A^{\rho(i)}, D)) \ . $$ Since $w(swap(A^i, A^{\rho(i)}, D)) = M_{i, \rho(i)}$, the sum $\sum_{i = 1}^n w(swap(A^i, A^{\rho(i)}, D))$ is already minimized to the optimum and no $\rho'$ can exist. \bigskip The neighborhood of a dimensionwise heuristic is as follows: \begin{equation} \label{eq:dv_neighborhood} N_\text{DV}(A) = \big\{ p_D(A, \rho) :\ D \in \mathcal{D} \text{ and $\rho$ is a permutation} \big\} \ , \end{equation} where $\mathcal{D}$ includes all dimension subsets acceptable by a certain heuristic. Observe that \begin{equation} \label{eq:p_D_symmetry} p_D(A, \rho) = p_{\overline{D}}(A, \rho^{-1}) \ , \end{equation} where $\rho^{-1}(\rho) = \rho(\rho^{-1}) = 1_s$ and $\overline{D} = \{ 1, 2, \ldots, s \} \setminus D$, and, hence, \begin{equation} \label{eq:P_D_symmetry} \big\{ p_D(A, \rho) :\ \text{$\rho$ is a permutation} \big\} = \big\{ p_{\overline{D}}(A, \rho) :\ \text{$\rho$ is a permutation} \big\} \end{equation} for any $D$. From (\ref{eq:P_D_symmetry}) and the obvious fact that $p_\varnothing(A, \rho) = p_{\{ 1, 2, \ldots, s \}}(A, \rho) = A$ for any $\rho$ we introduce the following restrictions for $\mathcal{D}$: \begin{equation} \label{eq:D_restrictions} D \in \mathcal{D} \Rightarrow\overline{D} \notin \mathcal{D} \text{\quad\ and \quad} \varnothing, \{ 1, 2, \ldots, s \} \notin \mathcal{D} \ . \end{equation} With these restrictions, one can see that for any pair of distinct sets $D_1, D_2 \in \mathcal{D}$ the equation $p_{D_1}(A, \rho_1) = p_{D_2}(A, \rho_2)$ holds if and only if $\rho_1 = \rho_2 = 1_n$. Hence, the size of the neighborhood $N_\text{DV}(A)$ is \begin{equation} \label{eq:dv_neighborhood_size} |N_\text{DV}(A)| = |\mathcal{D}| \cdot (n! - 1) + 1 \ . \end{equation} In~\cite{Bandelt2004} it is decided that the number of iterations should not be exponential with regards to neither $n$ nor $s$ while the size of the maximum $\mathcal{D}$ is $|\mathcal{D}| = 2^{s-1} - 1$. Therefore two heuristics, LS1 and LS2, are evaluated in~\cite{Bandelt2004}. LS1 includes only singleton values of $D$, i.e., $\mathcal{D} = \{ D :\ |D| = 1 \}$; LS2 includes only doubleton values of $D$, i.e., $\mathcal{D} = \{ D :\ |D| = 2 \}$. It is surprising but according to both~\cite{Bandelt2004} and our computational experience, the heuristic LS2 produces worse solutions than LS1 though it obviously has larger neighborhood and larger running times. We improve the heuristic by allowing $|D| \le 2$, i.e., $\mathcal{D} = \{ D :\ |D| \le 2 \}$. This does not change the theoretical time complexity of the algorithm but improves its performance. The heuristic LS1 is called {\heuristic{\OneDVPlain}}\ in our paper; LS2 with $|D| \le 2$ is called {\heuristic{\TwoDVPlain}}. We also assume (see Section~\ref{sec:introduction}) that the value of $s$ is a small fixed constant and, thus, introduce a heuristic {\heuristic{\MDVPlain}}\ which enumerates all feasible (recall (\ref{eq:D_restrictions})) $D \subset \{ 1, 2, \ldots, s \}$. The order in which the heuristics take the values $D \in \mathcal{D}$ in our implementations is as follows. For {\heuristic{\OneDVPlain}}\ it is $\{ 1 \}$, $\{ 2 \}$, \ldots, $\{ s \}$. {\heuristic{\TwoDVPlain}}\ begins as {\heuristic{\OneDVPlain}}\ and then takes all pairs of dimensions: $\{ 1, 2 \}$, $\{ 1, 3 \}$, \ldots, $\{ 1, s \}$, $\{ 2, 3 \}$, \ldots, $\{ s - 1, s \}$. Note that because of~(\ref{eq:D_restrictions}) it enumerates no pairs of vectors for $s = 3$, and for $s = 4$ it only takes the following pairs: $\{ 2, 3 \}$, $\{ 2, 4 \}$ and $\{ 3, 4 \}$. {\heuristic{\MDVPlain}}\ takes first all sets $D$ of size 1, then all sets $D$ of size 2 and so on up to $|D| = \lfloor s / 2 \rfloor$. If $s$ is even then we should take only half of the sets $D$ of size $s / 2$ (recall (\ref{eq:P_D_symmetry})); for this purpose we take all the subsets of $D \subset \{ 2, 3, \ldots, s \}$, $|D| = s / 2$ in the similar order as before. It is obvious that $N_\text{1DV}(A) \subseteq N_\text{2DV}(A) \subseteq N_\text{$s$DV}(A)$ for any $s$ however for $s = 3$ all the neighborhoods are equal and for $s = 4$ {\heuristic{\TwoDVPlain}}\ and {\heuristic{\MDVPlain}}\ also coincide. According to (\ref{eq:D_restrictions}) and (\ref{eq:dv_neighborhood_size}), the neighborhood size of {\heuristic{\OneDVPlain}}\ is $$ |N_\text{1DV}(A)| = s \cdot (n! - 1) + 1 \ , $$ of {\heuristic{\TwoDVPlain}}\ is $$ |N_\text{2DV}(A)| = \left\{ \begin{array}{ll} (2^{s-1} - 1) \cdot (n! - 1) + 1 & \text{if } s \in \{ 3, 4 \} \\ \left(\binom{s}{2} + s\right) \cdot (n! - 1) + 1 & \text{if } s \ge 5 \end{array} \right. \ , $$ and of {\heuristic{\MDVPlain}}\ is $$ |N_\text{$s$DV}(A)| = (2^{s-1} - 1) \cdot (n! - 1) + 1 \ . $$ The time complexity of every run of DV is $O(|\mathcal{D}| \cdot n^3)$ as every 2-AP takes $O(n^3)$ and, hence, the time complexity of {\heuristic{\OneDVPlain}}\ is $O(s \cdot n^3)$, of {\heuristic{\TwoDVPlain}}\ is $O(s^2 \cdot n^3)$ and of MDV is $O(2^{s-1} \cdot n^3)$. \subsection{$k$-opt} \label{sec:kopt} The \Opt{$k$}\ heuristic for 3-AP for $k = 2$ and $k = 3$ was first introduced by Balas and Saltzman~\cite{Balas1991} as a \emph{pairwise} and \emph{triple interchange heuristic}. \Opt{2}\ as well as its variations were also discussed in~\cite{Aiex2005,Clemons2004,Murphey1998,Oliveira2004,Pasiliao2005,Robertson2001} and some other papers. We generalize the heuristic for arbitrary values of $k$ and $s$. The heuristic proceeds as follows. For every subset of $k$ vectors taken in the assignment $A$ it removes all these vectors from $A$ and inserts some new $k$ vectors such that the assignment feasibility is preserved and its weight is minimized. Another definition is as follows: for every set of distinct vectors $e^1, e^2, \ldots, e^k \in A$ let $X'_j = \{ e_j^1, e_j^2, \ldots, e_j^k \}$ for $j = 1, 2, \ldots, k$. Let $A' = \{ e'^1, e'^2, \ldots, e'^k \}$ be the solution of this $s$-AP of size $k$. Replace the vectors $e^1, e^2, \ldots, e^k$ in the initial assignment $A$ with $e'^1, e'^2, \ldots, e'^k$. The time complexity of \Opt{$k$}\ is obviously $O\left( \binom{n}{k} \cdot k!^{s-1} \right)$; for $k \ll n$ it can be replaced with $O(n^k \cdot k!^{s-1})$. It is a natural question if one can use some faster solver on every iteration. Indeed, according to Section~\ref{sec:introduction} it is possible to solve $s$-AP of size $k$ in $O(k!^{s-2} \cdot k^3)$. However, it is easy to see that $k!^{s-1} < k!^{s-2} \cdot k^3$ for every $k$ up to 5, i.e., it is better to use the exhaustive search for any reasonable $k$. One can doubt that the exact algorithm actually takes $k!^{s-2} \cdot k^3$ operations but even for the lower bound $\Omega(k!^{s-2} \cdot k^2)$ the inequality $k!^{s-1} < k!^{s-2} \cdot k^2$ holds for any $k \le 3$, i.e., for all the values of $k$ we actually consider. Now let us find the neighborhood of the heuristic. For some set $\mathcal{I}$ and a subset $I \subset \mathcal{I}$ let a permutation $\rho$ of elements in $\mathcal{I}$ be an \emph{$I$-permutation} if $\rho(i) = i$ for every $i \in \mathcal{I} \setminus I$, i.e., if $\rho$ does not move any elements except elements from $I$\@. Let $E = \{ e^1, e^2, \ldots, e^k \} \subset A$ be a set of $k$ distinct vectors in $A$\@. For $j = 2, 3, \ldots, s$ let $\rho_j$ be an $E_j$-permutation, where $E_j = \{ e_j^1, e_j^2, \ldots, e_j^k \}$. Then a set $W(A, E)$ of all assignments which can be obtained from $A$ by swapping coordinates of vectors $E$ can be described as follows: $$ W(A, E) = \big\{ p(A, 1_n, \rho_2, \rho_3, \ldots, \rho_s) :\ \rho_j \text{ is an $E_j$-permutation for } j = 2, 3, \ldots, s \big\} \ . $$ Recall that $1_n$ is the identity permutation of size $n$ and $p(A, \rho_1, \rho_2, \ldots, \rho_s)$ is defined by (\ref{eq:p}). The neighborhood $N_{k\text{-opt}}(A)$ is defined as follows: \begin{equation} \label{eq:KoptNeighborhood} N_{k\text{-opt}}(A) = \bigcup_{E \subset A, |E| = k} W(A, E) \ . \end{equation} Let $Y, Z \subset A$ such that $|Y| = |Z| = k$. Observe that $W(A, Y) \cap W(A, Z)$ is nonempty and apart from the initial assignment $A$ this intersection may contain assignments which are modified only in the common vectors $Y \cap Z$. To calculate the size of the neighborhood of \Opt{$k$}{} let us introduce $W'(A, E)$ as a set of all assignments in $W(A, E)$ such that every vector in $E$ is modified in at least one dimension, where $E \subset A$ is the set of $k$ selected vectors in the assignment $A$: $$ W'(A, E) = \big\{ A' \in W(A, E) :\ |A \cap A'| = n - k \big\} \ . $$ Then the neighborhood $N_{k\text{-opt}}(A)$ of \Opt{$k$}{} is \begin{equation} N_{k\text{-opt}}(A) = \bigcup_{E \subset A, |E| \le k} W'(A, E) \end{equation} and since $W(A, Y) \cap W(A, Z) = \varnothing$ if $Y \neq Z$ we have \begin{equation} \label{eq:kopt_neighborhood_size} |N_{k\text{-opt}}(A)| = \sum_{E \subset A, |E| \le k} |W'(A, E)| = \sum_{i = 0}^k \binom{n}{i} N^i \ , \end{equation} where $N^i = |W(A, E)|$ for any $E$ with $|E| = i$. Observe that $$ W'(A, E) = W(A, E) \setminus \bigcup_{E' \subsetneq E} W'(A, E') $$ and $|W(A, E)| = k!^{s-1}$ for $|E| = k$ and, hence, \begin{equation} \label{eq:kopt_n_k} N^k = k!^{s-1} - \sum_{i = 0}^{k-1}\binom{k}{i} N^{i} \ . \end{equation} It is obvious that $N^0 = 1$ since one can obtain exactly one assignment (the given one) by changing no vectors. From this and~(\ref{eq:kopt_n_k}) we have $N^1 = 0$, $N^2 = 2^{s-1} - 1$ and $N^3 = 6^{s-1} - 3 \cdot 2^{s-1} + 2$. From this and~(\ref{eq:kopt_neighborhood_size}) follows \begin{equation} \label{eq:twoopt_neighborhood_size} |N_\text{2-opt}(A)| = 1 + \binom{n}{2} (2^{s-1} - 1) \ , \end{equation} \begin{equation} |N_\text{3-opt}(A)| = 1 + \binom{n}{2} (2^{s-1} - 1) + \binom{n}{3} (6^{s-1} - 3 \cdot 2^{s-1} + 2) \ . \end{equation} \bigskip In our implementation, we skip an iteration if the corresponding set of vectors $E$ either consists of the vectors of the minimal weight ($w(e) = \min_{e \in X} w(e)$ for every $e \in E$) or all these vectors have remained unchanged during the previous run of $k$-opt. \bigskip It is assumed in the literature~\cite{Balas1991,Pasiliao2005,Robertson2001} that \Opt{$k$}\ for $k > 2$ is too slow to be applied in practice. However, the neighborhood $N_{k\text{-opt}}$ do not only includes the neighborhood $N_{(k-1)\text{-opt}}$ but also grows exponentially with the growth of $k$ and, thus, becomes very powerful. We decided to include \Opt{2}\ and \Opt{3}\ in our research. Greater values of $k$ are not considered in this paper because of nonpractical time complexity (observe that the time complexity of \Opt{4} is $O(n^4 \cdot 24^{s-1})$) and even \Opt{3}\ with all the improvements described above still takes a lot of time to proceed. However, \Opt{3}\ is more robust when used in a combination with some other heuristic (see Section~\ref{sec:combined}). \bigskip It is worth noting that our extension of the pairwise (triple) interchange heuristic~\cite{Balas1991} is not typical. Many papers~\cite{Aiex2005,Clemons2004,Murphey1998,Pasiliao2005,Robertson2001} consider another neighborhood: $$ N_\text{\OptStarPlain{$k$}}(A) = \big\{ p_D(A, \rho) :\ D \subset \{ 1, 2, \ldots, s \}, |D| = 1 \text{ and $\rho$ moves at most $k$ elements} \big\} \ , $$ where $p_D$ is defined in~(\ref{eq:p_D}). The size of such neighborhood is $|N_\text{\OptStarPlain{$k$}}(A)| = s \cdot \binom{n}{k} \cdot (k! - 1) + 1$ and the time complexity of one run of \OptStar{$k$} in the assumption $k \ll n$ is $O(s \cdot n^k \cdot k!)$, i.e., unlike \Opt{$k$}, it is not exponential with respect to the number of dimensions $s$ which is considered to be important by many researchers. However, as it is stated in Section~\ref{sec:introduction}, we assume that $s$ is a small fixed constant and, thus, the time complexity of \Opt{$k$}\ is still reasonable. At the same time, observe that $N_{k\text{-opt*}}(A) \subset N_\text{1-DV}(A)$ for any $k \le n$, i.e., {\heuristic{\OneDVPlain}}\ performs as good as \OptStar{$n$} with the time complexity of \OptStar{3}. Only in the case of $k = 2$ the heuristic \OptStar{2} is faster in theory however it is known~\cite{Burkard1999} that the expected time complexity of AP is significantly less than $O(n^3)$ and, thus, the running times of \OptStar{2} and {\heuristic{\OneDVPlain}}\ are similar while {\heuristic{\OneDVPlain}}\ is definitely more powerful. Because of this we do not consider \OptStar{2} in our comparison. \subsection{Variable Depth Interchange (v-opt)} \label{sec:vopt} The \emph{Variable Depth Interchange} (VDI) was first introduced by Balas and Saltzman for 3-AP as a heuristic based on the well known Lin-Kernighan heuristic for the traveling salesman problem~\cite{Balas1991}. We provide here a natural extension \Opt{v}\ of the VDI heuristic for the $s$-dimensional case, $s \ge 3$, and then improve this extension. Our computational experiments show that the improved version of \Opt{v}\ is superior to the natural extension of VDI{} with respect to solution quality at the cost of a reasonable increase in running time. In what follows, \Opt{v}{} refers to the improved version of the heuristic unless otherwise specified. In~\cite{Balas1991}, the heuristic is described quite briefly. Our contribution is not only in the extending, improving and analyzing it but also in a more detailed and, we believe, clearer explanation of it. We describe the heuristic in a different way to the description provided in~\cite{Balas1991}, however, both versions of our algorithm are equal to VDI{} in case of $s = 3$. This fact was also checked by reproduction of the computational evaluation results reported in~\cite{Balas1991}. Further we will use function $U(u, v)$ which returns a set of swaps between vectors $u$ and $v$. The difference between the two versions of \Opt{v}{} is only in the $U(u, v)$ definition. For the natural extension of VDI{}, let $U(u, v)$ be a set of all the possible swaps (see~(\ref{eq:swap})) in at most one dimension between the vectors $u$ and $v$, where the coordinates in at most one dimension are swapped: $$ U(u, v) = \big\{ swap(u, v, D) :\ D \subset \{ 1, 2, \ldots, s \} \text{ and } |D| \le 1 \big\} \ . $$ For the improved version of \Opt{v}{}, let $U(u, v)$ be a set of all the possible swaps in at most $\lfloor s / 2 \rfloor$ dimensions between the vectors $v$ and $w$: $$ U(u, v) = \big\{ swap(u, v, D) :\ D \subset \{ 1, 2, \ldots, s \} \text{ and } |D| \le s / 2 \big\} \ . $$ The constraint $|D| \le s / 2$ guarantees that at least half of the coordinates of every swap are equal to the first vector coordinates. The computational experiments show that removing this constraint increases the running time and decreases the average solution quality. Let vector $\mu(u, v)$ be the minimum weight swap between vectors $u$ and $v$: $$ \mu(u, v) = \argmin_{e \in U(u, v)} w(e) \ . $$ Let $A$ be an initial assignment. \begin{enumerate} \item \label{item:vopt_mainloop} For every vector $c \in A$ do the rest of the algorithm. \item \label{item:vopt_loopbegin} Initialize the \emph{total gain} $G = 0$, the \emph{best assignment} $A_\text{best} = A$, and a set of available vectors $L = A \setminus \{ c \}$. \item \label{item:vopt_search} Find vector $m \in L$ such that $w(\mu(c, m))$ is minimized. Set $v = \mu(c, m)$ and $\overline{v}_j = \{ c_j, m_j\} \setminus \{ v_j \}$ for every $1 \le j \le s$. Now $v \in U(c, m)$ is the minimum weight swap of $c$ with some other vector $m$ in the assignment, and $\overline{v}$ is the complementary vector. \item \label{item:vopt_stop_condition} Set $G = G + w(c) - w(v)$. If now $G \le 0$, set $A = A_\text{best}$ and go to the next iteration (Step~\ref{item:vopt_mainloop}). \item Mark $m$ as an unavailable for the further swaps: $L = L \setminus \{ m \}$. Note that $c$ is already marked unavailable: $c \notin L$. \item Replace $m$ and $c$ with $v$ and $\overline{v}$. Set $c = \overline{v}$. \item If $w(A) < w(A_\text{best})$, save the new assignment as the best one: $A_\text{best} = A$. \item Repeat from Step~\ref{item:vopt_search} while the total gain is positive (see Step \ref{item:vopt_stop_condition}) and $L \neq \varnothing$. \end{enumerate} The heuristic repeats until no improvement is found during a run. The time complexity of one run of \Opt{v}{} is $O(n^3 \cdot 2^{s-1})$. The time complexity of the natural extension of VDI{} is $O(n^3 \cdot s)$, and the computation experiments also show a significant difference between the running times of the improved and the natural extensions. However, the solution quality of the natural extension for $s \ge 7$ is quite poor, while for the smaller values of $s$ it produces solutions similar to or even worse than {\heuristic{\MDVPlain}}{} solutions at the cost of much larger running times. The neighborhood $N_\text{\OptPlain{v}}(A)$ is not fixed and depends on the MAP instance and initial assignment $A$. The number of iterations (runs of Step~\ref{item:vopt_search}) of the algorithm can vary from $n$ to $n^2$. Moreover, there is no guarantee that the algorithm selects a better assignment even if the corresponding swap is in $U(c, m)$. Thus, we do not provide any results for the neighborhood of \Opt{v}{}. \subsection{Combined Neighborhood} \label{sec:combined} We have already presented two types of neighborhoods in this paper, let us say \emph{dimensionwise} (Section~\ref{sec:dv}) and \emph{vectorwise} (Sections~\ref{sec:kopt} and~\ref{sec:vopt}). The idea of the combined heuristic is to use the dimensionwise and the vectorwise neighborhoods togeather, combining them into so called Variable Neighborhood Search~\cite{Talbi2009}. The combined heuristic improves the assignment by moving it into the local optimum with respect to the dimensionwise neighborhood, then it improves it by moving it to the local minimum with respect to the vectorwise neighborhood. The procedure is repeated until the assignment occurs in the local minimum with respect to both the dimensionwise and the vectorwise neighborhoods. More formally, the combined heuristic \Comb{DV}{opt} consists of a dimensionwise heuristic $DV$ (either {\heuristic{\OneDVPlain}}, {\heuristic{\TwoDVPlain}}\ or {\heuristic{\MDVPlain}}) and a vectorwise heuristic $opt$ (either \Opt{2}, \Opt{3}\ or \Opt{v}). \Comb{DV}{opt} proceeds as follows. \begin{enumerate} \item \label{item:combined_first} Apply the dimensionwise heuristic $A = DV(A)$. \item Repeat: \begin{enumerate} \item Save the assignment weight $x = w(A)$ and apply the vectorwise heuristic $A = opt(A)$. \item If $w(A) = x$ stop the algorithm. \item Save the assignment weight $x = w(A)$ and apply the dimensionwise heuristic $A = DV(A)$. \item If $w(A) = x$ stop the algorithm. \end{enumerate} \end{enumerate} Step~\ref{item:combined_first} of the combined heuristic is the hardest one. Indeed, it is typical that it takes a lot of iterations to move a bad solution to a local minimum while for a good solution it takes just a few iterations. Hence, the first of the two heuristics should be the most efficient one, i.e., it should perform quickly and produce a good solution. In this case the dimensionwise heuristics are more efficient because, having approximately the same as vectorwise heuristics time complexity, they search much larger neighborhoods. The fact that the dimensionwise heuristics are more efficient than the vectorwise ones is also confirmed by experimental evaluation (see Section~\ref{sec:experiments}). It is clear that the neighborhood of a combined heuristic is defined as follows: \begin{equation} \label{eq:combined_neighborhood} N_\text{\CombPlain{DV}{opt}}(A) = N_\text{DV}(A) \cup N_\text{opt}(A) \ , \end{equation} where $N_\text{DV}(A)$ and $N_\text{opt}(A)$ are neighborhoods of the corresponding dimensionwise and vectorwise heuristics respectively. To calculate the size of the neighborhood $N_\text{\CombPlain{DV}{opt}}(A)$ we need to find the size of the intersection of these neighborhoods. Observe that \begin{equation} \label{eq:combined_intersection} N_\text{DV}(A) \cap N_{k\text{-opt}}(A) = \big\{ p_D(A, \rho) :\ D \in \mathcal{D} \text{ and $\rho$ moves at most $k$ elements} \big\} \ , \end{equation} where $p_D(A, \rho)$ is defined by~(\ref{eq:p_D}). This means that, if $r_k$ is the number of permutations on $n$ elements which move at most $k$ elements, the intersection (\ref{eq:combined_intersection}) has size \begin{equation} \label{eq:combined_intersection_size} |N_\text{DV}(A) \cap N_{k\text{-opt}}(A)| = |\mathcal{D}| \cdot (r_k - 1) + 1 \ . \end{equation} The number $r_k$ can be calculated as \begin{equation} \label{eq:combined_r_k} r_k = \sum_{i = 0}^k \binom{n}{i} \cdot d_i \ , \end{equation} where $d_i$ is the number of derangements on $i$ elements, i.e., permutations on $i$ elements such that none of the elements appear on their places; $d_i = i! \cdot \sum_{m = 0}^i (-1)^m / m!$~\cite{Harris2008}. For $k = 2$, $r_2 = 1 + \binom{n}{2}$; for $k = 3$, $r_3 = 1 + \binom{n}{2} + 2 \binom{n}{3}$. From (\ref{eq:dv_neighborhood_size}), (\ref{eq:kopt_neighborhood_size}), (\ref{eq:combined_neighborhood}) and (\ref{eq:combined_intersection_size}) we immediately have \begin{equation} \left|N_\text{\CombPlain{DV}{\OptPlain{$k$}}}(A)\right| = 1 + |\mathcal{D}| \cdot (n! - 1) + \left[\sum_{i = 2}^k \binom{n}{i} N^i\right] - |\mathcal{D}| \cdot (r_k - 1) \ , \end{equation} where $N^i$ and $r_k$ are calculated according to (\ref{eq:kopt_n_k}) and (\ref{eq:combined_r_k}) respectively. Substituting the value of $k$, we have: \begin{equation} \label{eq:combined_neighborhood_size_2} \left|N_\text{\CombPlain{DV}{\OptPlain{2}}}(A)\right| = 1 + |\mathcal{D}| \cdot (n! - 1) + \binom{n}{2} (2^{s-1} - 1) - |\mathcal{D}| \cdot \binom{n}{2} \qquad \text{and} \end{equation} \begin{multline} \label{eq:combined_neighborhood_size_3} \left|N_\text{\CombPlain{DV}{\OptPlain{3}}}(A)\right| = 1 + |\mathcal{D}| \cdot (n! - 1) + \binom{n}{2} (2^{s-1} - 1) \\ + \binom{n}{3} (6^{s-1} - 3 \cdot 2^{s-1} + 2) - |\mathcal{D}| \cdot \left[\binom{n}{2} + 2 \binom{n}{3}\right] \end{multline} One can easily substitute $|\mathcal{D}| = s$, $|\mathcal{D}| = \binom{s}{2}$ and $|\mathcal{D}| = 2^{s-1} - 1$ to (\ref{eq:combined_neighborhood_size_2}) or (\ref{eq:combined_neighborhood_size_3}) to get the neighborhood sizes of \heuristic{\OneDVtwoPlain}, \heuristic{\TwoDVtwoPlain}, \heuristic{\MDVtwoPlain}, \heuristic{\OneDVthreePlain}, \heuristic{\TwoDVthreePlain}\ and \heuristic{\MDVthreePlain}. We will only show the results for \heuristic{\MDVtwoPlain}: \begin{multline} \left|N_\text{\CombPlain{\MDVPlain}{2}}(A)\right| = 1 + (2^{s-1} - 1) \cdot (n! - 1) + \binom{n}{2} (2^{s-1} - 1) - (2^{s-1} - 1) \cdot \binom{n}{2} \\ = 1 + (2^{s-1} - 1) \cdot (n! - 1) \ , \end{multline} i.e., $\left|N_\text{\CombPlain{\MDVPlain}{2}}(A)\right| = \left|N_\text{$s$DV}(A)\right|$\@. Since $N_\text{$s$DV}(A) \subseteq N_\text{\CombPlain{\MDVPlain}{2}}(A)$ (see~(\ref{eq:combined_neighborhood})), we can conclude that $N_\text{\CombPlain{\MDVPlain}{2}}(A) = N_\text{$s$DV}(A)$. Indeed, the neighborhood of \Opt{2}\ can be defined as follows: $$ N_\text{\OptPlain{2}} = \big\{ p_D(A, \rho) :\ D \subset \{ 2, 3, \ldots, s \} \text{ and } \rho \text{ swaps at most two elements} \big\} \ , $$ which is obviously a subset of $N_\text{$s$DV}(A)$ (see (\ref{eq:dv_neighborhood})). Hence, the combined heuristic \heuristic{\MDVtwoPlain}{} is of no interest. For other combinations the intersection (\ref{eq:combined_intersection}) is significantly smaller than both neighborhoods $N_\text{DV}(A)$ and $N_{\OptPlain{$k$}}(A)$ (recall that the neighborhood $N_\text{\OptPlain{v}}$ has a variable structure). Indeed, $|N_\text{DV}(A)| \gg |N_\text{DV}(A) \cap N_{k\text{-opt}}(A)|$ because $|\mathcal{D}| \cdot (n! - 1) \gg |\mathcal{D}| \cdot (r_k - 1)$ for $k \ll n$. Similarly, $|N_\text{\OptPlain{2}}(A)| \gg |N_\text{DV}(A) \cap N_{k\text{-opt}}(A)|$ because $\binom{n}{2} (2^{s-1} - 1) \gg |\mathcal{D}| \cdot \binom{n}{2}$ if $|\mathcal{D}| \ll 2^{s-1}$, which is the case for {\heuristic{\OneDVPlain}}\ and {\heuristic{\TwoDVPlain}}\ if $s$ is large enough. Finally, $|N_\text{\OptPlain{3}}(A)| \gg |N_\text{DV}(A) \cap N_{k\text{-opt}}(A)|$ because $\binom{n}{2} (2^{s-1} - 1) + \binom{n}{3} (6^{s-1} - 3 \cdot 2^{s-1} + 2) \gg |\mathcal{D}| \cdot \left[ \binom{n}{2} + 2 \binom{n}{3} \right]$, which is true even for $|\mathcal{D}| = 2^{s-1}$, i.e., for {\heuristic{\MDVPlain}}. The time complexity of the combined heuristic is $O(n^k \cdot k!^{s-1} + |\mathcal{D}| \cdot n^3)$ in case of $opt = \text{\OptPlain{$k$}}$ and $O(n^3 \cdot (2^{s-1} + |\mathcal{D}|))$ if $opt = \text{\OptPlain{v}}$. The particular formulas are provided in the following table: \begin{center} \begin{tabular}{llll} \toprule & \OptPlain{2} & \OptPlain{3} & \OptPlain{v} \\ \cmidrule(){1-4} 1DV & $O(2^{s-1} \cdot n^2 + s \cdot n^3)$ & $O(6^{s-1} \cdot n^3)$ & $O(2^s \cdot n^3)$ \\ 2DV & $O(2^{s-1} \cdot n^2 + s^2 \cdot n^3)$ & $O(6^{s-1} \cdot n^3)$ & $O(2^s \cdot n^3)$ \\ $s$DV & (no interest) & $O(6^{s-1} \cdot n^3)$ & $O(2^s \cdot n^3)$ \\ \bottomrule \end{tabular} \end{center} Note that all the combinations with \Opt{3}\ and \Opt{v}\ have equal time complexities; this is because the time complexities of \Opt{3}\ and \Opt{v}\ are dominant. Our experiments show that the actual running times of \Opt{3}\ and \Opt{v}\ are really much higher then even the {\heuristic{\MDVPlain}}\ running time. This means that the combinations of these heuristics with {\heuristic{\MDVPlain}}{} are approximately as fast as the combinations of these heuristics with light dimensionwise heuristics {\heuristic{\OneDVPlain}}\ and {\heuristic{\TwoDVPlain}}\@. Moreover, as it was noticed above in this section, the dimensionwise heuristic, being executed first, simplifies the job for the vectorwise heuristic and, hence, the increase of the dimensionwise heuristic power may decrease the running time of the whole combined heuristic. At the same time, the neighborhoods of the combinations with {\heuristic{\MDVPlain}}\ are significantly larger than the neighborhoods of the combinations with {\heuristic{\OneDVPlain}}\ and {\heuristic{\TwoDVPlain}}\@. We can conclude that the `light' heuristics \heuristic{\OneDVthreePlain}, \heuristic{\TwoDVthreePlain}, \heuristic{\OneDVVPlain}\ and \heuristic{\TwoDVVPlain}\ are of no interest because the `heavy' heuristics \heuristic{\MDVthreePlain}\ and \heuristic{\MDVVPlain}, having the same theoretical time complexity, are more powerful and, moreover, outperformed the `light' heuristics in our experiments with respect to both solution quality and running time on average and in most of single experiments. \subsection{Other algorithms} Here we provide a list of some other MAP algorithms presented in the literature. \begin{itemize} \item A host of local search procedures and construction heuristics which often have some approximation guarantee (\cite{Bandelt2004,Burkard1996,Crama1992,Isler2005,Kuroki2007,Murphey1998} and some others) are proposed for special cases of MAP (usually with decomposable weights, see Section~\ref{sec:decomposable}) and exploit the specifics of these instances. However, as it was stated in Section~\ref{sec:introduction}, we consider only the general case of MAP, i.e., all the algorithms included in this paper do not rely on any special structure of the weight matrix. \item A number of construction heuristics are intended to generate solutions for general case MAP~\cite{Balas1991,Gutin2008,GK_MAP_Construction,Oliveira2004}. While some of them are fast and low quality, like {\heuristic{Greedy}}, some, like {\heuristic{Max-Regret}}, are significantly slower but produce much better solutions. A special class of construction heuristics, Greedy Randomized Adaptive Search Procedure (GRASP), was also investigated by many researchers~\cite{Aiex2005,Murphey1998,Oliveira2004,Robertson2001}. \item Several metaheuristics, including a simulated annealing procedure~\cite{Clemons2004} and a memetic algorithm~\cite{Huang2006}, were proposed in the literature. Metaheuristics are sophisticated algorithms intended to search for the near optimal solutions in a reasonably large time. Proceeding for much longer than local search and being hard for theoretical analysis of the running time or the neighborhood, metaheuristics cannot be compared straightforwardly to local search procedures. \item Some weak variations of \Opt{2}\ are considered in~\cite{Aiex2005,Murphey1998,Pasiliao2005,Robertson2001}. While our heuristic \Opt{2}\ tries all possible recombinations of a pair of assignment vectors, i.e., $2^{s-1}$ combinations, these variations only try the swaps in one dimension at a time, i.e., $s$ combinations for every pair of vectors. We have already decided that these variations have no practical interest, for details see Section~\ref{sec:kopt}. \end{itemize} \section{Test Bed} \label{sec:testbed} While the theoretical analysis can help in heuristic design, selection of the best approaches requires empirical evaluation~\cite{GK_Some_Theory,Rardin2001}. In this section we discuss the test bed and in Section~\ref{sec:experiments} the experimental results are reported and discussed. The question of selecting proper test bed is one of the most important questions in heuristic experimental evaluation~\cite{Rardin2001}. While many researchers focused on instances with random independent weights (\cite{Andrijich2001,Balas1991,Krokhmal2007,Pasiliao2005} and some others) and random instances with predefined solutions~\cite{Clemons2004,Grundel2005,GK_MAP_Construction}, several more sophisticated models are of greater practical interest~\cite{Bandelt2004,Burkard1996,Crama1992,Frieze1981,Kuroki2007}. There is also a number of papers which consider real-world and pseudo real-world instances~\cite{Bekker2005,Murphey1998,Pardalos2000b} but the authors of this paper suppose that these instances do not well represent all the instance classes and building a proper benchmark with the real-world instances is a subject for another research. In this paper we group all the instance families into two classes: instances with independent weights (Section~\ref{sec:independent}) and instances with decomposable weights (Section~\ref{sec:decomposable}). Later we show that the heuristics perform differently on the instances of these classes and, thus, this devision helps us in correct experimental analysis of the local search algorithms. \subsection{Instances With Independent Weights} \label{sec:independent} One of the most studied class of instances for MAP is \emph{Random Instance Family}. In {\ifamily{Random}}{}, the weight assigned to a vector is a random uniformly distributed integral value in the interval $[a, b - 1]$. {\ifamily{Random}}{} instances were used in~\cite{Aiex2005,Andrijich2001,Balas1991,Pierskalla1968} and some others. Since the instances are random and quite large, it is possible to estimate the average solution value for the Random Instance Family. The previous research in this area~\cite{Krokhmal2007} show that if $n$ tends to infinity than the problem solution approaches the bound $an$, i.e., the minimal possible assignment weight (observe that the minimal assignment includes $n$ vectors of weight $a$). Moreover, an estimation of the mean optimal solution is provided in~\cite{Grundel2004} but this estimation is not accurate enough for our experiments. In~\cite{GK_Some_Theory} we prove that it is very likely that every big enough {\ifamily{Random}}{} instance has at least one $an$-assignment, where \emph{$x$-assignment} means an assignment of weight $x$. Let $\alpha$ be the number of assignments of weight $an$ and let $c = b - a$. We would like to have an upper bound on the probability $\Pr(\alpha=0)$. Such an upper bound is given in the following theorem whose proof (see~\cite{GK_Some_Theory}) is based on the Extended Jansen Inequality (see Theorem 8.1.2 of~\cite{Alon2000}). \begin{theorem} \label{th:pr} For any $n$ such that $n\ge 3$ and \begin{equation} \label{eq:cbound} \left( \frac{n-1}{e} \right)^{s-1} \ge c \cdot 2^{\frac{1}{n-1}}, \end{equation} we have $\Pr(\alpha = 0) \le e^{-\frac{1}{2 \sigma}}$, where $\sigma = \sum\limits_{k = 1}^{n - 2} \frac{\binom{n}{k} \cdot c^k}{\left[n \cdot (n - 1) \cdots (n - k + 1) \right]^{s - 1}}.$ \end{theorem} The lower bounds of $\Pr(\alpha > 0)$ for different values of $s$ and $n$ and for $b - a = 100$, are reported below. \begin{center} \noindent\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}} *5{c}} \toprule{} $s = 4$ & $s = 5$ & $s = 6$ & $s = 7$ \\ \cmidrule(){1-4} \begin{tabular}[t]{r @{\quad} l} $n$ & $\Pr(\alpha > 0)$ \\ \cmidrule(){1-2} 15 & 0.575 \\ 20 & 0.823 \\ 25 & 0.943 \\ 30 & 0.986 \\ 35 & 0.997 \\ 40 & 1.000 \\ \end{tabular} & \begin{tabular}[t]{r @{\quad} l} $n$ & $\Pr(\alpha > 0)$ \\ \cmidrule(){1-2} 10 & 0.991 \\ 11 & 0.998 \\ 12 & 1.000 \\ \end{tabular} & \begin{tabular}[t]{r @{\quad} l} $n$ & $\Pr(\alpha > 0)$ \\ \cmidrule(){1-2} 8 & 1.000 \\ \end{tabular} & \begin{tabular}[t]{r @{\quad} l} $n$ & $\Pr(\alpha > 0)$ \\ \cmidrule(){1-2} 7 & 1.000 \\ \end{tabular} \\ \bottomrule{} \end{tabular*} \end{center} One can see that a 4-AP {\ifamily{Random}}{} instance has an $(an)$-assignment with the probability which is very close to 1 if $n \ge 40$; a 5-AP instance has an $(an)$-assignment with probability very close to 1 for $n \ge 12$, etc.; so, the optimal solutions for all the {\ifamily{Random}}{} instances used in our experiments (see Section~\ref{sec:experiments}) are very likely to be of weight $an$. For $s = 3$ Theorem~\ref{th:pr} does not provide a good upper bound, but we are able to use the results from Table~II in~\cite{Balas1991} instead. Balas and Saltzman report that in their experiments the average optimal solution of 3-AP for {\ifamily{Random}}{} instances reduces very quickly and has a small value even for $n = 26$. Since the smallest {\ifamily{Random}}{} instance we use in our experiments has size $n = 150$, we assume that all 3-AP {\ifamily{Random}}{} instances considered in our experiment are very likely to have an $an$-assignment. Useful results can also be obtained from (11) in~\cite{Grundel2004} which is an upper bound for the average optimal solution. Grundel, Oliveira and Pardalos~\cite{Grundel2004} consider the same instance family except the weights of the vectors are real numbers uniformly distributed in the interval $[a, b]$. However the results from~\cite{Grundel2004} can be extended to our discrete case. Let $w'(e)$ be a real weight of the vector $e$ in a continuous instance. Consider a discrete instance with $w(e) = \lfloor w'(e) \rfloor$ (if $w'(e) = b$, set $w(e) = b - 1$). Note that the weight $w(e)$ is a uniformly distributed integer in the interval $[a, b - 1]$. The optimal assignment weight of this instance is not larger than the optimal assignment weight of the continuous instance and, thus, the upper bound for the average optimal solution for the discrete case is correct. In fact, the upper bound $\bar{z}^*_u$ (see~\cite{Grundel2004}) for the average optimal solution is not accurate enough. For example, $\bar{z}^*_u \approx an + 6.9$ for $s = 3$, $n = 100$ and $b - a = 100$, and $\bar{z}^*_u \approx an + 3.6$ for $s = 3$, $n = 200$ and $b - a = 100$, so it cannot be used for $s = 3$ in our experiments. The upper bound $\bar{z}^*_u$ gives a better approximation for larger values of $s$, e.g., $\bar{z}^*_u \approx an + 1.0$ for $s = 4$, $n = 40$ and $b - a = 100$, however, Theorem~\ref{th:pr} provides stronger results ($\Pr(\alpha > 0) \approx 1.000$ for this case). \bigskip Another class of instances with almost independent weights is \emph{GP Instance Family} which contains pseudo-random instances with predefined optimal solutions. {\ifamily{GP}}{} instances are generated by an algorithm produced by Grundel and Pardalos~\cite{Grundel2005}. The generator is naturally designed for $s$-AP for arbitrary large values of $s$ and $n$. However, it is relatively slow and, thus, it was impossible to generate large {\ifamily{GP}}{} instances. Nevertheless, this is what we need since finally we have both small ({\ifamily{GP}}) and large ({\ifamily{Random}}) instances with independent weights with known optimal solutions. \subsection{Instances With Decomposable Weights} \label{sec:decomposable} In many cases it is not easy to define a weight for an $s$-tuple of objects but it is possible to define a relation between every pair of objects from different sets. In this case one should use \emph{decomposable weights}~\cite{Spieksma2000} (or \emph{decomposable costs}), i.e., the weight of a vector $e$ should be defined as follows: \begin{equation} \label{eq:decomposable_w} w(e) = f\left(d^{1, 2}_{e_1, e_2}, d^{1, 3}_{e_1, e_3}, \ldots, d^{s-1,s}_{e_{s-1}, e_s}\right) \ , \end{equation} where $d^{i,j}$ is a distance matrix between the sets $X_i$ and $X_j$ and $f$ is some function. The most natural instance family with decomposable weights is {\ifamily{Clique}}, which defines the function $f$ as the sum of all arguments: \begin{equation} \label{eq:clique_w} w_\text{c}(e) = \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} d^{i,j}_{e_i, e_j} \ . \end{equation} The {\ifamily{Clique}}\ instance family was investigated in~\cite{Bandelt2004,Crama1992,Frieze1981} and some others. It was proven~\cite{Crama1992} that MAP restricted to {\ifamily{Clique}}\ instances remains NP-hard. A special case of {\ifamily{Clique}}\ is \emph{Geometric Instance Family}. In {\ifamily{Geometric}}, the sets $X_1$, $X_2$, \ldots, $X_s$ correspond to sets of points in Euclidean space, and the distance between two points $u \in X_i$ and $v \in X_j$ is defined as Euclidean distance; we consider the two dimensional Euclidean space: $$ d_\text{g}(u, v) = \sqrt{(u_x - v_x)^2 + (u_y - v_y)^2} \ . $$ It is proven~\cite{Spieksma1996} that the {\ifamily{Geometric}}\ instances are NP-hard to solve for $s = 3$ and, thus, {\ifamily{Geometric}}\ is NP-hard for every $s \ge 3$. In this paper, we propose a new special case of the decomposable weights, {\ifamily{SquareRoot}}. It is a modification of the {\ifamily{Clique}}\ instance family. Assume we have $s$ radars and $n$ planes and each radar observes all the planes. The problem is to assign signals which come from different radars to each other. It is quite natural to define a distance function between each pair of signals from different radars, and for a set of signals which correspond to one plane the sum of the distances should be small so (\ref{eq:clique_w}) is a good choice. However, it is not actually correct to minimize the total distance between the assigning signals but one should also ensure that none of these distances is too large. Same requirements appear in a number of other applications. We propose a weight function which can leads to both small total distance between the assigned signals and small dispersion of the distances: \begin{equation} w_\text{sq}(e) = \sqrt{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \left(d^{i,j}_{e_i, e_j}\right)^2 } \ . \end{equation} Similar approach is used in~\cite{Kuroki2007} though they do not use square root, i.e., a vector weight is just a sum of squares of the edge weights in a clique. In addition, the edge weights in~\cite{Kuroki2007} are calculated as distances between some nodes in a Euclidean space. Another special case of the decomposable weights, {\ifamily{Product}}, is studied in~\cite{Burkard1996}. Burkard et al. consider 3-AP and define the weight $w(e)$ as $w(e) = a^1_{e_1} \cdot a^2_{e_2} \cdot a^3_{e_3}$, where $a^1$, $a^2$ and $a^3$ are random vectors of positive numbers. It is easy to show that the {\ifamily{Product}}\ weight function can be represented in the form~(\ref{eq:decomposable_w}). It is proven that the minimization problem for the {\ifamily{Product}}\ instances is NP-hard in case $s = 3$ and, thus, it is NP-hard for every $s \ge 3$. \section{Computational Experimentation} \label{sec:experiments} In this section, the results of empirical evaluation are reported and discussed. The experiments were conducted for the following instances (for instance family definitions see Section~\ref{sec:testbed}): \begin{itemize} \item {\ifamily{Random}}{} instances where each weight was randomly chosen in $\{ 1, 2, \ldots, 100 \}$, i.e., $a = 1$ and $b = 101$. According to Subsection~\ref{sec:independent}, the optimal solutions of all the considered {\ifamily{Random}}\ instances are very likely to be $an = n$. \item {\ifamily{GP}}{} instances with predefined optimal solutions (see Section~\ref{sec:independent}). \item {\ifamily{Clique}}\ and {\ifamily{SquareRoot}}\ instances, where the weight of each edge in the graph was randomly selected from $\{ 1, 2, \ldots, 100 \}$. Instead of the optimal solution value we use the best known solution value. \item {\ifamily{Geometric}}\ instances, where both coordinates of every point were randomly selected from $\{ 1, 2, \ldots, 100 \}$. The distances between the points are calculated precisely while the weight of a vector is rounded to the nearest integer. Instead of the optimal solution value we use the best known solution value. \item {\ifamily{Product}}\ instances, where every value $a_i^j$ was randomly selected from $\{ 1, 2, \ldots, 10 \}$. Instead of the optimal solution value we use the best known solution value. \end{itemize} An instance name consists of three parts: the number $s$ of dimensions, the type of the instance (`gp' for {\ifamily{GP}}{}, `r' for {\ifamily{Random}}{}, `c' for {\ifamily{Clique}}{}, `g' for {\ifamily{Geometric}}, `p' for {\ifamily{Product}}\ and 'sr' for {\ifamily{SquareRoot}}), and the size $n$ of the instance. For example, \texttt{5r40} means a five dimensional {\ifamily{Random}}\ instance of size 40. For every combination of instance size and type we generated 10 instances, using the number $seed = s + n + i$ as a seed of the random number sequences, where $i$ is an index of the instance of this type and size, $i \in \{ 1, 2, \ldots, 10 \}$. Thereby, every experiment is conducted for 10 different instances of some fixed type and size, i.e., every number reported in the tables below is average for 10 runs for 10 different instances. The sizes of all but {\ifamily{GP}}\ instances are selected such that every algorithm could process them all in approximately the same time. The {\ifamily{GP}}\ instances are included in order to examine the behavior of the heuristics on smaller instances (recall that {\ifamily{GP}}\ is the only instance set for which we know the exact solutions for small instances). All the heuristics are implemented in Visual C++. The evaluation platform is based on AMD Athlon 64 X2 3.0~GHz processor. Further, the results of the experiments of three different types are provided and discussed: \begin{itemize} \item In Subsection~\ref{sec:pure}, the local search heuristics are applied to the assignments generated by some construction heuristic. These experiments allow us to exclude several local searches from the rest of the experiments, however, the comparison of the results is complicated because of the significant difference in both the solution quality and the running time. \item In Subsection~\ref{sec:metaheuristics}, two simple metaheuristics are used to equate the running times of different heuristics. This is done by varying of number of iterations of the metaheuristics. \item In Subsection~\ref{sec:heuristics_comparison}, the results of all the discussed approaches are gathered in two tables to find the most successful solvers for the instance with independent and decomposable weights for every particular running time. \end{itemize} \subsection{Pure Local Search Experiments} \label{sec:pure} First, we run every local search heuristic for every instance exactly once. The local search is applied to solutions generated with one of the following construction heuristics: \begin{enumerate} \item {\heuristic{Trivial}}, which was first mentioned in~\cite{Balas1991} as \emph{Diagonal}. {\heuristic{Trivial}}\ construction heuristic simply assigns $A_j^i = i$ for every $i = 1, 2, \ldots, n$ and $j = 1, 2, \ldots, s$. \item {\heuristic{Greedy}}\ heuristic was discussed in many papers, see, e.g.~\cite{Balas1991,Burkard1996,Gutin2008,GK_Greedy_Chapter,GK_Some_Theory,GK_MAP_Construction}. It was proven~\cite{Gutin2008} that in the worst case {\heuristic{Greedy}}\ produces the unique worst solution; however, it was shown~\cite{GK_Greedy_Chapter} that in some cases {\heuristic{Greedy}}\ may be a good selection as a fast and simple heuristic. \item {\heuristic{Max-Regret}}\ was discussed in a number of papers, see, e.g.,~\cite{Balas1991,Burkard1996,Gutin2008,GK_MAP_Construction,Robertson2001}. As for {\heuristic{Greedy}}, it is proven~\cite{Gutin2008} that in the worst case {\heuristic{Max-Regret}}\ produces the unique worst solution however many researchers~\cite{Balas1991,GK_MAP_Construction} noted that {\heuristic{Max-Regret}}\ is quite powerful in practice. \item {\heuristic{ROM}}\ was first introduced in~\cite{Gutin2008} as a heuristic of a large domination number. On every iteration, the heuristic calculates the total weight for every set of vectors with the fixed first two coordinates: $M_{i,j} = \sum_{e \in X, e_1 = i, e_2 = j} w(e)$. Then it solves a 2-AP for the weight matrix $M$ and reorders the second dimension of the assignment according to this solution and the first dimension of the assignment. The procedure is repeated recursively for the subproblem where the first dimension is excluded. For details see~\cite{Gutin2008,GK_MAP_Construction}. \end{enumerate} We will begin our discussion from the experiments started from trivial assignments. The results reported in Tables~\ref{tab:ls_solutions_from_trivial} and~\ref{tab:ls_times_from_trivial} are averages for 10 experiments since every row of these tables corresponds to 10 instances of some fixed type and size but of different seed values (see above). The tables are split into two parts; the first part contains only the instances with independent weights ({\ifamily{GP}}\ and {\ifamily{Random}}) while the second part contains only the instances with decomposable weights ({\ifamily{Clique}}, {\ifamily{Geometric}}, {\ifamily{Product}}\ and {\ifamily{SquareRoot}}). The average values for different instance families and numbers of dimensions are provided at the bottom of each part of each table. The tables are also split vertically according to the classes of heuristics. The winner in every row and every class of heuristics is underlined. The value of the solution error is calculated as $\big(w(A) / w(A_\text{best}) - 1\big) \cdot 100\%$, where $A$ is the obtained assignment and $A_\text{best}$ is the optimal assignment (or the best known one, see above). \bigskip In the group of the vectorwise heuristics the most powerful one is definitely \Opt{3}. \Opt{v}{} outperforms it only in a few experiments, mostly three dimensional ones (recall that the neighborhood of \Opt{$k$}\ increases exponentially with the increase of the number of dimensions $s$). As it was expected, \Opt{2}\ never outperforms \Opt{3}\ since $N_\text{2-opt} \subset N_\text{3-opt}$ (see Section~\ref{sec:kopt}). The tendencies for the independent weight instances and for the decomposable weight instances are similar; the only difference which is worth to note is that all but \Opt{v}\ heuristics of this group solve the {\ifamily{Product}}\ instances very well. Note that the dispersion of the weights in {\ifamily{Product}}\ instances is really high and, thus, \Opt{v}, which minimizes the weight of only one vector in every pair of vectors while the weight of the complementary vector may increase arbitrary, cannot be efficient for them. As one can expect, {\heuristic{\MDVPlain}}\ is more successful than {\heuristic{\TwoDVPlain}}\ and {\heuristic{\TwoDVPlain}}\ is more successful than {\heuristic{\OneDVPlain}}\ with respect to the solution quality (obviously, all the heuristics of this group perform equally for 3-AP and {\heuristic{\TwoDVPlain}}\ and {\heuristic{\MDVPlain}}\ are also equal for 4-AP, see Section~\ref{sec:dv}). However, for the instances with decomposable weights all the dimensionwise heuristics perform very similarly and even for the large $s$, {\heuristic{\MDVPlain}}\ is not significantly more powerful than {\heuristic{\OneDVPlain}}\ or {\heuristic{\TwoDVPlain}}\ which means that in case of decomposable instances the most efficient iterations are when $|D| = 1$. We can assume that if $c$ is the number of edges connecting the fixed and unfixed parts of the clique, then an iteration of a dimensionwise heuristic is rather efficient when $c$ is small. Observe that, e.g., for {\ifamily{Clique}}\ the diversity of values in the weight matrix $[M_{i, j}]_{n \times n}$ (see~(\ref{eq:M})) decreases with the increase of the number $c$ and, hence, the space for optimization on every iteration is decreasing. Observe also that in the case $c = 1$ the iteration leads to the optimal match between the fixed and unfixed parts of the assignment vectors. All the combined heuristics show improvements in the solution quality over each of their components, i.e., over both corresponding vectorswise and dimensionwise local searches. In particular, \heuristic{\OneDVtwoPlain}\ outperforms both \Opt{2}\ and {\heuristic{\OneDVPlain}}, \heuristic{\TwoDVtwoPlain}\ outperforms both \Opt{2}\ and {\heuristic{\TwoDVPlain}}, \heuristic{\MDVthreePlain}\ outperforms both \Opt{3}\ and {\heuristic{\MDVPlain}}\ and \heuristic{\MDVVPlain}\ outperforms both \Opt{v}\ and {\heuristic{\MDVPlain}}\@. Moreover, \heuristic{\MDVthreePlain}\ is significantly faster than \Opt{3}\ and \heuristic{\MDVVPlain}\ is significantly faster than \Opt{v}. Hence, we will not discuss the single heuristics \Opt{3}\ and \Opt{v}\ in the rest of the paper. The heuristics \heuristic{\OneDVtwoPlain}\ and \heuristic{\TwoDVtwoPlain}, obviously, perform equally for 3-AP instances. While for the instances with independent weights the combination of the dimensionwise heuristics with the vectorwise ones significantly improves the solution quality, it is not the case for the instances with decomposable weights (observe that {\heuristic{\OneDVPlain}}\ performs almost as well as the most powerful heuristic \heuristic{\MDVthreePlain}) which shows the importance of the instances division. We conclude that the vectorwise neighborhoods are not efficient for the instances with decomposable weights. \bigskip Next we conducted the experiments starting from the other construction heuristics. But first we compared the construction heuristics themselves, see Table~\ref{tab:construction}. It is not surprising that {\heuristic{Trivial}}\ produces the worst solutions. However, one can see that {\heuristic{Trivial}}\ outperforms {\heuristic{Greedy}}\ and {\heuristic{Max-Regret}}\ for every {\ifamily{Product}}\ instance. The reason is in the extremely high dispersion of the weights in {\ifamily{Product}}. Both {\heuristic{Greedy}}\ and {\heuristic{Max-Regret}}\ construct the assignments by adding new vectors to it. The decision which vector should be added does not depend (or does not depend enough in case of {\heuristic{Max-Regret}}) on the rest of the vectors and, thus, at the end of the procedure only the vectors with huge weights are available. For other instance families, {\heuristic{Greedy}}, {\heuristic{Max-Regret}}\ and {\heuristic{ROM}}\ perform similarly though the running time of the heuristics is very different. {\heuristic{Max-Regret}}\ is definitely the slowest construction heuristic; {\heuristic{Greedy}}\ is very fast for the {\ifamily{Random}}\ instances (this is because of the large number of vectors of the weight $a$ and the implementation features, see~\cite{GK_MAP_Construction} for details) and relatively slow for the rest of the instances; {\heuristic{ROM}}'s running time almost does not depend on the instance and is constantly moderate. Starting from {\heuristic{Greedy}}\ (Table~\ref{tab:greedy}) significantly improves the solution quality. This mostly influenced the weakest heuristics, e.g., \Opt{2}\ average error decreased in our experiments from 59\% and 20\% to 15\% and 6\% for independent and decomposable weights respectively, though, e.g., the most powerful heuristic \heuristic{\MDVthreePlain}\ error also noticeably decreased (from 2.8\% and 5.8\% to 2.0\% and 2.5\%). As regards the running time, {\heuristic{Greedy}}\ is slower than most of the local search heuristics and, thus, the running times of all but \heuristic{\MDVthreePlain}\ and \heuristic{\MDVVPlain}\ heuristics are very similar. The best of the rest of the heuristics in this experiment is {\heuristic{\MDVPlain}}\ though \heuristic{\OneDVtwoPlain}\ and \heuristic{\TwoDVtwoPlain}\ perform similarly. Starting from {\heuristic{Max-Regret}}\ improves the solution quality even more but at the cost of very large running times. In this case the difference in the running time of the local search heuristics almost disappears and \heuristic{\MDVthreePlain}, the best one, reaches the average error values 1.3\% and 2.2\% for independent and decomposable weights respectively. Starting from {\heuristic{ROM}}\ improves the quality only for the worst heuristics. This is probably because all the best heuristics contain {\heuristic{\MDVPlain}}\ which does a good vectorwise optimization (recall that {\heuristic{ROM}}\ exploits a similar to the dimensionwise neighborhood idea). At the same time, starting from {\heuristic{ROM}}\ increases the running time of the heuristics significantly; the results for both {\heuristic{Max-Regret}}\ and {\heuristic{ROM}}\ are excluded from the paper; one can find them on the web~\cite{Karapetyan}. It is clear that the construction heuristics are quite slow comparing to the local search and we should answer the following question: is it worth to spend so much time on the initial solution construction or there is some way to apply local search several times in order to improve the assignments iteratively? It is known that the algorithms which apply local search several times are called metaheuristics. There is a number of different metaheuristic approaches such as tabu search or memetic algorithms, but this is not the subject of this paper. In what follows, we are going to use two simple metaheuristics, {\heuristic{Chain}}\ and {\heuristic{Multichain}}. \subsection{Experiments With Metaheuristics} \label{sec:metaheuristics} It is obvious that there is no sense in applying a local search procedure to one solution several times because the local search moves the solution to a local minimum with respect to its neighborhood, i.e., the second exploration of this neighborhood is useless. In order to apply the local search several times, one should perturb the solution obtained on the previous iteration. This idea immediately brings us to the first metaheuristic, let us say {\heuristic{Chain}}: \begin{enumerate} \item Initialize an assignment $A$; \item Set $A_\text{best} = A$; \item Repeat: \begin{enumerate} \item Apply local search $A = LS(A)$; \item If $w(A) < w(A_\text{best})$ set $A_\text{best} = A$; \item Perturb the assignment $A = Perturb(A)$. \end{enumerate} \end{enumerate} In this algorithm we use two subroutines, $LS(A)$ and $Perturb(A)$. The first one is some local search procedure and the second one is an algorithm which removes the given assignment from the local minimum by a random perturbation of it. The perturbation should be strong enough such that the assignment will not come back to the previous position on the next iteration every time though it should not be too strong such that the results of the previous search would be totally destroyed. Our perturbation procedure selects $p = \lceil n / 25 \rceil + 1$ vectors in the assignment and perturbs them randomly. In other words, $Perturb(A)$ is just a random move of the \Opt{p} heuristic. The parameters of the procedure are obtained empirically. One can doubt if {\heuristic{Chain}}\ is good enough for large running times and, thus, we introduce a little bit more sophisticated metaheuristic, {\heuristic{Multichain}}. Unlike {\heuristic{Chain}}, {\heuristic{Multichain}}\ maintains several assignments on every iteration: \begin{enumerate} \item Initialize assignment $A_\text{best}$; \item Set $P = \varnothing$ and repeat the following $c (c + 1) / 2$ times:\\ $P = P \cup \{ LS(Perturb(A_\text{best})) \}$\\ (recall that $Perturb(A)$ produces a different assignment every time); \item Repeat: \begin{enumerate} \item Save the best $c$ assignments from $P$ into $C_1, C_2, \ldots, C_c$ such that $w(C_i) \le w(C_{i+1})$; \item If $w(C_1) < w(C_\text{best})$ set $A_\text{best} = C_1$. \item Set $P = \varnothing$ and for every $i = 1, 2, \ldots, c$ repeat the following $c - i + 1$ times: $P = P \cup \{ LS(Perturb(C_i)) \}$. \end{enumerate} \end{enumerate} The parameter $c$ is responsible for the power of {\heuristic{Multichain}}; we use $c = 5$ and, thus, the algorithm performs $c (c + 1) / 2 = 15$ local searches on every iteration. \bigskip The results of the experiments with {\heuristic{Chain}}\ running for 5 and 10 seconds are provided in Tables~\ref{tab:chain5} and~\ref{tab:chain10} respectively. The experiments are repeated for three construction heuristics, {\heuristic{Trivial}}, {\heuristic{Greedy}}\ and {\heuristic{ROM}}\@. It was not possible to include {\heuristic{Max-Regret}}\ in the comparison because it takes much more than 10 seconds for some of the instances. The diversity in solution quality of the heuristics decreased with the usage of a metaheuristic. This is because the fast heuristics are able to repeat more times than the slow ones. Note that \heuristic{\MDVthreePlain}, which is the most powerful single heuristic, is now outperformed by other heuristics. The most successful heuristics for the instances with independent and decomposable weights are \heuristic{\MDVVPlain}\ and {\heuristic{\OneDVPlain}}\ respectfully, though \heuristic{\OneDVtwoPlain}\ and \heuristic{\TwoDVtwoPlain}\ are slightly more successful than \heuristic{\MDVVPlain}\ for the {\ifamily{GP}}\ instances. This result also holds for {\heuristic{Multichain}}, see Tables~\ref{tab:multichain5} and~\ref{tab:multichain10}. The success of {\heuristic{\OneDVPlain}}\ confirms again that a dimensionwise heuristic is most successful when $|D| = 1$ if the weights are decomposable and that it is more efficient to repeat these iterations many times rather than try $|D| > 1$. For the explanation of this phenomenon see Section~\ref{sec:pure}. The success of \heuristic{\OneDVtwoPlain}\ and \heuristic{\TwoDVtwoPlain}\ for {\ifamily{GP}}\ means existence of a certain structure in the weight matrices of these instances. One can see that the initialization of the assignment is not crucial for the final solution quality. However, using {\heuristic{Greedy}}\ instead of {\heuristic{Trivial}}\ clearly improves the solutions for almost every instance and local search heuristic. In contrast to {\heuristic{Greedy}}, using of {\heuristic{ROM}}\ usually does not improve the solution quality. It only influences \Opt{2}\ which is the only pure vectorwise local search in the comparison (recall that {\heuristic{ROM}}\ has a dimensionwise structure and, thus, it is good in combination with vectorwise heuristics). The {\heuristic{Multichain}}\ metaheuristic, given the same time, obtains better results than {\heuristic{Chain}}. However, {\heuristic{Multichain}}\ fails for some combinations of slow local search and hard instance because it is not able to complete even the first iteration in the given time. {\heuristic{Chain}}, having much easier iterations, do not have this disadvantage. Giving more time to a metaheuristic also improves the solution quality. Therefore, one is able to obtain high quality solutions using metaheuristics with large running times. \subsection{Solvers Comparison} \label{sec:heuristics_comparison} To compare all the heuristics and metaheuristics discussed in this paper we produced Tables~\ref{tab:heuristics_independent} and~\ref{tab:heuristics_decomposable}. These tables indicate which heuristics should be chosen to solve particular instances in the given time limitations. Several best heuristics are selected for every combination of the instance and the given time. A heuristic is included in the table if it was able to solve the problem in the given time, and if its solution quality is not worse than $1.1 \cdot w(A_\text{best})$ and its running time is not larger than $1.1 \cdot t_\text{best}$, where $A_\text{best}$ is the best assignment produced by the considered heuristics and $t_\text{best}$ is the time spent to produce $A_\text{best}$. The following information is provided for every solver in Tables~\ref{tab:heuristics_independent} and~\ref{tab:heuristics_decomposable}: \begin{itemize} \item Metaheuristic type (\textbf{C} for {\heuristic{Chain}}{}, \textbf{MC} for {\heuristic{Multichain}}{} or empty if the experiment is single). \item Local search procedure (\Opt{2}, {\heuristic{\OneDVPlain}}, {\heuristic{\TwoDVPlain}}, {\heuristic{\MDVPlain}}, \heuristic{\OneDVtwoPlain}, \heuristic{\TwoDVtwoPlain}, \heuristic{\MDVthreePlain}\, \heuristic{\MDVVPlain}\ or empty if no local search was applied to the initial solution). \item Construction heuristic the experiment was started with (\heuristic{Gr}, \heuristic{M-R} or empty if the assignment was initialized by {\heuristic{Trivial}}). \item The solution error in percent. \end{itemize} The following solvers were included in this experiment: \begin{itemize} \item Construction heuristics {\heuristic{Greedy}}, {\heuristic{Max-Regret}}\ and {\heuristic{ROM}}. \item Single heuristics \Opt{2}, {\heuristic{\OneDVPlain}}, {\heuristic{\TwoDVPlain}}, {\heuristic{\MDVPlain}}, \heuristic{\OneDVtwoPlain}, \heuristic{\TwoDVtwoPlain}, \heuristic{\MDVthreePlain}\ and \heuristic{\MDVVPlain} started from either {\heuristic{Trivial}}, {\heuristic{Greedy}}, {\heuristic{Max-Regret}}\ or {\heuristic{ROM}}. \item {\heuristic{Chain}}\ and {\heuristic{Multichain}}\ metaheuristics for either \Opt{2}, {\heuristic{\OneDVPlain}}, {\heuristic{\TwoDVPlain}}, {\heuristic{\MDVPlain}}, \heuristic{\OneDVtwoPlain}, \heuristic{\TwoDVtwoPlain}, \heuristic{\MDVthreePlain}\ or \heuristic{\MDVVPlain}\ and started from either {\heuristic{Trivial}}, {\heuristic{Greedy}}, {\heuristic{Max-Regret}}\ or {\heuristic{ROM}}\@. The metaheuristics proceeded until the given time limitations. \end{itemize} Note that for certain instances we exclude duplicating solvers (recall that all the dimensionwise heuristics perform equally for 3-AP as well as {\heuristic{\TwoDVPlain}}\ and {\heuristic{\MDVPlain}}\ perform equally for 4-AP, see Section~\ref{sec:dv}). The common rule is that we leave {\heuristic{\MDVPlain}}\ rather than {\heuristic{\TwoDVPlain}}\ and {\heuristic{\TwoDVPlain}}\ rather than {\heuristic{\OneDVPlain}}\@. For example, if the list of successful solvers for some 3-AP instance contains \textbf{C}~1DV~Gr, \textbf{C}~2DV~Gr and \textbf{C}~$s$DV~Gr, then only \textbf{C}~$s$DV~Gr will be included in the table. This is also applicable to the combined heuristics, e.g, having \CombPlain{\OneDVPlain}{2}~R and \CombPlain{\TwoDVPlain}{2}~R for a 3-AP instance, we include only \CombPlain{\TwoDVPlain}{2}~R in the final results. The last row in every table indicates the heuristics which are the most successful on average, i.e., the heuristics which can solve all the instances with the best average results. \bigskip Single construction heuristics are not presented in the tables; single local search procedures appear only for the small allowed times when all other heuristics take more time to run; the most of the best solvers are the metaheuristics. {\heuristic{Multichain}}\ seems to be more suitable than {\heuristic{Chain}}\ for large running times; however, {\heuristic{Multichain}}\ does not appear for the instances with small $n$. This is probably because the power of the perturbation degree increases with the decrease of the instance size (note that $perturb(A)$ perturbs at least two vectors in spite of $n$). The most successful heuristics for the assignment initialization are {\heuristic{Trivial}}\ and {\heuristic{Greedy}}; {\heuristic{Trivial}}\ is useful rather for small running times. {\heuristic{Max-Regret}}\ and {\heuristic{ROM}}\ appear only a few times in the tables. The success of a local search depends on the instance type. The most successful local search heuristic for the instance with independent weights is definitely \heuristic{\MDVVPlain}. The {\heuristic{\MDVPlain}}\ heuristic also appears several times in Table~\ref{tab:heuristics_independent}, especially for the small running times. For the instances with decomposable weights, the most successful are the dimensionwise heuristics and, in particular, {\heuristic{\OneDVPlain}}. \section{Conclusions} Several neighborhoods are generalized and discussed in this paper. An efficient approach of joining different neighborhoods is successfully applied; the yielded heuristics showed that they combine the strengths of their components. The experimental evaluation for a set of instances of different types show that there are several superior heuristic approaches suitable for different kinds of instances and running times. Two kinds of instances are selected: instances with independent weights and instances with decomposable weights. The first ones are better solvable by a combined heuristic \heuristic{\MDVVPlain}; the second ones are better solvable by {\heuristic{\OneDVPlain}}\@. In both cases, it is good to initialize the assignment with the {\heuristic{Greedy}}\ construction heuristic if there is enough time; otherwise one should use a trivial assignment as the initial one. The results can also be significantly improved by applying metaheuristic approaches for as log as possible. Thereby, it is shown in the paper that metaheuristics applied to the fast heuristics dominate slow heuristics and, thus, further research of some more sophisticated metaheuristics such as memetic algorithms is of interest. \bigskip {\small

proofpile-arXiv_069-11560

\section{Introduction} In the first two papers of this series \citep[][hereafter Paper I and Paper II]{DiehlGallery,DiehlAGN}, we conducted a comprehensive morphological analysis of the hot gas in normal elliptical galaxies. In Paper I, we introduced a technique to separate the hot gas emission from the contamination of unresolved point sources. We applied this technique to a {\it Chandra} archive sample of 54 elliptical galaxies and presented a gallery of adaptively binned gas-only images, which were photon-flux calibrated and background corrected. We used these gas maps to derive isophotal ellipticity profiles and conducted a systematic morphological analysis. A comparison between optical and X-ray ellipticities measured in the inner, stellar mass dominated regions shows no correlation, contrary to what would be expected if the gas were in perfect hydrostatic equilibrium. We modeled the expected correlation under various assumptions, and concluded that these systems, in general, are at best only approximately hydrostatic. Moreover, the gas morphologies almost always look disturbed. In Paper II, we introduced a quantitative measure of morphological asymmetry, and found it to be tightly correlated with radio continuum power and nuclear X-ray luminosity. We also found the gas morphology to be influenced, to a comparable degree, by the ambient intergalactic medium. But the AGN--morphology correlation forms a continuous trend down to the lowest detectable AGN luminosities, indicating the importance of AGN feedback, even in rather X-ray faint elliptical galaxies. In this third paper, we address the question of whether the central AGN is merely redistributing the gas, or heating it as well. We produce radial temperature profiles and find that they fall into a variety of distinct types. In particular, we confirm that negative (outwardly falling) temperature gradients \citep{HumphreyDarkmatter,FukazawaMassprofiles, RandallXMMNGC4649, PonmanNGC6482} are present, and relatively common, in low-luminosity systems. Outwardly {\em rising\/} (positive-gradient) temperature profiles, nearly ubiquitous in galaxy clusters, X-ray groups, and massive ellipticals, are usually understood as being the result of efficient radiative cooling in the dense central regions. Accreting gas at large radii can additionally shock-heat itself, amplifying the positive gradient. This interpretation is supported by the short central cooling times observed for galaxies, groups, and clusters, which can drop well below $100\,{\rm Myr}$. However, cooling times are equally short in galaxies with negative temperature gradients, i.e. with a warm center \citep{HumphreyDarkmatter}. Several solutions have been proposed to explain these counter-intuitive objects. \citet{FukazawaMassprofiles} suggest that the gradients are a function of environment, with outwardly rising (positive) gradients caused by the hotter ambient intracluster or intragroup gas surrounding the galaxy. Galaxies with negative gradients should instead be isolated. \citet{HumphreyDarkmatter}, on the other hand, propose a bimodal distribution of temperature gradients, and suggest that the total mass of the system is the decisive factor for the sign of the gradient. The division between their two groups happens at a virial mass of $\sim 10^{13}M_\sun$, implying a distinction between normal galaxies and groups. They hypothesize further that the temperature gradients could be related to a significant difference in the galaxies' evolutionary histories. \citet{HumphreyDarkmatter} also discuss the role of compressive (gravitational) heating, noting that during a slow inflow of relatively cool gas ($<1-2\mbox{$\rm\,keV$}$) the energy gain would exceed the radiative losses. For the inflow of hotter baryons, radiative cooling would dominate and one would observe a positive temperature gradient instead. However, they find no reason for hot baryons to be specific to systems above their break mass $\sim 10^{13}M_\sun$, and suspect the environment as a fuel source instead. \citet{PonmanNGC6482} observe a negative temperature gradient in the fossil group candidate NGC~6482 and argue along the same lines. They model NGC~6482's temperature profile successfully with a steady-state cooling flow with a reasonable cooling rate of $\dot M = 2M_\sun\,\mbox{$\rm\,yr$}^{-1}$, and adopt it as their preferred solution. They also estimate that type Ia supernovae (SN) may be responsible for balancing about $1/3$ of the radiative losses in this galaxy. They find the contribution from type II SN to be insignificant and argue against AGN feedback on grounds of the very relaxed appearance of NGC~6482. In this paper, we show that the distribution of temperature gradients is {\em not\/} bimodal. We further show that the temperature gradients within the inner 2 optical effective radii are {\em not\/} strongly influenced by the environment. Instead, we find evidence that these inner gradients owe their origins either to the specific nature of low-luminosity AGN feedback or to a declining importance of AGN relative to compressive heating or supernovae. In \S\ref{s5.dataanalysis}, we summarize our analyses and results from Papers I and II, and describe the methodology to derive radial temperature profiles. We then discuss the various types of temperature profiles seen in our sample in \S\ref{s5.temperature}. For a quantitative analysis, we split the radial range into two regions: the inner region extending out to 2 effective radii and an outer region between $2-4$ effective radii. We fit and analyze the gradients in these two regions separately, and demonstrate that the inner gradient is determined by galaxy properties, while the outer gradient is strongly influenced by the presence of neighboring galaxies and/or a hot ambient medium. In \S\ref{s5.discussion}, we discuss the implications of our findings for cooling flows, SN heating, and AGN feedback, before we briefly summarize in \S\ref{s5.conclusions}. \section{Data Analysis}\label{s5.dataanalysis} \subsection{Summary of Results from Papers I and II} \label{s5.preliminary} We make use of several parameters from Papers I and II. We list those essential to our analysis in Tables \ref{t.tempprop} and \ref{t.tempprop2} for completeness, along with some additional quantities. We extract absolute $K$ magnitudes $M_{\rm K}$ and $J$-band effective radii $\ensuremath{R_{\rm J}}$ from the 2MASS extended source catalog \citep{2MASS}. We adopt $20\mbox{$\rm\,cm$}$ radio continuum radio luminosities $L_{\rm NVSS}$ from the NRAO VLA Sky Survey \citep[NVSS,][]{NVSS} within $3\ensuremath{R_{\rm J}}$ (see Paper II). In addition, we extract $6\mbox{$\rm\,cm$}$ radio continuum luminosities $L_{\rm 6cm}$ from the GB6 catalog of radio sources \citep{GB6cm}, the Parkes-MIT-NRAO 4.85GHz Surveys \citep{PMN6cm}, and a $6\mbox{$\rm\,cm$}$ radio catalog by \citet{Becker6cm} in the same region. Central velocity dispersion values are taken from the Lyon--Meudon Extragalactic Database \citep[LEDA;][]{LEDA}. We also adopt the projected galaxy density parameter $\rho_{2MASS}$ from Paper II, which is based on the number of neighbors in the 2MASS extended source catalog, and corrected for incompleteness. As it is one of the few accessible parameters to describe the galaxy environment, we also list the \citet{TullyRho} galaxy density $\rho_{\rm Tully}$. \begin{deluxetable*}{lrrrrrrrrrrrrrrrr} \tablewidth{0pt} \tablecaption{{\it Chandra} X-ray luminosity and temperature profile parameters. \label{t.tempprop}} \tablehead{ \colhead{Name} & \colhead{} & \colhead{$L_{\rm X,Gas}$\tablenotemark{a}} & \colhead{$T_{\rm X}$\tablenotemark{b}} & \colhead{$\alpha_{02}$\tablenotemark{c}} & \colhead{$\alpha_{24}$\tablenotemark{c}} } \startdata IC1262 & & $ 2.0\pm 1.7\times 10^{43} $ & $ 1.30 \pm 0.01 $ & $ 0.29 \pm 0.02 $ & $ 0.21 \pm 0.07 $ \\ IC1459 & & $ 4.3\pm 3.2\times 10^{40} $ & $ 0.48 \pm 0.02 $ & $ -0.00 \pm 0.03 $ & $ -0.27 \pm 0.04 $ \\ IC4296 & & $ 1.1\pm 0.4\times 10^{41} $ & $ 0.88 \pm 0.02 $ & $ 0.23 \pm 0.04 $ & $ 0.08 \pm 0.07 $ \\ NGC0193 & & $ 2.5\pm 0.8\times 10^{41} $ & $ 0.77 \pm 0.01 $ & \nodata & \nodata \\ NGC0315 & & $ 9.4\pm 3.4\times 10^{40} $ & $ 0.64 \pm 0.01 $ & $ 0.02 \pm 0.02 $ & \nodata \\ NGC0383 & & $ < 7.5\times 10^{41} $ & $ 0.98 \pm 0.04 $ & $ 0.42 \pm 0.06 $ & $ 0.50 \pm 0.16 $ \\ NGC0404 & & $ < 2.1\times 10^{38} $ & $ 0.28 \pm 0.07 $ & \nodata & \nodata \\ NGC0507 & & $ > 5.7\times 10^{42} $ & $ 1.03 \pm 0.01 $ & $ 0.01 \pm 0.01 $ & $ 0.16 \pm 0.10 $ \\ NGC0533 & & $ 9.6\pm 3.5\times 10^{41} $ & $ 0.98 \pm 0.01 $ & $ 0.18 \pm 0.04 $ & \nodata \\ NGC0720 & & $ 9.3\pm 2.7\times 10^{40} $ & $ 0.57 \pm 0.01 $ & $ -0.05 \pm 0.03 $ & $ -0.04 \pm 0.15 $ \\ NGC0741 & & $ 3.2\pm 1.3\times 10^{41} $ & $ 0.96 \pm 0.02 $ & $ 0.31 \pm 0.04 $ & $ -0.10 \pm 0.12 $ \\ NGC0821 & & $ < 3.3\times 10^{40} $ & \nodata & \nodata & \nodata \\ NGC1132 & & $ > 9.1\times 10^{42} $ & $ 1.02 \pm 0.01 $ & $ 0.24 \pm 0.07 $ & $ -0.22 \pm 0.11 $ \\ NGC1265 & & $ < 1.1\times 10^{42} $ & $ 0.86 \pm 0.08 $ & \nodata & \nodata \\ NGC1316 & & $ 5.7\pm 2.1\times 10^{40} $ & $ 0.62 \pm 0.01 $ & $ -0.04 \pm 0.04 $ & $ 0.12 \pm 0.26 $ \\ NGC1399 & & $ > 7.9\times 10^{41} $ & $ 1.13 \pm 0.01 $ & $ 0.17 \pm 0.02 $ & $ 0.10 \pm 0.03 $ \\ NGC1404 & & $ 1.7\pm 0.4\times 10^{41} $ & $ 0.58 \pm 0.01 $ & $ -0.10 \pm 0.01 $ & $ 0.15 \pm 0.05 $ \\ NGC1407 & & $ 1.0\pm 0.3\times 10^{41} $ & $ 0.87 \pm 0.01 $ & $ 0.11 \pm 0.04 $ & $ 0.46 \pm 0.23 $ \\ NGC1549 & & $ > 2.0\times 10^{40} $ & $ 0.34 \pm 0.03 $ & \nodata & \nodata \\ NGC1553 & & $ 2.8\pm 2.6\times 10^{40} $ & $ 0.41 \pm 0.01 $ & $ -0.21 \pm 0.13 $ & $ -0.28 \pm 0.12 $ \\ NGC1600 & & $ > 1.2\times 10^{42} $ & $ 1.18 \pm 0.04 $ & \nodata & \nodata \\ NGC1700 & & $ > 3.2\times 10^{41} $ & $ 0.43 \pm 0.01 $ & $ -0.06 \pm 0.14 $ & \nodata \\ NGC2434 & & $ 2.6\pm 2.0\times 10^{40} $ & $ 0.53 \pm 0.03 $ & \nodata & \nodata \\ NGC2865 & & $ < 9.9\times 10^{40} $ & $ 0.66 \pm 0.09 $ & \nodata & \nodata \\ NGC3115 & & $ < 8.7\times 10^{39} $ & $ 0.50 \pm 0.04 $ & \nodata & \nodata \\ NGC3377 & & $ < 6.1\times 10^{39} $ & \nodata & \nodata & \nodata \\ NGC3379 & & $ < 6.3\times 10^{39} $ & $ 0.33 \pm 0.03 $ & $ -0.27 \pm 0.06 $ & \nodata \\ NGC3585 & & $ > 4.2\times 10^{39} $ & $ 0.33 \pm 0.01 $ & \nodata & \nodata \\ NGC3923 & & $ 4.3\pm 1.3\times 10^{40} $ & $ 0.48 \pm 0.02 $ & $ -0.07 \pm 0.01 $ & $ -0.37 \pm 0.11 $ \\ NGC4125 & & $ 7.2\pm 2.7\times 10^{40} $ & $ 0.44 \pm 0.01 $ & $ -0.06 \pm 0.02 $ & $ -0.15 \pm 0.16 $ \\ NGC4261 & & $ 4.8\pm 1.1\times 10^{40} $ & $ 0.78 \pm 0.01 $ & $ 0.25 \pm 0.05 $ & \nodata \\ NGC4365 & & $ > 3.8\times 10^{40} $ & $ 0.64 \pm 0.02 $ & $ 0.14 \pm 0.04 $ & \nodata \\ NGC4374 & & $ 5.9\pm 1.3\times 10^{40} $ & $ 0.71 \pm 0.01 $ & $ 0.16 \pm 0.02 $ & $ 0.44 \pm 0.08 $ \\ NGC4406 & & $ > 1.0\times 10^{42} $ & $ 0.78 \pm 0.01 $ & $ 0.06 \pm 0.01 $ & $ -0.05 \pm 0.04 $ \\ NGC4472 & & $ > 8.5\times 10^{41} $ & $ 0.97 \pm 0.01 $ & $ 0.14 \pm 0.01 $ & \nodata \\ NGC4494 & & $ < 2.1\times 10^{40} $ & \nodata & \nodata & \nodata \\ NGC4526 & & $ 8.8\pm 7.5\times 10^{39} $ & $ 0.35 \pm 0.03 $ & \nodata & \nodata \\ NGC4552 & & $ 2.1\pm 1.2\times 10^{40} $ & $ 0.57 \pm 0.01 $ & $ -0.21 \pm 0.04 $ & $ 0.42 \pm 0.16 $ \\ NGC4555 & & $ > 2.3\times 10^{41} $ & $ 0.97 \pm 0.03 $ & \nodata & $ -0.02 \pm 0.03 $ \\ NGC4564 & & $ > 2.0\times 10^{39} $ & \nodata & \nodata & \nodata \\ NGC4621 & & $ 1.1\pm 0.9\times 10^{40} $ & $ 0.23 \pm 0.03 $ & \nodata & \nodata \\ NGC4636 & & $ 2.7\pm 2.0\times 10^{41} $ & $ 0.69 \pm 0.01 $ & $ 0.11 \pm 0.01 $ & $ 0.04 \pm 0.02 $ \\ NGC4649 & & $ 1.3\pm 0.3\times 10^{41} $ & $ 0.80 \pm 0.01 $ & $ 0.02 \pm 0.01 $ & $ -0.01 \pm 0.01 $ \\ NGC4697 & & $ > 3.5\times 10^{40} $ & $ 0.32 \pm 0.01 $ & \nodata & \nodata \\ NGC5018 & & $ < 1.9\times 10^{41} $ & $ 0.45 \pm 0.09 $ & \nodata & \nodata \\ NGC5044 & & $ 2.6\pm 0.8\times 10^{42} $ & $ 0.91 \pm 0.01 $ & $ 0.09 \pm 0.01 $ & $ 0.12 \pm 0.02 $ \\ NGC5102 & & $ < 1.6\times 10^{39} $ & $ 0.38 \pm 0.08 $ & \nodata & \nodata \\ NGC5171 & & $ > 2.7\times 10^{42} $ & $ 0.80 \pm 0.05 $ & \nodata & \nodata \\ NGC5532 & & $ < 8.7\times 10^{41} $ & $ 0.61 \pm 0.02 $ & \nodata & $ -0.38 \pm 0.07 $ \\ NGC5845 & & $ < 5.2\times 10^{40} $ & $ 0.32 \pm 0.05 $ & \nodata & \nodata \\ NGC5846 & & $ 3.9\pm 0.9\times 10^{41} $ & $ 0.71 \pm 0.01 $ & $ 0.02 \pm 0.02 $ & $ 0.03 \pm 0.17 $ \\ NGC6482 & & $ 1.7\pm 1.3\times 10^{42} $ & $ 0.74 \pm 0.01 $ & $ -0.09 \pm 0.01 $ & $ -0.34 \pm 0.02 $ \\ NGC7052 & & $ > 1.1\times 10^{41} $ & $ 0.53 \pm 0.03 $ & \nodata & \nodata \\ NGC7618 & & $ 2.3\pm 0.9\times 10^{42} $ & $ 0.80 \pm 0.01 $ & $ -0.14 \pm 0.06 $ & $ -0.12 \pm 0.08 $ \\ \enddata \tablenotetext{a}{Total X-ray gas luminosity in ${\rm ergs\,s^{-1}}$ for the $0.3-5\mbox{$\rm\,keV$}$ band, see Paper I for more details}. \tablenotetext{b}{Luminosity weighted temperature within 3 optical radii.} \tablenotetext{c}{Temperature gradients, measured in $\log r/\ensuremath{R_{\rm J}} - \log T/\mbox{$\rm\,keV$}$ space, between $0-2\ensuremath{R_{\rm J}}$ (\ensuremath{\alpha_{\rm 02}}) and $2-4\ensuremath{R_{\rm J}}$ (\ensuremath{\alpha_{\rm 24}}).} \end{deluxetable*} \begin{deluxetable*}{lrrrrrrrrrrrrrrr} \tablewidth{0pt} \tablecaption{Optical, radio, and environmental parameters\label{t.tempprop2}} \tablehead{ \colhead{} & \colhead{} & \multicolumn{2}{c}{2MASS\tablenotemark{a}} & \colhead{} & \colhead{LEDA\tablenotemark{b}} & \colhead{} & \multicolumn{2}{c}{Radio\tablenotemark{c}} & \colhead{} & \multicolumn{2}{c}{Environment\tablenotemark{d}} \\ \cline{3-4} \cline{6-6} \cline{8-9} \cline{11-12}\\ \colhead{Name} & \colhead{} & \colhead{$M_{\rm K}$} & \colhead{$\ensuremath{R_{\rm J}}$} & \colhead{} & \colhead{$\sigma$} & \colhead{} & \colhead{$L_{\rm NVSS}$} & \colhead{$L_{\rm GB6}$} & \colhead{} & \colhead{$\log \rho_{\rm 2MASS}$} & \colhead{$\rho_{\rm Tully}$} } \startdata IC1262 & & $ -25.43 \pm 0.33 $ & $ 14.4 $ & & $ 266 \pm 36 $ & & $ 1.8\pm 0.5\times 10^{30} $ & $ < 4.5\times 10^{29} $ & & $ 3.47 \pm 0.22 $ & \nodata \\ IC1459 & & $ -25.53 \pm 0.28 $ & $ 29.1 $ & & $ 308 \pm 6 $ & & $ 1.3\pm 0.3\times 10^{30} $ & $ 1.2\pm 0.1\times 10^{30} $ & & $ 2.41 \pm 0.31 $ & $ 0.28 $ \\ IC4296 & & $ -26.06 \pm 0.33 $ & $ 25.5 $ & & $ 333 \pm 6 $ & & $ 6.0\pm 1.8\times 10^{30} $ & $ 6.0\pm 0.5\times 10^{30} $ & & $ 2.77 \pm 0.25 $ & \nodata \\ NGC0193 & & $ -24.71 \pm 0.33 $ & $ 14.5 $ & & \nodata & & $ 5.6\pm 1.7\times 10^{30} $ & $ 3.8\pm 0.3\times 10^{30} $ & & $ 2.66 \pm 0.31 $ & \nodata \\ NGC0315 & & $ -26.33 \pm 0.33 $ & $ 22.9 $ & & $ 296 \pm 21 $ & & $ 9.8\pm 2.9\times 10^{30} $ & $ 5.7\pm 0.5\times 10^{30} $ & & $ 2.43 \pm 0.43 $ & \nodata \\ NGC0383 & & $ -25.84 \pm 0.33 $ & $ 17.8 $ & & $ 277 \pm 6 $ & & $ 1.3\pm 0.4\times 10^{31} $ & $ 6.8\pm 0.6\times 10^{30} $ & & $ 3.52 \pm 0.13 $ & \nodata \\ NGC0404 & & \nodata & \nodata & & $ 38 \pm 3 $ & & $ 4.3\pm 0.6\times 10^{25} $ & $ < 2.3\times 10^{26} $ & & \nodata & $ 0.20 $ \\ NGC0507 & & $ -25.98 \pm 0.33 $ & $ 26.1 $ & & $ 315 \pm 9 $ & & $ 6.1\pm 1.8\times 10^{29} $ & $ < 1.1\times 10^{29} $ & & $ 3.03 \pm 0.22 $ & \nodata \\ NGC0533 & & $ -26.01 \pm 0.33 $ & $ 25.2 $ & & $ 275 \pm 6 $ & & $ 2.1\pm 0.6\times 10^{29} $ & $ < 1.3\times 10^{29} $ & & $ 3.17 \pm 0.19 $ & \nodata \\ NGC0720 & & $ -24.94 \pm 0.17 $ & $ 27.4 $ & & $ 242 \pm 5 $ & & $ < 2.3\times 10^{27} $ & $ < 3.8\times 10^{28} $ & & $ 2.57 \pm 0.25 $ & $ 0.25 $ \\ NGC0741 & & $ -26.19 \pm 0.33 $ & $ 25.9 $ & & $ 290 \pm 8 $ & & $ 6.4\pm 1.9\times 10^{30} $ & $ 2.0\pm 0.2\times 10^{30} $ & & $ 2.95 \pm 0.25 $ & $ 0.05 $ \\ NGC0821 & & $ -24.01 \pm 0.17 $ & $ 23.9 $ & & $ 199 \pm 2 $ & & $ < 1.7\times 10^{27} $ & $ < 1.2\times 10^{28} $ & & \nodata & $ 0.08 $ \\ NGC1132 & & $ -25.70 \pm 0.33 $ & $ 19.8 $ & & $ 247 \pm 13 $ & & $ 1.1\pm 0.3\times 10^{29} $ & $ < 4.6\times 10^{29} $ & & $ 3.07 \pm 0.25 $ & \nodata \\ NGC1265 & & \nodata & \nodata & & \nodata & & $ 3.3\pm 1.0\times 10^{31} $ & $ < 2.6\times 10^{29} $ & & \nodata & \nodata \\ NGC1316 & & $ -26.07 \pm 0.17 $ & $ 49.8 $ & & $ 227 \pm 4 $ & & $ 1.6\pm 0.3\times 10^{29} $ & $ < 2.2\times 10^{28} $ & & $ 3.13 \pm 0.13 $ & $ 1.15 $ \\ NGC1399 & & $ -25.19 \pm 0.16 $ & $ 36.9 $ & & $ 337 \pm 5 $ & & $ 3.0\pm 0.4\times 10^{29} $ & $ < 3.4\times 10^{28} $ & & $ 3.28 \pm 0.12 $ & $ 1.59 $ \\ NGC1404 & & $ -24.79 \pm 0.19 $ & $ 19.3 $ & & $ 233 \pm 3 $ & & $ 2.1\pm 0.5\times 10^{27} $ & $ < 3.8\times 10^{28} $ & & $ 3.27 \pm 0.11 $ & $ 1.59 $ \\ NGC1407 & & $ -25.60 \pm 0.26 $ & $ 36.4 $ & & $ 272 \pm 5 $ & & $ 9.7\pm 2.3\times 10^{28} $ & $ < 4.2\times 10^{28} $ & & $ 3.10 \pm 0.14 $ & $ 0.42 $ \\ NGC1549 & & $ -24.69 \pm 0.18 $ & $ 29.0 $ & & $ 203 \pm 3 $ & & \nodata & $ < 1.9\times 10^{28} $ & & $ 2.74 \pm 0.22 $ & $ 0.97 $ \\ NGC1553 & & $ -25.06 \pm 0.17 $ & $ 33.9 $ & & $ 177 \pm 4 $ & & \nodata & $ < 1.6\times 10^{28} $ & & $ 2.96 \pm 0.18 $ & $ 0.97 $ \\ NGC1600 & & $ -26.06 \pm 0.33 $ & $ 24.8 $ & & $ 335 \pm 6 $ & & $ 3.6\pm 1.1\times 10^{29} $ & $ < 2.1\times 10^{29} $ & & $ 3.09 \pm 0.19 $ & \nodata \\ NGC1700 & & $ -25.59 \pm 0.33 $ & $ 15.9 $ & & $ 235 \pm 3 $ & & $ < 8.8\times 10^{27} $ & $ < 1.4\times 10^{29} $ & & $ 2.31 \pm 0.43 $ & \nodata \\ NGC2434 & & $ -23.78 \pm 0.29 $ & $ 19.3 $ & & $ 188 \pm 5 $ & & \nodata & $ < 2.2\times 10^{28} $ & & $ 2.56 \pm 0.25 $ & $ 0.19 $ \\ NGC2865 & & $ -24.43 \pm 0.20 $ & $ 14.8 $ & & $ 170 \pm 2 $ & & $ < 4.3\times 10^{27} $ & $ < 7.2\times 10^{28} $ & & \nodata & $ 0.11 $ \\ NGC3115 & & $ -24.05 \pm 0.09 $ & $ 36.4 $ & & $ 257 \pm 5 $ & & $ < 2.8\times 10^{26} $ & $ < 4.5\times 10^{27} $ & & \nodata & $ 0.08 $ \\ NGC3377 & & $ -22.81 \pm 0.09 $ & $ 27.7 $ & & $ 139 \pm 2 $ & & $ < 3.8\times 10^{26} $ & $ < 2.7\times 10^{27} $ & & \nodata & $ 0.49 $ \\ NGC3379 & & $ -23.85 \pm 0.11 $ & $ 29.9 $ & & $ 205 \pm 2 $ & & $ 3.2\pm 0.7\times 10^{26} $ & $ < 2.4\times 10^{27} $ & & $ 3.82 \pm 0.10 $ & $ 0.52 $ \\ NGC3585 & & $ -24.81 \pm 0.18 $ & $ 32.3 $ & & $ 207 \pm 4 $ & & $ < 1.2\times 10^{27} $ & $ < 2.0\times 10^{28} $ & & $ 2.43 \pm 0.31 $ & $ 0.12 $ \\ NGC3923 & & $ -25.30 \pm 0.28 $ & $ 43.8 $ & & $ 247 \pm 6 $ & & $ < 1.6\times 10^{27} $ & $ < 2.6\times 10^{28} $ & & $ 2.89 \pm 0.16 $ & $ 0.40 $ \\ NGC4125 & & $ -25.03 \pm 0.25 $ & $ 33.0 $ & & $ 226 \pm 6 $ & & $ 1.7\pm 0.4\times 10^{28} $ & $ < 1.2\times 10^{28} $ & & $ 3.13 \pm 0.13 $ & $ 0.34 $ \\ NGC4261 & & $ -25.24 \pm 0.19 $ & $ 25.5 $ & & $ 320 \pm 8 $ & & $ 1.0\pm 0.2\times 10^{31} $ & $ 4.8\pm 0.1\times 10^{30} $ & & $ 3.09 \pm 0.14 $ & $ 0.84 $ \\ NGC4365 & & $ -24.91 \pm 0.17 $ & $ 40.7 $ & & $ 255 \pm 2 $ & & $ < 1.2\times 10^{27} $ & $ < 9.0\times 10^{27} $ & & $ 3.16 \pm 0.13 $ & $ 2.93 $ \\ NGC4374 & & $ -25.10 \pm 0.11 $ & $ 34.8 $ & & $ 280 \pm 2 $ & & $ 2.5\pm 0.3\times 10^{30} $ & $ 1.3\pm 0.0\times 10^{30} $ & & $ 3.18 \pm 0.14 $ & $ 3.99 $ \\ NGC4406 & & $ -25.07 \pm 0.14 $ & $ 59.7 $ & & $ 235 \pm 2 $ & & $ 1.1\pm 0.2\times 10^{27} $ & $ < 6.3\times 10^{27} $ & & $ 3.31 \pm 0.13 $ & $ 1.41 $ \\ NGC4472 & & $ -25.66 \pm 0.10 $ & $ 59.2 $ & & $ 288 \pm 2 $ & & $ 8.1\pm 0.8\times 10^{28} $ & $ 2.7\pm 0.0\times 10^{28} $ & & $ 3.41 \pm 0.12 $ & $ 3.31 $ \\ NGC4494 & & $ -24.16 \pm 0.11 $ & $ 30.8 $ & & $ 149 \pm 3 $ & & $ < 8.7\times 10^{26} $ & $ < 6.3\times 10^{27} $ & & \nodata & $ 1.04 $ \\ NGC4526 & & $ -24.67 \pm 0.20 $ & $ 43.8 $ & & $ 263 \pm 18 $ & & $ 6.6\pm 1.2\times 10^{27} $ & $ < 6.1\times 10^{27} $ & & $ 2.54 \pm 0.31 $ & $ 2.45 $ \\ NGC4552 & & $ -24.20 \pm 0.14 $ & $ 25.4 $ & & $ 253 \pm 2 $ & & $ 2.9\pm 0.4\times 10^{28} $ & $ 1.9\pm 0.0\times 10^{28} $ & & $ 3.62 \pm 0.09 $ & $ 2.97 $ \\ NGC4555 & & $ -25.78 \pm 0.33 $ & $ 10.9 $ & & \nodata & & $ < 2.8\times 10^{28} $ & $ < 2.0\times 10^{29} $ & & $ 3.19 \pm 0.22 $ & \nodata \\ NGC4564 & & $ -22.94 \pm 0.17 $ & $ 19.9 $ & & $ 158 \pm 2 $ & & $ < 6.7\times 10^{26} $ & $ < 4.8\times 10^{27} $ & & \nodata & $ 4.09 $ \\ NGC4621 & & $ -24.56 \pm 0.20 $ & $ 32.9 $ & & $ 225 \pm 3 $ & & $ < 10.0\times 10^{26} $ & $ < 7.2\times 10^{27} $ & & $ 2.66 \pm 0.25 $ & $ 2.60 $ \\ NGC4636 & & $ -24.41 \pm 0.13 $ & $ 59.3 $ & & $ 202 \pm 3 $ & & $ 2.7\pm 0.3\times 10^{28} $ & $ 1.8\pm 0.0\times 10^{28} $ & & $ 3.47 \pm 0.12 $ & $ 1.33 $ \\ NGC4649 & & $ -25.39 \pm 0.15 $ & $ 45.2 $ & & $ 334 \pm 3 $ & & $ 9.8\pm 1.4\times 10^{27} $ & $ 1.3\pm 0.0\times 10^{28} $ & & $ 3.39 \pm 0.12 $ & $ 3.49 $ \\ NGC4697 & & $ -23.98 \pm 0.14 $ & $ 42.4 $ & & $ 173 \pm 2 $ & & $ < 4.1\times 10^{26} $ & $ < 6.6\times 10^{27} $ & & $ 3.65 \pm 0.11 $ & $ 0.60 $ \\ NGC5018 & & $ -25.27 \pm 0.33 $ & $ 15.6 $ & & $ 214 \pm 8 $ & & $ < 4.8\times 10^{27} $ & $ < 8.0\times 10^{28} $ & & $ 2.50 \pm 0.31 $ & $ 0.29 $ \\ NGC5044 & & $ -24.76 \pm 0.28 $ & $ 25.3 $ & & $ 238 \pm 8 $ & & $ 4.0\pm 1.0\times 10^{28} $ & $ < 4.9\times 10^{28} $ & & $ 3.17 \pm 0.13 $ & $ 0.38 $ \\ NGC5102 & & $ -21.09 \pm 0.14 $ & $ 79.3 $ & & $ 66 \pm 4 $ & & $ 6.1\pm 1.5\times 10^{25} $ & $ < 1.4\times 10^{27} $ & & \nodata & $ 0.17 $ \\ NGC5171 & & $ -24.95 \pm 0.33 $ & $ 10.8 $ & & \nodata & & $ < 2.9\times 10^{28} $ & $ < 2.1\times 10^{29} $ & & \nodata & \nodata \\ NGC5532 & & $ -26.33 \pm 0.33 $ & $ 16.0 $ & & $ 293 \pm 18 $ & & $ 5.8\pm 1.7\times 10^{31} $ & $ 1.6\pm 0.1\times 10^{31} $ & & $ 3.11 \pm 0.25 $ & \nodata \\ NGC5845 & & $ -22.96 \pm 0.21 $ & $ 4.9 $ & & $ 234 \pm 8 $ & & $ < 2.0\times 10^{27} $ & $ < 1.4\times 10^{28} $ & & \nodata & $ 0.84 $ \\ NGC5846 & & $ -25.04 \pm 0.20 $ & $ 34.5 $ & & $ 239 \pm 3 $ & & $ 1.6\pm 0.3\times 10^{28} $ & $ < 1.3\times 10^{28} $ & & $ 2.97 \pm 0.15 $ & $ 0.84 $ \\ NGC6482 & & $ -25.48 \pm 0.33 $ & $ 12.6 $ & & $ 303 \pm 9 $ & & $ < 1.0\times 10^{28} $ & $ < 7.5\times 10^{28} $ & & $ 2.64 \pm 0.31 $ & \nodata \\ NGC7052 & & $ -25.66 \pm 0.33 $ & $ 21.8 $ & & $ 271 \pm 9 $ & & $ 1.2\pm 0.4\times 10^{30} $ & $ 6.7\pm 0.6\times 10^{29} $ & & $ 2.72 \pm 0.31 $ & \nodata \\ NGC7618 & & $ -25.40 \pm 0.33 $ & $ 11.6 $ & & \nodata & & $ 2.7\pm 0.8\times 10^{29} $ & $ < 1.3\times 10^{29} $ & & $ 2.76 \pm 0.31 $ & \nodata \\ \enddata \tablenotetext{a}{2MASS data from the extended source catalog \citep{2MASS}. $K$-band absolute magnitude $M_{\rm K}$ and $J$-band effective radius \ensuremath{R_{\rm J}}\ in $\mbox{$\rm\,kpc$}$.} \tablenotetext{b}{Velocity dispersion (in ${\rm km}\, \mbox{$\rm\,s$}^{-1}$) from LEDA \citep{LEDA}.} \tablenotetext{c}{\ensuremath{L_{\rm NVSS}} (in $\mbox{$\rm\,erg$} \, \mbox{$\rm\,s$}^{-1}$): $20\mbox{$\rm\,cm$}$ radio continuum luminosity from NVSS \citep{NVSS}; \ensuremath{L_{\rm 6cm}}\ (in $\mbox{$\rm\,erg$} \, \mbox{$\rm\,s$}^{-1}$): Combination of $6\mbox{$\rm\,cm$}$ radio continuum luminosities from \citet{GB6cm}, \citet{PMN6cm}, and \citet{Becker6cm}. } \tablenotetext{d}{Projected galaxy density \ensuremath{\rho_{\rm 2MASS}}\ is scaled logarithmically and in units of $\mbox{$\rm\,Mpc$}^{-2}$, Tully galaxy density \ensuremath{\rho_{\rm Tully}}\ is scaled linearly and in units of $\mbox{$\rm\,Mpc$}^{-3}$.} \end{deluxetable*} \subsection{Temperature Profiles}\label{s5.tempprofile} To produce radial temperature profiles, we divide the X-ray counts image of each galaxy into elliptical annuli, according to the X-ray ellipticity profiles computed in Paper I. For those galaxies with insufficient signal to fit ellipses, we revert to circular annuli. We find no evidence that this choice affects our results in any way. We adapt the width of our annuli to contain a minimum of 900 counts above the background level, which we determine by the appropriately rescaled Markevitch blank-sky background files\footnote{http://cxc.harvard.edu/cal/Acis/Cal\_prods/bkgrnd/acisbg/COOKBOOK}. We then extract a source and background spectrum for each annulus and fit them with a two-component model using the CIAO analysis package Sherpa. The first component consists of an APEC \footnote{Astrophysical Plasma Emission Code} plasma model to represent the hot gas emission. A quantitative comparison with its better-known predecessor, the Mekal model, shows nearly identical fitting results. We fix the gas metallicity at the solar abundance value. Unresolved point sources are represented by a power-law model with the power law index fixed at 1.6. This ``universal'' spectral model is an adequate representation for the emission of low-luminosity low-mass X-ray binaries, as demonstrated in Paper I and determined independently by \citet{Irwin03}. We also add a multiplicative absorption component, for which we fix the hydrogen column density to the Galactic value, evaluated at the target position with the CIAO tool {\it Colden}\footnote{http://cxc.harvard.edu/toolkit/colden.jsp}. We repeat our spectral analysis for a few objects with the gas abundance as a free parameter, and find that our choice to fix them to the solar value does not affect the fitted temperature. Since the metallicity is poorly constrained by the fits in low signal-to-noise systems, we fix the metallicity for all of our galaxies, in order not to introduce systematic differences in the analysis. \section{Results} \label{s5.temperature} \subsection{Radial Temperature Profile Types} We categorize the observed temperature profiles into four major groups, described below. Two examples from each are shown in Figure \ref{f.temptypes}. (Note that the distinctions between the groups are not always clear cut.) \paragraph{Positive Gradients.} These temperature profiles show a positive gradient at all radii, i.e. the temperature continuously rises outwardly. These profiles resemble those found in clusters of galaxies, which generally harbor cool cores. \paragraph{Negative Gradients.} The temperature profiles show a negative gradient at all radii, i.e. temperatures monotonically decline outward. This phenomenon is less well-known, and has been reported only recently \citep{RandallXMMNGC4649,FukazawaMassprofiles,HumphreyDarkmatter, PonmanNGC6482}. \paragraph{Hybrid.} These peculiar cases exhibit a dramatic change in the temperature gradients. The gradient changes its sign from negative to positive at some intermediate radius, generally between $1-3R_{\rm J}$. These galaxies have warm centers, outside of which their temperatures drop to a minimum level and rise back up again. These profiles have first been noted by \citet{HumphreyDarkmatter}. \paragraph{Quasi-Isothermal.} The radial temperature profiles in this category are consistent with being almost flat at all radii. These galaxies form the transition point between galaxies with positive and negative gradients. We observe only hybrid temperature profiles that change their gradient from negative to positive. Some cooling flow clusters have been found to exhibit the opposite behavior \citep{PiffarettiClusters}. Their temperature profiles show a cool center, then rise to a peak temperature and fall back down on the outskirts. This ``break'' usually happens at around 10\% of the virial radius, which is larger than the radii that we are probing in normal galaxies. A {\it ROSAT} study by \citet{OSullivan} exhibits similar trends for elliptical galaxies at larger radii. We split the profiles into two radial regions and analyze the inner and outer temperature gradients separately. As most hybrid profiles exhibit their turnover in slope somewhere around $2R_{\rm J}$, we use this radius as the boundary between our two regions. Accordingly, we define the inner region from outside the central point source extending out to $2R_{\rm J}$ and the outer region between $2-4R_{\rm J}$. We then fit each part of the profile with a power law to derive effective temperature gradients for each region. We will refer to the logarithmic gradients ${\rm d}\ln T /{\rm d} \ln R$ evaluated within 2\ensuremath{R_{\rm J}}\ and from $2-4\,\ensuremath{R_{\rm J}}$ as $\alpha_{02}$ and $\alpha_{24}$, respectively. The best-fit values for \ensuremath{\alpha_{\rm 02}}\ and \ensuremath{\alpha_{\rm 24}}\ are listed in Table \ref{t.tempprop}. The reported errors are the formal $1\sigma$ statistical errors obtained from the fitting procedure. For cases with only 2 valid data points within the fitting range, we use the difference between these two points to derive a gradient; the errors are derived by propagating the statistical errors of the individual temperature measurements. \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{f1} \end{center} \caption{Examples of different projected temperature profiles as a function of radius. Temperature profile types can be divided into 4 major groups (top to bottom rows): (1) Positive gradient (outwardly rising) at all radii; (2) Negative gradient (outwardly falling) at all radii; (3) Hybrid, negative gradient in the core and positive gradient at larger radii; (4) Quasi-isothermal, no apparent temperature change with radius. The complete set of temperature profiles is available online for all 36 galaxies with two or more valid temperature profile points. \label{f.temptypes}} \end{figure} \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{f2} \end{center} \caption{Combined plot of all projected temperature profiles, as a function of radius scaled by the $J$-band effective radius $R_{\rm J}$. Error bars are omitted for clarity; for typical error estimates, refer to Figure \ref{f.temptypes}. The profiles are colored according to the luminosity weighted average temperature of the galaxy within 3 optical radii \ensuremath{T_{\rm X}}, as indicated by the color scale bar. The temperature gradient changes continuously from positive gradients at the top to isothermal and hybrid profiles to negative gradient profiles at the bottom, along with the average temperature. \label{f.tempprofiles}} \end{figure} \subsection{The Inner Temperature Gradient $\alpha_{02}$} Figure \ref{f.tempprofiles} shows the compilation of all profiles overlaid in one plot, with the radial axes scaled by their $J$-band effective radius $\ensuremath{R_{\rm J}}$. This plot already clearly indicates the absence of any real bimodality in the temperature profiles. Each profile is colored according to its luminosity weighted temperature $T_{\rm X}$ within 3 optical radii. The coloring changes smoothly from top to bottom, indicating overall positive gradients for intrinsically hotter galaxies and negative gradients for cooler galaxies. We plot $\alpha_{02}$ as a function of $T_{\rm X}$ in Figure \ref{f.txtslope02}. Again, there is no sign of bimodality in $\alpha_{02}$. The plot shows a tight correlation, significant at the $9.1\sigma$ level, reflecting the fact that we observe only a very small range in central temperatures between $0.6-0.7\mbox{$\rm\,keV$}$ (Figure \ref{f.tempprofiles}). Thus, any average temperature will obviously be strongly correlated with the gradient as well. Consistently, the fit suggests that the transition from positive to negative inner temperatures gradient occurs at a mean temperature around $0.64\mbox{$\rm\,keV$}$. To establish the underlying cause for the negative inner temperature gradients, we perform a correlation analysis with various galaxy properties, the most interesting of which are listed in the upper half of Table \ref{t.gradcorrelations}: X-ray gas luminosities $L_{\rm X,gas}$, absolute $K$ magnitudes $M_{\rm K}$, central velocity dispersions $\sigma$, radio luminosities at 20cm ($L_{\rm NVSS}$) and 6cm ($L_{\rm6cm}$), and environmental measures of local galaxy density, \ensuremath{\rho_{\rm 2MASS}}\ and \ensuremath{\rho_{\rm Tully}}. We assess the correlation of $\alpha_{02}$ with each of these properties using the linear fitting algorithm \texttt{bandfit} (see Appendix of Paper II). This algorithm models the distribution of $(x,y)$ points as a linear band with a Gaussian intrinsic width. The model is fitted by maximizing the likelihood of the data; this is similar to fitting a straight line by minimizing the error-weighted perpendicular residuals. The standard error $\sigma$ in the best-fit slope is obtained from the covariance matrix, and the significance of the correlation is the number of $\sigma$ by which the slope differs from zero. In these fits, the quantities assigned as abscissae are scaled logarithmically (except for $M_{\rm K}$, which is already intrinsically logarithmic), while the ordinate \ensuremath{\alpha_{\rm 02}}\ is scaled linearly. The top half of Table \ref{t.gradcorrelations} lists the best-fit parameters and the statistical significance of the correlation. The fitted parameters are omitted where the significance is $<2\sigma$. The tabulated fits are obtained from the full data set including upper limits, but we also include in the last column the significance of the correlation obtained with the upper limits omitted.\footnote{The exception is $L_{\rm GB6}$, for which we give the parameters for the fit without upper limits. This is because the number of galaxies with upper limits only (49) greatly outweighs that of galaxies with actual detections (15).} The $x_0$ column indicates the transition point from negative to positive temperature gradients in each fit. As the table and Figure \ref{f.kmaglradiotslope02} show, the strongest correlations with \ensuremath{\alpha_{\rm 02}}\ are found with the 20 cm NVSS radio luminosity \ensuremath{L_{\rm NVSS}}, the velocity dispersion $\sigma$, and the absolute K-magnitude \ensuremath{M_{\rm K}}. All of these three correlations are of roughly the same significance, with comparable intrinsic widths, as Figure \ref{f.kmaglradiotslope02} shows. The correlation with the $6\mbox{$\rm\,cm$}$ radio luminosity $L_{\rm GB6}$ seems equally tight, but has a lower significance, due to the smaller sample size with $6\mbox{$\rm\,cm$}$ radio luminosity measurements. We do not find {\it any} evidence that \ensuremath{\alpha_{\rm 02}}\ is correlated with the environmental galaxy densities \ensuremath{\rho_{\rm 2MASS}}\ or \ensuremath{\rho_{\rm Tully}}. The \ensuremath{\rho_{\rm 2MASS}}--\ensuremath{\alpha_{\rm 02}}\ plot is shown in the bottom panel of Figure \ref{f.tempenvironment}. We conclude that the inner temperature gradients are neither the result of interactions with neighbor galaxies, nor with ambient intragroup or intracluster gas. Instead, we find that they are connected to intrinsic galaxy properties. We can generally characterize galaxies with negative inner temperature gradients as being smaller, optically fainter galaxies with lower velocity dispersions, lower X-ray gas luminosities, lower average temperatures, and lower radio luminosities than their positive gradient counterparts. Unfortunately, {\it all} of these galaxy properties are intimately connected with each other through well-known correlations such as the \ensuremath{T_{\rm X}}--$\sigma$ relation \citep[e.g.][]{OSullivan}, the Faber-Jackson relation \citep{FaberJackson}, the \ensuremath{L_{\rm X,gas}}--\ensuremath{T_{\rm X}}\ relation \citep[e.g.][]{OSullivan} and the $L_{\rm Radio}$--$\sigma$ relation \citep[e.g.][]{SnellenLsigma}. This makes it difficult to distinguish between fundamental correlations that are really responsible for determining the inner temperature structure and others that are simply ``riding along'' via other correlations. We check the robustness of our results by deriving inner temperature gradients for different cutoff radii and find that all of the observed trends are confirmed, as long as the cutoff-radius does not exceed $\sim 3\ensuremath{R_{\rm J}}$. In particular, we find that for smaller cutoff-radii (e.g. 1\ensuremath{R_{\rm J}}), the significance of the correlations with $\sigma$, \ensuremath{M_{\rm K}}\ and \ensuremath{L_{\rm X,gas}}\ slightly decreases, while the correlations with the radio luminosities \ensuremath{L_{\rm NVSS}}\ and \ensuremath{L_{\rm 6cm}}\ strengthen even further. This may suggest that the correlations with radio luminosities are intrinsically the strongest. Figure \ref{f.tempprofilesradio} shows a combined plot of all temperature profiles, similar to Figure \ref{f.tempprofiles}, but this time colored according to the NVSS radio luminosities. A trend with radio luminosity is clearly evident. We will discuss the implications of our results in \S\ref{s5.discussion}. \begin{deluxetable*}{llrrrcc} \tablewidth{0pt} \tablecaption{Correlations involving inner and outer temperature gradients. \label{t.gradcorrelations}} \tablehead{ \colhead{$y$} & \colhead{$x$} & \colhead{$a$} & \colhead{$b$} & \colhead{$x_0$} & \colhead{Significance} & \colhead{Significance} \\ & & & & & \colhead{(with limits)} & \colhead{(without limits)} } \startdata $\alpha_{02}$ & $\log L_{\rm NVSS}$ & $ 0.060\pm0.019$ & $-1.65\pm0.02$ & $ 27.4$ & $3.2\sigma$ & $4.3\sigma$ \\ $\alpha_{02}$ & $\log \sigma$ & $ 4.171\pm1.192$ & $-10.0\pm0.01$ & $ 2.40$ & $3.6\sigma$ & $3.6\sigma$ \\ $\alpha_{02}$ & $M_{\rm K}$ & $-0.141\pm0.044$ & $-3.50\pm0.02$ & $-24.8$ & $3.2\sigma$ & $3.2\sigma$ \\ $\alpha_{02}$ & $\log L_{\rm GB6}$\tablenotemark{*} & $ 0.083\pm0.036$ & $-2.35\pm0.04$ & $ 28.3$ & $0.4\sigma$ & $2.3\sigma$ \\ $\alpha_{02}$ & $\log L_{\rm X,gas}$ & \nodata & \nodata & \nodata & $1.0\sigma$ & $1.0\sigma$ \\ $\alpha_{02}$ & $\log \rho$ & \nodata & \nodata & \nodata & $0.7\sigma$ & $0.7\sigma$ \\ $\alpha_{02}$ & $\log \rho_{\rm Tully}$ & \nodata & \nodata & \nodata & $0.4\sigma$ & $0.4\sigma$ \\ \\%\tablevspace{0.25\baselineskip} $\alpha_{24}$ & $\log \rho$ & $ 0.735\pm0.223$ & $-2.28\pm0.03$ & $ 3.10$ & $4.4\sigma$ & $4.4\sigma$ \\ $\alpha_{24}$ & $\log \rho_{\rm Tully}$ & $ 0.272\pm0.102$ & $ 0.04\pm0.04$ & $ 0.15$ & $2.8\sigma$ & $2.8\sigma$ \\ $\alpha_{24}$ & $M_{\rm K}$ & \nodata & \nodata & \nodata & $1.6\sigma$ & $1.6\sigma$ \\ $\alpha_{24}$ & $\log \sigma$ & \nodata & \nodata & \nodata & $0.5\sigma$ & $0.5\sigma$ \\ $\alpha_{24}$ & $\log L_{\rm GB6}$ & \nodata & \nodata & \nodata & $<0.1\sigma$ & $0.6\sigma$ \\ $\alpha_{24}$ & $\log L_{\rm NVSS}$ & \nodata & \nodata & \nodata & $<0.1\sigma$ & $0.3\sigma $ \\ $\alpha_{24}$ & $\log L_{\rm X,gas}$ & \nodata & \nodata & \nodata & $<0.1\sigma$ & $<0.1\sigma$ \\ \enddata \tablecomments{Results are listed in order of decreasing correlation significance. Parameters refer to linear fits of the form $y=ax+b$ (for correlations of $>2\sigma$ significance). $x_0$ denotes the point where the fit yields 0, i.e. where the temperature gradients change sign. The last two columns quote the correlation significances with and without upper and lower limits included, respectively.} \tablenotetext{*}{The fitted parameters for the $L_{\rm GB6}$ -- $\alpha_{02}$ correlation are quoted for the fit without upper and lower limits included.} \end{deluxetable*} \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{f3} \end{center} \caption{Inner temperature gradient within $2\,R_{\rm J}$ as a function of the average luminosity weighted temperature within 3 optical radii. The dashed line indicates the best fit. \label{f.txtslope02}} \end{figure} \begin{figure*} \begin{center} \includegraphics[width=0.4\textwidth]{f4a} \includegraphics[width=0.4\textwidth]{f4b} \includegraphics[width=0.4\textwidth]{f4c} \includegraphics[width=0.4\textwidth]{f4d} \end{center} \caption{Inner temperature gradient within $2\,R_{\rm J}$ as a function of $20\mbox{$\rm\,cm$}$ NVSS radio luminosity (top left), central velocity dispersion (top right), absolute $K$ magnitude (or stellar mass, bottom left) and $6\mbox{$\rm\,cm$}$ radio luminosity (bottom right). The dashed lines indicate the best-fit correlations from bandfit, as given in Table \ref{t.gradcorrelations}. \label{f.kmaglradiotslope02} } \end{figure*} \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{f5} \end{center} \caption{Projected galaxy number density \ensuremath{\rho_{\rm 2MASS}}\ vs. outer (\ensuremath{\alpha_{\rm 24}}, top panel) and inner (\ensuremath{\alpha_{\rm 02}}, bottom panel) temperature gradients. Note that \ensuremath{\alpha_{\rm 02}}\ is evidently unaffected by environment, whereas \ensuremath{\alpha_{\rm 24}}\ depends strongly on the density of nearby systems, suggesting the influence of hot ambient gas.\label{f.tempenvironment}} \end{figure} \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{f6} \end{center} \caption{Temperature profiles, as in Figure \ref{f.tempprofiles}, but with colors indicating NVSS $20\mbox{$\rm\,cm$}$ continuum radio luminosity within $3 R_{\rm J}$, as shown by the color bar. The radio luminosity changes continuously as profiles change from positive to negative temperature gradients. \label{f.tempprofilesradio}} \end{figure} \subsection{The Outer Temperature Gradient $\alpha_{24}$}\label{s5.touter} We now look at the outer temperature gradient between $2$ and $4\,R_{\rm J}$, and with what it is correlated. Like $\alpha_{02}$, $\alpha_{24}$ is correlated with the average temperature within $3\,R_{\rm J}$ (Fig. \ref{f.txtslope24}), though less strongly ($2.5\sigma$, compared with $9.1\sigma$). This trend is such that galaxies with a hotter average temperature have stronger positive temperature gradients. Some correlation is expected because these quantities are not truly independent. As a consequence, the outer temperature gradient is also weakly correlated ($2.4\sigma$) with the inner temperature gradient. We now repeat the same analysis for the outer temperature gradient \ensuremath{\alpha_{\rm 24}} . The results of the correlation analysis are listed in the bottom half of Table \ref{t.gradcorrelations}. Unlike \ensuremath{\alpha_{\rm 02}}, \ensuremath{\alpha_{\rm 24}}\ does {\it not} depend on the intrinsic galaxy properties \ensuremath{L_{\rm NVSS}}, \ensuremath{L_{\rm 6cm}}, $\sigma$, or \ensuremath{M_{\rm K}}. Instead, we find strong evidence that \ensuremath{\alpha_{\rm 24}}\ depends only on the environmental density parameters \ensuremath{\rho_{\rm 2MASS}}\ and \ensuremath{\rho_{\rm Tully}}. This trend with environment is statistically even stronger than the one with the luminosity weighted temperature \ensuremath{T_{\rm X}}, even though those parameters are not independent measurements. To check the robustness of these results, we repeat our analysis using larger outer radial boundaries and confirm all trends. The significance of the environmental dependence gets even stronger when extending the analysis to larger radii. These relations are strongest, when one fits temperature gradients to all radii beyond 2\ensuremath{R_{\rm J}} without imposing an outer radial limit. However, since the gradients tend to get stronger with radius, and our galaxies have very different cutoff radii owing to different surface brightness profiles, we do not report the functional form of the fit, as it is driven by the brightest galaxies. Nevertheless, this strengthens the confidence in the observed correlation. We conclude that the inner and outer temperature gradients are essentially decoupled. While the inner gradient depends only on intrinsic galaxy properties, the outer gradient shows no correlations {\it but} with the environment. \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{f7} \end{center} \caption{Outer temperature gradient within $2-4\,R_{\rm J}$ as a function of the average luminosity weighted temperature within $3\,R_{\rm J}$. The dashed line indicates the best fit. \label{f.txtslope24} } \end{figure} \section{Discussion}\label{s5.discussion} \subsection{Implications for Cooling Flows} Steady-state cooling flow models have gone out of fashion recently due to extensive work on galaxy clusters, which show insufficient amounts of cooling gas at the center \citep[][]{PetersonCoolingspec}. These simple models are unlikely to apply to X-ray bright elliptical galaxies either. We showed in Paper II that the hot gas in these systems is almost always disturbed, and we see evidence linking the origin of these disturbances to the central AGN. However, it is far from proven that the same is true for low-luminosity galaxies, in which we find negative inner temperature gradients. Compressive heating during a gradually cooling inflow of relatively cool gas may be able to offset radiative losses for low-temperature gas in steep gravitational potentials \citep[e.g.][]{MathewsReview}. This counter-intuitively results in a cooling flow that gets heated during inflow and may even produce a hot center, i.e. a negative gradient. We also find in Paper II that these systems are generally less disturbed, which could be consistent with a steady state cooling flow solution. \citet{PonmanNGC6482} observe a falling temperature profile for the fossil group candidate NGC~6482 and successfully fit a steady-state cooling flow model with a reasonable cooling rate of $\dot M = 2M_\sun\,\mbox{$\rm\,yr$}^{-1}$. However, they derive the inner gravitational potential from the X-ray profiles themselves assuming hydrostatic equilibrium, which yields a steep inner potential gradient. They then use this potential to fit the cooling flow model to the negative temperature gradient. This could be circular reasoning: a central temperature peak together with the assumption of hydrostatic equilibrium implies a steep gravitational potential, which leads to increased compressional heating \citep[e.g.][]{MathewsReview} and a central temperature peak. In addition, we have argued in Paper I that an assumption of hydrostatic equilibrium is generally not secure. It would be much safer to derive the gravitational potential from independent stellar dynamics, as the inner region is most likely stellar-mass dominated, at least within two effective radii \citep[e.g.][]{MamonDarkmatterI}. In any event, NGC~6482 is the only negative-gradient object that has been successfully fitted with a cooling flow model so far. Only modeling a more complete sample will show if this idea can generally hold. \subsection{Implications for the Existence of Circumgalactic Gas} Negative temperature gradients have been recognized only recently \citep{HumphreyDarkmatter,FukazawaMassprofiles,RandallXMMNGC4649, PonmanNGC6482}. Because earlier observations revealed only positive gradients, many theoretical flow models have been dismissed on the grounds that they produce negative gradients \citep[e.g.][]{MathewsReview}. Instead, theoretical effort has focused on finding an explanation for the prevalence of positive gradients. \citet{BrighentiCircumgas} argue that a hot circumgalactic gas reservoir is able to reverse a negative temperature gradient. This explanation is consistent with our observations that the outer temperature gradient is correlated with the environment. Whether models with circumgalactic gas can quantitatively account for the more complex hybrid temperature profiles remains to be seen. \subsection{Implications for Supernova Feedback} Another means of producing negative temperature gradients involves supernova (SN) feedback. Since star formation should be negligible in elliptical galaxies, this mechanism would involve only contributions from type Ia SN. Early proposed wind models involving SN feedback \citep{BinneyCoolevolution} were later dismissed, since a main feature was a negative temperature gradient throughout their evolution, which had not been observed at that time. However, \citet{MathewsReview} point out that these models are able to reproduce observed gas profiles only for a very short period of time ($\sim 10^8\mbox{$\rm\,yr$}$) just before a cooling catastrophe sets in. Furthermore, these models are sensitive to the assumed SN rate, resulting in abrupt transitions to SN driven winds, and thus require fine-tuning. This fine-tuning problem can be circumvented by the presence of circumgalactic gas \citep{MathewsReview}. However, our observations of purely negative gradients without an outer rise in temperature may present a challenge to this model, though we cannot exclude the possibility of an external gas reservoir below our detection limit. In addition, the predicted metallicities in galactic wind models are generally super-solar, significantly exceeding the historically observed extremely low abundances in the hot gas \citep[][]{ArimotoIrondiscrepancy}. A {\it Chandra} spectral analysis of abundance gradients in a sample of 28 elliptical galaxies by \citet{Humphrey}, on the other hand, no longer shows strongly sub-solar abundances. They attribute this difference to previously imperfect modeling of the spectra, mainly caused by the neglect of the unresolved point source component and attempting to fit multi-temperature gas with a single-temperature model, the so-called iron-bias \citep[e.g.][]{BuoteIronbias}. Nevertheless, they conclude that their abundances are still far too low to be consistent with galactic wind models, and favor the circulation flow model of \citet{Circflow} instead. However, if we consider only the energy input by SNIa feedback, we find that SN feedback could play a role in heating the hot gas. The average SNIa rate for elliptical and S0 galaxies is $r_{\rm SN}=0.18\pm 0.06\, (100\mbox{$\rm\,yr$})^{-1}\, (10^{10} L_{B\sun})^{-1}$ \citep{CappellaroSNrates}. With an average energy injection of $\sim 10^{51}$ ergs per SN, this results in a SN heating rate of $L_{\rm SN}=5.7\times 10^{30} (L_{\rm B}/L_{\rm B, \sun})\, \mbox{$\rm\,erg$} \mbox{$\rm\,s$}^{-1} $. An inspection of the \ensuremath{L_{\rm X,gas}}--$L_{\rm B}$ diagram in Paper I shows that $L_{\rm SN}>\ensuremath{L_{\rm X,gas}}$ for galaxies below the blue luminosity where the inner temperature gradients turn from negative to positive ($\sim 4\times 10^{10} L_{\rm B\sun}$). Although this is somewhat suggestive, the large scatter in X-ray luminosity of almost 2 orders of magnitude at a given blue luminosity, combined with the also rather large scatter in the $L_{\rm B}$--\ensuremath{\alpha_{\rm 02}}\ relation, make it impossible to tell if this is simply coincidence. However, if SN heating were the main cause, we would expect a correlation between the inner temperature gradient and the SN heating to X-ray cooling ratio ($L_{\rm SN}/\ensuremath{L_{\rm X,gas}}$), in the sense that negative gradients would correspond to high ratios of heating to cooling. We do not find such a correlation. $L_{\rm SN}/\ensuremath{L_{\rm X,gas}}$ correlates with the inner temperature gradient very weakly on the less than $0.5\sigma$ level. We conclude that supernova feedback may be important for balancing part of the radiative losses in X-ray faint galaxies, but our analysis suggests that it is most likely not the dominant factor. \subsection{Implications for AGN feedback} In Paper II we have measured the amount of asymmetry in the hot gas, and find a strong correlation between asymmetry and AGN power. This correlation persists all the way down to the weakest AGN luminosities at the detection limit of the NVSS $20\mbox{$\rm\,cm$}$ survey. We now find that the temperature structure is also strongly correlated with the AGN luminosities, another hint toward the importance of AGN feedback throughout the elliptical galaxy population. However, we cannot completely rule out compressionally heated cooling flows or SN feedback to explain the prevalence of negative temperature gradients. Thus, we propose three possible scenarios involving AGN feedback to explain our results: \begin{enumerate} \item Weak AGN with smaller black holes heat the ISM locally, while higher-luminosity sources feed powerful jets that distribute the heat globally by blowing large cavities into the ISM. This is consistent with the observation that smaller elliptical galaxies have rather weak AGN and generally less extended radio emission, and also in agreement with our findings from Papers I and II that the amount of asymmetry correlates with AGN luminosity. In this scenario, weak AGN would still be disturbing the gas, but on a scale and surface brightness level that is simply less detectable, resulting in a lower asymmetry. Negative temperature gradients could then be a sign of very localized heating by the central AGN. \item AGN are responsible for globally heating the hot gas only in X-ray bright galaxies with positive temperature gradients. The onset of negative inner temperature gradients marks the point where AGN heating becomes unimportant, relative to other sources. These other sources could include compressional heating or supernovae. \item The observed temperature gradients are snapshots of different stages of a time-dependent flow, which cyclically reverses the temperature gradient over time. If such solutions exist, it will be challenging for theoretical models to explain the fact that none of our galaxies exhibit central temperatures below $\sim 0.6\mbox{$\rm\,keV$}$. Thus, any cyclic solution has to keep the central temperature rather constant, while reversing the temperature gradient by heating or cooling only at large radii. The best chance to achieve this may be for galaxies to cycle through wind and inflow phases, possibly intimately correlated with the time-dependent AGN activity of the central black hole. \end{enumerate} The possible importance of AGN heating for elliptical galaxies has also recently been pointed out by \citet{BestAGNcooling}. They combine two empirical results to derive an estimate of time-averaged heating by radio sources in galaxies. They use a result by \citet{BirzanCavities} for galaxy clusters that empirically links the $p{\rm d}V$ work associated with inflating X-ray cavities into the intracluster medium with the observed $20\mbox{$\rm\,cm$}$ radio continuum power of the associated radio source. Although this correlation exhibits significant scatter, Best et al. derive a linear fit and use it to convert their radio powers to mechanical energy. In an earlier study, \citet{BestRadioloudAGN} find that the fraction of elliptical galaxies hosting radio-loud AGN correlates with black hole mass and radio luminosity. Assuming that all elliptical galaxies have AGN at their centers, Best et al. interpret the fraction of galaxies with active AGN as the fraction of time that they are turned on. By combining the computed mechanical work per unit radio luminosity derived from \citet{BirzanCavities} with the fraction of time the radio source is turned on, Best et al. calculate the time-averaged mechanical heat input of the AGN. A comparison with the \ensuremath{L_{\rm X,gas}}--$L_{\rm B}$ relation for normal ellipticals shows a remarkable agreement between the time-average AGN heat input and the averaged radiative losses of elliptical galaxies \citep[Figure 2]{BestAGNcooling}. This good agreement is actually surprising, since the conversion factor from radio power to mechanical energy has a rather large scatter and is only based on observations of cluster cavities. Further support for AGN heating has been provided by \citet{FabianAccretionJet}, who measure the mechanical energy associated with X-ray cavities in 9 X-ray luminous elliptical galaxies. They compare this value to the Bondi accretion rate, which they derive from deprojected density and temperature profiles, evaluated at the accretion radius. Allen et al. find a tight correlation between the Bondi accretion rate and the mechanical energy injected into the ISM, and find that this energy input may be sufficient to prevent the gas from cooling. \subsection{What is so special about $\sim 0.6\mbox{$\rm\,keV$}$?} A close inspection of Figure \ref{f.tempprofiles} shows a remarkably small range in central temperature, which falls between $0.6$ to $0.7\mbox{$\rm\,keV$}$. The upper limit owes its origin to our explicit exclusion of brightest cluster galaxies, with higher temperatures, from our sample. Including cluster cDs in our sample would add the missing profiles, adding positive temperature gradients with higher central temperatures. However, the lower limit is quite mysterious. We find it unlikely that this is simply a {\it Chandra} sensitivity effect. We know that our temperature fits are sensitive to lower temperatures, as we can see them in fits to the outer regions of the same objects. Conceivably, this could represent a selection effect imposed on the {\it Chandra} archive through the proposal process, which disfavors observations of systems with lower temperature due to the drop in instrument sensitivity at lower energy. We find this explanation also difficult to believe, as galaxies with negative gradients would have been characterized simply as having a lower mean temperature, since {\it ROSAT} would not have been able to detect the rise in temperature toward the center. However, we do see that the faintest galaxies in our sample exclusively build the lower envelope in the temperature profiles, with luminosity weighted temperatures of $\sim 0.4\mbox{$\rm\,keV$}$. Thus, fainter galaxies could lower this envelope even further, and with it the central temperature. The lower envelope may also mark the transition to a galactic wind, which would render the temperature gradient for these galaxies unobservable due to low gas densities. Nevertheless, something {\it is} special about $\sim 0.6\mbox{$\rm\,keV$}$. First, we do not observe any central temperature below this value. Second, all hybrid temperature profiles drop below $0.6\mbox{$\rm\,keV$}$ at some intermediate radius and then rise back up again. And third, the best fit for the \ensuremath{T_{\rm X}}--\ensuremath{\alpha_{\rm 02}}\ relation puts the transition between negative and positive gradients at $0.64\mbox{$\rm\,keV$}$. Any flow model on the galaxy scale has to be able to reproduce these properties. \section{Conclusions}\label{s5.conclusions} We have reported on the shape of temperature profiles in 36 normal elliptical galaxies. These profiles show a variety of different profile types: purely positive gradients, purely negative gradients, quasi-isothermal and even hybrid profiles. To understand this complexity, we derive mean temperature gradients for an inner region within $2\ensuremath{R_{\rm J}}$, excluding the central point source, and an outer region between $2-4\ensuremath{R_{\rm J}}$. We find that the outer temperature gradient is independent of intrinsic galaxy properties, but a strong function of environment, such that positive outer temperature gradients are restricted to cluster and group environments. This suggests that the outer gradients are caused by interaction with hotter ambient gas, whereas galaxies with negative outer gradients are in less dense environments and lack this intergalactic gas reservoir. The inner temperature gradient, on the other hand, is completely independent of the environmental influence. Instead, we find that it is correlated with a number of intrinsic galaxy properties; in decreasing order of significance, the $20\mbox{$\rm\,cm$}$ radio luminosity, the central velocity dispersion, the absolute $K$ magnitude, and the $6\mbox{$\rm\,cm$}$ radio luminosity. The data cannot rule out the idea that negative gradients can be produced by compressional heating in low-temperature systems, during a slow cooling inflow in a steep gravitational potential. SN feedback may also provide sufficient energy to offset cooling in X-ray faint galaxies, but we find no direct evidence that SN heating dominates. Our preferred feedback model involves the central AGN. The inner temperature gradient is most strongly correlated with radio luminositiy and central velocity dispersion, which may be interpreted as a surrogate for black hole mass \citep{TremaineBHsigma}. The nature of these correlations is such that weak AGN hosts show negative temperature gradients, whereas more luminous AGN exclusively live in positive gradient systems. Thus, we propose three scenarios, to explain the observed features. (1) Weak AGN distribute their heat locally, whereas luminous AGN heat the gas more globally with their extended jets. (2) The onset of negative gradients marks the point where AGN heating becomes unimportant, and compressional heating or SN feedback becomes dominant. (3) A cyclic model in which the AGN drives an outflow, which shuts the AGN activity off until the flow reverses itself, fuels the black hole and starts another cycle. These findings are in agreement with the results from Paper I, which showed that precise hydrostatic equilibrium does not hold for the hot gas in elliptical galaxies, and established the prevalence of disturbances in the X-ray gas morphology. The results of Paper II indicate that the central AGN probably causes these disturbances. Combining these results with the connection between the temperature structure and the radio luminosity of the system produces a strong argument for the general importance of AGN feedback in nearly all normal elliptical galaxies. \acknowledgments We have made use of the HyperLEDA database (http://leda.univ-lyon1.fr). Support for this work was provided by the National Aeronautics and Space Administration (NASA) through Chandra Awards G01-2094X and AR3-4011X, issued by the {\em Chandra X-Ray Observatory Center}, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-39073, and by National Science Foundation grant AST0407152. \bibliographystyle{apj}

proofpile-arXiv_069-11586

\section{Introduction} The Markov chain Monte Carlo (MCMC) method, first proposed by \cite{metropolis}, is a commonly used device for numerical approximation of integrals of the type \[ \pi(f) = \int f(x) \pi(x) \,\ud x, \] where $\pi$ is a probability density function. Intuitively, the method is based on producing a sample $(X_k)_{k=1}^n$ of random variables from the distribution $\pi$ defines. The integral $\pi(f)$ is approximated with the average $I_n\defeq n^{-1}\sum_{k=1}^n f(X_k)$. In particular, the random variables $(X_k)_{k=1}^n$ are a realization of a Markov chain, constructed so that the chain has $\pi$ as the unique invariant distribution. One of the most commonly applied constructions of such a chain in $\R^d$ is to let $X_0\equiv x_0$ with some fixed point $x_0\in\R^d$, and recursively for $n\ge1$: \begin{enumerate} \item simulate $Y_{n} = X_{n-1} + U_n$, where $U_n$ is an independent random variable distributed according to some symmetric proposal distribution $q$, for example, a~zero-mean Gaussian, and \item with probability $\min\{1, \pi(Y_{n})/\pi(X_{n-1})\}$, the proposal is accepted and $X_{n} = Y_{n}$; otherwise the proposal is rejected and $X_{n} = X_{n-1}$. \end{enumerate} This symmetric random-walk Metropolis algorithm is often efficient enough, even in a relatively complex and high-dimensional situation, provided that the proposal distribution $q$ is selected properly. Finding a good proposal for a particular problem can, however, be a difficult task. Recently, there has been a number of publications describing different adaptation techniques aiming to find a good proposal automatically \cite {saksman-am,andrieu-robert,atchade-rosenthal,andrieu-moulines,roberts-rosenthal} (see also the review article \cite{andrieu-thoms}). It has been a common practice to perform trial runs, and determine the proposal from the outcome. The recently proposed methods are different in that they adapt on-the-fly, continuously during the estimation run. In this paper, we focus on the forerunner of these methods, the Adaptive Metropolis (AM) algorithm \cite{saksman-am}, which is a random-walk Metropolis sampler with a Gaussian proposal $q_v$ having a covariance $v$. The proposal covariance $v$ is updated continuously during the run, according to the history of the chain. In general, such an adaptation may, if carelessly implemented, destroy the correct ergodicity properties, that is, that $I_n$ does not converge to $\pi(f)$ as $n\to\infty$ (see, e.g., \cite{roberts-rosenthal}). For practical considerations of the AM algorithm, the reader may consult \cite{saksman-jrstatsoc,roberts-rosenthal-examples}. In the original paper \cite{saksman-am} presenting the AM algorithm, the first ergodicity result for such adaptive algorithms was obtained. More precisely, a strong law of large numbers was proved for bounded functionals, when the algorithm is run on a compact subset of $\R^d$. After that, several authors have obtained more general conditions under which an adaptive MCMC process preserves the correct ergodicity properties. Andrieu and Robert \cite{andrieu-robert} established the connection between adaptive MCMC and stochastic approximation, and proposed a general framework for adaptation. Atchad\'{e} and Rosenthal \cite{atchade-rosenthal} developed further the technique of \cite{saksman-am}. Andrieu and Moulines \cite{andrieu-moulines} made important progress by generalizing the Poisson equation and martingale approximation techniques to the adaptive setting. They proved the ergodicity and a central limit theorem for a class of adaptive MCMC schemes. Roberts and Rosenthal \cite{roberts-rosenthal} use an interesting approach based on coupling to show a weak law of large numbers. However, in the case of AM, all the techniques essentially assume that the adapted parameter is constrained to a predefined compact set, or do not present concrete verifiable conditions. The only result to overcome this assumption is the one by Andrieu and Moulines \cite{andrieu-moulines}. Their result, however, requires a modification of the algorithm, including additional re-projections back to some fixed compact set. This paper describes sufficient conditions under which the AM algorithm preserves the correct ergodicity properties, and $I_n\to\pi(f)$ almost surely as $n\to\infty$ for any function $f$ that is bounded on compact sets and grows at most exponentially as $\|x\|\to\infty$. Our main result (Theorem \ref{thm:am-ergodicity}) holds for the original AM process (without re-projections) having a target distribution supported on~$\R^d$. Essentially, the target density $\pi$ must have asymptotically lighter tails than $\pi(x) = c e^{-\|x\|^p}$ for some $p>1$, and for large enough $\|x\|$, the sets $A_x=\{y\in \R^d\dvtx\pi(y)\ge\pi(x)\}$ must have uniformly regular contours. Our assumptions are very close to the well-known conditions proposed by Jarner and Hansen \cite{jarner-hansen} to ensure the geometric convergence of a (nonadaptive) Metropolis process. By the techniques of this paper, one may also establish a central limit theorem (see Theorem \ref{thm:am-clt}). The ergodicity results for the AM process rely on three main contributions. First, in Section \ref{sec:notations}, we describe an adaptive MCMC framework, in which the adaptation parameter is constrained at each time to a feasible adaptation set. In Section \ref{sec:main-andrieu}, we prove a strong law of large numbers for such a process, through a modification of the technique of Andrieu and Moulines \cite{andrieu-moulines}. Second, we propose an independent estimate for the growth rate of a process satisfying a general drift condition in Section \ref{sec:grow-bound}. Third, in Section \ref{sec:am-ergodicity}, we provide an estimate for constants of geometric drift for a symmetric random-walk Metropolis process, when the target distribution has super-exponentially decaying tails with regular contours. The paper is essentially self-contained, and assumes little background knowledge. Only the basic martingale theory is needed to follow the argument, with the exception of Theorem \ref{th:m-t-computable} by Meyn and Tweedie \cite{meyn-tweedie-computable}, restated in Appendix \ref{sec:proof-lemma-drift-to-conv}. Even though we consider only the AM algorithm, our techniques apply also to many other adaptive MCMC schemes of similar type. \section{General framework and notation} \label{sec:notations} We consider an adaptive Markov chain Monte Carlo (MCMC) chain evolving in space $\mathbb{X}\times\mathbb{S}$, where $\mathbb{X}$ is the state space of the ``MCMC'' chain $(X_n)_{n\ge0}$ and the adaptation parameter $(S_n)_{n\ge0}$ evolves in $\mathbb{S}\subset\overline{\mathbb{S}}$, where $\overline{\mathbb {S}}$ is a separable normed vector space. We assume an underlying probability space $(\Omega,\F_\Omega,\P)$, and denote the expectation with respect to $\P$ by $\E$. The natural filtration of the chain is denoted with $\F\defeq(\F _k)_{k\ge 0}\subset\F_{\Omega}$ where $\F_k\defeq\sigma(X_j,S_j\dvtx0\le j\le k)$. We also assume that we are given an increasing sequence $K_0\subset K_1\subset\cdots\subset K_n \subset\mathbb{S}$ of subsets of the adaptation parameter space $\mathbb{S}$. The random variables $(X_n,S_n)_{n\ge0}$ form a stochastic chain, starting from $S_0\equiv s_0\in K_0\subset\mathbb{S}$ and $X_0 \equiv x_0\in \mathbb{X}$, and for $n\ge0$, satisfying the following recursion: \begin{eqnarray} \label{eq:state-eq} X_{n+1} &\sim& P_{S_n}(X_n, \uarg), \\ \label{eq:adapt-eq} S_{n+1} &=& \sigma_{n+1} (S_n, \eta_{n+1}H(S_n, X_{n+1}) ), \end{eqnarray} where $P_s$ is a transition probability for each $s\in\mathbb{S}$, $H\dvtx\mathbb{S}\times\mathbb{X}\to\overline{\mathbb{S}}$ is an adaptation function, and $(\eta_n)_{n\ge1}$ is a decreasing sequence of adaptation step sizes $\eta_n\in(0,1)$. The functions $\sigma_n\dvtx\mathbb{S}\times\overline{\mathbb {S}}\to \mathbb{S}$ are defined as \[ \sigma_n(s,s') \defeq\cases{ s+s',&\quad if $s+s'\in K_n$, \cr s,&\quad otherwise.} \] Thus, $\sigma_n$ ensures that $S_n$ lies in $K_n$ for each $n\ge0$. The recursion (\ref{eq:adapt-eq}) can also be considered as constrained Robbins--Monro stochastic approximation (see \cite{andrieu-moulines,sa-verifiable} and references therein). Let $V\dvtx\mathbb{X}\to[1,\infty)$ be a function. We define a $V$-norm of a function $f$ as \[ \|f\|_V \defeq\sup_{x\in\mathbb{X}} \frac{|f(x)|}{V(x)}. \] As usual, we denote the integration of a function $f$ with respect to a (signed) measure $\mu$ as $\mu(f)\defeq\int f( x) \mu(\ud x)$, and define $Pf(x) \defeq\int f(y) P(x,\ud y)$ for a transition probability $P$. The $V$-norm of a signed measure is defined as \[ \|\mu\|_V \defeq{\sup_{|f|\le V}} |\mu(f)|. \] The indicator function of a set $A$ is denoted as $\mathbh{1}_{A}(x)$ and equals one if $x\in A$ and zero otherwise. In addition, we use the notation $a\vee b\defeq\max\{a,b\}$ and $a \wedge b\defeq\min\{a,b\}$. Finally, we define the following regularity property for a family of functions $\{f_s\}_{s\in\mathbb{S}}$. \begin{definition} \label{def:poly-lip} Suppose $V\dvtx\mathbb{X}\to[1,\infty)$. Given an increasing sequence of subsets $K_n\subset\mathbb{S}$, $n\ge1$, we say that a family of functions $\{f_s\}_{s\in\mathbb{S}}$, with $f_s\dvtx \mathbb{X}\to\R $, is $(K_n,V)$-polynomially Lipschitz with constants $c\ge 1,\varepsilon\ge0$, if for all $s,s'\in K_n$, we have \[ \|f_s\|_V \le c n^\varepsilon\quad \mbox{and}\quad \|f_s - f_{s'}\|_V \le c n^\varepsilon|s-s'|. \] \end{definition} \section{Ergodicity of sequentially constrained adaptive MCMC} \label{sec:main-andrieu} This section contains general ergodicity results for a sequentially constrained process defined in Section \ref{sec:notations}. These results can be seen auxiliary to our results on Adaptive Metropolis in Section \ref{sec:am-ergodicity}, but may be applied to other adaptive MCMC methods as well. Suppose that the adaptation algorithm has the form given in (\ref{eq:state-eq}) and (\ref{eq:adapt-eq}), and the following assumptions are satisfied for some $c\ge1$ and $\varepsilon\ge0$. \begin{enumerate}[(A1)] \item[(A1)]\hypertarget{a:invariance} For each $s\in\mathbb{S}$, the transition probability $P_s$ has $\pi $ as the unique invariant distribution. \item[(A2)]\hypertarget{a:uniform-drift-mino} For each $n\ge1$, the following uniform drift and minorization condition holds for all $s\in K_n$: \begin{eqnarray} \label{eq:drift-ineq} P_s V(x) &\le&\lambda_n V(x) + b_n \mathbh{1}_{C_n}(x) \qquad \forall x\in\mathbb{X},\\ \label{eq:mino-ineq} P_s(x,A) &\ge&\delta_n \nu_s(A)\qquad \forall x\in C_n, \forall {A}\subset\mathbb{X}, \end{eqnarray} where $C_n\subset\mathbb{X}$ is a subset (a minorization set), $V\dvtx\mathbb{X}\to[1,\infty)$ is a drift function such that $\sup _{x\in C_n}V(x) \le b_n$, and $\nu_s$ is a probability measure on $\mathbb{X}$, concentrated on $C_n$. Furthermore, the constants $\lambda_n\in(0,1)$ and $b_n\in(0,\infty)$ are increasing, and $\delta_n\in(0,1]$ is decreasing with respect to $n$, and they are polynomially bounded so that \[ (1-\lambda_n)^{-1} \vee\delta_n^{-1} \vee b_n \le c n^{\varepsilon}. \] \item[(A3)]\hypertarget{a:kernel-lip} For all $n\ge1$ and any $r\in(0,1]$, there is $c'=c'(r)\ge1$ such that for all $s,s'\in K_n$, \[ \|P_s f - P_{s'} f \|_{V^r} \le c' n^{\varepsilon} \|f\|_{V^r} | s - s'| . \] \item[(A4)]\hypertarget{a:adapt-bound} There is a $\beta\in[0,1/2]$ such that for all $n\ge1$, $s\in K_n$ and $x\in\mathbb{X}$ \[ |H(s,x)| \le c n^\varepsilon V^\beta(x). \] \end{enumerate} \begin{theorem}\label{thm:slln-restricted} Assume \textup{\hyperlink{a:invariance}{(A1)}--\hyperlink{a:adapt-bound}{(A4)}} hold and let $f$ be a function with $\|f\|_{V^\alpha} < \infty$ for some $\alpha\in(0,1-\beta)$. Assume $\varepsilon< \kappa_*^{-1} [(1/2) \wedge(1-\alpha-\beta) ]$, where $\kappa_*\ge1$ is an independent constant, and that $\sum_{k=1}^\infty k^{\kappa_*\varepsilon-1} \eta_k < \infty$. Then \begin{equation} \label{eq:slln} \frac{1}{n} \sum_{k=1}^n f(X_k) \stackrel{n\to\infty}{\hbox to 1cm{\rightarrowfill}} \pi(f)\qquad \mbox{almost surely.} \end{equation} \end{theorem} The proof of Theorem \ref{thm:slln-restricted} is postponed to the end of this section. We start by the following lemma, whose proof is given in Appendix \ref{sec:proof-lemma-drift-to-conv}. It shows that if we have polynomially worse bounds for drift and minorization constants, then the speed of geometric convergence can get only polynomially worse. \begin{lemma} \label{lemma:drift-to-conv} Suppose \textup{\hyperlink{a:uniform-drift-mino}{(A2)}} holds. Then, one has for $r\in(0,1]$ that for all $s\in K_n$ and $k\ge1$, \[ \|P_s^k(x,\uarg)-\pi(\uarg)\|_{V^r} \le V^r(x) L_n \rho_n^k \] with bound \[ L_n \vee(1-\rho_n)^{-1} \le c_2 n^{\kappa_2\varepsilon}, \] where $\kappa_2>0$ is an independent constant, and $c_2=c_2(c,r)\ge1$. \end{lemma} Observe that the statement in Lemma \ref{lemma:drift-to-conv} entails that any function $\|f\|_{V} <\infty$ is integrable with respect to the measures $\pi$ and $P_s^k(x,\uarg)$, for all $x\in\mathbb{X}$, $k\ge1$ and $s\in\bigcup_{n\ge0} K_n$. The next three results are modified from Proposition 3, Lemma 5 and Proposition 6 of \cite{andrieu-moulines}, respectively. The first one bounds the regularity of the solutions $\hat{f}_s$ of the Poisson equation \begin{equation}\label{eq:poisson} \hat{f}_s - P_s \hat{f}_s = f_s - \pi(f_s) \end{equation} for a polynomially Lipschitz family of functions. \begin{proposition} \label{prop:poisson-regularity} Suppose that \textup{\hyperlink{a:invariance}{(A1)}--\hyperlink{a:kernel-lip}{(A3)}} hold, and the family of functions $\{f_s\}_{s\in\mathbb{S}}$ is $(K_n,V^r)$-polynomially Lipschitz with constants $(c,\varepsilon)$, for some $r\in(0,1]$. There is an independent constant $\kappa_3>0$ and a constant \mbox{$c_3 = c_3(c,c',r)\ge1$}, such that: \begin{longlist} \item The family $\{P_s f_s\}_{s\in\mathbb{S}}$ is $(K_n, V^r)$-polynomially Lipschitz with constants $(c_3,\kappa_3\varepsilon)$. \item Define, for any $s\in\mathbb{S}$, the function \begin{equation}\label{eq:poisson-solution} \hat{f}_s \defeq\sum_{k=0}^\infty[P_s^k f_s - \pi(f_s) ]. \end{equation} Then, $\hat{f}_s$ solves the Poisson equation (\ref{eq:poisson}), and the families $\{\hat{f}_s\}_{s\in\mathbb{S}}$ and $\{P_s \hat{f}_s\}_{s\in\mathbb{S}}$ are $(K_n, V^r)$-polynomially Lipschitz with constants $(c_3, \kappa_3\varepsilon)$. In other words, \begin{eqnarray} \label{eq:poisson-reg1} \|\hat{f}_s\|_{V^r} + \|P_s\hat{f}_s\|_{V^r} &\le& c_3 n^{\kappa_3\varepsilon}, \\ \label{eq:poisson-reg2} \|\hat{f}_s- \hat{f}_{s'}\|_{V^r} + \|P_s\hat{f}_s - P_{s'}\hat{f}_{s'}\|_{V^r} &\le& c_3 n^{\kappa_3\varepsilon} |s-s'| \end{eqnarray} for all $s,s'\in K_n$. \end{longlist} \end{proposition} \begin{pf} Throughout the proof, suppose $s,s'\in K_n$. The part (i) follows easily from Lemma \ref{lemma:drift-to-conv}, since \begin{eqnarray*} \|P_s f_s\|_{V^r} &\le& \|P_s f_s - \pi(f_s)\|_{V^r} + |\pi(f_s)|\le[c_2 n^{\kappa_2 \varepsilon} + \pi(V^r)] \| f_s\|_{V^r}, \\ \|P_s f_s - P_{s'} f_{s'}\|_{V^r} &\le& \|(P_s - P_{s'})f_s\|_{V^r} + \|P_{s'}(f_s-f_{s'})\|_{V^r} \\ &\le& c' n^{\varepsilon} \|f_s\|_{V^r} |s-s'| + \tilde{c} n^{\kappa_2\varepsilon}\|f_s-f_{s'}\|_{V^r} \le\tilde{c} n^{(\kappa_2+1)\varepsilon} |s-s'|. \end{eqnarray*} Consider then (ii). Estimate (\ref{eq:poisson-reg1}) follows by the definition of $\hat{f}_s$ and Lemma~\ref{lemma:drift-to-conv}, \begin{eqnarray*} \|\hat{f}_s\|_{V^r} &\le&\sum_{k=0}^\infty\|P_s^k f_s - \pi(f_s)\|_{V^r} \le L_n \|f_s\|_{V^r} \sum_{k=0}^\infty\rho_n^k\\ &=&\frac{L_n}{1-\rho_n} \|f_s\|_{V^r} \le(c_2 n^{\kappa_2\varepsilon})^2 c n^{\varepsilon} = c_2^2 c n^{(2\kappa_2 + 1)\varepsilon}. \end{eqnarray*} The above bound clearly applies also to $\|P_s \hat{f}_s\|_{V^r}$, and the convergence implies that $\hat{f}_s$ solves (\ref{eq:poisson}). For (\ref{eq:poisson-reg2}), define an auxiliary transition probability by setting $\Pi(x,{A})\defeq\pi(A)$, and write \[ P_s^k f - P_{s'}^k f = \sum_{j=0}^{k-1} (P_s^j - \Pi)(P_s-P_{s'})[P_{s'}^{k-j-1} f - \pi(f) ] \] since $\pi P_s = \pi$ for all $s$. By Lemma \ref{lemma:drift-to-conv} and assumption \hyperlink{a:kernel-lip}{(A3)}, we have for all $s,s'\in K_n$ and $j\ge0$ \begin{eqnarray*} &&\| (P_s^j - \Pi)(P_s-P_{s'}) [P_{s'}^{k-j-1} f - \pi(f) ] \|_{V^r} \\ &&\qquad\le L_n \rho_n^j \| (P_s-P_{s'}) [P_{s'}^{k-j-1} f - \pi(f) ]\|_{V^r} \\ &&\qquad\le L_n \rho_n^j c' n^{\varepsilon} |s-s'| \|P_{s'}^{k-j-1} f - \pi(f)\|_{V^r} \\ &&\qquad\le L_n^2 \rho_n^{k-1}, \end{eqnarray*} which gives that \begin{equation} \label{eq:kernel-diff-estimate} \| P_s^k f - P_{s'}^k f\|_{V^r} \le k L_n^2 \rho_n^{k-1} c' n^{\varepsilon} |s-s'| \|f\|_{V^r}. \end{equation} Write then \[ \hat{f}_s - \hat{f}_{s'} = \sum_{k=0}^\infty[ P_s^k f_s - P_{s'}^k f_s ] - \sum_{k=0}^\infty[ P_{s'}^k(f_{s'}-f_s)-\pi(f_{s'}-f_s) ]. \] By Lemma \ref{lemma:drift-to-conv} and estimate (\ref{eq:kernel-diff-estimate}) we have \begin{eqnarray*} \|\hat{f}_s - \hat{f}_{s'} \|_{V^r} &\le& L_n^2 c' n^{\varepsilon} |s-s'| \Biggl(\sum_{k=0}^\infty k\rho_n^{k-1} \Biggr) \|f_s\|_{V^r} + L_n \Biggl(\sum_{k=0}^\infty\rho_n^k \Biggr) \|f_{s'}-f_s\|_{V^r} \\ &\le&[ L_n^2 c' n^{\varepsilon} (1-\rho_n)^{-2} c n^{\varepsilon} + L_n (1-\rho_n)^{-1} c n^{\varepsilon} ] |s-s'| \\ &\le&[ (c_2 n^{\kappa_2\varepsilon})^2 c' n^{\varepsilon} (c_2 n^{\kappa_2\varepsilon})^{2} c n^{\varepsilon} + (c_2 n^{\kappa_2\varepsilon}) (c_2 n^{\kappa_2\varepsilon}) c n^{\varepsilon} ] |s-s'| \\ &\le&2 c_2^4 c' c n^{(4\kappa_2+2)\varepsilon} |s-s'|. \end{eqnarray*} The same bound applies, with a similar argument, to $P_s \hat{f}_s - P_{s'}\hat{f}_{s'}$. \end{pf} \begin{lemma} \label{lemma:stability} Assume that \textup{\hyperlink{a:uniform-drift-mino}{(A2)}} holds. Then, for all $r\in[0,1]$, any sequence $(a_n)_{n\ge1}$ of positive numbers, and $(x_0,s_0)\in\mathbb{X}\times K_0$, we have that \begin{eqnarray}\label{eq:stability1} \E[V^r(X_k) ] &\le& c_4^r k^{2r\varepsilon} V^r(x_0), \\ \label{eq:stability2} \E\Bigl[\max_{m \le j\le k} (a_j V(X_j))^r \Bigr] &\le& c_4^r \Biggl( \sum_{j=m}^k a_j j^{2\varepsilon} \Biggr)^r V^r(x_0), \end{eqnarray} where the constant $c_4$ depends only on $c$. \end{lemma} \begin{pf} For $(x_0,s_0)\in\mathbb{X}\times K_0$ and $k\ge1$, we can apply the drift inequality (\ref{eq:drift-ineq}) and the monotonicity of $\lambda_k$ and $b_k$ to obtain \begin{eqnarray} \label{eq:drift-estimate}\quad \mathbb{E} [V(X_k) ] &=& \mathbb{E} [\mathbb{E}[V(X_k) | \F_{k-1}] ] = \mathbb{E} [P_{S_{k-1}}V(X_{k-1}) ]\nonumber\\ &\le&\lambda_k \mathbb{E} [V(X_{k-1}) ] + b_k \le\cdots\le\lambda_k^k V(x_0) + b_k\sum_{j=0}^{k-1} \lambda _k^j\\ &\le&\Biggl(1+b_k\sum_{j=0}^\infty\lambda_k^j\Biggr)V(x_0) \le(1+c^2k^{2\varepsilon})V(x_0) \le c_4 k^{2\varepsilon} V(x_0).\nonumber \end{eqnarray} This estimate with Jensen's inequality yields for $r\in[0,1]$ that \[ \mathbb{E} [V^r(X_k) ] \le(\mathbb{E} [V(X_k) ] )^r \le c_4^r k^{2r\varepsilon} V^r(x_0). \] Similarly, we have \begin{eqnarray*} \mathbb{E} \Bigl[\max_{m\le j\le k} (a_j V(X_j))^r \Bigr] &\le&\Bigl( \mathbb{E} \Bigl[\max_{m\le j\le k} a_j V(X_j) \Bigr] \Bigr)^r\\ &\le&\Biggl( \sum_{j=m}^k a_j\mathbb{E} [V(X_j) ] \Biggr)^r\\ &\le& c_4^r \Biggl(\sum_{j=m}^k a_j j^{2\varepsilon} \Biggr)^r V^r(x_0) \end{eqnarray*} by Jensen's inequality and estimate (\ref{eq:drift-estimate}). \end{pf} Assume that $\{f_s\}_{s\in\mathbb{S}}$ is a regular enough family of functions. Consider the following decomposition, which is one of the key observations in \cite{andrieu-moulines}, \begin{equation} \label{eq:martingale-decomposition} \sum_{j=1}^k [f_{S_j}(X_j) - \pi(f_{S_j}) ] = M_k + R^{(1)}_k + R^{(2)}_k, \end{equation} where $(M_k)_{k\ge1}$ is a martingale with respect to $\F$, and $(R^{(1)}_k)_{k\ge1}$ and $(R^{(2)}_k)_{k\ge1}$ are ``residual'' sequences, given by \begin{eqnarray*} M_k &\defeq& \sum_{j=1}^k [ \hat{f}_{S_{j-1}}(X_j) - P_{S_{j-1}}\hat{f}_{S_{j-1}}(X_{j-1}) ], \\ R^{(1)}_k &\defeq& \sum_{j=1}^k [ \hat{f}_{S_{j}}(X_j) - \hat{f}_{S_{j-1}}(X_{j}) ], \\ R^{(2)}_k &\defeq& P_{S_0}\hat{f}_{S_{0}}(X_0) - P_{S_k}\hat{f}_{S_{k}}(X_{k}). \end{eqnarray*} Recall that $\hat{f}_s$ solves the Poisson equation (\ref{eq:poisson}). The following proposition controls the fluctuations of these terms individually. \begin{proposition} \label{prop:main-andrieu} Assume \textup{\hyperlink{a:invariance}{(A1)}--\hyperlink{a:adapt-bound}{(A4)}} hold, $(x_0,s_0)\in\mathbb{X}\times K_0$ and let $\{f_s\}_{s\in \mathbb{S}}$ be $(K_n, V^\alpha)$-polynomially Lipschitz with constants $(c,\varepsilon)$ for some $\alpha\in(0,1-\beta)$. Then, for any $p\in(1,(\alpha+\beta)^{-1}]$, for all $\delta>0$ and $\xi>\alpha$, there is a $c_* = c_*(c,p,\alpha,\beta,\xi)\ge1$, such that for all $n\ge1$, \begin{eqnarray} \label{eq:main-thm-m} \P\biggl[ \sup_{k\ge n} \frac{| M_k|}{k} \ge\delta\biggr] &\le& c_* \delta^{-p} n^{p\varepsilon_*-(p/2)\wedge(p-1)} V^{\alpha p}(x_0), \\ \label{eq:main-thm-r1} \P\biggl[ \sup_{k\ge n} \frac{|R_k^{(1)}|}{k^\xi}\ge\delta\biggr] &\le& c_* \delta^{-p} \Biggl(\sum_{j=1}^\infty (j\vee n)^{\varepsilon_*-\xi}\eta_j \Biggr)^p V^{(\alpha+\beta)p}(x_0), \\ \label{eq:main-thm-r2} \P\biggl[ \sup_{k\ge n} \frac{|R_k^{(2)}|}{k^\xi}\ge\delta\biggr] &\le& c_* \delta^{-p} n^{p\varepsilon_*-(\xi-\alpha)p} V^{\alpha p}(x_0), \end{eqnarray} whenever $\varepsilon>0$ is small enough to ensure that $\varepsilon_* \defeq\kappa_* \varepsilon< [\frac{1}{2} \wedge(1- \frac{1}{p} ) \wedge(\xi-\alpha) ]$, where $\kappa_*\ge1$ is an independent constant. \end{proposition} \begin{pf} In this proof, $\tilde{c}$ is a constant that can take different values at each appearance. By Proposition \ref{prop:poisson-regularity}, we have that $\|\hat{f}_s\|_{V^\alpha} + \| P_s\hat{f}_s\|_{V^\alpha} \le c_3 \ell^{\kappa_3\varepsilon}$ for all $s\in K_\ell$. Since $\alpha p \in [0,1]$, we can bound the martingale differences $\ud M_\ell\defeq M_\ell- M_{\ell-1}$ for $\ell\ge1$ as follows: \begin{eqnarray} \label{eq:dm-estimate} \E |\ud M_\ell|^p &=& \E| \hat{f}_{S_{\ell-1}}(X_\ell) - P_{S_{\ell-1}}\hat{f}_{S_{\ell-1}}(X_{\ell-1}) |^p\nonumber\\ &\le&\E\bigl| \|\hat{f}_{S_{\ell-1}}\|_{V^\alpha} V^\alpha(X_\ell) + \| P_{S_{\ell-1}}\hat{f}_{S_{\ell-1}}\|_{V^\alpha} V^\alpha (X_{\ell-1}) \bigr|^p\nonumber\\[-8pt]\\[-8pt] &\le& 2^p(c_3 \ell^{\kappa_3\varepsilon})^p \bigl( \mathbb{E} [V^{\alpha p}(X_\ell) ] + \mathbb{E} [V^{\alpha p}(X_{\ell-1}) ] \bigr)\nonumber\\ &\le& 2^{p+1}c_3^p c_4^{\alpha p} \ell^{p\kappa_3\varepsilon}\ell^{2\alpha p\varepsilon} V^{\alpha p}(x_0) \le\tilde{c} \ell^{(\kappa_3 + 2\alpha)p\varepsilon} V^{\alpha p}(x_0)\nonumber \end{eqnarray} by (\ref{eq:stability1}) of Lemma \ref{lemma:stability}. For $p\ge2$, we have, by Burkholder and Minkowski's inequalities, \begin{eqnarray*} \E|M_k|^p &\le& c_p \mathbb{E} \Biggl[\sum_{\ell=1}^k |\ud M_\ell|^2 \Biggr]^{p/2} \le c_p \Biggl[\sum_{\ell=1}^k (\E |\ud M_\ell|^p )^{2/p} \Biggr]^{p/2}\\ &\le& \tilde{c} k^{(\kappa_3 + 2\alpha)p\varepsilon+ p/2} V^{\alpha p}(x_0), \end{eqnarray*} where the constant $c_p$ depends only on $p$. For $1 < p \le2$, the estimate (\ref{eq:dm-estimate}) yields, by Burkholder's inequality, \begin{eqnarray*} \E|M_k|^p &\le& c_p \mathbb{E} \Biggl[\sum_{\ell=1}^k (|\ud M_\ell|^p)^{2/p} \Biggr]^{p/2} \le c_p \sum_{\ell=1}^k \E|\ud M_\ell|^p\\ &\le&\tilde{c} k^{(\kappa_3 +2\alpha)p\varepsilon+ 1} V^{\alpha p}(x_0). \end{eqnarray*} The two cases combined give that \begin{equation} \label{eq:m-ex-p} \E|M_k|^p \le\tilde{c} k^{(\kappa_3+2\alpha)p\varepsilon+ (p/2)\vee1} V^{\alpha p}(x_0). \end{equation} Now, by Corollary \ref{cor:birnbaum-marshall-martingale} of Birnbaum and Marshall's inequality in Appendix~\ref{sec:martingale}, \begin{eqnarray*} \P\biggl[ \max_{n\le k \le m} \frac{|M_k|}{k} \ge\delta \biggr] &\le&\delta^{-p} \Biggl[ m^{-p} \E|M_m|^p + \sum_{k=n}^{m-1} \bigl(k^{-p} - (k+1)^{-p} \bigr) \E|M_k|^p \Biggr] \\ &\le&\delta^{-p} \Biggl[ m^{-p} \E|M_m|^p + p\sum_{k=n}^{m-1} k^{-p-1} \E|M_k|^p \Biggr] \end{eqnarray*} for all $m\ge n$. By letting $\kappa_* \defeq\kappa_3 + 3$, we have from (\ref{eq:m-ex-p}) \[ m^{-p} \E|M_m|^p \le\tilde{c} m^{p(\kappa_*\varepsilon+ (1/2)\vee(1/p) - 1)} \stackrel{m\to\infty}{\hbox to 1cm{\rightarrowfill}}0, \] since $\kappa_*\varepsilon+ (1/2)\vee(1/p) < 1$. Now, (\ref{eq:main-thm-m}) follows by \begin{eqnarray*} \P\biggl[ \sup_{k\ge n} \frac{|M_k|}{k} \ge\delta \biggr] &\le&\tilde{c} \delta^{-p} \Biggl[ \sum_{k=n}^{\infty} k^{(\kappa_3+2\alpha)p\varepsilon+ (p/2)\vee1 - p - 1} \Biggr] V^{\alpha p}(x_0)\\ &\le&\tilde{c} \delta^{-p} n^{p\kappa_*\varepsilon- (p/2)\wedge(p - 1) } V^{\alpha p}(x_0) \end{eqnarray*} since we have that $p\kappa_*\varepsilon- (p/2)\wedge(p - 1)<0$. By Proposition \ref{prop:poisson-regularity}, $\|\hat{f}_s-\hat{f}_{s'}\|_{V^\alpha} \le c_3 \ell^{\kappa_3\varepsilon}|s -s'|$ for $s,s'\in K_\ell$. By construction, $|S_\ell-S_{\ell-1}|\le\eta _{\ell} |H(S_{\ell-1},X_\ell)|$, and assumption \hyperlink{a:adapt-bound}{(A4)} ensures that $|H(S_{\ell-1},X_\ell)|\le c \ell^\varepsilon V^\beta(X_\ell)$, so \[ |\hat{f}_{S_\ell}(X_\ell)-\hat{f}_{S_{\ell-1}}(X_\ell) | \le c_3 \ell^{\kappa_3\varepsilon}|S_\ell-S_{\ell-1}| V^\alpha (X_\ell) \le c_3 \ell^{\kappa_3\varepsilon}\eta_\ell c \ell^{\varepsilon} V^{\alpha+\beta}(X_\ell). \] Let $k\ge n$. Since $\ell^{(\kappa_3+1)\varepsilon}k^{-\xi} \le (\ell\vee n)^{(\kappa_3+1)\varepsilon-\xi}$ for $\ell\le k$, we obtain \begin{eqnarray*} k^{-\xi} \bigl|R_k^{(1)}\bigr| &\le& k^{-\xi} \sum_{\ell=1}^k | \hat{f}_{S_\ell}(X_\ell)-\hat {f}_{S_{\ell-1}}(X_\ell) | \\ &\le&\tilde{c} \sum_{\ell=1}^k (\ell\vee n)^{(\kappa _3+1)\varepsilon-\xi } \eta_\ell V^{\alpha+\beta}(X_\ell) \end{eqnarray*} and then by Minkowski's inequality and (\ref{eq:stability1}) of Lemma \ref{lemma:stability}, \begin{eqnarray} \label{eq:exp-estim-r1} && \E \Bigl[\max_{n\le k\le m} k^{-\xi p} \bigl|R^{(1)}_k\bigr|^p\Bigr] \nonumber\\ &&\qquad\le\E\Biggl[ \sum_{\ell=1}^m \tilde{c} (\ell\vee n)^{(\kappa_3+1)\varepsilon -\xi} \eta_\ell V^{(\alpha+\beta)p}(X_\ell) \Biggr]^p \nonumber\\[-8pt]\\[-8pt] &&\qquad\le\tilde{c} \Biggl[ \sum_{\ell=1}^m \bigl(\E \bigl[(\ell\vee n)^{(\kappa_3+1)\varepsilon-\xi} \eta_\ell V^{\alpha+\beta}(X_\ell) \bigr]^p \bigr)^{1/p} \Biggr]^p \nonumber\\ &&\qquad\le\tilde{c} \Biggl[\sum_{\ell=1}^\infty (\ell\vee n)^{(\kappa_3+1+2\alpha+2\beta)\varepsilon-\xi} \eta _\ell\Biggr]^p V^{(\alpha+\beta)p}(x_0).\nonumber \end{eqnarray} Finally, consider $R_k^{(2)}$. From Proposition \ref{prop:poisson-regularity}, we have that $\|P_{S_k}\hat{f}_{S_k}(X_k)\|_{V^\alpha} \le c_3 k^{\kappa_3\varepsilon}$, and by (\ref{eq:stability2}) of Lemma \ref{lemma:stability}, \begin{eqnarray*} \E\Bigl[ \max_{n\le k\le m} k^{-\xi p} |P_{S_k}\hat{f}_{S_k}(X_k)|^p \Bigr] &\le& c_3^p \E\Bigl[ \max_{n\le k\le m} \bigl(k^{(\kappa_3\varepsilon-\xi)/\alpha} V(X_k) \bigr)^{\alpha p} \Bigr] \\ &\le& c_3^{p}c_4^{\alpha p} \Biggl( \sum_{k=n}^m k^{(\kappa_3\varepsilon-\xi)/\alpha+2\varepsilon} \Biggr)^{\alpha p} V^{\alpha p}(x_0) \\ &\le& \tilde{c} n^{ (\kappa_3+2\alpha)p\varepsilon+(\alpha-\xi) p} V^{\alpha p}(x_0) \end{eqnarray*} since $(\kappa_3 + 2\alpha)\varepsilon- (\xi-\alpha) < 0$. So, we have that \begin{eqnarray} \label{eq:exp-estim-r2}\quad \mathbb{E} \Bigl[\sup_{k\ge n} k^{-\xi p} \bigl| R^{(2)}_k \bigr|^p \Bigr] &\le& 2^p \mathbb{E} \Bigl[\sup_{k\ge n} k^{-\xi p} \bigl( |P_{S_0}\hat{f}_{S_0}(X_0)|^p + |P_{S_k}\hat{f}_{S_k}(X_k)|^p \bigr) \Bigr] \nonumber\\ &\le& 2^p \mathbb{E} \Bigl[|P_{S_0}\hat{f}_{S_0}(X_0)|^p + \sup_{k\ge n} k^{-\xi p} |P_{S_k}\hat{f}_{S_k}(X_k)|^p \Bigr] \\ &\le& \tilde{c} n^{ (\kappa_3+2\alpha)p\varepsilon+(\alpha-\xi) p} V^{\alpha p}(x_0).\nonumber \end{eqnarray} The estimates (\ref{eq:main-thm-r1}) and (\ref{eq:main-thm-r2}) follow by Markov's inequality from (\ref{eq:exp-estim-r1}) and (\ref{eq:exp-estim-r2}). \end{pf} The proof of Theorem \ref{thm:slln-restricted} follows as a straightforward application of Proposition~\ref{prop:main-andrieu}. \begin{pf*}{Proof of Theorem \ref{thm:slln-restricted}} Let $\delta>0$, and denote \[ {B}_n^{(\delta)} \defeq \Biggl\{ \omega\in\Omega: \sup_{k\ge n} \frac{1}{k} \Biggl| \sum_{j=1}^k [f(X_j) -\pi(f) ] \Biggr| \ge\delta \Biggr\}. \] Since $\|f\|_{V^\alpha}<\infty$ by assumption, we may consider the family $\{f_s\}_{s\in\mathbb{S}}$ with $f_s \equiv f$ for all $s\in \mathbb{S}$. Then, we have by decomposition (\ref{eq:martingale-decomposition}) that \begin{equation} \label{eq:slln-estimation}\qquad \P\bigl(B_n^{(\delta)} \bigr) \le \P\biggl[\sup_{k\ge n} \frac{| M_k |}{k} \ge\frac{\delta}{3} \biggr] +\P\biggl[\sup_{k\ge n} \frac{| R_k^{(1)} |}{k} \ge\frac{\delta}{3} \biggr] +\P\biggl[\sup_{k\ge n} \frac{| R_k^{(2)} |}{k} \ge\frac{\delta}{3} \biggr]. \end{equation} We select $p\in(1,(\alpha+\beta)^{-1})$ so that $\kappa_*\varepsilon<(1-1/p)$, and let $\xi=1$. Then, Proposition \ref{prop:main-andrieu} readily implies that the first and the third terms in (\ref{eq:slln-estimation}) converge to zero as $n\to\infty$. For the second term, consider \[ \sum_{j=1}^\infty (j\vee n)^{\kappa_*\varepsilon-1}\eta_j = n^{\kappa_*\varepsilon-1}\sum_{j=1}^n \eta_j + \sum _{j=n+1}^\infty j^{\kappa_*\varepsilon-1}\eta_j, \] where the second term converges to zero by assumption, and the first term by Kronecker's lemma. There is an increasing sequence $(n_k)_{k\ge1}$ such that $\P({B}_{n_k}^{(1/k)})\le k^{-2}$. Denoting ${B}\defeq \bigcap_{m=1}^\infty\bigcup_{k=m}^\infty{B}_{n_k}^{(1/k)}$, the Borel--Cantelli lemma implies that $P(B^\complement)=1$, and for all $\omega\in{B}^\complement$, (\ref{eq:slln}) holds. \end{pf*} \section{Bound for the growth rate} \label{sec:grow-bound} In this section, we assume that $\mathbb{X}$ is a normed space, and establish a bound for the growth rate of the chain $(\|X_n\|)_{n\ge1}$, based on a general drift condition. The bound assumes little structure; one must have a drift function $V$ that grows rapidly enough, and that the expected growth of $V(X_n)$ is moderate. \begin{proposition} \label{prop:grow-bound} Suppose that there is $V:\mathbb{X}\to[1,\infty)$ such that the bound \begin{equation} \label{eq:drift-bound} P_s V(x) \le V(x) + b \end{equation} holds for all $(x,s)\in\mathbb{X}\times\mathbb{S}$, where $b<\infty $ is a constant independent of $s$. Suppose also that $V$ grows rapidly enough so that \begin{equation} \label{eq:norm-v-bound-1} \|x\| \ge u \quad\Longrightarrow\quad V(x)\ge r(u) \end{equation} for all $u\ge0$, where $r\dvtx[0,\infty)\to[0,\infty)$ is a function growing faster than any polynomial, that is, for any $p>0$ there is a $c=c(p)<\infty$ such that \begin{equation} \label{eq:norm-v-bound-2} \sup_{u\ge1}\frac{u^p}{r(u)} \le c. \end{equation} Then, for any $\varepsilon>0$, there is an a.s. finite $A=A(\omega,\varepsilon)$ such that \[ \|X_n\| \le An^{\varepsilon}. \] \end{proposition} \begin{pf} To start with, (\ref{eq:drift-bound}) implies for $n\ge1$ \begin{eqnarray*} \mathbb{E} [V(X_n) ] &=& \mathbb{E} [\mathbb{E}[V(X_n) | \F_{n-1}] ] = \mathbb{E} [P_{S_{n-1}}V(X_{n-1}) ] \le\mathbb{E} [V(X_{n-1}) ] + b \\ &\le&\cdots\le V(x_0) + b n \le\tilde{b}V(x_0) n, \end{eqnarray*} where $\tilde{b} \defeq b+1$. Now, with fixed $a\ge1$, we can bound the probability of $\|X_n\|$ ever exceeding $an^\varepsilon$ as follows \begin{eqnarray*} \P\biggl( \max_{1\le n\le m} \frac{\|X_n\|}{n^\varepsilon} \ge a \biggr) &\le& \sum_{n=1}^m \P( \|X_n\| \ge an^{\varepsilon} ) \le\sum_{n=1}^\infty \P\bigl( V(X_n) \ge r(an^{\varepsilon}) \bigr) \\ &\le& \sum_{n=1}^\infty \frac{\mathbb{E} [V(X_n) ]}{r(an^\varepsilon)} \le \tilde{b} V(x_0) \sum_{n=1}^\infty \frac{n}{r(an^\varepsilon)}\\ &\le& \frac{\tilde{b}V(x_0)c}{a^{3/\varepsilon}} \sum_{n=1}^\infty n^{-2} \stackrel{a\to\infty}{\hbox to 1cm{\rightarrowfill}}0, \end{eqnarray*} where we use Markov's inequality, and $c=c(3/\varepsilon)<\infty$ is from the application of (\ref{eq:norm-v-bound-2}). \end{pf} We record the following easy lemma, dealing with a particular choice of $V(x)$, for later use in Section \ref{sec:am-ergodicity}. \begin{lemma} \label{lemma:v-polybound} Assume that the target density $\pi$ is differentiable, bounded, bounded away from zero on compact sets, and satisfies the following radial decay condition: \[ \lim_{r\to\infty}\sup_{\|x\|\ge r} \frac{x}{\|x\|} \cdot\nabla\log\pi(x) < 0. \] Then, for $V(x)=c_V\pi^{-1/2}(x)$, the bound (\ref{eq:norm-v-bound-1}) applies with a function $r(u) \defeq c e^{\gamma u}$ for some $\gamma,c>0$, satisfying (\ref{eq:norm-v-bound-2}). \end{lemma} \begin{pf} Let $R\geq1$ be such that $\sup_{\|x\|\ge R} \frac{x}{\|x\|} \cdot\nabla\log\pi(x)\le-\gamma$ for some $\gamma>0$. Assume $y\in\R^d$ and $\|y\|\ge2R$, and write $y = (1+a)x$, where $\|x\| = R$ and $a=\frac{\|y\|}{R}-1\ge1$. Denote $h(x)\defeq\log\pi(x)$, and write \[ \log \frac{\pi(y)}{\pi(x)} = \int_1^{1+a} {x} \cdot\nabla h(t x)\,dt \le-\gamma a. \] We have that \[ V(y) = c_V \pi(x)^{-1/2} \biggl(\frac{\pi(y)}{\pi(x)} \biggr)^{-1/2} \ge c_V e^{{\gamma a}/{2}} \inf_{\|x\|=R} \pi(x)^{-1/2} \ge c e^{{\gamma}/({4R})\|y\|} \] and, since $\pi$ is bounded away from zero on $\{x\dvtx\|x\|< 2R\}$, we can select $c>0$ such that the bound applies to all $y\in\R^d$. \end{pf} \section{Ergodicity result for adaptive metropolis} \label{sec:am-ergodicity} We start this section by outlining the original Adaptive Metropolis (AM) algorithm \cite{saksman-am}. The AM chain starts from a point $X_0 \equiv x_0\in \R^d$, and we have an initial covariance $\Sigma_0\in\mathcal{C}^d$ where $\mathcal{C}^d\subset\R^{d\times d}$ stands for the symmetric and positive definite matrices. We generate, recursively, for $n\ge0$, \begin{eqnarray} X_{n+1} &\sim& P_{\theta\Sigma_n}(X_n, \uarg), \\ \label{eq:am-orig-s} \Sigma_{n+1} &=& \cases{v_0,&\quad $0\le n\le N_b-1$,\cr \operatorname{Cov}(X_0,\ldots,X_n)+\kappa I,&\quad $n\ge N_b$,} \end{eqnarray} where $\theta>0$ is a parameter, $N_b\ge2$ is the length of the burn-in, $\kappa>0$ is a small constant, $I$ is an identity matrix and $P_v(x,\uarg)$ is a Metropolis transition probability defined as \begin{eqnarray} \label{eq:metropolis-kernel} P_v(x,{A}) &\defeq& \mathbh{1}_{{A}}(x) \biggl[ 1- \int\biggl(1\wedge\frac{\pi(y)}{\pi(x)} \biggr) q_v(y-x) \,\ud y \biggr] \nonumber\\[-8pt]\\[-8pt] &&{}+ \int_{{A}} \biggl(1\wedge\frac{\pi(y)}{\pi(x)} \biggr) q_v(y-x) \,\ud y, \nonumber \end{eqnarray} where the proposal density $q_v$ is the Gaussian density with zero mean and covariance $v\in\mathcal{C}^d$. In this paper, just for notational simplicity (see Remark \ref {rem:convergence}), we consider a slight modification of the AM chain. First, we do not consider a burn-in period, that is, let $N_b=0$, and let $\Sigma_0\ge\kappa I$. Instead of (\ref{eq:am-orig-s}), we construct $\Sigma_{n}$ recursively for $n\ge1$ as \begin{equation} \label{eq:mod-recursion} \Sigma_{n} = \frac{n}{n+1} \Sigma_{n-1} + \frac{1}{n+1} [(X_{n}-\overline{X}_{n-1})(X_{n}-\overline{X}_{n-1})^T + \kappa I ], \end{equation} where $\overline{X}_n$ denotes the average of $X_0,\ldots,X_n$. \begin{remark} \label{rem:convergence} The original AM process uses the unbiased estimate of the covariance matrix. In this case, the recursion formula for $\Sigma_n$, when $n\ge N_b+2$, has the form \begin{equation} \label{eq:am-orig-crec} \Sigma_n = \frac{n-1}{n} \Sigma_{n-1} + \frac{1}{n+1} [ (X_n - \overline{X}_{n-1})(X_n - \overline{X}_{n-1})^T + \kappa I ]. \end{equation} This recursion can also be formulated in our framework described in Section \ref{sec:notations} by simply introducing a sequence of adaptation functions $H_n(s,x)$. Our proof applies with obvious changes. However, in the present paper, we prefer (\ref{eq:mod-recursion}) for simpler notation. Also, from a practical point of view, observe that (\ref {eq:mod-recursion}) differs from (\ref{eq:am-orig-crec}) by a factor smaller than $n^{-2} \Sigma_{n-1}$ whence it is mostly a matter of taste whether to use (\ref{eq:mod-recursion}) or (\ref{eq:am-orig-crec}). \end{remark} In the notation of the general adaptive MCMC framework in Section \ref{sec:notations}, we have the state space $\mathbb {X}\defeq\R ^d$. The adaptation parameter $S_n=(S_n^{(m)},S_n^{(v)})$ consists of the mean $S_n^{(m)}$ and the covariance $S_n^{(v)}$, having values in $(S_n^{(m)},S_n^{(v)}) \in\mathbb{S} \defeq\R^d\times\mathcal{C}^d$. The space $\overline{\mathbb{S}}\defeq\R^d\times\R^{d\times d} \supset \mathbb{S}$ is equipped with the norm $|s| \defeq\|s^{(m)}\| \vee \|s^{(v)}\|$ where we use the Euclidean norm, and the matrix norm $\|A\|^2\defeq\trace(A^T A)$, respectively. The Metropolis kernel $P_s$ is defined as in (\ref{eq:metropolis-kernel}), with the definition $q_s \defeq q_{s^{(v)}}$ for $s\in\mathbb{S}$. The adaptation function $H$ is defined for $s=(s^{(m)},s^{(v)})$ as \[ H(s, x) \defeq\left[\matrix{ x - s^{(m)} \cr \bigl(x-s^{(m)}\bigr)\bigl(x-s^{(m)}\bigr)^T - s^{(v)} + \kappa I}\right], \] and the adaptation weights are $\eta_n \defeq(n+1)^{-1}$. We now formulate our ergodicity result for the AM chain. \begin{theorem} \label{thm:am-ergodicity} Assume $\pi$ is positive, bounded, bounded from below on compact sets, differentiable and \begin{equation} \label{eq:super-pexp} \lim_{r\to\infty} \sup_{\|x\|\ge r} \frac{x}{\|x\|^\rho} \cdot\nabla\log\pi(x) = -\infty \end{equation} for some $\rho>1$. Moreover, assume that $\pi$ has regular contours \begin{equation} \label{eq:reg-contour} \lim_{r\to\infty} \sup_{\|x\|\ge r} \frac{x}{\|x\|} \cdot\frac{\nabla\pi(x)}{\|\nabla\pi (x)\|} < 0. \end{equation} Define $V(x)\defeq c_V \pi^{-1/2}(x)$ with $c_V = (\sup_x \pi(x))^{1/2}$. Then, for any $f$ with $\|f\|_{V^\alpha} < \infty$ where $0\le\alpha< 1$, \begin{equation} \label{eq:am-slln} \frac{1}{n}\sum_{k=1}^n f(X_k) \stackrel{n\to\infty}{\hbox to 1cm{\rightarrowfill}} \pi(f) \end{equation} almost surely. \end{theorem} \begin{remark} \label{rem:slln-moments} If the conditions of Theorem \ref{thm:am-ergodicity} are satisfied, the function $V(x)$ grows faster than an exponential, and hence (\ref{eq:am-slln}) holds for exponential moments. In particular, (\ref{eq:am-slln}) holds for power moments, that is, for $f(x) = \|x\|^p$ for any $p\ge0$, and therefore also $S_n \to(m_\pi, v_\pi+\kappa I)$ where $m_\pi$ and $v_\pi$ are the mean and covariance of $\pi$. \end{remark} The proof of Theorem \ref{thm:am-ergodicity} is postponed to the end of this section. We start by a simple lemma bounding the growth rate of the AM chain. \begin{lemma} \label{lemma:am-bound} If the conditions of Proposition \ref{prop:grow-bound} are satisfied for an AM chain, then for any $\varepsilon>0$, there is an a.s. finite $A=A(\omega,\varepsilon)$ such that \[ \bigl\| S_n^{(m)}\bigr\| \le An^\varepsilon,\qquad \bigl\| S_n^{(v)}\bigr\| \le An^{\varepsilon}. \] \end{lemma} \begin{pf} Since the AM recursion is a convex combination, this is a straightforward corollary of Proposition \ref{prop:grow-bound}. \end{pf} Next, we show that each of the Metropolis kernels used by the AM algorithm satisfy a geometric drift condition, and bound the constants of geometric drift. The result in Proposition \ref{prop:geom-bound} is similar to the results obtained in \cite{jarner-hansen,roberts-tweedie}, with the exception that we have a common minorization set $C$ for all proposal scalings. We start by two lemmas. We define $\ball(x,r) \defeq\{y\in\R^d\dvtx \|x-y\|\le r\}$. \begin{lemma} \label{lemma:boundary-estim} Assume $E\subset\R^d$ is measurable and $A\subset\R^d$ compact, given as \[ A \defeq\{ ru \dvtx u\in S^d, 0\le r\le g(u) \}, \] where $S^d\defeq\{u\in\R^d\dvtx\|u\|=1\}$ is the unit sphere, and $g\dvtx S^d\to[b,\infty)$ is a measurable function parameterising the boundary $\partial A$, with some $b>0$. For any $\varepsilon>0$, define $B_\varepsilon\defeq\{ru\dvtx u\in S^d, g(u)<r\le g(u)+\varepsilon\}$. Then, for all $\tilde{\varepsilon}>0$, there is a $\tilde{b}=\tilde{b}(\tilde{\varepsilon})\in(0,\infty)$ such that for all $0<\varepsilon<\tilde{\varepsilon}$ and for all $\lambda\ge3 \varepsilon$, it holds that \[ |E \cap B_\varepsilon| \le \bigl| \bigl(E \oplus\ball(0,\lambda) \bigr)\cap A \bigr|, \] whenever $b\ge\tilde{b}$. Above, $A\oplus B \defeq\{x+y\dvtx x\in A, y\in B\}$ stands for the Minkowski sum. \end{lemma} \begin{pf} See Figure \ref{fig:contour-est} for an illustration of the situation. Denote by $S^* \defeq\{u\in S^d\dvtx\exists r>0, ur\in E\cap B_\varepsilon\}$ the projection of the set $E\cap B_\varepsilon$ onto $S^d$. \begin{figure} \includegraphics{682f01.eps} \caption{Illustration of the boundary estimate. The set $A$ is in light grey, and the set $B_\varepsilon$ in dark gray.} \label{fig:contour-est} \end{figure} Then we have $E\cap B_{\varepsilon} \subset\{ru\dvtx u\in S^*, g(u)<r\le g(u)+\varepsilon\}$ and $A\supset\{ru\dvtx u\in S^*, 0\le r\le g(u)\}$. Now, for $\varepsilon\le\lambda\le g(u)$, we have \[ \bigl((E\cap B_\varepsilon)\oplus\ball(0,\lambda) \bigr)\cap A \supset\{ru\dvtx u\in S^*, g(u)-\lambda+\varepsilon\le r\le g(u)\} \eqdef G, \] for let $ru \in G$, then there is $g(u)< \tilde{r} \le g(u) + \varepsilon$ such that $\tilde{r}u \in E\cap B_\varepsilon$, and we can write $ru = \tilde {r}u + (r-\tilde{r})u$, where $(r-\tilde{r})u\in\ball(0,\lambda)$. Clearly, $E \oplus\ball(0,\lambda) \supset(E \cap B_\varepsilon) \oplus\ball(0,\lambda)$, and we can estimate \begin{eqnarray*} && \bigl| \bigl(E \oplus\ball(0,\lambda) \bigr)\cap A \bigr| - |E \cap B_\varepsilon| \\ &&\qquad\ge\int_{S^*} \int_{g(u)-2\varepsilon}^{g(u)} r^{d-1} \,\ud r - \int_{g(u)}^{g(u)+\varepsilon}r^{d-1} \,\ud r\, \mathcal{H}^{d-1}(\ud u) \\ &&\qquad=\frac{1}{d}\int_{S^*} 2 (g(u))^d - \bigl(g(u)-2\varepsilon\bigr)^d - \bigl(g(u)+\varepsilon\bigr)^d \mathcal{H}^{d-1}(\ud u), \end{eqnarray*} where $\mathcal{H}^{d-1}$ stands for the $(d-1)$-dimensional Hausdorff measure. This integral is nonnegative for all $0\le\varepsilon\le c_d b$, for some constant $c_d$ depending only on the dimension $d$, namely let $h(\varepsilon)\defeq(y-2\varepsilon)^d + (y+\varepsilon )^d$. The mean value theorem implies that for some $0\le\varepsilon'\le\varepsilon$, one has \[ h(0)-h(\varepsilon) = \varepsilon d (y-2\varepsilon')^{d-1} \biggl[ 2 - \biggl(\frac{y+\varepsilon'}{y-2\varepsilon'} \biggr)^{d-1} \biggr] \ge0, \] whenever $\varepsilon\le c_d y$. \end{pf} \begin{lemma} \label{lemma:exp-estim} Let $f(x) \defeq x e^{-{x^2}/{2}}$. For any $0<\varepsilon<1/8$, the following estimates hold: \[ 2f(x+\varepsilon) - f(x) \ge \frac{x}{8}\qquad \mbox{for all } 0<x\le \frac{1}{2} \] and \[ \int_0^\infty\bigl([2f(x+\varepsilon) - f(x)]\wedge0 \bigr) \,\ud x \ge - e^{-c \varepsilon^{-2}} \] for some constant $c>0$. \end{lemma} \begin{pf} We can write \[ 2 f(x+\varepsilon) - f(x) = e^{-{x^2}/{2}} [ 2(x+\varepsilon )e^{-x\varepsilon - {\varepsilon^2}/{2}} - x ], \] which is positive whenever $e^{-x\varepsilon - {\varepsilon^2}/{2}}\ge2/3$, holding at least for all $0\le x \le x^*$, with \[ x^* = \frac{\log(3/2)}{\varepsilon} - \frac{\varepsilon}{2} \ge\frac{1}{4\varepsilon}. \] Now, $x^*\ge1/2$ and we can estimate \[ 2 f(x+\varepsilon) - f(x) \ge\frac{1}{4} x e^{-{x^2}/{2}} \ge\frac{x}{8} \] for all $0<x\le1/2$. Also, \[ \int_0^\infty\bigl([2f(x+\varepsilon) - f(x)]\wedge0 \bigr) \,\ud x \ge-\int_{x^*}^\infty x e^{-{x^2}/{2}} \,\ud x = -e^{-c\varepsilon^{-2}} \] with $c = 1/32$. \end{pf} \begin{proposition} \label{prop:geom-bound} Assume that $\pi$ satisfies the conditions in Theorem \ref{thm:am-ergodicity} and $\kappa>0$. Then, there exists a compact set $C\subset\R^d$, a probability measure $\nu$ on $C$, and a constant $b\in[0,\infty)$ such that for the Metropolis transition probability $P_v$ in (\ref{eq:metropolis-kernel}) and for all $v\in\mathcal{C}^d$ with all eigenvalues greater than $\kappa>0$, it holds that \begin{eqnarray} \label{eq:drift-ineq-m} P_v V(x) &\le& \lambda_v V(x) + b \mathbh{1}_{C}(x)\qquad \forall x\in\mathbb{X},\\ \label{eq:mino-ineq-m} P_v(x,B) &\ge& \delta_v \nu({B}) \qquad \forall x\in C, \forall {B}\subset\mathbb{X}, \end{eqnarray} where $V(x)\defeq c_V \pi^{-1/2}(x)\ge1$ with $c_V \defeq(\sup_x \pi(x))^{1/2}$ and the constants $\lambda_v,\delta_v\in(0,1)$ satisfy the bound \[ (1-\lambda_v)^{-1} \vee\delta_v^{-1} \le c \det(v)^{1/2} \] for some constant $c\ge1$. \end{proposition} \begin{pf} Define the sets $A_x\defeq\{y\dvtx\pi(y)\ge\pi(x)\}$ and its complement $R_x\defeq\{y\dvtx\pi(y)< \pi(x)\}$, which are the regions of almost sure acceptance and possible rejection at $x$, respectively. Let $R>1$ be sufficiently large to ensure that for all $\|x\|\ge R$, it holds that \[ \sup_{\|x\|\ge R} \frac{x}{\|x\|} \cdot\frac{\nabla\pi(x)}{\|\nabla\pi (x)\|} < -\gamma \quad\mbox{and}\quad \sup_{\|x\|\ge R} \frac{x}{\|x\|} \cdot\nabla \log\pi(x) < -\|x\|^{\rho-1} \] for some $\gamma>0$. Suppose that the dimension $d\ge2$. Lemma \ref{lemma:contour} in Appendix \ref{sec:contour} implies that for $R$ sufficiently large, we have $\ball(0,M^{-1}\|x\|)\subset A_x\subset\ball(0,M\|x\|)$ for all $\|x\|\ge R$ with some constant $M\ge1$. Moreover, we can parameterize $A_x = \{ru\dvtx u\in S^d, 0\le r\le g(u)\}$ where $S^d \defeq\{u\in\R^d\dvtx\|u\|=1\}$ is the unit sphere, and $g\dvtx S^d\to[M^{-1}\|x\|,M\|x\|]$. Consider (\ref{eq:drift-ineq-m}). We may compute \begin{eqnarray}\label{eq:drift-fraction} \tau_v :\!&=& 1- \frac{P_v V(x)}{V(x)}\nonumber\\ &=& \int_{A_x} \Biggl( 1- \sqrt{\frac{\pi(x)}{\pi(y)}} \Biggr) q_v(y-x) \,\ud y \\ &&{}- \int_{R_x} \sqrt{\frac{\pi(y)}{\pi(x)}} \Biggl( 1 - \sqrt{\frac{\pi(y)}{\pi(x)}} \Biggr) q_v(y-x) \,\ud y.\nonumber \end{eqnarray} In what follows, unless explicitly stated, we assume $\|x\|\ge M(R + 1)$. Denote $\varepsilon_x \defeq\|x\|^{-\alpha} < 1$, where $\alpha= (\rho-1)/2>0$. Define $\tilde{A}_x \defeq\{ru\dvtx u\in S^d, 0\le r\le g(u)-\varepsilon_x\}\subset A_x$ and $\tilde{R}_x \defeq\{ru\dvtx u\in S^d, r\ge g(u)+\varepsilon_x \}\subset R_x$. From (\ref{eq:drift-fraction}), we can estimate \begin{eqnarray}\label{eq:tau-estim} \tau_v &\ge& \int\Biggl[ \Biggl( 1- \sqrt{\frac{\pi(x)}{\pi(y)}} \Biggr) \mathbh{1}_{\tilde{A}_x}(y) - \frac{1}{4} \mathbh{1}_{R_x\setminus\tilde{R}_x}(y) \Biggr]q_v(y-x) \,\ud y \nonumber\\[-8pt]\\[-8pt] &&{}- \sup_{z\in\R^d} q_v(z-x) \int_{\tilde{R}_x} \sqrt{\frac{\pi (y)}{\pi(x)}} \,\ud y.\nonumber \end{eqnarray} We estimate the two terms in the right-hand side separately, starting from the first. Let $h(x)\defeq\log\pi(x)$. Suppose $z\in\tilde{A}_x$, and write $z = (1-a/\|y\|)y$ for some $y\in\partial A_x$ and $\varepsilon_x \le a \le\|y\|$. Assume for a moment $\|z\|\ge R$. Then, $h$ is decreasing on the line segment from $z$ to $y$, and we can estimate \begin{eqnarray* \frac{\pi(x)}{\pi(z)} &=& \frac{\pi(y)}{\pi(z)} = e^{h(y)-h(z)} = e^{\int_{\|y\|-a}^{\|y\|} {y}/{\|y\|}\cdot\nabla h(t{y}/{\|y\|})\,\ud t} \le e^{\int_{\|y\|-\varepsilon_x}^{\|y\|} {y}/{\|y\|}\cdot\nabla h(t{y}/{\|y\|})\,\ud t} \\ &\le& e^{-\varepsilon_x(\|y\|-\varepsilon_x)^{\rho-1}} \le e^{-\varepsilon_x\|x\|^{\rho-1}/(2M)^{\rho-1}} = e^{-\|x\|^\alpha/(2M)^{\rho-1}}. \end{eqnarray*} Hence, in this case, $\pi(x)/\pi(z)\le1/4$ assuming $\|x\|\ge R_2$ for sufficiently large $R_2\ge R$. If $\|z\|<R$, then there is $z'$ such that $\|z'\|=R$ and the estimate above holds for $z'$. Consequently, \begin{equation} \label{eq:ax-estimate} \frac{\pi(x)}{\pi(z)} = \frac{\pi(y)}{\pi(z')}\frac{\pi (z')}{\pi(z)} \le e^{-\|x\|^\alpha/(2M)^{\rho-1}} \frac{\sup_{\|w\|\le R}\pi(w)}{\inf_{\|w\|\le R}\pi(w)} \le\frac{1}{4}, \end{equation} whenever $\|x\| \ge R_2$ by increasing $R_2$ if needed. In conclusion, we have shown that for $\|x\|\ge R_2$, it holds that $(1-\sqrt{\pi(x)/\pi(y)}) \ge1/2$ for all $y\in\tilde{A}_x$. By Fubini's theorem, we can write for positive $f$ that \begin{eqnarray*} \int f(z+x) q_v(z) \,\ud x &=& \frac{c_d}{\sqrt{\det(v)}} \int_0^1 \int_{\{e^{-{1/2}z^Tv^{-1}z}\ge t\}} f(z+x) \,\ud z \,\ud t \\ &=& \frac{c_d}{\sqrt{\det(v)}} \int_0^\infty \int_{E_u} f(y) \,\ud y\, u e^{-{u^2}/{2}}\,\ud u, \end{eqnarray*} where $c_d=(2\pi)^{-d/2}$ and $E_u \defeq\{z+x\dvtx z^Tv^{-1}z\le u^2\}$. Consequently, for $\|x\|\ge R_2$, we can estimate the first term of (\ref{eq:tau-estim}) from below by \begin{eqnarray*} &&\int_0^\infty \biggl(\frac{|E_u\cap\tilde{A}_x|}{2} - \frac{|E_u\cap(R_x\setminus \tilde{R}_x)|}{4} \biggr) u e^{-{u^2}/{2}}\,\ud u \\ &&\qquad\ge \frac{1}{4}\int_0^\infty 2|E_{u+a}\cap\tilde{A}_x| (u+a) e^{-{(u+a)^2}/{2}} - |E_u\cap(R_x\setminus \tilde{R}_x)| u e^{-{u^2}/{2}}\,\ud u \\ &&\qquad\ge \frac{1}{4}\int_0^\infty 2 \bigl| \bigl(E_{u}\oplus\ball(0,\kappa^{1/2}a) \bigr)\cap\tilde{A}_x \bigr| (u+a) e^{-{(u+a)^2}/{2}}\\ &&\qquad\quad\hspace*{26.4pt}{}- |E_u\cap B_\varepsilon| u e^{-{u^2}/{2}}\,\ud u \end{eqnarray*} for any $a\ge0$, since simple computation shows that $E_u \oplus\ball(0,\kappa^{1/2}a) = \{x+y\dvtx x\in E_u, y\in\ball(0,\kappa^{1/2}a)\} \subset E_{u + a}$, and as we may write $\tilde{A}_x = \{ru\dvtx u\in S^d, 0\le r\le\tilde{g}(u)\}$ where $\tilde{g}(u) = g(u)-\varepsilon_x$, we obtain that $R_x\setminus \tilde{R}_x \subset\{ru\dvtx u\in S^d, \tilde{g}(u)\le r\le\tilde {g}(u) + 2\varepsilon_x\} \eqdef B_\varepsilon$. We set $a=6\kappa^{-1/2}\varepsilon_x$ and apply Lemma \ref{lemma:boundary-estim} with the choice $\varepsilon =2\varepsilon_x$ and $\lambda=6\varepsilon_x$, \begin{eqnarray*} &&\int_0^\infty \biggl(\frac{|E_u\cap\tilde{A}_x|}{2} - \frac{|E_u\cap(R_x\setminus \tilde{R}_x)|}{4} \biggr) u e^{-{u^2}/{2}}\,\ud u \\ &&\qquad \ge\frac{1}{4}\int_0^\infty | [E_u\oplus\ball(0,6\varepsilon_x) ]\cap\tilde{A}_x | \bigl[2 (u+a) e^{-{(u+a)^2}/{2}} - u e^{-{u^2}/{2}} \bigr]\,\ud u \\ &&\qquad\ge\frac{1}{4}\int_{1/4}^{1/2} |E_{u}\cap\tilde{A}_x| \frac{u}{8}\,\ud u - |\tilde{A}_x| e^{-c_1\varepsilon_x^{-2}} \\ &&\qquad\ge c_2 |E_{1/4}\cap\tilde{A}_x| - M^d\|x\|^d e^{-c_1\|x\|^{\alpha}} \end{eqnarray*} by Lemma \ref{lemma:exp-estim}, for sufficiently large $\|x\|$, and since $E_u$ are increasing with respect to $u$. We have that $E_{1/4}\supset\ball(x,\kappa^{1/2}/4)$. If $\|x\|\to\infty$, then $\varepsilon_x\to0$ and also $|\ball(x,\kappa^{1/2}/4)\cap\tilde{A}_x| -|\ball(x,\kappa ^{1/2}/4)\cap A_x| \to0$. Moreover, it holds that $|\ball(x,\kappa^{1/2}/4)\cap A_x|\ge c_3>0$ (see the proof of Theorem 4.3 in \cite{jarner-hansen}). So, for large enough $\|x\|$, there is a $c_4>0$ so that $|E_{1/4}\cap\tilde{A}_x|\ge c_4$. To sum up, by choosing $R_3$ to be sufficiently large, we obtain that the first part of (\ref{eq:tau-estim}) is at least $c_5 (\det(v))^{-1/2}$ for all $\|x\|\ge R_3$, with a $c_5>0$. Next, we turn to the second term of (\ref{eq:tau-estim}). We obtain by polar integration that \begin{eqnarray*} \int_{\tilde{R}_x} \sqrt{\frac{\pi(y)}{\pi(x)}} \,\ud y &=&\int_{S^d} \int_{g(u)+\varepsilon_x}^\infty r^{d-1} e^{{1}/{2}h(r u)- {1}/{2}h(g(u)u)}\, \ud r\, \mathcal{H}^{d-1}(\ud u) \\ &\le& c_d' \sup_{M^{-1}\|x\|\le w\le M\|x\|} \int_{w+\varepsilon_x}^\infty r^{d-1} e^{-{1}/{2}\int_{w}^r t^{\rho-1} \,dt}\,\ud r, \end{eqnarray*} where $\mathcal{H}^{d-1}$ is the $(d-1)$-dimensional Hausdorff measure, and $c_d'=\mathcal{H}^{d-1}(S^d)$. Denote $T(w,r) \defeq r^{d-1} e^{-{1}/{4}\int_{w}^r t^{\rho-1}\,\ud t}$ and let us estimate the latter integral from above by \begin{eqnarray*} \int_{w+\varepsilon_x}^\infty e^{-{1}/{4}\int_{w}^r t^{\rho-1}\,\ud t} \,\ud r \sup_{r\ge w+\varepsilon_x} T(w,r) &\le& \int_{w}^\infty e^{-{w^{\rho-1}}/{4}(r-w)} \,\ud r \sup_{r\ge w+\varepsilon_x} T(w,r) \\ &\le& 4M^{\rho-1}\|x\|^{1-\rho} \sup_{r\ge w+\varepsilon_x} T(w,r) \end{eqnarray*} for any $w\ge M^{-1}\|x\|$. Suppose first $w+\varepsilon_x \le r \le2w$, then \[ T(w,r) \le(2w)^{d-1} e^{-{1}/{4}\varepsilon_x w^{\rho-1}} \le(2M)^{d-1} \|x\|^{d-1} e^{-{1}/{4}M^{1-\rho} \|x\|^\alpha} \le c_6 \] for any $M^{-1}\|x\| \le w \le M\|x\|$. For any $r>2w$ and $w\ge1$, we have \[ T(w,r) \le r^{d-1} e^{-{1}/{4} {r}/{2} w^{\rho-1}} \le r^{d-1} e^{-{r}/{8}} \le c_7. \] Put together, letting $R_4\ge R_3$ to be sufficiently large, we obtain that $\tau_v \ge c_8 (\det(v))^{-1/2}$ with $c_8 = c_5/2$ for all $\|x\| \ge R_4$. To sum up, by setting $C=\ball(0,R_4)$, we get that for all $v\in\mathcal{C}^d$ with eigenvalues bounded from below by $\kappa$, the estimate $P_v V(x) \le\lambda_v V(x)$ holds for $x\notin C$ with $\lambda_v \defeq1-c_8 \det(v)^{-1/2}$ satisfying $(1-\lambda_v)^{-1} \le c_8^{-1} \det(v)^{1/2}$. For $x\in{C}$, we have by (\ref{eq:drift-fraction}) that $P_v V(x) \le2 V(x) \le2 \sup_{z\in C} V(z) \le b < \infty$, so (\ref{eq:drift-ineq-m}) holds. In the one-dimensional case, the above estimates can be applied separately for the tails of the distribution. Finally, set $\nu({B})\defeq|C|^{-1}|{B}\cap{C}|$, and consider the minorization condition (\ref{eq:mino-ineq-m}) for $x\in C$, \begin{eqnarray*} P_v(x,{B}) &\ge&\int_{B\cap C} \biggl(1\wedge\frac{\pi(y)}{\pi(x)} \biggr) q_v(y-x) \,\ud y \\ &\ge&\frac{c_d}{\sqrt{\det(v)}} \int_{B\cap C} \biggl(1\wedge\frac{\pi(y)}{\pi(x)} \biggr) \inf_{x,y\in C} e^{-{1}/{2}(x-y)v^{-1}(x-y)} \,\ud y\\ &\ge&\frac{c_d}{\sqrt{\det(v)}} e^{-{1}/({2\kappa'})\operatorname{diam}(C)^2} \frac{\inf_{z\in{C}}\pi(z)}{\sup_z\pi(z)} \int_{B\cap C} \,\ud y. \end{eqnarray*} So (\ref{eq:mino-ineq-m}) holds with $\delta_v \defeq c_9\det(v)^{-1/2}$ for some $c_9>0$. Finally, the claim holds with $c\defeq c_8^{-1} \vee c_9^{-1}$. \end{pf} Finally, we are ready to prove the strong law of large numbers for the AM process. \begin{pf*}{Proof of Theorem \ref{thm:am-ergodicity}} We start by verifying the strong law of large numbers (\ref{eq:am-slln}). Fix $t\ge1$ and consider first the constrained process $(X_n^{(t)},S_n^{(t)})_{n\ge0}$ which is defined as the AM chain, but with the constraint sets $K_n^{(t)}$ defined as $K_n^{(t)} \defeq\{ s\in\mathbb{S}\dvtx|s|\le t n^{\varepsilon'} \} $, with $\varepsilon'=\varepsilon/(2d)$, and $\varepsilon\in(0,\kappa _*^{-1}[(1/2)\wedge (1-\alpha)])$, where $\kappa_*$ is the independent constant of Theorem \ref{thm:slln-restricted}. We check that assumptions \hyperlink{a:invariance}{(A1)}--\hyperlink{a:adapt-bound}{(A4)} are satisfied by the constrained process $(X_n^{(t)},S_n^{(t)})_{n\ge0}$ for all $t\ge 1$. Condition \hyperlink{a:invariance}{(A1)} is satisfied by construction of the Metropolis kernels $P_s$. Since $\det(v)\le\|v\|^d$, Proposition \ref{prop:geom-bound} ensures that there is a compact $C\subset\R^d$ such that \hyperlink{a:uniform-drift-mino}{(A2)} holds. For \hyperlink{a:kernel-lip}{(A3)}, we refer to \cite{andrieu-moulines}, Lemma 13, stating that $\|P_s f -P_{s'} f\|_{V^r} \le2 d\kappa^{-1} \|f\|_{V^r} |s^{(v)}-s'^{(v)}|$ for all $s^{(v)},s'^{(v)}\in\mathcal{C}^d$ with eigenvalues bounded from below by $\kappa$. Finally, we check that \hyperlink{a:adapt-bound}{(A4)} holds for any $\beta\in(0,1/2]$. Similarly to \cite{sa-verifiable}, we have that \begin{eqnarray*} && \sup_{s\in K_n^{(t)}} \|H(s,x)\|_{V^\beta}\\ &&\qquad= \sup_{s\in K_n^{(t)}} \sup_{x\in\R^d} \frac{|H(s,x)|}{V^\beta(x)} \\ &&\qquad\le\|\kappa I\| + \sup_{x\in\R^d} \sup_{s\in K_n^{(t)}} \frac{\|x\| + \|s^{(m)}\| + \|s^{(v)}\| + \| (x-s^{(m)})(x-s^{(m)})^T \| }{V^\beta(x)} \\ &&\qquad\le\sqrt{d}\kappa+ \sup_{x\in\R^d} \frac{\|x\| + \|x\|^2 + t^2n^{2\varepsilon'} + 2 t n^{\varepsilon'} + 2\|x\|t n^{\varepsilon'} }{V^\beta(x)} \\ &&\qquad\le\sqrt{d}\kappa+ 7t^2n^{2\varepsilon'} \sup_{x\in\R^d} \frac{\|x\|^2\vee1}{V^\beta(x)} \le\tilde{c} n^{\varepsilon} \end{eqnarray*} for any $\beta\in(0,1/2]$ by Lemma \ref{lemma:v-polybound}, where $\tilde{c} = \tilde{c}(t,\beta)$. So, assumption \hyperlink{a:adapt-bound}{(A4)} holds for any $\beta\in (0,1-\alpha)$. In particular, we can select $\beta$ so that $\varepsilon< \kappa _*^{-1}[(1/2)\wedge (1-\alpha-\beta)]$. Clearly, $\sum_k k^{\kappa_*\varepsilon-1} \eta_k < \sum_k k^{\kappa_*\varepsilon- 2}<\infty$, so all the conditions of Theorem \ref{thm:slln-restricted} are satisfied, implying that the strong law of large numbers holds for the constrained process $(X_n^{(t)},S_n^{(t)})$ for all $t\ge1$. Define $B^{(t)}\defeq\{\forall n\ge0\dvtx S_n\in K_n^{(t)}\}$. We can construct the constrained processes so that they coincide with the original process in $B^{(t)}$. That is, for $\omega\in B^{(t)}$ we have $(X_n(\omega),S_n(\omega)) = (X_n^{(t)}(\omega),S_n^{(t)}(\omega))$ for all $n\ge0$. Lemma \ref{lemma:am-bound} ensures that we have $\P(\forall n\ge 0\dvtx S_n\in K_n^{(t)})\ge g(t)$ where $g(t)\to1$ as $t\to\infty$. As in the proof of Theorem \ref{thm:slln-restricted}, we can use the Borel--Cantelli lemma to deduce that (\ref{eq:am-slln}) holds almost surely. \end{pf*} \begin{remark} Since $\varepsilon>0$ can be selected arbitrarily small in the proof of Theorem \ref{thm:am-ergodicity}, it is only required for (\ref{eq:am-slln}) to hold that the adaptation weights $\eta_n\in(0,1)$ are decreasing and that $\sum_k k^{\tilde{\varepsilon}-1} \eta_k <\infty$ holds for some $\tilde{\varepsilon}>0$. In particular, one can choose $\eta_n \defeq (n+1)^{-\gamma}$ for any $\gamma>0$. \end{remark} \begin{remark} Condition (\ref{eq:super-pexp}) implies the super-exponential decay of the tails of $\pi$: \begin{equation} \label{eq:super-exp} \lim_{r\to\infty} \sup_{\|x\|\ge r} \frac{x}{\|x\|} \cdot\nabla\log\pi(x) = -\infty. \end{equation} This condition, with the contour regularity condition (\ref{eq:reg-contour}), are common conditions to ensure geometric ergodicity of a random-walk Metropolis algorithm, and many standard distributions fulfil them \cite{jarner-hansen}. The decay condition (\ref{eq:super-pexp}) is only slightly more stringent than (\ref{eq:super-exp}). \end{remark} Finally, we formulate a central limit theorem for the AM algorithm. \begin{theorem} \label{thm:am-clt} Assume $\pi$ satisfies the conditions of Theorem \ref{thm:am-ergodicity}. For any $f$ with $\|f\|_{V^\alpha} < \infty$ for some $0\le\alpha< 1/2$, where $V(x)\defeq c_V \pi^{-1/2}(x)$ and $c_V = (\sup_x \pi(x))^{1/2}$, it holds that \[ \frac{1}{\sqrt{n}}\sum_{k=1}^n [f(X_k)-\pi(f) ] \stackrel{n\to\infty}{\hbox to 1cm{\rightarrowfill}} N(0,\sigma^2) \] in distribution, where $\sigma^2\in[0,\infty)$ is a constant. \end{theorem} The proof of Theorem \ref{thm:am-clt} follows by the techniques of the present paper applied to \cite {andrieu-moulines}, Theorem 9. A fully detailed proof can be found in the preprint \cite{saksman-vihola-preprint}. \begin{appendix} \section{\texorpdfstring{Proof of Lemma \protect\lowercase{\ref{lemma:drift-to-conv}}}{Proof of Lemma 3}} \label{sec:proof-lemma-drift-to-conv} We provide a restatement of a part of a theorem by Meyn and Tweedie \cite{meyn-tweedie-computable} before proving Lemma \ref{lemma:drift-to-conv}. For a more recent work on quantitative convergence bounds, we refer to \cite{baxendale-bounds}. \begin{theorem} \label{th:m-t-computable} Suppose that the following drift and minorization conditions hold: \begin{eqnarray*} P V(x) &\le& \lambda V(x) + b \mathbh{1}_{C}(x)\qquad \forall x\in\mathbb{X}, \\ P(x,A) &\ge& \delta\nu(A)\qquad \forall x\in C, \forall {A}\subset\mathbb{X} \end{eqnarray*} for constants $\lambda<1$, $b<\infty$ and $\delta>0$, a set ${C}\subset\mathbb{X}$ and a probability measure $\nu$ on ${C}$. Moreover, suppose that $\sup_{x\in{C}}V(x)\le b$. Then, for all $k\ge1$, \[ \|P_s^k(x,\uarg)-\pi(\uarg)\|_V \le V(x) (1+\gamma) \frac{\rho}{\rho-\vartheta}\rho^k \] for any $\rho>\vartheta=1-\tilde{M}^{-1}$, for \[ \tilde{M} = \frac{1}{(1-\check{\lambda})^2} \bigl[1-\check{\lambda}+\check {b}+\check{b}^2 + \bar{\zeta} \bigl( \check{b}(1-\check{\lambda}) \check{b}^2 \bigr) \bigr] \] defined in terms of \begin{eqnarray*} \gamma&=& \delta^{-2} [4b+2\delta\lambda b ] ,\\ \check{\lambda} &=& (\lambda+\gamma)/(1+\gamma)<1, \\ \check{b} &=& b + \gamma< \infty \end{eqnarray*} and the bound \[ \bar{\zeta} \le \frac{4-\delta^2}{\delta^5} \biggl(\frac{b}{1-\lambda} \biggr)^2. \] \end{theorem} \begin{pf} See \cite{meyn-tweedie-computable}, Theorem 2.3. \end{pf} \begin{pf*}{Proof of Lemma \ref{lemma:drift-to-conv}} Observe that $P_s V(x) = \mathbb{E}[V(X_{n+1}) | X_n=x,S_n=s]$, and therefore by Jensen's inequality, \hyperlink{a:uniform-drift-mino}{(A2)} implies for $x\notin{C}_n$ that \[ P_s V^r(x) \le(P_s V(x))^r \le\lambda_n^r V^r(x). \] We can bound $\tilde{\lambda}_n\defeq\lambda_n^r \le(1-c^{-1} n^{-\varepsilon})^r \le1 - r c^{-1}n^{-\varepsilon}$ implying \[ (1-\tilde{\lambda}_n)^{-1} \le r^{-1} c n^{\varepsilon}, \] whenever $r\in(0,1]$. Similarly, for $x\in{C}_n$, one has $P_s V^r(x) \le(\sup_{z\in{C}_n}V(z) + b_n)^r \le(2b_n)^r$, so by letting $\tilde{b}_n \defeq(2b_n)^r$, we obtain the drift inequality \[ P_s V^r(x) \le\tilde{\lambda}_n V^r(x) + \tilde{b}_n\mathbh{1}_{C_n}(x), \] and we can bound $\tilde{b}_n \le(2c n^\varepsilon)^r$. We have the bound $(1-\tilde{\lambda}_n)^{-1}\vee\tilde{b}_n \le\tilde{c} n^{\varepsilon}$ with some $\tilde{c}=\tilde{c}(c,r)\ge1$. Now, we can apply Theorem \ref{th:m-t-computable}, where we can estimate the constants \begin{eqnarray*} \gamma_n &=& \delta_n^{-2} [4\tilde{b}_n+2\delta_n\tilde{\lambda}_n \tilde{b}_n ] \le(c n^{\varepsilon})^2 6(\tilde{c}n^\varepsilon) = a_1 n^{3\varepsilon}, \\ \check{b}_n &=& \tilde{b}_n + \gamma_n \le(\tilde{c}+a_1)n^{3\varepsilon} \le a_2 n^{3\varepsilon} \end{eqnarray*} and consequently \[ 1-\check{\lambda}_n = \frac{1-\tilde{\lambda}_n}{1+\gamma_n} \ge\frac{\tilde{c}^{-1} n^{-\varepsilon}}{1+a_1 n^{3\varepsilon}} \ge\frac{\tilde{c}^{-1}}{1+a_1} n^{-4\varepsilon} = a_3^{-1} n^{-4\varepsilon}. \] Moreover, \[ \bar{\zeta}_n \le \frac{4-\delta_n^2}{\delta_n^5} \biggl(\frac{\tilde{b}_n}{1-\tilde {\lambda }_n} \biggr)^2 \le4 (c n^{\varepsilon})^5(\tilde{c}n^\varepsilon)^2 (\tilde {c}n^{\varepsilon})^2 = a_4 n^{9\varepsilon}, \] and then \begin{eqnarray*} \tilde{M}_n &=& \frac{1}{(1-\check{\lambda})^2} \bigl[ 1 - \check{\lambda}_n + \check{b}_n + \check{b}_n^2 + \bar{\zeta}_n\bigl(\check{b}_n(1-\check{\lambda}_n) + \check{b}_n^2\bigr) \bigr]\\ &\le& (a_3 n^{4\varepsilon})^2 [ 1 + \check{b}_n + \check{b}_n^2 + \bar{\zeta}_n(\check{b}_n + \check{b}_n^2) ] \\ &\le& (a_3 n^{4\varepsilon})^2(5 \bar{\zeta}_n\check{b}_n^2) \le5 a_3^2 n^{8\varepsilon} a_4 n^{9\varepsilon} a_2^2 n^{6\varepsilon } = a_5 n^{23\varepsilon} \end{eqnarray*} since we can assume that $\check{b}_n,\bar{\zeta}_n\ge1$. Now, \[ 1-\vartheta_n = \tilde{M}_n^{-1} \ge a_5^{-1} n^{-23\varepsilon} \] and we can choose $\rho_n \in(\vartheta_n,1)$ by letting $\rho_n \defeq\frac{1+\vartheta_n}{2}$. We have \[ \rho_n - \vartheta_n = 1-\rho_n = \tfrac{1}{2}(1-\vartheta_n) \ge\tfrac{1}{2}c_{9}^{-1} n^{-23\varepsilon} = (a_6 n^{23\varepsilon})^{-1}. \] Finally, from Theorem \ref{th:m-t-computable}, one obtains the bound \[ {\|P_s^k(x,\uarg)-\pi(\uarg)\|}_{V^r} \le V^r(x) L_n \rho_n^k, \] where \begin{eqnarray*} (1-\rho_n)^{-1} &\le& a_6 n^{23\varepsilon}, \\ L_n &=& (1+\gamma_n)\frac{\rho_n}{\rho_n-\vartheta_n} \le(1+a_1 n^{3\varepsilon})(a_6 n^{23\varepsilon}) \le a_7 n^{26\varepsilon} \end{eqnarray*} with $a_7 = (1+a_1)a_6$. This concludes the proof with $\kappa_2 = 26$ and $c_2 = a_7$. \end{pf*} \vspace*{-12pt} \section{Birnbaum and Marshall's inequality} \label{sec:martingale} \begin{theorem}[(Birnbaum and Marshall)] \label{th:birnbaum-marshall} Let $(X_k)_{k=1}^n$ be random variables, such that \[ \mathbb{E}[|X_k| | \F_{k-1}] \ge\psi_k|X_{k-1}|, \] where $\F_k\defeq\sigma(X_1,\ldots,X_k)$, and $\psi_k\ge0$. Let $a_k > 0$, and define \[ b_k \defeq\max\Biggl\{ a_k, a_{k+1}\psi_{k+1}, \ldots, a_n \prod_{j=k+1}^n \psi_j \Biggr\} \] for $1\le k\le n$, and $b_{n+1}\defeq0$. If $p\ge1$ is such that $\E|X_k|^p<\infty$ for all $1\le k \le n$, then \[ \P\Bigl(\max_{1\le k\le n} a_k|X_k| \ge1 \Bigr) \le\sum_{k=1}^n (b_k^p - \psi_{k+1}^p b_{k+1}^p)\E|X_k|^p. \] \end{theorem} \begin{pf} See \cite{birnbaum-marshall}, Theorem 2.1. \end{pf} \begin{corollary} \label{cor:birnbaum-marshall-martingale} Let $(M_k)_{k=1}^n$ be a martingale with respect to $(\F_k)_{k=1}^n$. Let $(a_k)_{k=1}^n$ be a strictly positive nonincreasing sequence. If $p\ge1$ is such that $\E|M_k|^p<\infty$ for all $1\le k\le n$, then for $1\le m\le n$, \[ \P\Bigl( \max_{m\le k\le n} a_k|M_k| \ge1 \Bigr) \le a_{n}^p \E|M_n|^p + \sum_{k=m}^{n-1} (a_k^p - a_{k+1}^p)\E|M_k|^p. \] \end{corollary} \begin{pf} By Jensen's inequality, \[ \mathbb{E}[|M_k| | \F_{k-1}] \ge| \mathbb{E}[M_k | \F_{k-1}] | = |M_{k-1}|. \] Define $\psi_k \defeq1$ for $1\le k\le n$, and $\tilde{a}_k \defeq a_m$ for $1\le k \le m$ and $\tilde{a}_k \defeq a_k$ for $m< k\le n$. The result follows from Theorem \ref{th:birnbaum-marshall}.\vadjust{\goodbreak} \end{pf} \section{Contour surface containment} \label{sec:contour} \begin{lemma} \label{lemma:contour} Suppose $A\subset\R^d$ is a smooth surface parameterized by the unit sphere $\mathcal{S}^d$, that is, $A = \{ug(u)\dvtx u\in\mathcal{S}^d\}$ with a continuously differentiable radial function $g\dvtx\mathcal{S}^d\to(0,\infty)$. Assume also that outer-pointing normal $n$ of $A$ satisfies $n(x) \cdot x/\|x\| \ge\beta$ for all $x\in A$ with some constant $\beta> 0$. There is a constant $M<\infty$ depending only on $\beta$ such that for any $x,y\in A$, it holds that $M^{-1} \le\|x\|/\|y\| \le M$. \end{lemma} \begin{pf} Consider first the two-dimensional case. Let $x$ and $y$ be two distinct points in $A$. We employ polar coordinates, thus let $u(\theta)r(\theta)\in A$ with $u(\theta) \defeq[\cos(\theta),\sin(\theta)]^T$ and $r(\theta) \defeq g(u(\theta))$ so that $u(\theta_1)r(\theta_1)=x$ and $u(\theta_2)r(\theta_2)=y$ with $\theta_1,\theta_2\in[0,2\pi)$. Let $\alpha(\theta)$ stand for the (smaller) angle between $u(\theta)$ and the normal of the curve~$A$, that is, the curve parametrized by $\theta\to u(\theta)r(\theta)$. Our assumption says that $|\alpha(t)|\leq\alpha_0 \defeq\arccos(\beta)<\pi/2$ for all $\theta\in[0,2\pi]$. On the other hand, an elementary computation shows that \[ \tan(\alpha(\theta) )=\frac{r'(\theta)}{r(\theta)}, \] and hence we have $|\frac{\ud}{\ud\theta}\log r(\theta))|=|r'(\theta)/r(\theta)|\leq\tan\alpha_0$ uniformly. We may estimate $|{\log}\|x\| - {\log}\|y\|| \le2\pi\tan(\alpha_0)$ yielding the claim with $M = e^{2\pi\tan{\alpha_0}}$. For $d\ge3$, take the plane $T$ containing the origin and the points $x$ and $y$. This reduces the situation to two dimensions, since $A\cap T$ inherits the given normal condition of the surface and the radius vector. \end{pf} \end{appendix} \section*{Acknowledgments} We thank the anonymous referees for a careful review and comments improving the paper significantly. We also thank Gersende Fort for useful comments.

proofpile-arXiv_069-11790

\section{Introduction} The fact, that the spacetime could be noncommutative \cite{michael,bala1} at very small length scale, received much attention in recent years. The typical length scale relevant for noncommutative physics is Planck scale. However, one usually looks for phenomenological consequence of noncommutativity in quantum mechanical regime. On the other hand, quantum mechanics is a simple platform for studying noncommutative physics and testing \cite{zhang1} the effect of noncommtativity. In noncommutative spacetime the ordinary product is replaced by a $star$ $product$ of the form \begin{eqnarray} \psi(x)\star\phi(x)=\psi(x)\exp\left\{-i\theta^{\mu\nu}\frac{\overleftarrow{\partial}}{\partial x^\mu}\frac{\overrightarrow{\partial}}{\partial y^\nu}\right\}\phi(y)\mid_{x=y}\,,\label{star1} \end{eqnarray} where the parameter $\theta^{\mu\nu}, \mu,\nu=1,2,3$, is antisymmetric with respect to $\mu$ and $\nu$. It is easy to check that, (\ref{star1}) leads to the coordinate noncommutativity $[x^\mu,x^\nu]= i\theta^{\mu\nu}$, by simply replacing $\psi=x^\mu, \phi=x^\nu$ in (\ref{star1}). This type of noncommutativity has been used in various fields like quantum field theory, string theory, quantum mechanics \cite{nair,pghosh1,Mikhail1}. On the other hand in recent years there have been a great deal of interest in the sudy of non Hermtian (as well as $\cal{PT}$ symmetric) quantum mechanics \cite{bender1,bender2,bender3,bender4,bender5}. Recently there has also been attempts to create $\cal{PT}$ symmetric systems in the laboratory \cite{bender6}. Here our objective is to study non Hermitian quantum mechanics in non commutative space. To be more specific we shall study models which are $\cal{PT}$ symmetric as well as those which are non Hermitian and non $\cal{PT}$ symmetric. It may be noted that noncommutative physics, defined by $[x^\mu,x^\nu]= i\theta^{\mu\nu}$, is not $\mathcal{PT}$-symmetric. We see that $\theta^{\mu\nu}$ does not change under $\mathcal{PT}$ transformation. The fact that $\theta^{\mu\nu}$ does not change under $\mathcal{P}$ and $\mathcal{T}$, hence under $\mathcal{PT}$ has been pointed out in \cite{jabbari}. However, in \cite{jabbari} certain transformation properties have been imposed so that the theory achieve a certain symmetry. In this article we shall first consider a $\cal{PT}$ symmetric deformation of non commutative space and examine some models in this space. Secondly we shall also examine non Hermitian (but not necessarily $\cal{PT}$ symmetric) models defined in standard non commutative space. It will be seen that in quite a few cases it is possible to obtain real spectrum despite the models being non Hermitian. We consider our problem on a plane. The $x_3$ coordinate commutes with the coordinates of the plane and the dynamics along the $x_3$ direction is taken to free throughout our discussion. The remainder of this article is organized as follows: In the next section we propose a $\mathcal{PT}$-symmetric generalization of the noncommutative quantum mechanics. As a simple example we then discuss the isotropic oscillator in this $\mathcal{PT}$-symmetric noncommutative space. In section \ref{ptoscillator} we discuss a $\mathcal{PT}$-symmetric displaced oscillator in our formulation of noncommutative space and solve it. Here we also discuss the same model in the standard non commutative space. A non-hermitian oscillator is then discussed in \ref{ani}. Finally, section \ref{conclu} is devoted to a conclusion. \section{$\mathcal{PT}$-symmetric deformation}\label{s3} In this section we discuss the noncommutative quantum mechanics (QM), where the noncommutativity respects $\mathcal{PT}$ symmetry, unlike the the standard formalism, where the noncommutative algebra \cite{bertolami} \begin{eqnarray} \nonumber \left[\hat{x_\mu},\hat{x_\nu}\right]= i2\theta_{\mu\nu},~ \left[\hat{p_\mu},\hat{p_\nu}\right]=i2\hat{\theta_{\mu\nu}}\,,\\ \left[\hat{x_\mu},\hat{p_\nu}\right]= i\hbar(\delta_{\mu\nu} +\theta_{\mu}^\beta\hat{\theta_{\nu\beta}})\,, \label{al4} \end{eqnarray} does not respect $\mathcal{PT}$-symmetry. Note that $\mathcal{PT}$ symmetry is broken by the first two relations of the algebra (\ref{al4}) but the third one respects the symmetry \begin{eqnarray} \nonumber\mathcal{PT}\left[\hat{x_\mu},\hat{x_\nu}\right]\mathcal{PT}^{-1}&\neq& \mathcal{PT}i2\theta_{\mu\nu}\mathcal{PT}^{-1}\,,\\ \nonumber \mathcal{PT}\left[\hat{p_\mu},\hat{p_\nu}\right]\mathcal{PT}^{-1}&\neq& \mathcal{PT}i2\hat{\theta_{\mu\nu}}\mathcal{PT}^{-1},\\ \mathcal{PT}\left[\hat{x_\mu},\hat{p_\nu}\right]\mathcal{PT}^{-1}&=& \mathcal{PT}i\hbar(\delta_{\mu\nu} +\theta_{\mu}^\beta\hat{\theta_{\nu\beta}})\mathcal{PT}^{-1}\,. \label{al4pt} \end{eqnarray} It is also clear from the representation \begin{eqnarray} \hat{x_\mu}=x_\mu -\theta_{\mu\nu} p^\nu,~~ \hat{p_\mu}=p_\mu +\hat{\theta_{\mu\nu}} x^\nu\,, \label{rep3} \end{eqnarray} of the noncommutative coordinates on the phase space that they do not behave as the commutative coordinates \begin{eqnarray} \mathcal{PT}\hat{x_\mu}\mathcal{PT}^{-1}\neq -\hat{x_\mu},~~ \mathcal{PT}\hat{p_\mu}\mathcal{PT}^{-1}\neq \hat{p_\mu}\,, \label{rep3npt} \end{eqnarray} under $\mathcal{PT}$ transformation. We can however consider a noncommutative formulation where the commutator algebra are $\mathcal{PT}$-symmetric. This can be done by replacing $\theta_{\mu\nu}\to i\theta_{\mu\nu}$, $\hat{\theta_{\mu\nu}}\to i\hat{\theta_{\mu\nu}}$ in the algebra (\ref{al4}) \begin{eqnarray} \nonumber \left[\hat{x_\mu},\hat{x_\nu}\right]= -2\theta_{\mu\nu},~ \left[\hat{p_\mu},\hat{p_\nu}\right]=-2\hat{\theta_{\mu\nu}}\,,\\ \left[\hat{x_\mu},\hat{p_\nu}\right]= i\hbar(\delta_{\mu\nu} -\theta_{\mu}^\beta\hat{\theta_{\nu\beta}})\,, \label{al4pt} \end{eqnarray} It is useful to consider the same representation (\ref{rep3}) but with the replacement $\theta_{\mu\nu}\to i\theta_{\mu\nu}$, $\hat{\theta_{\mu\nu}}\to i\hat{\theta_{\mu\nu}}$ as \begin{eqnarray} \hat{x_\mu}&=&x_\mu -i\theta_{\mu\nu} p^\nu,~~ \hat{p_\mu}=p_\mu +i\hat{\theta_{\mu\nu}} x^\nu \,, \label{rep3pt} \end{eqnarray} which transforms in the same way as the ordinary coordinate and momentum under $\mathcal{PT}$-transformation. It is now obvious that any quantum mechanical system which is $\mathcal{PT}$-symmetric in commutative space will remain $\mathcal{PT}$-symmetric when considered in the noncommutative space given by the algebra (\ref{al4pt}). To discuss the QM in this formulation we consider the isotropic oscillator on a plane. \begin{eqnarray} H = \frac{1}{2m}(p_1^2+p_2^2)+ \frac{1}{2}m\omega^2(x_1^2+x_2^2)\,, \label{2dh} \end{eqnarray} Eq. (\ref{2dh}) is both Hermitian and $\mathcal{PT}$-symmetric. The corresponding noncommutative counterpart \begin{eqnarray} \hat{H} = \frac{1}{2m}(\hat{p_1}^2 +\hat{p_2}^2)+ \frac{1}{2}m\omega^2(\hat{x_1}^2+\hat{x_2}^2)\,, \label{2dhn} \end{eqnarray} then takes the form \begin{eqnarray} \nonumber H_\mathcal{PT} &=& \frac{1}{2M_\mathcal{PT}}(p_1^2+p_2^2)+ \frac{1}{2}M_\mathcal{PT}\Omega_{\mathcal{PT}}^2(x_1^1+x_2^2)\\& -& iS_{\mathcal{PT}}L_z\,, \label{2dh1pt} \end{eqnarray} where $1/M_\mathcal{PT}= 1/m -m\omega^2\theta^2$, $S_\mathcal{PT}=(m\omega^2\theta +\hat{\theta}/m)$, and $\Omega_\mathcal{PT}= \sqrt{(1/m -m\omega^2\theta^2)(m\omega^2 -\hat{\theta}^2/m)}$. Note that, in order to get (\ref{2dh1pt}) we used the relation (\ref{rep3pt}) with $\theta_{12}=\theta$, $\hat{\theta_{12}}= \hat\theta$ and $\mu,\nu=1,2$. It is possible to evaluate the spectrum of the Hamiltonian (\ref{2dh1pt}) for a special case $S_\mathcal{PT}=0$, which makes it isotropic. The eigenvalues are \begin{eqnarray} E_\mathcal{PT}= \Omega_\mathcal{PT}\left(n^++ n^- +1\right)\,, \label{eigenpt} \end{eqnarray} where $n^\pm \in \mathbb{N}$, $\Omega_\mathcal{PT}= \omega(1-m^2\omega^2\theta^2)= \omega(1+\theta\hat\theta)$. However for general case, $S_\mathcal{PT}\neq 0$, although the Hamiltonian (\ref{2dh1pt}) is $\mathcal{PT}$-symmetric the solution of the corresponding eigenvalue problem can be complex. It is possible to write (\ref{2dh1pt}) as a sum of two separate one dimensional harmonic oscillators with frequencies $\Omega_{\mathcal{PT}} + iS_{\mathcal{PT}}$ and $\Omega_{\mathcal{PT}}-iS_{\mathcal{PT}}$, \begin{eqnarray} \nonumber H_{\mathcal{PT}}= \Omega_{\mathcal{PT}}\left(a_{\mathcal{PT}}^\dagger a_{\mathcal{PT}} + b_{\mathcal{PT}}^\dagger b_{\mathcal{PT}} +1\right) \\+ iS_{\mathcal{PT}}\left(a_{\mathcal{PT}}^\dagger a_{\mathcal{PT}} -b_{\mathcal{PT}}^\dagger b_{\mathcal{PT}}\right)\,. \label{2dosh} \end{eqnarray} Here the annihilation operators $a_{\mathcal{PT}}$ and $b_{\mathcal{PT}}$ are \begin{eqnarray} a_{\mathcal{PT}}=\frac{\left[(p_1+ip_2) - iM_{\mathcal{PT}}\Omega_{\mathcal{PT}}(x_1+ix_2) \right]}{2\sqrt{M_\mathcal{PT}\Omega_{\mathcal{PT}}}}\,,\\ b_{\mathcal{PT}}=\frac{\left[(p_1-ip_2) - iM_{\mathcal{PT}}\Omega_{\mathcal{PT}}(x_1-ix_2) \right]}{2\sqrt{M_{\mathcal{PT}}\Omega_{\mathcal{PT}}}}\,. \label{annihi1} \end{eqnarray} The corresponding creation operators $a_{\mathcal{PT}}^\dagger, b_{\mathcal{PT}}^\dagger$ together with the annihilation operators satisfy the usual commutation relations ($\hbar=1$ unit is used) \begin{eqnarray} \left[a_{\mathcal{PT}}, a_{\mathcal{PT}}^\dagger\right]=\left[b_{\mathcal{PT}}, b_{\mathcal{PT}}^\dagger\right] =1\,,\label{2doshac} \end{eqnarray} all other commutators are zero. The number operators $\mathcal{N}_{\mathcal{PT}}^+=a_{\mathcal{PT}}^\dagger a_{\mathcal{PT}}$ and $\mathcal{N}_{\mathcal{PT}}^-=b_{\mathcal{PT}}^\dagger b_{\mathcal{PT}}$ satisfy the eigenvalue equation $\mathcal{N}_{\mathcal{PT}}^\pm|n^+,n^-\rangle = n^\pm|n^+,n^-\rangle$ with $n^\pm \in \mathbb{N}_0$. The exact eigenvalue for the Hamiltonian (\ref{2dosh}) can be found as \begin{eqnarray} E_{\mathcal{PT}}= \Omega_{\mathcal{PT}}(n^++ n^- +1) +iS_{\mathcal{PT}}(n^+- n^-)\,. \label{2dosei} \end{eqnarray} Note that the eigenvalue is real only for $n^+= n^-= n$. In that case $E_{\mathcal{PT}}= \Omega_{\mathcal{PT}}(2n +1)$. It would be informative at this point to mention the result of isotropic oscillator (\ref{2dh}) studied on the standard noncommutative space \cite{agni} defined by (\ref{al4}). The $2$-dimensional isotropic oscillator on a plane with only noncommutative coordinates becomes anisotropic with two different frequencies. Even for both the coordinates and momenta to be noncommutative like (\ref{al4}), the oscillator becomes anisotropic with completely real eigenvalues \cite{giri1} \begin{eqnarray} E_{NC} = \Omega_{NC}(n^++ n^- +1) +S_{NC}(n^+- n^-)\,. \label{2dosein} \end{eqnarray} where, $\Omega_{NC}= \sqrt{(1/m +m\omega^2\theta^2)(m\omega^2 +\hat{\theta}^2/m)}$ and $S_{NC}=(m\omega^2\theta +\hat{\theta}/m)$ and $1/M_{NC}= 1/m +m\omega^2\theta^2$. To obtain the above result for $E_{NC}$, the relation (\ref{rep3}), with $\theta_{12}=\theta$, $\hat{\theta_{12}}=\hat\theta$ and $\mu,\nu=1,2$ has been used. \section{$\mathcal{PT}$-symmetric oscillator}\label{ptoscillator} In this section, we consider a $\mathcal{PT}$-symmetric oscillator given by the Hamiltonian \begin{eqnarray} \nonumber H_{\mathcal{PT}} &=& \frac{1}{2m}\left(p_1^2+p_2^2\right)+ \frac{1}{2}m\omega^2\left(x_1^2+x_2^2\right)\\ &+&iC \left(x_1+x_2\right)\,. \label{pt2dh} \end{eqnarray} where the real constant $C$ is the strength of $\mathcal{PT}$-symmetric interaction. It has real eigenvalues \cite{bender1} with the ground state shifted. The noncommutative version defined as \begin{eqnarray} \nonumber \hat{H_{\mathcal{PT}}} &=& \frac{1}{2m}\left(\hat{p_1}^2+\hat{p_2}^2\right)+ \frac{1}{2}m\omega^2\left(\hat{x_1}^2+\hat{x_2}^2\right)\\ &+&iC \left(\hat{x_1}+\hat{x_2}\right)\,, \label{pt2dhn} \end{eqnarray} is $\mathcal{PT}$-symmetric, $\mathcal{PT}\hat{H_{\mathcal{PT}}}\mathcal{PT}^{-1}=\hat{H_{\mathcal{PT}}}$. Note that here we used the algebra (\ref{al4pt}) and the representation (\ref{rep3pt}). In terms of commutative coordinates (\ref{pt2dhn}) can be written as \begin{eqnarray} \nonumber \hat{H_\mathcal{PT}} &=& \frac{1}{2M_{\mathcal{PT}}}\left(p_1^2+p_2^2\right)+ \frac{1}{2}M_{\mathcal{PT}}\Omega_{\mathcal{PT}}^2\left(x_1^2+x_2^2\right) \\&-& iS_{\mathcal{PT}}L_z +iC\left(x_1+x_2\right)-C\theta\left(p_1-p_2\right)\,. \label{pt2dh1pt} \end{eqnarray} We first solve the problem (\ref{pt2dh1pt}) for a special case $S_{\mathcal{PT}}=0$. We use the following transformation \begin{eqnarray} \nonumber Q_1 &=&x_1 + \frac{iC}{M_{\mathcal{PT}}\Omega_{\mathcal{PT}}^2},~~ Q_2=x_2 + \frac{iC}{M_{\mathcal{PT}}\Omega_{\mathcal{PT}}^2}\\ P_1 &=& p_1-C\theta M_{\mathcal{PT}},~~ P_2=p_2+ C\theta M_{\mathcal{PT}}\label{trans} \end{eqnarray} on (\ref{pt2dh1pt}), which then becomes an isotropic oscillator with a constant shift, \begin{eqnarray} \hat{H_\mathcal{PT}} = \frac{1}{2M_{\mathcal{PT}}}\left(P_1^2+P_2^2\right)+ \frac{1}{2}M_{\mathcal{PT}}\Omega_{\mathcal{PT}}^2\left(Q_1^2+Q_2^2\right) +E_0\,, \label{pt2dh1pt2} \end{eqnarray} where the constant is $E_0= \frac{C^2}{M_{\mathcal{PT}}\Omega^2_{\mathcal{PT}}}- C^2\theta^2M_{\mathcal{PT}}$. The eigenvalue is readily obtained to be \begin{eqnarray} \hat{E_\mathcal{PT}}= \Omega_\mathcal{PT}\left(n^++ n^- +1\right) +E_0\,, \label{eigenpt2} \end{eqnarray} However for general case, $S_\mathcal{PT}\neq 0$, although the Hamiltonian (\ref{pt2dh1pt}) is $\mathcal{PT}$-symmetric, the solution of the corresponding eigenvalue problem can be complex. To solve the eigenvalue problem we make the following transformation \begin{eqnarray} Q_1 &=&x_1 + i\lambda\,, Q_2 =x_2 + i\lambda\,,\\ P_1 &=& p_1+\beta \,, P_2 = p_2- \beta\,,\label{trans1} \end{eqnarray} where $\lambda=\frac{C(\theta M_{\mathcal{PT}}S_{\mathcal{PT}} +1)}{M_{\mathcal{PT}}( \Omega_{\mathcal{PT}}^2 + S^2_{\mathcal{PT}})}$ and $\beta=\frac{C(S_{\mathcal{PT}}-\theta M_{\mathcal{PT}}\Omega^2_{\mathcal{PT}})}{S^2_{\mathcal{PT}}+\Omega^2_{\mathcal{PT}}} $. After this transformation Eq. (\ref{pt2dh1pt}) takes the form \begin{eqnarray} \nonumber \hat{H_\mathcal{PT}} &=& \frac{1}{2M_{\mathcal{PT}}}\left(P_1^2+P_2^2\right)+ \frac{1}{2}M_{\mathcal{PT}}\Omega_{\mathcal{PT}}^2\left(Q_1^2+Q_2^2\right)\\&-&iS_{\mathcal{PT}}(Q_1P_2-Q_2P_1) +\hat{E_0}\,, \label{pt2dh1pt2} \end{eqnarray} where $\hat{E_0}= \beta^2/M_{\mathcal PT} - M_{\mathcal PT}\Omega^2_{\mathcal PT}\lambda^2 -2S_{\mathcal PT}\lambda\beta + 2C\lambda +2C\theta\beta$. The exact eigenvalues for the Hamiltonian (\ref{pt2dh1pt}) can be found as \begin{eqnarray} \hat{E_{\mathcal{PT}}}= \Omega_{\mathcal{PT}}(n^++ n^- +1) +iS_{\mathcal{PT}}(n^+- n^-) +\hat{E_0}\,. \label{2dosei1} \end{eqnarray} Note that the eigenvalues are real only for $n^+= n^-= n$, in that case $\hat{E_{\mathcal{PT}}}= \Omega_{\mathcal{PT}}(2n +1) + \hat{E_0}$. One can see that the limit $\hat{\theta} \to 0$ of Eq. (\ref{2dosei1}) can be taken smoothly. Note that although (\ref{2dosei1}) respects $\mathcal{PT}$-symmetry, it becomes complex in general. It would now be interesting to see what happens if we consider the standard noncommutativity, given by (\ref{al4}), to study the $\mathcal{PT}$-symmetric Hamiltonian (\ref{pt2dh}). This can be simply achieved by replacing $i\theta \to \theta$ and $i\hat\theta \to \hat\theta$ from Eq. (\ref{pt2dh1pt}) onwards. It may be seen that in this case the Hamiltonian (\ref{pt2dh}) is non Hermitian and non $\cal{PT}$ symmetric. The eigenvalue are then given by \begin{eqnarray} \hat{E_{NC}}= \Omega_{NC}(n^++ n^- +1) + S_{NC}(n^+- n^-) +\epsilon_{NC}\,. \label{NCeigen1} \end{eqnarray} where, $\epsilon_{NC}$ \begin{eqnarray} \nonumber &=& -\frac{C^2(S_{NC}-\theta M_{NC}\Omega^2_{NC})^2}{(-S^2_{NC}+\Omega^2_{NC})^2M_{NC}} - \frac{C^2\Omega_{NC}^2(1-\theta M_{NC}S_{NC})^2}{M_{NC}( \Omega_{NC}^2 -S^2_{NC})^2}\\ \nonumber & +&\frac{2C^2S_{NC}(1-\theta M_{NC}S_{NC})(S_{NC}-\theta M_{NC}\Omega_{NC})}{M_{NC}(\Omega^2_{NC}-S_{NC}^2)} \\ \nonumber &+& \frac{2C^2(1-\theta M_{NC}S_{NC})}{M_{NC}(\Omega^2_{NC}-S^2_{NC})} -\frac{2C^2\theta(S_{NC}-\theta M_{NC}\Omega^2_{NC})} {(\Omega^2_{NC}-S^2_{NC})} \end{eqnarray} Note that, this time the eigenvalues (\ref{NCeigen1}) are completely real. \section{Anisotropic oscillator with non-hermitian coupling}\label{ani} We consider anisotropic oscillator on a plane with a non-hermitian coupling of the form $\sim ix_1x_2$. The Hamiltonian we consider is of the form \cite{asiri} \begin{eqnarray} H= \frac{1}{2}\sum_{i=1}^{2}\left(p_i^2 + C_ix_i^2\right) + \frac{1}{2}iC_3x_1x_2\label{ham} \end{eqnarray} The Hamiltonian (\ref{ham}) is not hermitian $H^\dagger \neq H$. We now consider this Hamiltonian in noncommutative space. It is however better to follow the transformation of Ref. \cite{asiri}, which is of the form \begin{eqnarray} \nonumber X &=& \alpha_1x_1 + x_2\,,~~~ Y= \alpha_2x_1 + x_2\,,\\ P_X &=& \frac{p_1-\alpha_2p_2}{\alpha_1-\alpha_2}\,,~P_Y= \frac{p_1-\alpha_1p_2}{\alpha_2-\alpha_1}\,. \label{trans} \end{eqnarray} where $\alpha_1=\left((C_1^2-C_2^2)-\sqrt{(C_1^2-C_2^2)^2- C_3^2}\right)/(iC_3)$ and $\alpha_1=\left((C_1^2-C_2^2)+\sqrt{(C_1^2-C_2^2)^2- C_3^2}\right)/(iC_3)$. Note that the new variables satisfy the usual commutation rules $[X,P_X]= [Y,P_Y]= i\hbar$. All other commutators being zero. In terms of the new canonical variables $X, Y, P_X, P_Y$, the Hamiltonian (\ref{ham}) reads as \begin{eqnarray} H= e^{K}H_K + e^{-K}H_{-K}\,,\label{ham12} \end{eqnarray} with the two separate one dimensional Hamiltonians \begin{eqnarray} H_K= \frac{\tilde{P_X}^2}{2m} + \frac{1}{2}m\Omega^2\tilde{X}^2\,, H_{-K}= \frac{\tilde{P_Y}^2}{2m} + \frac{1}{2}m\Omega^2\tilde{Y}^2\,, \label{ham123} \end{eqnarray} where the new canonical phase space coordinates are \begin{eqnarray} \nonumber \tilde{P_X} &=& e^{-K/2}\sqrt[4]{\frac{1+\alpha_1^2}{1+\alpha_2^2}}P_X\,, \tilde{P_Y}= e^{K/2}\sqrt[4]{\frac{1+\alpha_2^2}{1+\alpha_1^2}}P_Y,\\ \tilde{X} &=& e^{K/2}\sqrt[4]{\frac{1+\alpha_2^2}{1+\alpha_1^2}}X\,, \tilde{Y}= e^{-K/2}\sqrt[4]{\frac{1+\alpha_1^2}{1+\alpha_2^2}}Y\,. \label{newcoor} \end{eqnarray} The two oscillators of (\ref{ham12}) share the same mass $m= 1/\sqrt{(1+\alpha_1^2)(1+\alpha_2^2)}$, and have the frequencies \begin{eqnarray} \nonumber \Omega e^K=\sqrt{1/2\left[(C_1^2+C_2^2)-\sqrt{(C_1^2-C_2^2)^2-C_3^2}\right]} ,\\ \Omega e^{-K}=\sqrt{1/2\left[(C_1^2+C_2^2)+\sqrt{(C_1^2-C_2^2)^2-C_3^2}\right]}\,. \label{freq11} \end{eqnarray} The explicit form of the parameters $\Omega$ and $K$ can be found from (\ref{freq11}) as $\Omega=\sqrt[4]{( C_1^2C_2^2+ C_3^2/4)}$ and $e^K=\sqrt{\frac{\left[(C_1^2+C_2^2)-\sqrt{(C_1^2-C_2^2)^2-C_3^2}\right]}{ \left[(C_1^2+C_2^2)+\sqrt{(C_1^2-C_2^2)^2-C_3^2}\right]}}$. Note that in oreder to keep two frequencies of (\ref{freq11}) real, the condition $|C_1^2-C_2^2| \geq C_3$ needs to be satisfied. Thus (\ref{ham12}) is an anisotropic oscillator and its eigenvalues are \cite{asiri} \begin{eqnarray} E=\Omega\left[e^{K}(n_K +1/2) + e^{-K} (n_{-K} +1/2)\right]\,, \label{eigen123} \end{eqnarray} where $n_K, n_{-K}\in \mathbb{N}$. Before discussing the $\mathcal{PT}$-symmetric noncommutative quantum mechanics for (\ref{ham12}), we first discuss the problem in standard noncommutative space. We consider only coordinates to be noncommutative of the form \begin{eqnarray} \left[\hat{\tilde{X}},\hat{\tilde{Y}}\right]= 2i\underline\theta\,,\label{nonX1} \end{eqnarray} with the representation $\hat{\tilde{X}}=\tilde{X}-\underline\theta\tilde{P_Y}, \hat{\tilde{Y}}=\tilde{Y}+\underline\theta\tilde{P_X}$. Note that consistency with (\ref{al4}) and (\ref{rep3}) demands that $\underline\theta= (\alpha_1-\alpha_2)\theta$. The noncommutative version of (\ref{ham12}) is \begin{eqnarray} \nonumber H &=& e^{K}H_K + e^{-K}H_{-K} \\ \nonumber &+& e^K\left(m^2\Omega^2\underline\theta^2\tilde{P_Y}^2-m\Omega^2\underline\theta \tilde{X}\tilde{P_Y}\right)\\ &+&e^{-K}\left(m^2\Omega^2\underline\theta^2\tilde{P_X}^2+m\Omega^2\underline\theta \tilde{Y}\tilde{P_X}\right)\label{ham12N} \end{eqnarray} The perturbative spectrum of (\ref{ham12N}) is given by \begin{eqnarray} \nonumber E &=&\Omega\left[e^{K}(n_K +1/2) + e^{-K} (n_{-K} +1/2)\right]\\ &+&\Omega^3 m^3\underline{\theta}^2\left[e^{K}(n_K +1/2) + e^{-K} (n_{-K} +1/2)\right]. \label{eigen123p1} \end{eqnarray} Now let us consider the $\mathcal{PT}$-symmetric non-commutative case. In noncommutative phase space, the coordinates $X, Y, P_X, P_Y$ of (\ref{ham12}) are replaced by the corresponding noncommutative counterparts $\hat{X}, \hat{Y}, \hat{P_X}, \hat{P_Y}$, which lead to \begin{eqnarray} \nonumber \hat{H}= e^{K}H_K + e^{-K}H_{-K} \hspace{5cm}\\ \nonumber - e^{K}\left(\frac{\underline{\hat\theta}^2}{2m} \tilde{Y}^2 +\frac{m\Omega^2\underline{\theta}^2}{2}\tilde{P_Y}^2 +i{\underline{\hat\theta}}{m}\tilde{Y}\tilde{P_X}-i m\Omega^2\underline{\theta}\tilde{X}\tilde{P_Y}\right) \\- e^{-K}\left(\frac{\underline{\hat\theta}^2}{2m} \tilde{X}^2 +\frac{m\Omega^2\underline{\theta}^2}{2}\tilde{P_X}^2 -i{\underline{\hat\theta}}{m}\tilde{X}\tilde{P_Y}+i m\Omega^2\underline{\theta}\tilde{Y}\tilde{P_X}\right) \label{ham1hat} \end{eqnarray} Note that the noncommutative operators $\hat{X}, \hat{Y}, \hat{P_X}, \hat{P_Y}$, now satisfy the commutation algebra \begin{eqnarray} \nonumber \left[\hat{X},\hat{Y}\right]&=& -2\underline{\theta}\,, \left[\hat{P_X},\hat{P_Y}\right]= -2\underline{\hat\theta}\,,\\ \left[\hat{X},\hat{P_X}\right]&=& \left[\hat{P},\hat{P_Y}\right]= i\hat\hbar\,,\label{communew} \end{eqnarray} where $\underline{\theta}= (\alpha_1-\alpha_2)\theta$, $\underline{\hat\theta}= \hat\theta/(\alpha_1-\alpha_2)$ and the representation \begin{eqnarray} \nonumber \hat{X} &=& X +i\underline{\theta}P_Y\,, \hat{Y}= Y -i\underline{\theta}P_X\\ \hat{P_X}&=& P_X -i\underline{\hat\theta}Y\,, \hat{P_Y}= P_Y +i\underline{\hat\theta}X\,, \label{repX} \end{eqnarray} which can be obtained from (\ref{rep3pt}), is used to get (\ref{ham1hat}). It is possible to solve (\ref{ham1hat}) by perturbation. The eigenvalues are given by \begin{eqnarray} \nonumber E &=&\Omega\left[e^{K}(n_K +1/2) + e^{-K} (n_{-K} +1/2)\right]\\ \nonumber &+&\frac{1}{2}\Omega^3 m^2\underline{\theta}^2\left[e^{K}(n_K +1/2) + e^{-K} (n_{-K} +1/2)\right]\\ &+&\frac{1}{2}\frac{\hat{\underline{\theta}}^2}{m^2\Omega}\left[e^{K}(n_K +1/2) + e^{-K} (n_{-K} +1/2)\right]\,. \label{eigen123p2} \end{eqnarray} Note that all the complex terms of (\ref{ham1hat}) has the zero expectation values, which makes the spectrum (\ref{eigen123p2}) real. Importantly, as mentioned before, the condition for reality of frequencies, $|C_1^2-C_2^2| \geq C_3$, must have to be satisfied by (\ref{eigen123p1}) and (\ref{eigen123p2}) in order to keep the spectrum real. Othewise, it will generate complex eigenvalues in conjugate pairs. \section{Conclusion and discussion}\label{conclu} In this article $\mathcal{PT}$-symmetric quantum mechanical model is studied in noncommutative space. In order to keep the $\mathcal{PT}$-symmetry intact, we needed a non-commutative formulation which itself respects $\mathcal{PT}$-symmetry. This is done by replacing the non-commutativity parameter with pure imaginary parameter. However the systems on the $\mathcal{PT}$-symmetric non-commutative spaces generates complex eigen-values in some cases. On the other hand non Hermitian systems in standard noncommutative space has been found to have entirely real spectrum.

proofpile-arXiv_069-11881

\section*{References\@mkboth {REFERENCES}{REFERENCES}}\typeout{references:} \list {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\hskip .11em plus .33em minus .07em{\hskip .11em plus .33em minus .07em} \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \makeatother \def\bibtoc{\addcontentsline{toc}{section}{References}} \def\beginliteral{% \message{------------------------------------------------------------}% \message{LITERAL.TEX: for TeXing visible ascii character material}% \message{NB:}% \message{..1. Control-A and Control-B should not appear in material.}% \message{...... Used for escape and whitespace, respectively.}% \message{..2. `endliteral' should not appear in material.}% \message{..3. Should `untabify' material (e.g., using emacs command)}% \message{...... to avoid misalignments from improper tab processing by TeX.}% \message{..4. Should compare TeX output w/ actual ascii to double-check}% \message{...... for bugs in this code.}% \message{------------------------------------------------------------}% \begingroup \catcode`\\=12 \catcode`\{=12 \catcode`\}=12 \catcode`\$=12 \catcode`\&=12 \catcode`\#=12 \catcode`\^=12 \catcode`\_=12 \catcode`\ =12 \catcode`\~=12 \catcode`\%=12 \catcode`\*=12 \catcode`\"=12 \catcode`\@=12 \catcode`\^^A=0 \catcode`\^^B=10 \catcode`\^^C=12 \catcode`\^^D=12 \catcode`\^^E=12 \catcode`\^^F=12 \catcode`\^^G=12 \catcode`\^^H=12 \catcode`\^^I=12 \catcode`\^^J=12 \catcode`\^^K=12 \catcode`\^^L=12 \catcode`\^^M=12 \catcode`\^^N=12 \catcode`\^^O=12 \catcode`\^^P=12 \catcode`\^^Q=12 \catcode`\^^R=12 \catcode`\^^S=12 \catcode`\^^T=12 \catcode`\^^U=12 \catcode`\^^V=12 \catcode`\^^W=12 \catcode`\^^X=12 \catcode`\^^Y=12 \catcode`\^^Z=12 \obeyspaces \losenolines \parskip0pt \parindent0pt \tt \thatisit } {\obeyspaces\gdef {\hglue.5em\relax}} {\catcode`\^^M=\active % \gdef\losenolines{\catcode`\^^M=\active \def^^M{\leavevmode\endgraf}}} {\catcode`\
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ignorespaces}} \begin{document} \title{Optimal hash functions for approximate matches on the $n$-cube} \author{Daniel~M.~Gordon, Victor~S.~Miller and Peter~Ostapenko \thanks{D. Gordon and P. Ostapenko are with the IDA Center for Communications Research, 4320 Westerra Court, San Diego, 92121}% \thanks{V. Miller is with the IDA Center for Communications Research, 805 Bunn Drive, Princeton, New Jersey 08540}} \maketitle \begin{abstract} One way to find near-matches in large datasets is to use hash functions \cite{broder}, \cite{kwz}. In recent years locality-sensitive hash functions for various metrics have been given; for the Hamming metric projecting onto $k$ bits is simple hash function that performs well. In this paper we investigate alternatives to projection. For various parameters hash functions given by complete decoding algorithms for error-correcting codes work better, and asymptotically random codes perform better than projection. \end{abstract} \maketitle \section{Introduction} \label{sec:intro} Given a set of $M$ $n$-bit vectors, a classical problem is to quickly identify ones which are close in Hamming distance. This problem has applications in numerous areas, such as information retrieval and DNA sequence comparison. The nearest-neighbor problem is to find a vector close to a given one, while the closest-pair problem is to find the pair in the set with the smallest Hamming distance. Approximate versions of these problems allow an answer where the distance may be a factor of $(1+\varepsilon)$ larger than the best possible. One approach (\cite{broder}, \cite{gim}, \cite{kwz}) is {\em locality-sensitive hashing} (LSH). A family of hash functions ${\cal H}$ is called $(r,cr,p_1,p_2)$-sensitive if for any two points $\mathbf{x}$, $\mathbf{y} \in {\cal V}$, \begin{itemize} \item if $d(\mathbf{x},\mathbf{y}) \leq r$, then $\prob(h(\mathbf{x}) = h(\mathbf{y})) \geq p_1$, \item if $d(\mathbf{x},\mathbf{y}) \geq cr$, then $\prob(h(\mathbf{x}) = h(\mathbf{y})) \leq p_2$. \end{itemize} Let $\rho = \log(1/p_1)/\log(1/p_2)$. An LSH scheme can be used to solve the approximate nearest neighbor problem for $M$ points in time $O(M^\rho)$. Indyk and Motwani \cite{im} showed that projection has $\rho = 1/c$. The standard hash to use is projection onto $k$ of the $n$ coordinates \cite{gim}. An alternative family of hashes is based on minimum-weight decoding with error-correcting codes \cite{be}, \cite{w}. A $[n,k]$ code ${\cal C}$ with a complete decoding algorithm defines a hash $h^{\cal C}$, where each $\mathbf{v} \in {\cal V} :=\Fnum_2^n$ is mapped to the codeword $\mathbf{c} \in {\cal C} \subset {\cal V}$ to which $\mathbf{v}$ decodes. Using linear codes for hashing schemes has been independently suggested many times; see \cite{be}, \cite{dhlnp}, and the patents~\cite{patent2} and \cite{w}. In \cite{be} the binary Golay code was suggested to find approximate matches in bit-vectors. Data is provided that suggests it is effective, but it is still not clear when the Golay or other codes work better than projection. In this paper we attempt to quantify this, using tools from coding theory. Our model is somewhat different from the usual LSH literature. We are interested in the scenario where we have collection of $M$ random points of ${\cal V}$, one of which, $\mathbf{x}$, has been duplicated with errors. The error vector $\mathbf{e}$ has each bit nonzero with probability $\gamma$. Let ${\hbox{\sf\slshape P}}^{\cal C}(p)$ be the probability that $h^{\cal C}(\mathbf{x})=h^{\cal C}(\mathbf{x}+\mathbf{e})$. Then the probability of collision of two points $\mathbf{x}$ and $\mathbf{y}$ is \begin{itemize} \item if $\mathbf{y}=\mathbf{x}+\mathbf{e}$, then $\prob(h(\mathbf{x}) = h(\mathbf{y})) = \tilde{p}_1 = {\hbox{\sf\slshape P}}^{\cal C}(p)$, \item if $\mathbf{y} \neq \mathbf{x}+\mathbf{e}$, then $\prob(h(\mathbf{x}) = h(\mathbf{y})) = \tilde{p}_2 = 2^{-k}$. \end{itemize} Then the number of elements that hash to $h(\mathbf{x})$ will be about $M/2^k$, and the probability that one of these will be $\mathbf{y}=\mathbf{x}+\mathbf{e}$ is ${\hbox{\sf\slshape P}}^{\cal C}(p)$. If this fails, we may try again with a new hash, say the same one applied after shifting the $M$ points by a fixed vector, and continue until $\mathbf{y}$ is found. Let $\rho=\log(1/\tilde{p}_1)/\log(1/\tilde{p}_2)$ as for LSH. Taking $2^k \approx M$, we expect to find $\mathbf{y}$ in time $$ \frac{M}{2^k {\hbox{\sf\slshape P}}^{\cal C}(p)} = O(M^\rho). $$ As with LSH, we want to optimize this by minimizing $\rho$, i.e. finding a hash function minimizing ${\hbox{\sf\slshape P}}^{\cal C}(p)$. For a linear code with a complete translation-invariant decoding algorithm (so that $h(\mathbf{x})=\mathbf{c}$ implies that $h(\mathbf{x}+\mathbf{c}')=\mathbf{c}+\mathbf{c}'$), studying ${\hbox{\sf\slshape P}}^{{\cal C}}$ is equivalent to studying the properties of the set ${\cal S}$ of all points in ${\cal V}$ that decode to ${\bf 0}$. In Section~\ref{sec:sets} and the appendix we systematically investigate sets of size $\leq 64$. Suppose that we pick a random $\mathbf{x} \in {\cal S}$. Then the probability that $\mathbf{y} = \mathbf{x} + \mathbf{e}$ is in ${\cal S}$ is \begin{equation} \label{eq:pue} {\hbox{\sf\slshape P}}_{\cal S}(\gamma) = \frac{1}{|{\cal S}|} \sum_{\mathbf{x},\mathbf{y} \in {\cal S}} \gamma^{d(\mathbf{x},\mathbf{y})} (1-\gamma)^{n-d(\mathbf{x},\mathbf{y})}. \end{equation} This function has been studied extensively in the setting of error-detecting codes \cite{kk}. In that literature, ${\cal S}$ is a code, ${\hbox{\sf\slshape P}}_{\cal S}(\gamma)$ is the probability of an undetected error, and the goal is to minimize this probability. Here, on the other hand, we will call a set {\em optimal} for $\gamma$ if no set in ${\cal V}$ of size $|{\cal S}|$ has greater probability. As the error rate $\gamma$ approaches $1/2$, this coincides with the definition of {\em distance-sum optimal sets}, which were first studied by Ahlswede and Katona \cite{ak}. The error exponent of a code ${\hbox{\sf\slshape C}}$ is $$ {\hbox{\sf\slshape E}}^{\cal C}(\gamma) = -\frac{1}{n} \lg {\hbox{\sf\slshape P}}^{\cal C}(\gamma). $$ In this paper $\lg$ denotes log to base 2. We are interested in properties of the error exponent over codes of rate $R=k/n$ as $n \to \infty$. Note that $\rho = E^{\cal C}(p)/R$, so minimizing the error exponent will give us the best code to use for finding closest pairs. In Section~\ref{sec:shannon} we will show that hash functions from random (nonlinear) codes have a better error exponent than projection. \section{Hash Functions From Codes} \label{sec:codes} For a set ${\cal S} \subset {\cal V}$, let $$ A_i = \# \{ (\mathbf{x},\mathbf{y}) : \mathbf{x},\mathbf{y} \in {\cal S} \hbox{ and } d(\mathbf{x},\mathbf{y}) = i \} $$ count the number of pairs of words in ${\cal S}$ at distance $i$. The {\em distance distribution function} is \begin{equation} \label{eq:gf} A({\cal S},\zeta) := \sum_{i=0}^n A_i \zeta^i . \end{equation} This function is directly connected to ${\hbox{\sf\slshape P}}_{\cal S}(\gamma)$ \cite{kk}. If $\mathbf{x}$ is a random element of ${\cal S}$, and $\mathbf{y} = \mathbf{x} +e$, where $e$ is an error vector where each bit is nonzero with probability $\gamma$, then the probability that $\mathbf{y} \in {\cal S}$ is \begin{eqnarray} \label{eq:pr_ue} {\hbox{\sf\slshape P}}_{\cal S}(\gamma) & := & \frac{1}{|{\cal S}|} \sum_{\mathbf{x},\mathbf{y} \in {\cal S}} \gamma^{d(\mathbf{x},\mathbf{y})} (1-\gamma)^{n-d(\mathbf{x},\mathbf{y})} \\ \nonumber & = & \frac{1}{|{\cal S}|} \sum_{i=0}^n A_i \gamma^i (1-\gamma)^{n-i} \\ \nonumber & = & \frac{(1-\gamma)^n}{|{\cal S}|} A\left({\cal S} ,\frac{\gamma}{1-\gamma} \right). \end{eqnarray} In this section we will evaluate (\ref{eq:pr_ue}) for projection and for perfect codes, and then consider other linear codes. \subsection{Projection} \label{subsec:projection} The simplest hash is to project vectors in ${\cal V}$ onto $k$ coordinates. Let \emph{$k$-projection} denote the $[n,k]$ code ${\cal P}_{n,k}$ corresponding to this hash. The associated ${\cal S}$ of vectors mapped to $\mathbf{0}$ is an $2^{n-k}$-subcube of ${\cal V}$. The distance distribution function is \begin{equation} \label{eq:proj_ddf} A({\cal S},\zeta) = (2(1+\zeta))^{n-k} \, , \end{equation} so the probability of collision is \begin{equation} \label{eq:proj_prob} {\hbox{\sf\slshape P}}^{{\cal P}_{n,k}}(\gamma) = \frac{(1-\gamma)^n}{2^{n-k}} \left(\frac{2}{1-\gamma}\right)^{n-k} = (1-\gamma)^k. \end{equation} ${\cal P}_{n,k}$ is not a good error-correcting code, but for sufficiently small error probability its hash function is optimal. \begin{theorem} \label{thm:gamma_small} Let ${\cal S}$ be the $2^{n-k}$-subcube of ${\cal V}$. For any error probability $\gamma \in (0, \, 2^{-2(n-k)})$, ${\cal S}$ is an optimal set, and so $k$-projection is an optimal hash. \end{theorem} \begin{proof} The distance distribution function for ${\cal S}$ is $$ A({\cal S},\zeta) = 2^{n-k}(1+\zeta)^{n-k}. $$ The edge isoperimetric inequality for an $n$-cube \cite{harper2} states that \begin{lemma} \label{lem:iso} Any subset $S$ of the vertices of the $n$-dim\-en\-sion\-al cube $Q_n$ has at most $$ \frac{1}{2} |S| \lg |S| $$ edges between vertices in $S$, with equality if and only if $S$ is a subcube. \end{lemma} Any set ${\cal S}'$ with $2^{n-k}$ points has distance distribution function $$A({\cal S}',\zeta) = \sum_{i=0}^k c_i \zeta^i,$$ where $c_0 = 2^{n-k}$, $c_1 < (n-k) 2^{n-k}$ by Lemma~\ref{lem:iso}, and the sum of the $c_i$'s is $2^{2(n-k)}$. By (\ref{eq:proj_prob}) the probability of collision is $(1-\gamma)^n 2^{n-k}A({\hbox{\sf\slshape S}}',\gamma/(1-\gamma))$. \begin{eqnarray*} A({\cal S}',\zeta) & \leq & 2^{n-k} + \zeta((n-k)2^{n-k}-1)\\ && + \zeta^2 \left(2^{2(n-k)}-({n-k}+1)2^{n-k}+1\right), \end{eqnarray*} and \begin{eqnarray*} \lefteqn{A({\cal S},\zeta) - A({\cal S}',\zeta)} \quad \\ &\geq& \zeta - \zeta^2 \left(2^{2(n-k)}+ 2^{{n-k}-1}\left({n-k}^2+{n-k}+2\right)+1 \right) \\ &>& \zeta-\zeta^2(2^{2(n-k)}-1). \end{eqnarray*} This is positive if $\gamma<1/2$ and $(1-\gamma)/\gamma > 2^{2(n-k)}-1$, i.e., for $\gamma<2^{-2(n-k)}$. \end{proof} \subsection{Concatenated Hashes} \label{subsec:concat} Here we show that if $h$ and $h'$ are good hashes, then the concatenation is as well. First we identify ${\cal C}$ with $\Fnum_2^k$ and treat $h^{\cal C}$ as a hash $h$ from $\Fnum_2^n \to \Fnum_2^k$. We denote ${\hbox{\sf\slshape P}}^{\cal C}$ by ${\hbox{\sf\slshape P}}^h$. From $h : \Fnum_2^n \to \Fnum_2^k$ and $h' : \Fnum_2^{n'} \to \Fnum_2^{k'}$, we get a concatenated hash $(h,h') : \Fnum_2^{n+n'} \to \Fnum_2^{k+k'}$. \begin{lemma} Fix $\gamma \in (0,1/2)$. Let $h$ and $h'$ be hashes. Then $$ \min \{ {\hbox{\sf\slshape E}}^h(\gamma), {\hbox{\sf\slshape E}}^{h'}(\gamma) \} \leq {\hbox{\sf\slshape E}}^{(h,h')(\gamma)} \leq \max \{ {\hbox{\sf\slshape E}}^{h}(\gamma), {\hbox{\sf\slshape E}}^{h'}(\gamma) \} \, , $$ with strict inequalities if ${\hbox{\sf\slshape E}}^{h}(\gamma) \not= {\hbox{\sf\slshape E}}^{h'}(\gamma)$. \end{lemma} \begin{proof} Since $\gamma$ is fixed, we drop it from the notation. Suppose ${\hbox{\sf\slshape E}}^h \leq {\hbox{\sf\slshape E}}^{h'}$. Then $$ \frac{\lg {\hbox{\sf\slshape P}}^h}{n} \leq \frac{\lg {\hbox{\sf\slshape P}}^h + \lg {\hbox{\sf\slshape P}}^{h'}}{n+n'} \leq \frac{\lg {\hbox{\sf\slshape P}}^{h'}}{n'}. $$ Since ${\hbox{\sf\slshape P}}^{(h,h')} = {\hbox{\sf\slshape P}}^h \, {\hbox{\sf\slshape P}}^{h'}$, we have ${\hbox{\sf\slshape E}}^h \leq {\hbox{\sf\slshape E}}^{(h,h')} \leq {\hbox{\sf\slshape E}}^{h'}$. \end{proof} \subsection{Perfect Codes} \label{subsec:perfect} An {\em $e$-sphere} around a vector $\mathbf{x}$ is the set of all vectors $\mathbf{y}$ with $d(\mathbf{x},\mathbf{y}) \leq e$. An $[n,k,2e+1]$ code is {\em perfect} if the $e$-spheres around codewords cover ${\cal V}$. Minimum weight decoding with perfect codes is a reasonable starting point for hashing schemes, since all vectors are closest to a unique codeword. The only perfect binary codes are trivial repetition codes, the Hamming codes, and the binary Golay code. Repetition codes do badly, but the other perfect codes give good hash functions. \subsubsection{Binary Golay Code} \label{subsec:golay} The $[23,12,7]$ binary Golay code ${\cal G}$ is an important perfect code. The 3-spheres around each code codeword cover $\Fnum_2^{23}$. The 3-sphere around $\mathbf{0}$ in the 23-cube has distance distribution function \begin{eqnarray*} \lefteqn{2048 + 11684 \zeta + 128524 \zeta^2 + 226688 \zeta^3} \qquad\qquad \\ && \hbox{} + 1133440 \zeta^4 + 672980 \zeta^5 + 2018940 \zeta^6 \, . \end{eqnarray*} From this we find ${\hbox{\sf\slshape E}}^{\cal G}(\gamma)>{\hbox{\sf\slshape E}}^{{\cal P}_{23,12}}(\gamma)$ for $\gamma \in (0.2555, \, 1/2)$. \subsubsection{Hamming Codes} \label{subsec:hamming} Aside from the repetition codes and the Golay code, the only perfect binary codes are the Hamming codes. The $[2^m-1,\, 2^m-m-1,\, 3]$ Hamming code ${\cal H}_m$ corrects one error. The distance distribution function for a 1-sphere is \begin{equation} \label{eq:hamming_ddf} 2^m + 2(2^m-1) \zeta + (2^m-1)(2^m-2) \zeta^2, \end{equation} so the probability of collision ${\hbox{\sf\slshape P}}^{{\cal H}_m}(\gamma)$ is \begin{eqnarray} \label{eq:ham_prob} \frac{(1-\gamma)^{2^m-1}}{2^m}(2^m &+& 2(2^m-1) \frac{\gamma}{1-\gamma}\\ &+& (2^m-1) (2^m-2) \frac{\gamma^2}{(1-\gamma)^2} ) \nonumber \end{eqnarray} Table~\ref{tab:ham} gives the crossover error probabilities where the first few Hamming codes become better than projection. \begin{table}[!t] \caption{Crossover error probabilities $\gamma$ for Hamming codes ${\cal H}_m$.} \label{tab:ham} \centering \begin{tabular}{|cc|c|} \hline $m$ & $k$ & $\gamma$ \\ \hline 4 & 11 & $0.2826$ \\ 5 & 26 & $0.1518$ \\ 6 & 57 & $0.0838$ \\ 7 &120 & $0.0468$ \\ \hline \end{tabular} \end{table} \begin{theorem} \label{thm:hamming_wins} For any $m>4$ and $p>m/(2^m-m)$, the Hamming code ${\cal H}_m$ beats $(2^m-m-1)$-projection. \end{theorem} \begin{proof} The difference between the distribution functions of the cube and the 1-sphere in dimension $2^m-1$ is \begin{eqnarray} \label{eq:fm} f_m(\zeta) &:=& A({\cal S},\zeta) - A({\cal H}_m,\zeta) \\ &=& 2^m(1+\zeta)^m \nonumber \\ && - (2^m + 2(2^m-1) \zeta + (2^m-1)(2^m-2) \zeta^2).\nonumber \end{eqnarray} We will show that, for $m \ge 4$, $f_m(\zeta)$ has exactly one root in $(0,1)$, denoted by $\alpha_m$, and that $\alpha_m \in \left( (m-2)/2^m,m/2^m\right)$. We calculate \begin{eqnarray*} f_m(\zeta) &=& ((m-2)2^m+1)\zeta \\ &&-\left(2^{2m} -\left(3+\binom{m}{2}\right)2^m + 2\right)\zeta^2 \\ &&+ 2^m\sum_{i=3}^m \binom{m}{i} \zeta^i. \end{eqnarray*} All the coefficients of $f_m(\zeta)$ are non-negative with the exception of the coefficient of $\zeta^2$, which is negative for $m \ge 2$. Thus, by Descartes' rule of signs $f(\zeta)$ has 0 or 2 positive roots. However, it has a root at $\zeta=1$. Call the other positive root $\alpha_m$. We have $f_m(0) = f_m(1) = 0$, and since $f'(0) = (m-2) 2^m+2 >0$ and $f'(1) = 2^{2m-1}(m-4)+2^{m+2}-2>0$ for $m\geq 4$, we must have $\alpha_m<1$ for $m\geq 4$. For $p>\alpha_m$ the Hamming code ${\cal H}_m$ beats projection. Using (\ref{eq:fm}) and Bernoulli's inequality, it is easy to show that $f_m(\zeta)>0$ for $\zeta < c(m-2)/2^m$ for any $c<1$ and $m \geq 4$. For the other direction, we may use Taylor's theorem to show $$ 2^m\left(1+\frac{m}{2^m}\right)^m < 2^m + m^2 + \frac{m^4}{2^{m+1}} \left(1+\frac{m}{2^m}\right)^{m-2}. $$ Plugging this into (\ref{eq:fm}), we have that $f_m(m/2^m)<0$ for $m>6$. \end{proof} \subsection{Other Linear Codes} \label{subsec:linear} The above codes give hashing strategies for a few values of $n$ and $k$, but we would like hashes for a wider range. For a hashing strategy using error-correcting codes, we need a code with an efficient {\em complete decoding} algorithm; that is a way to map every vector to a codeword. Given a translation invariant decoder, we may determine ${\cal S}$, the set of vectors that decode to ${\bf 0}$, in order to compare strategies as the error probability changes. \Magma/~\cite{magma} has a built-in database of linear codes over $\Fnum_2$ of length up to 256. Most of these do not come with efficient complete decoding algorithms, but \magma/ does provide syndrome decoding. Using this database new hashing schemes were found. For each dimension $k$ and minimum distance $d$, an $[n,k,d]$ binary linear code with minimum length $n$ was chosen for testing.\footnote{The \magma/ call {\tt BLLC(GF(2),k,d)} was used to choose a code.} (This criterion excludes any codes formed by concatenating with a projection code.) For each code there is an error probability above which the code beats projection. Figure~\ref{fig:cross} shows these crossover probabilities. Not surprisingly, the $[23,12,7]$ Golay code ${\cal G}$ and Hamming codes ${\cal H}_4$ and ${\cal H}_5$ all do well. The facts that concatenating the Golay code with projection beats the chosen code for $13 \leq k \leq 17$ and concatenating ${\cal H}_m$ with projection beats the chosen codes for $27 \leq k \leq 30$ show that factors other than minimum length are important in determining an optimal hashing code. \begin{figure}[!t] \centering \leavevmode \begin{tikzpicture}[xscale=0.23,yscale=9] \draw (0,0) -- (30,0); \draw (30,0) -- (30,0.5); \draw (0,0) -- (0,0.5); \draw (0,0.5) -- (30,0.5); \draw (14,-0.07) node [left] {$k$}; \draw (-5,0.25) node [below] {$\gamma$}; \path[draw=black] (23,0.45) ellipse (5pt and 0.1pt) -- (25,0.45) ellipse (5pt and 0.1pt) node [right] {\footnotesize $d=3$}; \path[draw=black] (8 ,0.456) ellipse (5pt and 0.1pt) node (3_8) [] {} -- (9 ,0.393) ellipse (5pt and 0.1pt) node (3_9) [] {} -- (10 ,0.333) ellipse (5pt and 0.1pt) node (3_10) [] {} -- (11 ,0.2826) ellipse (5pt and 0.1pt) node (3_11) [left] {\footnotesize ${\cal H}_4$} -- (12 ,0.28656) ellipse (5pt and 0.1pt) node (3_12) [] {} -- (13 ,0.28731) ellipse (5pt and 0.1pt) node (3_13) [] {} -- (14 ,0.28503) ellipse (5pt and 0.1pt) node (3_14) [] {} -- (15 ,0.279944) ellipse (5pt and 0.1pt) node (3_15) [] {} -- (16 ,0.272384) ellipse (5pt and 0.1pt) node (3_16) [] {} -- (17 ,0.262737) ellipse (5pt and 0.1pt) node (3_17) [] {} -- (18 ,0.25145) ellipse (5pt and 0.1pt) node (3_18) [] {} -- (19 ,0.239) ellipse (5pt and 0.1pt) node (3_19) [] {} -- (20 ,0.2259) ellipse (5pt and 0.1pt) node (3_20) [] {} -- (21 ,0.2125) ellipse (5pt and 0.1pt) node (3_21) [] {} -- (22 ,0.1992) ellipse (5pt and 0.1pt) node (3_22) [] {} -- (23 ,0.1864) ellipse (5pt and 0.1pt) node (3_23) [] {} -- (24 ,0.1741) ellipse (5pt and 0.1pt) node (3_24) [] {} -- (25 ,0.1626) ellipse (5pt and 0.1pt) node (3_25) [] {} -- (26 ,0.1518) ellipse (5pt and 0.1pt) node (3_26) [below] {\footnotesize ${\cal H}_5$} -- (27 ,0.1529) ellipse (5pt and 0.1pt) node (3_27) [] {} -- (28,0.1535) ellipse (5pt and 0.1pt) node (3_28) [] {} -- (29,0.1538) ellipse (5pt and 0.1pt) node (3_29) [] {} -- (30,0.1537) ellipse (5pt and 0.1pt) node (3_30) [] {}; \path[draw=red] (23,0.42) ellipse (5pt and 0.1pt) -- (25,0.42) ellipse (5pt and 0.1pt) node [right] {\footnotesize $d=5$}; \path[draw=red] (9 ,0.453) ellipse (5pt and 0.1pt) node (5_9) [] {} -- (10 ,0.45346) ellipse (5pt and 0.1pt) node (5_10) [] {} -- (11 ,0.424057) ellipse (5pt and 0.1pt) node (5_11) [] {} -- (12 ,0.399166) ellipse (5pt and 0.1pt) node (5_12) [] {} -- (13 ,0.37273) ellipse (5pt and 0.1pt) node (5_13) [] {} -- (14 ,0.34225) ellipse (5pt and 0.1pt) node (5_14) [] {} -- (15 ,0.355389) ellipse (5pt and 0.1pt) node (5_15) [] {} -- (16 ,0.335497) ellipse (5pt and 0.1pt) node (5_16) [] {} -- (17 ,0.318722) ellipse (5pt and 0.1pt) node (5_17) [] {} -- (18 ,0.302035) ellipse (5pt and 0.1pt) node (5_18) [] {} -- (19 ,0.28666) ellipse (5pt and 0.1pt) node (5_19) [] {} -- (20 ,0.273291) ellipse (5pt and 0.1pt) node (5_20) [] {} -- (21 ,0.260697) ellipse (5pt and 0.1pt) node (5_21) [] {} -- (22 ,0.248295) ellipse (5pt and 0.1pt) node (5_22) [] {} -- (23 ,0.236963) ellipse (5pt and 0.1pt) node (5_23) [] {} -- (24 ,0.2611) ellipse (5pt and 0.1pt) node (5_24) [] {} -- (25 ,0.2542) ellipse (5pt and 0.1pt) node (5_25) [] {}; \path[draw=green] (23,0.39) ellipse (5pt and 0.1pt) -- (25,0.39) ellipse (5pt and 0.1pt) node [right] {\footnotesize $d=7$}; \path[draw=green] (8 ,0.48534) ellipse (5pt and 0.1pt) node (7_8) [] {} -- (9 ,0.42879) ellipse (5pt and 0.1pt) node (7_9) [] {} -- (10 ,0.372251) ellipse (5pt and 0.1pt) node (7_10) [] {} -- (11 ,0.315547) ellipse (5pt and 0.1pt) node (7_11) [] {} -- (12 ,0.2555) ellipse (5pt and 0.1pt) node (7_12) [below] {\footnotesize ${\cal G}$} -- (13 ,0.34646) ellipse (5pt and 0.1pt) node (7_13) [] {} -- (14 ,0.32713) ellipse (5pt and 0.1pt) node (7_14) [] {} -- (15 ,0.32515) ellipse (5pt and 0.1pt) node (7_15) [] {} -- (16 ,0.309727) ellipse (5pt and 0.1pt) node (7_16) [] {} -- (17 ,0.29765) ellipse (5pt and 0.1pt) node (7_17) [] {} -- (18 ,0.29002) ellipse (5pt and 0.1pt) node (7_18) [] {} -- (19 ,0.279135) ellipse (5pt and 0.1pt) node (7_19) [] {} -- (20 ,0.2675) ellipse (5pt and 0.1pt) node (7_20) [] {} -- (21 ,0.2577) ellipse (5pt and 0.1pt) node (7_21) [] {}; \foreach \x/\xtext in {0/0, 5/5, 10/10, 15/15, 20/20, 25/25, 30/30} \draw[shift={(\x,0)},gray,very thin] (0,0.5) -- (0,0) node[below] {\footnotesize $\xtext$}; \foreach \y/\ytext in {0/0,0.05/0.05,0.1/0.1,0.15/0.15,0.2/0.2,0.25/0.25,0.3/0.3,0.35/0.35,0.4/0.4,0.45/0.45,0.5/0.5} \draw[shift={(0,\y)},gray,very thin] (30,0) -- (0,0) node[left] {\footnotesize $\ytext$}; \end{tikzpicture} \caption{Crossover error probabilities for minimum length linear codes.}\label{fig:cross} \end{figure} As linear codes are subspaces of $\Fnum_2^n$, lattices are subspaces of $\Rnum^n$. The 24-dimensional Leech lattice is closely related to the Golay code, and also has particularly nice properties. It was used in \cite{ai} to construct a good LSH for $\Rnum^n$. \section{Optimal Sets} \label{sec:sets} In the previous section we looked at the performances of sets associated with various good error-correcting codes. However, the problem of determining optimal sets ${\cal S} \subset \Fnum_2^n$ is of independent interest. The general question of finding an optimal set of size $2^t$ in ${\cal V}$ for an error probability $\gamma$ is quite hard. In this section we will find the answer for $t \leq 6$, and look at what happens when $\gamma$ is near $1/2$. \subsection{Optimal Sets of Small Size} \label{subsec:nauty} For a vector $\mathbf{x} = (x_1,\ldots,x_n) \in {\cal V}$, let $r_i(\mathbf{x})$ be $\mathbf{x}$ with the $i$-th coordinate complemented, and let $s_{ij}(\mathbf{x})$ be $\mathbf{x}$ with the $i$-th and $j$-th coordinates switched. \begin{definition} Two sets are isomorphic if one can be gotten from the other by a series of $r_i$ and $s_{ij}$ transformations. \end{definition} \begin{lemma} If ${\cal S}$ and ${\cal S}'$ are isomorphic, then ${\hbox{\sf\slshape P}}_{\cal S}(p) = {\hbox{\sf\slshape P}}_{{\cal S}'}(p)$ for all $p \in [0,1]$. \end{lemma} The corresponding non-invertible transformation are: \begin{eqnarray} \label{eq:rho} \rho_i(\mathbf{x}) &:=& (x_1,x_2,\ldots,x_{i-1},0,x_{i+1},\ldots x_n) \, , \\ \nonumber \sigma_{ij}(\mathbf{x}) &:=& \left\{ \begin{array}{ll} \mathbf{x}, & x_{\min(i,j)}=0, \\ \label{eq:si} s_{ij}(\mathbf{x}), & x_{\min(i,j)}=1. \end{array} \right. \end{eqnarray} \begin{definition} A set ${\cal S} \subset {\cal V}$ is a {\em down-set} if $\rho_i({\cal S}) \subset {\cal S}$ for all $i \leq n$. \end{definition} \begin{definition} A set ${\cal S} \subset {\cal V}$ is {\em right-shifted} if $\sigma_{ij}({\cal S}) \subset {\cal S}$ for all $i,j \leq n$. \end{definition} \begin{theorem}\label{thm:rightdown} If a set ${\cal S}$ is optimal, then it is isomorphic to a right-shifted down-set. \end{theorem} \begin{proof} We will show that any optimal set is isomorphic to a right-shifted set. The proof that it must be isomorphic to a down-set as well is similar. A similar proof for distance-sum optimal sets (see Section~\ref{subsec:large_garble}) was given by K\"{u}ndgen in \cite{kundgen}. Recall that $${\hbox{\sf\slshape P}}_{\cal S}(\gamma) = \frac{(1-\gamma)^n}{|{\cal S}|} \sum_{\mathbf{x},\mathbf{y} \in {\cal S}} \zeta^{d(\mathbf{x},\mathbf{y})},$$ where $\zeta = \gamma/(1-\gamma) \in (0,1)$. If ${\cal S}$ is not right-shifted, there is some $\mathbf{x}\in {\cal S}$ with $x_i=1$, $x_j=0$, and $i<j$. Let $\varphi_{ij}({\cal S})$ replace all such sets $\mathbf{x}$ with $s_{ij}(\mathbf{x})$. We only need to show that this will not decrease ${\hbox{\sf\slshape P}}_{\cal S}(\gamma)$. Consider such an $\mathbf{x}$ and any $\mathbf{y} \in {\cal S}$. If $y_i = y_j$, then $d(\mathbf{x},\mathbf{y}) = d(s_{ij}(\mathbf{x}),\mathbf{y})$, and $P_{\cal S}(\gamma)$ will not change. If $y_i = 0$ and $y_j=1$, then $d(\mathbf{x},\mathbf{y}) = d(s_{ij}(\mathbf{x}),\mathbf{y})-2$, and since $\zeta^{l-2} \geq \zeta^l$, that term's contribution to $P_{\cal S}(\gamma)$ increases. Suppose $y_i = 1$ and $y_j=0$. If $s_{ij}(\mathbf{y}) \in {\cal S}$, then $d(\mathbf{x},\mathbf{y})+d(\mathbf{x},s_{ij}(\mathbf{y}))=d(s_{ij}(\mathbf{x}),\mathbf{y})+d(s_{ij}(\mathbf{x}),s_{ij}(\mathbf{y}))$, and $P_{\cal S}(\gamma)$ is unchanged. Otherwise, $\varphi_{ij}({\cal S})$ will replace $\mathbf{y}$ by $s_{ij}(\mathbf{y})$, and $d(\mathbf{x},\mathbf{y}) = d(s_{ij}(\mathbf{x}),s_{ij}(\mathbf{y}))$ means that $P_{\cal S}(\gamma)$ will again be unchanged. \end{proof} Let $R_{s,n}$ denote an optimal set of size $s$ in $\Fnum_2^n$. By computing all right-shifted down-sets of size $2^t$, for $t \leq 6$, we have the following result: \begin{theorem} \label{thm:nauty_results} The optimal sets $R_{2^t,n}$ for $t \in \{1,\dots,6\}$ correspond to Tables~\ref{tab:opt} [pg.~\pageref{tab:opt}] and \ref{tab:opt64} [pg.~\pageref{tab:opt64}]. \end{theorem} These figures, and details of the computations, are given the Appendix. Some of the optimal sets for $t=6$ do better than the sets corresponding to the codes in Figure~\ref{fig:cross}. \subsection{Optimal Sets for Large Error Probabilities} \label{subsec:large_garble} Theorem~\ref{thm:gamma_small} states that for any $n$ and $k$, for a sufficiently small error probability $\gamma$, a $2^{n-k}$-subcube is an optimal set. One may also ask what an optimal set is at the other extreme, a large error probability. In this section we use existing results about minimum average distance subsets to list additional sets that are optimal as $\gamma \to 1/2^-$. We have \begin{eqnarray*} {\hbox{\sf\slshape P}}_{\cal S}(\gamma) & := & \frac{(1-\gamma)^n}{|{\cal S}|} A\left({\cal S} ,\frac{\gamma}{1-\gamma} \right) \\ &=& \frac{1}{|{\cal S}|} \sum\nolimits_i A_i \gamma^i (1-\gamma)^{n-i} \, . \end{eqnarray*} Letting $\gamma = 1/2-\ep$ and $s = |{\cal S}|$, ${\hbox{\sf\slshape P}}_{\cal S}(\gamma)$ becomes \begin{eqnarray*} \lefteqn{s^{-1} \sum\nolimits_i A_i \left( 1/2 - \ep \right)^i \left( 1/2 + \ep \right)^{n-i}} \qquad && \\ &=& \frac{1}{s \, 2^n} \left( \sum\nolimits_i A_i + \ep \left( \sum\nolimits_i 2 (n-2i) A_i \right) + O(\ep^2) \right) \\ &=& \frac{s}{2^n} ( 1 + 2 n \ep ) - \frac{4 \ep}{s \, 2^n} \sum\nolimits_i i A_i + O(\ep^2 ) \, . \end{eqnarray*} Therefore, an optimal set for $\gamma \to 1/2^-$ must minimize the \emph{distance-sum} of ${\cal S}$ \begin{eqnarray} d({\cal S}) &:=& \frac{1}{2} \sum_{\mathbf{x},\mathbf{y} \in {\cal S}} d(\mathbf{x},\mathbf{y}) = \frac{1}{2} \sum\nolimits_i i A_i \, . \end{eqnarray} Denote the minimal distance sum by \begin{eqnarray*} f(s,n) := \min \left\{ d({\cal S}): {\cal S} \subset \Fnum_2^n, \, |{\cal S}|=s \right\} \, . \end{eqnarray*} If $d({\cal S}) = f(s,n)$ for a set ${\cal S}$ of size $s$, we say that ${\cal S}$ is \emph{distance-sum optimal}. The question of which sets are distance-sum optimal was proposed by Ahlswede and Katona in 1977; see K\"{u}ndgen \cite{kundgen} for references and recent results. This question is also difficult. K\"{u}ndgen presents distance-sum optimal sets for small $s$ and $n$, which include the ones of size 16 from Table~\ref{tab:opt}. Jaeger et al. \cite{jaeger} found the distance-sum optimal set for $n$ large. \begin{theorem} {} (Jaeger, et al.~\cite{jaeger}, cf.~\cite[pg.~151]{kundgen}) For $n \geq s-1$, a generalized 1-sphere (with $s$ points) is distance-sum optimal unless $s \in \{ 4, 8 \}$ (in which case the subcube is optimal). \end{theorem} From this we have: \begin{corollary} For $n \geq 2^t-1$, with $t\geq 4$ and $\gamma$ sufficiently close to $1/2$, a $(2^t-1)$-dimensional 1-sphere is hashing optimal. \end{corollary} \section{Hashes from Random Codes} \label{sec:shannon} In this section we will show that hashes from random codes under minimum weight decoding\footnote{Ties arising in minimum weight decoding are broken in some unspecified manner.} perform better than projection. Let $R=k/n$ be the rate of a code. The error exponent for $k$-projection, $E^{{\cal P}_{n,k}}(p)$, is \begin{equation} \label{eq:proj_asymp} -\frac{1}{n} \lg {{\cal P}^{n,k}}(p) = -\frac{1}{n} \lg (1-p)^k = -R \lg (1-p). \end{equation} Theorem~\ref{thm:hamming_wins} shows that for any $\gamma>0$ there are codes with rate $R \approx 1$ which beat projection. For any fixed $R$, we will bound the expected error exponent for a random code ${\cal R}$ of rate $R$, and show that it beats (\ref{eq:proj_asymp}). Let $H$ be the binary entropy \begin{equation} \label{eq:entropy} H(\delta) := - \delta \lg \delta - (1-\delta) \lg (1-\delta) \, . \end{equation} Fix $\delta \in [0,1/2)$. Let $d := \lfloor \delta n \rfloor$, let ${\cal S}_d(\mathbf{x})$ denote the sphere of radius $d$ around $\mathbf{x}$, and let $V(d) := |{\cal S}_d(\mathbf{x})|$. It is elementary to show (see \cite{gallager}, Exercise 5.9): \begin{lemma}\label{lem:gv} Let ${\cal R}$ be a random code of length $n$ and rate $R$, where $n$ is sufficiently large. For $\mathbf{c}\in {\cal R}$, the probability that a given vector $\mathbf{x} \in {\cal S}_d(\mathbf{c})$ is closer to another codeword than $\mathbf{c}$ is at most $$ 2^{n (H(\delta)-1+R)}. $$ \end{lemma} Lemma~\ref{lem:gv} implies that if $H(\delta) < 1-R$ (the Gilbert-Varshamov bound), then with high probability, any given $\mathbf{x} \in {\cal S}_d(\mathbf{c})$ will be decoded to $\mathbf{c}$. For the rest of this section we will assume this bound, so that Lemma~\ref{lem:gv} applies. Let ${\hbox{\sf\slshape P}}^{\cal R}(\gamma)$ be the probability that a random point $\mathbf{x}$ and $\mathbf{x}+\mathbf{e}$ both hash to $\mathbf{c}$. This is greater than the probability that $\mathbf{x}+\mathbf{e}$ has weight exactly $d$, so $$ {\hbox{\sf\slshape P}}^{\cal R}(p) > \sum_{i=0}^d {d \choose i} {{n-d} \choose i} p^{2i} (1-p)^{n-2i}. $$ Theorem 4 of \cite{ackl} gives a bound for this: \begin{theorem}\label{thm:emax} For any $\epsi \leq 1/2$ and $\delta$ such that $H(\delta)<1-R$ and $\epsi \leq 2\delta$, \begin{eqnarray*} -{\hbox{\sf\slshape E}}^{\cal R}(\gamma) &\geq & \epsi \lg \gamma + (1-\epsi) \lg (1-\gamma) \\ & + & \delta H \left( \frac{\epsi}{2\delta} \right) + (1-\delta) H \left( \frac{\epsi}{2(1-\delta)} \right) \end{eqnarray*} for any $\epsi \leq 1/2$. The right hand side is maximized at ${\epsi_{\min}}$ satisfying \begin{eqnarray*} \frac{(2\delta-{\epsi_{\min}})(2(1-\delta)-{\epsi_{\min}})}{{\epsi_{\min}}^2} &=& \frac{(1-\gamma)^2}{\gamma^2} . \end{eqnarray*} \end{theorem} Define \begin{eqnarray*} D(\gamma,\delta,\epsi) &:=& \epsi \lg \gamma + (1-\epsi) \lg (1-\gamma) + \delta H \left( \frac{\epsi}{2\delta} \right)\\ && \hbox{} + (1-\delta) H \left( \frac{\epsi}{2(1-\delta)} \right) \\ && \hbox{} -(1-H(\delta)) \lg(1-p) \, . \end{eqnarray*} Then $E^{{\cal P}_{n,k}}(p)-{\hbox{\sf\slshape E}}^{\cal R}(\gamma) \geq D(\gamma,\delta,\epsi)$. \begin{theorem}\label{thm:Dpos} $D(\gamma,\delta,{\epsi_{\min}}) > 0$ for any $\delta, \gamma \in (0,1/2)$. \end{theorem} \begin{proof} Fix $\delta \in (0,1/2)$, and let $f(\gamma) := D(\gamma,\delta,{\epsi_{\min}})$. It is easy to check that: \begin{eqnarray*} && \lim_{\gamma \rightarrow 0^+} f(\gamma) = 0, \\ && \lim_{\gamma \rightarrow 1/2^-} f(\gamma) = 0, \\ && \lim_{\gamma \rightarrow 0^+} f'(\gamma) > 0, \\ && \lim_{\gamma \rightarrow 1/2^-} f'(\gamma) < 0, \\ \end{eqnarray*} Therefore, it suffices to show that $f'(\gamma)$ has only one zero in $(0,1/2)$. Observe that ${\epsi_{\min}}$ is chosen so that $\frac{\partial D}{\partial \epsi}(\delta,\gamma,{\epsi_{\min}}) = 0.$ Hence \begin{eqnarray*} f'(p) &=& \frac{\partial D}{\partial \gamma}(\delta,\gamma,{\epsi_{\min}})\\ &=& \frac{{\epsi_{\min}}}{\gamma \log(2)} - \frac{1-{\epsi_{\min}}}{(1-\gamma) \lg(2)} + \frac{1-H(\delta)}{(1-\gamma) \log(2)}, \end{eqnarray*} so \begin{eqnarray*} \log(2) f'(\gamma) &=& \frac{{\epsi_{\min}}}{\gamma} - \frac{1-{\epsi_{\min}}}{1-\gamma} + \frac{1-H(\delta)}{1-\gamma}. \end{eqnarray*} Therefore $f'(\gamma)=0$ when ${\epsi_{\min}} = \gamma H(\delta)$. From Theorem~\ref{thm:emax} we find $$ \gamma = \frac{ 4\delta(1-\delta) - H(\delta)^2 }{2 (H(\delta) - H(\delta)^2) }. $$ \end{proof} Thus we have $E^{{\cal P}_{n,k}}(p) > {\hbox{\sf\slshape E}}^{\cal R}(\gamma)$, and so: \begin{corollary}\label{thm:codes_win} For any $\gamma \in (0, 1/2)$, $R \in (0,1)$ and $n$ sufficiently large, the expected probability of collision for a random code of rate $R$ is higher than projection. \end{corollary} \section*{Acknowledgements.} The authors would like to thank William Bradley, David desJardins and David Moulton for stimulating discussions which helped initiate this work. Also, Tom Dorsey and Amit Khetan provided the simpler proof of Theorem~\ref{thm:Dpos} given here. The anonymous referees made a number of good suggestions that improved the paper, particularly the exposition in the introduction.

proofpile-arXiv_069-11903

\section{Introduction} \label{s:intro} Optimal control of Markov control processes (MCP) up to an exit time is a problem with a long and rich history. It has mostly been studied as the minimization of an expected undiscounted cost until the first time that the state enters a given target set, see e.g.,~\cite[Chapter~II]{ref:borkarTopicsControlledMC},~\cite[Chapter~8]{ref:hernandez-lerma2}, and the references therein. In particular, if a unit cost is incurred as long as the state is outside the target set, then the problem of minimizing the cost accumulated until the state enters the target is known variously as the \textsl{pursuit problem}~\cite{ref:eatonzadeh62}, \textsl{transient programming}~\cite{ref:whittleOptimization}, the \textsl{first passage problem}~\cite{ref:dermanMDP, ref:kushnerIntroStochControl}, the \textsl{stochastic shortest path problem}~\cite{ref:bertsekasDP2}, and \textsl{control up to an exit time}~\cite{ref:borkarConvexAnalyticApproach, ref:borkarTopicsControlledMC, ref:kestenMCP}. These articles deal with at most countable state and action spaces. The problem of optimally controlling a system until an exit time from a given set has gained significance in financial and insurance mathematics, see, e.g., \cite{ref:boda04, ref:schmidliInsurance}. Our interest in this problem stems from our attempts to develop a general theory of stochastic model-predictive control (MPC). In its bare essentials, deterministic MPC~\cite{ref:maciejowskibk} consists of two steps: (i) solving a finite-horizon optimal control problem with constraints on the state and the controlled inputs to get an optimal policy, and (ii) applying a controller derived from the policy obtained in step (i) in a rolling-horizon fashion. Theoretical foundation of stochastic MPC is still in its infancy, see~\cite{ref:PrimbsSung09, ref:bertsimas2007, ref:vanHessem2006, ref:kouvaritakissMPCIneqconstraints, ref:batinaPhDthesis} and the references therein for some related work. In view of its close relationship with applications, any satisfactory theory of stochastic MPC must necessarily take into account its practical aspects. In this context an examination of a standard linear system with constrained controlled inputs affected by independent and identically distributed (i.i.d.)\ unbounded (e.g., Gaussian) disturbance inputs shows that no control policy can ensure that with probability one the state stays confined to a bounded \emph{safe set} for all instants of time. This is because the noise is unbounded and the samples are independent of each other. Although disturbances are not likely to be unbounded in practice, assigning an a priori bound seems to demand considerable insight. In case a bounded-noise model is adopted, existing robust MPC techniques~\cite{ref:bemporad1999rmp, ref:blanchini1999sic} may be applied, in which the central idea is to synthesize a controller based on the bounds of the noise such that the target set becomes invariant with respect to the closed-loop dynamics. However, since the optimal policy is based on a worst-case analysis, it usually leads to rather conservative controllers and sometimes even to infeasibility. Moreover, complexity of the optimization problem grows rapidly (typically exponentially) with the optimization horizon. An alternative is to replace the hard constraints by probabilistic (soft) ones. The idea is to find a policy that guarantees that the state constraints are satisfied with high probability over a sufficiently long time horizon. While this approach may improve feasibility aspects of the problem, it does not address the issue of what actions should be taken once the state violates the constraints. See~\cite{ref:hokayemcdc09, ref:smpcbnddu, ref:accl08} for recent results in this direction. In view of the above considerations, developing recovery strategies appears to be a necessary step. Such a strategy is to be activated once the state violates the constraints and to be deactivated whenever the system returns to the safe set. In general, a recovery strategy must drive the system quickly to the safe set while simultaneously meeting other performance objectives. In the context of MPC, two merits are immediate: (a) once the constraints are transgressed, appropriate actions can be taken to bring the state back to the safe set quickly and optimally, and (b) if the original problem is posed with hard constraints on the state, in view of (a) they may be relaxed to probabilistic ones to improve feasibility. In this article we address the problem of synthesizing optimal recovery strategies. We formulate the problem as the minimization of an expected discounted cost until the state enters the safe set. An almost customary assumption in the literature (see, e.g.,~\cite{ref:hindererAbsorbingSet} and the references therein,) concerned with stochastic optimal control up to an exit time is that the target set is absorbing. That is, there exists a control policy that makes the target set invariant with respect to the closed-loop stochastic dynamics. This is rather restrictive for MPC problems---it is invalid, for instance, in the very simple case of a linear controlled system with i.i.d.\ Gaussian noise inputs. We do not make this assumption, for, as mentioned above, our primary motivation for solving this problem is precisely to deal with the case that the target set is not absorbing. As a result of this, it turns out that the dynamic programming equations involve integration over subsets of the state-space and therefore are difficult to solve. At present there is no established method to solve such equations in uncountable state-spaces. However, in finite state-space cases tractable approximate dynamic programming methods~\cite{ref:bertsekasNDP, ref:powellADP} may be employed to arrive at suboptimal but efficient policies. This article unfolds as follows. In~\secref{s:prelims} we define the general setting of the problem, namely, Markov control processes on Polish spaces, their transition kernels and the main types of control strategies. In~\secref{s:EDC} we establish our main Theorem~\ref{t:EDC} under standard mild hypotheses. This result guarantees the existence of a deterministic stationary policy that leads to the minimal cost and also provides a Bellman equation that the value function must satisfy. A contraction mapping approach to the problem is pursued in~\secref{s:contr} under the (standard) assumption that the cost-per-stage function satisfies certain growth-rate conditions. The main result (Proposition~\ref{p:Tfp}) of this section asserts both the existence and uniqueness of the optimal value function. Asymptotic discount-optimality of the value-iteration policy is investigated in~\secref{s:ado} under two different sets of hypotheses; in particular, the results of this section show that rolling-horizon strategy approaches optimality as the length of the horizon window increases to infinity. A rolling-horizon strategy corresponding to our optimal control problem is developed in~\secref{s:rh}; in Theorem~\ref{t:rh} we establish quantitative bounds on the degree of sub-optimality of the rolling-horizon strategy with respect to the optimal policy. We conclude in~\secref{s:concl} with a discussion of future work. The state and control/action sets are assumed to be Borel subsets of Polish spaces. \section{Preliminaries} \label{s:prelims} We employ the following standard notations. Let $\ensuremath{\mathbb{N}}$ denote the natural numbers $\{1, 2, \ldots\}$, and $\ensuremath{\mathbb{N}_0}$ denote the nonnegative integers $\{0\}\cup\ensuremath{\mathbb{N}}$. Let $\indic{A}(\cdot)$ be the standard indicator function of a set $A$, i.e., $\indic{A}(\xi) = 1$ if $\xi\in A$ and $0$ otherwise. For two real numbers $a$ and $b$, let $a\ensuremath{\wedge} b \coloneqq \min\{a, b\}$. Given a nonempty Borel set $X$ (i.e., a Borel subset of a Polish space), its Borel $\sigma$-algebra is denoted by $\Borelsigalg{X}$. By convention ``measurable'' means ``Borel-measurable'' in the sequel. If $X$ and $Y$ are nonempty Borel spaces, a \emph{stochastic kernel} on $X$ given $Y$ is a mapping $Q(\cdot|\cdot)$ such that $Q(\cdot|y)$ is a probability measure on $X$ for each fixed $y\in Y$, and $Q(B|\cdot)$ is a measurable function on $Y$ for each fixed $B\in\Borelsigalg X$. We let $\mc P(X|Y)$ be the family of all stochastic kernels on $X$ given $Y$ We briefly recall some standard definitions. \begin{defn} \label{d:mcm} A \emph{Markov control model} is a five-tuple \begin{equation} \label{e:mmodel} \bigl(X, A, \{A(x)\mid x\in X\}, Q, c\bigr) \end{equation} consisting of a nonempty Borel space $X$ called the \emph{state space}, a nonempty Borel space $A$ called the \emph{control} or \emph{action set}, a family $\{A(x)\mid x\in X\}$ of nonempty measurable subsets $A(x)$ of $A$, where $A(x)$ denotes the set of \emph{feasible controls} or \emph{actions} when the system is in state $x\in X$, and with the property that the set $\mathbb K \coloneqq \bigl\{(x, a)\big|x\in X, a\in A(x)\bigr\}$ of feasible state-action pairs is a measurable subset of $X\times A$, a stochastic kernel $Q$ on $X$ given $\mathbb K$ called the \emph{transition law}, and a measurable function $c:\mathbb K\ensuremath{\longrightarrow} \ensuremath{\mathbb{R}}$ called the \emph{cost-per-stage function}.\hspace{\stretch{1}}{$\Diamond$} \end{defn} \begin{assumption} \label{a:basic} The set $\mathbb K$ of feasible state-action pairs contains the graph of a measurable function from $X$ to $A$.\AssumptionEn \end{assumption} We let $\Pi$, $\Pi_{RM}$, $\Pi_{DM}$ and $\Pi_{DS}$ denote the set of all randomized and history-dependent admissible policies, randomized Markov, deterministic Markov and deterministic stationary policies, respectively. For further details and notations on policies see, e.g.,~\cite{ref:hernandez-lerma1}. Consider the Markov control model~\eqref{e:mmodel}, and for each $i=0, 1, \ldots,$ define the space $H_i$ of \emph{admissible histories} up to time $i$ as $H_0 \coloneqq X$, and $H_i \coloneqq \mathbb K^i\times X = \mathbb K\times H_{i-1}$ for $i\in \ensuremath{\mathbb{N}}$. A generic element $h_i$ of $H_i$, called an admissible $i$-history is a vector of the form $h_i = (x_0, a_0, \ldots, x_{i-1}, a_{i-1}, x_i)$, with $(x_j, a_j)\in\mathbb K$ for $j=0, \ldots, i-1$ and $x_i\in X$. Hereafter we let the $\sigma$-algebra generated by the history $h_i$ be denoted by $\ensuremath{\mathfrak{F}}_i$, $i\in\ensuremath{\mathbb{N}_0}$. Let $(\Omega, \ensuremath{\mathfrak{F}})$ be the measurable space consisting of the (canonical) sample space $\Omega \coloneqq \overline H_\infty = (X\times A)^\infty$, and $\ensuremath{\mathfrak{F}}$ is the corresponding product $\sigma$-algebra. Let $\pi = (\pi_i)_{i\in\ensuremath{\mathbb{N}_0}}$ be an arbitrary control policy and $\nu$ an arbitrary probability measure on $X$, referred to as the initial distribution. By a theorem of Ionescu-Tulcea~\cite[Chapter 3, \S4, Theorem~5]{ref:raoProbTheo}, there exists a unique probability measure $\mathsf P_\nu^\pi$ on $(\Omega, \ensuremath{\mathfrak{F}})$ supported on $H^\infty$, and such that for all $B\in\Borelsigalg X$, $C\in\Borelsigalg A$, and $h_i\in H_i$, $i\in\ensuremath{\mathbb{N}_0}$, $\mathsf P_\nu^\pi\bigl(\ensuremath{x_0}\in B\bigr) = \nu(B)$ and \begin{subequations} \begin{align} \label{e:actiontrans} \mathsf P_\nu^\pi\bigl(a_i\in C\,\big|\, h_i\bigr) & = \pi_i\bigl(C\,\big|\, h_i\bigr)\\ \label{e:statetrans} \mathsf P_\nu^\pi\bigl(x_{i+1}\in B\,\big|\, h_i, a_i\bigr) & = Q\bigl(B\,\big|\, x_i, a_i\bigr). \end{align} \end{subequations} The stochastic process $\bigl(\Omega, \ensuremath{\mathfrak{F}}, \mathsf P_\nu^\pi, (x_i)_{i\in\ensuremath{\mathbb{N}_0}}\bigr)$ is called a discrete-time \emph{Markov control process}. Let $\Phi$ denote the set of stochastic kernels $\ensuremath{\varphi}$ in $\mathcal P(A| X)$ such that $\ensuremath{\varphi}(A(x)| x) = 1$ for all $x\in X$, and let $\mathbb F$ denote the set of all measurable functions $f:X\ensuremath{\longrightarrow} A$ satisfying $f(x)\in A(x)$ for all $x\in X$. The functions in $\mathbb F$ are called \emph{selectors} of the set-valued mapping $X\ni x\mapsto A(x)\subset A$. The transition kernel $Q$ in~\eqref{e:statetrans} under a policy $\pi \coloneqq (\ensuremath{\varphi}_i)_{i\in\ensuremath{\mathbb{N}_0}}\in\Pi_{RM}$ is given by $\bigl(Q(\cdot|\cdot, \ensuremath{\varphi}_i)\bigr)_{i\in\ensuremath{\mathbb{N}_0}}$, defined as $\Borelsigalg{X}\times X\ni (B, x)\mapsto Q(B|x, \ensuremath{\varphi}_i(x)) \coloneqq \int_{A(x)}\ensuremath{\varphi}_i(\mrm da|x) Q(B|x, a)$. Occasionally we suppress the dependence of $\ensuremath{\varphi}_i$ on $x$ and write $Q(B|x, \ensuremath{\varphi}_i)$ in place of $Q(B|x, \ensuremath{\varphi}_i(x))$. The cost-per-stage function at the $j$-th stage under a policy $(\ensuremath{\varphi}_i)_{i\in\ensuremath{\mathbb{N}_0}}$ is written as $c(x_j, \ensuremath{\varphi}_j) \coloneqq \int_{A(x_j)} \ensuremath{\varphi}_j(\mrm da|x_j)c(x_j, a)$. We simply write $\ensuremath{\varphi}^\infty$ and $f^\infty$, respectively, for policies $(\ensuremath{\varphi}, \ensuremath{\varphi}, \ldots)\in\Pi_{RS}$ and $(f, f, \ldots)\in \Pi_{DS}$. Since we shall be exclusively concerned with Markov policies and its subclasses, in the sequel we use the notation $\Pi$ for the class of all randomized Markov strategies. \section{Expected Discounted Cost up to the first Exit Time} \label{s:EDC} Let $K\subset X$ be a measurable set, $x_0 = x\in X$ and let $\tau \coloneqq \inf\bigl\{i\in\ensuremath{\mathbb{N}_0}\big| x_i\in K\bigr\}$.\footnote{As usual the infimum over an empty set is taken to be $+\infty$.} We note that $\tau$ is an $(\ensuremath{\mathfrak{F}}_i)_{i\in\ensuremath{\mathbb{N}_0}}$-stopping time. Let us define \[ V(\pi, x) \coloneqq \mathsf E_x^\pi\!\left[\sum_{i=0}^{\tau-1} \alpha^i c(x_i, a_i)\right], \qquad \alpha\in\:]0, 1[, \] as the \emph{$\alpha$-discounted expected cost} under policy $\pi\in\Pi$ corresponding to the Markov control process $\bigl(\Omega, \ensuremath{\mathfrak{F}}, \mathsf P_\nu^\pi, (x_i)_{i\in\ensuremath{\mathbb{N}_0}}\bigr)$.\footnote{We employ the standard convention that a summation from a higher to a lower index is defined to be $0$.} Our objective is to minimize $V(\pi,x)$ over a class of control policies $\Pi$, i.e., find the $\alpha$-discount value function \begin{align} \label{e:problem} V^\star(x) \coloneqq \inf_{\pi\in\Pi} V(\pi, x) = \inf_{\pi\in\Pi}\mathsf E_x^\pi\!\left[\sum_{i=0}^{\tau-1} \alpha^i c(x_i, a_i)\right], \qquad \alpha\in\:]0, 1[. \end{align} A policy that attains the infimum above is said to be \emph{$\alpha$-discount optimal}. \begin{remark} As mentioned in the introduction, the optimization problem~\eqref{e:problem} with $\alpha = 1$ and the cost-per-stage function $c(x, a) = \indic{X\ensuremath{\!\smallsetminus} K}(x)$ is known as the stochastic shortest path problem. The objective of this problem is to drive the state to a desired set ($K$ in our case) as soon as possible, and the expected cost $V_{\text{ssp}}(\pi, x)$ for a policy $\pi$ corresponding to the above cost-per-stage function is readily seen to be $\mathsf E^\pi_x\bigl[\tau\bigr]$. In this light we observe that the minimization problem in~\eqref{e:problem} with the cost-per-stage function $c(x, a) = \indic{X\ensuremath{\!\smallsetminus} K}(x)$ can be viewed as a discounted stochastic shortest path problem. It follows immediately that the corresponding expected cost $V_{\text{dssp}}(\pi, x)$ is $\bigl(1-\mathsf E^\pi_x\bigl[\alpha^\tau\bigr]\bigr)/(1-\alpha)$. Note that the minimization of $V_{\text{dssp}}(\pi, x)$ over a class of policies is always well-defined for $\alpha < 1$. Moreover, because of the monotonic behavior of the map $]0, 1[\;\ni\alpha\mapsto \bigl(1-\mathsf E^\pi_x\bigl[\alpha^\tau\bigr]\bigr)/(1-\alpha)$, one may hope to get a good approximation of the original stochastic shortest path problem. However, pathological examples can be constructed to show that a solution to the stochastic shortest path problem may not exist, whereas minimization of $V_{\text{dssp}}(\pi, x)$ is always well defined, although in either case the state may never reach the desired set $K$ almost surely-$\ensuremath{\mathsf{P}}^\pi_x$.\hspace{\stretch{1}}{$\vartriangleleft$} \end{remark} \begin{remark} \label{r:diffc} Given a cost-per-stage function $c$ on $\mathbb K$, one can redefine it to be $c'(x, a) \coloneqq c(x, a)\indic{X\ensuremath{\!\smallsetminus} K}(x)$ to turn the problem~\eqref{e:problem} into the minimization of $\mathsf E_x^\pi\!\left[\sum_{i=0}^\tau \alpha^i c'(x_i, a_i)\right]$ for $\alpha\in\:]0, 1[$. This cost functional can be equivalently written as an infinite horizon cost functional, as in $\mathsf E_x^\pi\!\left[\sum_{i=0}^\infty \alpha^i c'(x_i, a_i)\indic{\{i \le \tau\}}\right]$, or as in $\mathsf E_x^\pi\!\left[\sum_{i=0}^\infty \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\right]$. However, the absence of a policy that guarantees that $(x_i)_{i\in\ensuremath{\mathbb{N}_0}}$ stays inside $K$ for all time after $\tau$ necessarily means that the problem~\eqref{e:problem} corresponding to the Markov control model in Definition~\ref{d:mcm} is not equivalent to the minimization of the infinite horizon cost functional $\mathsf E_x^\pi\!\left[\sum_{i=0}^\infty \alpha^i c'(x_i, a_i)\right]$.\hspace{\stretch{1}}{$\vartriangleleft$} \end{remark} \begin{prgr} \label{pgr:policies} \emph{A word about admissible policies.} It is clear at once that the class of admissible policies for the problem~\eqref{e:problem} is different from the classes considered in~\secref{s:prelims}. Indeed, since the process is killed at the stopping time $\tau$, it follows that the class of admissible policies should also be truncated at the stage $\tau-1$. For a given stage $t\in\ensuremath{\mathbb{N}_0}$ we define the $t$-th policy element $\pi_t$ only on the set $\{t < \tau\}$. Note that with this definition $\pi_t$ becomes a $\ensuremath{\mathfrak{F}}_{t\ensuremath{\wedge}\tau}$-measurable randomized control (in general). It is also immediate from the definition of $\tau$ that if the initial condition $x$ is inside $K$, then the set of admissible policies is empty; indeed, in this case $\tau = 0$, and there is no control needed. In other words, the domain of $\pi_t$ is contained in the ``spatial'' region $\bigl\{(x, a)\in\mathbb K\,\big|\,x\in X\ensuremath{\!\smallsetminus} K, a\in A(x)\bigr\}$; since $\pi_t$ is not defined on $K$, this is equivalent to $\pi_t$ being well-defined on $\{t < \tau\}$. \end{prgr} \begin{prgr} \label{pgr:convention} \emph{Some re-definitions.} To simplify the formulas from now on we let the cost-per-stage function to be defined on $X\ensuremath{\!\smallsetminus} K$. With this convention in place our problem~\eqref{e:problem} can be posed as the minimization of $\ensuremath{\mathsf{E}}^\pi_x\bigl[\sum_{i=0}^{\tau-1} \alpha^i c(x_{i}, a_{i})\bigr]$ over admissible policies. Also, henceforth we redefine the set $\mathbb K$ of state-action pairs to be $\mathbb K \coloneqq \bigl\{(x, a)\in X\times A\,\big|\,x\in X\ensuremath{\!\smallsetminus} K, a\in A(x)\bigr\}$, and we note that this new set is a measurable subset of the original set of state-action pairs. Also, we let $\mathbb F$ be the set of selectors of the set-valued mapping $X\ensuremath{\!\smallsetminus} K\ni x\mapsto A(x)\subset A$. \end{prgr} Recall that a function $g:\mathbb K\ensuremath{\longrightarrow} \ensuremath{\mathbb{R}}$ is said to be \emph{inf-compact on $\mathbb K$} if for every $x\in X$ and $r\in \ensuremath{\mathbb{R}}$ the set $\bigl\{a\in A(x)\big| g(x, a) \le r\bigr\}$ is compact. A transition kernel $Q$ on a measurable space $X$ given another measurable space $Y$ is said to be \emph{strongly Feller} (or \emph{strongly continuous}) if the mapping $y\mapsto \int_X g(x) Q(\mrm dx| y)$ is continuous and bounded for every measurable and bounded function $g:X\ensuremath{\longrightarrow}\ensuremath{\mathbb{R}}$. A function $g:\mathbb K\ensuremath{\longrightarrow}\ensuremath{\mathbb{R}}$ is \emph{lower semicontinuous} (l.s.c.) if for every sequence $(x_j, a_j)_{j\in\ensuremath{\mathbb{N}}}\subset\mathbb K$ converging to $(x, a)\in\mathbb K$, we have $\liminf_{j\ensuremath{\rightarrow}\infty} g(x_j, a_j) \ge g(x, a)$; or, equivalently, if for every $r\in\ensuremath{\mathbb{R}}$, the set $\bigl\{(x, a)\in\mathbb K\big| g(x, a) \le r\bigr\}$ is closed in $\mathbb K$. \begin{assumption} \label{a:key} In addition to Assumption~\ref{a:basic}, we stipulate that \begin{enumerate}[align=right, leftmargin=*, widest=iii, label=(\roman*)] \item the set $A(x)$ is compact for every $x\in X$, \item the cost-per-stage $c$ is lower semicontinuous, nonnegative, and inf-compact on $\mathbb K$, and \item the transition kernel $Q$ is strongly Feller.\hspace{\stretch{1}}{$\diamondsuit$} \end{enumerate} \end{assumption} The following is our main result on expected discounted cost up to the first time $\tau$ to hit $K$; a proof is presented later in this section. \begin{theorem} \label{t:EDC} Suppose that Assumption {\rm \ref{a:key}} holds. Then \begin{enumerate}[label=\emph{(\roman*)}, align=right, leftmargin=*, widest=iii] \item The $\alpha$-discount value function $V^\star$ is the (positive) minimal measurable solution to the $\alpha$-discounted cost optimality equation ($\alpha$-DCOE) \begin{equation} \label{e:alphadcoe} \xi(x) = \min_{A(x)}\left[c(x, a) + \alpha\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy| x, a)\:\xi(y)\right]\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K. \end{equation} \item There exists a selector $f_\star\in\mathbb F$ such that $f_\star(x)\in A(x)$, $x\in X\ensuremath{\!\smallsetminus} K$, attains the minimum in~\eqref{e:alphadcoe}, i.e., \begin{equation} \label{e:alphado} V^\star(x) = c(x, f_\star) + \alpha\int_{X\ensuremath{\!\smallsetminus} K}Q(\mrm dy| x, f_\star)\:V^\star(y)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K, \end{equation} and the deterministic stationary policy $f_\star^\infty$ is $\alpha$-discount optimal; conversely, if $f_\star^\infty\in\Pi_{DS}$ is $\alpha$-discount optimal, then it satisfies~\eqref{e:alphado}. \end{enumerate} \end{theorem} We observe that Theorem~\ref{t:EDC} does not assert that the optimal value function $V^\star$ is unique in any sense. In~\secref{s:contr} we prove a result (Proposition~\ref{p:Tfp}) under additional hypotheses that guarantees uniqueness of $V^\star$. Since we do not assume that the cost-per-stage function $c$ is bounded, a useful approach is to consider the $\alpha$-\emph{value iteration} ($\alpha$-VI) \emph{functions} defined by \begin{equation} \label{e:VI} \begin{cases} v_0(x) = 0,\\ v_n(x) = \displaystyle{\min_{A(x)}\left[c(x, a) + \alpha \int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a)\: v_{n-1}(y)\right]}, \end{cases} n\in\ensuremath{\mathbb{N}},\;\; x\in X\ensuremath{\!\smallsetminus} K. \end{equation} Of course we have to demonstrate that $V^\star(x) = \lim_{n\ensuremath{\rightarrow}\infty} v_n(x)$ for all $x\in X$. The functions $v_n$, $n\in\ensuremath{\mathbb{N}}$, may be identified with the optimal cost function for the minimization of the process stopped at the $n\ensuremath{\wedge}(\tau-1)$-th step, i.e., \[ v_n(x) = \inf_{\pi\in\Pi} \mathsf E_x^\pi\!\left[\sum_{i=0}^{(n-1)\ensuremath{\wedge}(\tau-1)} \alpha^i c(x_i, a_i)\right]. \] To get an intuitive idea, fix a deterministic Markov policy $\pi = (\pi_i)_{i\in\ensuremath{\mathbb{N}_0}}$, and take the first iterate $v_1$. From~\eqref{e:VI} it is immediately clear that $v_1(x) = \min_{a\in A(x)} c(x, a)$ if $x\not\in K$, and not defined otherwise. For the second iterate, we have \begin{align*} v_2(x) & = \inf_{\pi\in\Pi} \mathsf E_x^\pi\!\left[\sum_{i=0}^{1\ensuremath{\wedge} (\tau-1)} \alpha^i c(x_i, a_i)\right]\\ & = \inf_{\pi\in\Pi}\left(c(x, \pi_0(x)) + \alpha\!\int_X \! Q(\mrm d\xi_1|x, \pi_1(x))\indic{X\ensuremath{\!\smallsetminus} K}(\xi_1) c(\xi_1, \pi_1)\right). \end{align*} Note that only those sample paths that do not enter $K$ at the first step contribute to the cost at the second stage. This property is ensured by the indicator function that appears on the right-hand side of the last equality above. \begin{example} Let $(x_i)_{i\in\ensuremath{\mathbb{N}_0}}$ be a Markov chain with state-space $X = \{1, 2, \ldots, m\}$ and transition probability matrix $Q = [q_{ij}(a)]_{m\times m}$, where the argument of $q_{ij}$ depicts the dependence on the action $a\in A$ with $A$ being a compact subset of $\ensuremath{\mathbb{R}}$. Let $K = \{1, 2, \ldots, m'\}$ for $m' < m$, fix $\alpha\in\;]0, 1[$ and let $c(x, a) \coloneqq \indic{X\ensuremath{\!\smallsetminus} K}(x)$. Suppose further that $\inf_{a}q_{ij}(a) > 0$ for all $i, j\in X$; this means, in particular, that the target set $K$ cannot be absorbing for any deterministic stationary policy. Our objective is to find an optimal policy corresponding to the the minimal cost~\eqref{e:alphado}. The optimal value function $V^\star$ is $0$ on $K$ and for every $i\in \{m'+1, \ldots, m\}$ we have $V^\star(i) = \min_{a\in A(i)}\bigl[\indic{X\ensuremath{\!\smallsetminus} K}(i) + \alpha\int_{X\ensuremath{\!\smallsetminus} K}Q(\mrm dy|i, a) V^\star(y)\bigr] = 1 + \alpha\min_{a\in A(i)}\sum_{j=m'+1}^m q_{ij}(a) V^\star(j)$. The most elementary case is that of $m' = m - 1$; then $V^\star(m) = 1 + \alpha\min_{a\in A(m)}q_{mm}(a)V^\star(m)$, and given a sufficiently regular function $q_{mm}(\cdot)$ this can be solved at once to get $V^\star(m)$, which characterizes the function (vector) $V^\star$ completely. The optimal policy in this case is $f(m) \in \argmin_{a\in A(m)} q_{mm}(a)V^\star(m)$; if the function $q_{mm}(\cdot)$ is convex, then the minimum is attained on $A$ and thus leads to a unique optimal policy.\hspace{\stretch{1}}{$\triangle$} \end{example} \subsection*{Proof of Theorem~\ref{t:EDC}} Recall from paragraph~\ref{pgr:convention} that $c$ is defined on $X\ensuremath{\!\smallsetminus} K$, $\mathbb K = \bigl\{(x, a)\in X\times A\,\big|\, x\in X\ensuremath{\!\smallsetminus} K, a\in A(x)\bigr\}$ and $\mathbb F$ is the set of selectors of the set-valued map $X\ensuremath{\!\smallsetminus} K\ni x\mapsto A(x)\subset A$. We begin with a sequence of Lemmas. \begin{lemma}[{\cite[Lemma~4.2.4]{ref:hernandez-lerma1}}] \label{l:keyconvergence} Let the functions $u:\mathbb K\ensuremath{\longrightarrow}\ensuremath{\mathbb{R}}$ and $u_i:\mathbb K\ensuremath{\longrightarrow}\ensuremath{\mathbb{R}}$, $i\in\ensuremath{\mathbb{N}}$, be l.s.c., inf-compact and bounded below. If $u_i\uparrow u$, then \[ \lim_{i\ensuremath{\rightarrow}\infty}\min_{A(x)} u_i(x, a) = \min_{A(x)} u(x, a)\qquad \ensuremath{\forall\,} x\in X. \] \end{lemma} \begin{lemma}[Adapted from \cite{ref:riederselectors}] \label{l:basicselector} Suppose that \begin{itemize}[label=\textbullet, align=right, leftmargin=*] \item $A(x)$ is compact for each $x\in X\ensuremath{\!\smallsetminus} K$ and $\mathbb K$ is a measurable subset of $(X\ensuremath{\!\smallsetminus} K)\times A$, and \item $v:\mathbb K\ensuremath{\longrightarrow}\ensuremath{\R_{\ge 0}}$ is a measurable inf-compact function, $v(x, \cdot)$ is l.s.c.\ on $A(x)$ for each $x\in X$. \end{itemize} Then there exists a selector $f_\star\in\mathbb F$ such that \[ v(x, f_\star(x)) = v^\star(x) \coloneqq \min_{A(x)}v(x, a)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K, \] and $v^\star$ is a measurable function. \end{lemma} \begin{defn} Let $\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ denote the convex cone of nonnegative extended real-valued measurable functions on $X\ensuremath{\!\smallsetminus} K$, and for every $u\in \Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ let us define the map $T u$ by \begin{equation} \label{e:Tdef} X\ensuremath{\!\smallsetminus} K\ni x\mapsto T u(x) \coloneqq \inf_{A(x)}\left[c(x, a) + \alpha\int_{X\ensuremath{\!\smallsetminus} K}Q(\mrm dy|x, a)\:u(y)\right]. \end{equation} The map $T$ is the \emph{dynamic programming operator} corresponding to our problem~\eqref{e:problem}.\hspace{\stretch{1}}{$\Diamond$} \end{defn} Having defined the dynamic programming operator $T$ above, it is important to distinguish conditions under which the function $T u$ is measurable for $u\in\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$. We have the following lemma. \begin{lemma} \label{l:selector} Under Assumption {\rm \ref{a:key}}, the mapping $T$ in~\eqref{e:Tdef} takes $\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ into itself. Moreover, there exists a selector $f\in \mathbb F$ such that $T u$ defined in~\eqref{e:Tdef} satisfies \begin{equation} \label{e:Tviaselector} T u(x) = c(x, f) + \alpha\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, f)\:u(y)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K. \end{equation} \end{lemma} \begin{proof} Fix $u\in\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$. The strong-Feller property of $Q$ on $\mathbb K$ and lower-semicontinuity of the cost-per-stage function $c$ defined on $K$ show that the map \[ \mathbb K\ni (x, a)\mapsto T'u(x, a) \coloneqq c(x, a) + \alpha\int_{X}Q(\mrm dy|x, a)\;\indic{X\ensuremath{\!\smallsetminus} K}(y) u(y) \] is lower-semicontinuous. From nonnegativity of $u$ it follows that for every $x\in X\ensuremath{\!\smallsetminus} K$ and $r \in \ensuremath{\mathbb{R}}$, \begin{equation} \label{e:selector1} K' \coloneqq \bigl\{a\in A(x) \big| T'u(x, a) \le r\bigr\}\subset\bigl\{a\in A(x)\big| c(x, a) \le r\bigr\}, \end{equation} and the set $\bigl\{a\in A(x)\big| c(x, a) \le r\bigr\}$ is compact by inf-compactness of $c$. Since by definition $T u(x) = \inf_{A(x)} T'u(x, a)$, by Lemma \ref{l:basicselector} it would follow that a selector $f$ exists such that $T u(x) = T'u(x, f(x))\;\;\ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K$ once we verify the hypotheses of this Lemma. For this we only have to verify that $T'u$ is l.s.c.\ (which implies it is measurable) and inf-compact on $\mathbb K$. We have seen above that $T'u$ is a l.s.c.\ function on $\mathbb K$. Therefore, for each $x\in X\ensuremath{\!\smallsetminus} K$ the map $T'u(x, \cdot)$ is also l.s.c.\ on $A(x)$. Thus, by definition of lower semicontinuity, the set $K'$ in~\eqref{e:selector1} is closed for every $x\in X\ensuremath{\!\smallsetminus} K$ and $r\in\ensuremath{\mathbb{R}}$. Since a closed subset of a compact set is compact, it follows that $K'$ is compact, which in turn shows inf-compactness of $T'u$ on $\mathbb K$ and proves the assertion. \end{proof} The following lemma shows how functions $u\in\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ satisfying $u\ge T u$ relate to the optimal value function. \begin{lemma} \label{l:Tineq} Suppose that Assumption {\rm \ref{a:key}} holds. If $u\in \Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ is such that $u\ge T u$, then $u\ge V^\star$. \end{lemma} \begin{proof} Suppose $u\in\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ satisfies $u \ge T u$, and let $f$ be a selector (whose existence is guaranteed by Lemma \ref{l:selector}) that attains the infimum in~\eqref{e:Tdef}. Fix $x\in X\ensuremath{\!\smallsetminus} K$. We have \[ u(x) \ge T u(x) = c(x, f) + \alpha \int_X Q(\mrm d\xi_1|x, f)\:\indic{X\ensuremath{\!\smallsetminus} K}(\xi_1)u(\xi_1). \] The operator $T$ in~\eqref{e:Tdef} is monotone, for if $u, u'\in \Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ are two functions with $u \le u'$, then clearly $T u \le T u'$ due to nonnegativity of $c$. Therefore, iterating the above inequality for a second time we obtain \begin{equation*} \begin{aligned} u(x) & \ge c(x, f) + \alpha \int_X Q(\mrm d\xi_1|x, f)\:\indic{X\ensuremath{\!\smallsetminus} K}(\xi_1) c(\xi_1, f)\\ & \qquad + \alpha^2\int_X Q(\mrm d\xi_1|x, f)\:\indic{X\ensuremath{\!\smallsetminus} K}(\xi_1)\int_X Q(\xi_2|\xi_1,f)\: \indic{X\ensuremath{\!\smallsetminus} K}(\xi_2)u(\xi_2). \end{aligned} \end{equation*} After $n$ such iterations we arrive at \[ u(x) \ge \mathsf E^{f^\infty}_x\!\left[\sum_{i=0}^{(n-1)\ensuremath{\wedge}(\tau-1)} \alpha^i c(x_i, f)\right] + \mathsf E^{f^\infty}_x\bigl[\alpha^{n} u(x_{n})\indic{\{n < \tau\}}\bigr]. \] Since $u\ge 0$, letting $n\ensuremath{\rightarrow}\infty$ we get \[ u(x) \ge V(f, x) \ge V^\star(x). \] Since $x\in X\ensuremath{\!\smallsetminus} K$ is arbitrary, the assertion follows. \end{proof} The next lemma deals with convergence of the value iterations to the optimal value function. \begin{lemma} \label{l:VIconv} Suppose that Assumption {\rm \ref{a:key}} holds. Then $v_n\uparrow V^\star$ on $X\ensuremath{\!\smallsetminus} K$, and the function $V^\star$ satisfies the $\alpha$-DCOE \eqref{e:alphadcoe}. \end{lemma} \begin{proof} Note that since $v_n(x) = \inf_{\pi\in\Pi} \mathsf E_x^\pi\left[\sum_{i=0}^{(n-1)\ensuremath{\wedge}(\tau-1)} \alpha^i c(x_i, a_i)\right]$ for $x\in X\ensuremath{\!\smallsetminus} K$, it follows that \[ v_n(x) \le \mathsf E_x^\pi\!\left[\sum_{i=0}^{(n-1)\ensuremath{\wedge}(\tau-1)} \alpha^i c(x_i, a_i)\right] \le \mathsf E_x^\pi\!\left[\sum_{i=0}^{\tau-1} \alpha^i c(x_i, a_i)\right], \] and therefore, taking the infimum over all policies $\pi\in\Pi$ on the right hand side, we get \begin{equation} \label{e:vnbound} v_n(x) \le V^\star(x)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K. \end{equation} Since the cost-per-stage function is nonnegative, $T$ is a monotone operator. Therefore, since $v_0 \coloneqq 0$ and $v_n = T v_{n-1}$ for $n\in\ensuremath{\mathbb{N}}$, it follows that the $\alpha$-VI functions form a nondecreasing sequence in $\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$, which implies that $v_n\uparrow v^\star$ for some function $v^\star \in \Lp 0(X\ensuremath{\!\smallsetminus} K)^+$. For $n\in\ensuremath{\mathbb{N}}$ we define \begin{align*} \mathbb K\ni (x, a)& \mapsto T'v_n(x, a) \coloneqq c(x, a) + \alpha \int_{X} Q(\mrm dy|x, a)\indic{X\ensuremath{\!\smallsetminus} K}(y)v_n(y)\in\ensuremath{\mathbb{R}},\\ \mathbb K\ni (x, a)& \mapsto T'v^\star(x, a) \coloneqq c(x, a) + \alpha \int_{X} Q(\mrm dy|x, a)\indic{X\ensuremath{\!\smallsetminus} K}(y) v^\star(y)\in\ensuremath{\mathbb{R}}. \end{align*} The monotone convergence theorem guarantees that $T'v_n\uparrow T'v^\star$ pointwise on $\mathbb K$. As in the proof of Lemma~\ref{l:selector} one can establish inf-compactness and lower semicontinuity of $T'v_n$, and $T'v^\star$ on $\mathbb K$. From Lemma~\ref{l:keyconvergence} it now follows that for every $x\in X\ensuremath{\!\smallsetminus} K$ we have \begin{align*} v^\star(x) & = \lim_{n\ensuremath{\rightarrow}\infty}v_n(x) = \lim_{n\ensuremath{\rightarrow}\infty} T v_{n-1}(x)\\ & = \lim_{n\ensuremath{\rightarrow}\infty} \min_{A(x)}T'v_{n-1}(x, a) = \min_{A(x)} T'v^\star(x, a)\\ & = T v^\star(x). \end{align*} This shows that $v^\star$ satisfies the $\alpha$-DCOE, $v^\star = T v^\star$. It remains to show that $v^\star = V^\star$. But by Lemma \ref{l:Tineq}, $v^\star = T v^\star$ implies that $v^\star \ge V^\star$ and the reverse inequality follows from~\eqref{e:vnbound} by taking limits as $v^\star = \lim_{n\ensuremath{\rightarrow}\infty} v_n \le V^\star$. \end{proof} \begin{lemma} \label{l:adcoestat} For every deterministic stationary policy $f^\infty$ we have \begin{equation} \label{e:adcoepolicy} V(f^\infty, x) = c(x, f) + \alpha\int_X Q(\mrm dy|x, f)\:\indic{X\ensuremath{\!\smallsetminus} K}(y)V(f^\infty, y)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K. \end{equation} \end{lemma} \begin{proof} Fix a deterministic stationary policy $f^\infty$ and $x\in X\ensuremath{\!\smallsetminus} K$. The $\alpha$-discounted cost $V(f^\infty, x)$ corresponding to this policy satisfies, in view of the definition of $\tau$ and the fact that $x\in X\ensuremath{\!\smallsetminus} K$, \begin{align} V(f^\infty, x) & \coloneqq \mathsf E_x^{f^\infty}\!\left[\sum_{i=0}^{\tau-1} \alpha^i c(x_i, f)\right] = \mathsf E_x^{f^\infty}\!\left[ c(x, f) + \sum_{i=1}^{\tau-1} \alpha^i c(x_i, f)\right]\nonumber\\ & = c(x, f) + \alpha\mathsf E_x^{f^\infty}\!\left[\sum_{i=1}^{\tau-1} \alpha^{i-1} c(x_i, f)\right].\label{e:spolicy1} \end{align} But then by the Markov property, \begin{align*} \mathsf E_x^{f^\infty}\!\left[\sum_{i=1}^{\tau-1} \alpha^{i-1} c(x_i, f)\right] & = \mathsf E^{f^\infty}\!\left[\mathsf E^{f^\infty}\!\left[\sum_{i=1}^{\tau-1} \alpha^{i-1} c(x_i, f)\left.\left.\vphantom{\sum_i^\tau}\right|x_{1\ensuremath{\wedge}(\tau-1)}\right]\right|x_0 = x\right]\\ & = \int_X \indic{X\ensuremath{\!\smallsetminus} K}(y) Q(\mrm dy|x, f)\; \mathsf E^{f^\infty}\!\left[\sum_{i=1}^{\tau-1} \alpha^{i-1} c(x_i, f)\left.\vphantom{\sum_i^\tau}\right|x_1 = y\right]\\ & = \int_X \indic{X\ensuremath{\!\smallsetminus} K}(y) Q(\mrm dy|x, f)\; V(f^\infty, y). \end{align*} This substituted back in~\eqref{e:spolicy1} gives~\eqref{e:adcoepolicy}. \end{proof} \begin{proof}[Proof of Theorem {\rm \ref{t:EDC}}] (i) That $V^\star$ is a solution of the $\alpha$-DCOE follows from Lemma \ref{l:VIconv}, and that $V^\star$ is the minimal solution follows from Lemma~\ref{l:Tineq}, since $u = T u$ implies $u \ge V^\star$. (ii) Lemma~\ref{l:selector} guarantees the existence of a selector $f_\star\in\mathbb F$ such that~\eqref{e:alphado} holds. Fix $n\in\ensuremath{\mathbb{N}}$ and $x\in X\ensuremath{\!\smallsetminus} K$. As in the proof of Lemma~\ref{l:Tineq}, iterating equation~\eqref{e:alphado} $n$-times we arrive at \begin{align*} V^\star(x) & = \mathsf E_x^{f_\star^\infty}\!\left[\sum_{i=0}^{(n-1)\ensuremath{\wedge}(\tau-1)} \alpha^i c(x_i, f_\star)\right] + \mathsf E^{f_\star^\infty}_x\bigl[\alpha^{n} V^\star(x_{n})\indic{\{n < \tau\}}\bigr] \ge \mathsf E_x^{f_\star^\infty}\!\left[\sum_{i=0}^{(n-1)\ensuremath{\wedge}(\tau-1)} \alpha^i c(x_i, f_\star)\right]. \end{align*} By the monotone convergence theorem we have \[ V^\star(x) \ge \lim_{n\ensuremath{\rightarrow}\infty} \mathsf E_x^{f_\star^\infty}\!\left[\sum_{i=0}^{(n-1)\ensuremath{\wedge}(\tau-1)} \alpha^i c(x_i, f_\star)\right] = \mathsf E_x^{f_\star^\infty}\!\left[\sum_{i=0}^{\tau-1} \alpha^i c(x_i, f_\star)\right], \] which shows that $V^\star(x) \ge V(f_\star^\infty, x)$, and since $x\in X\ensuremath{\!\smallsetminus} K$ is arbitrary, it follows that $V^\star(\cdot) \ge V(f_\star^\infty, \cdot)$. The reverse inequality follows from the definition of $V^\star$ in~\eqref{e:problem}. We conclude that $V^\star(\cdot) = V(f_\star^\infty, \cdot)$, and that $f_\star^\infty$ is an optimal policy. For the converse, if $f_\star^\infty$ is an optimal deterministic stationary policy, then by Lemma \ref{l:adcoestat}, equation~\eqref{e:adcoepolicy} becomes \[ V^\star(x) = V(f_\star^\infty, x) = c(x, f_\star) + \alpha \int_X Q(\mrm dy|x, f_\star)\:\indic{X\ensuremath{\!\smallsetminus} K}(y)V(f_\star^\infty, y) \] for $x\in X\ensuremath{\!\smallsetminus} K$, which is identical to~\eqref{e:alphado}. \end{proof} \section{A Contraction Mapping Approach} \label{s:contr} For the purposes of this section we let $\Lp 0(X\ensuremath{\!\smallsetminus} K)$ denote the real vector space of real-valued measurable functions on $X$, and $\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ be the convex cone of nonnegative elements of $\Lp 0(X\ensuremath{\!\smallsetminus} K)$. (Note that according to paragraph~\ref{pgr:convention} we let the elements of $\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ take the value $+\infty$.) Given a measurable \emph{weight function} $w:X\ensuremath{\!\smallsetminus} K\ensuremath{\longrightarrow}[1, \infty[$ in $\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$, we define the weighted norm $\norm{u}_w \coloneqq \sup_{x\in X} \abs{u(x)}/w(x)$. It is well-known that $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K), \norm{\cdot}_w\bigr)$ is a Banach space. \begin{assumption} \label{a:further} In addition to Assumption~\ref{a:key}, we require that there exist $\overline c > 0$, $\beta\in[1, 1/\alpha[$, and a measurable weight function $w:X\ensuremath{\!\smallsetminus} K\ensuremath{\longrightarrow}[1, \infty[$ such that for every $x\in X\ensuremath{\!\smallsetminus} K$ \begin{enumerate}[label=(\roman*), leftmargin=*, align=right, widest=iii] \item $\displaystyle{\sup_{A(x)} c(x, a) \le \overline c w(x)}$; \item $\displaystyle{\sup_{A(x)} \int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a) w(y) \le \beta w(x)}$.\hspace{\stretch{1}}{$\diamondsuit$} \end{enumerate} \end{assumption} \begin{remark} If $c$ is bounded, the weight function $w$ may be taken to be $\indic{X\ensuremath{\!\smallsetminus} K}$. Also, if $x$ and $x^+$ are the current and the next states of the Markov control process, respectively, then Assumption~\ref{a:further}(ii) implies that \[ \sup_{A(x)}\mathsf E\bigl[w(x^+)\indic{\{x^+\in X\ensuremath{\!\smallsetminus} K\}}\big|(x, a)\bigr] \le \beta w(x)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K. \] We observe that this bears a resemblance with classical Lyapunov-like stability criteria, more specifically, the Foster-Lyapunov conditions~\cite[Chapter~8]{ref:meynCTCN}, \cite{ref:foss04}. However, the condition in Assumption~\ref{a:further}(ii) is uniform over the set of actions $A(x)$ pointwise in $x$. It connects the growth of the cost-per-stage function $c$ with a contraction induced by the discount factor $\alpha$.\hspace{\stretch{1}}{$\vartriangleleft$} \end{remark} Recall that a mapping $f:Y\ensuremath{\longrightarrow} Y$ on a nonempty complete metric space $(Y, \rho)$ is a \emph{contraction} if there exists a constant $\gamma\in[0, 1[$ such that $\rho(f(x_1), f(x_2)) \le \gamma\rho(x_1, x_2)$ for all $x_1, x_2\in Y$. The constant $\gamma$ is said to the the \emph{modulus} of the map $f$. A contraction has a unique fixed point $x^\star\in Y$ satisfying $f(x^\star) = x^\star$. \begin{proposition}[{\cite[Proposition~7.2.9]{ref:hernandez-lerma2}}] \label{p:contr} Let $T$ be a monotone map from the Banach space $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K), \norm{\cdot}_w\bigr)$ into itself. If there exists a $\gamma\in[0, 1[$ such that \begin{equation} \label{e:contr} T(u+rw) \le T(u) + \gamma rw\qquad\text{whenever}\quad u\in\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K), \norm{\cdot}_w\bigr),\quad r\in\ensuremath{\mathbb{R}}, \end{equation} then $T$ is a contraction with modulus $\gamma$. \end{proposition} We have the following lemma. \begin{lemma} \label{l:Tcontr} Under Assumption {\rm \ref{a:further}}, the map $T$ in~\eqref{e:Tdef} is a contraction on $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{\cdot}_w\bigr)$ with modulus $\gamma = \alpha\beta < 1$. \end{lemma} \begin{proof} Fix $u\in\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ with $\norm{u}_w < \infty$. As in the proof of Lemma~\ref{l:selector}, the mapping \[ \mathbb K\ni(x, a)\mapsto T'u(x, a) = c(x, a) + \alpha\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a) u(y)\in\ensuremath{\R_{\ge 0}} \] is well-defined and l.s.c.\ in $a\in A(x)$ for all $x\in X\ensuremath{\!\smallsetminus} K$. By the same Lemma we also know that $T$ maps $\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$ into $\Lp 0(X\ensuremath{\!\smallsetminus} K)^+$. For every $(x, a)\in\mathbb K$, by Assumption~\ref{a:further}, \begin{align*} \abs{T'(x, a)} & \le c(x, a) + \alpha\int_{X\ensuremath{\!\smallsetminus} K}Q(\mrm dy|x, a) \frac{u(y)}{w(y)} w(y) \le \overline cw(x) + \alpha \norm{u}_w\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a) w(y)\\ & \le \bigl(\overline c + \alpha\beta\norm{u}_w\bigr) w(x), \end{align*} which shows that $\norm{T'u}_w \le \overline c + \alpha\beta\norm{u}_w$. Therefore, $T$ maps $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{\cdot}_w\bigr)$ into itself. Since $c\ge 0$, it is clear that $T$ is a monotone map on $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{\cdot}_w\bigr)$. By Assumption~\ref{a:further}(ii), for $r\in\ensuremath{\mathbb{R}}$ and $x\in X\ensuremath{\!\smallsetminus} K$ we have \begin{align*} T(u+rw)(x) & = \min_{A(x)}\left(c(x, a) + \alpha\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a)\bigl(u(y) + rw(y)\bigr)\right)\\ & \le \min_{A(x)}\left(c(x, a) + \alpha\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a)u(y)\right) + r\alpha\beta w(x)\\ & \le Tu(x) + r\alpha\beta w(x). \end{align*} This shows that~\eqref{e:contr} holds with $\gamma = \alpha\beta$, and Proposition~\ref{p:contr} implies that $T$ is a contraction on $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{\cdot}_w\bigr)$. \end{proof} The following proposition establishes bounds for the distance between the optimal value function $V^\star$ and the $\alpha$-VI functions $(v_n)_{n\in\ensuremath{\mathbb{N}_0}}$ by employing the contraction mapping $T$ of Lemma~\ref{l:Tcontr}. \begin{proposition} \label{p:Tfp} Suppose that Assumption {\rm \ref{a:further}} holds, and let $\gamma \coloneqq \alpha\beta$. Then: \begin{enumerate}[label=\emph{(\roman*)}, align=right, leftmargin=*, widest=iii] \item The $\alpha$-discount value function $V^\star$ satisfies $\norm{V^\star}_w \le \overline c/(1-\gamma)$. \item The $\alpha$-VI functions $(v_n)_{n\in\ensuremath{\mathbb{N}_0}}$ satisfy \[ V^\star(x) - v_n(x) \le \overline c w(x)\left(\frac{\gamma^n}{1-\gamma}\right)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K, \quad \ensuremath{\forall\,} n\in\ensuremath{\mathbb{N}}. \] In particular, $\norm{v_n - V^\star}_w \le \overline c\gamma^n/(1-\gamma)\;\;\; \ensuremath{\forall\,} n\in\ensuremath{\mathbb{N}_0}$. \item The optimal value function $V^\star$ is the unique function in $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{\cdot}_w\bigr)$ that solves the $\alpha$-DCOE~\eqref{e:alphadcoe}. \end{enumerate} \end{proposition} \begin{proof} (i) Let $\pi$ be an arbitrary Markov policy. Trivially we have $\mathsf E^\pi_x\bigl[w(x_0)\bigr] \le w(x)$. Fix $i\in\ensuremath{\mathbb{N}}$, and a history $h_i\in\ensuremath{\mathfrak{F}}_{i\ensuremath{\wedge}\tau}$. In view of Assumption~\ref{a:further}(ii), on the event $\{i < \tau\}$ we have \begin{align*} \mathsf E^\pi_x\bigl[w(x_i)\big|h_{i-1}, a_{i-1}\bigr] = \int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x_{i-1}, a_{i-1}) w(y) \le \beta w(x_{i-1})\quad \ensuremath{\forall\,} a_i\in A(x_i), \end{align*} which shows that $\mathsf E^\pi_x\bigl[w(x_i)\indic{\{i < \tau\}}\bigr] \le \beta \mathsf E^\pi_x\bigl[w(x_{i-1})\indic{\{i < \tau\}}\bigr]$. Iterating this inequality we arrive at $\mathsf E^\pi_x\bigl[w(x_i)\indic{\{i < \tau\}}\bigr] \le \beta^i w(x)$. Also, by Assumption~\ref{a:further}(i) we have $c(x_i, a_i) \le \overline c w(x_i)$ for all $i\in\ensuremath{\mathbb{N}_0}$ such that $i < \tau$, which in conjunction with the above inequality gives \begin{equation} \label{e:Tfp1} \mathsf E^\pi_x\bigl[c(x_i, a_i)\indic{\{i < \tau\}}\bigr] \le \overline c\beta^i w(x). \end{equation} By the monotone convergence theorem and~\eqref{e:Tfp1} we have \begin{equation} \label{e:Tfp2} \begin{aligned} V(\pi, x) & = \mathsf E^\pi_x\!\left[\sum_{i=0}^\infty \alpha^i c(x_i, a_i) \indic{\{i < \tau\}}\right] \le \sum_{i=0}^\infty \alpha^i \mathsf E^\pi_x\bigl[c(x_i, a_i)\indic{\{i < \tau\}}\bigr]\\ & \le \overline c\sum_{i=0}^\infty (\alpha\beta)^i w(x) \le w(x)\cdot\frac{\overline c}{1-\gamma}. \end{aligned} \end{equation} It follows immediately that $\norm{V^\star}_w = \norm{\inf_{\Pi} V(\pi, x)}_w \le \overline c/(1-\gamma)$. (ii) By definition, the $\alpha$-VI functions $(v_n)_{n\in\ensuremath{\mathbb{N}_0}}$ satisfy $v_n = T v_{n-1} = T^n v_0$, with $v_0 \coloneqq 0$. Since $T$ is a contraction on $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{\cdot}_w\bigr)$ by Lemma \ref{l:Tcontr}, it follows that $T$ has a unique fixed point, which, by definition is $V^\star$, since $\norm{V^\star}_w < \infty$ by (i). A standard property of contraction maps implies that \[ \norm{T^n v_0 - V^\star}_w \le \gamma^n\norm{v_0 - V^\star}_w \qquad\ensuremath{\forall\,} u\in\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{u}_w < \infty,\quad \ensuremath{\forall\,} n\in\ensuremath{\mathbb{N}_0}. \] With the bound on $\norm{V^\star}_w$ obtained in (i), we get $\norm{v_n - V^\star}_w \le \overline c\cdot\gamma^n/(1-\gamma)$. Since $T$ is also a contraction on $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{\cdot}_w\bigr)$, $v_n|_{K} = 0$, and $v_n\uparrow V^\star$, the last inequality yields $V^\star(x) - v_n(x) \le \overline cw(x)\gamma^n/(1-\gamma)$ for every $x\in X\ensuremath{\!\smallsetminus} K$. (iii) Of course $V^\star$ solves the $\alpha$-DCOE~\eqref{e:alphadcoe}. Uniqueness follows from the facts that the operator $T$ in~\eqref{e:Tdef} is a contraction by Lemma \ref{l:Tcontr}, and that the fixed point of a contraction mapping in a Banach space (or more generally, in a complete metric space) is unique. \end{proof} Note that the conditions in Assumption~\ref{a:further} are automatic if $c$ is bounded. This gives the following straightforward result. \begin{corollary} \label{c:Tfp} Suppose that Assumption {\rm \ref{a:key}} holds, and $\widetilde c \coloneqq \sup_{\mathbb K}c(x, a) < \infty$. Then: \begin{enumerate}[label=\emph{(\roman*)}, align=right, leftmargin=*, widest=iii] \item The $\alpha$-discount value function $V^\star$ satisfies $\norm{V^\star} \le \widetilde c/(1-\alpha)$. \item The $\alpha$-VI functions $(v_n)_{n\in\ensuremath{\mathbb{N}_0}}$ satisfy \[ V^\star(x) - v_n(x) \le \widetilde c \left(\frac{\alpha^n}{1-\alpha}\right)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K,\quad \ensuremath{\forall\,} n\in\ensuremath{\mathbb{N}}. \] In particular, $\norm{v_n - V^\star} \le \widetilde c\:\alpha^n/(1-\alpha)\;\;\; \ensuremath{\forall\,} n\in\ensuremath{\mathbb{N}_0}$. \item The optimal value function $V^\star$ is the unique function in $\bigl(\Lp 0(X\ensuremath{\!\smallsetminus} K)^+, \norm{\cdot}_w\bigr)$ that solves the $\alpha$-DCOE~\eqref{e:alphadcoe}. \end{enumerate} \end{corollary} \section{Asymptotic Discount Optimality of the $\alpha$-VI Policy} \label{s:ado} We have seen that the $\alpha$-value iteration functions $(v_n)_{n\in\ensuremath{\mathbb{N}_0}}$ defined in~\eqref{e:VI} converge to $V^\star$ by Lemma~\ref{l:VIconv}. In this section we address the question whether the $\alpha$-VI policies converge in some sense to a policy $f_\star^\infty$ as $n\ensuremath{\rightarrow}\infty$. \begin{defn} \label{d:avip} Let $(v_n)_{n\in\ensuremath{\mathbb{N}_0}}$ be the sequence of $\alpha$-VI functions in~\eqref{e:VI}, and let $\widehat\pi = (\widehat f_n)_{n\in\ensuremath{\mathbb{N}_0}}\in\Pi_{DM}$ be a deterministic Markov policy such that $\widehat f_0\in\mathbb F$ is arbitrary, and for $n\in\ensuremath{\mathbb{N}}$, \[ v_n(x) = c\bigl(x, \widehat f_n\bigr) + \alpha \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat f_n\bigr) v_{n-1}(y)\qquad \ensuremath{\forall\,} x\in X\ensuremath{\!\smallsetminus} K. \] Then $\widehat\pi$ is called an \emph{$\alpha$-VI policy}.\hspace{\stretch{1}}{$\Diamond$} \end{defn} Under Assumption~\ref{a:key} we get the following basic existential result. \begin{proposition} \label{p:ado} Suppose that Assumption {\rm \ref{a:key}} holds, the action space $A$ is locally compact, and let $\widehat\pi = \bigl(\widehat f_n\bigr)_{n\in\ensuremath{\mathbb{N}_0}}\in\Pi_{DM}$ be an $\alpha$-VI policy as defined in Definition~\ref{d:avip}. Then there exists a selector $\widehat f\in\mathbb F$ such that for every $x\in X\ensuremath{\!\smallsetminus} K$, $\widehat f(x)\in A(x)$ is an accumulation point of $\bigl(\widehat f_n(x)\bigr)_{n\in\ensuremath{\mathbb{N}_0}}$, and the corresponding deterministic stationary policy $\widehat f^\infty\in\Pi_{DS}$ is $\alpha$-discount optimal. \end{proposition} The proof is based on the following immediate adaptation of \cite[Lemma 4.6.6]{ref:hernandez-lerma1}. \begin{lemma} \label{l:ado} Let $u$ and $u_n$, $n\in\ensuremath{\mathbb{N}}$, be l.s.c.\ functions, bounded below, and inf-compact on $\mathbb K$. For every $n\in\ensuremath{\mathbb{N}}$ let $u_n^\star(x) \coloneqq \min_{A(x)} u_n(x, a)$ and $u^\star(x) \coloneqq \min_{A(x)} u(x, a)$, let $\widehat f_n\in\mathbb F$ be a selector such that $u_n^\star(x) = u_n\bigl(x, \widehat f_n(x)\bigr)$ for all $x\in X\ensuremath{\!\smallsetminus} K$. If $A$ is locally compact and $u_n\uparrow u$, then there exists a selector $\widehat f\in\mathbb F$ such that $\widehat f(x)\in A(x)$ is an accumulation point of the sequence $\bigl(\widehat f_n(x)\bigr)_{n\in\ensuremath{\mathbb{N}}}$ for every $x\in X\ensuremath{\!\smallsetminus} K$, and $u^\star(x) = u\bigl(x, \widehat f(x)\bigr)$. \end{lemma} \begin{proof}[Proof of Proposition {\rm \ref{p:ado}}] For $(x, a)\in\mathbb K$ we define $u(x, a) \coloneqq c(x, a) + \alpha \int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a) V^\star(y)$, and \begin{equation} \label{e:unconv} u_n(x, a) \coloneqq c(x, a) + \alpha \int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a) v_{n-1}(y). \end{equation} Since $c\ge 0$, the functions $u_n$ and $u$ are nonnegative. Since $v_n\uparrow V^\star$ by Lemma~\ref{l:VIconv}, the monotone convergence theorem implies that \[ \int_{X\ensuremath{\!\smallsetminus} K}Q(\mrm dy|x, a) v_n(y) \ensuremath{\longrightarrow} \int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a) V^\star(y) \] pointwise on $\mathbb K$. It is clear that $u_n\uparrow u$, and the assertion follows at once from Lemma~\ref{l:ado}. \end{proof} Under the stronger Assumption~\ref{a:further} we get quantitative estimates of the rate at which the $\alpha$-VI policy defined in Definition~\ref{d:avip} converges to an optimal one. \begin{defn} The function $D:\mathbb K\ensuremath{\longrightarrow}\ensuremath{\R_{\ge 0}}$ defined by \[ \mathbb K\ni (x, a) \mapsto D(x, a) \coloneqq c(x, a) + \alpha\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, a) V^\star(y) - V^\star(x) \] is called the \emph{$\alpha$-discount discrepancy function}. The $\alpha$-VI policy $\widehat\pi = \bigl(\widehat f_n\bigr)_{n\in\ensuremath{\mathbb{N}_0}}$ defined in Definition~\ref{d:avip} is called \emph{pointwise asymptotically discount optimal} if for every $x\in X\ensuremath{\!\smallsetminus} K$ we have $\lim_{n\ensuremath{\rightarrow}\infty} D\bigl(x, \widehat f_n\bigr) = 0$.\hspace{\stretch{1}}{$\Diamond$} \end{defn} It is clear that for $x\in X\ensuremath{\!\smallsetminus} K$ and a selector $f\in\mathbb F$ (see paragraph~\ref{pgr:convention}), the $\alpha$-discount discrepancy function $D(x, f(x))$ is $0$ if and only if $f^\infty$ is an optimal policy. The function $D$ measures closeness to an optimal selector in a weak sense. \begin{proposition} Suppose that Assumption {\rm \ref{a:further}} holds, and let $\gamma \coloneqq \alpha\beta$. Then the $\alpha$-VI policy $\widehat\pi = \bigl(\widehat f_n\bigr)_{n\in\ensuremath{\mathbb{N}_0}}$ is pointwise asymptotically discount optimal, and for every $x\in X\ensuremath{\!\smallsetminus} K$ and $n\in\ensuremath{\mathbb{N}}$, \[ 0 \le D\bigl(x, \widehat f_n\bigr) \le 2\overline c\left(\frac{\gamma^{n+1}}{1-\gamma}\right) w(x). \] \end{proposition} \begin{proof} The first inequality follows directly from the definition of $V^\star$. To prove the second inequality fix $x\in X\ensuremath{\!\smallsetminus} K$. We see that by the definition of the discrepancy function, \begin{equation} \label{e:adiscop1} \begin{aligned} D\bigl(x, \widehat f_n\bigr) & = c\bigl(x, \widehat f_n\bigr) + \alpha \int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x, \widehat f_n\bigr) V^\star(y) - V^\star(x)\\ & = \bigl(v_{n+1}(x) - V^\star(x)\bigr) + \alpha\int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x, \widehat f_n\bigr)\bigl(V^\star(y) - v_{n}(y)\bigr). \end{aligned} \end{equation} By Proposition~\ref{p:Tfp}(ii) we have \begin{equation} \label{e:adiscop2} \abs{v_{n+1}(x) - V^\star(x)} \le \overline cw(x) \frac{\gamma^{n+1}}{1-\gamma}, \end{equation} and in the light of Assumption~\ref{a:further}(ii) we arrive at \begin{equation} \label{e:adiscop3} \begin{aligned} \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat f_n\bigr)\bigl(V^\star(y) - v_{n}(y)\bigr) & \le \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat f_n\bigr)\bigl(V^\star(y) - v_{n}(y)\bigr)\\ & \le \frac{\gamma^{n}}{1-\gamma}\beta w(x). \end{aligned} \end{equation} The assertion follows immediately after substituting~\eqref{e:adiscop2} and~\eqref{e:adiscop3} in~\eqref{e:adiscop1}. \end{proof} For bounded costs we have the following straightforward conclusion. \begin{corollary} Suppose that Assumption {\rm \ref{a:key}} holds, and $\widetilde c \coloneqq \sup_{\mathbb K}c(x, a) < \infty$. Then the $\alpha$-VI policy $\widehat\pi = \bigl(\widehat f_n\bigr)_{n\in\ensuremath{\mathbb{N}_0}}$ is pointwise asymptotically discount optimal, and for every $x\in X\ensuremath{\!\smallsetminus} K$ and $n\in\ensuremath{\mathbb{N}}$, \[ 0 \le D\bigl(x, \widehat f_n\bigr) \le 2\widetilde c\:\left(\frac{\alpha^{n+1}}{1-\alpha}\right). \] \end{corollary} \section{Average cost of recovery} As mentioned in \secref{s:intro}, a motivation for this work was to come up with a suitable recovery strategy for MPC. Tracing our development of the MPC methodology in \secref{s:intro}, one sees that in the presence of state and/or action constraints, one seeks a deterministic stationary policy $g_\star^\infty$ that is active whenever the state is inside the safe set $K$, and a recovery strategy outside $K$. Let us assume that for a given problem we have determined such a policy, and we have also determined a deterministic stationary policy $f_\star^\infty$ corresponding to the recovery strategy corresponding to a cost-per-stage function defined on $X\ensuremath{\!\smallsetminus} K$ for the same problem as described in the preceding sections. One of the natural questions at this stage is whether one can find estimates of the average cost of recovery To this end let us define two constants: \begin{equation} \label{e:betadef} \beta_1 \coloneqq \inf_{x\in K}\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, g_{\star}) V^\star(y),\quad \beta_2 \coloneqq \sup_{x\in K}\int_{X\ensuremath{\!\smallsetminus} K} Q(\mrm dy|x, g_{\star}) V^\star(y), \end{equation} where $V^\star$ is as defined in \eqref{e:problem}. Let $\widetilde f^\infty$ be the deterministic stationary policy defined by \begin{equation} \label{e:overallpolicy} \widetilde f(x) \coloneqq f_\star(x)\indic{X\ensuremath{\!\smallsetminus} K}(x) + g_\star(x)\indic{K}(x); \end{equation} to wit, $\widetilde f^\infty$ consists of concatenation of $f_\star^\infty$ and $g_\star^\infty$ between exit and entry times to $K$. We have the following result: \begin{proposition} Let $g_\star^\infty$ be a deterministic stationary policy that is active whenever the state is inside the set $K$, and let $f_\star^\infty$ be a recovery strategy corresponding to the problem \eqref{e:problem}. Let the initial condition $x$ be in $X\ensuremath{\!\smallsetminus} K$. We define the average cost of recovery \[ \widetilde W(x) \coloneqq \lim_{n\to\infty} \frac{1}{n+1} \ensuremath{\mathsf{E}}^{\widetilde f^\infty}_x\Biggl[\sum_{i=0}^{n}\sum_{t=\tau_{2i}}^{\tau_{2i+1}-1} \alpha^{t-\tau_{2i}} c(x_t, a_t)\Biggr], \] where $\widetilde f^\infty$ is as defined in \eqref{e:overallpolicy}, $\tau_0 \coloneqq 0$, $\tau_1$ is the first entry time to $K$, $\tau_2$ is the first exit time from $K$ after $\tau_1$, and so on. Suppose that from any initial condition in $X\ensuremath{\!\smallsetminus} K$ the first hitting time of $K$ is finite almost surely under $f_\star^\infty$, and from any initial condition in $K$ the first hitting time of $X\ensuremath{\!\smallsetminus} K$ is finite almost surely under $g_\star^\infty$. Then we have $\beta_1 \le \widetilde W(x) \le \beta_2$, where $\beta_1, \beta_2$ are as defined in \eqref{e:betadef}. \end{proposition} Note that an identical bound holds if the initial condition $x\in K$, with an obvious relabelling of the stopping times $(\tau_i)_{i\in\ensuremath{\mathbb{N}_0}}$. \begin{proof} First of all, note that the policy $\widetilde f^\infty$ is deterministic stationary, and under this policy the controlled process is stationary Markov. Now we have for a fixed $n\in \ensuremath{\mathbb{N}}$: \begin{equation} \label{e:costofrecovery} \begin{aligned} \ensuremath{\mathsf{E}}^{\widetilde f^\infty}_x & \Biggl[\sum_{i=0}^n \sum_{t=\tau_{2i}}^{\tau_{2i+1}-1} \alpha^{t-\tau_{2i}} c(x_t, a_t)\Biggr] = \sum_{i=0}^n \ensuremath{\mathsf{E}}^{\widetilde f^\infty}_x\Biggl[\sum_{t=\tau_{2i}}^{\tau_{2i+1}-1} \alpha^{t-\tau_{2i}} c(x_t, a_t)\Biggr]\\ & = \sum_{i=0}^n \ensuremath{\mathsf{E}}^{\widetilde f^\infty}_x\Biggl[\ensuremath{\mathsf{E}}^{f_\star^\infty}\Biggl[\sum_{t=\tau_{2i}}^{\tau_{2i+1}-1} \alpha^{t-\tau_{2i}} c(x_t, a_t)\Bigg|\ensuremath{\mathfrak{F}}_{\tau_{2i}}\Biggr]\Biggr] = \sum_{i=0}^n \ensuremath{\mathsf{E}}^{\widetilde f^\infty}_x\Bigl[\ensuremath{\mathsf{E}}^{f_\star^\infty}\bigl[V^\star(x_{\tau_{2i}})\big|\ensuremath{\mathfrak{F}}_{\tau_{2i}}\bigr]\Bigr], \end{aligned} \end{equation} where the first equality follows from monotone convergence and the last equality from the strong Markov property. Appealing to the strong Markov property once again we see that $\ensuremath{\mathsf{E}}^{f_\star^\infty}\bigl[V^\star(x_{\tau_{2i}})\big|\ensuremath{\mathfrak{F}}_{\tau_{2i}}\bigr] = \ensuremath{\mathsf{E}}^{f_\star^\infty}\bigl[V^\star(x_{\tau_{2i}})\big|x_{\tau_{2i}}\bigr]$. Finally, from the definition of $\tau_{2i}$ it follows that \[ \ensuremath{\mathsf{E}}^{f_\star^\infty}\bigl[V^\star(x_{\tau_{2i}})\big|x_{\tau_{2i}}\bigr] \le \sup_{\xi\in K}\int_X Q(\mrm dy|\xi, f_{\mathrm{in}}) V^\star(y)\indic{X\ensuremath{\!\smallsetminus} K}(y) = \beta_2. \] It is not difficult to arrive at the lower bound $\ensuremath{\mathsf{E}}^{f_\star^\infty}\bigl[V^\star(x_{\tau_{2i}})\big|x_{\tau_{2i}}\bigr]\ge \beta_1$ by following the same steps as above. Substituting in~\eqref{e:costofrecovery} and taking limits we arrive at the assertion. \end{proof} \section{A Rolling Horizon Implementation} \label{s:rh} The \emph{rolling-horizon} procedure can be briefly described as follows. Fix a horizon $N\in\ensuremath{\mathbb{N}}$ and set $n = 0$. Then \begin{enumerate}[label=(\alph*), leftmargin=*, align=right] \item we determine an optimal control policy, say $\pi^\star_{n:n+N}$, for the $(N+1)$-period cost function starting from time $n$, given the (perfectly observed) initial condition $x_n$; standard arguments lead to a realization of this policy as a sequence of $(N+1)$ selectors $\bigl\{\widehat f_{n, n+N-j}\big|j=n, n+1, \ldots, n+N\bigr\}$ \item we increase $n$ to $n+1$, and go back to step (a). \end{enumerate} Accordingly, the $n$-th step of this procedure consists of minimizing the stopped $(N+1)$-period cost function starting at time $n$, namely, the objective is to find a control policy that attains \begin{equation} \label{e:rhcf} \begin{aligned} \inf_{\pi\in\Pi} V_{n, n+N}(\pi, x) \coloneqq \;\inf_{\pi\in\Pi}\mathsf E^\pi\!\left[\left.\sum_{i=n\ensuremath{\wedge}(\tau-1)}^{(n+N)\ensuremath{\wedge}(\tau-1)} \alpha^{i-n\ensuremath{\wedge}(\tau-1)} c(x_i, a_i)\right|x_{n\ensuremath{\wedge}(\tau-1)} = x\right] \end{aligned} \end{equation} for $x\in X\ensuremath{\!\smallsetminus} K$. By stationarity and Markovian nature of the control model, it is enough to consider the control problem of minimizing the cost for $n = 0$, i.e., the problem of minimizing $V_{0, N}(\pi, x)$ over $\pi\in\Pi$. The corresponding policy $\pi$ is given by the policy that minimizes the $(N+1)$-stage $\alpha$-VI function $v_{N+1}$ in~\eqref{e:VI}. This particular policy is realized as a sequence of $(N+1)$ selectors $\bigl(\widehat f_N, \ldots, \widehat f_0\bigr)$. Thus, in the light of the above discussion, the rolling-horizon procedure yields the stationary suboptimal control policy $\widehat\pi \coloneqq \widehat f_N^\infty$ for the original problem~\eqref{e:problem}. Let $V\bigl(\widehat f_N^\infty, x\bigr)$ be the value function corresponding to the deterministic stationary policy $\widehat f_N^\infty \coloneqq \bigl(\widehat f_N, \widehat f_N, \ldots\bigr)$, $x\in X\ensuremath{\!\smallsetminus} K$. Observe that $\norm{V\bigl(\widehat f_N^\infty, x\bigr)}_w < \infty$, which follows from the more general estimate in~\eqref{e:Tfp2}. Our objective in this section is to give quantitative estimates of the extent of sub-optimality of the rolling-horizon policy $\widehat\pi$, compared to the optimal policy $\pi^\star$ that attains the infimum in~\eqref{e:problem}. We shall follow the notations of~\secref{s:contr} above. \begin{theorem} \label{t:rh} Suppose that Assumption {\rm \ref{a:further}} holds, and let $\gamma \coloneqq \alpha\beta$. For every $N\in\ensuremath{\mathbb{N}_0}$ and $x\in X\ensuremath{\!\smallsetminus} K$ we have \begin{equation} \label{e:keyrh} 0 \le V\bigl(\widehat f_N^\infty, x\bigr) - v_{N+1}(x) \le \overline c w(x) \left(\frac{\gamma^{N+1}}{1-\gamma}\right), \end{equation} where $v_{N+1}$ is the $(N+1)$-th $\alpha$-VI function defined in~\eqref{e:VI}. In particular, \begin{equation} \label{e:sloppyrh} V\bigl(\widehat f_N^\infty, x\bigr) - V^\star(x) \le \overline c w(x) \left(\frac{\gamma^{N+1}}{1-\gamma}\right). \end{equation} \end{theorem} A proof of Theorem \ref{t:rh} is given in the Appendix, if follows the arguments in~\cite[Theorem~1]{ref:aldenRH} for finite state-space Markov decision processes and bounded costs. It is of interest to note that the bound in~\eqref{e:keyrh} is identical to the bound between $V^\star(\cdot)$ and $v_{N+1}(\cdot)$ that appears in Proposition~\ref{p:Tfp}. If the cost-per-stage function $c$ is bounded on $\mathbb K$, we have the following immediate corollary: \begin{corollary} Suppose the Markov control process satisfies Assumption {\rm \ref{a:key}}. Let the cost-per-stage function $c:\mathbb K\ensuremath{\longrightarrow}\ensuremath{\R_{\ge 0}}$ be bounded, with $\widetilde c \coloneqq \sup_{\mathbb K} c(x, a) < \infty$. Then $V\bigl(\widehat f_N^\infty, x\bigr) \ge V^\star(x)$ for every $x\in X\ensuremath{\!\smallsetminus} K$, and \[ \sup_{x\in X\ensuremath{\!\smallsetminus} K}\left(V\bigl(\widehat f_N^\infty, x) - V^\star(x)\right) \le \frac{\widetilde c\cdot\alpha^{N+1}}{1-\alpha}. \] \end{corollary} \section{Application} \label{s:appl} In this section we give a numerical example concerning fishery management. The example is motivated by~\cite[Chapter~7]{hastings1989introduction}. The example considers a fishery modeled in discrete-time with the time period representing a fishing season. The state of the controlled Markov chain is the population of the fish species of interest. Fishermen might on the one hand want to harvest all that they can manage in order to increase their short-run profit, but on the other hand this might lead to very low levels of the population. Our goal is to design a recovery strategy for the case that the population gets over-fished and goes below a critical level. For doing so, we consider a simple model, with four possible fish population levels, 1 (almost extinct), 2, 3, and 4 (the target set). We assume that we can accurately measure the population size at the beginning of each season $k$, $X_k$. During a season the following set of actions are available: {Harvest (1), Harvest less (2), Do nothing (3), Import fish (4), Import less (5)}. We also take as given the following transition probabilities between the Markov States, where $T_a(i,j)$ denotes the probability that the population level at the beginning of the next season will be $j$, given that the current population is $i$ and action $a$ is applied during this season. \begin{alignat*}{2} T_1 & = \begin{bmatrix} 1 & 0 & 0 &0\\ 0.7 & 0.3 & 0 & 0\\ 0.1 & 0.6 & 0.3 & 0\\ - & - & - & - \end{bmatrix} & \quad & T_2 = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0.35 & 0.65 & 0 & 0\\ 0.04 & 0.5 & 0.46 & 0\\ - & - & - & - \end{bmatrix}\\ T_3 & = \begin{bmatrix} 0.99 & 0.01 & 0 &0\\ 0.01 & 0.7 & 0.28 & 0.01\\ 0 & 0.03 & 0.65 & 0.32\\ - & - & - & - \end{bmatrix} & \quad & T_4 = \begin{bmatrix} 0.4 & 0.6 & 0 & 0\\ 0 & 0.3 & 0.65 & 0.05\\ 0 & 0 & 0.25 & 0.75\\ - & - & - & - \end{bmatrix}\\ T_5 & = \begin{bmatrix} 0.6 & 0.4 & 0 & 0\\ 0 & 0.45 & 0.54 & 0.01\\ 0 & 0 & 0.45 & 0.55\\ - & - & - & - \end{bmatrix} & & \end{alignat*} The costs incurred at each state are $c(x_i,\alpha_i) = C(x_i) + A(x_i,\alpha_i)$, where \begin{equation*}C(x_i)=\begin{bmatrix} 300 & 150 & 100 & - \end{bmatrix}^\textrm{T} \end{equation*} represents a cost incurred for being at the current state and \begin{equation*}A(x_i,\alpha_i) = \begin{bmatrix} -20 & -10 & 0 & 150 & 75\\ -40 & -20 & 0 & 150 & 75\\ -80 & -40 & 0 & 150 & 75\\ - & - & - & - & - \end{bmatrix}\end{equation*} the action cost associated with each action and state. We assume a discount factor $\alpha = 0.9$. Using this setting, one can compute the policy that attains the $\alpha$-discount value function~\eqref{e:problem}. This turns out to be to import fish when in state $(1)$, to import fewer fish in state $(2)$, and do nothing at state $(3)$. Next, we search for the optimum policy, while using a rolling horizon control scheme, i.e., finding the policy that attains~\eqref{e:rhcf}. We solve the problem for horizon lengths between $1$ and $10$, in order to compare the results with the infinite horizon optimal policy. \begin{figure}[h] \centering \includegraphics[width=0.49\textwidth]{avg_cost.pdf} \includegraphics[width=0.49\textwidth]{std_cost.pdf} \caption{Accumulated cost average and standard-deviation} \label{f:cost} \end{figure} \begin{figure}[h] \centering \includegraphics[width=0.49\textwidth]{avg_hitting_time.pdf} \includegraphics[width=0.49\textwidth]{std_hitting_time.pdf} \caption{Hitting time average and standard-deviation} \label{f:time} \end{figure} Figure~\ref{f:cost} shows the average and the standard-deviation of the accumulated costs over $2\times 10^5$ Monte Carlo runs, with the initial population level at state $1$. Similarly, Figure~\ref{f:time} shows the average and the standard-deviation of the time steps needed for the recovery into the target state $4$. The results suggest that for the rolling horizon policy to match the optimal infinite horizon one, a horizon length of at least $8$ should be used. Smaller horizons provide sub-optimal policies (with respect to the infinite horizon one), with the sub-optimality gap reducing as the horizon length increases. Note that the case of $N = 1$ is not included in the data; this is because for horizon length of $1$ the optimal policy is to harvest while the system is at state $1$, leading to an $\infty$ cost and recovery time, which does not allow the system to ever recover to state $4$. \section{Future Work} \label{s:concl} We established in~\secref{s:EDC} that the optimal value function $V^\star$ is the minimal solution of the $\alpha$-discounted cost optimality equation~\eqref{e:alphadcoe}. However, obtaining analytical expression of the optimal value function $V^\star$ is difficult, particularly due to the integration over a subset $X\ensuremath{\!\smallsetminus} K$ of the state space. Obtaining good approximations of $V^\star$ is of vital importance, and will be reported in subsequent articles. It is interesting to note that our basic framework of stochastic model-predictive control (described in~\secref{s:intro}) naturally leads to a partitioning of the state-space with different dynamics in each partition; thus, the controlled system may be viewed as a stochastic hybrid system. One of the basic questions in this context is that of stability of the controlled system, and in view of the fact that in general there will be infinitely many excursions of the state outside the safe set, establishing any stability property is a challenging task. Classical Lyapunov-based methods are difficult to apply directly precisely because of the infinitely many state-dependent switches between multiple regimes, each with different dynamics. However, excursion-theory of Markov processes~\cite{ref:blumenthal1992} enables us to establish certain stability properties of quite general stochastic hybrid systems with state-dependent switching; some of these results are reported in~\cite{ref:palExcur}. \section*{Acknowledgments} The authors are grateful to Vivek S. Borkar, On\'esimo Hern\'andez-Lerma, and Sean P. Meyn for illuminating discussions and pointers to relevant literature. They also thank the anonymous reviewers for their helpful comments. \begin{appendix} \section{Proof of Theorem \ref{t:rh}} \begin{proof}[Proof of Theorem {\rm \ref{t:rh}}] For brevity of notation in this proof, we let $\widehat\pi \coloneqq \widehat f_N^\infty$, and let $\widehat\pi_{i:j}$ denote the (ordered) elements of the policy $\widehat\pi$ from stage $i$ through $j$ for $j > i$. The first inequality in~\eqref{e:keyrh} is trivial because $v_{N+1}(x) \le V^\star(x) \le V\bigl(\widehat f_N^\infty, x\bigr)$ for all $x\in X\ensuremath{\!\smallsetminus} K$. Before the proof of the second inequality in~\eqref{e:keyrh}, let us fix some notation. Pick $N\in\ensuremath{\mathbb{N}_0}$. For $n\in\ensuremath{\mathbb{N}_0}$, a policy $\pi_{n:n+N}$ for stages $n$ through $n+N$, and $i\in\{n, \ldots, n+N\}$, let $\pi_{n:n+N}(i)$ denote $i$-th element of the policy $\pi_{n:n+N}$. Also, let $Q\bigl(\cdot\big| x, \pi_{n:n+N}\bigr)$ denote the sub-stochastic kernel\footnote{Recall that $Q(\cdot|\cdot)$ is a \emph{sub-stochastic kernel} on $X\ensuremath{\!\smallsetminus} K$ given $Y$ if $Q(B|\cdot)$ is a measurable function on $Y$ for each $B\in\Borelsigalg{X}$, and $Q(\cdot|y)$ is a measure on $X$ with $Q(X|y) \le 1$ for each $y\in Y$.} defined for $x\in X\ensuremath{\!\smallsetminus} K$ by \begin{align*} Q\bigl(B\big|x, \pi_{n:n+N}\bigr) \coloneqq &{} \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm d\xi_0\big|x, \pi_{n:n+N}(n)\bigr)\cdots\int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm d\xi_N\bigr|\xi_{N-1}, \pi_{n:n+N}(n+N)\bigr)\indic{B}(\xi_N) \end{align*} for $B\in\Borelsigalg{X\ensuremath{\!\smallsetminus} K}$. Let $\pi^\star_{n:n+N}$ be an optimal policy for stages $n$ through $n+N$, i.e., let $\pi^\star_{n:n+N}$ attain the infimum in~\eqref{e:rhcf}. Fix $x\in X\ensuremath{\!\smallsetminus} K$. Let $\zeta_{n+1:n+N+1}$ be an $(N+1)$-period policy starting from stage $n+1$, such that its first $N$ elements are identical to the last $N$ elements of $\pi^\star_{n:n+N}$, i.e., $\zeta_{n+1:n+N+1}(j) = \pi^\star_{n:n+N}(j)$ for $j=n+1, \ldots, n+N$. By optimality of $\pi^\star_{n:n+N}$ we have \begin{multline*} \mathsf E^{\zeta_{n+1:n+N+1}}_x\!\left[\sum_{i=n+1}^{n+N+1} \alpha^i c(x_i, a_i)\indic{\{i< \tau\}}\left.\vphantom{\sum_{i=n+1}^{n+N+1}}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}\right]\\ \ge \mathsf E^{\pi^\star_{n+1:n+N+1}}_x\!\left[\sum_{i=n+1}^{n+N+1} \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_{i=n+1}^{n+N+1}}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}\right]. \end{multline*} Since $\widehat\pi_{n:n+N}(n) = \pi^\star_{n:n+N}(n)$ by construction, conditional on $x_{n\ensuremath{\wedge}(\tau-1)} = x'\in X\ensuremath{\!\smallsetminus} K$, \begin{multline} \label{e:keyineq} \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x', \pi^\star_{n:n+N}(n)\bigr)\mathsf E^{\zeta_{n+1:n+N+1}}\!\left[\sum_{i=n+1}^{n+N+1} \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_{i=n+1}^{n+N+1}}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)} = y\right]\ge\\ \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x', \widehat\pi_{n:n+N}(n)\bigr)\mathsf E^{\pi^\star_{n+1:n+N+1}}_x\!\left[\sum_{i=n+1}^{n+N+1} \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_{i=n+1}^{n+N+1}}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)} = y\right]. \end{multline} By definition of $\zeta$ we have \begin{align*} \mathsf E^{\zeta_{n+1:n+N+1}}\!& \left[\sum_{i=n+1}^{n+N+1} \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_{i=n+1}^{n+N+1}}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}\right]\\ & = \mathsf E^{\zeta_{n+1:n+N+1}}\!\left[\sum_{i=n+1}^{n+N} \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_{i=n+1}^{n+N+1}}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}\right] \\ & \qquad+ \mathsf E^{\zeta_{n+1:n+N+1}}\!\left[\alpha^{n+N+1} c(x_{n+N+1}, a_{n+N+1})\indic{\{n+N+1 < \tau\}}\big|x_{(n+1)\ensuremath{\wedge}(\tau-1)}\right], \end{align*} and the right-hand side equal \begin{multline*} \mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n+1}^{n+N} \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_{i=n+1}^{n+N+1}}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}\right] \\ + \mathsf E^{\zeta_{n+1:n+N+1}}\!\left[\alpha^{n+N+1} c(x_{n+N+1}, a_{n+N+1})\indic{\{n+N+1 < \tau\}}\big|x_{(n+1)\ensuremath{\wedge}(\tau-1)}\right]. \end{multline*} In conjunction with~\eqref{e:keyineq} and conditional on $x_{n\ensuremath{\wedge}(\tau-1)} = x'\in X\ensuremath{\!\smallsetminus} K$, we have \begin{multline*} \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x', \pi^\star_{n:n+N}(n)\bigr)\mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n+1}^{n+N} \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_{i}^N}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\right]\\ + \int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x', \pi^\star_{n:n+N}(n)\bigr)\mathsf E^{\zeta_{n+1:n+N+1}}\bigl[\alpha^{n+N+1} c(x_{n+N+1}, a_{n+N+1})\indic{\{n+N+1 < \tau\}}\big|x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\bigr]\\ \ge \int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x', \widehat\pi_{n:n+N}(n)\bigr)\mathsf E^{\pi^\star_{n+1:n+N+1}}\!\left[\sum_{i=n+1}^{n+N+1}\alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\right]. \end{multline*} To wit, conditional on $x_{n\ensuremath{\wedge}(\tau-1)} = x'\in X\ensuremath{\!\smallsetminus} K$, \begin{multline*} \mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n}^{n+N} \alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_{i}^N}\right|x_{n\ensuremath{\wedge}(\tau-1)} = x'\right] - \mathsf E^{\pi^\star_{n:n+N}}\!\left[\alpha^n c(x_n, a_n)\indic{\{i < \tau\}}\big|x_{n\ensuremath{\wedge}(\tau-1)} = x'\right]\\ + \alpha^{n+N+1} \int_{X\ensuremath{\!\smallsetminus} K} \!\!Q\bigl(\mrm dy\big|x', \pi^\star_{n:n+N}(n)\bigr)\mathsf E^{\zeta_{n+1:n+N+1}}\bigl[c(x_{n+N+1}, a_{n+N+1})\indic{\{n+N+1 < \tau\}}\big|x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\bigr]\\ \ge \int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x', \widehat\pi_{n:n+N}(n)\bigr)\mathsf E^{\pi^\star_{n+1:n+N+1}}\!\left[\sum_{i=n+1}^{n+N+1}\alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\right]. \end{multline*} Let $\zeta_{n+1:n+N+1}(n+N+1)(\cdot)$ be a selector that attains the minimal value of \[ \alpha^{n+N+1}\int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x', \pi^\star_{n:n+N}(n)\bigr)\mathsf E^{\zeta_{n+1:n+N+1}}\bigl[c(x_{n+N+1}, a_{n+N+1})\indic{\{n+N+1 < \tau\}}\big|x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\bigr] \] whenever $x'\in X\ensuremath{\!\smallsetminus} K$, and let the corresponding minimal value be denoted by $e_n(x')$; clearly $e_n$ is well-defined on $X\ensuremath{\!\smallsetminus} K$, and is a measurable function of $x'$. With this notation, the last inequality becomes \begin{multline*} \mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n}^{n+N}\alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right|x_{n\ensuremath{\wedge}(\tau-1)} = x'\right] - \mathsf E^{\pi^\star_{n:n+N}}\bigl[\alpha^n c(x_n, a_n)\indic{\{n < \tau\}}\big|x_{n\ensuremath{\wedge}(\tau-1)} = x'\bigr]\\ + e_n(x') \ge \int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x', \widehat\pi_{n:n+N}(n)\bigr) \mathsf E^{\pi^\star_{n+1:n+N+1}}\!\left[\sum_{i=n+1}^{n+N+1}\alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\right] \end{multline*} whenever $x'\in X\ensuremath{\!\smallsetminus} K$. Therefore, \begin{multline*} \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr) \mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n}^{n+N}\alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right|x_{n\ensuremath{\wedge}(\tau-1)}=y\right]\\ - \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr) \mathsf E^{\pi^\star_{n:n+N}}\bigl[\alpha^n c(x_n, a_n)\indic{\{n < \tau\}}\big|x_{n\ensuremath{\wedge}(\tau-1)}\bigr] + \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr) e_n(y)\\ \ge \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n}\bigr) \mathsf E^{\pi^\star_{n+1:n+N+1}}\!\left[\sum_{i=n+1}^{n+N+1}\alpha^i c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right|x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\right]. \end{multline*} Rearranging and summing over $n$ we arrive at \begin{multline} \label{e:ineq1} \sum_{n=0}^\infty \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr) \mathsf E^{\pi^\star_{n:n+N}}\!\left[\alpha^n c(x_n, a_n)\indic{\{n < \tau\}}\left.\vphantom{\sum}\right|x_{n\ensuremath{\wedge}(\tau-1)}=y\right]\\ \le \sum_{n=0}^\infty\left(\alpha^n\int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr)\mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n}^{n+N}\alpha^{i-n} c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right| x_{n\ensuremath{\wedge}(\tau-1)}=y\right]\right.\\ - \left.\alpha^{n+1}\int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n}\bigr)\mathsf E^{\pi^\star_{n+1:n+N+1}}\!\left[\sum_{i=n+1}^{n+N+1}\alpha^{i-n-1} c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right| x_{(n+1)\ensuremath{\wedge}(\tau-1)}=y\right]\right)\\ + \sum_{n=0}^\infty \int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr) e_n(y). \end{multline} In~\eqref{e:ineq1} we have employed the notation $\int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy|x, \pi_{0:-1}\bigr) g(y) \coloneqq g(x)$ for any policy $\pi$. We observe that the left-hand side of~\eqref{e:ineq1} is just $\mathsf E^{\widehat\pi}_x\!\left[\sum_{i=0}^{\tau-1} \alpha^i c(x_i, a_i)\right]$. By Assumption~\ref{a:further}(i), \[ \mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n}^{n+N}\alpha^{i-n} c(x_i, a_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right| x_{n\ensuremath{\wedge}(\tau-1)}=y\right] \le \overline c\mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n}^{n+N}\alpha^{i-n} w(x_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right| x_{n\ensuremath{\wedge}(\tau-1)}=y\right], \] and by Assumption~\ref{a:further}(ii), \[ \mathsf E^{\pi^\star_{n:n+N}}\!\left[\sum_{i=n}^{n+N}\alpha^{i-n} w(x_i)\indic{\{i < \tau\}}\left.\vphantom{\sum_i^N}\right| x_{n\ensuremath{\wedge}(\tau-1)}=y\right] \le w(y)\sum_{i=n}^{n+N}\gamma^{i-n}. \] We notice that since $c\ge 0$, the first series on the right-hand side of~\eqref{e:ineq1} is at most \begin{equation} \label{e:ineq2} \begin{aligned} \overline c \sum_{n=0}^\infty \alpha^n\sum_{i=n}^{n+N}\gamma^{i-n} \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr) w(y). \end{aligned} \end{equation} For a fixed $n\in\ensuremath{\mathbb{N}_0}$, the quantity $\int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy|x, \widehat\pi_{0:n}\bigr)w(y)$ is at most $\beta^{n+1} w(x)$ in view of Assumption~\ref{a:further}(ii) and the definition of the stochastic kernel $Q\bigl(\cdot\big|x, \pi_{n:n+N}\bigr)$ at the beginning of this proof. Therefore, \begin{align*} \sum_{n=0}^\infty & \alpha^n\sum_{i=n}^{n+N}\gamma^{i-n} \int_{X\ensuremath{\!\smallsetminus} K}Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr) w(y) \le \sum_{n=0}^\infty\alpha^n\sum_{i=n}^{n+N}\gamma^{i-n}\beta^n w(x)\\ & \le w(x) \left(\frac{1-\gamma^{N+1}}{1-\gamma}\right) < \infty. \end{align*} This shows that series in~\eqref{e:ineq2} is summable. Hence, cancellations of the telescopic terms in the first series on the right-hand side of~\eqref{e:ineq1} are justified. The inequality in~\eqref{e:ineq1} now simplifies to \begin{multline} \label{e:ineq3} \mathsf E^{\widehat\pi}_x\!\left[\sum_{i=0}^{\tau-1} \alpha^i c(x_i, a_i)\right] \le \mathsf E^{\pi^\star_{0:N}}_x\!\left[\sum_{i=0}^{(N+1)\ensuremath{\wedge}(\tau-1)}\!\!\alpha^{i} c(x_i, a_i)\right] + \sum_{n=0}^\infty \int_{X\ensuremath{\!\smallsetminus} K} Q\bigl(\mrm dy\big|x, \widehat\pi_{0:n-1}\bigr) e_n(y). \end{multline} By Assumption~\ref{a:further}(ii) and the definition of $e_n$, conditional on $x_{n\ensuremath{\wedge}\tau} = x'\in X\ensuremath{\!\smallsetminus} K$, \begin{align*} & e_n(x')\\ & \le \alpha^{n+N+1}\int_{X\ensuremath{\!\smallsetminus} K} \!\!Q\bigl(\mrm dy\big| x', \pi^\star_{n:n+N}(n)\bigr)\mathsf E^{\zeta_{n+1:n+N+1}}\bigl[c(x_{n+N+1}, a_{n+N+1})\indic{\{n+N+1 < \tau\}}\big|x_{(n+1)\ensuremath{\wedge}(\tau-1)} = y\bigr]\\ & \le \overline c w(x')\alpha^n \gamma^{N+1}. \end{align*} Substituting the last inequality in~\eqref{e:ineq3} we arrive at \[ \mathsf E^{\widehat\pi}_x\!\left[\sum_{i=0}^{\tau-1} \alpha^i c(x_i, a_i)\right] \le \mathsf E^{\pi^\star_{0:N}}_x\!\left[\sum_{i=0}^{(N+1)\ensuremath{\wedge}(\tau-1)}\!\!\alpha^{i} c(x_i, a_i)\right] + \frac{\overline c\gamma^{N+1}}{1-\gamma}w(x), \] which is the second bound in~\eqref{e:keyrh}. The inequality~\eqref{e:sloppyrh} follows immediately from the fact that $V^\star \ge v_n$ for every $n\in\ensuremath{\mathbb{N}}$. \end{proof} \end{appendix} \def$'${$'$}

proofpile-arXiv_069-11986

\section{Introduction} In the last decade substantial attention has been devoted to entanglement because it is considered to be the physical resource on which quantum technologies are based \cite{nielsen}. For this reason, entanglement dynamics have been widely studied, with the goal of finding ways to manipulate entangled states in a practical way, and in addition, to get a fundamental understanding of entanglement and all its embodiments. When interacting with a reservoir the fragile quantum states are easily destroyed. E.g., several years ago it was shown that a pair of initially entangled qubits can loss their entanglement in a finite time and not asymptotically as one would naively expect \cite{karol}. This phenomena was eventually called entanglement sudden death \cite{esd} (ESD) and several investigations have followed, see \cite{esd}-\cite{eberly}. Recently ESD was demonstrated in a linear optics experiment \cite{almeida}. Several schemes have been proposed in order to preserve fragile quantum states when interacting with a reservoir, for example decoherence-free subspaces \cite{dfs} and quantum error correction (QEC) (see \cite{shor}-\cite{Fletcher}). A recent worry expressed in \cite{eberly} is that if an entangled state suffers a ``sudden death'' the entanglement is irrevocably lost, and therefore no QEC scheme can help to restore the state. Recently, using the $[4,1]$ code which encodes a single qubit into four physical qubits and which was specifically constructed to deal with an amplitude damping channel \cite{leung}, it has been demonstrated that when a pair of initially entangled qubits {\it locally} encoded with a $[4,1]\times[4,1]$ code, and subjected to {\it local} recovery operations similar to the ones proposed in \cite{Fletcher}, QEC can delay ESD in some cases, but may cause ESD for certain states that will not succumb to ESD without QEC \cite{QEC}. We would like to stress that by {\it local} we mean that each qubit is encoded, measured and recovered separately. In the same work \cite{QEC}, it is mentioned that even using the {\it non-local} code $[6,2]$ which encodes a pair of logical qubits into six physical qubits \cite{Fletcher}, the obtained results does not differ qualitatively from the shown in \cite{QEC}. In this work we demonstrate this explicitly, which means that even when the QEC protocol is {\it non-local}, in all the stages, there is not a qualitatively difference with the {\it local} code studied before \cite{QEC}. We also find that the success of {\it non-local} coding and error recovery has a rather substantial state dependence. \section{The Model} As a simple example we will consider an amplitude damping channel, that for one qubit can be represented in terms of the Kraus operators : \begin{displaymath} \hat{E}_0=\left( \begin{array}{cc} 1&0\\ 0&\sqrt{1-\gamma} \end{array}\right),\qquad \hat{E}_1=\left( \begin{array}{cc} 0&\sqrt{\gamma}\\ 0&0 \end{array}\right), \end{displaymath} where $\gamma$ is the jump probability from the excited ($\ket{1}$) to the ground ($\ket{0}$) state, $\hat{E}_0$ is the one qubit no-jump operator that leaves the ground state unchanged, but decreases the excited state probability with the factor $1-\gamma$, while $\hat{E}_1$ represents the jump operator that transforms the state $\ket{1}$ into the state $\ket{0}$ with probability $\gamma$. To study the possibility of protecting a two qubit state and particularly entanglement by {\it non-local} coding, where the pair of qubits are encoded together, and to compare it with previous results obtained by {\it local} coding \cite{QEC}, we will use the $[6,2]$ code introduced in \cite{Fletcher}. This code is specially made for protecting two qubits against this kind of disturbance. It encodes an arbitrary two-qubit pure state \begin{equation}\ket{\varphi_0}=\cos\alpha\cos\delta\ket{11}+\sin\alpha\cos\delta e^{i\epsilon_1}\ket{00}+\cos\beta\sin\delta e^{i\epsilon_2}\ket{10}+\sin\beta\sin\delta e^{i\epsilon_3}\ket{01}\nonumber\end{equation} into the corresponding logical state \begin{eqnarray}\ket{\varphi_0}_L&=&\cos\alpha\cos\delta\ket{11}_L+\sin\alpha\cos\delta e^{i\epsilon_1}\ket{00}_L\nonumber\\ &&+\cos\beta\sin\delta e^{i\epsilon_2}\ket{10}_L+\sin\beta\sin\delta e^{i\epsilon_3}\ket{01}_L\nonumber,\end{eqnarray} were the codewords $\ket{00}_L$, $\ket{01}_L$, $\ket{10}_L$ and $\ket{11}_L$, are given by,\cite{Fletcher} \begin{eqnarray} \ket{00}_L&=&\left(\ket{000000}+\ket{111111}\right)/\sqrt{2},\label{basis00}\\ \ket{01}_L&=&\left(\ket{001001}+\ket{110110}\right)/\sqrt{2},\label{basis01}\\ \ket{10}_L&=&\left(\ket{000110}+\ket{111001}\right)/\sqrt{2},\label{basis10}\\ \ket{11}_L&=&\left(\ket{110000}+\ket{001111}\right)/\sqrt{2}.\label{basis11}\end{eqnarray} Above, we have used the notation $\ket{000000}=\ket{0}\otimes\ldots\otimes\ket{0}$, etc. It is worth noticing that the codewords (\ref{basis00})-(\ref{basis11}) in principle can be mapped in any way onto the physical states on the right-hand side. With different mappings we will obtain different results for a given state $\ket{\varphi_0}$, where the labeling of the physical state $\left(\ket{000000}+\ket{111111}\right)/\sqrt{2}$ with the possible codewords $\ket{ij}_L$, $i,j=0,1$ is the main cause of such differences. We will suppose that every physical qubit is interacting with its own environment, a situation that leads to ESD (see \cite{karol}-\cite{eberly} and references therein). This implies that the amplitude damping channels can be considered to be independent, which means that the many-qubit amplitude-damping channel will be described by the tensorial product of the corresponding Kraus operators as in \cite{QEC}. For simplicity we will also assume that all the qubits are damped with the same probability $\gamma$. As was pointed out in \cite{Fletcher}, the one physical-qubit damping-errors are spanned by orthogonal subspaces. Nevertheless, in this channel we have conditional evolution (the no-damping error introduces some distortion in the original state). However, the distortion generated by this channel in this code is of the order of $\gamma^2$, so it satisfies the approximate QEC conditions \cite{leung}. Therefore the no-damping evolution is ``approximately'' spanned by the codewords (\ref{basis00})-(\ref{basis11}). Every one of the subspaces related to no-damping and one-qubit damping are spanned by four 64-dimensional vectors $\{{\ket{R_{k00}},\ket{R_{k01}},\ket{R_{k10}},\ket{R_{k11}}}\}$, where $k=0$ denotes no damping, $k=1,\ldots,6$ means damping in the first to the sixth physical qubit respectively, and the last two subscripts ($ij$) in each $\ket{R_{kij}}$ refer to the corresponding two-qubit state. In total we have 28 vectors spanning the no-damping and one-qubit damping. For no damping we have the codewords (\ref{basis00})-(\ref{basis11}), \begin{equation} \ket{R_{000}}=\ket{00}_L,\quad\ket{R_{001}}=\ket{01}_L,\quad\ket{R_{010}}=\ket{10}_L,\quad\ket{R_{011}}=\ket{11}_L.\nonumber \end{equation} Meanwhile, for the one-qubit damping one obtains \begin{eqnarray} \ket{R_{100}}=\ket{011111}\quad\ket{R_{101}}=\ket{010110}\quad\ket{R_{110}}=\ket{011001}\quad\ket{R_{111}}=\ket{010000}\nonumber,\\ \ket{R_{200}}=\ket{101111}\quad\ket{R_{201}}=\ket{100110}\quad\ket{R_{210}}=\ket{101001}\quad\ket{R_{211}}=\ket{100000}\nonumber,\\ \ket{R_{300}}=\ket{110111}\quad\ket{R_{301}}=\ket{000001}\quad\ket{R_{310}}=\ket{110001}\quad\ket{R_{311}}=\ket{000111}\nonumber,\\ \ket{R_{400}}=\ket{111011}\quad\ket{R_{401}}=\ket{110010}\quad\ket{R_{410}}=\ket{000010}\quad\ket{R_{411}}=\ket{001011}\nonumber,\\ \ket{R_{500}}=\ket{111101}\quad\ket{R_{501}}=\ket{110100}\quad\ket{R_{510}}=\ket{000100}\quad\ket{R_{511}}=\ket{001101}\nonumber,\\ \ket{R_{600}}=\ket{111110}\quad\ket{R_{601}}=\ket{001000}\quad\ket{R_{610}}=\ket{111000}\quad\ket{R_{611}}=\ket{001110}\nonumber.\end{eqnarray} It is worth noticing that the vectors given above are 28 basis vectors out of 64. The rest of the basis vectors can be found in the following manner: first we should add vectors similar to (\ref{basis00})-(\ref{basis11}), but with a minus instead of a plus. For the rest of the basis' elements there are several ways. For example, we can add the rest of the vectors of the computational basis with three excitations and, we can arrange the vectors of the computational basis with two and four excitations not considered in (\ref{basis00})-(\ref{basis11}), in a similar way as in (\ref{basis00})-(\ref{basis11}) and consider not only the addition, but the substraction too. However, to list one or several of the complete bases lies outside the scope of this paper. In order to make the syndrome measurement more transparent, let us introduce the following unitary transformation, that changes the basis given above to the computational one \begin{equation}\hat{S}=\sum_{k=0}^{15}\sum_{i,j=0}^1\ket{ij\textrm{Bin}(k)}\bra{R_{kij}},\end{equation} where the $\textrm{Bin}(k)$ is the binary representation of the $k=0,\ldots,15$ for the last four qubits. The error syndrome detection is completed when we measure the last four (physical) qubits, i.e., $\bin{k}$. This allows us to detect possible errors in a way that preserves any superposition in the two first qubits. When measuring $\bin{k}$, if the outcome is, for example $0000$ (see Table \ref{SynTable}), we know that no qubit was damped. If, on the other hand, the results are $0001,0010,0011,0100,0101$ or $0110$ we conclude that there is a jump in the qubit one to six, respectively. It is important to notice that this assumption is only true until a certain ``degree'' $\sim\gamma^2$, because more than one-qubit damping could lead the same state as one qubit damping. Similarly, some two or four-qubit damping could lead to the state $\ket{000000}$, which is part of the no-damping basis state. \begin{center} \begin{table} \caption[10pt]{Syndrome outcomes and corresponding recovery operations. \label{SynTable}} {\begin{tabular}{|c|c|} \hline Syndrome & Recovery operation\\ \hline $\textrm{Bin}(0)=0000$&${\hat{R}}_0=\sum_{i,j=0}^1\ket{ij}_L\bra{ij\textrm{Bin}(0)}$\\ $\textrm{Bin}(1)=0001$&${\hat{R}}_1=\sum_{i,j=0}^1\ket{ij}_L\bra{ij\textrm{Bin}(1)}$\\ $\textrm{Bin}(2)=0010$&${\hat{R}}_2=\sum_{i,j=0}^1\ket{ij}_L\bra{ij\textrm{Bin}(2)}$\\ $\textrm{Bin}(3)=0011$&${\hat{R}}_3=\sum_{i,j=0}^1\ket{ij}_L\bra{ij\textrm{Bin}(3)}$\\ $\textrm{Bin}(4)=0100$&${\hat{R}}_4=\sum_{i,j=0}^1\ket{ij}_L\bra{ij\textrm{Bin}(4)}$\\ $\textrm{Bin}(5)=0101$&${\hat{R}}_5=\sum_{i,j=0}^1\ket{ij}_L\bra{ij\textrm{Bin}(5)}$\\ $\textrm{Bin}(6)=0110$&${\hat{R}}_6=\sum_{i,j=0}^1\ket{ij}_L\bra{ij\textrm{Bin}(6)}$\\ $\textrm{Bin}(7)=0111$& Measure the first two qubits\\ $\textrm{Bin}(8)=1000$& and project onto $\hat{I}_L/4$\\ $\textrm{Bin}(9)=1001$&\\ $\textrm{Bin}(10)=1010$&\\ $\textrm{Bin}(11)=1011$&\\ $\textrm{Bin}(12)=1100$&\\ $\textrm{Bin}(13)=1101$&\\ $\textrm{Bin}(14)=1110$&\\ $\textrm{Bin}(15)=1111$&\\\hline \end{tabular}} \end{table} \end{center} Once that we have measured the syndrome we should apply a proper recovery operation, summarized in Table \ref{SynTable}. If the outcome is ``$\textrm{Bin}(k)$'' for $k=0,\ldots,6$ we will preserve the superpositions between the two first qubits by applying the operation $\hat{R}_k$. If we measure ``$\textrm{Bin}(k)$'' for $k=7,\ldots,15$ we know that more than one qubit was damped, and we cannot correct the state, but we will try to keep some features of the original state by projecting the system to the state $\hat{I}_L/4$ for reasons of symmetry, where $\hat{I}_L=\ket{00}_L\bra{00}_L+\ket{01}_L\bra{01}_L+\ket{10}_L\bra{10}_L+\ket{11}_L\bra{11}_L$ is the identity in the codeword space. In this scheme, it is always possible to reconstruct a two-qubit density matrix from the six-qubit recovered state, because the state after recovering is in the four-dimensional codeword sub-space, which enables us to compute (measure) not only the exact fidelity but the pairwise entanglement by means of the standard concurrence. \section{Entanglement Sudden Death and Quantum Error Correction} As was pointed out in the Introduction, ESD is present in different scenarios for some bipartite states. In this context the so-called $X$-states \cite{esd1} have been widely studied \cite{xstates}. They have the property that when interacting with independent environments, the corresponding composite density matrix preserves the $X$ form in the computational basis. They also merit studying because they include the Bell-states and the Werner-states. Furthermore, among them one can find states that are subject to ESD and states that are not, when evolving under dissipation. Particularly, we will work with the Bell-like states of the form \begin{eqnarray}\label{phi} \ket{\phi_0}=\cos\alpha\ket{11}+e^{i\beta}\sin\alpha\ket{00},\\ \label{psi} \ket{\psi_0}=\cos\alpha\ket{10}+e^{i\beta}\sin\alpha\ket{01}.\end{eqnarray} It is worth noticing that state $\ket{\phi_0}$ will succumb to ESD for $\alpha$ such that $\vert\tan\alpha\vert<1$, this was experimentally shown \cite{almeida}, meanwhile $\ket{\psi_0}$ will not become disentangled for any finite amount of dissipation. In order to show the effect of different mappings of the codeword space we will also consider the separable states \begin{eqnarray}\label{zeta}\ket{\zeta_0}&=&\cos\alpha\ket{01}+e^{i\beta}\sin\alpha\ket{00},\\ \label{xi} \ket{\xi_0}&=&\cos\alpha\ket{11}+e^{i\beta}\sin\alpha\ket{10}.\end{eqnarray} We will try to protect the information carried by the states (\ref{phi})-(\ref{xi}) by encoding these states using the codewords given by the set of equations (\ref{basis00})-(\ref{basis11}), so that the corresponding logical states are \begin{eqnarray}\label{phil}\ket{\phi_0}_L&=&\cos\alpha\ket{11}_L+e^{i\beta}\sin\alpha\ket{00}_L,\\ \label{psil}\ket{\psi_0}_L&=&\cos\alpha\ket{10}_L+e^{i\beta}\sin\alpha\ket{01}_L,\\ \label{zetal}\ket{\zeta_0}_L&=&\cos\alpha\ket{01}_L+e^{i\beta}\sin\alpha\ket{00}_L,\\ \label{xil}\ket{\xi_0}_L&=&\cos\alpha\ket{11}_L+e^{i\beta}\sin\alpha\ket{10}_L.\end{eqnarray} The standard way to quantify the efficiency of quantum information processes and protocols, particularly QEC is the fidelity, that for an initially pure state is just the overlap between the original/desirable state $\ket{\varphi_0}$ and the obtained one $\hat{\rho}$ after dissipation and application of the considered protocol. That is ${\cal F}=\bra{\varphi_0}\hat{\rho}\ket{\varphi_0}$. In this case the fidelities between the initial encoded states (\ref{phil})-(\ref{zetal}) and the corresponding ones after amplitude damping modeled by the Kraus operators and QEC are given by \begin{eqnarray} {\cal F}_{\phi}={\cal F}_{\zeta}&=&1-\frac{\gamma^2}{4}\left(21-9\cos2\alpha-\sin^22\alpha\cos^2\beta\right)+O(\gamma^3)\label{fidphi}\\ {\cal F}_{\psi}={\cal F}_{\xi}&=&1-\frac{\gamma^2}{4}\left(12-\sin^22\alpha\cos^2\beta\right)+O(\gamma^3).\label{fidpsi} \end{eqnarray} Here, the {\it non-locality} and the codeword labeling is manifested in the fact that the fidelities when considering the initial states $\ket{\phi_0}_L$ and $\ket{\zeta_0}_L$ are exactly the same (not only the first terms in the series expansion given above). This also happens for the initial states $\ket{\psi_0}_L$ and $\ket{\xi_0}_L$. This is because when considering independent damping $\gamma$, there is no difference in the behavior between the physical states in (\ref{basis01})-(\ref{basis11}), while if we use the $[4,1]\times[4,1]$ code (encoded with the same labels in each qubit), the unique logical states that will behave in the same way, which is a desirable feature, are $\ket{01}_L$ and $\ket{10}_L$. Disregarding this, let us study the effect of {\it non-local} coding for a pair of qubits and the possibility of obtaining better results, particularly protection against ESD, than with the {\it local} coding considered in \cite{QEC}. In order to see the effect of the proposed QEC protocol, we should compare the fidelities given above with the fidelities given by the initial (not encoded) corresponding states, and the fidelities obtained with the $[4,1]\times[4,1]$ code for a pair of qubits which are given by\cite{QEC} \begin{eqnarray} {\cal F}_{\phi_1}&=&1-\frac{\gamma^2}{2}\left(8-3\cos2\alpha-2\cos^22\alpha\right)+O(\gamma^3),\label{fidphi1}\\ {\cal F}_{\zeta_1}&=&1-\frac{\gamma^2}{4}\left(15-3\cos2\alpha-2\sin^22\alpha\cos^2\beta\right)+O(\gamma^3),\label{fidzeta1}\\ {\cal F}_{\psi_1}&=&1-\gamma^2\left(4-\cos^22\alpha\right)+O(\gamma^3),\label{fidpsi1}\\ {\cal F}_{\xi_1}&=&1-\frac{\gamma^2}{4}\left(9-3\cos2\alpha-2\sin^22\alpha\cos^2\beta\right)+O(\gamma^3),\label{fidxi1}\end{eqnarray} where we use the subscript ``1'' to indicate the use of the $[4,1]\times[4,1]$ in both the fidelities and in the (coded) states ($\ket{\varphi}_{L_1}$). For the uncoded states (\ref{phi})-(\ref{xi}) we will similarly label the fidelities with the subscript ``0'', and these fidelities are given by \begin{eqnarray} {\cal F}_{\phi_0}&=&1-2\gamma\cos^2\alpha+\gamma^2\cos^2\alpha,\label{fidphi0}\\ {\cal F}_{\zeta_0}&=&1-\gamma\cos^4\alpha-\frac{\gamma^2}{16}\sin^22\alpha+O(\gamma^3),\label{fidzeta0}\\ {\cal F}_{\psi_0}&=&1-\gamma,\label{fidpsi0}\\ {\cal F}_{\xi_0}&=&1-\frac{\gamma}{2}\left(5+2\cos2\alpha+\cos^22\alpha\right) +\frac{\gamma^2}{8}\cos^2\alpha\left(3+5\cos2\alpha\right)+O(\gamma^3),\label{fidxi0}\end{eqnarray} From Equations (\ref{fidphi})-(\ref{fidxi1}) is easy to see that with the $[6,2]$ code, the loss of the fidelity is of the same order as for the $[4,1]\times[4,1]$ code, in both cases decreases as $\sim\gamma^2$, while the decrease is linear for the uncoded states (\ref{phi})-(\ref{xi}). A first-sight difference between the $[6,2]$ code and the $[4,1]\times[4,1]$ is that the fidelity of the entangled states in the first case depends on the parameter $\beta$ already in the second term in the series expansion for the Bell-like states, meanwhile for the second case this dependence is weak \cite{QEC}. Since the fidelity for the states $\ket{\phi}_L$ and $\ket{\zeta}_L$ (using the $[6,2]$ code) is the same, let us analyze its behavior and compare it with the fidelities ${\cal F}_{\phi_1}$ and ${\cal F}_{\zeta_1}$. Even if ${\cal F}_{\phi}$ and ${\cal F}_{\zeta_1}$ depend on the parameter $\beta$ already in the term of the order of $\gamma^2$, the differences introduced by this parameter (which reach their maximum when $\alpha=\pi/4$) are not very significant compared with the ones introduced by $\alpha$, and we will disregard this parameter, setting it to zero, in the following. In Fig. \ref{Figfidphi} we compare the (exact) fidelities under dissipation for the states $\ket{\phi}$ and $\ket{\zeta}$ uncoded, and for both codes. In (a) it is shown that when $\alpha=\pi/6$ the fidelities do not differ too much between the states and codes. This difference is increased as $\alpha$ becomes larger, and around $\alpha=\pi/4$, Fig. \ref{Figfidphi}(b), the separable state $\ket{\zeta}_{L_1}$ achieves a higher fidelity than the Bell-state $\ket{\phi}$, and the fidelity for this state is roughly the same for both codes. The difference between ${\cal F}_{\phi}$ and ${\cal F}_{\zeta_1}$ remains more or less equal until $\alpha=\pi/2$, but ${\cal F}_{\phi_1}\rightarrow{\cal F}_{\zeta_1}$ as $\alpha\rightarrow\pi/2$. This is because the state $\ket{\zeta}_{L_1}$ has a high probability amplitude for the physical state $\ket{0000}$, which is not evolving (see the codewords in \cite{QEC}). \begin{center} \begin{figure} \includegraphics[width=0.45\textwidth]{FidPhiA6.eps}(a) \includegraphics[width=0.45\textwidth]{FidPhiA4.eps}(b) \caption[10pt]{Plot of the fidelities against the damping parameter $\gamma$, for the state $\vert\phi\rangle$, without QEC (dot-dashed), after recovering for the $[4,1]\times[4,1]$ code (dotted) and the code $[6,2]$ (continuous), and for the separable state $\vert\zeta\rangle$, without QEC (dot-dot-dashed), after recovering for the $[4,1]\times[4,1]$ code (dashed) and the code $[6,2]$ (continuous), when (a) $\alpha=\pi/6$, $\beta=0$ and (b) $\alpha=\pi/4$, $\beta=0$.}\label{Figfidphi} \end{figure} \end{center} Nevertheless, in order to see how the QEC protocol is working we should compare not only the results obtained with the different codes, but with the fidelities without making use of any QEC. The uncoded state $\ket{\zeta_0}$, given by Eq. (\ref{zeta}), after damping will have such a high fidelity that both codes are practically useless (see Fig. \ref{Figfidphi}(b)). However, in this paper we are concerned about preventing ESD, and for the Bell-like state $\ket{\phi}$ QEC seems to work fine in terms of the fidelities, and we can say that for moderately large damping parameter ($\gamma\leq0.3$) the fidelity obtained by {\it local} coding will be higher or equal (in around $\alpha=\pi/4$) than the one obtained by {\it non-local} coding. As was pointed out in \cite{QEC}, the fidelity is insufficient to characterize the remaining entanglement. Therefore we will study the concurrence as well. The analytic expressions of the concurrence are cumbersome and therefore we only present the exact, numerical plots shown in Fig. \ref{FigConphi}, where the concurrences are almost the same for values of $\gamma$ where it is worth to apply QEC. It is important to notice that even for {\it non-local} QEC, the protocol will introduce ESD when is not present in the uncoded state. For example, it is known that for the Bell-like states (\ref{phi}) the entanglement disappears under dissipation only when $\gamma=1$ if $|\cos\alpha|\leq|\sin\alpha|$.\cite{almeida} However, in Fig. \ref{FigConphi}(b) is easy to see that this is not true when we use either of the QEC protocols. Moreover, we do not find any significative improvement by applying the $[6,2]$ code when comparing with the results presented in \cite{QEC} for this states if we consider both, the fidelity and the concurrence. \begin{center} \begin{figure} \includegraphics[width=0.45\textwidth]{ConPhiA6.eps}(a) \includegraphics[width=0.45\textwidth]{ConPhiA4.eps}(b) \caption[10pt]{Plot of the concurrences against the damping parameter $\gamma$, for the state $\vert\phi\rangle$, without QEC (dot-dashed), after recovering for the $[4,1]\times[4,1]$ code (dotted) and the code $[6,2]$ (continuous), when (a) $\alpha=\pi/6$, $\beta=0$ and (b) $\alpha=\pi/4$, $\beta=0$.}\label{FigConphi} \end{figure} \end{center} As a final step we will do a similar analysis for the states $\ket{\psi}$ and $\ket{\xi}$, and in this case the differences between the fidelities are more notable, ${\cal F}_{\xi_1}$ being the higher and ${\cal F}_{\psi_1}$ being the lower in general, meanwhile for the $[6,2]$ code the fidelity ${\cal F}_{\psi}={\cal F}_{\xi}$ is between them. As $\alpha\rightarrow\pi/2$ the fidelities are roughly the same with the value, up to second order of $\gamma$, given by $1-3\gamma^2$, being in this case the one for {\it non-local} coding a little higher than the other two. In Fig. \ref{figfidpsi} we plot the exact fidelities for this states for $\beta=0$ and (a) $\alpha=\pi/4$, (b) $\alpha=\pi/6$. We can conclude that, for this particular Bell-like state, encoding both qubits together leads to a significative higher fidelity than encoding separately for certain values of $\alpha$ and $\beta$, this in contrast with the previous example. In summary, we can say that in terms of the fidelity the {\it non-local} QEC protocol presented here leads to better results in this case. \begin{center} \begin{figure} \includegraphics[width=0.45\textwidth]{FidPsiA6.eps}(a) \includegraphics[width=0.45\textwidth]{FidPsiA4.eps}(b) \caption[10pt]{Plot of the fidelities against the damping parameter $\gamma$, for the state $\vert\psi\rangle$, without QEC (dot-dashed), after recovering for the $[4,1]\times[4,1]$ code (dotted) and the code $[6,2]$ (continuous), and for the separable state$\vert\xi\rangle$, without QEC (dot-dot-dashed), after recovering for the $[4,1]\times[4,1]$ code (dashed) and the code $[6,2]$ (continuous), when (a) $\alpha=\pi/6$, $\beta=0$ and (b) $\alpha=\pi/4$, $\beta=0$. }\label{figfidpsi} \end{figure} \end{center} Now let us complement our study by means of the concurrence plotted in Fig. \ref{FigConpsi}. By this figure we see that the {\it non-local} encoding will also leads to better results for the concurrence. Nevertheless, as it was pointed out in our previous work \cite{QEC} the differences between the codes is more quantitative than qualitative, since for large values of $\gamma$ the coded state $\ket{\psi}$ loses its entanglement in a finite time, while the uncoded state loses it asymptotically, which means that this QEC protocol will also introduce ESD to states that originally (uncoded) do not succumb to it. An explanation for this behavior is given in \cite{terra}. The asymptotic state under dissipation for the two uncoded states is the vacuum state $\ket{00}\bra{00}$ which is a pure separable state. Therefore it lies on the border between the sets of inseparable and separable states. The asymptotic (logical) state obtained using QEC corresponds to the vacuum state after applying the correction protocol, and with the two protocols considered these will be separable mixed states. These states lie inside the set of separable states. Hence, a continuous evolution from an initial entangled state ($\gamma = 0$) to the asymptotic state ($\gamma = 1$) must hence cross the border between inseparable to separable states for a finite amount of loss. However, for small values of $\gamma$ both the fidelity and concurrence achieved by {\it non-local} coding are higher than when using {\it local} coding for this Bell-like state. \begin{center} \begin{figure} \includegraphics[width=0.45\textwidth]{ConPsiA6.eps}(a) \includegraphics[width=0.45\textwidth]{ConPsiA4.eps}(b) \caption[10pt]{Plot of the concurrences against the damping parameter $\gamma$, for the state $\vert\psi\rangle$, without QEC (dot-dashed), after recovering for the $[4,1]\times[4,1]$ code (dotted) and the code $[6,2]$ (continuous), when (a) $\alpha=\pi/6$, $\beta=0$ and (b) $\alpha=\pi/4$, $\beta=0$.}\label{FigConpsi} \end{figure} \end{center} \section{Conclusions} With some examples we have studied the possibility of protecting two-qubit Bell-like states making use of a {\it non-local} code, we also have compared the results with previous obtained with some {\it local} coding \cite{QEC}. We can conclude that for the Bell-like states $\ket{\phi}$ the {\it non-local} QEC protocol presented here does not lead to significant better results in terms of either, the fidelity and the concurrence, while for the Bell-like states labeled by $\ket{\psi}$ the results obtained for {\it non-local} coding are significantly better in comparison with {\it local} coding. However, this comparison is not completely fair because the {\it local} code, used here, is employing eight qubits in total and is able to correct some two-qubit errors (when each error is in different logical qubit) whereas the {\it non-local} code, due to its shorter length, can only correct a single error. However, as was pointed out in \cite{QEC}, the implementation of a $[6,2]$ code requires creation of entanglement over the same physical distances as the entangled qubits exist over. Yet, qualitatively the results are the same, that is, QEC can delay ESD but it can also cause it for states that, uncoded, are not disentangled in a finite time. This behavior can be understood in terms of the asymptotic state formalism \cite{terra}, which suggest that, in general, QEC will produce ESD in the logical qubits for any entangled state. \section*{Acknowledgments} The authors thank Dr. Marcelo Terra Cunha for insightful comments. This work was supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), the Swedish Research Council (VR), and the Swedish Foundation for Strategic Research (SSF).

proofpile-arXiv_069-12277

\section{Introduction} Vision-language alignment learning for video-text retrieval becomes an emerging requirement with the increasing of videos and short videos uploaded online, and has attracted great attention in recent years. In the research field, it returns the most relevant videos for a given text query to facilitate large-scale videos searching and management. In the recommendation field, it recommends the most relevant text queries for the video that the user is watching to prompt more searching and browsing. Although a lot of recent works \cite{zhu2020actbert, gabeur2020multi, dzabraev2021mdmmt, lei2021less, liu2021hit, luo2021clip4clip, fang2021clip2video, cheng2021improving, chen2021mind, cao2022visual} have made remarkable progress, the cross-modal alignment between video and text remains a challenging task. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{model_show_new_0802} \captionsetup{font={small}} \caption{The multi-modal information of a video can be extracted and converted to explicit tags to motivate the video-text alignment. As shown, with multi-modal tags, the joint video embedding is closer to caption embedding than pure visual embedding.} \label{fig1} \end{figure} Most of the existing video-text retrieval models use multi-layer transformers to learn generic representations from massive video-text pairs, which can be roughly divided into two categories. The first category uses only frame features to transfer the knowledge of image-text pretrained model to video-text retrieval task without fully exploring the multi-modal information of videos \cite{lei2021less, luo2021clip4clip, fang2021clip2video, cheng2021improving, cao2022visual}. A representative method is CLIP4Clip \cite{luo2021clip4clip}, which utilizes the knowledge of the CLIP (Contrastive Language-Image Pretraining) \cite{radford2021learning} model to visually encode multi-frame information as an overall representation of the video. The model achieves good performance on multiple benchmarks, demonstrating the effectiveness of the knowledge transferring. The second category is to jointly encode the information of various modalities in the video, such as objects, actions, scenes, audio, etc. \cite{gabeur2020multi, dzabraev2021mdmmt, wang2021t2vlad, hao2021multi}. A typical representative method is MMT (Multi-Modal Transformer) \cite{gabeur2020multi}, which uses multiple pretrained experts to extract the embeddings of different modalities of the video, and builds a multi-modal transformer for feature fusion. The performance of this model demonstrates the advantage of using multi-modal information over using only single visual information of the video. However, most of these methods simply fuse multi-modal features in a brute force manner without explicit guidance, which increases the difficulty of the learning process. In this paper, we introduce a novel approach which not only utilizes image-text pretrained model for knowledge transfer, but also fully exploits the multi-modal information of video in an explicit manner to guide the visual-language alignment. A set of pretrained experts are utilized to extract the features from a diversity of modalities of videos, including object, person, scene, motion, and audio. Object and person focus on instance features or local features of the video; scene focuses on background or global features of the video. The motion features reveal the behavior or action happening in the video, which extract temporal information from consecutive frames. The audio provides supplementary descriptions of the video, which may not appear in visual features. The most challenging part of the proposed approach is how to fuse the features of different modalities effectively and efficiently. In this work, we utilize tagging as the bridge, whose advantages lie in the following two aspects. First, embeddings from various modalities generated by different experts are not compatible from each other; while tagging generates unified representations for different modalities. Second, multi-modal tags can be treated as anchor points to make the visual-language alignment learn the local feature, the global feature, the temporal feature, and other supplementary features comprehensively, which eases the learning process significantly. For example, in Fig. \ref{fig1}, through object detector, we can identify objects appearing in the video, and label them as {\it bowl}, {\it bottle}, and {\it cup}; the people in the video can also be precisely identified as {\it woman}. Through image classification on multiple frames, we can obtain the scene tag, i.e., {\it kitchen}, and some supplementary tags, like {\it apron}. Through action recognition, we can recognize the ongoing behavior in the video and get the motion tag, i.e., {\it cooking}. Further, we can convert the audio of the video to text by Automatic Speech Recognition (ASR), and extract the keywords from it as audio tags, i.e., {\it marinating a chicken}, {\it washed}, and {\it cleaned}. The above tags provide abundant information from multi-modality of the video, which bridge the gap between vision and language, making the alignment learning more efficiently and precisely. Specifically, we construct a TABLE (TAgging Before aLignmEnt) network, which utilizes a visual encoder to transfer the knowledge of image-text pretrained model to extract the frame features, a tag encoder to extract information of multi-modal tags, and a cross-modal encoder to jointly learn multi-frame visual features and multi-modal tags features of the video. The multi-modal tags are used as anchors to guide the visual-text alignment. The temporal information between frames is also learned by the cross encoder. Further, we form a triplet input consisting of [vision, tag, text] to perform two auxiliary tasks, namely Video Text Matching (VTM) and Masked Language Modeling (MLM). The VTM task treats the vision and tag information as a whole, and trains the model to identify whether it matches the text query. We perform in-batch hard negative mining by the contrastive similarity distribution. The objective of the MLM task is to recover the randomly masked words according to the triplet of [vision, tag, unmasked text]. To summarize, the contributions of this work lie in four-fold: \begin{itemize} \item We propose a novel method for video-text retrieval, which not only transfers image-text knowledge, but also fully exploits the multi-modal information of video, including object, person, scene, motion and audio. \item We integrate multi-modal information by an explicit and unified method, tagging, and use them as anchors for guiding visual-text alignment. \item We build a TABLE network to jointly encode multi-frame visual features and multi-modal tag information. And we introduce VTM and MLM as auxiliary supervisions to strengthen video-text interaction. \item The proposed TABLE model achieves SOTA performance on several video-text retrieval benchmarks, including MSR-VTT, MSVD, LSMDC and DiDeMo. \end{itemize} \begin{figure*}[h] \centering \includegraphics[width=0.7\linewidth]{model_structure_v4} \captionsetup{font={small}} \caption{Overall framework of our proposed TABLE (TAgging Before aLignmEnt) model.} \label{fig2} \end{figure*} \section{Related Work} \subsection{Video representation learning} A lot of approaches have been proposed for video representation learning, which can be classified into convolution-based methods and transformer-based methods. Many previous works \cite{tran2015learning, xie2018rethinking, feichtenhofer2019slowfast} employed 2D or 3D convolutional network to encode spatial and temporal information of the videos. Recently, the Vision Transformer (ViT) was proposed and attained excellent results on image classification compared to convolutional methods. Since then, many recent works applied ViT to encode the visual features of videos, such as ViViT \cite{arnab2021vivit} and TimeSformer \cite{bertasius2021space}, which generally decoupled the spatial and temporal information by designing two-stage transformer structures. Most of the above approaches exploited temporal information by improving image representation, however, the abundant multi-modal information of videos were not fully utilized. Our work performs temporal encoding with cross-modal interaction and exploits comprehensive information from multiple modalities, which are extracted by various pretrained experts, including object detection \cite{ultralytics2020yolov5}, action recognition \cite{xie2018rethinking}, speech recognition \cite{stewart2013robust}, etc. \subsection{Video-Text Retrieval} Recent video-text retrieval approaches can be divided into two types according to whether the multi-modal information of videos is utilized or not. The first type simply adopted the visual features for video-text alignment, where CLIP-based methods \cite{luo2021clip4clip, fang2021clip2video, cheng2021improving, cao2022visual} show obvious advantages in recent years. CLIP4Clip \cite{luo2021clip4clip} firstly transferred the knowledge of large-scale image-text pretraining to the task of video-text retrieval with fine-tuning. Based on it, CLIP2VIDEO \cite{fang2021clip2video} proposed a temporal different block to capture video’s motion feature, and a temporal alignment block to re-align the tokens of video clips and phrases. The second type used multi-modal information contained in videos for enhancing video-text alignment \cite{gabeur2020multi, dzabraev2021mdmmt, wang2021t2vlad, hao2021multi}. MMT \cite{gabeur2020multi} exploited multi-modal information extracted by seven pretrained experts but only fused them in a brute force manner without explicit guidance. MDMMT \cite{dzabraev2021mdmmt} emphasized the importance of the motion information on the basis of MMT. Wang et al. \cite{wang2021t2vlad} and Hao et al. \cite{hao2021multi} performed local alignment between multi-modal features and text features, but the gap between the embeddings of different modalities is too huge to bridge. To solve this problem, we propose to integrate multi-modal information by an explicit method, i.e., tagging. Tagging generates unified representations to erase the gap between different modalities, and can be treated as anchors to guide the visual-language alignment more explicitly. Some image-text retrieval methods \cite{zhen2019deep, qian2021dual} also used tag to boost performance, but our work is different in many ways. Firstly, these work only used the unimodal tag of image, while our work fully extracts the multi-modal tags of video, which is more comprehensive. Secondly, these work used classification loss as extra supervision, while our method does not directly use tag as training target, but regards it as anchor for video-text alignment. \section{Methodology} Given a set of videos and texts, our goal is to learn accurate representations and calculate their similarities. The video (or text) candidates are then ranked by their similarities with the query text (or video) in the field of text-to-video (or video-to-text) retrieval. To achieve this goal, video and text need to be aligned in a joint embedding space, which is difficult due to the independence of the two feature extraction processes. Therefore, we propose to fully employ the multi-modal tags of the video and utilizes them as anchors to explicitly motivate the semantic alignment between visual and text. Specifically, we construct a TABLE (TAgging Before aLignmEnt) network to jointly encode multi-frame visual information and multi-modal tag information, meanwhile the temporal information is also modeled. Moreover, we incorporate additional Video Text Matching (VTM) and Masked Language Modeling (MLM) losses. The training target of VTM is to judge whether the text matches the fusion of visual and tag features. The objective of MLM is recovering the masked text word according to video, tag and the context information. As shown in Fig. \ref{fig2}, our model consists of a visual encoder, a tag encoder, a text encoder, a tag-guiding (\textbf{TG}) cross-modal encoder and a joint cross-modal encoder. The two cross-modal encoders share paramteters with each other and performs multi-modal information integration. \subsection{Multi-modal Tag Mining} In this paper, multiple pretrained experts are used to extract related tag information of individual modality. Firstly, we adopt the yolov5s \cite{ultralytics2020yolov5} model pretrained on the COCO \cite{lin2014microsoft} dataset for video object detection. Main objects appearing in multiple frames with high confidence score are selected with object tags \bm{$t_{obj}$}. In particular, we choose the detection model pretrained on Open Image dataset \cite{kuznetsova2020open} to distinguish person into man, woman, boy and girl, obtaining specific person tags \bm{$t_{per}$}. Secondly, we employ the COCO pretrained ViT-B-16 \cite{dosovitskiy2020image} model to perform image classification of video frames, where predicted categories with high confidence score are regarded as the scene tags \bm{$t_{sce}$}. Object and person tags focus on instance or local features, while scene tags put attention to global information of video frames. Thirdly, we adopt the S3D \cite{xie2018rethinking} network pretrained on Kinetics-400 dataset \cite{carreira2017quo} to obtain the motion tags \bm{$t_{mot}$}, which is important for video understanding. Furthermore, we apply the Automatic Speech Recognition (ASR) API of iFLYTEK to get video transcripts and then employ the KeyBert \cite{grootendorst2020keybert} to extract transcript keywords as audio tags \bm{$t_{aud}$}. The audio modality usually provides complementary information which is always not included in visual features. At last, we concatenate the above tags and obtain the whole multi-modal tag information of the video, \bm{$t_{mul}}=[\bm{t_{obj}}, \bm{t_{per}}, \bm{t_{sce}}, \bm{t_{mot}}, \bm{t_{aud}$}]. \subsection{Visual and Text Encoder} \textbf{Visual Encoder.} We uniformly sample $N$ frames to form a sequence as the video representation, $f_i=\left\{f_i^1, f_i^2, ..., f_i^N\right\}$. Inspired by the successful transferring of the image-text pretraining knowledge to video-text learning, we directly adopt the ViT model in CLIP \cite{radford2021learning} to extract visual features, which processes non-overlapping image patches of individual frame and linearly project them to 1D token sequence. The patch tokens are then passed through a 12-layer transformer to realize self-attention procedure. The [CLS] embedding is projected into a normalized lower-dimensional embedding space with a MLP layer to obtain the overall representation of each frame. For simple presentation, we omit the process of taking [CLS] token of ViT in Fig. \ref{fig2}, and thus the output of the visual encoder is the sequential representations of multiple frames, \bm{$v_i}=\left\{\bm{v_i^1}, \bm{v_i^2}, ..., \bm{v_i^N}\right\}$. \textbf{Text Encoder.} We apply the BERT-base encoder in CLIP \cite{radford2021learning} to obtain tag and caption embedding, which is a 12-layer transformer with 512 dimensional width and 8 attention heads. The transformer output of tag and caption can be expressed as \bm{$t_i}=\left\{\bm{t_i^1}, \bm{t_i^2}, ..., \bm{t_i^K}\right\}$ and \bm{$c_i}=\left\{\bm{c_i^1}, \bm{c_i^2}, ..., \bm{c_i^M}\right\}$ respectively, where $K$ and $M$ represents the token length of tag and caption. The [EOS] embedding of the last layer is passed through a linear projection layer and regarded as the overall representation of the text (e.g., tag and caption). Particularly, the parameters of transformer blocks are shared between tag and caption encoders, except the linear projection layer. \subsection{Tag-guiding cross-modal encoder} To bridge the visual-text semantic gap for alignment, we introduce multi-modal tags as anchors. As shown in Fig. \ref{fig2}, we concatenate multi-frame visual features and multi-modal tag embeddings as the input of the cross-modal encoder. Specifically, the concatenated input can be represented by $\left\{\bm{v_i^1}, \bm{v_i^2}, ..., \bm{v_i^N}; \bm{t_i^e}\right\}$, where $\bm{t_i^e}$ represents the [EOS] embedding of the tag encoder output and the overall information of the multi-modal tags. The \textbf{TG} cross-modal encoder is composed of 4-layer transformer with 512 dimensional width and 8 cross-attention heads. The position embedding and transformer parameters are initialized by the weights of first four layers of the CLIP’s text encoder. The \textbf{TG} cross-modal encoder conducts deep fusion of multi-frame visual features with multi-modal tag embeddings, meanwhile modelling the temporal information of the video. The fused output can be expressed as $\left\{\bm{g_i^1}, \bm{g_i^2}, ..., \bm{g_i^N}; \bm{g_i^{e}}\right\}$. We then apply an average pooling layer to acquire the overall representation, which can be formulated as $\bm{g_i^o}=\bm{\rho}(\left\{\bm{g_i^1}, \bm{g_i^2}, ..., \bm{g_i^N}; \bm{g_i^{e}}\right\})$. In addition, to fully exploit the pretrained image-text knowledge, we adopt residual connection between the pooled visual features and the overall cross-modal representation, which can be represented by $\bm{\hat{g}_i^o}=\lambda\bm{g_i^o}+ \bm{\rho}(\left\{\bm{v_i^1}, \bm{v_i^2}, ..., \bm{v_i^N}\right\})$. $\lambda$ is a learnable weight factor. The output of text encoder can be denoted by \bm{$c_i}=\left\{\bm{c_i^1}, \bm{c_i^2}, ..., \bm{c_i^N}\right\}$, where the [EOS] embedding $\bm{c_i^e}$ is selected as the overall representation of caption. We define a similarity function as $s(V_i, T_i) = \phi(\bm{\hat{g}_i^o})^T \psi(\bm{c_i^e}) $, where $\phi(\cdot)$ and $\psi(\cdot)$ are the linear projection functions that project visual and caption embedding into the shared semantic space. Then, we construct in-batch similarity between video and text: \begin{equation} \label{tv_loss} L_{t2v} = -\frac{1}{B}\sum_{i}^{B}log\frac{exp(s(V_i, T_i)/\tau)}{\sum_{j=1}^{B}exp(s(V_j, T_i)/\tau)}, \end{equation} \begin{equation} \label{vt_loss} L_{v2t} = -\frac{1}{B}\sum_{i}^{B}log\frac{exp(s(V_i, T_i)/\tau)}{\sum_{j=1}^{B}exp(s(V_i, T_j)/\tau)}, \end{equation} \begin{equation} \label{total_loss} \centering L_{con}=\frac{1}{2}(L_{t2v}+L_{v2t}), \end{equation} where $\tau$ is a learnable temperature parameter, $B$ is batch size, $L_{t2v}$ and $L_{v2t}$ represents the text-to-video loss and video-to-text loss. $L_{con}$ is the total contrastive loss. \subsection{Video Text Matching and Masked Language Modeling} \label{s3.4} In addition to contrastive loss for visual-text alignment, Video Text Matching (\textbf{VTM}) and Masked Language Modeling (\textbf{MLM}) are incorporated to motivate fine-grained interaction between video and text. \textbf{VTM} determines whether the pair of [visual, tag] and [text] is matched, where [visual, tag] is the \textbf{TG} cross-modal fusion output of multi-frame visual features and multi-modal tag information, [text] is the overall representation of text. This pair is then fed into a joint cross-modal encoder, which shares parameters with the \textbf{TG} cross-modal encoder. The first output token embedding is regarded as the joint video-text representation $V_i$, and passed through a fully connected layer for binary prediction (match or not match with the text representation $T_i$). Defining the precition as $p^{vtm}(V_i,T_i)$, the VTM loss is: \begin{equation} \label{vtm_loss} L_{vtm} = - \frac{1}{O}\sum_{i}^{O}\sum_{t=0}^{1}y_{it}^{vtm}log_2(p_t^{vtm}(V_i, T_i)), \end{equation} where $y_{it}^{vtm}$is a sign function which has the value of 1 if $t$=1 else 0 if $t=0$. $(V_i, T_i)$ is a positive pair when $t$=1, and a negative pair otherwise. $p_t^{vtm}(V_i, T_i)$ denotes the prediction probability of $t$. $O$ is the total number of video-text pairs for VTM task, which is composed of positive and negative pairs. We perform hard negative mining strategy when constructing the triplet sample. For each [visual, tag] in mini-batch, we sample a negative [text] according to the in-batch similarity matrix calculated in Equation \ref{tv_loss}. Likewise, for each [text], we sample a negative [visual, tag] in the mini-batch. \begin{table*}[ht] \centering \resizebox{0.8\textwidth}{0.1\textheight}{ \begin{tabular}{ccc|ccccc|ccccc} \toprule \ & \ & \ & \multicolumn{5}{c|}{Text-to-Video (T2V)} & \multicolumn{5}{c}{Video-to-Text (V2T)}\\ \midrule Type & Method & Pretrained Dataset & R@1 & R@5 & R@10 & MdR & MnR & R@1 & R@5 & R@ 10 & MdR & MnR \\ \midrule NO-CLIP & JSFusion \cite{yu2018joint} &- & 10.2 & 31.2 & 43.2 & 13.0 & - & - & - & - & - & - \\ NO-CLIP & HT-Pretrained \cite{miech2019howto100m} &HowTo100M \cite{miech2019howto100m} & 14.9 & 40.2 & 52.8 & 9.0 & - & - & - & - & - & - \\ NO-CLIP & CE \cite{liu2019use} &- & 20.9 & 48.8 & 62.4 & 6.0 & 28.2 & 20.6 & 50.3 & 64.0 & 5.3 & 25.1 \\ NO-CLIP & MMT-Pretrained \cite{gabeur2020multi} &HowTo100M & 26.6 & 57.1 & 69.6 & 4.0 & 24.0 & 27.0 & 57.5 & 69.7 & 3.7 & 21.3 \\ NO-CLIP & TACo \cite{yang2021taco} &HowTo100M & 26.7 & 54.5 & 68.2 & 4.0 & - & - & - & - & - & - \\ NO-CLIP & SUPPORT-SET \cite{patrick2020support} &- & 27.4 & 56.3 & 67.7 & 3.0 & - & 26.6 & 55.1 & 67.5 & 3.0 & - \\ NO-CLIP & FROZEN \cite{bain2021frozen} &WebVid-2M\cite{bain2021frozen} & 31.0 & 59.5 & 70.5 & 3.0 & - & - & - & - & - & - \\ NO-CLIP & HIT-Pretrained \cite{liu2021hit} &HowTo100M & 30.7 & 60.9 & 73.2 & 2.6 & - & 32.1 & 62.7 & 74.1 & 3.0 & - \\ \hline\hline CLIP-based & CLIP \cite{portillo2021straightforward} &WIT \cite{radford2021learning} & 31.2 & 53.7 & 64.2 & 4.0 & - & 27.2 & 51.7 & 62.6 & 5.0 & - \\ CLIP-based & MDMMT \cite{dzabraev2021mdmmt} &WIT+HowTo100M & 38.9 & 69.0 & 79.7 & 2.0 & 16.5 & - & - & - & - & - \\ CLIP-based & CLIP4Clip-meanP \cite{luo2021clip4clip} &WIT & 43.1 & 70.4 & 80.8 & 2.0 & 16.2 & 43.1 & 70.5 & 81.2 & 2.0 & 12.4 \\ CLIP-based & CLIP4Clip-seqTransf \cite{luo2021clip4clip} &WIT & 44.5 & 71.4 & 81.6 & 2.0 & 15.3 & 42.7 & 70.9 & 80.6 & 2.0 & 11.6 \\ CLIP-based & VCM \cite{cao2022visual} &WIT & 43.8 & 71.0 & 80.9 & 2.0 & 14.3 & 45.1 & 72.3 & 82.3 & 2.0 & 10.7 \\ CLIP-based & CLIP2VIDEO \cite{fang2021clip2video} &WIT & 45.6 & 72.6 & 81.7 & 2.0 & 14.6 & 43.5 & 72.3 & 82.1 & 2.0 & 10.2 \\ CLIP-based & CAMoE \cite{cheng2021improving} &WIT & 44.6 & 72.6 & 81.8 & 2.0 & 13.3 & 45.1 & 72.4 & 83.1 & 2.0 & 10.0 \\ CLIP-based & \textbf{TABLE (ours)} &WIT & 47.1 & 74.3 & 82.9 & 2.0 & 13.4 & 47.2 & 74.2 & 84.2 & 2.0 & 11.0 \\ \hline\hline CLIP-based & CAMoE$\ast$ \cite{cheng2021improving} &WIT & 47.3 & 74.2 & 84.5 & 2.0 & 11.9 & 49.1 & 74.3 & 84.3 & 2.0 & 9.9 \\ CLIP-based & CAMoE-online$\ast$ \cite{cheng2021improving} &WIT & 48.8 & 75.6 & \bf{85.3} & 2.0 & 12.4 & 50.3 & 74.6 & 83.8 & 2.0 & \bf{9.9} \\ CLIP-based & \textbf{TABLE$\ast$ (ours)} &WIT & \bf{52.3} & \bf{78.4} & 85.2 & \bf{1.0} & \bf{11.4} & \bf{51.8} & \bf{77.5} & \bf{85.1} & \bf{1.0} & 10.0 \\ \bottomrule \end{tabular}} \captionsetup{font={small}} \caption{Retrieval results on the MSR-VTT dataset. $\ast$ indicates methods with inference strategy (as in subsequent tables).} \label{tab2} \end{table*} \textbf{MLM} predicts the masked word based on the [visual, tag] representations and the unmasked context, which is also a classification task. To be specific, we randomly mask the text tokens with a probability of 15\%, and the replacement is the [MASK] token with an 80\% probability, a random token with a 10\% probability, original token with a 10\% probability. Supposing $\hat{T}_i$ denotes the masked text and $p^{mlm}(V_i,\hat{T}_i)$ denotes the prediction result of the masked word, the loss function of MLM can be expressed as: \begin{equation} \label{mlm_loss} L_{mlm} = -\frac{1}{Q} \sum_{i}^{Q}\sum_{v=1}^{V}y_{iv}^{mlm}log_2(p_v^{mlm}(V_i,\hat{T}_i)), \end{equation} where $y_{iv}^{mlm}$ is a sign function which has the value of 1 if the masked word of sample $i$ is $v$. $V$ is the vocabulary size and $Q$ is the total number of video-text pairs for MTM task. Finally, the overall loss function of the TABLE model is: \begin{equation} \label{model_loss} \centering L=L_{con} + L_{vtm} + L_{mlm} \end{equation} \subsection{Inference Strategy} \label{s3.5} During the inference stage, the joint cross-modal encoder with VTM and MLM loss is discarded, and the similarity confidences between video and text are only calculated by the \textbf{TG} encoder and the text encoder. To further improve the performance, we also adopt the inference strategy proposed in CAMoE \cite{cheng2021improving}, which conduct SoftMax operation to revise the similarity matrix. \section{Experiments} \subsection{Datasets} \subsubsection{MSR-VTT} \cite{xu2016msr} dataset is widely studied in video-text retrieval task, which contains 10,000 videos of 10-32 seconds, each corresponding to 20 captions. We follow the data split in previous method \cite{gabeur2020multi}, i.e., 9000 videos for training and 1,000 videos for testing. \subsubsection{MSVD} \cite{chen2011collecting} dataset contains 1,970 videos, each video has nearly 40 titles. Train, validation and test sets have 1,200, 100 and 670 videos respectively. Following previous works \cite{luo2021clip4clip}, we report the result on testing set with multiple captions per video. The limited training data makes the learning on this dataset challenging. \subsubsection{LSMDC} \cite{rohrbach2015long} dataset consists of 118,081 short videos clips extracted from 202 movies. Similar to previous work \cite{luo2021clip4clip}, we validate the performance on the test set containing 1,000 videos. \subsubsection{DiDeMo} \cite{anne2017localizing} dataset involves about 10,000 videos and each video has about 4 annotated sentences. Following previous works \cite{luo2021clip4clip}, we conduct video-paragraph retrieval task on the test split (with 1,004 videos) by concatenating all sentences of each video into a single query. The challenge of this dataset lies in the alignment of long videos and long texts. \subsection{Metrics} We evaluate the model performance with standard retrieval metrics, i.e., Recall at rank K (R@K, K=1,5,10), Median Rank (MdR) and Mean Rank (MnR). R@K represents the proportion of the ground-truth result included in the top-K recalled results. MdR and MnR represent the median and mean rank of correct results respectively. Therefore, higher R@K, lower MdR and MnR indicates better performance. \subsection{Implementation Details} The visual encoder, tag and text encoder of our TABLE model are initialized by the pretrained CLIP (ViT-B/32) \cite{radford2021learning} model. The parameters of tag and text encoder are shared in the transformer blocks but individually learned in the linear projection layer. The position embedding and the blocks of the cross-modal encoders are initialized by the first four layers of the CLIP’s text encoder. The initial learning rate is 1e-7 for visual and text encoder, and 1e-4 for cross-modal encoders. The TABLE model is trained for 5 epoches by Adam optimizer and warmup scheduler. The batch size is 128, except on DiDeMo is 48. For MSR-VTT, MSVD and LSMDC, the max token length of caption and tag, and the frame length are set to 32, 32, 12, respectively. For DiDeMo, the value of the above parameters are 64, 32, 32, respectively. To prevent the tag information from being overwhelmed by a larger number of video frames on DiDeMo, we adopt all token output of the tag encoder instead of the [EOS] output as the input of \textbf{TG} cross-modal encoder. We adopt 5 pretrained experts (object, person, scene, motion and audio) to extract multi-modal tags on MSR-VTT and DiDeMo dataset, and 4 experts (without audio) on MSVD and LSMDC dataset. \begin{table}[t] \centering \resizebox{!}{16mm}{ \begin{tabular}{ccc|ccccc} \toprule \ & \ & \ & \multicolumn{5}{c}{Text-to-Video (T2V)} \\ \midrule Type & Method & Pre-D & R@1 & R@5 & R@10 & MdR & MnR \\ \midrule NO-CLIP & VSE \cite{kiros2014unifying} &- & 12.3 & 30.1 & 42.3 & 14.0 & - \\ NO-CLIP & CE &- & 19.8 & 49.0 & 63.8 & 6.0 & 23.1 \\ NO-CLIP & SSML \cite{amrani2020noise} &HowTo100M & 20.3 & 49.0 & 63.3 & 6.0 & - \\ NO-CLIP & SUPPORT-SET &- & 28.4 & 60.0 & 72.9 & 4.0 & - \\ NO-CLIP & FROZEN &WebVid-2M & 33.7 & 64.7 & 76.3 & 3.0 & - \\ \hline\hline CLIP-based & CLIP &WIT& 37.0 & 64.1 & 73.8 & 3.0 & - \\ CLIP-based & CLIP4Clip-seqTransf &WIT& 45.2 & 75.5 & 84.3 & 2.0 & 10.3 \\ CLIP-based & CLIP4Clip-meanP &WIT & 46.2 & 76.1 & 84.6 & 2.0 & 10.0 \\ CLIP-based & CLIP2VIDEO &WIT & 47.0 & 76.8 & 85.9 & 2.0 & 9.6 \\ CLIP-based & CAMoE &WIT & 46.9 & 76.1 & 85.5 & - & 9.8 \\ CLIP-based & \textbf{TABLE} &WIT& 49.9 & 79.3 & 87.4 & 2.0 & \bf{9.1} \\ \hline\hline CLIP-based & CAMoE$\ast$ &WIT & 49.8 & 79.2 & 87.0 & - & 9.4 \\ CLIP-based & \textbf{TABLE$\ast$ (ours)} &WIT & \bf{52.3} & \bf{80.5} & \bf{87.9} & \bf{1.0} & 9.8 \\ \bottomrule \end{tabular}} \captionsetup{font={small}} \caption{Retrieval results on the MSVD dataset. Pre-D represents Pretrained Dataset (as in subsequent tables).} \label{tab3} \end{table} \begin{table}[t] \centering \resizebox{!}{19mm}{ \begin{tabular}{ccc|ccccc} \toprule \ & \ & \ & \multicolumn{5}{c}{Text-to-Video (T2V)} \\ \midrule Type & Method & Pre-D & R@1 & R@5 & R@10 & MdR & MnR \\ \midrule NO-CLIP & JSFusion &- & 9.1 & 21.2 &34.1 & 36.0 & - \\ NO-CLIP & CE &- & 11.2 & 26.9 & 34.8 & 25.3 & 96.8 \\ NO-CLIP & HIT-Pretrained &HowTo100M & 14.0 & 31.2 & 41.6 & 18.5 & - \\ NO-CLIP & FROZEN &WebVid-2M & 15.0 & 30.8 & 39.8 & 20.0 & - \\ \hline\hline CLIP-based & CLIP &WIT& 11.3 & 22.7 & 29.2 & 56.5 & - \\ CLIP-based & MDMMT &WIT & 18.8 & 38.5 & 47.9 & 12.3 & 58.0 \\ CLIP-based & CLIP4Clip-meanP &WIT & 20.7 &38.9 & 47.2 & 13.0 & 65.3 \\ CLIP-based & CLIP4Clip-seqTransf &WIT & 22.6 & 41.0 & 49.1 & 11.0 & 61.0 \\ CLIP-based & CAMoE &WIT & 22.5 & 42.6 & 50.9 & - & 56.5 \\ CLIP-based & \textbf{TABLE (ours)} &WIT & 24.3 & 44.9 & 53.7 & 8.0 & 52.7 \\ \hline\hline CLIP-based & CAMoE$\ast$ &WIT & 25.9 & \bf{46.1} & 53.7 & - & 54.4 \\ CLIP-based & \textbf{TABLE$\ast$ (ours)} &WIT & \bf{26.2} & 45.9 & \bf{55.0} & \bf{7.0} & \bf{51.0} \\ \bottomrule \end{tabular}} \captionsetup{font={small}} \caption{Retrieval results on the LSMDC dataset.} \label{tab4} \end{table} \begin{table}[t] \centering \resizebox{!}{15mm}{ \begin{tabular}{ccc|ccccc} \toprule \ & \ & \ & \multicolumn{5}{c}{Text-to-Video (T2V)} \\ \midrule Type & Method & Pre-D & R@1 & R@5 & R@10 & MdR & MnR \\ \midrule NO-CLIP & S2VT \cite{venugopalan2014translating} &HowTo100M & 11.9 & 33.6 & - & 13.0 & - \\ NO-CLIP & FSE \cite{zhang2018cross} &Kinetics \cite{kay2017kinetics} & 13.9 & 36.0 & - & 11.0 & - \\ NO-CLIP & CE &- & 16.1 & 41.1 & - & 8.3 & 43.7 \\ NO-CLIP & TT-CE \cite{croitoru2021teachtext} &- & 21.6 & 48.6 & 62.9 & 6.0 & - \\ NO-CLIP & FROZEN &WebVid-2M & 34.6 & 65.0 & 74.7 & 3.0 & - \\ \hline\hline CLIP-based & ClipBERT \cite{lei2021less} &COCO \cite{chen2015microsoft} + & 20.4 & 48.0 & 60.8 & 6.0 & - \\ \ & \ & VGC \cite{krishna2017visual} & \ & \ &\ & \ & \ \\ CLIP-based & CLIP4Clip-seqTransf &WIT & 42.8 & 68.5 & 79.2 & 2.0 & 18.9 \\ CLIP-based & CLIP4Clip-meanP &WIT & 43.4 & 70.2 & 80.6 & 2.0 & 17.5 \\ CLIP-based & \textbf{TABLE (ours)} &WIT & 47.9 & 74.0 & 82.1 & 2.0 & \bf{14.3} \\ \hline\hline CLIP-based & CAMoE$\ast$ &WIT & 43.8 & 71.4 & 79.9 & 2.0 & 16.3 \\ CLIP-based & \textbf{TABLE$\ast$ (ours)} &WIT & \bf{49.1} & \bf{75.6} & \bf{82.9} & \bf{2.0} & 14.8 \\ \bottomrule \end{tabular}} \captionsetup{font={small}} \caption{Retrieval results on the DiDeMo dataset.} \label{tab5} \end{table} \begin{table}[t] \centering \resizebox{0.47\textwidth}{0.033\textheight}{ \begin{tabular}{c|ccccc|ccccc|ccccc} \toprule \ & \multicolumn{5}{c|}{Tag Modalities} & \multicolumn{5}{c|}{Text-to-Video (T2V)} & \multicolumn{5}{c}{Video-to-Text (V2T)}\\ \midrule Method & Object & Person & Motion & Scene & Audio & R@1 & R@5 & R@10 & MdR & MnR & R@1 & R@5 & R@ 10 & MdR & MnR \\ \midrule Baseline & - & - \ & - \ & - \ & - & 45.2 & 72.0 & 81.3 & 2.0 & 14.3 & 45.4 & 73.8 & 83.8 & 2.0 & 11.5 \\ TABLE & \ding{52} & - & - & - & - & 45.6 & 72.8 & 81.7 & 2.0 & 13.8 & 45.3 & 73.6 & 83.9 & 2.0 & 11.1 \\ TABLE & \ding{52} & \ding{52} & - & - & - & 45.6 & 72.8& 81.9 & 2.0 & 14.1 & 45.9 & 74.9 & 83.4 & 2.0 & 11.1 \\ TABLE & \ding{52} & \ding{52} & \ding{52} & - & - & 46.2 & 73.0 & 82.2 & 2.0 & \bf{13.8} & 45.9 & 74.7 & 83.8 & 2.0 & \bf{11.0} \\ TABLE & \ding{52} & \ding{52} & \ding{52} & \ding{52} & - & 46.5 & 73.4 & 82.2 & 2.0 & 13.9 & 45.6 & 74.8 & 83.6 & 2.0 & 11.2 \\ TABLE & \ding{52} & \ding{52} & \ding{52} & \ding{52} & \ding{52} & \bf{46.8} & \bf{73.5} & \bf{82.2} & \bf{2.0} & 13.9 & \bf{45.9} & \bf{75.1} & \bf{84.1} & \bf{2.0} & 11.1 \\ \bottomrule \end{tabular}} \captionsetup{font={small}} \caption{Effects of multi-modal tags on MSR-VTT dataset.} \label{tab1} \end{table} \begin{table}[b] \centering \resizebox{0.47\textwidth}{0.035\textheight}{ \begin{tabular}{c|cc|ccccc|ccccc} \toprule \ & \multicolumn{2}{c|}{Task} & \multicolumn{5}{c|}{Text-to-Video (T2V)} & \multicolumn{5}{c}{Video-to-Text (V2T)}\\ \midrule Method & VTM & MLM & R@1 & R@5 & R@10 & MdR & MnR & R@1 & R@5 & R@ 10 & MdR & MnR \\ \midrule Baseline & - & - \ & 45.2 & 72.0 & 81.3 & 2.0 & 14.3 & 45.4 & 73.8 & 83.8 & 2.0 & 11.5 \\ TABLE & - & - & 46.8 & 73.5 & 82.2 & 2.0 & 13.9 & 45.9 & 75.1 & 84.1 & 2.0 & 11.1 \\ TABLE & \ding{52} & - & 46.7 & 73.9 & 82.1 & 2.0 & 13.6 & 46.6 & 75.0 & \bf{84.7} & 2.0 & \bf{10.6} \\ TABLE & \ding{52} & \ding{52} & \bf{47.1} & \bf{74.3} & \bf{82.9} & \bf{2.0} & \bf{13.4} &\bf{47.2} & 74.2 & 84.2 & \bf{2.0} & 11.0 \\ \bottomrule \end{tabular}} \captionsetup{font={small}} \caption{Effects of VTM and MLM on MSR-VTT dataset.} \label{tab6} \end{table} \subsection{Comparisons to the State of The Art} In this subsection, we compare our model with state-of-the-art methods on four representative benchmarks. We divide existing methods into CLIP-based and NO-CLIP, where CLIP-based methods usually have better performance. As shown in Table \ref{tab2}, without inference strategy, our TABLE model achieves great performance on the MSR-VTT dataset. For example, our method surpasses CAMoE \cite{cheng2021improving} by a large margin of 2.5 and 2.1 on R@1 in text-to-video (T2V) and video-to-text (V2T) task respectively. It not only demonstrates the importance of multi-modal information, but also proves the superiority of our tagging method for enabling cross-modal alignment. Furthermore, with the inference strategy, our method achieves SOTA performance, i.e., 52.3 R@1 in T2V and 51.8 R@1 in V2T task. For the MSVD dataset, as illustrated in Table \ref{tab3}, our TABLE method surpasses CAMoE by 3.0 on R@1 in T2V task without inference strategy, and finally achieves SOTA performance of 52.3 R@1. We believe that on scale-limited datasets, the explicit guidance of multi-modal tags is more important for visual-text alignment. LSMDC dataset contains the most videos and each video matches only one caption, and thus most methods perform pooly. But our method still improves 1.8 on R@1 and achieves SOTA performance (26.2 R@1) in T2V task, as shown in Table \ref{tab4}. Finally, for the video-paragraph retrieval task on DiDeMo, our method obtains 5.3 R@1 gains over SOTA method CAMoE, as presented in Table \ref{tab5}. The alignment of long videos and long text meets greater challenge compared to short video-text pairs, and thus the introduction of multi-modal tags shows greater advantages on this dataset. In conclusion, our TABLE model proves to be effective and superior by achieving SOTA performance on various benchmarks. \subsection{Abalation Studies} \subsubsection{Effects of Multi-Modal Tags.} The multi-modal information of the video are fully exploited in this work by an explicit way of tagging. In this subsection, we delve into the impact of multi-modal tags on model's performance. As shown in Table \ref{tab1}, our backbone model achieves R@1 of 45.2 for text-to-video (T2V) retrieval, which is already a comparable performance on the MSR-VTT dataset. The R@1 of T2V is improved to 45.6 after the object tags are introduced by our TABLE model, and the R@1 of V2T is further improved to 45.9 with object and person tags. It proves that the visual-text alignment benefits from additional tag information. The object and person tags contain local information, which can guide the local attention of visual features. But in video retrieval task, the motion modality plays an important role. With motion tags, the performance is obviously improved, i.e., the T2V R@1 reaches 46.2. It is further promoted to 46.5 after introducing scene tags, which usually describe the background information. Finally, the T2V R@1 is improved to 46.8 with audio tags, which contain information not included in visual features, which is helpful for the retrieval of videos like news report. To sum up, introducing multi-modal tags is beneficial for video-text retrieval task. \begin{figure*}[ht] \centering \includegraphics[width=0.8\linewidth]{visual_show_total_v2} \captionsetup{font={small}} \caption{Visualization results of TABLE. We visualize the spatial attention in each frame, where brighter regions represent higher spatial attention. And, we visualize the temporal attention through the color bar below the frame list, where redder color represents higher attention. } \label{fig3} \end{figure*} \subsubsection{Effects of VTM and MLM.} VTM and MLM are proposed as auxiliary supervisions with a joint cross-modal encoder, both for facilitating the video-text interaction. As shown in Table \ref{tab6}, with the help of VTM, the video-to-text retrieval performance is improved from 45.9 to 46.6 at R@1. Using both VTM and MLM, the model performance is further promoted. The R@1 of text-to-video reaches 47.1, and the R@1 of video-to-text achieves 47.2. Although VTM is a binary classification task, the model can only make well judgment on hard negative with profound understanding of video and text content. MLM is a more complex task which requires fine-grained alignment between the visual features and text tokens for inferring masked words. In general, these two tasks strengthen the interaction between video and text, and thus effectively improves the model performance. \subsection{Visual Analysis} \label{s4.4} To clearly reveal the anchor role of multi-modal tags, we present some visualization results of the \textbf{TG} cross-modal encoder and the joint cross-modal encoder using the Attention Rollout \cite{abnar2020quantifying} method. In Fig. \ref{fig3}, we simultaneously visualize the spatial attention and temporal attention, where the temporal attention is calculated by the cross attention between frame features and the overall representation of multi-modal tags (or specific word token of the caption). Higher temporal attention indicates higher weights in representing the whole video. In Fig. \ref{fig3}(a) and Fig. \ref{fig3}(b), the multi-modal tags are treated as an entirety for calculating cross attention with visual features. As shown, the cross attention focuses on some frames that highly correlated with multi-modal tags. For example, in Fig. \ref{fig3}(a), the 1st frame (highly correlated with ``man'' and ``ballplayer'') and the 10th frame (highly correlated with ``catching or throwing baseball'' and ``baseball glove'') are highlighted, while the 2rd-4th frames describing distant view of the stadium are suppressed. For each frame, important spatial regions like head, body, baseball glove are getting more attention than unimportant backgrounds. Although the multi-modal tags might contain noise, e.g., ``bird'' in Fig. \ref{fig3}(a), they can still guide the model to pay more attention to key frames and key regions. We also find that the motion modality show dominant role in the cross-modal attention. For example, in Fig. \ref{fig3}(b), the 9th and 12th frames describing standard ``dancing ballet'' are highlighted, while the first few frames depicting the scene and person are suppressed. To sum up, the \textbf{TG} cross-modal encoder is capable of selecting key frames and key regions from redundant visual features with the guidance of multi-modal tags, which is beneficial for accurate video-text retrieval. To further investigate the capabilities of our cross-modal encoder, we also visualize the cross attention in the joint cross-modal encoder. In detail, here we visualize the cross attention between visual features and specific word token of the caption rather than the overall representation. For example, in Fig. \ref{fig3}(c), given the ``car'' word, the 1st frame is highlighted as it gives the whole picture of the car. The spatial attention also focuses on the car body. The last few frames are judged to be less correlated as they just focus on seat belts. In Fig. \ref{fig3}(d), the 10th and 11th frames are determined as the most relevant frames to the word ``monkey'' because they describe the monkey with a close view. The other frames describing about ``people'' or ``swiming'' are suppressed. Thus, it can be concluded that the proposed joint cross-modal encoder is capable of modeling the fine-grained correlation between video and a single word of text, which is inherited from VTM and MLM tasks. Moreover, it reveals the strong capability of \textbf{TG} cross-modal encoder as the parameters are shared between it and the joint cross-modal encoder. \section{Conclusion} In this paper, we propose to integrate multi-modal tags as anchors to motivate the video-text alignment. Specifically, we construct a TABLE model to jointly encode multi-frame visual features and multi-modal information (including object, person, scene, motion and audio). To further enhance the video-text interaction, we introduce VTM and MLM tasks on the triplet input of [visual, tag, text] as auxiliary supervisions. Our proposed TABLE model achieves SOTA performance on various benchmarks, including MSR-VTT, MSVD, LSMDC and DiDeMo, which indicates the superiority of introducing multi-modal tags as anchors for video-text retrieval task. The ablation studies and visualization results further reveal the anchor role of multi-modal tags in guiding visual-text alignment. \vspace{.2em}

proofpile-arXiv_069-12434

\section{Introduction} Since the demonstration that spontaneous breaking of Lorentz symmetry is allowed in the high energy context of string theories \cite{Samuel}, Lorentz-violating theories have been extensively studied in\ diverse low energy systems and used as an effective probe to test the limits of Lorentz covariance with direct consequences on the Planck scale physics. The great majority of such investigations take place in the framework of the Extended Standard Model (ESM), conceived by Colladay and Kosteletck\'{y} \cite{Colladay}\ as a extension of the minimal Standard Model of the fundamental interactions. The ESM admits Lorentz and CPT violation in all sectors of interactions by incorporating tensor terms (generated possibly as vacuum expectation values of a more fundamental theory) that account for such a breaking. Actually, the ESM model sets out as an effective model that keeps unaffected the $SU(3)\times SU(2)\times U(1)$ gauge structure of the underlying fundamental theory while it breaks Lorentz symmetry at the particle frame. There is a large amount of work in the literature \cite{Coleman, Electro} searching for experimental and observational evidence of Lorentz breaking, in an attempt to put lower bound on the parameters. Topological defects are actually one of the most active research branches of theoretical physics, with profound connections with condensed matter physics, gravitation and field theory \cite{topol}. In particular, kink solutions are the simplest topological defects which appear after symmetry breaking of models with one or more scalar fields. Explicit solutions have been worked in several models involving one or more scalar fields \cite{bnrt}, including gravity \cite{grav}, nested defects \cite{nest}, junctions \cite{junct} or connections with other areas of physics \cite{inter}. The influence of Lorentz violation on kink solutions was investigated in ref. \cite{bbm}, where the first-order framework was applied for some class of models finding explicit solutions in some interesting cases. Here we pursue this problem studying another class of models with some new results including explicit solutions, phase space analysis, stability analysis and perturbative solutions. This paper is outlined as follows. In Sec. II, we present our model with Lorentz violation and we apply the Bogomol'nyi method to find first-order equations that minimize the energy and solve the equations of motion. The behavior of numerical solutions are studied in the phase space for several values of the Lorentz breaking parameter. In Sec. III we prove that our solutions are stable and we find that the model support traveling waves. Further, in Sec. IV we find explicit perturbative solutions for the fields. Our final conclusions are presented in Sec. V. \section{The model and first-order equations} We consider a class of models with two real scalar fields \begin{equation} \mathcal{L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi+\frac{1}% {2}\partial^{\mu}\chi\partial_{\mu}\chi+k^{\alpha\beta}\partial_{\alpha}% \phi\partial_{\beta}\chi-V(\phi,\chi), \label{L1}% \end{equation} where the coupling tensor $k^{\mu\nu}$ characterizes Lorentz violation. The tensor coefficient is a $2\times2$ matrix, \[ k^{\alpha\beta}=\left[ \begin{array} [c]{cc}% \beta\ & \alpha\\ \alpha & \beta \end{array} \right] , \] and is an element that couples the scalar fields, introducing Lorentz violation and inducing an asymmetry in the propagation of the traveling waves present in these models. The Euler-Lagrange equations can be written as \begin{subequations} \begin{align} \ddot{\phi}-\phi^{\prime\prime}+\beta(\ddot{\chi}+\chi^{\prime\prime}% )+2\alpha\dot{\chi}^{\prime}+V_\phi & =0,\label{D1}\\ \ddot{\chi}-\chi^{\prime\prime}+\beta(\ddot{\phi}+\phi^{\prime\prime}% )+2\alpha\dot{\phi}^{\prime}+V_\chi & =0, \label{D2}% \end{align} \end{subequations} where $V_\phi=\partial V /\partial\phi$, $V_\chi=\partial V /\partial\chi$, and prime and dot stand for space and time derivative, respectively. In the case of static solutions, we get \begin{subequations} \begin{align} -\phi^{\prime\prime}+\beta\chi^{\prime\prime}+V_\phi & =0,\label{E1}\\ -\chi^{\prime\prime}+\beta\phi^{\prime\prime}+V_\chi & =0. \label{E2}% \end{align} \end{subequations} The energy-momentum tensor, \begin{equation} T^{\mu\nu}=\frac{1}{2}\left[ \partial^{\mu}\phi\partial^{\nu}\phi +\partial^{\mu}\chi\partial^{\nu}\chi-k^{\mu\alpha}\partial_{\alpha}% \chi\partial^{\nu}\phi+k^{\alpha\mu}\partial_{\alpha}\phi\partial^{\nu}% \chi-g^{\mu\nu}\mathcal{L}\right] , \end{equation} provides the energy density of the static solutions: \begin{equation} T^{00}=\frac{1}{2}\left[ \phi^{\prime2}+\chi^{\prime2}-2\beta\phi^{\prime }\chi^{\prime}+2V(\phi,\chi)\right] . \label{T00}% \end{equation} A useful form for representing the potential $V(\phi,\chi)$ can be attained from Eqs. (\ref{E1}) and (\ref{E2}), as we get \begin{equation} \frac{d}{dx} \biggl( -\frac12{\phi'}^2 -\frac12{\chi'}^2 + \beta\phi'\chi' + V\biggr) = 0. \end{equation} Then a potential satisfying \begin{equation} \label{V} V=\frac12{\phi'}^2 + \frac12{\chi'}^2 - \beta\phi'\chi' \end{equation} would lead to the second-order equations of motion for the fields $\phi$ and $\chi$. This shows that the gradient and potential portions of the energy do not contribute equally, the difference being proportional to the Lorentz violating coefficient $\beta$. In order to obtain first-order differential equations whose solutions are also solutions of the equations of motion we use the Bogomol'nyi approach \cite{bog}, looking at the energy density $T^{00}$ and trying to extract simple conditions that minimize the total energy. We introduce a smooth function $W=W(\phi,\chi)$ and consider the {\it ansatz} \begin{subequations}\label{A} \begin{equation} \label{A1} \phi'=W{_\phi} + s_1\chi' , \end{equation} \begin{equation} \label{A2} \chi'=W{_\chi} + s_2\phi' . \end{equation} \end{subequations} Substituting this into Eqs. (\ref{T00}) and (\ref{V}) leads to \begin{eqnarray} T^{00}&=&\frac{1}{2}( \phi^{\prime2}+\chi^{\prime2}) -\beta\phi^{\prime }\chi^{\prime} + \frac12(W_\phi+s_1\chi')^2 + \frac12(W_\chi+s_2\phi')^2 - \beta\phi^{\prime }\chi^{\prime} \nonumber \\ &=& (\phi'W_\phi + \chi'W\chi) + \phi'\chi'(s_1+s_2-2\beta) + \frac12(\phi'-W_\phi-s_1\chi')^2 + \frac12(\chi'-W_\chi-s_2\phi')^2. \nonumber \end{eqnarray} For $s_1+s_2=2\beta$ and when the fields $\phi$ and $\chi$ satisfy the {\it ansatz} given by Eq.~(\ref{A}), we have $T^{00}\equiv\epsilon=dW/dx$, and the total energy $E$ is minimized to the value \begin{equation} E=\int_{-\infty}^{\infty} \epsilon(x) dx=|\Delta W|. \end{equation} In the remaining of this section we take $s_1=s_2=\beta$. Explicitly, the {\it ansatz} given by Eq.~(\ref{A}) results in two first-order differential equations that minimize the energy \begin{subequations}\label{App} \begin{eqnarray}\label{A1p} \phi'&=& \frac{1}{1-\beta^2}(W_\phi+\beta W_\chi)\\ \label{A2p} \chi'&=& \frac{1}{1-\beta^2}(W_\chi+\beta W_\phi) \end{eqnarray} \end{subequations} Note that Eq. (\ref{V}) can be written as \begin{equation} V(\phi,\chi)=\frac1{1-\beta^2}\biggl( \frac12W_\phi^2 + \frac12W_\chi^2 + \beta W_\phi W_\chi\biggr), \end{equation} and for $\beta^2<1$, which concur with the fact that any Lorentz violating parameter should be very small, the potential is positive. The simplest solutions for the equation \eqref{App} are the homogeneous solution. These solution can be found using the $W_\phi=W_\chi=0$ that are minima of potential. \subsection{An Example} Now we will consider a particular potential. We choose the function $W$ introduced in Ref.~\cite{bnrt} \begin{equation} \label{W_bnrt} W(\phi,\chi)=\phi-\frac13\phi^3-r\phi\chi^2 \end{equation} corresponding to the potential \begin{equation} V(\phi,\chi)=\frac1{1-\beta^2}\biggl[ \frac12(1-\phi^2-r\chi^2)^2 + 2r^2\phi^2\chi^2 - 2r\beta\phi\chi (1-\phi^2-r\chi^2)\biggr] \end{equation} This potential depends explicitly on $\beta$, but the minima are $(\phi,\chi)=(0,\pm\sqrt{1/r})$ and $(\phi,\chi)=(\pm 1,0)$ and they do not depend of $\beta$. These points correspond to the following values of $W$: $P_1=(\phi,\chi)=(-1,0)\Rightarrow W_1=-2/3$; $P_2=(1,0)\Rightarrow W_2=2/3$; $P_3=(0,\sqrt{1/r})\Rightarrow W_3=0$; $P_4=(0,-\sqrt{1/r})\Rightarrow W_4=0$. In this way we can possibly have five BPS sectors connecting $P_1 \leftrightarrow P_2$, $P_1\leftrightarrow P_3$, $P_1\leftrightarrow P_4$, $P_2\leftrightarrow P_3$, $P_2\leftrightarrow P_4$ and one non-BPS sector connecting $P_3\leftrightarrow P_4$. Now we analyze Eqs.~(\ref{App}) on the $(\phi,\chi)$ plane, looking for explicit solutions. Those equations reduce, in the model considered here, to the following first-order equations for the fields: \begin{subequations}\label{PCbeta} \begin{eqnarray} \label{phibeta} \phi^\prime&=& \frac1{1-\beta^2}[1-\phi^2-r\chi^2-2\beta r\phi\chi]\\ \label{chibeta} \chi^\prime&=& \frac1{1-\beta^2}[-2 r\phi\chi+\beta(1-\phi^2-r\chi^2)] \end{eqnarray} \end{subequations} The case $\beta=0$ lead to \begin{subequations} \begin{eqnarray} \label{phibeta0} \phi^\prime &=& 1-\phi^2-r\chi^2, \\ \label{chibeta0} \chi^\prime &=& -2 r\phi\chi, \end{eqnarray} \end{subequations} which can be integrated to give the general orbit constraint \begin{equation} \phi^2=\frac{r}{2r-1}\chi^2 + C \chi^{\frac1r} +1 \end{equation} A special case is the elliptic orbit for $C=0$, with $\phi^2+\frac r{1-2r}\chi^2=1$. As this particular orbit connects the minimum energy points $P_1$ and $P_2$, it is of particular interest with explicit solutions \begin{equation} \phi(x)=\tanh(2rx),\;\;\;\;\; \chi(x)=\sqrt{\frac1r-2}\,\,\,\text{sech}(2rx) \end{equation} Another special orbit is the line $\chi=0$ ($C\to \infty$). In this case, we take $\phi(x)=\tanh(x)$. We have been unable to find explicit solutions for general nonvanishing values of $\beta$. However, numerically we could obtain orbits and the corresponding solutions $\phi(x)$ and $\chi(x)$. In Fig.~\ref{orbit}, we plot the orbit profile of the Eqs.~\eqref{PCbeta}. The green (solid) lines connect the minima $P_1$ and $P_2$. The blue (dashed) lines connect the minima $P_1$ and $P_2$ to $P_3$ and $P_4$. Note that, in contrast with the case of vanishing $\beta$, there is no solution with one constant field. \begin{figure}[h!] \includegraphics[{angle=0,width=12.0cm,height=5cm}]{rr.eps} \caption{Profile of the orbits in the $(\phi,\chi)$ plane, given by Eqs.~\eqref{PCbeta}, for $\beta=1/3$ and $r=1/3$. The orbit in black (dot-dashed line) corresponds to odd solutions, which have the symmetry $\phi(-x)=-\phi(x)$ and $\chi(-x)=-\chi(x)$.} \label{orbit} \end{figure} We plot three pair of solutions in the next three figures, and in Fig.~{\ref{phichi1}} we show that the $\phi(x)$ (left panel) and $\chi(x)$ (center panel) solutions are related to the central orbit highlighted in black in Fig.~\ref{orbit}, which connects $P1$ and $P2$. The right panel shows the energy density, and we note that there are two small energy peaks in addition to the large central peak. These solutions have energy $E_B=4/3$. In Fig.~\ref{phichi2}, we show the solutions and the respective energy density related to the orbit connecting $P1$ and $P3$. The $\phi$ solution is not a monotonic function anymore, with a minimum at $x\approx3.5$. Finally, in Fig.~\ref{phichi3}, we show the solutions that connect $P2$ and $P3$. We note that, in contrast with the $\beta=0$ case, the BPS solutions that connects $P1$ to $P3$ do not have the same form as the BPS solution that connects $P2$ to $P3$, as we can see in Fig.~\ref{phichi2} and in Fig.~\ref{phichi3}. Nevertheless, both solutions have the same energy $E_B=2/3$. \begin{figure}[h!] \includegraphics[{angle=0,width=5.4cm}]{Phi_1.eps} \includegraphics[{angle=0,width=5.4cm}]{Chi_1.eps} \includegraphics[{angle=0,width=5.4cm}]{Rho_1.eps} \caption{The $\phi(x)$ (left panel) and $\chi(x)$ (right panel) solutions and energy density (right panel), for $\beta=1/3$ and $r=1/3$. These solutions connect the points $P1$ and $P2$.} \label{phichi1} \end{figure} In the case of $\beta$ being very small, we can find approximate results. However , we will present a perturbative analysis for $\beta\ll1$ in Sect.~\ref{sect_perturb}. \section{Stability Analysis} In order to analyze the stability of the solutions, one should start with the full partial differential equations (\ref{D1}) and (\ref{D2}). For $\alpha=0$, for simplicity, we get to \begin{subequations} \label{EQ12} \begin{align} \ddot{\phi}-\phi^{\prime\prime}+\beta(\ddot{\chi}+\chi^{\prime\prime })+dV/d\phi & =0,\label{Eq1}\\ \ddot{\chi}-\chi^{\prime\prime}+\beta(\ddot{\phi}+\phi^{\prime\prime })+dV/d\chi & =0, \label{Eq2}% \end{align} \end{subequations} It is interesting to observe that the Lorentz violation does not jeopardize the linear stability of the differential equations, which turns stable the associated kinks solutions. Such demonstration is performed according to the usual way. With this purpose, we consider general fluctuations, $\eta,\xi,$ on the field configurations: \begin{equation} \phi(x,t)=\phi(x)+\eta(x,t),\;\;\;\;\chi(x,t)=\chi(x)+\xi(x,t). \end{equation} With this, the differential equations (\ref{EQ12}) take the form \begin{subequations} \begin{align} \alpha_{2}\ddot{\eta}-\alpha_{1}\eta^{\prime\prime}+V_{\varphi_{1}\varphi_{1}% }\eta+V_{\varphi_{1}\varphi_{2}}\xi & =0,\label{Dif1}\\ \alpha_{1}\ddot{\xi}-\alpha_{2}\xi^{\prime\prime}+V_{\varphi_{1}\varphi_{2}% }\eta+V_{\varphi_{2}\varphi_{2}}\xi & =0. \label{Dif2}% \end{align} \end{subequations} where $\alpha_{1}=(1-\beta),\alpha_{2}=(1+\beta)$ and \begin{equation} \varphi_{1} =\frac{1}{\sqrt{2}}(\phi+\chi),\,\,\,\,\,\,\,\,\,\,\,\varphi_{2} =\frac{1}{\sqrt{2}}(\phi-\chi), \label{T2}% \end{equation} \begin{figure}[h!] \includegraphics[{angle=0,width=5.2cm}]{Phi_2.eps} \includegraphics[{angle=0,width=5.2cm}]{Chi_2.eps} \includegraphics[{angle=0,width=5.2cm}]{Rho_2.eps} \caption{The same as in Fig.~\ref{phichi1}, for solutions that connect the points $P1$ and $P3$.} \label{phichi2} \end{figure} \begin{figure}[h!] \includegraphics[{angle=0,width=5.2cm}]{Phi_3.eps} \includegraphics[{angle=0,width=5.2cm}]{Chi_3.eps} \includegraphics[{angle=0,width=5.2cm}]{Rho_3.eps} \caption{The same as in Fig.~\ref{phichi1}, for solutions that connect $P2$ and $P3$.} \label{phichi3} \end{figure} Such system may be written in a matrix form: we use $\eta(x,t)=\eta(x)\cos(\omega t)$ and $\xi(x,t)=\xi(x)\cos(\omega t)$ to get \begin{equation} H\left[ \begin{array} [c]{c}% \eta\\ \xi \end{array} \right] =\omega^{2}\left[ \begin{array} [c]{c}% \alpha_{2}\eta\\ \alpha_{1}\xi \end{array} \right] \label{H1}% \end{equation} where \begin{equation} H=\left[ \begin{array} [c]{cc}% -\alpha_{1}\frac{d^{2}}{dx^{2}}+V_{\varphi_{1}\varphi_{1}} & V_{\varphi _{1}\varphi_{2}}\\ V_{\varphi_{1}\varphi_{2}} & -\alpha_{2}\frac{d^{2}}{dx^{2}}+V_{\varphi _{2}\varphi_{2}}% \end{array} \right] . \end{equation} Finally, it is possible to show that $H=S^{\dagger}S,$ with:% \begin{equation} S=\left[ \begin{array} [c]{cc}% -\sqrt{\alpha_{1}}\frac{d}{dx}+p_{1}W_{\varphi_{1}\varphi_{1}} & p_{2}W_{\varphi_{1}\varphi_{2}}\\ p_{1}W_{\varphi_{1}\varphi_{2}} & -\sqrt{\alpha_{2}}\frac{d}{dx}% +p_{2}W_{\varphi_{2}\varphi_{2}}% \end{array} \right] . \end{equation} with $p_{1}=a\sqrt{\alpha_{1}},$ $p_{2}=b\sqrt{\alpha_{2}}.$ Eq. (\ref{H1}) is not an eigenvalue equation but its structure is enough to demonstrate stability. Indeed, such equation may be written as $H|\Psi\rangle=\omega_{n}^{2}M|\Psi\rangle,$ where $M$ is a diagonal $2\times2$ matrix with $M_{11}=\alpha_{2}% ,M_{22}=\alpha_{1}.$ Considering the internal product $\langle\Psi |H|\Psi\rangle=\langle\Psi|S^{\dagger}S|\Psi\rangle=|S|\Psi\rangle|^{2},$ we obtain $|S|\Psi\rangle|^{2}=$ $\omega_{n}^{2}(\alpha_{2}|\eta|^{2}+\alpha _{1}|\xi|^{2}),$ which assures positivity of $\omega^{2}$, since $\alpha_{1},\alpha_{2}>0.$ The positivity of $\omega^{2}$ implies real values for $\omega$, which yields well behaved solutions for any time, meaning stability. Another study concerns the propagation of traveling waves in this model of two coupled fields. First of all, we note from Eqs.~(\ref{EQ12}) that \begin{subequations} \label{possible} \begin{align} \alpha_{2}\ddot{\varphi}_{1}-\alpha_{1}\varphi_{1}^{\prime\prime}% +dV/d\varphi_{1} & =0,\label{phi1}\\ \alpha_{1}\ddot{\varphi}_{2}-\alpha_{2}\varphi_{2}^{\prime\prime}% +dV/d\varphi_{2} & =0, \label{phi2}% \end{align} \end{subequations} If such a model entails the presence of traveling waves, then there should exist moving waves endowed with the same form as the static solutions, which satisfy $\alpha_{1}\varphi_{1}^{\prime\prime}=dV/d\varphi_{1},\alpha_{2}\varphi_{2}^{\prime\prime}=dV/d\varphi_{2}.$ In order to properly examine this issue, the fields are written in the form $\varphi_{1}=\varphi_{1}(u_{1}),\varphi_{2}=\varphi_{2}(u_{2}),$ with $u_{1}=\gamma_{1}(x-v_{1}t),u_{2}=\gamma_{2}(x-v_{2}t),$ where $\gamma_{1},\gamma_{2}$ being constants and $v_{1},$ and $v_{2}$ represent the propagation velocities. Replacing $\varphi_{1}(u_{1}),\varphi_{2}(u_{2})$ into eqs.~(\ref{possible}), one gets \begin{subequations} \begin{align} \alpha_{1}d^{2}\varphi_{1}/du^{2} & =dV/d\varphi_{1},\label{TW1}\\ \alpha_{2}d^{2}\varphi_{2}/du^{2} & =dV/d\varphi_{2},\label{TW2}% \end{align} \end{subequations} after making the following identification \begin{equation} \gamma_{1}=[1-(\alpha_{2}/\alpha_{1})v_{1}^{2}]^{-1/2},\text{ }\gamma _{2}=[1-(\alpha_{1}/\alpha_{2})v_{2}^{2}]^{-1/2}. \end{equation} The attainment of Eqs.~(\ref{TW1},\ref{TW2}) confirms that this model supports traveling waves. The phase velocity $\left(v_{ph}=\omega/k\right)$ of such waves is given by the dispersion relations coming from the free version of Eqs.~(\ref{phi1},\ref{phi2}), $-\alpha_{2}k^{2}+\alpha_{1}% \omega^{2}=0,$ $-\alpha_{1}k^{2}+\alpha_{2}\omega^{2}=0$, which implies $v_{1}=\pm\sqrt{\alpha_{2}/\alpha_{1}},$ $v_{2}=\pm\sqrt{\alpha_{1}/\alpha _{2}}.$ Considering the redefinitions (\ref{T2}), the stationary wave configurations associated with the fields $\phi,\chi$ are composed by two travelling waves of velocities $v_{1},$ $v_{2}.$ It is interesting to note that $v_{2}<1$ while $v_{1}>1$, and that this outcome prevails for the group velocity as well, with $v_{g_{2}}<1$ while $v_{g_{1}}>1.$ \section{Analytical method for perturbed solutions} \label{sect_perturb} Taking into account that Lorentz violation is a small effect, a perturbative approach is well suited to deal with it. In this sense, an analytical method may be used to construct explicit first-order solutions for the differential equations (\ref{EQ12}) constrained to the situation where the LV parameter $\beta$ is small (since its presence is addressed as a perturbation on the Lorentz symmetric case). In order to obtain first order analytical solutions, we should first define the unperturbed system $(\beta=0)$ as \begin{subequations} \begin{align} \ddot{\phi}-\phi^{\prime\prime}+dV/d\phi &=0,\\ \ddot{\chi}-\chi^{\prime\prime}+dV/d\chi &=0, \end{align} \end{subequations} which provides unperturbed solutions (labeled as $\phi_{0}$ and $\chi_{0})$, whose form depend on the adopted potential $V.$ In this section we consider $V(\phi,\chi)= V_1(\phi)+V_2(\chi),$ which makes the above equations uncoupled. In the static perturbed case, the differential equations are: \begin{subequations} \begin{align} -\phi^{\prime\prime}+\beta\chi^{\prime\prime}+dV/d\phi & =0,\label{AE1}\\ -\chi^{\prime\prime}+\beta\phi^{\prime\prime}+dV/d\chi & =0. \label{AE2}% \end{align} \end{subequations} Note that these equations are coupled only by $\beta$-dependent terms. In order to solve it, we propose the following ansatz \begin{equation} \phi(x)=\phi_{0}(x)+\beta g(x),\;\;\;\;\;\chi(x)=\chi_{0}(x)+\beta f(x), \label{Qui3B}% \end{equation} where $g$ and $f$ are functions to be determined. Replacing such ansatz in eqs.~(\ref{AE1},\ref{AE2}), it results \begin{subequations} \begin{align} \label{eqg} -g^{\prime\prime}+\chi_{0}^{\prime\prime}+gd^{2}V/d\phi^{2}|_{\phi_{0}}&=0,\\ \label{eqf} -f^{\prime\prime}+\phi_{0}^{\prime\prime}+fd^{2}V/d\chi^{2}|_{\chi_{0}}&=0, \end{align} \end{subequations} where $\phi_{0}$ and $\chi_{0}$ are the corresponding unperturbed solutions. These are second-order differential equations, whose solutions may be achieved in an analytical form for certain potentials. Aa a first example, let us consider the potential such that \begin{subequations} \begin{eqnarray} \label{Pot} V_1(\phi)=(1-\phi^{2})^{2}/2,\;\;\;\;\;V_2(\chi)=(1-\chi^{2})^{2}/2, \end{eqnarray} \end{subequations} which provides the well-known solutions, $\phi_{0}(x)=\pm\tanh(x),$ $\chi_{0}(x)=\pm\tanh(x),$ besides the homogeneous ones: $\phi_{0}(x)=\chi_{0}(x)=\pm1,0.$ In this way, Eqs.~(\ref{eqg})-(\ref{eqf}) give \begin{subequations} \begin{align} -g^{\prime\prime}+\chi_{0}^{\prime\prime}+g(-2+6\phi_{0}^{2}) & =0,\\ -f^{\prime\prime}+\phi_{0}^{\prime\prime}+f(-2+6\chi_{0}^{2}) & =0. \end{align} \end{subequations} A first situation consists in choosing $\phi_{0}(x)=\pm\tanh(x),$ $\chi_{0}(x)=\pm1,$ which leads to the system of equations \begin{subequations} \begin{align} -g^{\prime\prime}+g(6\tanh^{2}x-2) & =0,\label{g2A}\\ -f^{\prime\prime}\mp2\tanh x\operatorname*{sech}\nolimits^{2}x+4f & =0. \label{f2A}% \end{align} \end{subequations} Under the boundary conditions $(dg/dx)|_{x\to\infty}=(df/dx)|_{x\to\infty}=0,$ $g(0)=a,$ $f(0)=b,$ such system supports an exact analytical solution given as follows \begin{subequations} \begin{align} g(x)&=a\operatorname*{sech}\nolimits^{2}x,\label{S1A}\\ f(x)&=be^{-2x}\pm2[1-\ln(2\cosh x)]\sinh2x\pm2x\cosh2x\mp\tanh x(1+2\cosh2x). \label{S1B} \end{align} \end{subequations} With this, the classical solutions for the Lorentz-violating system are determined at first order in accordance with eqs.~(\ref{Qui3B}). A second case is $\phi_{0}(x)=\pm\tanh(x),$ $\chi_{0}(x)=\tanh(x)$ with the conditions $(dg/dx)|_{x\to\infty}=(df/dx)|_{x\to\infty}=0,$ $g(0)=a,$ $f(0)=b,$ which gives \begin{equation} g(x)=(a+x/2)\operatorname*{sech}\nolimits^{2}x,\;\;\;\;\;f(x)=(b\pm x/2)\operatorname*{sech}\nolimits^{2}x. \label{S2B} \end{equation} Once we have obtained first-order analytical solutions, it is also possible to carry out the corresponding first-order energy corrections for the solutions already known. The starting point is the energy density given by eq.~(\ref{T00}), written for eqs.~(\ref{Qui3B}), which implies $T^{00}=T_{0}^{00}+T_{\beta}^{00},$ where: \begin{subequations} \begin{align} T^{00} & =\frac{1}{2}\left[ \phi_{0}^{\prime2}+\chi_{0}^{\prime2}% +2V(\phi,\chi)\right] \\ T_{\beta}^{00} & =\beta\lbrack\phi_{0}^{\prime}g^{\prime}+\chi_{0}^{\prime }f^{\prime}-\phi_{0}^{\prime}\chi_{0}^{\prime}+V_{\phi}g+V_{\chi}f]. \end{align} \end{subequations} Here, $T_{0}^{00}$ is the usual energy density (without Lorentz-violation) and $T_{\beta}^{00}$ is the first-order correction due to Lorentz violation. Now, using the potentials of Eqs.~(\ref{Pot})a-(\ref{Pot})b, the energy correction takes the form \begin{equation} T_{\beta}^{00}=\beta\lbrack\phi_{0}^{\prime}g^{\prime}+\chi_{0}^{\prime }f^{\prime}-\phi_{0}^{\prime}\chi_{0}^{\prime}-2\phi_{0}(1-\phi_{0}% ^{2})g-2\chi_{0}(1-\chi_{0}^{2})f]. \end{equation} A first case whose energy is to be analyzed is the one in which $\phi_{0}(x)=\pm\tanh(x),$ $\chi_{0}(x)=\pm1.$ In this case, the corresponding solutions are given by eqs.~(\ref{S1A},\ref{S1B}), which lead to the following result \begin{subequations} \begin{align} T_{0}^{00} & =\operatorname*{sech}\nolimits^{2}x,\\ T_{\beta}^{00} & =\mp\beta\lbrack6a\tanh x\operatorname*{sech}% \nolimits^{4}x]. \end{align} \end{subequations} The energy associated with this density, $E=\int_{-\infty}^{\infty}T_{\beta}^{00}dx,$ is zero, since one has to integrate an odd function. Hence, the energy is simply the one of the usual case (without LV): $E=\int_{-\infty}^{\infty}T_{0}^{00}dx=4/3.$ This means that the energy does not change up to first-order in $\beta$. The second case is given by $\phi_{0}(x)=\pm\tanh(x)$ and $\chi_{0}(x)=\tanh(x),$ whose solutions are given by eqs.~(\ref{S2A},\ref{S2B}). In this case, one gets \begin{subequations} \begin{align} T_{0}^{00}&=2\operatorname*{sech}\nolimits^{2}x,\\ T_{\beta}^{00}&=\mp\frac{4\beta}{3}[x\tanh x\operatorname*{sech} \nolimits^{4}x]. \end{align} \end{subequations} The associated energy correction is then equal to $\mp4\beta/3,$ with total energy being given by $E=4(2\mp\beta)/3.$ Here the energy receives first-order correction in $\beta$. \section{Ending comments} In this work, we have studied the effects of a Lorentz-violating tensor term on the topological defects associated with models described by two real scalar fields. We applied the Bogomol'nyi approach to successfully attain first-order differential equations whose solutions minimize the energy and solve the equations of motion. This simplification is extremely useful to study numerically the solutions in phase space. We studied a model where the potential describing the two scalar fields presents nonlinear couplings between the fields, with a rich structure of minima. As we have shown, for general values of the parameter which implements the Lorentz breaking, a numerical investigation was performed, and we have found numerical solutions connecting minima of the potential. For general models, we performed a stability analysis to show that the solutions of the equations of motion are well behaved in time. Also, we have shown that the models support traveling waves in general. Since we where not able to find explicit solutions for general values of $\beta$, we studied the case with $\beta\ll1$, which concur with the fact that the Lorentz breaking should be very small. In this case, we proposed an ansatz for the perturbed solutions, finding explicit solutions for the special case where the potential is described by the two fields $\phi$ and $\chi$, that do not interact with each other. \begin{acknowledgments} The authors express their gratitude to CAPES, CNPq, FAPEMA and PADCT-MCT-CNPq for financial support. \end{acknowledgments}

proofpile-arXiv_069-12613

\section{Introduction} There is a wide range of colloidal particles with platelet-like shape, including materials such as gibbsite \cite{vanderkooij.f.m:2000.a} and certain clays including montmorillonite, laponite and hydrotalcite \cite{dijkstra.m:1995.a,dijkstra.m:1997.a, pizzey.c:2004.a, vanduijneveldt.js:2005.a, leach.esh:2005.a,mourad.m.c.d.:2008.b}. Clays are some of the most abundant minerals on the Earth's surface and are used as pharmaceuticals, cosmetics and catalysts. There is much current interest in the use of platelets in nanocomposite materials, e.g. the nematic phase of sterically stabilised gibbsite platelets may be used as a template for gibbsite-polymer nanocomposites with nematic order \cite{maurice_thesis}. Interest in platelet dispersions is also present in geophysics \cite{maitland.gc:2000.a}, biomedicine \cite{mason.tg:2002.a} and liquid crystal display (LCD) technology \cite{majumdar.a:2007.a}. Understanding the liquid crystalline phase behaviour of systems of non-spherical particles \cite{chandrasekhar:1990.a, kumar:2004.a, bushby:2002.a} is an important topic in modern condensed matter physics. One of the most celebrated cases of a phase transition in such systems is the isotropic-nematic (\textit{I}-\textit{N}) transition. For athermal model systems, where the particle interactions are hard core, phase transitions arise purely from entropic contributions to the free energy and the phase behaviour is governed solely by density and is independent of temperature. Such models can be used to describe lyotropic liquid crystals and phase transitions such as the \textit{I}-\textit{N} transition. Onsager showed how the formation of liquid crystalline phases can be understood on the basis of pair interactions between the constituents of the material \cite{onsager.l:1949.a,Hansen06,gray.cg:1984.a}. He considered the hard platelet fluid but we know that unlike the case of rod-like particles, his second-virial theory does not produce quantitatively correct results for the equation of state and the \textit{I}-\textit{N} coexistence densities. Onsager himself noted that higher virial contributions are important for obtaining reliable results, estimating the ratio $B_{3}/B_{2}^{2}$ at $\mathcal{O} (1)$, with $B_{2}$ and $B_{3}$ being the second and third virial coefficients, respectively. Nevertheless, the second-virial theory has been employed to investigate the monodisperse platelet system \cite{forsyth.pa:1977.a, forsyth.pa:1978.a}. In Ref.~\cite{forsyth.pa:1977.a} a numerical approach was used to calculate the phase diagram of platelets for varying thickness, including the case of zero thickness. This is complemented by a calculation for the equation of state in both the \textit{I} and \textit{N} phases for vanishingly thin platelets in Ref.~\cite{forsyth.pa:1978.a}. The first off-lattice simulation study of the \textit{I}-\textit{N} transition for infinitely thin platelets was carried out in Ref.~\cite{eppenga.r:1982.a}. This showed that the phase diagram differs substantially from the Onsager prediction and that the \textit{I}-\textit{N} transition is actually much more weakly first order and occurs at lower densities than predicted theoretically. The authors also carried out a fifth order virial calculation for the equation of state. More accurate predictions of the higher virial contributions for disks were presented in Ref.~\cite{veerman.jac:1992.a}, where simulation results are reported for hard cut spheres and more recently in Ref.~\cite{fartaria.rps:2009.a}. Later simulation work was carried out on polydisperse platelet systems \cite{bates.ma:1999.a} and systems of platelets with different polygonal shapes (e.g. hexagons, triangles) \cite{bates.ma:1999.b}. Further simulation results of model circular platelets were reported in Ref.~\cite{reich.h:2007.a} and simulations alongside an integral equation approach for mixtures of rods and disks were carried out in Ref.~\cite{cheung.dl:2008.e}. Simulations of binary platelet systems have not yet been carried out. Binary mixtures of particles of different shape and/or size are interesting due to the richness of the phase diagrams they may exhibit. Binary rod mixtures form a prominent example. Studies include mixtures of thick and thin rods \cite{vanroij.r:1998.a} and long and short rods \cite{lekkerkerker.hnw:1984.a} using Onsager theory as well as using Parsons-Lee scaling \cite{varga.s:2003.a}. The phase behaviour in binary mixtures can include: the fractionation effect, whereby the larger particles go preferentially into the nematic phase; widening of the biphasic region; a re-entrant $I\rightarrow N \rightarrow I$ phenomenon on increasing density; the possibility of demixing into two different isotropic states and/or two different nematic states and triphasic equilibria (see e.g. Ref.~\cite{vanroij.r:1998.a} for examples of these phenomena). Nematic-nematic (\textit{N}-\textit{N}) demixing, at high enough pressures, can be viewed as a result of competition between orientational entropy of the smaller platelets favouring mixing, and the entropy of mixing \cite{vanroij.r:1998.a}. The \textit{N}-\textit{N} phase separation for binary mixtures of rods, including the high density regime, is studied in detail in Ref.~\cite{varga.s:2005.a}. Binary mixtures of thin and thick platelets have been investigated \cite{wensink.hh:2001.a, wensink.hh:2004.a} with the Parsons-Lee scaling of the Onsager functional \cite{parsons.jd:1979.a, lee.sd:1987.a, lee.sd:1989.a}. Studies based on the Zwanzig model for binary hard platelets, where the particles are restricted to occupy only three mutually perpendicular directions, have been carried out for the bulk and interfacial properties of the demixed phases. Rich phase diagrams, involving isotropic and nematic phases have been reported in Refs. \cite{bier.m:2004.a, harnau.l:2002.c, harnau.l:2002.d}. A recent review \cite{harnau.l:2008.f} of platelet fluids contains a summary of these results. Recently Verhoeff \textit{et al.} \cite{verhoeff.a.a:2009.a} have investigated experimentally and theoretically the phase behaviour of colloidal platelets with bimodal shape distribution. Their theory is based on the Onsager-Parsons free energy and a cell approach for the columnar (\textit{Col}) state \cite{wensink.hh:2004.c}. The authors find agreement between their experimental findings and theoretical predictions for sufficiently large thickness ratios. The phase diagram features an \textit{I}-\textit{N} density inversion and triphasic \textit{I}-\textit{N}-\textit{Col} equilibrium. Fundamental measure theory (FMT) is an approximate non-perturbative density functional theory (DFT) \cite{evans.r:1979.a}, originally proposed by Rosenfeld for additive hard sphere mixtures \cite{rosenfeld.y:1989.a, rosenfeld.y:1997.a}. The approach was later generalised to other convex shapes \cite{rosenfeld.y:1994.a, rosenfeld.y:1995.a}, which led to subsequent work \cite{cinacchi.g:2002.a, hansen-goos.h:2009.a}. The bulk \textit{I}-\textit{N} coexistence densities (scaled by the cube of the platelet radius) and nematic order parameter at the transition ($c_{I},c_{N}$ and $S_{N}$, respectively) were calculated by Frenkel and Eppenga in Ref.~\cite{eppenga.r:1982.a} by simulation; for more recent simulation results see Ref.~\cite{reich.h:2007.b}. The values previously obtained from FMT are $c_{I}=0.418$, $c_{N}=0.46$ and $S_{N}=0.53$ \cite{reich.h:2007.b}, which are in agreement with the present study. Recently \cite{hendrik_communication} these values were improved using the same method but with increased resolution to $c_{I}=0.419$, $c_{N}=0.469$ and $S_{N}=0.533$ \cite{fmt_note}. The FMT functional for pure platelets was later utilised to study inhomogeneous situations including the \textit{I}-\textit{N} interface and wetting at a hard wall \cite{vanderbeek.d:2006.a, reich.h:2007.a} and capillary nematisation of platelets between two parallel walls \cite{reich.h:2007.b}. Generalising the theory for the corresponding one-component system \cite{esztermann.a:2006.a}, we here propose a functional to describe binary mixtures of vanishingly thin circular platelets. Our theory features the exact virial second order term in density and an approximate term of third order in density. We investigate three types of demixing phase behaviour in the case of binary platelets with varying size ratio, finding \textit{I}-\textit{N} and \textit{N}-\textit{N} phase coexistence. We do not find stable \textit{I}-\textit{I} demixing (as could be driven by the depletion effect \cite{vanroij.r:1996.a}) for the regimes considered in the present work. We restrict our attention to uniaxial arrangements of the (uniaxial) platelets, as we do not expect biaxial arrangement of the particles to occur. We study a range of size ratios in this investigation, ranging between $\lambda=1.1$ and $5$. We present the phase diagrams in different representations to facilitate comparison with simulations which may be performed in different ensembles or experiments. We expect the phase diagrams from FMT to be quantitatively more accurate than those from Onsager theory, which we calculate as a reference. The topologies of the phase diagrams are the same in both theories for our chosen values of the size ratio between two species. Although the integral kernel, which represents the pair excluded volume term, is the same for long thin rods as it is for platelets of vanishing thickness, the results from Onsager theory cannot be obtained by simple scaling of literature results for binary mixtures of rods. This paper is organised as follows. In Sec.\ \ref{dft} we define the model, outline the density functional theory and the conditions for thermodynamic stability and phase coexistence. In Sec.\ \ref{results} we present results for the phase behaviour of binary platelet mixtures. In Sec.\ \ref{outlook} we provide conclusions and an outlook on possible future work. \section{Density Functional Theory for Binary Hard Platelet Mixtures} \label{dft} \subsection{Pair Interactions and Model Parameters} \label{model} We consider a binary mixture of hard circular platelets with vanishing thickness and continuous positional and orientational degrees of freedom. Species 1 and species 2 have radii $R_{1}$ and $R_{2}$ respectively, with $R_{2}>R_{1}$. The pair potential $u_{ij}$ between two particles $i$ and $j$, where $i,j=1,2$, is infinite if the geometrical shapes of the two platelets overlap and zero otherwise and is hence given by \begin{equation} \label{eq:potential} {u_{ij}(\bf{r}-\bf{r}', \boldsymbol{\omega},\boldsymbol{\omega}')=\begin{cases}\infty&\text{if particles overlap} \\0&\text{otherwise, } \end{cases}} \end{equation} where $\textbf{r}$ and $\textbf{r}'$ are the positions of the particle centres and $\boldsymbol{\omega}$ and $\boldsymbol{\omega}'$ are unit vectors indicating the particle orientations (normal to the particle surface). The size ratio \begin{equation} \label{eq:size_ratio} \lambda=\frac{R_{2}}{R_{1}} > 1 \end{equation} characterises the radial bidispersity and is the only control parameter in the model. We characterise the thermodynamic state by two dimensionless densities $c_{1} = \rho_{1}R_{1}^{3}$ and $c_{2} = \rho_{2}R_{1}^{3}$, where $\rho_{1}$ and $\rho_{2}$ are the number densities of the two species, $\rho_{i}=N_{i}/V$, where $N_{i}$ is the number of particles of species $i=1,2$ and $V$ is the system volume. The composition (mole fraction) of the (larger) species 2 is $x=\rho_{2}/(\rho_{1}+\rho_{2})$ and the total dimensionless concentration is $c=R_{1}^{3}(\rho_{1}+\rho_{2})=c_{1}+c_{2}$. \subsection{Grand Potential Functional and Minimisation Principle} Density functional theory (DFT) is formulated on the one-body level of the density distributions $\rho_{i}(\bf{r},\boldsymbol{\omega})$. The variational principle \cite{evans.r:1979.a} states that the true equilibrium density profile is the one which minimises the grand potential functional $\Omega$ and so obeys \begin{equation} \label{eq:minimization_principle} \frac{\delta\Omega([\rho_{1},\rho_{2}],\mu_{1},\mu_{2},V,T)}{\delta \rho_{i}(\textbf{r},\boldsymbol{\omega})}=0, \end{equation} where $\mu_{i}$ is the chemical potential of species $i=1,2$ and $T$ is absolute temperature. The grand potential functional can be decomposed as \begin{align} \label{eq:grand_potential} \Omega([\rho_{1},&\rho_{2}],\mu_{1},\mu_{2},V,T)= F_{\textrm{id}}([\rho_{1},\rho_{2}],V,T)\notag \\ &+ F_{\textrm{exc}}([\rho_{1},\rho_{2}],V,T) + \sum_{i=1}^{2}\int d \textbf{r} \int d \boldsymbol{\omega} \rho_{i} (V_{\textrm{ext}}^{i}(\textbf{r},\boldsymbol{\omega})-\mu_{i}), \end{align} where the spatial integral (over $\textbf{r}$) is over the system volume $V$ and the angular integral (over $\boldsymbol{\omega}$) is over the unit sphere; $V_{\textrm{ext}}^{i}(\textbf{r},\boldsymbol{\omega})$ is an external potential acting on species $i$; $F_{\textrm{exc}}([\rho_{1},\rho_{2}],V,T)$ is the excess (over ideal gas) contribution to the Helmholtz free energy and describes the inter-particle interactions. The free energy functional for a binary ideal gas of uniaxial rotators is given by \begin{align} \label{eq:ideal} \beta F_{\textrm{id}}([\rho_{1},\rho_{2}],V,T) = \sum_{i=1}^{2}&\int{d\textbf{r}}\int d\boldsymbol{\omega}\rho_{i}(\textbf{r},\boldsymbol{\omega})\notag \\ &\times[\ln(\rho_{i}(\textbf{r},\boldsymbol{\omega})\Lambda_{i}^{3})-1], \end{align} where $\Lambda_{i}$ is the (irrelevant) thermal wavelength of species $i$ and $\beta=1/(k_{B}T)$, where $k_{B}$ is the Boltzmann constant. We let $\Lambda_{i}=R_{1}$, which is equivalent to fixing an additive constant to the chemical potential and hence does not affect any observable properties of the system. One systematic way to express the excess free energy functional is to expand it in a virial series in density \cite{Hansen06}. Onsager theory is based entirely on the second-virial level. FMT (as described in Sec.~\ref{fmt} below) approximates higher order terms using single particle geometries. Nevertheless we find it useful to give the terms in the virial expansion up to third order in density explicitly: the second and third order contributions to the (exact) virial series for the excess free energy $\beta {F}_{\textrm{exc}}([\rho_{1},\rho_{2}],V,T)$ are given respectively by \begin{eqnarray} \label{eq:virial_expansion} -\frac{1}{2}\left[\begin{picture}(10,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$1$}} \put(3,17){\small{$1$}} \put(5,-1){\line(0,1){12}} \put(5,12){\circle*{5}} \end{picture}+ \begin{picture}(10,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$2$}} \put(3,17){\small{$2$}} \put(5,-1){\line(0,1){12}} \put(5,12){\circle*{5}} \end{picture}+ 2\begin{picture}(10,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$1$}} \put(3,17){\small{$2$}} \put(5,-1){\line(0,1){12}} \put(5,12){\circle*{5}} \end{picture} \right], \end{eqnarray} \begin{eqnarray} \label{eq:virial_expansion_2} -\frac{1}{6}\Big[\begin{picture}(25,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$1$}} \put(3,17){\small{$1$}} \put(19,-15){\small{$1$}} \put(5,-2){\line(0,1){12}} \put(5,12){\circle*{5}} \put(7,-4){\line(1,0){12}} \put(19,-4){\circle*{5}} \put(17.8,-1.5){\line(-1,1){11}} \end{picture}+ \begin{picture}(25,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$2$}} \put(3,17){\small{$2$}} \put(19,-15){\small{$2$}} \put(5,-2){\line(0,1){12}} \put(5,12){\circle*{5}} \put(7,-4){\line(1,0){12}} \put(19,-4){\circle*{5}} \put(17.8,-1.5){\line(-1,1){11}} \end{picture}+ 3\begin{picture}(25,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$1$}} \put(3,17){\small{$2$}} \put(19,-15){\small{$1$}} \put(5,-2){\line(0,1){12}} \put(5,12){\circle*{5}} \put(7,-4){\line(1,0){12}} \put(19,-4){\circle*{5}} \put(17.8,-1.5){\line(-1,1){11}} \end{picture}+ 3\begin{picture}(25,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$1$}} \put(3,17){\small{$2$}} \put(19,-15){\small{$2$}} \put(5,-2){\line(0,1){12}} \put(5,12){\circle*{5}} \put(7,-4){\line(1,0){12}} \put(19,-4){\circle*{5}} \put(17.8,-1.5){\line(-1,1){11}} \end{picture} \Big], \end{eqnarray}\\ where each line in the diagrams represents a Mayer function $f_{ij}({\textbf r}-{\textbf r }', \boldsymbol{\omega}, \boldsymbol{\omega}')= \exp \left(-\beta u_{ij}({\textbf r}-{\textbf r }', \boldsymbol{\omega}, \boldsymbol{\omega}')\right)-1$, which equals $-1$ if the two particles overlap and zero otherwise. The shaded circles, {\em field points}, indicate multiplication by the one-body density $\rho_{i}({\textbf r},\boldsymbol{\omega})$ and integration over the coordinates $\textbf r$ and $\boldsymbol{\omega}$ \cite{Hansen06}. The number alongside each field point represents the species $i$. Here we consider only spatially homogeneous states, such that the one-body density does not depend on position $\textbf{r}$ and, for the case of uniaxial nematic states considered in this paper, depends only on the polar angle $\theta$ of $\boldsymbol{\omega}$ with respect to the nematic director. Hence the one-body densities factorise as $\rho_{i}(\textbf{r}, \boldsymbol{\omega}) = \rho_{i}\Psi_{i}(\theta)$ where $\Psi_{i}(\theta)$ is the orientational distribution function (ODF) and $\rho_{i}$ the number density of species $i$. Before laying out the FMT we focus on the second-virial level. \subsection{Onsager Second-Virial Theory for Binary Platelets} \label{onsager} The diagrams in Eq.~(\ref{eq:virial_expansion}), using the fact that $\rho_{i}(\textbf{r}, \boldsymbol{\omega}) = \rho_{i}\Psi_{i}(\theta)$ for spatially homogeneous states, become\\ \begin{eqnarray} \label{eq:virial_expansion_example} -\frac{1}{2}\begin{picture}(10,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-16){\small{$i$}} \put(3,18){\small{$j$}} \put(5,-1){\line(0,1){12}} \put(5,12){\circle*{5}} \end{picture} =-\frac{1}{2}\rho_{i}\rho_{j}\int d \boldsymbol{\omega} \int d \boldsymbol{\omega}' \int d\textbf{r} f_{ij}(\textbf{r}, \boldsymbol{\omega}, \boldsymbol{\omega}') \Psi_{i}(\theta)\Psi_{j}(\theta ') , \end{eqnarray} where we have renamed ${\textbf r}-{\textbf r'}\rightarrow \textbf{r}$. The spatial integral over the Mayer bond yields (minus) the excluded volume $-\mathcal{E}_{ij}(\boldsymbol{\omega},\boldsymbol{\omega}')$ between two particles of species $i$ and $j$, as a function of the angle $\gamma$ between $\boldsymbol \omega$ and $\boldsymbol \omega '$. Hence \begin{equation} \label{eq: second_virial_part} \mathcal{E}_{ij}(\boldsymbol{\omega},\boldsymbol{\omega}')=-\int d\textbf{r} f_{11}(\textbf{r}, \boldsymbol{\omega}, \boldsymbol{\omega}')=2\pi (R_{i}^{2}R_{j}+R_{j}^{2}R_{i})\sin\gamma , \end{equation} Therefore \begin{eqnarray} \label{eq:virial_expansion_example_1} -\frac{1}{V}\begin{picture}(10,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-16){\small{$i$}} \put(3,18){\small{$j$}} \put(5,-1){\line(0,1){12}} \put(5,12){\circle*{5}} \end{picture} =16\pi^{2} (R_{i}^{2}R_{j}+R_{j}^{2}R_{i})\rho_{i}\rho_{j} \int_{0}^{\frac{\pi}{2}} d\theta \sin \theta \end{eqnarray} \vspace{-5mm} $$ \hspace{10mm}\times \int_{0}^{\frac{\pi}{2}} d \theta ' \sin \theta ' K(\theta, \theta') \Psi_{i}(\theta) \Psi_{j}(\theta '). $$ The integrals in Eq.~(\ref{eq:virial_expansion_example_1}) (omitting the prefactor) can be written as \begin{align} \label{eq:virial_expansion_example_2} \int_{0}^{2\pi} d \phi \int_{0}^{\pi} d \theta \sin \theta &\int_{0}^{2\pi} d \phi '\int_{0}^{\pi} d \theta \sin \theta ' \notag \\ & \times \Psi_{1}(\theta)\Psi_{1}(\theta ')\sin\gamma, \end{align} where $\theta$ and $\theta'$ are the polar angles of two platelets with respect to the nematic director and $\phi$ and $\phi '$ are the azimuthal angles. Due to the inversion symmetry of the nematic state $\int_{0}^{\pi} d \theta =2\int_{0}^{\frac{\pi}{2}} d \theta$. In order to deal with the azimuthal integral we introduce the kernel $K(\theta,\theta')$ via \begin{align} \label{eq: kernel_k} &K(\theta,\theta')=\int_{0}^{2\pi}d\phi \sin\gamma=\int_{0}^{2\pi}d\phi \sqrt{1- (\boldsymbol{\omega} \cdot \boldsymbol{\omega}')^{2}} \notag \\ & =\int_{0}^{2\pi}d\phi \sqrt{1-(\cos\theta \cos\theta' + \sin\theta \sin\theta' \cos\phi)^{2}}, \end{align} where we have renamed $\phi-\phi ' \rightarrow \phi$ as the difference between the azimuthal angles of the two platelets. Adding all three terms and multiplying by $R_{1}^{3}$ yields the Onsager contribution to the excess free energy in the dimensionless form \begin{align} \label{eq: free_energy_2} \frac{\beta F_{\textrm{exc}}^{(2)}}{V}R_{1}^{3} =16 &\pi^{2}c^{2} \int_{0}^{\frac{\pi}{2}} d\theta \sin \theta \int_{0}^{\frac{\pi}{2}} d \theta ' \sin \theta ' K(\theta, \theta') \notag \\ &\times\Big[(1-x)^{2}\Psi_{1}(\theta)\Psi_{1}(\theta ') +x^{2}\lambda^{3}\Psi_{2}(\theta)\Psi_{2}(\theta ') \notag \\ &\hspace{6mm}+x(1-x)(\lambda^{2}+\lambda)\Psi_{1}(\theta)\Psi_{2}(\theta') \Big], \end{align} where the superscript of $F^{(\alpha)}_{\textrm{exc}}$ represents the order in density of the excess free energy. \subsection{Fundamental Measure Theory for Binary Platelet Mixtures} \label{fmt} We generalise the monodisperse functional to the case of binary mixtures using an approximate term at third order in density which is based on the FMT developed in Ref.~\cite{esztermann.a:2006.a}. Our theory contains the exact second order Onsager term and instead of using any higher order terms from the series, such as the {\em exact} third virial level (\ref{eq:virial_expansion_2}), an approximate term which is of third order in density is used \cite{Sollich_communication}. This term is nonvanishing (and constant) for cases with common triple intersection of the three platelets involved. There are no higher order terms due to the scaled-particle roots \cite{barker.ja:1976.a} of the approach; the vanishing volume of the platelets truncates the series. Global prefactors are used to compensate for \textit{lost cases} \cite{cuesta.ja:2002.a, tarazona.p:1997.a}. We postulate the excess free energy \begin{align} \label{eq: free_energy_fmt} \beta F_{\textrm{exc}}([\rho_{1},\rho_{2}],V)=\int d& \textbf{r} \int d \boldsymbol{\omega} \int d \boldsymbol{\omega}' \Big[n^{\textrm{DD}}_{1}(\textbf{r},\boldsymbol{\omega}) n_{2}^{\textrm{D}}(\textbf{r},\boldsymbol{\omega}') \notag \\ &+ \frac{1}{24 \pi} n_{2}^{\textrm{D}}(\textbf{r},\boldsymbol{\omega}) n_{2}^{\textrm{DDD}}(\textbf{r},\boldsymbol{\omega},\boldsymbol{\omega}') n_{2}^{\textrm{D}}(\textbf{r},\boldsymbol{\omega}') \Big], \end{align} (where the right hand side is independent of $T$). The first term of the sum in Eq.~(\ref{eq: free_energy_fmt}) is equivalent to the Onsager contribution to the excess free energy and the second is the FMT contribution. The weighted densities are related to the bare one-body densities, $\rho_{i}(\textbf{r}, \boldsymbol{\omega})$, via \begin{align} \label{eq: density_1dd} n_{1}^{\textrm{DD}}(\textbf{r}, \boldsymbol{\omega}) &= \sum_{i=1}^{2} \int d \boldsymbol{\omega}'w_{1}^{\textrm{DD}i}(\textbf{r}, \boldsymbol{\omega}',\boldsymbol{\omega})*\rho_{i}(\textbf{r}, \boldsymbol{\omega}'),\\ \label{eq: density_2d} n_{2}^{\textrm{D}}(\textbf{r}, \boldsymbol{\omega})&= \sum_{i=1}^{2}w_{2}^{\textrm{D}i}(\textbf{r},\boldsymbol{\omega})*\rho_{i}(\textbf{r}, \boldsymbol{\omega}),\\ \label{eq: density_2ddd} n_{2}^{\textrm{DDD}}(\textbf{r}, \boldsymbol{\omega},\boldsymbol{\omega}')&= \sum_{i=1}^{2} \int d \boldsymbol{\omega}''w_{2}^{\textrm{DDD}i}(\textbf{r}, \boldsymbol{\omega},\boldsymbol{\omega}',\boldsymbol{\omega}'')*\rho_{i}(\textbf{r}, \boldsymbol{\omega}''), \end{align} where $*$ represents the three-dimensional convolution $h(\textbf{r})*g(\textbf{r})=\int d^{3}x h(\textbf{x})g(\textbf{x}-\textbf{r})$. We have kept the notation of Ref.~\cite{esztermann.a:2006.a} where the upper index $\textrm{D}$ (disk) is indicative of the number of particle orientations that appear in the weight function (below) or weighted density. The weight functions for species $i$ are given by \begin{align} \label{eq: weight_1d} w_{1}^{\textrm{D}i}(\textbf{r}, \boldsymbol{\omega})&= \delta(R_{i}-|\textbf{r}|)\delta(\textbf{r} \cdot \boldsymbol{\omega})/8,\\ \label{eq: weight_2d} w_{2}^{\textrm{D}i}(\textbf{r}, \boldsymbol{\omega}) &= 2 \Theta(R_{i}- |\textbf{r}|) \delta (\textbf{r} \cdot \boldsymbol{\omega}),\\ \label{eq: weight_1dd} w_{1}^{\textrm{DD}i}(\textbf{r}, \boldsymbol{\omega}, \boldsymbol{\omega}') &= \frac{2}{R_{i}} | \boldsymbol{\omega} \cdot ( \boldsymbol{\omega}' \times \textbf{r})| w_{1}^{\textrm{D}i}(\textbf{r}, \boldsymbol{\omega}),\\ \label{eq: weight_1ddd} w_{2}^{\textrm{DDD}i}(\textbf{r}, \boldsymbol{\omega}, \boldsymbol{\omega}',\boldsymbol{\omega}'') &= \frac{8}{\pi} | \boldsymbol{\omega} \cdot ( \boldsymbol{\omega}' \times \boldsymbol{\omega}'')| w_{2}^{\textrm{D}i}(\textbf{r}, \boldsymbol{\omega}), \end{align} \\ \noindent where $\delta(\cdot)$ is the Dirac delta distribution, $\Theta(\cdot)$ is the Heaviside step function and $\times$ denotes the vector product, such that $\boldsymbol{\omega} \cdot ( \boldsymbol{\omega}' \times \boldsymbol{\omega}'')$ is the triple scalar product. Note the modulus in Eqs.~(\ref{eq: weight_1dd}) and (\ref{eq: weight_1ddd}). For spatially homogeneous states, Eq.~(\ref{eq: free_energy_fmt}) becomes \begin{align} \label{eq: free_energy_fmt_bulk} \frac{\beta F_{\textrm{exc}} ([\rho_{1},\rho_{2}])}{V} = \int d & \boldsymbol{\omega} \int d \boldsymbol{\omega}' \Big[n_{1}^{\textrm{DD}}(\boldsymbol{\omega}) n_{2}^{\textrm{D}}(\boldsymbol{\omega}') \notag \\ &+ \frac{1}{24 \pi} n_{2}^{\textrm{D}}(\boldsymbol{\omega}) n_{2}^{\textrm{DDD}}(\boldsymbol{\omega},\boldsymbol{\omega}') n_{2}^{\textrm{D}}(\boldsymbol{\omega}') \Big]. \end{align} Inserting the definitions of the weighted densities (\ref{eq: density_1dd})-(\ref{eq: density_2ddd}) into the excess free energy (\ref{eq: free_energy_fmt}) we obtain \begin{eqnarray} \label{eq: free_energy_fmt_alternative} \beta F_{\textrm{exc}}([\rho_{1},\rho_{2}])= -\frac{1}{2}\left[\begin{picture}(10,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$1$}} \put(3,17){\small{$1$}} \put(5,-1){\line(0,1){12}} \put(5,12){\circle*{5}} \end{picture}+ \begin{picture}(10,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$2$}} \put(3,17){\small{$2$}} \put(5,-1){\line(0,1){12}} \put(5,12){\circle*{5}} \end{picture}+ 2\begin{picture}(10,10)(0,0) \put(5,-4){\circle*{5}} \put(3,-15){\small{$1$}} \put(3,17){\small{$2$}} \put(5,-1){\line(0,1){12}} \put(5,12){\circle*{5}} \end{picture} \right] +\int d \textbf{x} \int d \boldsymbol{\omega} \int d \boldsymbol{\omega '} \int d \boldsymbol{\omega ''} \frac{| \boldsymbol{\omega} \cdot ( \boldsymbol{\omega}' \times \boldsymbol{\omega}'')|}{3\pi^{2}} \end{eqnarray} $$ \hspace{55mm}\times n_{2}^{\textrm{D}}(\textbf{x},\boldsymbol{\omega})n_{2}^{\textrm{D}}(\textbf{x},\boldsymbol{\omega}')n_{2}^{\textrm{D}}(\textbf{x},\boldsymbol{\omega}''). $$ The fundamental measures of a platelet of species $i$ are the integral mean curvature $\xi_{i}^{\textrm{IMC}}=\pi R_{i}/4$ and the surface $\xi_{i}^{\textrm{S}}=2\pi R_{i}^{2}$. The first term of Eq.~(\ref{eq: free_energy_fmt_bulk}) may be expressed as \begin{align} \label{eq: measures_interpreation} A'(\boldsymbol{\omega},\boldsymbol{\omega}')=\frac{4}{\pi}&\Big[\xi_{1}^{\textrm{IMC}}\xi_{1}^{\textrm{S}}\rho_{1}^{2}\Psi_{1}(\theta)\Psi_{1}(\theta') +\xi_{2}^{\textrm{IMC}}\xi_{2}^{\textrm{S}}\rho_{2}^{2}\Psi_{2}(\theta)\Psi_{2}(\theta')\notag \\ &+\left(\xi_{2}^{\textrm{IMC}}\xi_{1}^{\textrm{S}}+\xi_{1}^{\textrm{IMC}}\xi_{2}^{\textrm{S}}\right)\Psi_{1}(\theta)\Psi_{2}(\theta')/2\Big]\sin \gamma, \end{align} remembering that $\Psi_{i}(\boldsymbol{\omega})=\Psi_{i}(\theta)$ for uniaxial nematics. Eq.~(\ref{eq: measures_interpreation}) is simply the fundamental measures interpretation \cite{esztermann.a:2006.a} of the Onsager contribution to the free energy. Note that the term which represents the excluded volume between two particles of different species is given by $\xi_{2}^{\textrm{IMC}}\xi_{1}^{\textrm{S}}+\xi_{1}^{\textrm{IMC}}\xi_{2}^{\textrm{S}}$. This leads to a scaling of the excluded volume by $\lambda^{2} + \lambda$, which is quite different from the scaling that occurs for binary mixtures of rods. The second term in Eq.~(\ref{eq: free_energy_fmt_bulk}) is the FMT contribution to the excess free energy. This is given by \begin{align} \frac{\beta F^{(3)}_{\textrm{exc}}}{V}= \int d \boldsymbol{\omega} \int d \boldsymbol{\omega}' \int d \boldsymbol{\omega}'' B'(\boldsymbol{\omega},\boldsymbol{\omega}',\boldsymbol{\omega}'') \end{align} \noindent where \begin{align} B'(\boldsymbol{\omega},\boldsymbol{\omega}',\boldsymbol{\omega}'')=\frac{| \boldsymbol{\omega} \cdot ( \boldsymbol{\omega}' \times \boldsymbol{\omega}'')|}{3\pi^{2}}&\Big[(\xi_{1}^{\textrm{S}})^{3}\rho_{1}^{3}\Psi_{1}(\theta)\Psi_{1}(\theta')\Psi_{1}(\theta'') \notag \\ &+(\xi_{2}^{\textrm{S}})^{3}\rho_{2}^{3}\Psi_{2}(\theta)\Psi_{2}(\theta')\Psi_{2}(\theta'') \notag \\ &+3(\xi_{1}^{\textrm{S}})^{2}\xi_{2}^{\textrm{S}}\rho_{1}^{2}\rho_{2}\Psi_{1}(\theta)\Psi_{1}(\theta')\Psi_{2}(\theta'') \notag \\ &+3\xi_{1}^{\textrm{S}}(\xi_{2}^{\textrm{S}})^{2}\rho_{1}\rho_{2}^{2}\Psi_{1}(\theta)\Psi_{2}(\theta')\Psi_{2}(\theta'')\Big]. \end{align} We choose coordinates such that $\boldsymbol{\omega}=(\sin\theta,0,\cos\theta)$, $\boldsymbol{\omega}'=(\cos\theta'\sin\theta',\sin\phi'\sin\theta',\cos\theta')$ and $\boldsymbol{\omega}''=(\cos\theta''\sin\theta'',\sin\phi''\sin\theta'',\cos\theta'')$ where $\theta, \theta '$ and $\theta ''$ are the polar angles of three platelets. The third order contribution in density to the FMT excess free energy in these coordinates is given by \begin{widetext} \begin{align} \label{eq: free_energy_3} \frac{\beta F_{\textrm{exc}}^{(3)}}{V} R_{1}^{3}= \frac{128\pi^{2}}{3}&c^{3} \int_{0}^{\frac{\pi}{2}} d\theta \sin \theta \int_{0}^{\frac{\pi}{2}} d \theta ' \sin \theta ' \int_{0}^{\frac{\pi}{2}} d\theta'' \sin \theta'' L(\theta, \theta',\theta'') \notag \\ &[(1-x)^{3}\Psi_{1}(\theta)\Psi_{1}(\theta')\Psi_{1}(\theta'') +x^{3}\lambda^{6}\Psi_{2}(\theta)\Psi_{2}(\theta')\Psi_{2}(\theta'') \notag \\ & +3x(1-x)^{2}\lambda^{2}\Psi_{1}(\theta)\Psi_{1}(\theta')\Psi_{2}(\theta'') +3x^{2}(1-x)\lambda^{4}\Psi_{1}(\theta)\Psi_{2}(\theta')\Psi_{2}(\theta'') ], \end{align} \end{widetext} where the kernel $L(\theta,\theta', \theta'')$ is \begin{align} \label{eq: kernel_l} L(\theta,\theta', \theta'')=&\int_{0}^{2\pi} \int_{0}^{2\pi} d\phi' d\phi'' | \boldsymbol{\omega} \cdot ( \boldsymbol{\omega}' \times \boldsymbol{\omega}'')| \notag \\ &\hspace{-4.2mm}=\int_{0}^{2\pi} \int_{0}^{2\pi} d\phi' d\phi'' |\sin \theta(\sin \phi' \sin \theta' \cos \theta'' \notag \\ &\hspace{5mm}+\cos\theta' \sin \phi'' \sin \theta'' )+\cos\theta(\cos\phi' \sin\theta' \sin\phi '' \sin \theta '' \notag \\ &\hspace{5mm}- \sin \phi' \sin \theta' \cos \phi '' \sin \theta '')|. \end{align} The full form of the excess free energy, as used in the calculations described below, is given by the sum of Eqs.~(\ref{eq: free_energy_2}) and (\ref{eq: free_energy_3}). In practice, along with Eqs.~(\ref{eq: kernel_k}) and (\ref{eq: kernel_l}), these require numerical computation on a grid as described in the following. \subsection{Self-consistency equations for the orientational distribution functions} The minimisation principle (\ref{eq:minimization_principle}) together with the FMT approximation (\ref{eq: free_energy_fmt_bulk}) for $F_{\textrm{exc}}([\rho_{1},\rho_{2}])$ leads to two coupled Euler-Lagrange equations for the ODFs: \begin{widetext} \begin{align} \label{eq:self_consistency_1} \Psi_{1}(\theta) = \frac{1}{Z_{1}} \exp \bigg[ &-8\pi c \int_{0}^{\frac{\pi}{2}} d \theta ' \sin \theta ' K(\theta, \theta ')[(1-x) \Psi_{1}(\theta ') + \frac{1}{2}x(\lambda^{2} + \lambda)\Psi_{2}(\theta ') ] \notag \\ & -32\pi c^{2} \int_{0}^{\frac{\pi}{2}} d \theta ' \sin \theta ' \int_{0}^{\frac{\pi}{2}} d \theta '' \sin \theta '' L(\theta, \theta ', \theta'') \notag \\ & \times[(1-x)^{2}\Psi_{1}(\theta')\Psi_{1}(\theta'')+2x(1-x)\lambda^{2}\Psi_{1}(\theta')\Psi_{2}(\theta'')+x^{2}\lambda^{4}\Psi_{2}(\theta')\Psi_{2}(\theta'') ] \bigg], \end{align} \begin{align} \label{eq:self_consistency_2} \Psi_{2}(\theta) = \frac{1}{Z_{2}} \exp \bigg[ &-8 \pi c \int_{0}^{\frac{\pi}{2}} d \theta ' \sin \theta ' K(\theta, \theta ')[x \lambda^{3}\Psi_{2}(\theta ') + \frac{1}{2}(1-x) (\lambda^{2} + \lambda)\Psi_{1}(\theta ')] \notag \\ & -32\pi c^{2} \int_{0}^{\frac{\pi}{2}} d \theta ' \sin \theta ' \int_{0}^{\frac{\pi}{2}} d \theta '' \sin \theta '' L(\theta, \theta ', \theta'') \notag \\ & \times[x^{2} \lambda^{6}\Psi_{2}(\theta')\Psi_{2}(\theta'')+2x(1-x)\lambda^{4}\Psi_{1}(\theta')\Psi_{2}(\theta'')+(1-x)^{2}\lambda^{2}\Psi_{1}(\theta')\Psi_{1}(\theta'') ] \bigg], \end{align} \end{widetext} where the constants $Z_{1}$ and $Z_{2}$ are such that the normalisation $\int d \boldsymbol{\omega}\Psi_{i}(\boldsymbol{\omega})=1$, for $i=1,2$. Neglecting terms of order $c^{2}$ in the exponentials, the equations for FMT reduce to the Euler-Lagrange equations of Onsager theory. We solve Eqs.~(\ref{eq:self_consistency_1}) and (\ref{eq:self_consistency_2}) numerically with a straightforward extension to the iterative procedure given in Ref.~\cite{herzfeld84} and numerical techniques similar to those described in Ref.~\cite{vanroij.r.:2005.d}. The $\theta$- and $\phi$-grids are defined on $[0,\pi/2]$ and $[0,2\pi]$ respectively. The $\theta$-grid is divided into 200 equal steps on [0,$\pi/8$], where the ODF changes most rapidly in the nematic phase and into 50 equal steps on [$\pi/8$,$\pi/2$] where the ODF is almost zero. The $\phi$-grid is divided into 200 equally spaced intervals on [0,$2\pi$]. We start the iteration with two initial trial distributions, for example a normalized Gaussian $(c/\pi^{2})\exp \left[-2c^{2}\theta^{2}/\pi\right]$ or a constant distribution $1/4\pi$. The choice of the constant distribution is more efficient at low densities where the system is expected to be isotropic. These guesses are substituted into the right hand sides of Eqs.~(\ref{eq:self_consistency_1}) and~(\ref{eq:self_consistency_2}) to obtain a new pair of ODFs, $\Psi_{i,\textrm{new}}(\theta)$. This procedure is repeated until $\max|\Psi_{i,\textrm{new}}(\theta)-\Psi_{i,\textrm{old}}(\theta)|<t$, $i=1,2$, where $t$ is the tolerance given by the magnitude of the largest acceptable deviation to the ODF of the previous step: $\Psi_{i,\textrm{old}}(\theta)$. We set $t=10^{-9}$. For FMT the solutions take a longer time to converge than for Onsager theory, as each step has a slower execution time due to the increased complexity of the coupled equations (\ref{eq:self_consistency_1}) and (\ref{eq:self_consistency_2}). \subsection{Conditions for Phase Coexistence} Once we have found the ODFs we solve the phase coexistence equations. The requirements for phase coexistence between two phases \textit{A} and \textit{B} are the mechanical and chemical equilibria between the phases as well as the equality of temperature in the two coexisting phases (which is trivial in hard-body systems). Hence we have the non-trivial conditions \begin{equation} \label{eq: p_equality} p^{\textit{A}}=p^{\textit{B}} \end{equation} and \begin{equation} \label{eq: mu_equality} \mu_i^{\textit{A}}=\mu_i^{\textit{B}}, \end{equation} where $i=1,2$ again labels the species. We calculate the the total Helmholtz free energy $F$ numerically by inserting $\Psi_{i}(\theta)$ into Eqs.~(\ref{eq:ideal}), (\ref{eq: free_energy_2}) and (\ref{eq: free_energy_3}). Likewise, the pressure can be obtained numerically as \begin{equation} \label{eq: pressure} p = -\frac{F}{V}+\sum_{i=1}^{2}\rho_{i}\frac{\partial(F/V)}{\partial \rho_{i}} \end{equation} and the chemical potentials as \begin{equation} \label{eq: chem_pot} \mu_{i}= \frac{\partial (F/V)}{\partial \rho_{i}}. \end{equation} We define a reduced pressure $p^{*}=\beta p R_{1}^{3}$ and reduced chemical potentials $\mu_{i}^{*}=\beta \mu_{i}$. Eqs.~(\ref{eq: p_equality}) and (\ref{eq: mu_equality}) are three equations for four unknowns (two state points each characterised by two densities). Therefore, regions of two-phase coexistence depend parametrically on one free parameter. Eqs.~(\ref{eq: pressure}) and (\ref{eq: chem_pot}) are solved numerically with a Newton-Raphson procedure \cite{press.wh:2007.a}. The resulting set of solutions yields the binodal. \textit{I}-\textit{N}-\textit{N} triple points are located where the \textit{I}-\textit{N} and \textit{N}-\textit{N} coexistence curves cross. Therefore, regions of two-phase coexistence depend parametrically on one free parameter (which can be chosen arbitrarily, e.g. as the value of concentration $x$ in one of the phases). \subsection{Equation of State for the Isotropic Phase} Analytic expressions for the free energy, pressure and chemical potentials for the isotropic phase may be found on insertion of $\Psi(\theta)=1/(4\pi)$ into the ideal (\ref{eq:ideal}) and excess (\ref{eq: free_energy_fmt_bulk}) parts of free energy functional. For this purpose, we use \begin{equation} \int_{0}^{\frac{\pi}{2}}d\theta \sin \theta\int_{0}^{\frac{\pi}{2}}d\theta' \sin \theta'K(\theta,\theta')= \frac{\mathcal{I}_{1}}{8\pi}=\frac{\pi^{2}}{2} \end{equation} and \begin{equation} \int_{0}^{\frac{\pi}{2}}d\theta \sin \theta\int_{0}^{\frac{\pi}{2}}d\theta' \sin \theta'\int_{0}^{\frac{\pi}{2}}d\theta'' \sin \theta'' L(\theta,\theta', \theta'')\\= \frac{\mathcal{I}_{2}}{16\pi} =\frac{\pi^{3}}{2} \end{equation} where the integrals $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are calculated in Appendix A. The expressions for the FMT isotropic free energy, $\beta F_{\textrm{iso}}/V$, pressure, $p_{\textrm{iso}}^{*}$ and chemical potentials $\mu_{i,\textrm{iso}}^{*}$ are \begin{align} \label{eq: isotropic_free_energy} \frac{\beta F_{\textrm{iso}}}{V}R_{1}^{3} = c&(1-x)\ln \left( \frac{c(1-x)}{4\pi}\right) + cx\ln \Big( \frac{cx}{4\pi}\Big)-c \notag \\ & +\frac{\pi^{2}}{2}c^{2}\left[(1-x)^{2}+x^{2}\lambda^{3}+x(1-x)(\lambda^{2}+\lambda) \right] \notag \\ & + \frac{\pi^{2}}{3}c^{3}\left[(1-x)^{3}+x^{3}\lambda^{6}+3x(1-x)^{2}\lambda^{2}+3x^{2}(1-x)\lambda^{4} \right], \end{align} \begin{align} \label{eq: isotropic_pressure} p^{*}_{\textrm{iso}}= c &+ \frac{c^{2}\pi^{2}}{2}\big[(1-x)^{2}+x^{2}\lambda^{3}+(\lambda^{2}+\lambda)(x-x^{2}) \big] \notag \\ & +\frac{2\pi^{2}c^{3}}{3}\Big[x^{3}\lambda^{6}+3(x^{2}-x^{3})\lambda^{4}\notag \\ & +3(x-2x^{2}+x^{3})\lambda^{2} +(1-x)^{3} \Big], \end{align} \begin{align} \label{eq: isotropic_mu_1} \mu^{*}_{1,\textrm{iso}} = \ln& \left(\frac{c(1-x)}{4\pi}\right) +\frac{c\pi^{2}}{2}\big[2(1-x)+x(\lambda^{2}+\lambda)\big] \notag \\ & + c^{2}\pi^{2}[(1-x)^{2}+2x(1-x)\lambda^{2}+x^{2}\lambda^{4}], \end{align} \begin{align} \label{eq: isotropic_mu_2} \mu^{*}_{2,\textrm{iso}}= \ln& \left(\frac{cx}{4\pi}\right) +\frac{c\pi^{2}}{2}\big[2x\lambda^{3}+(1-x)(\lambda^{2}+\lambda)\big] \notag \\ & +c^{2}\pi^{2}[x^{2}\lambda^{6}+2x(1-x)\lambda^{4}+(1-x)^{2}\lambda^{2}]. \end{align} The Onsager versions of these equations are given by the same expressions but without the final bracketed term in $c^{3}$ for $\beta F_{\textrm{iso}}/V$ and $p^{*}_{\textrm{iso}}$ and $c^{2}$ for $\mu^{*}_{i,\textrm{iso}}$. The authors of Ref.~\cite{eppenga.r:1984.a} claim that inclusion of the exact third virial term along with the Onsager term, at least for monodisperse platelets, would give a {\em worse} equation of state in the isotropic phase than the Onsager term alone. \subsection{Isotropic-Nematic Bifurcation Analysis} On increasing the density in the isotropic state, a point is reached known as the bifurcation density, where an infinitesimal nematic perturbation destabilises the system. The first \textit{I}-\textit{N} bifurcation analysis for a liquid crystalline system was performed in Ref.~\cite{kayser.rf:1978}. This was extended to a class of liquid crystal models in Ref.~\cite{mulder89}. The bifurcation concentration lies inside the coexistence region for the monodisperse case of platelets \cite{cheung.dl:2008.e} but as we will see, this is not always true in the binary case. We insert $\Psi_{1}(\theta)= [1+\epsilon_{1}P_{2}(\cos \theta)]/4\pi$ and $\Psi_{2}(\theta)= [1+\epsilon_{2}P_{2}(\cos \theta)]/4\pi$ into the free energy (\ref{eq:ideal}) and (\ref{eq: free_energy_fmt_bulk}), where $\epsilon_{1}$ and $\epsilon_{2}$ are small parameters measuring the strengths of the nematic perturbation and $P_{2}(\cos \theta)=(3\cos^{2}(\theta)-1)/2$ is the second degree Legendre polynomial in $\cos\theta$. We then extract all second order terms, i.e. those proportional to $\epsilon_{1}^{2}$, $\epsilon_{2}^{2}$ and $\epsilon_{1}\epsilon_{2}$, respectively. The coefficients of these terms are denoted by $a_{1}(c)$, $a_{2}(c)$ and $a_{12}(c)$, respectively. These involve integrals of Legendre polynomials and are obtained using the integrals $\mathcal{I}_{3},\mathcal{I}_{4},\mathcal{I}_{5}$ defined in Appendix A. One then solves $\det \textbf{M}=0$ \cite{vanroij.r:1996.a} where \begin{equation} \label{eq: bifurcation_matrix} \textbf{M}=\left( \begin{array}{cc} a_{1}(c) & a_{12}(c)/2 \\ a_{12}(c)/2 & a_{2}(c) \end{array} \right). \end{equation} Here \begin{align} \label{eq: matrix_element_11} a_{1}(c) = & \frac{c}{10}(1-x)-\frac{\pi^{2}}{80}c^{2}(1-x)^{2}\notag \\ &-\frac{\pi^{2}}{40}c^{3}(1-x)^{3}-\frac{\pi^{2}}{40}c^{3}x(1-x)^{2}\lambda^{2}, \end{align} \begin{align} \label{eq: matrix_element_22} a_{2}(c) = & \frac{cx}{10}-\frac{\pi^{2}}{80}c^{2}\lambda^{3}x^{2}\notag \\ &-\frac{\pi^{2}}{40}\lambda^{6}c^{3}x^{3}-\frac{\pi^{2}}{40}c^{3}x^{2}(1-x)\lambda^{4}, \end{align} \begin{align} \label{eq: matrix_element_12} \frac{a_{12}(c)}{2}=&-\frac{\pi^{2}}{160}c^{2}x(1-x)(\lambda^{2}+\lambda)\notag \\ &-\frac{\pi^{2}}{40}c^{3}x(1-x)^{2}\lambda^{2}-\frac{\pi^{2}}{40}c^{3}x^{2}(1-x)\lambda^{4}. \end{align} The results for the spinodals were checked by running the self-consistency program for the solutions of the coupled ODFs (\ref{eq:self_consistency_1}) and (\ref{eq:self_consistency_2}) with the trial functions $1/4\pi$; the locus of $c$ values (for a given value of $\lambda$) for which the maximum number of iterations occured was found to agree numerically very well to the \textit{I}-\textit{N} spinodal. To calculate the spinodals in the $(x,p^{*})$ representation we insert the $(x,c)$ values which form the spinodal into the expression for the isotropic pressure (\ref{eq: isotropic_pressure}). In the monodisperse limit, one needs to solve the following cubic polynomial to calculate the bifurcation point $c_{*}$ \begin{equation} c_{*}-\frac{\pi^{2}}{8}c_{*}^{2}-\frac{\pi^{2}}{4}c_{*}^{3}=0, \end{equation} where the bifurcation concentration is $c_{*}=8/\pi^{2}=0.811$ for Onsager theory and $c_{*}=0.434$ for FMT, the latter much closer to the limit of stability of the isotropic phase as observed in simulations \cite{cheung.dl:2008.e}. \subsection{Symmetry-Conserved Demixing Spinodals} A thermodynamic phase is locally stable if the determinant of the Hessian matrix of the Helmholtz free energy density with respect to the species densities is positive. The spinodal is the limit of stability, defined by $\det \textbf{N}=0$, where \begin{equation} \label{eq: hessian} \textbf{N}=\left( \begin{array}{cc} \frac{\partial^{2}(F/V)}{\partial \rho_{1}^{2}} & \frac{\partial^{2}(F/V)}{\partial \rho_{1}\partial \rho_{2}} \\ \frac{\partial^{2}(F/V)}{\partial \rho_{1}\partial \rho_{2}} & \frac{\partial^{2}(F/V)}{\partial \rho_{2}^{2}} \end{array} \right). \end{equation} For the \textit{I}-\textit{I} spinodals, we insert $\Psi_{1}(\theta)=\Psi_{2}(\theta)=1/4\pi$ into the free energy (\ref{eq: isotropic_free_energy}). This yields an analytic solution of Eq.~(\ref{eq: hessian}) for Onsager theory, given by \begin{equation} \left(\pi^{2}+\frac{1}{\rho_{1}} \right) \left(\pi^{2}\lambda^{3}+\frac{1}{\rho_{2}} \right) - \frac{\pi^{4}}{4} \left(\lambda^{2}+\lambda \right)^{2}=0 \end{equation} and for FMT by \begin{align} \label{eq:i-i_fmt} &\Big(\pi^{2}+2\pi^{2}\rho_{1}+2\rho_{2}\pi^{2}{\lambda}^{2}+\frac{1}{\rho_{1}} \Big) \notag \\ & \times\Big(\pi^{2}\lambda^{3}+2\rho_{2}\pi^{2}\lambda^{6}+2\rho_{1}\pi^{2}\lambda^{4}+\frac{1}{\rho_{2}}\Big)\notag \\ & - \left( 2\rho_{1} \pi^{2}\lambda^{2}+2\rho_{2}\pi^{2}{\lambda}^{4} \right) ^{2}=0. \end{align} \textit{I}-\textit{I} demixing never occurs for the values of $\lambda$ considered here; indeed solutions of Eq.~(\ref{eq:i-i_fmt}) only begin to exist at about $\lambda=20$. For the \textit{N}-\textit{N} spinodals, there are no such analytic equations (for the Zwanzig model, see Ref.~\cite{harnau.l:2002.d}). For practical reasons we rather solve \begin{equation} \left(\frac{\partial p^{*}}{\partial \rho_{2}}\right)_{\mu_{1}}=0, \label{eq: spin} \end{equation} which is equivalent \cite{vanroij.r:1996.a} to solving $\det \textbf{N}=0$. In order to calculate the \textit{N}-\textit{N} spinodals we have to numerically evaluate the left hand side of Eq.~(\ref{eq: spin}). Exchanging the species labels in Eq.~(\ref{eq: spin}) one obtains the same results. The \textit{N}-\textit{N} spinodals are calculated to give an idea of the location of the \textit{N}-\textit{N} phase boundaries. For cases where there is \textit{N}-\textit{N} coexistence closed by critical point, the spinodal and the binodal coincide at the critical point. \section{Results} \label{results} We have calculated the phase diagrams of binary platelet mixtures for seven different size ratios, $\lambda = 1.1,1.4,1.7,2,2.5,4$ and $5$. Fig.~1a shows the results for the slightly asymmetric case $\lambda = 1.1$ in the $(x,c)$ representation. The \textit{I}-\textit{N} transition is the only type of transition which we find for this size ratio. The FMT results show that the \textit{I}-\textit{N} transition concentrations at $x=0$ are $c_{I}=0.418$ and $c_{N}=0.46$ respectively, in agreement with the monodisperse results \cite{reich.h:2007.a}. At $x=1$, the coexistence values are $c_{I}=0.418/\lambda^{3}=0.314$ and $c_{N}=0.46/\lambda^{3}=0.346$. The coexistence curves interpolate smoothly from $x=0$ to $x=1$. The tie lines joining coexisting isotropic and nematic phases are naturally vertical at $x=0,1$ (corresponding to the cases of the pure systems of small and big platelets, respectively) whereas between $x=0$ and $x=1$ they vary in gradient, leaning with large positive gradient as $x$ increases from zero composition, less so at about 50\% composition and then more so again on approaching $x=1$. Therefore, there is stronger fractionation at 50\% composition than towards $x=0$ and $x=1$, such that the isotropic phase is dominated by particles of species 1 (smaller species) and the nematic phase by particles of species 2 (larger species). The biphasic region, where there is coexistence between the isotropic and nematic phase, is very narrow in FMT. At zero composition, Onsager theory predicts a density jump of $\thicksim22\%$ whereas this is only $\thicksim9\%$ for FMT, which agrees more closely to simulation results of 8\% \cite{eppenga.r:1984.a}. The bifurcation concentration for FMT is $c_{*}=0.434$ at $x=0$ and $c_{*}=0.434/\lambda^{3}=0.326$ at $x=1$. The spinodal for FMT lies closer to the isotropic phase boundary than the nematic phase boundary (whereas the converse is true for Onsager theory). Fig.~1b shows the same results but in the $(c_{1},c_{2})$ representation. The results for both theories again interpolate smoothly between the two pure limits. The two branches of the binodal in this representation move from the $c_{1}$-axis to the $c_{2}$-axis with increasing composition. Hence the tie lines move from being horizontal on the $c_{1}$-axis to vertical on the $c_{2}$-axis. In the $(x,p^{*})$ representation (Fig.~1c) the tie lines are horizontal due to the requirement of equal pressure in the coexisting phases. The spinodal lies above the nematic branch of the binodal in this representation because it is obtained by inserting the bifurcation densities into the isotropic equation of state, which yields a higher pressure than the coexistence value. The binodal obtained from FMT is located at significantly smaller densities as compared to that from Onsager theory. The isotropic end of the tie line is at a lower composition than the nematic end of the tie line. For Onsager theory at zero composition, the isotropic branch of the binodal intersects the $c$-axis at $c_{I}=0.666$ and the nematic branch intersects at $c_{N}=0.849$, in agreement with the monodisperse limit found in earlier work \cite{esztermann.a:2006.a}. The binodals interpolate smoothly from $x=0$ to $x=1$ where the values of $c$ corresponding to the \textit{I}-\textit{N} coexistence concentrations are $c_{I}=0.5$ and $c_{N}=0.638$. The nature of the tie lines is similar to FMT. The \textit{I}-\textit{N} spinodal lies between the two branches of the binodal and interpolates smoothly from zero composition, where $c=8/\pi^{2}=0.811$ to $x=1$ where $c_{*}=0.609$. The nematic phase of a mixture of two components can be characterised by two partial nematic order parameters, $S_{1}$ and $S_{2}$ defined by \begin{equation} \label{eq: order_parameter} S_{i} = 4\pi \int_{0}^{\frac{\pi}{2}}d\theta\sin\theta\Psi_{i}(\theta)P_{2}(\cos\theta). \end{equation} The total nematic order parameter is the weighted average \begin{equation} S_{\textrm{tot}}= (1-x)S_{1}+xS_{2}. \end{equation} \label{eq: total_order} Here we investigate the behaviour of these quantities at \textit{I}-\textit{N} and \textit{N}-\textit{N} coexistence. We have chosen $S_{\textrm{tot}}$ to be a simple weighted average of the $S_{i}$. As such, each particle contributes to $S_{\textrm{tot}}$ independently of its size. Of course there are other suitable choices which may be appropriate to certain applications; the $S_{i}$ in the sum for the total order may be weighted by the surface area of the platelet, for example. Due to the absence of particles of species 2 at $x=0$, $S_{\textrm{tot}}$ and $S_{1}$ take the same value. Similarly at $x=1$, $S_{\textrm{tot}}$ and $S_{2}$ take the same value. As $x$ increases from $0$ to $1$, $S_{\textrm{tot}}$ changes smoothly. In Fig.~1d the FMT values for $S_{1}$, $S_{2}$ and $S_{\textrm{tot}}$ are smaller than those obtained from Onsager theory, which places the values much higher. However FMT predicts that the difference between $S_{1}$ and $S_{2}$ to be bigger than Onsager theory does. $S_{\textrm{tot}}$ at $x=0$ takes the same value $S_{\textrm{tot}}=0.531$ as previous work (and agrees well with simulation \cite{reich.h:2007.a}) in the monodisperse case and the the order parameters vary smoothly in the same manner as for Onsager theory. In Fig.~2 we show the results for $\lambda=1.4$. There is a widening of the biphasic region between the two pure components, clearly seen in Fig.~2a where we show the phase diagram in the $(x,c)$ representation. FMT again predicts that the mixture undergoes \textit{I}-\textit{N} phase separation at lower densities than Onsager theory. The tie lines become less steep with a smaller positive gradient than for $\lambda=1.1$ for intermediate values of composition indicating that there is larger difference in mole fraction between coexisting isotropic and nematic states. The widening of the biphasic gap is even more noticeable in the ($c_{1},c_{2}$) representation (Fig.~2b), especially for the case of Onsager theory. In the $(x,p^{*})$ representation (Fig.~2c), the spinodal again lies above the nematic binodal for Onsager theory. However, for FMT we see that while the spinodal is above the nematic branch of the binodal close to $x=0$ and $x=1$ the curve enters the biphasic region in between about $x=0.1$ and $x=0.6$. $S_{1}$ is smaller than $S_{2}$ and their difference has increased for both theories, with FMT still possessing the larger difference suggesting that the particles of species 2 are significantly more ordered than species 1 for a given mole fraction at coexistence. In Fig.~3 we plot the graphs for $\lambda=1.7$. The \textit{I}-\textit{N} biphasic gap becomes even more pronounced. The results for FMT (Fig.~3a) show that tracing along the nematic branch of the binodal as $x$ increases leads to an increase in $c$. At approximately $x=0.2$, $c$ then decreases and near $x=0.6$ bends back on itself before approaching $x=1$. This bending of the binodal constitutes a re-entrant phenomenon. There is also a large range of compositions between about $x=0.1$ and $x=0.6$ for which the biphasic \textit{I}-\textit{N} phase overlaps for both theories, which is also seen clearly in the ($c_{1},c_{2}$) representation (Fig.~3b). In the $(x,p^{*})$ representation (Fig.~3c) both theories predict a region of composition values for which the spinodal lies inside the biphasic region. For illustration of the re-entrant part of the phase diagram one should keep a constant fluid composition near $x=0.6$ but increases the pressure from $p^{*}=0$ to $p^{*}=1$: the state changes from \textit{I} $\rightarrow$ \textit{I}+\textit{N$_{2}$} $\rightarrow$ \textit{N} $\rightarrow$ \textit{I}+\textit{N$_{2}$} $\rightarrow$ \textit{N} where \textit{N$_{2}$} is a nematic phase composed mostly of particles of species 2. The order parameters along the nematic branch of the binodal are shown in Fig.~3d. $S_{1}$ remains lower than $S_{2}$ for a given composition, again suggesting that the particles of species 2 are more ordered than those of species 1 at coexistence. When the re-entrant feature occurs at about $x=0.6$, $S_{1}$, $S_{2}$ and $S_{\textrm{tot}}$ drop sharply. The difference between $S_{1}$ and $S_{2}$ along the nematic branch of the binodal has become much larger than for $\lambda=1.4$. In Fig.~4 we present the phase diagrams for $\lambda=2$. The most striking feature after increasing the size ratio to $\lambda=2$ is the stable \textit{N}-\textit{N} coexistence region. For FMT the $(c_{1},c_{2})$ representation (Fig.~4a) reveals that this region is in the form of an almost symmetric hump suggesting that the fractionation between the two distinct nematic phases becomes less with increasing density. There is also a triple point which is in the form of a triangle connecting a low composition isotropic phase, a nematic phase composed mostly of particles of species 1 (\textit{N$_{1}$}) and a nematic phase composed mostly of particles of species 2 (\textit{N$_{2}$}). The symmetric \textit{N}-\textit{N} coexistence region is more clearly seen in the $(x,p^{*})$ representation (Fig.~4c) between $x$ just greater than 0 and $x=0.45$ ending in a critical point at $(x,p^{*})=(0.19,2.72)$. In the $(x,p^{*})$ representation the triple point collapses onto a line and the pressure at the triple point is approximately $p^{*}=1.75$. The \textit{I}-\textit{N} biphasic region has become increasingly pronounced. The topology of the phase diagrams is the same as that obtained in Onsager theory results (Fig.~4b and Fig.~4d). The re-entrant feature obtained from Onsager theory is less pronounced than in FMT. The \textit{N}-\textit{N} phase separation occurs over a larger range of composition values (up to near $x=0.5$) and the triple point occurs at approximately a unit of reduced pressure higher than the triple point predicted by FMT, nevertheless, as we emphasise, the topologies predicted by both theories are the same. The \textit{I}-\textit{N} spinodal enters the biphasic region for both theories. We postpone the discussion of the order parameters for $\lambda=2$ and higher size ratios until the end of this section. In Fig.~5 we show the results for $\lambda=2.5$. There is \textit{N}-\textit{N} demixing in the system, as is observed for $\lambda=2$, however there is a big difference in the toplogy of the phase diagrams in that the demixing now does not end in a critical point. For FMT in the ($c_{1},c_{2}$) representation (Fig.~5a) the \textit{N}-\textit{N} region opens up and the two branches of the binodal extend outwards suggesting that the demixing extends to arbitrarily high density. No critical point is observed up to the densities we examine; we follow the phase boundaries up to $c=1$ for FMT and $c=1.9$ for Onsager theory. A similar splaying of the phase boundaries is observed in Fig.~5b in the ($c_{1},c_{2}$) for Onsager theory, suggesting that the topology obtained with FMT is correct. Also, the \textit{I}-\textit{N}$_{1}$ region has shrunk to a tiny region close to zero composition, making the triple region mostly dominated by the nematic state rich in species 2. In the $(x,p^{*})$ representation for FMT (Fig.~5c) the re-entrant feature has become extremely pronounced, extending as low as approximately $x=0.5$ in the $(x,p^{*})$ representation and the triple point occurs at approximately $p^{*}=1.8$. In the ($x,p^{*}$) representation for Onsager theory (Fig.~5d), the re-entrant feature has also become more pronounced but less so than for FMT. The triple line extends over approximately the same composition range ($x \leq 0.8$) as for FMT but occurs at just over a unit of reduced pressure higher than for FMT. In Fig.~6 we present the phase diagrams for $\lambda = 4$. In the $(c_{1},c_{2})$ representation for FMT (Fig.~6a) the strength of the \textit{I}-\textit{N}$_{2}$ fractionation effect becomes very large. The \textit{N}-\textit{N} separation is very wide giving a huge immiscibility gap, hence we do not show it. Fig.~6b also shows a large immiscibility gap in the ($c_{1},c_{2}$) representation for Onsager theory. We again have confidence that the results from FMT are quantitatively more accurate than those from Onsager theory. At concentrations $c_{1},c_{2}<0.1$ the coexisting compositions already approach closely the $c_{1},c_{2}$ axes highlighting that coexisting compositions are close to $x=0,1$ as is clearly seen in the $(x,p^{*})$ representation (Fig.~6c for FMT and Fig.~6d for Onsager theory). The general trend of the re-entrant bend moving to lower composition with increasing $\lambda$ has continued here, reaching as low as about $x=0.3$ for FMT and near $x=0.5$ for Onsager theory. In Fig.~7 we plot the results for $\lambda=5$. In the ($c_{1},c_{2}$) representation for FMT (Fig.~7a) the \textit{I}-\textit{N}$_{2}$ coexistence region is very pronounced. The larger platelets have an area twenty-five times that of the smaller platelet so the asymmetry of the mixture is large. The \textit{I}-\textit{N} spinodal remains close to the isotropic branch of the binodal; more so than in Fig.~7b, the ($c_{1},c_{2}$) representation of Onsager theory. The $(x,p^{*})$ representation for FMT (Fig.~5c) shows that the general trend of the re-entrant bend moving to lower composition has continued, here reaching as low as about $x=0.2$ for FMT. There is a very narrow region of coexistence between the isotropic and nematic state of species 2 up to about $p^{*}=0.06$, suggesting that up to this pressure there is only a small fractionation effect. This fractionation becomes wider on increasing the pressure beyond this point, with the phase boundaries almost reaching $x=0,1$ by $p^{*}=0.1$. The re-entrant bend is also a dominating feature of the phase diagram in Fig.~7d for Onsager theory suggesting the FMT phase behaviour is correct, however it is not as pronounced as for FMT and the narrow \textit{I}-\textit{N}$_{2}$ \textit{handle} present at low pressures for FMT is wider for Onsager theory. In Fig.~8 we show the partial nematic order parameters $S_i$ in the coexisting nematic phase(s) for $\lambda = 2, 2.5, 4$ and $5$, as obtained from FMT. For $\lambda=2$ (Fig. 8a) $S_1$ and $S_2$ both rise rapidly near zero composition to near unity. $S_1$ takes on smaller values than $S_2$ for a given mole fraction as has already been observed for the lower size ratios. Hence the (smaller) particles of species 1 are much less ordered than particles of species 2 at a given composition, since the particles of species 2 are high in number and there is more freedom for the smaller particles to rotate in the dense system. Both partial order parameters remain close to unity as the composition $x$ is increased from 0 to about 0.5. The system is at \textit{N}-\textit{N} coexistence over this range (see Fig. 4c for the corresponding phase diagram). $S_{\textrm{tot}}$ reaches as low as about 0.5 and $S_{1}$ becomes as small as 0.2 at $x=1$. For $\lambda= 2.5$ (Fig. 8b) the \textit{N}-\textit{N} coexistence does not end at a critical point, leaving an interval (in composition) that the curves $S_i(x)$ do not enter. For $\lambda=4$ (Fig. 8c) $S_{\textrm{tot}}$ varies between about 0.3 and almost 1. This range of $S_{\textrm{tot}}$ increases further for $\lambda=5$ (Fig. 8d) where it reaches as low as about 0.2. For $\lambda=4,5$, $S_{\textrm{tot}}$ also increases more rapidly as the composition increases than is the case for the lower size ratios. At high compositions, the (small) particles of species 1 possess very low nematic order, reaching less than 0.1 due to the larger size difference between the two species. Our phase diagrams share several features with those reported in previous studies of binary mixtures of anisometric hard core particles. Widening of the \textit{I}-\textit{N} phase coexistence region upon increasing the bidispersity parameter was found in binary mixtures of thick and thin rods \cite{vanroij.r:1998.a}, long and short rods \cite{lekkerkerker.hnw:1984.a}, mixtures of rods and platelets \cite{wensink.hh:2001.b}, as well as in binary mixtures of platelets using both the Parons-Lee scaling of the Onsager functional \cite{wensink.hh:2001.a, wensink.hh:2004.a} and the Zwanzig model \cite{harnau.l:2002.c, harnau.l:2002.d}. \textit{N}-\textit{N} phase coexistence ending in an upper critical point, as we find for an intermediate range of values of size ratios $\lambda$, occurs in certain regimes for binary mixtures of rods and platelets. A notable difference to other studies of platelet mixtures \cite{wensink.hh:2004.a} is that we do not find \textit{N}-\textit{N} coexistence ending in a lower critical point, at least not for the range of densities and size ratios that we explored. Similar results to ours, where \textit{N}-\textit{N} coexistence ends in an upper critical point were obtained using Onsager theory for mixtures of thick and thin rods \cite{vanroij.r:1998.a}. This system of rods also displays isotropic-isotropic phase coexistence at high enough values of the diameter ratio, which we do not find to be stable in the present study. \section{Conclusions and Outlook} \label{outlook} We have studied the bulk phase behaviour of binary mixtures of hard platelets, including isotropic and nematic states. The platelets are assumed to have circular shape and vanishing thickness. We have not considered positionally ordered phases such as columnar and crystalline phases, which are not expected to occur for the present model of particles with zero volume and zero packing fraction. Note that a first order \textit{N}-\textit{Col} transition was found at non-zero packing fraction in the limit of vanishing thickness \cite{bates.m:1998.a}; simulations in this limit are possible because the model can be mapped onto a system of particles with finite volume but variable shape. Also, platelets with non-zero thickness exhibit a \textit{N}-\textit{Col} transition \cite{wensink.h:2009.a}; this was also reported for the Zwanzig model \cite{bier.m:2004.a}. For a variety of size ratios, we have compared the results of Onsager theory and FMT. In the monodisperse limit we find that Onsager theory overestimates the \textit{I}-\textit{N} transition densities and predicts a larger biphasic gap than FMT; the results for FMT in this limit compare quantitatively well with those from simulation. Both theories predict a first order \textit{I}-\textit{N} phase transition. We expect that Onsager theory overestimates the density jump at coexistence and also overestimates the size of the biphasic gap. The FMT results show the appearance of re-entrant phenomena at a lower size ratio than for Onsager theory. FMT results also show a larger range of compositions for which \textit{N}-\textit{N} demixing at a given size ratio occurs. The \textit{N}-\textit{N} demixing occurs at $\lambda = 2$, where it is closed by an upper critical point. For $\lambda \geq 2.5$ there is no critical point up to the densities we consider. In an experimental system of platelets with nonzero volume, one would expect a positionally ordered phase to be favoured before any possible remixing into a homogeneous nematic phase. We also examine the degree of nematic ordering along the phase boundaries for a selection of size ratios up to $\lambda = 2$ where \textit{N}-\textit{N} demixing ends in an upper critical point. The partial nematic order parameters $S_{1}$, $S_{2}$ vary smoothly with increasing composition and hence so does $S_{\textrm{tot}}$. Where a re-entrant feature occurs, all three order parameters along the nematic branch of the binodal drop sharply. As $\lambda$ increases, $S_{2}$ becomes smaller at $x=1$, reaching $<0.2$ for $\lambda = 2$. We do not find any stable \textit{I}-\textit{I} demixing for the range of $\lambda$ in this investigation. Through the range of $\lambda$ values considered, a striking observation is that whilst FMT predicts the occurence of phase boundaries at locations quantitatively different from Onsager theory, the topology of the phase diagrams for a particular choice of $\lambda$ is the same as for Onsager theory. Coupled with the fact that the \textit{I}-\textit{N} transition for the monodisperse case agrees well with simulation results \cite{reich.h:2007.a}, we gain confidence that the phase diagrams predicted by FMT are quantitatively more reliable since we assume that Onsager theory predicts the correct physics qualitatively. Furthermore, we have confidence that the order parameter profiles predicted by FMT are closer to those that would be observed from simulation studies, since the monodisperse limit in the present theory yields results for the \textit{I}-\textit{N} transition and nematic order parameter that is in good agreement with simulation results. Whether the differences to the true transition densities for the case of binary mixtures increases remains to be seen. The results from Onsager theory cannot be obtained by some scaling of the results that exist for binary mixtures of rods (thick and thin \cite{vanroij.r:1998.a} or long and short \cite{lekkerkerker.hnw:1984.a}). For example, for there to be a mapping between thick and thin rods \cite{vanroij.r:1998.a} and the present system, we would require simultaneously that $(1+d)=\lambda^{2}+\lambda$ and $d=\lambda^{3}$, where $d=D_{2}/D_{1}>1$ is the diameter ratio of the thick and thin rods. These requirements are obtained from consideration of the free energy for thick and thin rods (compare Eq.~(2) of Ref.~\cite{vanroij.r:1998.a} with Eq.~(\ref{eq: free_energy_2}) of the present investigation). Clearly these conditions cannot be satisfied simultaneously except for the trivial case $\lambda=d=1$. In the monodisperse limit, the mapping of concentrations from rods to platelets is $c_{\textrm{rod}}=(\pi^{2}/2)c_{\textrm{platelet}}$ where $c_{\textrm{rod}}=(\pi/4)L^{2}D$ is the conventional dimensionless concentration for rods ($L$ is the rod length, $D$ is the rod diameter). Clearly, simulation results for the systems studied in the present paper are most desirable. The predictions of the present work could also be tested experimentally. For example, for gibbsite (having typical experimental radius of approximately 100nm) and hydrotalcite (typically between about 25 and 75 nm) \cite{maurice_thesis} one has values of $\lambda$ between $1.3$ and $4$, which are in the range of the present work. We have taken mean radii here. Of course in experiments there will be effects due to polydispersity and our model ignores effects of finite platelet thickness. Interesting future work could involve examining the phase behaviour of binary mixtures of polarizable platelets, that is particles that interact with some external applied magnetic field, as has been investigated for binary rod mixtures \cite{varga.s.:2000.a, dobra.s:2006.a}. Also, binary mixtures of rods and platelets could be investigated \cite{vanderkooij.f.m.:2000.b, varga.s.:2002.a, varga.s.:2002.b, varga.s.:2002.c, wensink.h.h.:2002.b}. Another line of investigation would be to incorporate polydispersity into the model since many platelet systems often have a significant polydispersity, see for example \cite{vanderkooij.f.m:2000.a}. This would lead to an extension of the work by Speranza and Sollich and others on rod-like particles, \cite{speranza.a:2002.a, speranza.a:2003.a, speranza.a:2003.b} and by Wensink and Vroege on thickness-polydisperse platelets \cite{wensink.h.h:2002.a}. \\

proofpile-arXiv_069-12677

\section{Introduction} {SAX J1753.5$-$2349}\ is a neutron star Low Mass X-ray Binary (LMXB) discovered in 1996 by {\it BeppoSAX}/Wide Field Camera (WFC) during a single type-I X-ray burst \cite{zand99}. However, no steady emission was detected from the source leading to an upper limit of about 5 mCrab (2--8 keV) for a total exposure of 300 ks \cite{zand99}. Cornelisse et al. (2004) proposed {SAX J1753.5$-$2349}\ being member of a possible non-homogeneous class of LMXBs, the so-called ``burst-only'' sources (see also Cocchi et al. 2001). These are a group of nine bursters discovered by {\it BeppoSAX}/WFC when exhibiting a type-I burst without any detectable persistent X-ray emission. Recently, {\it INTEGRAL}\ identified two new members of this class. In fact, photospheric radius expansion (PRE) bursts have been caught in two previously unclassified sources, namely XMMU~J174716.1--281048\ \cite{brandt06} and \ax\ \cite{chelo07}. Afterwards, both have been classified as ``quasi-persistent'' Very Faint X-ray Transients (VFXTs), since they undergo prolonged accretion episodes of many years at low $\dot{M}$\ (Del Santo et al. 2007, Bassa et al. 2008). VFXTs are transients showing outbursts with low peak luminosity ($10^{34}$--10$^{36}$ erg s$^{-1}$\ in 2--10 keV), mainly discovered with high sensitivity instruments on-board {\it Chandra}\ and {\it XMM-Newton}\ during surveys of the Galactic Center region \cite{wij06}. They are believed to be the faintest known accretors, and are very likely a non homogeneous class of sources. A significant fraction ($\sim 1/3$) of VFXTs are X-ray bursters (Degenaar \& Wijnands 2009, Del Santo et al. 2007, Del Santo et al. 2008, Cornelisse et al. 2004); thus they can be identified with neutron stars accreting matter from a low mass companion (M $\lesssim$ 1M$_\odot$). \begin{table} \begin{center} \caption{Log of the {\it INTEGRAL}\ observations of the {SAX J1753.5$-$2349}\ region: orbit number (Rev.), start and end time of the observations, exposures time for each orbit taking into account the whole data-set, and number of pointings (SCW) are reported. Observations within a single orbit are not continuous. The first {\it INTEGRAL}\ detection of {SAX J1753.5$-$2349}\ occurred in rev. 732. A data sub-set from rev. 732 to 736 has been used to compute the averaged spectra. The last column reports the exposures of spectra in each orbit.} \scriptsize \begin{tabular}{lccccc} \hline Rev. & Start & End & Total Exp. & SCW & Spec. Exp.\\ & (MJD) & (MJD) & (ks) & & (ks)\\ \hline 724 & 54727.50 & 54728.23 & 58 & 17 & - \\ 725 & 54729.11 & 54731.52 & 198 & 56 & - \\ 726 & 54732.52 & 54734.46 & 160 & 45 & - \\ 729 & 54741.37 & 54741.86 & 42 & 12 & - \\ 731 & 54749.22 & 54749.55 & 20 & 8 & -\\ 732 & 54749.90 & 54750.85 & 83 & 32 & 26.2 \\ 733 & 54754.96 & 54755.46 & 38 & 11 & 10.8\\ 734 & 54756.87 & 54758.54 & 128 & 48 & 36.5\\ 735 & 54760.91 & 54761.53 & 43 & 13 & 23.2 \\ 736 & 54762.03 & 54763.63 & 38 & 49 & 30.0\\ \hline \\ \end{tabular}\\ \label{tab:log} \end{center} \end{table} In 2002 observations with {\it Chandra}\ and {\it XMM-Newton}\ allowed to reveal the nature of four {\it BeppoSAX}\ ``burst-only'' sources: one persistent very-faint source, two faint transient systems (with 2--10 keV peak luminosity in the range $10^{36}$--10$^{37}$ erg s$^{-1}$), and one VFXT (see Wijnands et al. 2006 and reference therein). For the other five bursters, including {SAX J1753.5$-$2349}, only the quiescent emission could be derived ($\sim$10$^{32}$ erg s$^{-1}$; Cornelisse et al. 2004). Wijnands et al. (2006) proposed these systems, as good candidates to be classified as VFXTs (see also Campana 2009). In 2008 October 11, {\it RXTE}/PCA, {\it Swift}/BAT \cite{mark08} and {\it INTEGRAL}/IBIS \cite{cadol08} detected an outburst from {SAX J1753.5$-$2349}\ at 10 mCrab flux level. Then, {\it Swift}/XRT pointed {SAX J1753.5$-$2349}\ on October 23 \cite{degewij08}, during the decline phase of the outburst (Fig. \ref{fig:lc}). An improvement in the source position, R.A.(J2000)=$17^{h} 53^{m} 31.90^{s}$, Dec(J2000)=$-23^{\circ} 48' 16.7''$, has been provided \cite{starl08}. On 2009 March 13, it was re-pointed by {\it Swift}\ and a 3$\sigma$ upper-limit derived. This translates in a luminosity level $\lesssim$ $5 \times 10^{32}$ erg s$^{-1}$\ \cite{delsanto09}. In this paper we present the hard X-ray outburst of {SAX J1753.5$-$2349}\ observed by {\it INTEGRAL}/IBIS, as well as the first broad-band spectral analysis of the steady emission of a ``burst-only''. We estimate the long-term mass-accretion rate and discuss the nature of the transient system. \begin{figure} \centering \includegraphics[height=6cm]{delsanto10_fig1.ps} \caption{{SAX J1753.5$-$2349}\ BAT (top) and IBIS/ISGRI (bottom) count rate evolution in the 15--50 keV and 18-40 keV energy ranges, respectively. The XRT detection time is also shown on the bottom plot. The public BAT light curve starts from 54754 MJD; after MJD=54764 {SAX J1753.5$-$2349}\ was no longer pointed by {\it INTEGRAL}. \label{fig:lc}} \end{figure} \begin{figure*} \centering \includegraphics[height=8cm,angle=-90]{delsanto10_fig2.ps} \includegraphics[height=8cm,angle=-90]{delsanto10_fig3.ps} \caption{XRT and IBIS/ISGRI count rate spectra fitted with a simple power-law ({\it{left}}); the total \texttt{bb+comptt} model (continuous line) and the two single components (dashed lines) ({\it{right}}) \label{fig:model}} \end{figure*} \section{Observation and data analysis} \subsection{{\it INTEGRAL}} This paper is based on {\it INTEGRAL}\ observations of the Galactic Centre region carried out in the framework of the AO6 Key-Programme. Moreover, we used data from a public ToO on the source H 1743-322, at 8.6$^\circ$ from {SAX J1753.5$-$2349}, performed on 2008 October, for a total exposure time of 800 ks (see Tab. \ref{tab:log}). We reduced the data of the IBIS \cite{ube03} low energy detector ISGRI \cite{lebrun03}, and JEM-X \cite{lund03} data using the {\it INTEGRAL}\ Off-Line Scientific Analysis, release 8.0. Due to the source weakness, no signal was found in the JEM-X data. On October 10, the first IBIS detection of {SAX J1753.5$-$2349}\ was found (rev. 732). We extracted the IBIS/ISGRI light curves from each revolution as reported in Tab.\ref{tab:log} (binning size as the Total Exposure column) in the energy range 18--40 keV, 40--80 keV, 80--150 keV. For the spectral extraction, we used a sub-set of the data reported in Tab. \ref{tab:log}, selecting only pointings including {SAX J1753.5$-$2349}\ in the IBIS FOV up to 50\% coding (15$^\circ \times$15$^\circ$). We obtained four averaged spectra from revolutions 732, 733, 734 and 735-736 (the latests have been added together because of the poor statistics). Spectral fits were performed using the spectral X-ray analysis package XSPEC v. 11.3.1. \subsection{{\it Swift}} A {\it Swift}\ ToO was performed on October 23 (Degenaar \& Wijnands 2008). The {\it Swift}/XRT data of observation 00035713002 were collected in photon counting (PC) mode between 2008-10-23 17:48:53 and 21:08:57 UT, for a total on-source net exposure of 1 {\rm{ks}}. They were processed with standard procedures ({\tt xrtpipeline} v0.12.1), filtering and screening criteria by using the {\tt Heasoft} package (v.6.6.1). Moderate pile-up was present, so source events were extracted from an annular region (radii of 20 and 3 pixels; 1 pixel $\sim 2\farcs36$), while background events were extracted from an annular region (radii 120 and 80 pixels) away from background sources. An XRT spectrum was extracted and ancillary response files were generated with {\tt xrtmkarf}, to account for different extraction regions, vignetting and PSF corrections. We used the spectral redistribution matrices v011 in the Calibration Database maintained by HEASARC. All spectra were rebinned with a minimum of 20 counts per energy bin. We retrieved the BAT daily light curves (15--50 keV) available starting from MJD=54754, from the {\it Swift}/BAT transient monitor (Krimm et al. 2006, 2008; http://heasarc.gsfc.nasa.gov/docs/swift/results/transients/) page. \section{Results} The IBIS/ISGRI and BAT count rate of {SAX J1753.5$-$2349}\ are shown in Fig. \ref{fig:lc}. Based on the IBIS data, the hard X-ray outburst started on October 10 at a flux level of 10 mCrab (18--40 keV) and lasted at least 14 days (last pointing at 4 mCrab). This outburst is hence characterised by a fast increase of the flux and a linear decay with a slope of $-$0.13$\pm$0.01. An {\it INTEGRAL}\ pointing with no {SAX J1753.5$-$2349}\ detection was performed eight hours before the outburst started. We also averaged all our data (from rev 724 to 731) collected before the first source detection for a total of 500 ks, resulting in a 3$\sigma$ upper limit of 1 mCrab (Fig. \ref{fig:lc}). In order to look for any possible spectral variability, we fitted the four averaged IBIS spectra with a simple power law. We obtained a constant value (within the errors) of the photon index ($\Gamma$ $\sim$ 2) which indicates, in spite of the flux variation, a steady spectral state. The lack of spectral parameter variation led us to average the IBIS spectra of different revolutions. The 18-100 keV averaged spectrum is well described by a simple power law model with a slope as $2.2 \pm 0.3$. A mean 18--100 keV flux of $1.5\times 10^{-10}$ erg cm$^{-2}$ s$^{-1}$ can be derived. The XRT spectrum can be fitted by an absorbed power law model with a Hydrogen column density of N${\rm _H}=1.8 (\pm 0.6) \times 10^{22}$ cm$^{-2}$. The photon index is $\Gamma = 2.0 \pm 0.5$ and the resulting 2--10 keV absorbed and unabsorbed fluxes are $\sim$4.4 and $\sim$5.2 $\times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$ , respectively. We note that the derived N${\rm _H}$ is higher than the absorption column of $0.83 \times 10^{22}$ cm$^{-2}$ \cite{corne02} found by interpolating the HI maps of Dickey \& Lockman (1990). In fact, the two values are perfectly consistent within the errors, given the large range of values (about $0.4-1.5 \times 10^{22}$ cm$^{-2}$) obtained in the box adopted to calculate the Weighted Average N${\rm _H}$ (with the nH Column Density Tool)\footnote{http://heasarc.gsfc.nasa.gov/docs/tools.html} from the HI maps. The joint IBIS and XRT spectrum (0.3--100 keV) was then fitted with different models. First we used an empirical model such as the power law (Fig. \ref{fig:model}, {\it left}), then the more physical Comptonisation model. Indeed, the 1--200 keV spectrum of X-ray bursters in low/hard state is most likely produced by the upscattering of soft seed photons by a hot optically thin electron plasma (i.e. Barret et al. 2000 and references therein). Moreover, a black-body emission from the neutron star surface is also expected to be observed in the low/hard states of bursters (i.e. Natalucci et al. 2000 and references therein). We tried to add a \texttt{BB} component to the two models. The best fit parameters and mean fluxes are reported in Tab. \ref{tab:fit_sim}. Thus, using a physical thermal Comptonisation model, \texttt{COMPTT} \cite{tita94} in XSPEC, the electron temperature is not constrained, while a lower limit of $\sim$24 keV (at $90\%$) can be inferred (see Tab. \ref{tab:fit_sim} and contour levels in Fig. \ref{fig:cont}). This is consistent with the electrons temperature observed in burster systems, even brighter than {SAX J1753.5$-$2349}\ \cite{barret00}. With the addition of the \texttt{BB} component to the thermal Comptonisation, a typical value of the black-body temperature (kT$_{\rm BB}$ $\sim$0.3 keV) is obtained (Fig. \ref{fig:model}, {\it right}), even though this component is not requested by the Ftest probability ($7 \times 10^{-2}$). We may argue that the high absorption observed in {SAX J1753.5$-$2349}\ could be a strong obstacle to the firm detection of this component. As a firts approximation, the accretion luminosity L$_{\rm acc}$ is coincident with the bolometric luminosity of the source (0.1--100 keV). Using the mean 0.1--100 keV flux obtained with the \texttt{COMPTT} model fit and assuming a distance of 8 kpc (Galactic Centre), a value of L$_{\rm acc}=4.3\times10^{36}$ erg s$^{-1}$\ ($\sim$0.02 L$_{Edd} $) is derived. The averaged mass-accretion rate ($\langle \dot{M}_{\mathrm{ob}} \rangle=R L_{\mathrm{acc}}/GM$, where $G$ is the gravitational constant, $M=1.4~\mathrm{M_{\odot}}$ and $R=10$~km for a neutron star accretor) during the outburst is $6.7 \times 10^{-10}$ M$_\odot$\ yr$^{-1}$. \begin{table*} \begin{center} \caption{The parameters the XRT/IBIS spectra fitted four different models.} \vspace{1em} \renewcommand{\arraystretch}{1.5} \begin{tabular}{lrccccccc} \hline Model & N$_{H}$ & $kT_{BB}$ & $\Gamma$ & $E_{c}$ & $kT_{e}$ & $\tau$ & $\chi^2_{\nu}$(dof) & $F_{\rm bol}^{\mathrm a}$ \\ & $10^{22}$ ($\rm {cm}^{-2}$) & (keV) & & (keV) & (keV) & & & (erg cm$^{-2}$ s$^{-1}$ ) \\ \hline POW & 2.2$^{+0.5}_{-0.4}$ & - & $2.3 \pm 0.3$ & - & - & - & 0.91(19)& $1.3\times 10^{-9}$ \\ BB+POW & 2.8$^{+2.0}_{-1.0}$ & $0.4^{+0.3}_{-0.1}$ & $2.1 \pm 0.3$ & - & & & 0.82(17) & $5.6\times 10^{-10}$ \\ Comptt & 1.9$\pm 0.4$ & - & - & - & $> 24$ & $0.2^{+1.3}_{-0.1}$ & 1.07(18) & $1.1\times 10^{-9}$ \\ BB+Comptt & 2.7$^{+2.0}_{-1.0}$ & $0.4^{+0.3}_{-0.2}$ & - & - & $> 17$ & $0.8^{+2.2}_{-0.6}$ & 0.86(16) & $6.3\times 10^{-10}$ \\ \hline \\ \end{tabular} \label{tab:fit_sim} \end{center} \vspace{-0.6cm} \begin{list}{}{} \item[$^{\mathrm{a}}$] The bolometric flux of the unabsorbed best-fit model spectrum. \end{list} \end{table*} \section{Discussion} We report here for the first time the broad-band spectrum, from soft to hard X-rays, of the persistent emission from a so-called ``burst-only'' source. In particular, none of these sources have ever been studied above 10 keV during their persistent emission. The outburst from {SAX J1753.5$-$2349}\ observed with {\it INTEGRAL}/IBIS has a duration of at least 14 days, without any evidence for type-I X-ray bursts, all along the performed {\it INTEGRAL}\ observations of the Galactic Centre region started in 2003. From the {\it RXTE}/PCA flux detection at 8 mCrab \cite{mark08} we can derive an absorbed 2--10 keV peak flux of about $1.7 \times 10^{-10}$ erg cm$^{-2}$ s$^{-1}$ \ which translates in an unabsorbed luminosity higher than $1.3 \times 10^{36}$ erg s$^{-1}$. This value seems to indicate {SAX J1753.5$-$2349}\ being a hybrid system (such as AX J1745.6--2901 and GRS 1741.9--2853, see Degenaar \& Wijnands 2009) which displays very-faint outbursts with 2--10 keV peak luminosity $L_{X} < 10^{36}$ erg s$^{-1}$\ (as resulted from WFC observations in 1996), as well as outbursts with luminosities in the range $10^{36-37}$ erg s$^{-1}$, which are classified as faint (FXT; Wijnands et al. 2006). However, it is worth to know that the $L_{X}$ boundary as $10^{36}$ erg s$^{-1}$\ is somewhat arbitrary (such as the VFXT/FXT classification). Nevertheless, our result reinforces the hypothesis that the so-called ``burst-only'' sources belong to the class of the subluminous neutron star X-ray binaries. A rough estimate of the duty cycle (as the ratio of $t_{\mathrm{ob}}/t_{\mathrm{rec}}$) can be obtained. The time interval between the two 2008 measurements of the quiescence (February 2008-March 2009) is about 13 months while the outburst recurrence ($t_{\mathrm{rec}}$) is about 12 years (from the burst event in 1996). However, it is possible that we missed other outbursts of {SAX J1753.5$-$2349}\ that occurred between 1996 and 2008 whithin periods not covered by Galactic Centre monitoring. The outburst duration ($t_{\mathrm{ob}}$) ranges from a minimum of 14 days (as observed) and a maximum of 13 months, since there are not any other X-ray observations but the ones in October. In fact, we cannot exclude that the hard X-ray outburst may be part of a longer outburst occurred at a lower luminosity level, only detectable by high-sensitivity X-ray telescopes. \begin{figure} \centering \includegraphics[height=7cm,angle=-90]{delsanto10_fig4.ps} \caption{Confidence contour levels of electron temperature and plasma optical depth for the {\texttt{comptt}} model fitting the broad-band spectrum. \label{fig:cont}} \end{figure} This translates into a duty cycle ranging from a minimum of 0.3$\%$ to a maximum of 9$\%$ and into a long-term time-averaged accretion rate ($\langle \dot{M}_{\mathrm{long}} \rangle=\langle \dot{M}_{\mathrm{ob}} \rangle \times t_{\mathrm{ob}} / t_{\mathrm{rec}}$) ranging from 2.2$\times$$10^{-12}$ to 6.0$\times$$10^{-11}$ M$_\odot$ yr$^{-1}$. King \& Wijnands (2006) suggested that neutron star in transient LMXBs with low time-averaged mass accretion rate might pose difficulties explaining their existence without invoking exotic scenarios such as accretion from a planetary donor. However, the regime of $\langle \dot{M}_{\mathrm{long}} \rangle$ estimated for {SAX J1753.5$-$2349}\ can be well explained within current LMXB evolution models. In spite of the flux variability along the outburst, the spectral state of {SAX J1753.5$-$2349}\ remains steady, in low/hard state. This is in agreement with the fact that a really low X-ray luminosity, $L_{X}$ $\lesssim$ $0.01 L_{Edd}$ or so, produces a hard state in most sources \cite{klis06}. Following in't Zand et al. (2007), we have estimated the hardness ratio 40--100/20--40 keV within each {\it INTEGRAL}\ revolutions. We find a value consistent with 1 which confirms the hard nature of the system. This is also consistent with the low mass accretion rate inferred (see also Paizis et al. 2006), i. e. {SAX J1753.5$-$2349}\ is not a fake faint system and there would be no reason to assume that the system is obscured to explain the low $\dot{M}$. Moreover, King (2000) argued that the faint low-mass X-ray transients are mainly neutron star X-ray binaries in very compact binaries with orbital periods lower than 80 min. We suggest that the {SAX J1753.5$-$2349}\ system is a good candidate to harbor an accreting neutron star in a very compact system. In conclusion, {SAX J1753.5$-$2349}\ joins a sample of low-luminosity transient LMXBs \cite{degewij09}, which display different behaviour in terms of peak luminosity, outburst duration and recurrence time from year to year. Up to now, it is not understood whether these variations should be interpreted as being due to changes in the mass-transfer rate or as results of instabilities in the accretion disc (Degenar \& Wijnands 2009 and reference therein). \section*{Acknowledgments} Data analysis is supported by the Italian Space Agency (ASI), via contracts ASI/INTEGRAL I/008/07/0, ASI I/088/06/0. MDS thanks Memmo Federici for the {\it INTEGRAL}\ data archival support at IASF-Roma. We thank the anonymous referee for his quick response and useful suggestions.

proofpile-arXiv_069-12856

\section{Introduction} Most conservation biologists recognize the utilitarian value of biodiversity for human existence \cite{cardinale2012biodiversity, newbold2015global}. Therefore, numerous research studies have focused on protecting and restoring wildlife by monitoring their population distribution, identifying threats to their survival, and eliminating the perils. Our collaboration primarily focused on the conservation of the endangered Florida panther (Puma concolor coryi), which is one of the most endangered species of mammals in the United States, with less than 230 individual adults estimated to be living in South Florida \cite{mcclintock2015endangered}. \\ Measuring abundance is central to endangered species recovery and essential for management, but accurate estimates of population size are difficult to obtain for rare species using traditional field methods. While researchers have relied on field studies involving direct observation, mark-recapture, and radio-telemetry to monitor species population and their behavior in the past; technological advancements and ease of access to non-invasive inexpensive trail cameras have drawn the attention of present-day investigators. Trail camera imagery has a wide variety of biological and ecological applications in conservation \cite{o2010camera, yu2013automated, miguel2016finding, swanson2015snapshot}. For our project, two different locations in South Florida, namely Corkscrew Swamp (Corkscrew) and Okaloacoochee Slough State Forest (OKSSF), were surveyed between January 2018 to late 2019. Each trail camera took up to 40,000 images during the first couple of months, and by the end of the survey, over three-quarters of a million images were generated. The images captured using these cameras, first, had to be curated for efficient information retrieval. It took many volunteers several months to manually sort and organize the data into file systems, eventually creating a clean dataset of more than 100k images and 22 classes, including the endangered Florida panther. \\ Wildlife populations can be monitored via camera trap studies, yet processing the data generated in these studies is an enormous task. Due to the time-consuming nature of manually annotating camera trap images, scientists have begun to experiment with different methods of automatically classifying images using deep neural networks. There is a growing literature base demonstrating the power of AI in recognizing images of wildlife \cite{willi2019identifying, swanson2015snapshot, gomez2016animal, villa2017towards, norouzzadeh2018automatically, tabak2019machine}. Recent studies have demonstrated that deep neural networks can automatically identify images of different wildlife species with accuracy on par of human volunteers, saving over 8.4 years at 40 hours per week of volunteer labeling effort \cite{norouzzadeh2018automatically, schneider2019past}. AI can dramatically improve the speed and efficiency of species identification and data management. \\ Contemporary models used to classify images from trail cameras, however, do not perform well when used on images from a different location compared to the training data, which implies that the models are non-transferable \cite{tabak2020improving}. Model transferability is vital if the model is to be used by biologists for more than one study area. We introduce a challenging real-world condition-based wildlife dataset, which will promote the need for better generalization, thereby promoting advances in this area. Using a ResNet-50 architecture, we train a preliminary transferable model on images of the Florida panther and other species available in the dataset. \begin{figure*} \begin{center} \includegraphics[width=0.75\textwidth,height=0.45\textheight]{images/Examples.png} \end{center} \caption{Examples of images from the dataset. The first two columns exhibit samples from the Corkscrew location, while the next set of columns contain specimens from the OKSSF location.} \label{fig:Examples} \end{figure*} \section{Dataset Overview} \subsection{Trail Camera Images} Our dataset is composed of 104,495 images from both of our trail camera study locations combined (see Table \ref{table:Dataset}). The dataset contains over 2500 images of Florida panthers. Due to the rarity of this species, this dataset has the potential to be invaluable as a conservation tool for this species. The dataset has images with varying backgrounds taken from several sites at the two primary locations surveyed. The locations of survey were chosen due to their proximity to major roads and the relatively high level of panther activity in these areas. Sample images from the dataset can be viewed in Figure \ref{fig:Examples}. \begin{figure}[t] \begin{center} \fbox{\includegraphics[width=1\linewidth]{images/Distribution.png}} \end{center} \caption{Location-based distribution of images per species.} \label{fig:long} \label{fig:Distribution} \end{figure} \setlength{\tabcolsep}{3.9pt} \begin{table} \renewcommand{\arraystretch}{1.06} \begin{center} \begin{tabular}{|l|c|c|c|c|} \hline {\bf {Classes}} & {\bf {Corkscrew}} & {\bf {OKSSF}} & {\bf {Per-Class}} & {\bf {\% of}} \\ & & & {\bf {Total}} & {\bf {Overall}} \\ \hline \hline Alligators & 18 & 9 & 27 & 0.03 \\ \hline Armadillos & 119 & 11 & 130 & 0.12 \\ \hline Bears & 149 & 363 & 512 & 0.49 \\ \hline Birds\_Carrion & 1028 & 318 & 1346 & 1.29 \\ \hline Birds\_Other & 2146 & 304 & 2450 & 2.34 \\ \hline Birds\_Raptors & 54 & 19 & 73 & 0.07 \\ \hline Birds\_Wading & 3741 & 80 & 3821 & 3.66 \\ \hline Boar & 117 & 0 & 117 & 0.11 \\ \hline Bobcats & 966 & 1497 & 2463 & 2.36 \\ \hline Cattle & 5 & 22297 & 22302 & 21.34 \\\hline Coyotes & 28 & 118 & 146 & 0.14 \\ \hline Deer & 5415 & 30780 & 36195 & 34.64 \\\hline Dogs & 48 & 148 & 196 & 0.19 \\ \hline Negatives & 8999 & 9000 & 17999 & 17.22 \\\hline Opossum & 882 & 519 & 1401 & 1.34 \\ \hline Otters & 18 & 6 & 24 & 0.02 \\ \hline Panthers & 965 & 1593 & 2558 & 2.45 \\ \hline Rabbits & 589 & 1031 & 1620 & 1.55 \\ \hline Raccoons & 661 & 359 & 1020 & 0.98 \\ \hline Sandhill\_Crane & 39 & 1120 & 1159 & 1.11 \\ \hline Squirrels & 501 & 21 & 522 & 0.50 \\ \hline Turkey & 1070 & 7344 & 8414 & 8.05 \\ \hline \hline {\bf {Overall}} & {\bf {27558}} & {\bf {76937}} & {\bf {104495}} & {\bf {100}} \\ \hline \end{tabular} \end{center} \caption{A comprehensive list providing the number of images collected per species from two different locations in our dataset.} \label{table:Dataset} \end{table} \subsection{Data Challenges} Similar to other trail camera datasets, such as Beery et al.’s iWildCam 2018 Challenge Dataset \cite{beery2019iwildcam}, our dataset exhibits a range of inherent challenges, as illustrated in Figure \ref{fig:Challenges}. Since the trail camera captures images throughout the day, multiple samples suffer from illumination variations (either over or under saturated). Frequently, images turn blurry when the subject moves too quickly, and several times blur could be present from moisture on the lens. In the case of some cameras, IR system malfunction has resulted in “pink-washed” daytime photos. Additionally, the distance between animal and camera does not remain consistent; hence scale, focus, and visibility vary considerably even in subsequent images. Sometimes, animals tend to be almost entirely obscured by vegetation in the background, or they could be so close to the camera only an indiscernible part of their body would be visible. In addition, the image backgrounds are non-static, and vegetation could shift quite a lot from one image to the next. While all these complexities add real-world challenges to the dataset, these limitations occasionally make it impossible for the image to be identified. Such unidentifiable images have been omitted from the dataset. There is also a margin of error for what has been sorted, around 2\%. \begin{figure}[t] \begin{center} \includegraphics[width=1\linewidth]{images/Challenges.png} \end{center} \caption{Specimen images depicting the highlighted challenges associated with the dataset: (1) Motion blur, (2) Occlusion, (3) Camouflage, (4) Illumination issues, (5) Excessive proximity of the animals to the camera traps making it impossible to identify salient features, (6) Imbalance in the dataset due to substantial difference in the numbers of images for classes such as cattle/deer, (7/8) Multiple images with hardly any difference captured by continuous camera triggers, and (9) View-point based deception resulting in bobcats appearing similar to panthers.} \label{fig:Challenges} \end{figure} \section{Baseline Experiments} We train a baseline model using ResNet-50 architecture. The dataset comprising 104,495 samples is split into train, validation, and test set in 70\%-10\%-20\% proportion. We have also employed random cropping, horizontal flip, and rotation for data augmentation. Images resized to $224\times224$px are processed through the model in a batch size of 64, using a learning rate of 1e-3. We achieve a baseline test accuracy of 78.75\% on our ResNet-based model. For the test images, we do a t-SNE visualization of the output embeddings, as shown in Figure \ref{fig:TSNE}, to demonstrate the effectiveness of our method. \begin{figure}[t] \begin{center} \fbox{\includegraphics[width=1\linewidth]{images/Confusion_1.png}} \end{center} \caption{Confusion matrices illustrating results on the test data for 22 classes.} \label{fig:Confusion} \end{figure} \begin{figure}[t] \begin{center} \includegraphics[width=1\linewidth]{images/TSNE.png} \end{center} \caption{t-SNE visualization of randomly initialized features (left) and learned embeddings (right) for test images in the dataset.} \label{fig:TSNE} \end{figure} \iffalse \begin{figure}[t] \begin{center} \fbox{\includegraphics[width=1\linewidth]{images/Confusion.png}} \end{center} \caption{Confusion matrices illustrating results on the test data for 22 classes.} \label{fig:Challenges} \end{figure} \setlength{\tabcolsep}{4.5pt} \begin{table}[h] \renewcommand{\arraystretch}{1.2} \begin{center} \begin{tabularx}{8.8cm}{l*{5}{Y}c@{}} \toprule {\bf {Classes}} & {\bf {Corkscrew}} & {\bf {OKSSF}} & {\bf {Per-Class}} & {\bf {\% of}} \\ & & & {\bf {Total}} & {\bf {Overall}} \\ \midrule Alligators & 18 & 9 & 27 & 0.03 \\ Armadillos & 119 & 11 & 130 & 0.12 \\ Bears & 149 & 363 & 512 & 0.49 \\ Birds\_Carrion & 1028 & 318 & 1346 & 1.29 \\ Birds\_Other & 2146 & 304 & 2450 & 2.34 \\ Birds\_Raptors & 54 & 19 & 73 & 0.07 \\ Birds\_Wading & 3741 & 80 & 3821 & 3.66 \\ Boar & 117 & 0 & 117 & 0.11 \\ Bobcats & 966 & 1497 & 2463 & 2.36 \\ Cattle & 5 & 22297 & 22302 & 21.34 \\ Coyotes & 28 & 118 & 146 & 0.14 \\ Deer & 5415 & 30780 & 36195 & 34.64 \\ Dogs & 48 & 148 & 196 & 0.19 \\ Negatives & 8999 & 9000 & 17999 & 17.22 \\ Opossum & 882 & 519 & 1401 & 1.34 \\ Otters & 18 & 6 & 24 & 0.02 \\ Panthers & 965 & 1593 & 2558 & 2.45 \\ Rabbits & 589 & 1031 & 1620 & 1.55 \\ Raccoons & 661 & 359 & 1020 & 0.98 \\ Sandhill\_Crane & 39 & 1120 & 1159 & 1.11 \\ Squirrels & 501 & 21 & 522 & 0.50 \\ Turkey & 1070 & 7344 & 8414 & 8.05 \\ \midrule {\bf {Overall}} & {\bf {27558}} & {\bf {76937}} & {\bf {104495}} & {\bf {100}} \\ \bottomrule \end{tabularx} \end{center} \caption{A comprehensive list providing the number of images collected per species from two different locations in our dataset.} \label{table:Youtube-VIS-Ablation} \end{table} \begin{figure}[t] \begin{center} \fbox{\includegraphics[width=1\linewidth]{images/Model.png}} \end{center} \caption{Classification results on training, validation and test split.} \label{fig:long} \label{fig:onecol} \end{figure} \fi \section{Conclusions and Future Work} In this paper, we have introduced a camera trap dataset comprising 100k+ wildlife samples belonging to 22 different categories. The complexity of the landscape, adverse lighting conditions, unconventional subject behavior captured in the dataset, using trail cameras, has aimed to improve the ability of automated systems to generalize to new environments. While the dataset provides a good platform to work on domain adaptation and data imbalance issues, we also have some additional unlabeled data which will allow us to explore the area of semi-supervised learning. Although this project originally focused only on monitoring the ecosystem concerning the endangered Florida panther, the curated trail camera images have presented avenues to explore several areas of biodiversity conservation. The dataset will be released publicly to enable access to a broader community and foster development in biodiversity research. {\small \bibliographystyle{ieee_fullname}

proofpile-arXiv_069-12969

\section{Introduction} \label{sec:introduction} Consequential decisions compel individuals to react in response to the specifics of the decision rule. This individual-level response in aggregate can disrupt both statistical patterns and social facts that motivated the decision rule, leading to unforeseen consequences. A similar conundrum in the context of macroeconomic policy making fueled the \emph{microfoundations} program following the influential critique of macroeconomics by Lucas in the 1970s~\citep{lucas1976econometric}. Microfoundations refers to a vast theoretical project that aims to ground theories of aggregate outcomes and population forecasts in microeconomic assumptions about individual behavior. Oversimplifying a broad endeavor, the hope was that if economic policy were \emph{microfounded}, it would anticipate more accurately the response that the policy induces. Predominant in neoclassical economic theory is the assumption of an agent that exhaustively maximizes a utility function on the basis of perfectly accurate information. This modeling assumption about agent behavior underwrites many celebrated results on markets, mechanisms, and games. Although called into question by behavioral economics and related fields (e.g. see \citep{behavioralecon}), the assumption remains central to economic theory and has become standard in computer science, as well. When reasoning about incentives and strategic behavior in the context of classification tasks, it is tempting to combine the predominant modeling assumptions from microeconomic theory with the statistical decision theory underlying classification. In the resulting model, agents have perfect information about the decision rule and compute best-response feature changes according to their utility function with the goal of achieving a more favorable classification outcome. We refer to this agent model as \textit{standard microfoundations}. Building on the assumption that agents follow standard microfoundations, the decision-maker then chooses the decision rule that maximizes their own objective in anticipation of the resulting agent response. This is the conceptual route taken in the area of \emph{strategic classification}, but similar observations may apply more broadly to the intersection of economics and learning. \subsection{Our work} We argue that standard microfoundations are a poor basis for studying strategic behavior in binary classification problems. We make this point through three observations that illustrate the limited descriptive power of the standard model and the problematic solution concepts it implies. In response, we explore the space of alternative agent models for strategic classification, and we identify desirable properties that when satisfied by microfoundations lead to more realistic and robust insights. Guided by these desiderata, we propose \emph{noisy response} as a promising alternative to the standard model. \subsubsection{Limitations of standard microfoundations} In strategic classification, agents respond strategically to the deployment of a binary decision rule $f_\theta$ specified by classifier weights~$\theta$. The decision-maker assumes that agents follow standard microfoundations: that is, agents have full information about $f_\theta$ and change their features so as to maximize their utility function. The utility function captures the benefit of a positive classification outcome, as well as the cost of feature change. Consequently, an agent only invests in changing their features if the cost of feature change does not exceed the benefit of positive classification. Our first observation concerns the \textit{aggregate response}---the distribution $\cD(\theta)$ over feature, label pairs induced by a classifier $f_\theta$. We show that in the standard model, the aggregate response necessarily exhibits discontinuities that we often do not observe in empirical settings. The problem persists even if we assume an approximate best response or allow for heterogeneous cost functions. Our second observation reveals that, apart from lacking descriptive power, the standard model also leads to brittle conclusions about the solution concept of \textit{performative stability}. Performative stability \citep{PZMH20} refers to decision rules that are optimal on the particular distribution they entail. Stable points thus represent fixed points of \textit{retraining methods}, which repeatedly update the classifier weights to be optimal on the data distribution induced by the previous classifier. We show that the existence of performatively stable classifiers breaks down whenever a positive fraction of randomly chosen agents in the population are non-strategic. This brittleness of the existence of fixed points suggests that the standard model does not constitute a reliable basis for investigating dynamics of retraining algorithms. Our last observation concerns the solution concept of \textit{performative optimality}. Performative optimality \citep{PZMH20} refers to a decision rule that exhibits the highest accuracy on the distribution it induces. The global nature of this solution concept means that finding performatively optimal points requires the decision-maker to anticipate strategic feedback effects. We prove that relying on standard microfoundations to model strategic behavior leads to extreme decision rules that maximize a measure of negative externality called \textit{social burden} within a broad class of alternative models. Social burden, proposed in recent work, quantifies the expected cost that positive instances of a classification problem have to incur in order to be accepted. Thus, standard microfoundations produce optimal solutions that are unfavorable for the population being classified. \subsubsection{Alternative microfoundations} Recognizing the limitations of standard microfoundations, we systematically explore alternatives to the standard model. We investigate alternative assumptions on agent responses, encompassing general agent behavior that need not be fully informed, strategic, or utility maximizing. We formalize microfoundations as a randomized mapping $M:X\times Y\rightarrow \AllTypes$ that assigns each agent to a response type $t\in\AllTypes$. The response type $t$ is associated with a response function $\Response_t\colon X\times\Theta\to X$ specifying how agents of type~$t$ change their features $x$ in response to each decision rule $f_{\theta}$. Letting $\DBase$ be the base distribution over features and labels prior to any strategic adaptation, the \emph{aggregate response} to a classifier~$f_\theta$ is given by the distribution $\DMap(\theta;M)$ over induced feature, label pairs $(\Response_t(x,\theta), y)$ for a random draw $(x, y)\sim\DBase$ and $t= M(x,y)$. In this sense, the mapping $M$ \emph{microfounds} the distributions induced by decision rules, endowing the distributions with structure that allows the decision-maker to deduce the aggregate response from a model of individual behavior. To guide our search for more appropriate microfoundations for binary classification, we introduce a collection of properties that are desirable for a model of agent responses to satisfy. The first condition, that we call \textit{aggregate smoothness}, rules out the discontinuities arising from standard microfoundations. Conceptually, it requires that varying the classifier weights slightly must change the aggregate response smoothly. We find that this property alone is sufficient to guarantee the robust existence of stable points under mixtures with non-strategic agents. The second condition, that we call the \textit{expenditure constraint}, helps ensure that the model encodes realistic agent-level responses $\Response_t$. At a high level, it requires that each agent does not spend more on changing their features than the utility of a positive outcome. This natural constraint gives rise to a large set of potential models. For any such model that satisfies an additional weak assumption, the social burden of the optimal classifier is no larger than the social burden of the optimal classifier deduced from the standard model. Moreover, the optimal points are determined by local behavior. This frees the decision-maker from fully understanding the aggregate response $\cD(\theta)$ and makes the task of finding an approximately optimal classifier more tractable. \subsubsection{Noisy response---an alternative model} Using the properties described above as a compass to navigate the large space of alternative models, we identify \emph{noisy response} as a compelling model of microfoundations. In this model, each agent best responds with respect to $\theta+\xi$, where $\xi$ is an independent sample from a zero mean noise distribution. This model is inspired by \textit{smoothed analysis} \citep{smoothedanalysis} and encodes imperfection in the population's response to a decision rule by perturbing the manipulation targets of individual agents. We show that noisy response satisfies a number of desirable properties that make it a promising model of microfoundations for classification in strategic settings, both for theoretical analyses and from a practical standpoint. First, noisy response satisfies aggregate smoothness, and thus leads to the robust existence of stable points. Moreover, the model satisfies the expenditure constraint, and thus encodes natural agent-level responses which can be used to reason about metrics such as social burden. When used to anticipate strategic feedback effects and compute optimal points, noisy response leads to strictly less pessimistic acceptance thresholds than those computed under standard microfoundations, given the same constraints on manipulation expenditure. In fact, we show via simulations that a larger variance $\sigma$ of the noise in the manipulation target leads to a more conservative optimal threshold, and for $\sigma\rightarrow 0$, we approximate the extreme case of standard microfoundations. Finally, from a practical perspective, we demonstrate that the distribution map $\cD(\theta)$ for noisy response can be estimated by gathering information about individuals to learn the parameter $\sigma$ and the cost function $c$. This has the desirable consequence that the decision-maker can compute performatively optimal points without ever deploying a classifier. \subsection{Related work} \label{ref:relatedwork} Existing work on strategic classification in machine learning has mostly followed standard microfoundations for modeling agent behavior in response to a decision rule, e.g.,~\citep{DDMSV04, BS11, HMPW16, Khaje19opt,tsirtsis2020decisions} to name a few. This includes works that focus on minimizing Stackelberg regret~\citep{dong18revealedpref, CLP20}, quantify the price of transparency~\citep{akyol16transp}, and investigate the benefits of randomization in the decision rule~\citep{BG20}. Investigations on externalities such as social cost~\citep{MMDH19, HIW19} and whether classifiers incentivize improvement as opposed to gaming~\citep{Kleinberg19strat, MMH20, SEA20, HILW20} have also mostly built on the standard assumption of best-responding agents with perfect information. Recent work by~\cite{levanon2021strategic} studied practical implications of the optimization problem resulting from standard microfoundations and how to make it more amenable to practical optimization methods. A handful of works have suggested potential limitations of the standard strategic classification framework. \citet{bruckner12pred} recognized that the standard model leads to very conservative Stackelberg solutions, and proposed resorting to Nash equilibria as an alternative solution concept. We instead take a different route and advocate for altogether rethinking standard microfoundations that lead to these conservative acceptance thresholds. Concurrent and independent work by \citet{ghalme2021strategic} and \citet{bechavod2021information} relaxed the perfect information assumption in the standard model and studied strategic classification when the classifier is not fully revealed to the agents. In this work, we argue that agents often do not perfectly respond to the classifier even when the decision rule is fully transparent. Therefore, we propose incorporating imperfection into the model of microfoundations in order to anticipate natural deviations from the standard assumptions. Related work in economics also investigates strategic responses to decision rules. This line of work, initiated by \cite{Spence73}, has shown that information about individuals can become \emph{muddled} as a result of heterogeneous gaming behavior \citep{FK19}, investigated the role of commitment power of the decision-maker \citep{FK20}, considered the impact of an intermediary who aggregates the agents' multi-dimensional features \citep{Ball2020}, and considered the performance of different training approaches in strategic environments \citep{HG20}. A notable work by~\cite{BBK20} investigates strategic behavior through a field experiment in the micro-lending domain, with a focus on evaluating approaches for designing strategy-robust classifiers. An important distinction is that these works tend to study regression, while we focus on classification. These settings appear to be qualitatively different in the context of strategic feedback effects (e.g. see note in~\citep{HG20}). Our work is conceptually related to recent work in economics that has recognized mismatches between the predictions of standard models and empirical realities, for example in macroeconomic policy \citep{S18, KV18, CGK18} and in mechanism design \citep{L17}. These works, and many others, have explored incorporating richer behavioral and informational assumptions into typical models used in economic settings. Although our work also explores alternatives to standard microfoundations, we focus on algorithmic decision-making, where the limitations of the standard model had not been previously identified. We believe that our approach of navigating the entire space of potential models using a collection of properties could be of broader interest when developing alternative microfoundations. \subsection{Setup and basic notation}\label{sec:background} Let $X \subseteq \mathbb{R}^m$ denote the feature space, and let $Y = \left\{0,1 \right\}$ be the space of binary outcomes. Each agent is associated to a feature vector $x \in X$ and a binary outcome $y \in Y$ which represents their true label. A feature, label pair $(x,y)$ need not uniquely describe an agent, and many agents may be associated to the same pair $(x,y)$. The base distribution $\DBase$ is a joint distribution over $X \times Y$ describing the population prior to any strategic adaption. Throughout the paper we assume that $\DBase$ is continuous and has zero mass on the boundary of $X$. We focus on binary classification where each classifier $f_{\theta}: X \rightarrow \left\{0,1\right\}$ is parameterized by $\theta\in \mathbb{R}^d$, and the decision-maker selects classifier weights $\theta$ from $\Theta\subseteq \mathbb{R}^d$ which is a compact, convex set. We assume that for every $\theta \in \Theta$, the set $\left\{x \in X \mid f_{\theta}(x) = 1 \right\}$ is closed, and the decision boundary is measure $0$. We adopt the notion of a distribution map $\cD(\theta)$ from~\citep{PZMH20} to describe the distribution over $X\times Y$ induced by strategic adaptation of agents drawn from the base distribution in response to the classifier $f_{\theta}$. \section{Limitations of standard microfoundations} \label{sec:limitations} In the strategic classification literature, the typical agent model is a \textit{rational agent with perfect information}. At the core of this model lies the assumption that agents have perfect knowledge of the classifier and maximize their utility given the classifier weights. The utility consists of two terms: a reward for obtaining a positive classification and a cost of manipulating features. The reward is denoted $\gamma>0$ and the manipulation cost is represented by a function $c: X \times X \rightarrow \mathbb{R}$ where $c(x,x')$ reflects how much agents need to expend to change their features from $x$ to $x'$. A valid cost function satisfies a natural monotonicity requirement as stated in Assumption~\ref{assumption:validcost}. Given a feature vector $x$ and a classifier $f_{\theta}$, agents solve the following utility maximization problem: \begin{equation} \label{eq:SM} \argmax_{x'\in X} \;\left[\gamma f_{\theta}(x') - \costfn(x, x')\right]. \end{equation} \begin{assumption} \vspace{-0.5cm} \label{assumption:validcost} A cost function $c: X \times X \rightarrow \mathbb{R}$ is \emph{valid}, if it is continuous in both arguments, it holds that $\costfn(x, x')=0$ for $x=x'$, and $c$ increases with distance in the sense that $\costfn(x, \bar x) < \costfn(x, x')$ and $\costfn(\bar x, x) < \costfn(x', x)$ for every $\bar x\in X$ that lies on the line segment connecting the two points $x,x'\in X$.\footnote{We model a non-zero cost for all modifications to features, regardless of whether they result in positive classification or not. Generalizing beyond standard microfoundations, this accounts for how agents may erroneously expend effort on changing their features in an incorrect direction, as empirically demonstrated by \cite{BBK20}.} \end{assumption} \noindent We will refer to this response model as \emph{standard microfoundations}. \subsection{Discontinuities in the aggregate response} \label{sec:degeneracy} A striking property of distributions induced by standard microfoundations in response to a binary classifier is that they are necessarily either trivial or discontinuous. The underlying cause is that agents behaving according to standard microfoundations either change their features exactly up to the decision boundary, or they do not change their features at all. \begin{restatable}{proposition}{propdegenerate} \label{prop:degenerate} \propositionlabel{prop:degenerate} Given a base distribution $\DBase$, let $\DMap(\theta)$ be the distribution induced by a classifier $f_\theta$. Then, if $\DMap(\theta)$ is continuous and $\DMap(\theta) \neq \DBase$, there does not exist a valid cost function $c$ such that $\DMap(\theta)$ is an aggregate of agents following standard microfoundations. \end{restatable} In addition to the discontinuities implied by Proposition \ref{prop:degenerate}, the aggregate response induced by standard microfoundations faces additional degeneracies. Namely, a similar argument shows that any non-trivial distribution arising from the standard model must have a region of zero density below the decision boundary. These properties are unnatural in a number of practical applications, as we discuss in the following examples. \begin{example}[Lending decisions and credit scores] \label{ex:FICO} Consider banking lending decisions and the corresponding distribution over FICO credit scores. If lending decisions are based on FICO scores, then under standard microfoundations, the distribution over credit scores should exhibit a discontinuity at the threshold. However, this is not what we observe empirically. In particular, previous work \citep{HPS16} studied a FICO dataset from 2003, where credit scores range from 300 to 850, and a cutoff of 620 was commonly used for prime-rate loans. The observed distribution over credit scores appears continuous and is supported across the full range of scores.\footnote{In practice, lending decisions might be based on additional features beyond credit scores, and different lenders might use different cutoffs. In any case, this example demonstrates that the observed aggregate response cannot be captured by agents behaving according to standard microfoundations in response to a decision rule that is a threshold function of the credit score. } \end{example} \begin{example}[Yelp online ratings and rounding thresholds] \label{ex:yelp} Restaurant ratings on Yelp are rounded to the nearest half star, and the star rating of a restaurant can significantly influence restaurant customer flows. In this setting, strategic adaptation arises from restaurants leaving fake reviews. Under standard microfoundations, the distribution of restaurant ratings would exhibit discontinuities at the rounding thresholds. However, previous work \citep{AM12} examined the distribution of restaurant ratings, and showed that there is no significant discontinuity in the density of the restaurants at the rounding thresholds (see Figure 4 in their work). \end{example} \begin{example}[High school exit exams and score cutoffs] \label{ex:testscores} New York high school exit exams have important stakes for students, teachers, and schools, based on whether students meet designated score cutoffs. Interestingly, test score distributions did exhibit discontinuities prior to reforms on teacher grading procedures in 2012 \citep{dee19}. These discontinuities resulted from teachers deliberately adjusting student scores to be just above the cutoff during grading. This example demonstrates that sharp discontinuities can arise when there is perfect information about the decision rule and perfect control over manipulations. However, following grading reforms that largely eliminated the possibility for strategic manipulation by \emph{teachers}, these discontinuities disappeared and test score distributions appeared continuous \citep{dee19}. Similar to observations in Examples~\ref{ex:FICO}-\ref{ex:yelp}, strategic adaptation by \textit{students} cannot be described by standard microfoundations. \end{example} It is important to note that the degeneracies in the aggregate response induced by standard microfoundations arise from the fact that classification decisions are discrete and based on a \emph{hard} decision. Agents who are not classified positively receive no reward: it does not matter how close to the decision boundary the agent is. This discontinuity in the utility is specific to classification and does not arise in regression problems that are predominantly studied in the economics literature. However, in machine learning and statistical decision theory, binary classification is ubiquitous, and degeneracies that we have identified pertain to general settings where the decisions are binary. The reader might imagine that common variations and generalizations of standard microfoundations can mitigate these issues. Unfortunately, the two variations of standard microfoundations that are typically considered---\textit{heterogeneous cost functions} \citep{HIW19}, and \textit{approximate best response} \citep{MMH20}---result in similar degeneracies. Heterogeneity in the cost (or utility) function can only change whether or not an agent decides to change their features, but it does not change their target of manipulation. If agents approximately best-respond, and thus move to features $x'$ that approximately maximize their utility, the model no longer leads to point masses at the decision boundary, but agents will never \textit{undershoot} the decision boundary. This means that any nontrivial aggregate distribution must have a region of zero density below the decision boundary to comply with standard microfoundations and any of these variants. In fact, agent behavior that is not consistent with standard microfoundations or variants has been observed in field experiments. In particular, agents both overshoot and undershoot the decision boundary as well as generally exhibit noisy responses, even if the classifier is fully transparent. \begin{example}[Field Experiment \citep{BBK20}] \label{ex:FE} The authors developed an app that mimicked aspects of ``digital credit'' applications, and deployed it in Kenya in order to empirically investigate strategic behavior. Participants were rewarded if the app guessed that they were a high-income earner. When the participants were given access to the coefficients of the decision rule, they tended to change their features in the right direction, but a high variance in their responses was observed---see Table 5 in their work. The noise in the response was even more pronounced when participants were only given opaque access to the decision rule. In this case, agents often did not even change their features in the right direction. \end{example} \subsection{Brittleness under natural model misspecifications} \label{sec:misspecifications} We describe two scenarios where the behavior of standard microfoundations is undesirable under natural model misspecifications. In particular, the existence of stable solutions crucially relies on \emph{all} agents being perfectly strategic, and the optimal solutions associated with the standard model cause extreme negative externalities within a broad class of alternative approaches to model agent behavior. \subsubsection{Stability as a fragile solution concept} \label{subsubsec:fragilestability} Performatively stable solutions are guaranteed to exist under standard microfoundations (see \cite{MMDH19}). Our first result demonstrates that this no longer holds if any positive fraction of randomly chosen individuals are non-strategic. Recall that performative stability \citep{PZMH20} requires that a classifier is optimal on the data distribution that it induces: that is, that $\thetaPS$ is a global optimum of the following optimization problem: \begin{equation} \min_{\theta \in \Theta} \;\mathbb{E}_{(x,y)\sim \cD(\thetaPS)} \;\Indicator\left\{y\neq f_\theta(x) \right\}. \label{eq:localPS} \end{equation} Since we focus on the 0-1 loss in this work, the objective in \eqref{eq:localPS} need not be convex, and it is natural to consider a local relaxation of performative stability. We say $\thetaPS$ is \emph{locally stable} if $\thetaPS$ is a local minimum or a stationary point of \eqref{eq:localPS}. Note that any performatively stable solution is locally stable. Performative stability defines the fixed points of \textit{repeated risk minimization}---the retraining method where the decision-maker repeatedly updates the classifier to a global optimum on the data distribution induced by the previous classifier. Thus, there is no incentive for the decision-maker to deviate from a stable model based on the observed data. Similarly, when the objective in \eqref{eq:localPS} is differentiable, locally stable points correspond to fixed points of \textit{repeated gradient descent}~\citep{PZMH20}. Another interesting property of performatively stable points is that they closely relate to the concept of a pure strategy (local)\textit{ Nash equilibrium} in a simultaneous game between the strategic agents who respond to the classifier $f_\theta$ and the decision-maker who responds to the observed distribution $\cD(\theta)$.\footnote{More precisely, when agents follow standard microfoundations, any pure strategy local Nash equilibrium is locally stable.} To showcase that the existence of locally stable classifiers under standard microfoundations crucially relies on all agents following the modeling assumptions, we focus on the following simple 1-dimensional setting. \begin{setup}[1-dimensional] \label{ex:stablepointsnotexist} \setuplabel{ex:stablepointsnotexist} Let $X \subseteq \mathbb{R}$ and consider a threshold functions $f_{\theta}(\noargument) =\mathds{1}\{\noargument \ge \theta\}$ with $\theta\in \Theta\subseteq\mathbb R$. Let $\mu(x)$ be the conditional probability over $\DBase$ of the true label being $1$ given features $x$. Suppose that $\mu(x)$ is strictly increasing in $x$ and there is an $\theta\in \Int(\Theta)$ such that $\mu(\theta)=0.5$. \end{setup} \begin{restatable}{proposition}{propstablepoints} \vspace{-0.1cm} \label{propo:stablepointsnotexist} Consider Setup~\ref{ex:stablepointsnotexist}. Suppose that a $p$ fraction of agents drawn from $\DBase$ do not ever change their features, and a $1-p$ fraction of agents drawn independently from $\DBase$ follow standard microfoundations with a valid cost function $c$. Then, we have the following properties: \begin{itemize}[itemsep=0ex, topsep=-0.2ex] \item[a)] For $p \in\{ 0,1\}$, locally stable points exist. \item[b)] For $p\in (0,1)$, locally stable points do not exist. \item[c)] Let $\thetaPS^{\mathrm{SM}}$ denote the smallest locally stable point when all agents follow standard microfoundations, and let $\theta_{\mathrm{SL}}$ denote the optimal classifier for the base distribution $\DBase$. For $p\in (0,1)$, repeated risk minimization will oscillate between $\theta_{\mathrm{SL}}$ and a threshold $\tau(p) \in (\theta_{\mathrm{SL}}, \thetaPS^{\mathrm{SM}})$.The threshold $\tau(p)$ is decreasing in $p$, approaching $\theta_{\mathrm{SL}}$ as $p \rightarrow 1$ and $\thetaPS^{\mathrm{SM}}$ as $p \rightarrow 0$. \end{itemize} \end{restatable} \begin{proof}[Proof Sketch] For $p = 1$, it is easy to see that $\theta_{\mathrm{SL}}$ is the unique locally stable point, and for $p = 0$, the claim follows from an argument similar to Lemma 3.2 in~\citep{MMDH19}. For $p\in(0,1)$, the core observation is that for any $\theta$ the distribution $\cD(\theta)$ contains no \emph{strategic} agents in the interval $\mathsf{Gap}(\theta):=[\theta-\Delta,\theta]$ for some $\Delta>0$. Furthermore, for any $\theta>\theta_{\mathrm{SL}} $ the misclassification rate on \emph{non-strategic} agents could be improved by reducing the threshold to $\theta_{\mathrm{SL}}$. Thus, it is not hard to see that or any $\theta>\theta_{\mathrm{SL}}$, the threshold $\max(\theta - \Delta, \theta_{\mathrm{SL}})$ achieves smaller loss than $\theta$, and thus $\theta$ cannot be stable. We formalize this argument and the case for $\theta\leq\theta_{\mathrm{SL}}$ in Appendix~\ref{app:nostablepoints}. \end{proof} \begin{figure*}[t!] \subfigure[\footnotesize{SM mixed with non-strategic agents}]{ \includegraphics[width = 0.48\textwidth]{figures/oscillation.pdf} \label{fig:oscillationBR}} \subfigure[\footnotesize{NR mixed with non-strategic agents}]{ \includegraphics[width = 0.48\textwidth]{figures/oscillation-fuzzy.pdf} \label{fig:oscillation-fuzzy}} \caption{Convergence of retraining algorithm in a 1d-setting for different values of $p$ with $\epsilon=10^{-2}$. The population consists of $10^{5}$ individuals. Half of the individuals are sampled from $x\sim\mathcal N(1,0.33)$ with true label $1$ and the other half is sampled from $x\sim\mathcal N(0,0.33)$ with true label $0$. $\thetaPS^{\mathrm{SM}}$ and $\theta_{\mathrm{SL}}$ are defined as in Proposition~\ref{propo:stablepointsnotexist}, and $\thetaPS^{\mathrm{NR}}$ is defined to be the smallest locally stable point when all agents follow noisy response (NR). The parameter of the noisy response in (b) is taken to be $\sigma^2 = 0.1$. } \label{fig:oszillation} \end{figure*} Proposition~\ref{propo:stablepointsnotexist} implies that not only does the existence of locally stable points break down if a positive fraction $p\in(0,1)$ of randomly chosen agents are non-strategic, but also repeated risk minimization oscillates between two extreme points. To illustrate this, we have implemented a simple instantiation of Setup~\ref{ex:stablepointsnotexist}, and we visualize the trajectories of repeated risk minimization for different values of $p$ in Figure~\ref{fig:oscillationBR}. The main insight is that repeated risk minimization starts oscillating substantially even when $p$ is very close to $0$ (only an $\epsilon$ fraction of agents are not following standard microfoundations), even though this method converges when $p = 0$. This sensitivity of the trajectory to natural deviations from the modeling assumptions suggests that standard microfoundations do not constitute a reliable model to study algorithm dynamics. \begin{remark} Unlike repeated risk minimization, repeated gradient descent is not well-defined for standard microfoundations. Because of the induced discontinuities, the optimization objective in \eqref{eq:localPS} for Setup \ref{ex:stablepointsnotexist} with $p < 1$ is not differentiable in the classifier weights. Thus, standard microfoundations do not serve as a useful basis to study repeated gradient descent. \end{remark} \subsubsection{Maximal negative externalities at optimality}\label{subsec:extreme} Our next result describes a natural scenario where performatively optimal classifiers computed under standard microfoundations lead to the highest negative externalities within a broad class of alternative models for agent responses. Recall that a performatively optimal solution corresponds to the best classifier for the decision-maker from a global perspective, but it is not necessarily stable under retraining. Formally, a classifier $\thetaPO$ is \textit{performatively optimal}~\citep{PZMH20} if it minimizes the performative risk: \begin{equation} \thetaPO:=\argmin_{\theta\in\Theta} \mathbb{E}_{(x,y)\sim\cD(\theta)} \,\mathds{1}\{y\neq f_\theta(x)\}. \label{eq:PO} \end{equation} \noindent Performative optimality is closely related to the concept of a \textit{Stackelberg equilibrium} in a leader-follower game, where the decision-maker plays first and the agents respond. The key challenge of computing performative optima is that optimizing \eqref{eq:PO} requires the decision-maker to anticipate the population's response $\cD(\theta)$ to any classifier $f_\theta$. A natural approach to model this response is to build on microfoundations and deduce properties of the distribution map from individual agent behavior. Different models for agent behavior can lead to solutions with qualitatively different properties. While the decision-maker is unlikely to have a fully specified model for agent behavior at hand, we outline a few natural criteria that agent responses could reasonably satisfy. To formalize these criteria, we again focus on the 1-dimensional setting. \begin{property}[Expenditure monotonicity] \label{property:SB} For every feature vector $x \in X$, any agent $a$ with true features $x$ must have manipulated features $\Response_a(x; \theta)$ in response to each classifier $f_{\theta}$ that satisfy: \begin{enumerate}[itemsep=-0.5ex, topsep=0.5ex] \item[a)] $c(x,\mathcal R_a(x;\theta)) < \gamma $ for every $\theta \in \Theta$. \item[b)] $f_{\theta}(\mathcal R_a(x;\theta)) = 1$ $\implies$ $f_{\theta'}(\mathcal R_a(x;\theta')) = 1$ $\forall \theta' \le \theta$. \end{enumerate} \end{property} \noindent Property~\ref{property:SB} describes agents that a) do not expend more on gaming than their utility from a positive outcome, and b) do not have their outcome worsened if the threshold is lowered. However, agents complying with Property \ref{property:SB} do not necessarily behave according to standard microfoundations. For example, Property \ref{property:SB} is satisfied by non-strategic agents who do not ever change their features and by the imperfect agents that we describe in Section \ref{sec:framework}. We now show that within the broad class of microfoundations that exhibit Property~\ref{property:SB}, the standard model leads to an extreme acceptance threshold. For the formal statement, see Appendix~\ref{app:socialburden}. \begin{proposition}[Informal] \label{prop:socialburdeninformal} Consider Setup \ref{ex:stablepointsnotexist}. Let $\mathscr{D}$ be the class of distribution maps $\cD:\Theta\rightarrow \Delta(X\times Y)$ that can be represented by a population of agents who all satisfy Property~\ref{property:SB}. Then under mild assumptions, for every distribution map $\cD\in\mathscr{D}$, it holds that \vspace{-0.2cm} \begin{align*} &\quad\quad\quad\thetaPO(\DMap_{\text{SM}}) \geq \thetaPO(\DMap) \end{align*} where $\DMap_{\text{SM}}$ is the distribution map induced by standard microfoundations. \end{proposition} A problematic implication of Proposition~\ref{prop:socialburdeninformal} is that standard microfoundations also maximize the negative externality called \emph{social burden}~\citep{MMDH19}: \[\SocialBurden{\theta}:=\mathbb{E}_{(x,y) \in \DBase} \left[\min_{x' \in X} \left\{c(x, x') \mid f_{\theta}(x') = 1\right\} \mid y = 1\right].\] Social burden quantifies the average cost that a positively labeled agent has to expend in order to be positively classified by $f_\theta$. While previous work introduced and studied social burden within standard microfoundations, and showed that Nash equilibria lead to smaller social burden than Stackelberg equilibria, we instead use social burden as a metric to study implications of different modeling assumptions on agent behavior. In particular, the following corollary demonstrates that standard microfoundations lead to \textit{worst possible social burden} across all microfoundations that satisfy Property~\ref{property:SB}. \begin{corollary} \label{coro:socialburdeninformal} Under the same assumptions as Proposition~\ref{prop:socialburdeninformal}, for every distribution map $\cD\in\mathscr{D}$, it holds that \vspace{-0.1cm} \begin{align*} &\SocialBurden{\thetaPO(\DMap_{\text{SM}})} \geq \SocialBurden{\thetaPO(\DMap)}. \end{align*} where $\DMap_{\text{SM}}$ is the distribution map induced by standard microfoundations. \end{corollary} This result has implications for the natural situation where standard microfoundations do not exactly describe agent behavior. In particular, relative to the performative optimal point of the true agent responses, the solutions computed using standard microfoundations would not only experience suboptimal performative risk but also would cause unnecessarily high social burden. This makes it hard for the decision-maker to justify the use of standard microfoundations as a representative model for agent behavior. Implicit in our argument is the following moral stance: \textit{given a set of criteria for what defines a plausible model for microfoundations, the decision-maker should not select the one that maximizes negative externalities}. \section{Alternative microfoundations}\label{sec:agnostic} We now depart from the classical approach and systematically search for models that are more appropriate for binary classification. We define the space of all possible alternative microfoundations and collect a set of useful properties that we show are desirable for microfoundations to satisfy. \subsection{Defining the space of alternatives} The principle behind microfoundations for strategic classification is to equip the distribution map with structure by viewing the distribution induced by a decision rule as an aggregate of the responses of individual agents. We consider agent responses in full generality by introducing a family of response types $\AllTypes$ that represents the space of all possible ways that agents can perceive and react to the classifier $f_\theta$. The response type fully determines agent behavior through the \textit{agent response function} $\Response_t: X \times \Theta \rightarrow X$. In particular, an agent with true features $x$ and response type $t$ changes their features to $x'=\Response_t(x, \theta)$ when the classifier $f_\theta$ is deployed. \begin{remark} Using the language of agent response functions, non-strategic agents correspond to a response type $t_{\text{NS}}$ such that $\Response_{t_{\text{NS}}}(x, \theta) = x$ for all $\theta \in \Theta$, and standard microfoundations correspond to a response type $t_{\text{SM}}$ where $\Response_{t_{\text{SM}}}(x, \theta)$ is given by \eqref{eq:SM} for all $\theta \in \Theta$. Note that a population of agents could be heterogeneous and exhibit a mixture of different types, or even be described by a \textit{continuum} of response types. \end{remark} We formalize microfoundations through a \textit{mapping} $M: X \times Y \rightarrow \AllTypes$ from agents to response types. We denote the set of possible mappings $M$ by the collection $\cM$ that consists of all\footnote{These mappings are subject to mild measurability constraints that we describe in detail in Appendix \ref{subsec:requirements}.} possible randomized functions $X \times Y \rightarrow \AllTypes$. For example, standard microfoundations correspond to the mapping $M_{\textsc{SM}} \in \cM$ such that $M_{\textsc{SM}}(x, y) = t_{\textsc{SM}}$ for all $x, y \in X \times Y$, and a population of non-strategic agents corresponds to the mapping $M_{\textsc{NS}} \in \cM$ such that $M_{\textsc{NS}}(x, y) = t_{\textsc{NS}}$ for all $x, y \in X \times Y$. While these two homogeneous populations can be captured by deterministic mappings, randomization is necessary to capture heterogeneity in agent responses across agents with the same original features. For example, randomness allows us to capture a \textit{mixed} population of agents where some agents behave according to standard microfoundations and other agents are non-strategic. Conceptually, the mapping $M\in\cM$ sets up the rules of agent behavior. One aspect that distinguishes our framework from typical approaches to microfoundations in the economics literature is that it directly specifies agent responses, rather than specifying an underlying behavioral mechanism. An advantage of this approach is that responses can be observed, whereas the behavioral mechanism is harder to infer. Importantly, the mapping $M$ coupled with the base distribution $\DBase$ provides all the necessary information to specify the population's response to a classifier $f_\theta$. In particular, for each $\theta \in \Theta$, the \textit{aggregate response} $\cD(\theta; M)$ is the distribution over $(\Response_t(x, \theta), y)$ where $(x, y) \sim \DBase$ and $t=M(x,y)$. We use the notation $\cD(\cdot; M): \Theta \rightarrow \Delta(X \times Y)$ to denote the aggregate response map induced by $M$. By defining the distribution map, the mapping $M$ thus provides sufficient information to reason about the performative risk; in addition, it also provides sufficiently fine-grained information about individuals to reason about metrics such as social burden. Naturally, with such a flexible model, \textit{any} distribution map can be microfounded, albeit with complex response types, as long as feature manipulations do not change the fraction of positively labeled agents in the population. We refer to Appendix \ref{appendix:microfound} for an explicit construction of $M$. \begin{restatable}{proposition}{microfound} \label{prop:microfound} Let $\DBase$ be a non-atomic distribution. Let $\DMap(\theta)$ be any distribution map that preserves the marginal distribution over $Y$ of $\DBase$. Then, there exists a mapping $M \in \mathcal{M}$ such that $\cD(\noargument; M)$ is equal to $\cD(\noargument)$. \end{restatable} This result primarily serves as an existence result to show that our general framework for microfoundations can capture continuous distributions that are observed empirically (e.g. Examples \ref{ex:FICO}-\ref{ex:testscores}). In the following subsections, we focus on narrowing down the space of candidate models and describe two properties that we believe microfoundations should satisfy. \subsection{Aggregate smoothness} The first property pertains to the induced distribution and its interactions with the function class. This aggregate-level property rules out unnatural discontinuities in the distribution map. We call this property \emph{aggregate smoothness}, and formalize it in terms of the \textit{decoupled performative risk}~\citep{PZMH20}. \begin{property}[Aggregate smoothness] \label{prop:AS} Define the decoupled performative risk induced by $M$ to be $\PRDecoupledM{\theta}{ \theta'} := \mathbb{E}_{(x,y)\sim \cD(\theta;M)} [\;\Indicator\{y\neq f_{\theta'}(x)\}]$. For a given base distribution $\DBase$, a mapping $M$ satisfies \textit{aggregate smoothness} if the derivative of the decoupled performative risk with respect to $\theta'$ exists and is continuous in $\theta$ and $\theta'$ across all of $\Theta$. \end{property} Intuitively, the existence of the partial derivative of $\PRDecoupledM{\theta}{ \theta'}$ with respect to $\theta'$ guarantees that \textit{each distribution $\DMap(\theta; M)$ is sufficiently continuous (and cannot have a point mass at the decision boundary)}, and assuming continuity of the derivative we guarantee that \textit{$\DMap(\theta;M)$ changes continuously in $\theta$}. This connection between aggregate smoothness and continuity of the distribution map can be made explicit in the case of 1-dimensional features: \begin{restatable}{proposition}{onedsmooth} \label{prop:1dsmooth} Suppose that $X \subseteq \mathbb{R}$, and let $\Theta \subseteq \mathbb{R}$ be a function class of threshold functions. Then, if the distribution map $\DMap(\noargument; M)$ has the following properties, the mapping $M$ satisfies aggregate smoothness w.r.t. $\Theta$: \begin{enumerate}[itemsep=-0.5ex, topsep=-0.2ex] \item For each $\theta$, the probability density $p_{\theta}(x,y)$ of $\DMap(\theta; M)$ exists everywhere and is continuous in $x$. \item For each $x, y$, the probability density $p_{\theta}(x,y)$ is continuous in $\theta$. \end{enumerate} \end{restatable} We believe that these two continuity properties are natural and likely to capture practical settings, given the empirical evidence in Examples \ref{ex:FICO}-\ref{ex:FE}. A consequence of aggregate smoothness is that it is sufficient to guarantee the existence of locally stable points. \begin{restatable}{theorem}{existencethm} \label{lemma:existence} Given a base distribution $\DBase$ and function class $\Theta$, for any mapping $M$ that satisfies aggregate smoothness, there exists a locally stable point. \end{restatable} In fact, this result implies that stable points exist under deviations from the model, as long as aggregate smoothness is preserved. Our next result shows that under weak assumptions on the base distribution this is the case for any mixture with non-strategic agents. For ease of notation, we formalize such a mixture through the operator $\Phi_p(M)$, where for $p \in [0,1]$, we let $\Phi_p(M(x,y))$ be equal to $t_{\text{NS}}$ with probability $p$ and equal to $M(x, y)$ otherwise. \begin{restatable}{proposition}{mixtures} \label{cor:mixtures} Suppose that the non-performative risk $\mathrm R(\theta) := \mathbb{E}_{(x,y) \in \DBase} \mathds{1}\{f_{\theta}(x) = y\}$ is continuously differentiable for all $\theta\in\Theta$. Then, for any $p \in [0,1]$, aggregate smoothness of a mapping $M$ is preserved under the operator $\Phi_p(M)$ . \end{restatable} Proposition \ref{cor:mixtures}, together with Theorem \ref{lemma:existence}, implies the robust existence of locally stable points under mixtures with non-strategic agents, for any microfoundations model that satisfies aggregate smoothness. Conceptually, our investigations in this section have been inspired by the line of work on performative prediction~\citep{PZMH20,MPZH20} that demonstrated that regularity assumptions on the aggregate response alone can be sufficient to guarantee the existence of stable points for smooth, strongly convex loss functions. However, our results differ from these previous analyses of performative stability in that we instead focus on the 0-1 loss. In Appendix \ref{appendix:lipschitz}, we provide an example to show why the Lipschitzness assumptions on the distribution map used in prior work are not sufficient to guarantee the existence of stable points in a binary classification setting. \subsection{Constraint on manipulation expenditure} \label{sec:tractability} \label{subsec:microassumptions} While aggregate smoothness focused on the population-level properties of the induced distribution, a model for microfoundations must also be descriptive of realistic agent-level responses in order to yield useful qualitative insights about metrics such as social burden or accuracy on subgroups. A minimal assumption on agent responses is that an agent never expends more on manipulation than the utility of a positive outcome. \begin{property}[Expenditure constraint] \label{def:weakmicro} Given a function class $\Theta$ and a cost function $c$, a mapping $M \in \mathcal{M}$ is \textit{expenditure-constrained} if $c(x, \Response_t(x,\theta))\leq\gamma$ for every $ \theta \in \Theta$ and every $t \in \text{Image}(M)$. \end{property} This constraint is implicitly encoded in standard microfoundations and many of its variants. We have previously encountered the expenditure constraint in Section~\ref{sec:misspecifications}, where we showed that if $c$ is a valid cost function, then this property, together with a basic monotonicity requirement on agent's feature manipulations, defines a set of microfoundations models among which the standard model achieves extreme social burden at optimality. In Section \ref{sec:framework} we will describe on one particular model for microfoundations within this set which results in a \emph{strictly} lower social burden than the standard model. \paragraph{Reducing the complexity of estimating the distribution map.} Apart from defining a natural class of feasible microfoundations models, an additional advantage of Property~\ref{def:weakmicro} is that it naturally constrains each agent's range of manipulations. This can significantly reduce the complexity of estimating the distribution map for a decision-maker who wants to compute a strategy robust classifier offline. Assume the decision-maker follows a two-stage estimation procedure to estimate a performatively optimal point, similar to \citep{MPZ21}. First, they compute an estimate $\tilde M$ of the true mapping $M$ and infer $ \cD(\noargument; \tilde M)$ from the base distribution $\DBase$. Second, they assume the model reflects the true decision dynamics and approximate optimal points as follows: \begin{align} \label{eq:PO_M} \thetaPO(\tilde M) &:=\argmin_{\theta\in\Theta} \mathbb E_{(x,y)\sim\cD(\theta;\tilde M)} \left[\mathds{1}\{y\neq f_\theta(x)\}\right]. \end{align} Using a naive bound (see Lemma \ref{lemma:TVbound}) it is not difficult to see that it suffices to compute an estimate $\tilde M$ of $M$, such that $\sup_\theta \TV(\cD(\theta;\tilde M),\cD(\theta;M))\le \xi$ to guarantee that $\PR(\thetaPO(M))-\PR(\thetaPO(\tilde M) )\leq 2\xi$. However, achieving this level of accuracy fundamentally requires a full specification of the response types for every agent in the population. The expenditure constraint helps to make this task more tractable, in that the decision-maker only needs to estimate responses for a small fraction of the agents to achieve the same bound on the suboptimality of the obtained performative risk. To formalize this, let's assume the decision-maker can define a set $\Theta_0\subseteq \Theta$ that contains the performatively optimal classifier $\thetaPO(M)$. Then, given the implied restriction in the search space in \eqref{eq:PO_M}, the expenditure constraint enables us to restrict the set of covariates that are relevant for the optimization problem to \begin{align} \label{eq:S} S(\Theta_0,c)&:= \cup_{\theta \in \Theta_0} \{x\in X: \exists x' \in X : f_{\theta}(x') \neq f_{\theta}(x) \wedge c(x, x') \le \gamma\}. \end{align} The salient part $S(\Theta_0,c) \subseteq X$ captures all agents who are sufficiently close to the decision boundary for some $\theta\in\Theta_0$ so they are able to cross it without expending more than $\gamma$ units of cost. The subset $S(\Theta_0,c)$ can be entirely specified by the cost function $c$ and can be much smaller than $X$. To see this, consider the following example setting. \begin{example}[Informal] \label{ex:salient} Consider Setup \ref{ex:stablepointsnotexist}. Define the cost function to be linear: $c(x,x') = \alpha |x - x'|$ for some $\alpha>0$. Then, if $M$ satisfies the expenditure constraint and Assumption \ref{assumption:gamingbehavior}, then: \[\thetaPO \in \Theta_0 := [\theta_{\text{SL}} - 3/\alpha, \theta_{\text{SL}} + 3/\alpha] \quad S(\Theta_0, c) = \left\{ x \mid \mu(x) \in [\theta_{\text{SL}}- 4/\alpha, \theta_{\text{SL}} + 4/\alpha]\right\},\] where $\theta_{\text{SL}}$ is such that $\mu(\theta) = 0.5$ (where $\mu$ is defined in Setup \ref{ex:stablepointsnotexist}). \end{example} \noindent Example \ref{ex:salient} demonstrates the salient part consists of agents who are sufficiently close to the supervised learning threshold, where closeness is measured by the cost function. We formalize this example in Appendix~\ref{appendix:construction}. We now describe the implications of constraining to the salient part for a 1-dimensional setting where $X \subseteq \mathbb{R}$ and $f_\theta$ is a threshold function.\footnote{Proposition \ref{prop:samplecomplexity} directly extends to \textit{posterior threshold functions} \citep{MMDH19}.} Let us define an \textit{agent response oracle} that given $x$ and $\theta$, outputs a draw $x'$ from the response distribution $(\Response_t(x, \theta), y)$ where $(x, y) \sim \DBase$. We show with few calls to the oracle, the decision-maker can build an sufficiently precise estimate of $M$. \begin{restatable}{proposition}{oracle} \label{prop:samplecomplexity} Let $X \subseteq \mathbb{R}$, let $\Theta \subseteq \mathbb{R}$ be the function class of threshold functions. Suppose that $M$ satisfies the expenditure constraint, the distribution map $\DMap(\cdot; M)$ is 1-Lipschitz with respect to TV distance\footnote{That is, for any $\theta, \theta' \in \Theta$, we have that $\TV\left(\DMap(\theta'; M), \DMap(\theta; M)\right) \le \norm{\theta - \theta'}_2$.}, and $\Theta_0\subseteq \Theta:\thetaPO(M)\in \Theta_0$. We further assume that an agent's type does not depend on their label, i.e., $M(x,0)=M(x,1)$ for all $x\in X$. Then, with $O\left(\zeta^2 \frac{\ln(1/\epsilon)}{2\epsilon^3}\right)$ calls to the agent response oracle, where $\zeta:= \mathbb{P}_{\DBase}[x \in S(\Theta_0, c)]$, the decision-maker can create an estimate $\tilde{M}$ so that: \[\PR(\thetaPO(\tilde M)) \le \PR(\thetaPO(M)) + \epsilon.\] with probability $0.9$ for any $\epsilon>0$.\footnote{We note that the decision-maker will only search over $\Theta_0$ in \eqref{eq:PO_M} when computing $\thetaPO(\tilde{M})$. In particular, they compute $\argmin_{\theta\in\Theta_0} \mathbb E_{(x,y)\sim\cD(\theta;\tilde M)} \left[\mathds{1}\{y\neq f_\theta(x)\}\right]$ rather than $\argmin_{\theta\in\Theta} \mathbb E_{(x,y)\sim\cD(\theta;\tilde M)} \left[\mathds{1}\{y\neq f_\theta(x)\}\right]$.} \end{restatable} \noindent The number of necessary calls to the response function oracle for estimating $M$ decays with $\zeta:=\mathbb{P}_{\DBase}[x \in S(\Theta_0, c)]$. Without any assumption on agent responses we have $S(\Theta_0,c) = X$ and $\zeta=1$. However, when the decision-maker is able to constrain $S(\Theta_0, c)$ to a small part of the input space by relying on the expenditure constraint, domain knowledge, or stronger assumptions on agent behavior, $\zeta$ and thus the number of oracle calls can be reduced significantly. The concept of a salient part bears resemblance to the approaches by \citet{ZC21}; \citet{ZCC21}, which directly specify the set of feature changes that an agent may make, rather than implicitly specifying agent actions through a cost function. While these models assume that agents best-respond, our key finding is that constraining agent behavior alone can lessen the empirical burden on the decision-maker. \section{Microfoundations based on imperfect agents} \label{sec:framework} Using the properties established in the previous section as a guide, we propose an alternate model for microfoundations that naturally allows agents to undershoot or overshoot the decision boundary, while complying with aggregate smoothness and expenditure monotonicity. Furthermore, we show that this model, called \textit{noisy response}, leads to strictly smaller social burden than the standard model while retaining analytical tractablility. \subsection{Noisy response} Noisy response captures the idea of an \textit{imperfect agent} who does not perfectly best-respond to the classifier weights. This imperfection can arise from many different sources---including interpretability issues, imperfect control over actions, or opaque access to the classifier. Inspired by \emph{smoothed analysis}~\citep{smoothedanalysis}, we do not directly specify the source of imperfection but instead capture imperfection in an agnostic manner, by adding small random perturbations to the classifier weights targeted by the agents. Since smoothed analysis has been successful in explaining convergence properties of algorithms in practical (instead of worst case) situations, we similarly hope to better capture empirically observed strategic phenomena. We define the relevant set of types $T_{\text{noisy}}\subset\AllTypes$ so that each type $t \in T_{\text{noisy}}$ is associated with a noise vector $\eta_t \in \mathbb{R}^m$. An agent of type $t\in T_{\text{noisy}}$ perceives $\theta$ as $\theta + \eta_t$ and responds to the classifier $f_\theta$ as follows: \begin{equation} \Response_t(x, \theta) := \label{eq:RFP} \argmax_{x'\in X'} \; \left[\gamma \cdot f_{\theta + \eta_t}(x') - \costfn(x, x')\right], \end{equation} where $c$ denotes a valid cost function, $\gamma>0$ denotes the utility of a positive outcome, and $X' \subseteq \mathbb{R}^d$ is a compact, convex set containing $X$.\footnote{We assume that $c$ is defined on all of $X'\times X'$, and $c(x, x') > \gamma$ for all $x \in X$ and all $x'$ that are on the boundary of $X'$.} For each $(x,y) \in X \times Y$, we model the distribution over noise across all agents with feature, label pair $(x,y)$ as a multivariate Gaussian. To formalize this, we define a \textit{randomized} mapping $M_{\sigma}: X \times Y \rightarrow \AllTypes$ as follows. For each $(x,y)$, the random variable $M_\sigma(x,y)$ is defined so that if $t \sim M_\sigma(x,y)$, then $\eta_t$ is distributed as $\mathcal N (0, \sigma^2 I)$. This model results in the perceived values of $\theta$ across all agents with a given feature, label pair following a Gaussian distribution centered at $\theta$. The noise level $\sigma$ reflects the degree of imperfection in the population. Conceptually, our model of noisy response bears similarities to models of \textit{incomplete information} \citep{harsanyi68incomplete} that are standard in game theory (but that have not been traditionally considered in the strategic classification literature). However, a crucial difference is that we advocate for modeling agents actions as imperfect even if the classifier is fully transparent, because we believe that imperfection can also arise from other sources. This is supported by the empirical study of \cite{BBK20} discussed in Example \ref{ex:FE} where agents act imperfectly even when the classifier weights are revealed. We want to emphasize that we instantiate imperfection by adding noise to the manipulation targets, instead of directly adding noise to the \textit{responses}. While both approaches would mitigate the discontinuities in the aggregate distribution, the approach of adding noise directly to the responses results in a less natural model for agent behavior that violates the expenditure constraint. \subsection{Aggregate-level properties of noisy response} \label{sec:limitA} Intuitively, the noise in the manipulation target of noisy response smooths out the discontinuities of standard microfoundations, eliminating the point mass at the decision boundary and region of zero density below the decision boundary. We show this explicitly in a 1-dimensional setting. \begin{restatable}{proposition}{continuous} \label{thm:continuous} Let $\Theta \subseteq \mathbb{R}$ be a function class of threshold functions, and suppose also that $X \subseteq \mathbb{R}$. For any $\sigma \in (0, \infty)$, the distribution map $\DMap(\noargument; \Fuzzy{\sigma})$ satisfies the continuity properties in Proposition \ref{prop:1dsmooth}, and thus the mapping $\Fuzzy{\sigma}$ satisfies aggregate smoothness. \end{restatable} \begin{remark} This result implies that noisy response inherits the robust existence of stable points under mixtures with non-strategic agents from Theorem~\ref{lemma:existence}. Furthermore, we illustrate in Figure~\ref{fig:oscillation-fuzzy} how noisy response mitigates the large oscillations of repeated retraining that we observed for standard microfoundations. We observe that in the case of $p = 0$, noisy response results in a lower stable point than standard microfoundations. \end{remark} \begin{figure*}[t!] \subfigure[\footnotesize{$\cD(\theta)$ of NR vs. SM}]{ \includegraphics[width = 0.31\textwidth]{figures/response.pdf} \label{fig:response}} \subfigure[\footnotesize{$\cD(\theta)$ of NR for different $\sigma$}]{ \includegraphics[width = 0.31\textwidth]{figures/response-sig.pdf} \label{fig:sig}} \subfigure[\footnotesize{$\cD(\theta)$ of NR for different $\theta$}]{ \includegraphics[width = 0.31\textwidth]{figures/response-theta.pdf} \label{fig:theta}} \vspace{-0.2cm} \caption{Probability density of the aggregate response $\DMap(\theta)$ in a 1d-setting, where the base distribution $\cD_{X}$ is a Gaussian with $x\sim\mathcal N(0,0.5)$. We illustrate (a) $\DMap(\theta)$ for a population of agents that follow noisy response (NR) compared to standard microfoundations (SM), (b) how $\DMap(\theta)$ of NR changes for different $\theta$, (c) variations in $\DMap(\theta)$ of NR for different values of $\sigma$.} \label{fig:pdf} \end{figure*} To visualize the aggregate-level properties of noisy response and contrast them with standard microfoundations we depict the respective density functions for a 1-dimensional setting with a Gaussian base distribution in Figure~\ref{fig:response}. The distribution $\cD(\theta)$ can be bimodal, since agents closer to the threshold $\theta$ are more likely to change their features. The shape of the response distribution changes with $\sigma$ as illustrated in Figure~\ref{fig:sig}. As $\sigma \rightarrow 0$, the aggregate response of a population of noisy response agents approaches that of standard microfoundations. This means that noisy response maintains continuity while also being able to approximate the aggregate response of standard microfoundations to arbitrary accuracy. Finally, we note that the distribution map of noisy response changes \textit{continuously} with $\theta$, as visualized in Figure~\ref{fig:theta}. In fact, the distribution map induced by noisy response is Lipschitz in total-variation distance, where the Lipschitz constant grows with $1 / \sigma$. \begin{restatable}{lemma}{tvlipschitz} \label{lemma:tvlipschitz} Given $\sigma\in(0,\infty)$, the distribution map $\DMap(\theta; M_{\sigma})$ is continuous in TV distance and supported on all of $X$. Moreover, the distribution map is Lipschitz in TV distance. That is, for any $\theta, \theta' \in \Theta$, we have that $\TV\left(\DMap(\theta'; \Fuzzy{\sigma}), \DMap(\theta; \Fuzzy{\sigma})\right) \le \frac{1}{2 \sigma} \norm{\theta - \theta'}_2$. \end{restatable} \noindent This result highlights a favorable property of noisy response compared to standard microfoundations, in that the performative risk changes smoothly with changes in the classifier weights. \begin{remark}[Implications beyond 0-1 loss] Lemma~\ref{lemma:tvlipschitz} implies that noisy response induces a distribution map that is Lipschitz in Wasserstein distance \textit{for any cost function}, where the Lipschitz constant depends on the diameter of the set $X'$. For smooth and strongly convex loss functions, this readily implies convergence of repeated retraining~\citep{PZMH20, MPZH20}. \end{remark} \subsection{Trade-off between imperfection and social burden}\label{sec:limitC} Apart from satisfying desirable aggregate-level properties, noisy response also satisfies the expenditure monotonicity requirement in Property~\ref{property:SB} (see Proposition \ref{prop:expenditurerationality} for a proof). By Corollary~\ref{coro:socialburdeninformal}, this implies that in Setup \ref{ex:stablepointsnotexist} the social burden of the optimal classifier computed under noisy response is no larger than that of standard microfoundations. That is, $\SocialBurden{\thetaPO(\Fuzzy{\sigma})} \le \SocialBurden{\thetaPO(M_{\mathrm{SM}})}$. In certain cases, we can obtain a stronger result and show that the social burden of noisy response is \emph{strictly} lower than that of standard microfoundations. \begin{restatable}{corollary}{socialburdenpos} \label{cor:socialburden} Consider Setup \ref{ex:stablepointsnotexist}. Let $M_{\mathrm{SM}}$ be the mapping associated with standard microfoundations, let $\sigma \in (0, \infty)$, and let the cost function be of the form $c(x_1, x_2) = |x_1 - x_2|$. Suppose that $[\theta_\mathrm{SL}, \theta_{\mathrm{SL}} + 1] \in \Theta \cap X$, where $\theta_{\mathrm{SL}}$ is defined so that $\mu(\theta_{\mathrm{SL}}) = 0.5$. Then, it holds that: \[\SocialBurden{\thetaPO(\Fuzzy{\sigma})} < \SocialBurden{\thetaPO(M_{\mathrm{SM}})}.\] \end{restatable} \begin{figure*}[t!] \centering \subfigure[\footnotesize{Optimal points}]{ \includegraphics[height = 0.245\textwidth]{figures/threshold.pdf} \label{fig:thresholds}}\hspace{0.5cm} \subfigure[\footnotesize{Social burden of optimal points}]{ \includegraphics[height = 0.245\textwidth]{figures/burden.pdf} \label{SB}} \vspace{-0.2cm} \caption{Optimal points in a 1d-setting evaluated for different values of $\sigma$ (noise in the response model) and $p$ (fraction of non-strategic agents). The population consists of $10^{5}$ individuals. Half of the individuals are sampled from $x\sim\mathcal N\big(1,\tfrac 1 3\big)$ with true label $1$ and the other half is sampled from $x\sim\mathcal N\big(0,\tfrac 1 3\big)$ with true label $0$. } \label{fig:socialburdenthresholds} \end{figure*} In fact, the social burden for fuzzy perception can be well below the social burden of standard microfoundations. To demonstrate this we visualize the threshold and social burden across a variety of different values of the noise parameter $\sigma$ and the fraction of non-strategic agents $p$ in Figure \ref{fig:socialburdenthresholds}. The dashed lines indicate the respective reference values for standard microfoundations (SM) and for a population of non-strategic agents (NS). We observe that the threshold as well as the social burden decrease with the fraction $p$ of non-strategic agents in the population. Furthermore, if every agent follows noisy response ($p = 0$), the threshold and the social burden are decreasing with $\sigma$ (and hence the degree of imperfection in agents response). Overall, the acceptance threshold and the social burden of optimal classifiers derived under our new microfoundations is significantly lower than for standard microfoundations. \subsection{Approximating the distribution map for noisy response}\label{subsec:practical} Microfoundations are useful in practice to model and anticipate how the population responds to a deployed classifier. This tool is particularly powerful if the decision-maker can estimate agent behavior without needing to expose the population to potentially inaccurate and harmful classifiers to explore and learn about agents responses. An appealing aspect of assuming a parameterized model of the agent responses is that the complex task of learning agent behavior is reduced to a parameter estimation problem. For noisy response, the aggregate response $\DMap(\theta)$ is parameterized by the variance $\sigma$ of the target noise. In practice such parameters of individual responses can often be estimated via \textit{individual experiments}, i.e., by gathering information about individuals without ever deploying a classifier. For example, the decision-maker can randomly sample agents in the population, ask survey questions to learn about their perceptions of the deployed classifier, and infer the variance $\sigma$ from these samples. We refer to \citep{BBK20} for an actual field experiment that shows an example procedure for how to similarly obtain a reasonable estimate of the cost function $c$. We now show that an error in parameter estimation can then be translated into an error in the aggregate response. \begin{restatable}{lemma}{tvboundfuzzy} \label{lemma:tvboundfuzzy} Given a population of agents following noisy response with parameter $\sigma\in(0,\infty)$, and an estimate $\tilde{\sigma}$ of $\sigma$, it holds that $\TV\big(\DMap(\theta; \Fuzzy{\sigma}), \DMap(\theta; \Fuzzy{\tilde{\sigma}})\big) \le \frac 1 2 \sqrt{\frac {|\sigma^2 - \tilde{\sigma}^2| m}{\min(\sigma^2, \tilde{\sigma}^2)}}$ where $m$ is the dimension of the feature space. \end{restatable} Combining Lemma \ref{lemma:tvboundfuzzy} with a bound on the performative risk, we obtain the following robustness guarantee of the performative risk to estimation errors in the noise parameter $\sigma$. \begin{restatable}{corollary}{fuzzyTV} \label{cor:fuzzyTV} Let the population be the aggregate of agents following noisy response with parameter $\sigma\in(0,\infty)$, and let $\tilde{\sigma} \in (0,\infty)$ be an estimate of the perception parameter ${\sigma}$. Then the suboptimality of the estimated performative risk of $\thetaPO( M_{\tilde \sigma})$ on the true population represented by $M_\sigma$ is bounded by: \[\PR(\thetaPO({M}_{\tilde \sigma}))- \PR(\thetaPO(M_\sigma))\leq \sqrt{\frac {|\sigma^2 - \tilde{\sigma}^2| m}{\min(\sigma^2, \tilde{\sigma}^2)}}. \] \end{restatable} \noindent Hence, if the true distribution map is sufficiently close to some parameterization of noisy response, estimating the noise parameter $\sigma$ provides a robust procedure to infer an estimate of performative optima in practice. Overall, noisy response offers a more descriptive and prescriptive model of agent behavior compared to standard microfoundations, and still maintains analytical tractability. While we have focused on Gaussian noise in the perception function throughout this work, the outlined benefits of noisy response also apply to other \textit{parameterized} noise distributions, as long as they are sufficiently smooth and continuous on all of $\mathbb{R}^m$. Hence, depending on the application, the decision-maker might prefer to pick a different noise model that can better capture the expected particularities of agents imperfections. The inference procedure via individual experimentation can then be adapted to obtain performative risk estimates that depend on the parameters of the noise distribution. \section{Discussion} To anticipate the strategic behavior of a population in response to a classifier, the decision-maker needs to understand and model the distribution shifts induced by decision rules. Traditional approaches for modeling these distribution shifts are either purely individual-level or purely population-level: \emph{strategic classification} typically builds on standard microfoundations as a specification of individual-level behavior to deduce aggregate-level responses, whereas \textit{performative prediction} directly works with a population-level characterization of distribution shifts. In this work, we provided a fresh perspective on microfoundations for strategic classification that is inspired by performative prediction. While microfoundations can provide valuable structure to distribution shifts, we illustrated that the standard model for agent responses can lead to degenerate behavior in aggregate. These unanticipated degeneracies motivated us to consider population-level properties, rather than only individual-level properties, when searching through the space of alternative models for agent responses. This approach helped us identify noisy response as a promising candidate model for reasoning about strategic behavior in response to a binary decision rule. While we have focused on strategic classification in this work, we believe that striving for a synergy between individual-level and aggregate-level considerations of agent behavior can more broadly lead to interesting insights about algorithmic decision-making in a social context. \section*{Acknowledgments} We would like to thank Jacob Steinhardt and Tijana Zrnic for feedback on this manuscript. MJ acknowledges support from the Paul and Daisy Soros Fellowship. CMD acknowledges support from the Swiss National Science Foundation Postdoc.Mobility fellowship program. MH acknowledges support from NSF Award 1750555.

proofpile-arXiv_069-12974

\section{Introduction} The Schramm-Loewner evolution (SLE) introduced by Oded Schramm is a one-parameter ($\kappa\in(0,\infty)$) family of random fractal curves growing in plane domains, which satisfy conformal invariance and domain Markov property. Due to its close relation with two-dimensional statistical lattice models, Brownian motion, Gaussian free field and Liouville quantum gravity, SLE has received a lot of attention over the past two decades. We refer the reader to Lawler's textbook \cite{Law-SLE} for basic properties of SLE. Hausdorff dimension and Minkowski content are central objects in the study of the fractal properties of SLE. It is known that the Hausdorff dimension of an SLE$_\kappa$ curve is $2\wedge(1+\frac\kappa 8)$ (\cite{RS,Bf}), and the corresponding Minkowski content of any bounded SLE$_\kappa$ arc exists (\cite{LR}). The Hausdorff dimension of the intersection of an SLE$_\kappa$ curve with the domain boundary is $0\vee(1\wedge (2-\frac 8\kappa))$ (\cite{dim-real}), and the corresponding Minkowski content of the intersection restricted to any compact boundary arc also exists (\cite{Mink-real}). The Minkowski content of SLE agrees with the natural parametrization of SLE developed in \cite{LS,LZ}, which was constructed using the Doob-Meyer decomposition. The authors of \cite{cov} used that idea to construct a conformally covariant measure on the intersection of a {\it chordal} SLE$_\kappa$ (one important type of SLE) curve with $\R$, and conjectured that their measure has close relation with the Minkowski content of the intersection. In this paper, we work on chordal SLE$_\kappa(\ulin\rho)$, which are natural variants of chordal SLE$_\kappa$. The growth of a chordal SLE$_\kappa(\ulin\rho)$ curve is affected by the force values $\ulin\rho=(\rho_1,\dots,\rho_m)$ and force points $(v_1,\dots,v_m)$ in addition to the initial point and target point of the curve. One-force-point SLE$_\kappa(\rho)$ was introduced in \cite{LSW-8/3} for the construction of conformal restriction measures. Multi-force-point SLE$_\kappa(\ulin\rho)$ was extensively studied in \cite{MS1} in the framework of imaginary geometry. Using imaginary geometry, the authors of \cite{MW} proved that for a (one-force-point) chordal SLE$_\kappa(\rho)$ curve $\eta$ in $\HH$ started from $w$ with force point $v<w$, the Hausdorff dimension of $\eta\cap (-\infty,v)$ is $\frac{(\rho +4)(\kappa-4-2\rho )}{2\kappa}$, when $\rho\in((-2)\vee (\frac\kappa 2-4),\frac\kappa 2-2)$. The goal of this paper is to prove the existence of the Minkowski content of the intersection of a multi-force-point chordal SLE$_\kappa(\ulin\rho)$ curve $\eta$ with some boundary arc of the domain. We assume that the curve $\eta$ grows in $\HH=\{z\in\C:\Imm z>0\}$ from a point $w\in\R$ to $\infty$, and the force points lie on the same side of $w$, say $v_m<\cdots<v_1<w$. Moreover, we assume that the force values are such that a.s.\ $\eta$ intersects but does not contain $(-\infty,v_m)$. By the work of \cite{MW}, the Hausdorff dimension of $\eta\cap(-\infty,v_m]$ is $d:=\frac{(\rho_\Sigma +4)(\kappa-4-2\rho_\Sigma )}{2\kappa}$, where $\rho_\Sigma:=\sum_{j=1}^m\rho_j$. We are interested in the $d$-dimensional Minkowski content of $\eta\cap I$ for intervals $I\subset (-\infty,v_m]$. To establish the Minkowski content, we follow the standard approach in \cite{LR} and \cite{Mink-real} to study the existence of the one-point Green's function $$G_1(z_1):=\lim_{r_1\to 0^+} r_1^{d-1} \PP[\dist(z_1,\eta\cap \R)\le r_1]$$ and the two-point Green's function $$G_2(z_1,z_2):=\lim_{r_1,r_2\to 0^+} r_1^{d-1} r_2^{d-1}\PP[\dist(z_j,\eta\cap \R)\le r_j,j=1,2],$$ for $z_1\ne z_2\in (-\infty,v_m)$, and derive some estimates. The first result on SLE Green's function in the literature (cf.\ \cite{Law-Green}) is about the one-point Green's function for a chordal SLE at an interior point, but with the Euclidean distance replaced by the conformal radius. The two-point conformal radius Green's function for a chordal SLE at interior points was proved to exist in \cite{LW}. The existence of the Eucliean distance version (the original definition) of these Green's functions was proved in \cite{LR}. The one- and two-point Green's functions for a chordal SLE at boundary points were derived in \cite{Mink-real}. We use the ideas in \cite{Mink-real} to study the boundary Green's functions of a chordal SLE$_\kappa(\ulin\rho)$ curve $\eta$. Due to the force points, the proofs here involve more technical details than those in \cite{Mink-real}. We prove the existence of the one-point and two-point Green's functions for $\eta$ on $(-\infty,v_m)$ as well as some upper bounds and convergence rates. We also obtain the exact formula of the one-point Green's function up to a multiplicative constant. Using the estimates of these Green's functions, we then prove that, for any bounded measurable set $S\subset (-\infty,v_m]$, the limit $\lim_{r\to 0^+} r^{1-d} \lambda(\{u\in S: \dist(u,\eta\cap\R)\le r\}) $ converges both a.s.\ and in $L^2$, where $\lambda$ is the Lebesgue measure on $\R$. We let $\xi_S$ denote the limit and prove that whenever $\pa S$ has zero Minkowski content, $\xi_S$ is the Minkowski content of $\eta\cap I$. Furthermore, we use these $\xi_S$ to prove that there is a random measure $\mu_\eta$ supported by $\eta\cap (-\infty,v_m]$ such that for any compact interval $I\subset (-\infty, v_m]$, $\mu_\eta(I)=\xi_I$. We call such $\mu_\eta$ the Minkowski content measure on $\eta\cap (-\infty,v_m]$. The Minkowski content measures could be defined on a more general family of subsets of Euclidean spaces. We prove that they satisfy the conformal covariance and restriction property, and use these properties to show that the increasing function $\mu_\eta(\eta[0,t])$ could act as one component in the Doob-Meyer decomposition of some supermartingale. When all $\rho_j=0$, the SLE$_\kappa(\ulin\rho)$ reduces to the standard chordal SLE$_\kappa$. The above statement shows that the $\mu_\eta$ in this special case agrees with the covariant measure developed in \cite{cov} for chordal SLE$_\kappa$, and so proves the conjecture there. The rest of the paper is organized as follows. In Section \ref{Section 2} we review chordal Loewner equations, SLE$_\kappa(\ulin\rho)$ processes, extremal length, and a family of Bessel-like diffusion processes, which will be needed later in this paper. In Section \ref{Section 3} we derive some one-point estimates of the SLE$_\kappa(\ulin\rho)$ curve $\eta$, i.e., some upper bounds of the probability that $\eta$ gets near a marked point in $(-\infty,v_m]$. In Sections \ref{Section 4} and \ref{Section 5} we obtain the existence and some estimates of the one- and two-point Green's functions for $\eta$ on $(-\infty,v_m)$. In Section \ref{Section 6.1} we define the notion of Minkowski content measure, and develop a framework of this new concept. Finally, in Section \ref{Section 6.2} we use the Green's functions from Sections \ref{Section 4} and \ref{Section 5} to construct the Minkowski content measure on $\eta\cap (-\infty,v_m]$ and prove the conjecture of \cite{cov}. \section{Preliminaries} \label{Section 2} \subsection{Chordal Loewner equation}\label{Section 2.1} A bounded relatively closed subset $K$ of $\HH$ is called an $\HH$-hull if $\HH\sem K$ is a simply connected domain. For an $\HH$-hull $K$, there is a unique conformal map $g_K$ from $\HH\sem K$ onto $\HH$ such that $g_K(z)=z+ O(\frac 1{z })$ as $z\to \infty$. If $K=\emptyset$, $g_K=\id$. Suppose $K\ne\emptyset$. Let $K^{\doub}=\lin K\cup\{\lin z:z\in {\lin K}\}$, where $\lin K$ is the closure of $K$, and $\lin z$ is the complex conjugate of $z$. By Schwarz reflection principle, $g_K$ extends {conformally} to $\C\sem K^{\doub}$. {Let $S_K=\C\sem g_K(\C\sem K^{\doub})$. Then $S_K$ is a compact subset of $\R$.} Let $a_K=\min(\lin{K}\cap \R)$, $b_K=\max(\lin{K}\cap\R)$, $c_K=\min S_K$, $d_K=\max S_K$. Then $g_K$ maps $\C\sem (K^{\doub}\cup [a_K,b_K])$ conformally onto $\C\sem [c_K,d_K]$. Let $W \in C([0,T),\R)$ for some $T\in(0,\infty]$. The chordal Loewner equation driven by $W$ is $$\pa_t g_t(z)= \frac{2}{g_t(z)-W(t)},\quad 0\le t<T;\quad g_0(z)=z.$$ For every $z\in\C$, let $\tau^*_z$ be the first time that the solution $t\mapsto g_t(z)$ blows up; if such time does not exist, then set $\tau^*_z=\infty$. For $t\in[0,T)$, let $K_t=\{z\in\HH:\tau^*_z\le t\}$. It turns out that for each $t\ge 0$, $K_t$ is an $\HH$-hull, $K_0=\emptyset$, $K_t\ne\emptyset$ for $t>0$, and $g_t$ agrees with $g_{K_t}$. We call $g_t$ and $K_t$ the chordal Loewner maps and hulls, respectively, driven by $W$. We write $g^W_t$ and $K^W_t$ to emphasize the dependence of $g_t$ and $K_t$ on $W$, when necessary. Let $c^W_t=c_{K^W_t}$ and $d^W_t=d_{K^W_t}$ for $t>0$; and $c^W_0=d^W_0=W(0)$. \begin{Definition} For $w\in\R$, let $\R_{w}$ be the set $\R\sem\{w\}\cup \{w^+,w^-\}$ endowed with the obvious order. Let $W,g^W_t,\tau^*_z,c^W_t,d^W_t$ be as above. The modified chordal Loewner maps $\ha g^W_t$ driven by $W$ are functions defined on $\R_{W(0)}$ for $0\le t<T$ such that $$\ha g^W_t(v)=\left\{\begin{array}{ll} g^W_t(v), &\mbox{if }0\le t<\tau^*_v; \\ d^W_t, & \mbox{if }t\ge \tau^*_v \mbox{ and }v\ge W(0)^+;\\ c^W_t , & \mbox{if }t\ge \tau^*_v\mbox{ and }v\le W(0)^-. \end{array} \right.$$ Here $\tau^*_{W(0)^\pm}$ are understood as $0$, and so $\ha g^W_t(W(0)^+)=d^W_t$ and $\ha g^W_t(W(0)^-)=c^W_t$. \label{Def-Rw} \end{Definition} For $z_0\in \C$ and $S\subset\C$, let $\rad_{z_0}(S)=\sup\{|z_0-z|:z\in S\cup \{z_0\}\}$. The following proposition is well known (cf.\ \cite{Law-SLE}). \begin{Proposition} Let $K_t$ and $g_t$, $0\le t\le \delta$, be chordal Loewner hulls and maps driven by $W$. Suppose $\rad_{\ha W(0)}( K_\delta)\le r$ for some $x_0\in\R$ and $r>0$. Then we have \begin{enumerate} \item [(i)] $| W(\delta) - W(0)|\le 2r$; \item [(ii)] $|g_{\delta}(z)-z|\le 3r$ for any $z\in\lin\HH\sem \lin {\ha K_\delta}$; \item [(iii)] $|\ln (g_\delta'(z))|\le 5 (\frac r{|z-x_0|})^2$ for any $z\in\lin\HH$ with $|z-x_0|\ge 10 r$. \end{enumerate} \label{basic-chordal} \end{Proposition} We use $f_t=f^W_t$ to denote the continuation of $(g^W_t)^{-1}$ from $\HH$ to $\lin\HH$, when the continuation exists. If for every $t\in[0,T)$, $f^W_t$ exists, and $\eta(t):=f^W_t({W(t)})$ is continuous on $[0,T)$, then we say that $\eta$ is the chordal Loewner curve driven by $W$. Such $\eta$ may not exist in general; and when it exists, $\eta$ is a curve in $\lin\HH$ started from $W(0)$, and $\HH\sem K_t$ is the unbounded connected component of $\HH\sem \eta[0,t]$ for all $t$. \subsection{Chordal SLE$_\kappa({\protect\ulin\rho})$ processes}\label{Section 2.2} When the driving function $W$ equals $\sqrt\kappa B(t)$, where $\kappa>0$ and $B$ is a standard Brownian motion, the chordal Loewner curve driven by $W$ is known to exist (\cite{RS}) and is called a chordal SLE$_\kappa$ curve in $\HH$ from $0$ to $\infty$. The chordal SLE$_\kappa(\ulin\rho)$ processes are natural variants of chordal SLE$_\kappa$, where one keeps track of additional marked points. We now review the definition and properties of chordal SLE$_\kappa(\ulin \rho)$ developed in \cite{MS1} that will be needed in this paper. Let $\kappa>0$, $m\in\N$, $\rho_1,\dots,\rho_m\in\R$, $w\in\R$. Let $v_1,\dots,v_m\in \R_w=\R\sem\{w\}\cup\{w^+,w^-\}$. We require that for $\sigma\in\{+,-\}$, $\sum_{j:v_j=w^\sigma}\rho_j>-2$. We write $\ulin\rho$ and $\ulin v$ respectively for $(\rho_1,\dots,\rho_m)$ and $(v_1,\dots,v_m)$. The (chordal) SLE$_\kappa(\ulin)$ process in $\HH$ started from $w$ with force points $\ulin v$ is the chordal Loewner process driven by the function $W(t)$, $0\le t<T$, which solves the SDE $$dW(t)=\sqrt\kappa dB(t)+\sum_{j=1}^m{\bf 1}_{\{W(t)\ne \ha g^W_t(v_j)\}} \frac{\rho_j}{W(t)- \ha g^W_t(v_j)}\,dt, \quad W(0)=w,$$ where $B(t)$ is a standard Brownian motion, and $\ha g^W_t$ are as in Definition \ref{Def-Rw}. The SDE should be understood as an integral equation, i.e., $W-w-\sqrt\kappa B$ is absolutely continuous with the derivative being the summation in the SDE. The solution exists uniquely up to the first time $T$ (called a continuation threshold) that $ \sum_{j: g^W_t(v_j)=W(t)} \rho_j\le -2$. If there does not exist a continuation threshold, then the lifetime is $\infty$. We call $V_j(t):=\ha g^W_t(v_j)$ the force point process started from $v_j$. It is absolutely continuous with initial value $V_j(0)=v_j$ and derivative ${\bf 1}_{\{W(t)\ne \til g^W_t(v_j)\}}\frac 2{V_j(t)-W(t)}$. In fact, the set $\{t:W(t)=\til g^W_t(v_j)\}$ has Lebesgue measure zero. So we will omit the factors ${\bf 1}_{\{W(t)\ne \til g^W_t(v_j)\}}$. Thus, $W$ and $V_1,\dots,V_m$ together solve the following system of SDE-ODE: \BGE dW=\sqrt\kappa dB+\sum_{j=1}^m \frac{\rho_j dt}{W-V_j},\quad W_j(0)=w_j;\label{dW}\EDE \BGE dV_j=\frac{2 dt}{V_j-W},\quad V_j(0)=v_j,\quad 1\le j\le m.\label{dVU}\EDE Here we omit ``$(t)$'' for simplicity. Moreover, $V_j\ge W$ if $v_j\ge w^+$; and $V_j\le W$ if $v_j\le w^-$. If $V_j(t_0)=V_k(t_0)$ at some point $t_0$, then $V_j(t)=V_k(t)$ for all $t\ge t_0$. If $v_j\ge v_k$, then $V_j\ge v_k$. If $v_j=v_k$ for some $j\ne k$, then $V_j=V_k$, and so $v_j$ and $v_k$ may be treated as a single force point with force value $\rho_j+\rho_k$. An SLE$_\kappa(\ulin\rho)$ process a.s.\ generates a chordal Loewner curve $\eta$, which satisfies the following DMP. Let $\F=(\F_t)_{t\ge 0}$ be the complete filtration generated by $\eta$. We write $\ulin V$ for $(V_1,\dots,V_m)$. If $\tau$ an $\F$-stopping time, then on the event $\{\tau<T\}$, there is a curve $\eta^\tau$ in $\lin\HH$ such that $\eta(\tau+\cdot)=f_\tau(\eta^\tau)$, and the law of $\eta^\tau$ conditionally on $\F_\tau$ is an SLE$_\kappa(\ulin\rho)$ curve started from $W(\tau)$ with force points $\ulin V(\tau)$, where when $V_j(\tau)=W(\tau)$, then $V_j(\tau)$ as a force point is treated as $W(\tau)^+$ (resp.\ $W(\tau)^-$) if $v_j\ge w^+$ (resp.\ $v_j\le w^-$). \begin{comment} For the generality needed in this paper, we assume that all force points lie on the left side of $w$. By merging and reordering the force points, we assume that $v_m<\cdots<v_1\le w^-$. Then $V_m\le \cdots\le V_1\le W$. If for any $1\le k\le m$, $\sum_{j=1}^k \rho_j>-2$, then the process has lifetime $\infty$, and a.s.\ $\lim_{t\to\infty} \eta(t)=\infty$. For any $k_0\in\{1,\dots,m\}$, if $\sum_{j=1}^{k_0} \rho _j> \frac{\kappa}{2}-4$, then for any $u\in (v _{k_0+1},v_{k_0})$, a.s.\ $\eta$ does not visit $u$; and if $\sum_{j=1}^{k_0} \rho_j\ge \frac{\kappa}{2}-2$, then a.s.\ $\eta$ does not hit the whole interval $(v _{k_0+1},v_{k_0})$, where $v_{m+1}$ is understood as $-\infty$. \end{comment} \subsection{Extremal length} \label{Section 2.3} Extremal length is a nonnegative quantity $\lambda(\Gamma)$ associated with a family of (piecewise $C^1$) curves (cf.\ \cite[Definition 4-11]{Ahl}). The extremal distance between $E$ and $F$ in $\Omega$, denoted by $d_\Omega(E,F)$ , is the extremal length of the family of curves in $\Omega$ connecting $E$ and $F$. Extremal length satisfies conformal invariance (cf.\ \cite[Section 4-1]{Ahl}) and comparison principle (\cite[Theorems 4.1 and 4.2]{Ahl}). If $\Gamma$ is a family of curves in a domain $D$, and $f$ is a conformal map on $D$, then $\lambda(f(\Gamma))=\lambda(\Gamma)$. If every curve in $\Gamma$ contains a curve in $\Gamma'$, then $\lambda(\Gamma)\ge \lambda(\Gamma_1)$. If every curve in $\Gamma$ contains a curve in $\Gamma_1$ and a curve in $\Gamma_2$, and $\Gamma_1$ and $\Gamma_2$ are respectively contained in $\Omega_1$ and $\Omega_2$, which are disjoint open sets, then $\lambda(\Gamma)\ge \lambda(\Gamma_1)+\lambda(\Gamma_2)$. Suppose $R$ is a rectangle of dimension $a\times b$, and $I_1,I_2$ are sides of $R$ with length $a$. Then $d_R(I_1,I_2)=b/a$ (cf.\ \cite[Section 4-2]{Ahl}). By conformal invariance, we get the following result. \begin{Example} For $x\in\R$ and $0<r_1<r_2$, let $A_{\HH}(x,r_1,r_2)$ denote the semi-annulus $\{z\in\HH:r_1<|z-x|<r_2\}$. Let $I_j=\{z\in\HH:|z-x|=r_j\}$, $j=1,2$. Then $d_{A_{\HH}(x,r_1,r_2)}(I_1,I_2)=\frac 1\pi\ln(\frac{r_2}{r_1})$. \end{Example} \begin{Proposition} Let $S_1$ and $S_2$ be a disjoint pair of connected nonempty closed subsets of $\lin\HH$ that intersect $\R$. Then we have $$\prod_{j=1}^2 \Big(\frac{\diam(S_j)}{\dist(S_1,S_2)}\wedge 1\Big)\le 144 e^{-\pi d_{\HH}(S_1,S_2)}.$$ If $S_2$ is unbounded, then for any $z_0\in S_1$, $$\frac{\rad_{z_0}(S_1)}{\dist(z_0,S_2)}\wedge 1 \le 144 e^{-\pi d_{\HH}(S_1,S_2)}.$$ \label{extrem-prop} \end{Proposition} \begin{proof} The first statement in the special case that $S_1$ and $S_2$ are bounded is exactly \cite[Lemma 2.9]{existence}. The general case follows from the special case by replacing $S_j$ with $S_j\cap \{|z|\le n\}$ and letting $n\to \infty$. The second statement follows immediately from the first one. \end{proof} \subsection{Transition density of some diffusion processes} \label{Section 2.4} In this subsection we review some results about the transition density of a family of Bessel-like diffusion processes. We first briefly review Jacobi polynomials (cf.\ \cite[Chapter 18]{NIST:DLMF}). For indices $\alpha_+,\alpha_->-1$, Jacobi polynomials $P_n^{(\alpha_+,\alpha_-)}(x)$, $n\in\N\cup\{0\}$, are a class of orthogonal polynomials w.r.t.\ the weight $w(s):=(1-s)^{\alpha_+}(1+s)^{\alpha_-}$. They satisfy \BGE P_0^{(\alpha_+,\alpha_-)}\equiv 1;\label{P0}\EDE \BGE \int_{-1}^1 w(s) P_n^{(\alpha_+,\alpha_-)}(s) P_m^{(\alpha_+,\alpha_-)}(s) ds =0,\quad \mbox{if }n\ne m;\label{orthogonal}\EDE \BGE \int_{-1}^1 w(s) P_n^{(\alpha_+,\alpha_-)}(s)^2 ds=\frac{2^{\alpha_++\alpha_-+1}\Gamma(n+\alpha_++1)\Gamma(n+\alpha_-+1)} {n!(2n+\alpha_++\alpha_-+1)\Gamma(n+\alpha_++\alpha_-+1)}=:{\cal A}_n^{(\alpha_+,\alpha_-)};\label{Jacobi-norm}\EDE \BGE \sup_{x\in[-1,1]} |P_n^{(\alpha_+,\alpha_-)}(x)|\le \frac{\Gamma((\alpha_+\vee \alpha_-)+n+1)}{n!\Gamma((\alpha_+\vee \alpha_-)+1)} ,\quad \mbox{if }\alpha_+\vee \alpha_->-\frac 12.\label{Jacobi-bound}\EDE $$\frac d{dx} P_n^{(\alpha_+,\alpha_-)}(x)=\frac {n+\alpha_++\alpha_-+1} 2 P_{n-1}^{(\alpha_++1,\alpha_-+1)}(x),\quad \mbox{if }n\ge 1;$$ $$P_n^{(\alpha_+,\alpha_-)}(1)=\frac{\Gamma(\alpha_++n+1)}{n!\Gamma(\alpha_++1)}.$$ When $n\ge 1$, using the last two displayed formulas, and applying (\ref{Jacobi-bound}) to $P_{n-1}^{(\alpha_++1,\alpha_-+1)}$, we find that for all $\alpha_+,\alpha_->-1$, \BGE \sup_{x\in[-1,1]} |P_n^{(\alpha_+,\alpha_-)}(x)|\le \frac{\Gamma(\alpha_++n+1)}{n!\Gamma(\alpha_++1)} + \frac{n(n+\alpha_++\alpha_-+1)\Gamma((\alpha_+\vee \alpha_-)+n+1)}{n!\Gamma((\alpha_+\vee \alpha_-)+2)}.\label{Jacobi-bound-general} \EDE This inequality trivially holds for $n=0$. \begin{Proposition} Let $\delta_+,\delta_-\in\R$. Let $B$ be a standard Brownian motion. Suppose $Z(t)$, $0\le t<T$, is a diffusion process that lies on an interval $I$ and satisfies the SDE \BGE dZ=\sqrt{1-Z^2}dB-\frac{\delta_+}4(Z+1)dt-\frac{\delta_-}4(Z-1)dt.\label{Bessel-SDE}\EDE We consider the following four cases: \begin{itemize} \item [(i)] $\delta_+>0,\delta_->0$, $I=[-1,1]$, and $T=\infty$; \item [(ii)] $\delta_+<2,\delta_-<2$, $I=(-1,1)$, and $\lim_{t\uparrow T} Z(t)\in \{-1,1\}$; \item [(iii)] $\delta_+>0$, $\delta_-<2$, $I= (-1,1]$, and $\lim_{t\uparrow T} Z(t)=-1$; \item [(iv)] $\delta_->0$, $\delta_+<2$, $I= [-1,1)$, and $\lim_{t\uparrow T} Z(t)=1$. \end{itemize} Then in each of the four cases, $Z$ has a transition density $p_t(x,y)$ in the sense that if $Z$ starts from $x\in I$, then for any $t>0$ and any measurable set $A\subset I$, $$\PP_x[ \{T>t\}\cap \{Z(t)\in A\}]=\int_A p_t(x,y)dy.$$ Moreover, $p_t(x,y)$ can be expressed by \BGE p_t(x,y)= f(x)g(y)\sum_{n=0}^\infty \frac{P_n^{(\alpha_+,\alpha_-)}(x) P_n ^{(\alpha_+,\alpha_-)}(y)}{{\cal A}^{(\alpha_+,\alpha_-)}_n} e^{-\beta_n t}.\label{Jacobi-series}\EDE where $P_n^{(\alpha_+,\alpha_-)}$ are Jacobi polynomials with indices $(\alpha_+,\alpha_-)$, ${\cal A}^{(\alpha_+,\alpha_-)}_n $ are normalization constants given by (\ref{Jacobi-norm}), and $\alpha_+,\alpha_-,f,g,\beta_n$, $n\ge 0$, are given by the following formulas depending on cases. \begin{itemize} \item In Case (i), $\alpha_+ =\frac{\delta_+}2-1$, $\alpha_- =\frac{\delta_-}2-1$, $f(x)= 1$, $g(y)=(1-y)^{\alpha_+}(1+y)^{\alpha_-}$, and $\beta_n=\frac 1 2 n(n+1+\alpha_++\alpha_-)$. \item In Case (ii), $\alpha_+=1-\frac{\delta_+}2$, $\alpha_-=1-\frac{\delta_-}2$, $f(x)=(1-x)^{\alpha_+}(1+x)^{\alpha_-}$, $g(y)= 1$, and $\beta_n=\frac 1 2 (n+1)(n+\alpha_++\alpha_-)$. \item In Case (iii), $\alpha_+=\frac{\delta_+}2-1$, $\alpha_-=1-\frac{\delta_-}2$, $f(x)=(1+x)^{\alpha_-} $, $g(y)=(1-y)^{\alpha_+}$, and $\beta_n=\frac 1 2 (n+1+\alpha_+)(n+\alpha_-)$. \item In Case (iv), $\alpha_-=\frac{\delta_-}2-1$, $\alpha_+=1-\frac{\delta_+}2$, $f(x)=(1-x)^{\alpha_+}$, $g(y)=(1+y)^{\alpha_-}$, and $\beta_n=\frac 1 2 (n+1+\alpha_-)(n+\alpha_+)$. \end{itemize} \label{Bessel-transition} \end{Proposition} \begin{Remark} Cases (i) and (ii) of the proposition in the special case $\delta_+=\delta_-$ were proven in \cite[Appendix B]{tip}. The general case was mentioned in the remark after \cite[Corollary 8.5]{tip}. We state the proposition here for future reference, but omit the proof since it is similar to the proofs of the special cases. \end{Remark} In each case of Proposition \ref{Bessel-transition}, we define a probability density $p_\infty$ by \BGE {\cal Z}=\int_{-1}^1 g(x)dx,\quad p_\infty(x)= \frac{{\bf 1}_{[-1,1]}(x) g(x)} {\cal Z}.\label{invariant-def}\EDE Note that ${\cal Z}\in(0,\infty)$. Using (\ref{P0},\ref{orthogonal},\ref{Jacobi-bound-general}), we get \BGE \int_{-1}^1 p_\infty(x) p_t(x,y) dy=p_\infty(y) e^{-\beta_0 t},\quad y\in[-1,1],\quad t>0.\label{invariant=}\EDE In Case (i), since $\beta_0=0$, $ p_\infty$ is the invariant density of $Z$. In Cases (ii-iv), since $\beta_0>0$, $ p_\infty$ is the quasi-invariant density of $Z$ with decay rate $\beta_0$. Because of (\ref{P0}), the first term ($n=0$) in the series (\ref{Jacobi-series}) is $ {\cal Z} f(x) p_\infty (y)e^{-\beta_0 t}/{\cal A}_0^{(\alpha_+,\alpha_-)}$. Since $\beta_n\ge \beta_1>\beta_0$ for $n\ge 1$, using (\ref{P0},\ref{Jacobi-norm},\ref{Jacobi-bound-general}) and Stirling's formula, we find that there are constants $C,L\in(0,\infty)$ depending only on $\alpha_+,\alpha_-$ such that \BGE |p_t(x,y)- {\cal Z} f(x) p_\infty(y)e^{-\beta_0 t}/{\cal A}_0^{(\alpha_+,\alpha_-)}|\le C f(x)g(y) e^{-\beta_1 t}, \quad\mbox{if } t>L.\label{transition-rate}\EDE This formula describes the convergence rate of the transition density to the invariant/quasi-invariant density. A simpler argument shows that there are constants $C,L\in(0,\infty)$ depending only on $\alpha_+,\alpha_-$, such that \BGE |p_t(x,y)|\le C f(x) g(y) e^{-\beta_0 t},\quad \mbox{if }t\ge L.\label{upper-bound-pt}\EDE In Cases (ii)-(iv), we have $\beta_0>0$. So \BGE \PP_x[T>t]=\int_{-1}^1 p_t(x,y)dy\le C {\cal Z} f(x) e^{-\beta_0 t}\le C {\cal Z} e^{-\beta_0 t},\quad \mbox{if }t\ge L.\label{T>t}\EDE This shows that the lifetime of $Z$ in Cases (ii)-(iv) has an exponential tail. If $X$ is a diffusion process that lies on $[0,1]$ and satisfies the SDE \BGE dX=\sqrt{X(1-X)}dB-\frac{\delta_+}4 Xdt-\frac{\delta_-}4(X-1)dt\label{Bessel-SDE-X}\EDE for some standard Brownian motion $B$, then $Z:=2X-1$ satisfies SDE (\ref{Bessel-SDE}), and we may use Proposition \ref{Bessel-transition} to get a transition density of $X$. \begin{Lemma} Let $B$ be a standard Brownian motion. Let $\delta_-,\delta_+\ge 2$. Suppose a continuous semimartingale $X(t)$, $0\le t<\infty$, lie on $(0,1)$, and satisfies (\ref{Bessel-SDE-X}), and another continuous semimartingale $\til X(t)$, $0\le t<T$, lie on $(0,1]$, and satisfies the SDE \BGE d\til X=\sqrt{\til X(1-\til X)} dB-Pdt-\frac{\delta_-}4 (\til X-1) dt.\label{Bessel-SDE-til-X}\EDE Assume that $P\le \frac{\delta_+}4\til X$, $X(0)\le \til X(0)$, and the $B$ in (\ref{Bessel-SDE-X}) and (\ref{Bessel-SDE-til-X}) are the same Brownian motion. Then a.s.\ $X(t)\le \til X(t)$ for all $t\in [0,T)$. \label{dominated} \end{Lemma} \begin{proof} Let $f(x)=\arccos(1-2x)$, which is strictly increasing and maps $[0,1]$ onto $[0,\pi]$. Let $\Theta=f(X)\in (0,\pi)$ and $\til \Theta=f(\til X)\in (0,\pi]$. It suffices to show that a.s.\ $\Theta(t)\le \til \Theta(t)$ for all $t\in[ 0,T)$. Since $f$ is $C^2$ on $(0,1)$, by It\^o's formula (cf.\ \cite{RY}) $\Theta$ satisfies the following SDE on $[0,\infty)$: $$d\Theta=dB+\frac{\delta_--1}4\cot_2(\Theta)dt-\frac{\delta_+-1}4 \tan_2(\Theta)dt,$$ where $\cot_2:=\cot(\cdot/2)$ and $\tan_2:=\tan(\cdot/2)$. Thus, $\Theta-B$ is $C^1$ with derivative $\frac{\delta_--1}4\cot_2(\Theta)-\frac{\delta_+-1}4 \tan_2(\Theta)$. The above argument does not apply to $\til\Theta$ since $f$ is not $C^2$ at $1$. Let $$\Psi={\bf 1}_{\til X\in (0,1)}\cdot \Big(\frac{P}{\sqrt{\til X(1-\til X)}}-\frac 14 \tan_2(\til\Theta)\Big).$$ Since $P\le \frac{\delta_+}4\til X$, we have $\Psi\le \frac{\delta_+-1 }4\tan_2(\til \Theta)$. For a (random) interval $I$, we say that $\til \Theta$ is {\it nice} on $I$ if $\til\Theta-B$ is $C^1$ on $I$ with derivative $\frac{\delta_--1}4\cot_2(\til\Theta)-\Psi$. Suppose $\tau$ is any finite stopping time. Let $T_1(\tau)=\inf(\{t\ge \tau: \til X(t)=1\}\cup \{\infty\})$. Then $\til X\in(0,1)$ on $[\tau,T_1(\tau))$. Since $f$ is $C^2$ on $(0,1)$, we may apply It\^o's formula to conclude that $\til\Theta$ satisfies the SDE $$d\til\Theta=dB+\frac{\delta_--1}4\cot_2(\til\Theta)dt -\Psi dt$$ on $[\tau,T_1(\tau))$. Thus, a.s.\ $\til\Theta$ is nice on $[\tau,T_1(\tau))$. Let $\sigma_1$ and $\sigma_2$ be two random times taking values in $[0,\infty]$, which may not be stopping times. Let $E$ be any event on which $\sigma_1<\sigma_2$ and $\til X$ stays in $(0,1)$ on $(\sigma_1,\sigma_2)$. We claim that, a.s.\ on the event $E$, $\til\Theta$ is nice on $(\sigma_1,\sigma_2)$. First, the claim holds if on the event $E$, $\sigma_1 $ and $\sigma_2$ are deterministic times $t_1<t_2$, and $\til X(t_1)\in (0,1)$. To see this, we apply the result of the last paragraph to $\sigma_1=t_1$ and use and the fact that $E\subset\{T_1(t_1)>t_2\}=\{[t_1,t_2]\subset [t_1,T_1(t_1))\}$. Second, the claim holds if on the event $E$, $\sigma_1$ and $\sigma_2$ take values in $\Q_+$, and $\til X(\sigma_1)\in (0,1)$. This is true because for any $p<q\in \Q_+$, a.s.\ on the event $ E\cap \{\sigma_1=p\}\cap \{\sigma_2=q\}$, $\til\Theta$ is nice on $(\sigma_1,\sigma_2)$. Finally, for the general case, we construct two sequences of $\Q_+$-valued random times $(\sigma_1^{(n)})$ and $(\sigma_2^{(n)})$ such that $\sigma_1^{(n)}\downarrow \sigma_1$ and $\sigma_2^{(n)}\uparrow \sigma_2$. Let $E_n=E\cap \{\sigma_1^{(n)}<\sigma_2^{(n)}\}$, $n\in\N$. Then $E_n\uparrow E$. We have known that, for each $n\in\N$, a.s.\ on $E_n$, $\til\Theta$ is nice on $(\sigma_1^{(n)},\sigma_2^{(n)})$. Since $ (\sigma_1^{(n)},\sigma_2^{(n)})\uparrow (\sigma_1,\sigma_2)$ and $E_n\uparrow E$, the claim is proved. Since $X(0)\le \til X(0)$, $\Theta(0)\le \til\Theta(0)$. Now we prove that a.s.\ $\Theta \le \til \Theta $ on $[0,T)$. Let $$E=\{\exists t\in [0,T):\Theta(t)>\til\Theta(t)\},\quad E_1=\{\exists t\in [0,T):\til\Theta(t)=0\},\quad E_2=E\sem E_1.$$ Define random times $\sigma_1$ and $\sigma_2$ as follows. On $E_1$, let $\sigma_2=\inf\{t\ge 0: \til\Theta(t)=0\}$; on $E_2$, let $$\sigma_2=\inf\{t\ge 0: \til \Theta(t)-\Theta(t)= \inf \{\til\Theta(s)-\Theta(s):0\le s<T\}/2\};$$ on $E=E_1\cup E_2$, let $\sigma_1=\max\{t\in [0,\sigma_2]: \til\Theta(t)\ge \Theta(t)\}$; and on $E^c$, let $\sigma_1=0$ and $\sigma_2=\infty$. Then on the event $E$, $\sigma_1<\sigma_2<\infty$, $\til\Theta(\sigma_1)=\Theta(\sigma_1)$, and $\til\Theta\in (0,\Theta)$ on $(\sigma_1,\sigma_2]$. By the claim, a.s.\ on the event $E$, $$(\til\Theta(\sigma_2)-B(\sigma_2))-(\til\Theta(\sigma_1)-B(\sigma_1))=\int_{\sigma_1}^{\sigma_2} \Big(\frac{\delta_--1}4\cot_2(\til\Theta(t))-\Psi(t)\Big)dt.$$ The fact that $\Theta-B$ is $C^1$ on $[0,\infty)$ with derivative being $\frac{\delta_--1}4\cot_2(\Theta)-\frac{\delta_+-1}4 \tan_2(\Theta)$ implies $$(\Theta(\sigma_2)-B(\sigma_2))-(\Theta(\sigma_1)-B(\sigma_1))=\int_{\sigma_1}^{\sigma_2} \Big(\frac{\delta_--1}4\cot_2(\Theta(t))-\frac{\delta_+-1}4 \tan_2(\Theta(t))\Big)dt.$$ Combining the above two displayed formulas and using that $\til\Theta(\sigma_1)=\Theta(\sigma_1)$, we get a.s.\ on $E$, $$ \til\Theta(\sigma_2)-\Theta(\sigma_2)=\int_{\sigma_1}^{\sigma_2} \Big(\frac{\delta_--1}4(\cot_2(\til\Theta(t))-\cot_2(\Theta(t)))-\Psi(t)+\frac{\delta_+-1}4 \tan_2(\Theta(t))\Big)dt>0.$$ Here we use the facts that $0<\til\Theta<\Theta<\pi$ on $(\sigma_1,\sigma_2)$, $\delta_->1$, and $\Psi\le \frac{\delta_+-1}4\tan_2(\til \Theta)$. However, since $\til\Theta(\sigma_2)<\Theta(\sigma_2)$, $E$ must be a null event. So a.s.\ $\Theta \le \til \Theta $ on $[0,T)$. \end{proof} \section{One-point Estimate} \label{Section 3} Let $\kappa>0$, $m\in\N$, $\ulin\rho=( \rho_1,\dots,\rho_m) \in\R^n $, $w\in\R$, and $\ulin v=(v_1,\dots,v_m)\in\R_w^m$ (Definition \ref{Def-Rw}). Assume that $v_m\le v_{m-1}\le \cdots \le v_1\le w^-$. Let $\eta$ be an SLE$_\kappa(\ulin\rho)$ curve in $\HH$ started from $w$ with force points $\ulin v$. Let $\F$ be the complete filtration generated by $\eta$. Let $W$ be the driving process for $\eta$, and let $V_j$ be the force point process started from $v_j$, $1\le j\le m$. We write $\ulin V$ for $(V_1,\dots,V_m)$. Then $W\ge V_1\ge\cdots\ge V_m$, and they satisfy the $\F$-adapted SDE/ODE (\ref{dW},\ref{dVU}). Let $g_t$ and $K_t$ be the chordal Loewner maps and hulls generated by $\eta$. For $u\in\R\sem \{w\}$, let \BGE U(t)=g_t(u),\quad D_u(t)=g_t'(u),\quad 0\le t<\tau_u^*.\label{U-Du}\EDE Then $U$ and $D_u$ satisfy the following ODE on $[0,\tau_u^*)$: \BGE dU=\frac{2 dt}{U-W},\quad \frac{d D_u}{D_u}=\frac{-2dt}{(U-W)^2}.\label{dUD}\EDE The purpose of this section is to prove the following theorem. \begin{Theorem} Suppose that $\ulin\rho$ satisfies that $\rho_\Sigma:=\sum_{k=1}^m \rho_k >\frac\kappa 2-4$ and $\sigma_j:=2+\sum_{k=1}^j \rho_k\in(0, \sigma^*]$, $0\le j\le m-1$, for some $\sigma^*\ge 2$. Then there is a constant $C\in(0,\infty)$ depending only on $\kappa,\rho_\Sigma,\sigma^*$ such that the following holds. Suppose $v_m<w^-$. Let $u=v_m$ and $v_0=w^-$. Let $r\in (0,|w-u|)$ and $j_0=\max \{0\le j\le m: v_j\ge u+r\} $. Let $\gamma=\frac{2\rho_\Sigma+8-\kappa}{2\kappa}$.Then \BGE \PP[\dist(\eta,u)\le r]\le C \Big(\frac{r}{|v_{j_0}-u|}\Big)^{ {\gamma\sigma_{j_0}}} \prod_{j=0}^{j_0-1}\Big(\frac{|v_{j+1}-u|}{|v_j-u|}\Big) ^{{\gamma\sigma_{j}}}.\label{est1}\EDE \label{Thm-est1} \end{Theorem} \begin{proof} Throughout the proof, a constant is a number in $(0,\infty)$ depending only on $\kappa,\rho_\Sigma,\sigma^*$, whose value may vary. We write $X\lesssim Y$ if there is a constant $C$ such that $X\le CY$, and write $X\asymp Y$ if $X\lesssim Y$ and $Y\lesssim X$. By adding force points with force value $0$, we may assume that there is $j_0\in \{1,\dots,m-1\}$ such that $v_{j_0}-u=r$, and that $v_0=w^-$ is a force point. Let $V_0(t)$ be the force point process started from $v_0$, which also satisfies (\ref{dVU}). By merging force points, we assume that $v_{m-1}>v_m$. Let $U$ and $D_u$ be defined by (\ref{U-Du}). Then $W\ge V_0\ge \cdots \ge V_{m-1}>V_m=U$ on $[0,\tau_u^*)$. Since the part of $\eta$ after $\tau_u^*$ is disconnected from $u$ by $\eta[0,\tau_u^*)$, $\dist(\eta,u)=\dist(\eta[0,\tau_u^*),u)$. So (\ref{est1}) becomes \BGE \PP[\dist(\eta[0,\tau_u^*),u)\le r]\lesssim \prod_{j=0}^{j_0-1}\Big(\frac{|v_{j+1}-u|}{|v_j-u|}\Big) ^{\gamma \sigma_j}.\label{est1''}\EDE For $0\le t<\tau_u^*$, $u$ lies in $\C\sem (K_t^{\doub}\cup [w,\infty))$; $g_t$ maps $\C\sem (K_t^{\doub}\cup [w,\infty))$ conformally onto $\C\sem [V_0(t),\infty)$, and sends $u$ to $U(t)$. Define $ H_j$ and $Q_j$ on $[0,\tau_u^*)$ by \BGE H_j=\frac{V_j-U}{V_0-U},\quad Q_j=\frac{W-V_0}{W-V_j}.\label{DHQ}\EDE Here if $\tau^*_{v_j}$ happens before $\tau^*_u$, then we define $Q_j$ to be constant $1$ on $[\tau^*_{v_j},\tau^*_u)$. Then $0=H_m<H_{m-1}\le \cdots \le H_0=1$ and $0\le Q_m\le\cdots\le Q_0=1$. By (\ref{dVU}), $H_j$ satisfies the ODE \BGE \frac{d H_j}{H_j}=\frac{2(V_0-V_j)dt}{(W-U)(W-V_0)(W-V_j)}.\quad 0\le j\le m-1.\label{dH}\EDE Define $R=1-Q_m=\frac{V_0 -U }{W -U }\in(0,1]$ on $[0,\tau_u^*)$. Then $R$ starts from $1$ (because $V_0(0)=W(0)=w$), and by (\ref{dW},\ref{dVU}) satisfies the SDE: \BGE dR=R\Big[-\frac{\sqrt\kappa dB}{W-U}+\frac{(\kappa-2-\rho_m)dt}{(W-U)^2} -\frac{2dt}{(W-U)(W-V_{0})} -\sum_{j=0}^{m-1} \frac{\rho_j dt }{(W-U)(W-V_j)} \Big] .\label{dR}\EDE Define functions $\Up_j =\frac{|V_j -U |}{D_u }$ on $[0,\tau_u^*)$. By Koebe $1/4$ theorem, \BGE \Up_j(t)\asymp |v_j-u|\wedge \dist(\eta[0,t],u).\label{Upj}\EDE Define $\Phi =-\frac \kappa2 \ln\Big(\frac{\Upsilon_0 }{|w-u|}\Big)$ on $[0,\tau_u^*)$. Then $\Phi$ stars from $0$, and by (\ref{dVU},\ref{dUD}) satisfies the ODE \BGE d\Phi(t)=\frac{\kappa Rdt}{(W-U)(W-V_{0})}>0\quad\mbox{on }[0,\tau_u^*).\label{dPhi}\EDE So $\Phi$ is strictly increasing on $[0,\tau_u^*)$. Since $v_0=w=\eta(0)$, by (\ref{Upj}) we have \BGE \dist(u,\eta[0,t]) \asymp \Up_0(t)=|w-u| e^{-\frac 2\kappa \Phi(t)}, \quad 0\le t<\tau_u^*\label{dist(u,eta)}\EDE Let $\ha \tau_u^*=\sup \Phi[0,\tau_u^*)$. Define $\ha W,\ha U,\ha R,\ha D_u, \ha V_j,\ha Q_j,\ha H_j$ on $[0,\ha \tau_u^*)$, to be respectively the time-changes of $W,U, R, D_u, V_j,Q_j,H_j$ via $\Phi$. For example, $\ha R(t)=R(\Phi^{-1}(t))$, $0\le t<\ha \tau_u^*$. By (\ref{dH},\ref{dPhi}), $\ha H_j$ satisfies the following ODE on $[0,\ha \tau_u^*)$: \BGE \frac{d\ha H_j}{\ha H_j}=\frac 2\kappa \cdot \frac 1{\ha R}\cdot \frac{\ha V_0-\ha V_j}{\ha W-\ha V_j}dt=\frac 2\kappa\cdot \frac{1-\ha Q_j}{\ha R}dt,\quad 0\le j\le m-1. \label{dhaH}\EDE By (\ref{Upj},\ref{dist(u,eta)}), we have \BGE \ha H_j(t) =\frac{\Up_j(\Phi^{-1}(t))}{\Up_0(\Phi^{-1}(t))}\asymp \frac{|v_j-u|\wedge (|w-u|e^{-\frac 2\kappa t})}{|w-u| e^{-\frac 2\kappa t}} \label{Koebe-H-hat}\EDE From (\ref{dR},\ref{dPhi}), for some standard Brownian motion $\ha B$, $\ha R$ satisfies the following SDE: \BGE d\ha R=\sqrt{\ha R(1-\ha R)}d\ha B+\frac{\kappa-2-\rho_m}{\kappa}(1-\ha R)dt -\frac 2\kappa dt-\sum_{j=0}^{m-1} \frac{\rho_j}\kappa \ha Q_j dt. \label{dhaR}\EDE By It\^o's formula, for any $c\in\R$, $$\frac{d\ha R^c}{\ha R^c}=c \sqrt{\frac{1-\ha R}{\ha R}}d\ha B+\Big(c\frac{\kappa -2-\rho_m}\kappa+\frac {c(c-1)}2 \Big)\frac{1-\ha R}{\ha R} dt-\frac{2c}{\kappa \ha R} dt - \frac c{\ha R}\sum_{j=0}^{m-1} \frac{\rho_j}\kappa \ha Q_j dt$$ $$=c \sqrt{\frac{1-\ha R}{\ha R}}d\ha B+\Big(c\frac{\kappa -4-\rho_\Sigma}\kappa+\frac {c(c-1)}2 \Big)\frac{dt}{\ha R} -\Big(c\frac{\kappa -2-\rho_m}\kappa+\frac {c(c-1)}2 \Big)dt + c\sum_{j=0}^{m-1} \frac{\rho_j}\kappa \frac{1- \ha Q_j}{\ha R}dt.$$ Let $\ha E(t)= e^{-\frac 2 \kappa t}$. By taking $c=2\gamma$ in the above SDE and using (\ref{dhaH}), we find that $$\ha M:=\ha E^{-\gamma\sigma_{m-1}} \ha R^{2\gamma}\prod_{j=0}^{m-1} \ha H_j^{-\gamma\rho_j}$$ is a positive local martingale on $[0,\ha\tau^*_u)$ and satisfies the SDE \BGE \frac{d \ha M}{\ha M =\frac{2\rho_\Sigma+8-\kappa}{\kappa} \frac{1-\ha R}{\sqrt{\ha R(1-\ha R)}} d\ha B.\label{dhaM}\EDE Let $\ha H_{j,j+1}=\frac{\ha H_{j+1}}{\ha H_j}=\frac{\ha V_{j+1}-\ha U}{\ha V_j-\ha U}\in(0,1]$, $0\le j\le m-2$. By (\ref{dhaH}), each $\ha H_{j,j+1}$ satisfies $\frac{d\ha H_{j,j+1}}{\ha H_{j,j+1}}=\frac 2\kappa \frac{\ha Q_j-\ha Q_{j+1}}{\ha R}\ge 0$, and so is increasing. Since $\sigma_j-\sigma_{j-1}=\rho_j$ and $\ha H_0=1$, we have \BGE \ha M=\ha E^{-\gamma\sigma_{m-1}} \ha R^{2\gamma} \prod_{j=0}^{m-2}\ha H_{j,j+1}^{\gamma\sigma_j-\gamma\sigma_{m-1}}. \label{haM-re}\EDE We will apply Girsanov theorem. Let $\PP$ denote the current measure. Fix $n\in\N$. Let $\tau_n$ be the maximum element in $[0,\ha \tau_u^*]$ such that $|\ln(\frac{\ha M(t)}{\ha M(0)})|\le n$ on $[0,\tau_n)$. Then $\ha M(\cdot\wedge \tau_n)$ is a uniformly bounded martingale on $[0,\infty)$. Here when $\tau_n=\ha \tau_u^*$ and $t\ge \tau_n$, $\ha M(t\wedge \tau_n)$ is understood as $\lim_{t\uparrow \ha \tau_u^*} \ha M(t)$, which a.s. converges on the event $\{\tau_n=\ha \tau_u^*\}$. So we get $\EE[\frac{\ha M(\tau_n)}{\ha M(0)}]=1$. Now we define a new probability measure $\til\PP_n$ by $\frac{d\til\PP_n}{d\PP}=\frac{\ha M(\tau_n)}{\ha M(0)}$. By Girsanov theorem and (\ref{dhaR},\ref{dhaM}), under the new measure $\til\PP_n$, $\ha X$ satisfies the following SDE on $[0,\tau_n)$: $$ d\ha R=\sqrt{\ha R(1-\ha R)}d\til B -\frac 2\kappa dt-\sum_{j=0}^{m-1} \frac{\rho_j}\kappa \ha Q_j dt-\frac{2\rho_\Sigma +6-\rho_m}{\kappa}(\ha R-1)dt, $$ where $\til B$ is a standard Brownian motion under $\til\PP_n$. Since $\sigma_j-\sigma_{j-1}=\rho_j$, $\sigma_0=2$, $\ha Q_0=1$, and $\ha Q_{m-1}=1-\ha R$, we may rewrite the above SDE as \BGE d\ha R=\sqrt{\ha R(1-\ha R)}d\til B -\sum_{j=0}^{m-2} \frac{\sigma_j}\kappa (\ha Q_j-\ha Q_{j+1}) dt-\frac{\rho_\Sigma +4}{\kappa}(\ha R-1)dt. \label{dtilR}\EDE Since $\ha Q_j\ge \ha Q_{j+1}$ and $\sigma_j\le \sigma^*$, we get $\sum_{j=0}^{m-2} \frac{\sigma_j}\kappa (\ha Q_j-\ha Q_{j+1}) \le \frac{\sigma^*}\kappa (1-\ha Q_{m-1})\le \Big(\frac{\sigma^*}\kappa\vee \frac 12\Big) \ha R$. Thus, under $\til\PP$, $\ha R$ up to $\tau_n$ satisfies the property of $\til X$ in Lemma \ref{dominated} with $\delta_-:=\frac 4\kappa(\rho_\Sigma +4)>2$ and $\delta_+:=\frac 4\kappa\sigma^* \vee 2\ge 2$. By Lemma \ref{dominated}, under $\til\PP_N$, up to $\tau_n$, $\ha R$ stochastically dominates the process $X$, which starts from $1$, and satisfies the SDE (\ref{Bessel-SDE-X}). By Proposition \ref{Bessel-transition} (i), $X$ has a transition density $p_t(x,y)$, which by (\ref{upper-bound-pt}) satisfies \BGE p_t(x,y)\lesssim y^{\frac{\delta_-}2-1}=y^{2\gamma}.\label{ptupper}\EDE Let $L>0$ be such that $|w-u|e^{-\frac 2\kappa L}=\frac r4$. By (\ref{dist(u,eta)}), we get (\ref{est1''}) if \BGE \PP[\ha \tau_u^*>L]\lesssim \prod_{j=0}^{j_0-1} \Big(\frac{|v_{j+1}-u|}{|v_j-u|}\Big)^{\gamma \sigma_j}. \label{est1'''}\EDE Since $\frac{d\til\PP_n}{d\PP}=\frac{\ha M(\tau_n)}{\ha M(0)}$ and $\ha M(\cdot\wedge \tau_n)$ is a uniformly bounded martingale under $\PP$, we get $$ \PP[L<\tau_n\wedge \ha \tau_u^*]=\til\EE_n [{\bf 1}_{\{L<\tau_n\wedge \ha \tau_u^*\}} {\ha M(0)}/{\ha M(L\wedge \tau_n\wedge L)} ]=\til\EE_n [{\bf 1}_{\{L<\tau_n\wedge \ha \tau_u^*\}} {\ha M(0)}/{\ha M(L )} ].$$ Define $\ha N(t)=\prod_{j=0}^{m-2}\Big( \frac{\ha H_{j,j+1}(t)}{\ha H_{j,j+1}(0)} \Big)^{ \gamma\sigma_{m-1}-\gamma\sigma_j} $. Since $\ha E(L)=e^{-\frac 2\kappa L}\asymp \frac r{|w-u|}$, by (\ref{haM-re}), $$ \frac{\ha M(0)}{\ha M(L )}=\ha E(L)^{\gamma\sigma_{m-1}} \ha R(L)^{-2 \gamma} \ha N(L)\asymp \Big(\frac r {|w-u|}\Big) ^{\gamma\sigma_{m-1}} \ha R(L)^{-2 \gamma} \ha N(L).$$ Since $0<\ha H_{j,j+1}(0)\le \ha H_{j,j+1}(L)\le 1$ and $\sigma^*\ge \sigma_j\ge 0$, by (\ref{Koebe-H-hat}) and the fact that $|v_{j_0}-u|=r\asymp |w-u|e^{-\frac 2\kappa L}$, we have $$\ha N(L)\le \prod_{j=0}^{j_0-1}\Big( \frac{1}{\ha H_{j,j+1}(0)}\Big)^{ \gamma\sigma_{m-1}} \cdot \prod_{j=j_0}^{m-2}\Big( \frac{\ha H_{j,j+1}(L)}{\ha H_{j,j+1}(0)}\Big)^{ \gamma\sigma_{m-1}} \cdot \prod_{j=0}^{j_0-1}\Big( \frac{\ha H_{j,j+1}(L)}{\ha H_{j,j+1}(0)}\Big)^{ -\gamma\sigma_j}$$ $$\le \ha H_{m-1}(0)^{-\gamma\sigma_{m-1}}\Big(\frac{\ha H_{m-1}(L)}{\ha H_{j_0}(L)}\Big)^{\gamma\sigma_{m-1}} \ha H_{j_0}(L)^{-\gamma\sigma_*}\prod_{j=0}^{j_0-1} \ha H_{j,j+1}(0)^{\gamma\sigma_j}$$ $$\asymp \Big(\frac{r}{|w-u|}\Big)^{-\gamma\sigma_{m-1}} \prod_{j=0}^{j_0-1} \Big(\frac{|v_{j+1}-u|}{|v_j-u|}\Big)^{\gamma\sigma_j}.$$ Here we used that $\ha H_{m-1}(L)/\ha H_{m-1}(0)\asymp |w-u|/r$ and $\ha H_{j_0}(L)\asymp 1$. Since $\ha R$ under $\til \PP$ up to $\tau_n$ stochastically dominates the process $X$ with transition density $p_t(x,y)$, and $-2\gamma<0$, by (\ref{ptupper}) we get $$\til \EE_n\Big[{\bf 1}_{\{L<\tau_n\wedge \ha \tau_u^*\}}\ha R(L)^{-2\gamma}\Big] \le \int_0^1 y^{-2\gamma} p_L(1,y) dy\lesssim 1.$$ By the last four displayed formulas we get $ \PP[L<\tau_n\wedge \ha \tau_u^*]\lesssim \prod_{j=0}^{j_0-1} \Big(\frac{|v_{j+1}-u|}{|v_j-u|}\Big)^{\gamma \sigma_j}$, which implies the desired estimate (\ref{est1'''}) as $n$ tends to $\infty$. \end{proof} \begin{comment} \begin{Remark} The assumption in Theorem \ref{Thm-est1} is quite mild. The assumption that $\sigma_j>0$, $0\le j\le m-1$, guarantee that $\eta$ does note encounter continuation threshold before it disconnects $u$ from $\infty$. The assumption that $\sum_{k=0}^m \rho_k>\frac\kappa 2-4$ implies that the probability that $\eta$ hits $u$ is $0$. As long as all force points of $\eta$ lie between $w^-$ and $u$, we may always assume that $w^-$ and $u$ are force points by adding additional force points with eigenvalue $0$, if necessary. Suppose now $u<v_m$ and $\sigma_m>0$. Let $\rho_\Sigma=\sum_{k=0}^m \rho_k$, $\sigma_*=\min\{\sigma_j:0\le j\le m\}$, \BGE \gamma=\frac{2\rho_\Sigma+8-\kappa}{2\kappa},\quad \alpha=(\rho_\Sigma+2)\gamma,\quad \alpha_*=\sigma_*\gamma,\quad \alpha^*=\sigma^*_\gamma.\label{alpha}\EDE Since $\rho_\Sigma>\frac \kappa 2-4$, $\gamma>0$. From $\sigma^*\ge \sigma_m=\rho_\Sigma+2\ge \sigma_*>0$ we get $\alpha^*\ge \alpha\ge \alpha_*>0$. By (\ref{est1'}), for $r\in (0,|v_m-u|)$, \BGE \PP[\dist(\eta,u)\le r]\le C \Big(\frac{|v_{m}-u|}{|w-u|}\Big)^{\alpha_*} \Big(\frac{r}{|v_{m}-u|}\Big)^{\alpha} \le C \Big(\frac{r}{|v_{m}-u|}\Big)^{\alpha},\label{est1'-u}\EDE where $C\in(0,\infty)$ depends only on $\kappa$, $\rho_\Sigma$, and $\sigma^*:=\max\{\sigma_j:0\le j\le m\}$. \end{Remark} \end{comment} \section{One-point Green's Function} \label{Section 4} Let $\kappa,\rho_\Sigma,\sigma^*,\sigma_*\in\R$ satisfy that $\kappa>0$, $\rho_\Sigma>(-2)\vee(\frac\kappa 2-4)$ and $\sigma^*\ge 2\ge \sigma_*>0$. In this section, a constant depends only on $\kappa,\rho_\Sigma,\sigma^*,\sigma_*$. We will use the following positive constants: \BGE \gamma=\frac{2\rho_\Sigma+8-\kappa}{2\kappa},\quad \alpha=(\rho_\Sigma+2)\gamma,\quad \alpha_*=\sigma_*\gamma,\quad \alpha^*=\sigma^*\gamma.\label{alpha}\EDE We write $X\lesssim Y$ or $Y\gtrsim X$ if there is a constant $C>0$ such that $X\le C Y$; and write $X\asymp Y$ if $X\lesssim Y$ and $Y\lesssim X$. We write $X=O(Y)$ as $R\to 0^+$ if there are constants $C,c>0$ such that when $0<R<c$, we have $X\le C Y$. Let $m\in\N$ and $\ulin\rho=(\rho_1,\dots,\rho_m)\in\R^m$ satisfy that $\sum_{j=1}^m \rho_j=\rho_\Sigma$ and $\sigma_j:=2+\sum_{k=1}^j\rho_k\in [\sigma_*,\sigma^*]$, $0\le j\le m-1$. We also define $\sigma_m=2+\rho_\Sigma=2+\sum_{j=1}^m\rho_j$, but do not require $\sigma_m\in[\sigma_*,\sigma^*]$. Define \BGE \alpha_0=2\gamma,\quad \alpha_j=\rho_j \gamma,\quad 1\le j\le m.\label{alphaj}\EDE Note that $\alpha=\sum_{j=0}^m \alpha_j$. Let $w\in\R$ and $v_m\le \cdots\le v_1\in(-\infty,w)\cup\{ w^-\}\subset\R_w$. We write $\ulin v$ for $(v_1,\dots,v_m)$. Let $\eta$ be an SLE$_\kappa(\ulin\rho)$ curve in $\HH$ started from $w$ with force points $\ulin v$. The assumption that $\sum_{k=1}^j\rho_k=\sigma_j-2\ge \sigma_*-2>-2$, $1\le j\le m-1$, and $\sum_{k=1}^m \rho_k=\rho_\Sigma>-2$ implies that the continuation threshold is never met, and the lifetime of $\eta$ is $\infty$. Define $G:\{(w,\ulin v,u)\in\R^{m+2}:w\ge v_1\ge \cdots\ge v_m>u\}\to (0,\infty)$ by \BGE G(w,\ulin v;u)= |w-u|^{-\alpha_0} \prod_{j=1}^m |v_j-u|^{-\alpha_j}\label{G-univ}\EDE \BGE = \Big(\frac{|v_1-u|}{|w-u|}\Big)^{\sigma_0\gamma}\cdot \Big(\frac{|v_2-u|}{|v_1-u|}\Big)^{\sigma_1\gamma} \cdots \Big(\frac{|v_m-u|}{|v_{m-1}-u|}\Big)^{\sigma_{m-1}\gamma}\cdot\Big(\frac 1{|v_m-u|}\Big)^{\sigma_m\gamma} .\label{G-rewrite}\EDE For $u\in\R$, $r>0$, and $\FF\in\{\C,\R\}$, let $B_{\FF}(u,r)=\{z\in\FF:|z-u|\le r\}$, and $\tau^u_{r;\FF}=\inf(\{t: \eta(t)\in B_{\FF}(u,r)\}\cup \{\infty\})$. Then $\eta\cap B_{\FF}(u,r)\ne\emptyset$ iff $\tau^u_{r;\FF}<\infty$. We also define $\tau^u_r=\tau^u_{r;\C}$. The purpose of this section is to prove the following theorem. \begin{Theorem} [One-point Green's Function] Suppose either $\rho_\Sigma\in ((-2)\vee (\frac\kappa 2-4),\frac\kappa 2-2)$ and $\FF\in \{\R,\C\}$, or $\rho_\Sigma\ge \frac\kappa 2-2$ and $\FF=\C$. There are a constant $C_{\FF}>0$ depending only on $\FF,\kappa,\rho_\Sigma$, and a constant $\beta\in(0,1)$ depending only on $\kappa,\rho_\Sigma,\sigma_*$ such that, for any $u\in(-\infty,v_m)$, as $ r/{|v_m-u|}\to 0^+$, \BGE \PP[\eta\cap B_{\FF}(u,r)\ne \emptyset]=C_{\FF} G(w,\ulin v;u) r^{\alpha} + O\Big(\Big(\frac{|v_m-u|}{|w-u|}\Big)^{\alpha_*} \Big(\frac r{|v_m-u|}\Big)^{\alpha+\beta}\Big).\label{Green-1pt}\EDE Recall that the implicit constants in the $O$ symbol depend only on $\kappa,\rho_\Sigma,\sigma^*,\sigma_*$. \label{main-thm1} \end{Theorem} \begin{Remark} The theorem does not include the case that $\FF=\R$ and $\rho_\Sigma\ge \frac\kappa 2-2$ because in that case $\eta$ a.s.\ does not hit $(-\infty,v_m)$. We do not have exact formulas of $C_{\FF}$ in general (unless $\FF=\R$ and $\rho=0$), as happened in \cite{Mink-real,LR}. We do have an exact formula of $\beta$: $\frac 1\beta=\frac{\rho_\Sigma+4}{\rho_\Sigma+3}+\frac{\alpha_*+1}{\alpha_*}$, but it is very unlikely to be sharp. \end{Remark} \begin{comment} The proof of the following lemma is straightforward. \begin{Lemma} Define $G:\{(w,\ulin v,u)\in\R^{m+2}:w\ge v_1\ge \cdots\ge v_m>u\}\to (0,\infty)$ by \BGE G(w,\ulin v;u)= |w-u|^{-\alpha_0} \prod_{j=1}^m |v_j-u|^{-\alpha_j}.\label{G-univ}\EDE Define $M$ on $[0,\tau_u^*)$ by \BGE M=D_u^\alpha G(W,\ulin V;U)=D_u^{\alpha} |W-U|^{-\alpha_0} \prod_{j=1}^m |V_j-U|^{-\alpha_j} .\label{M-first}\EDE Then $M$ is a positive local martingale. \label{M1-lemma} \end{Lemma} \end{comment} From now on, we fix $u\in(-\infty, v)$. Let $L_0=|w-v_m|\ge 0$ and $L_1=|v_m-u|>0$. Using that $\sigma^*\ge \sigma_k\ge \sigma_*>0$, $|w-u|\ge |v_1-u|\ge \cdots \ge |v_m-u|>0$, and (\ref{alpha},\ref{G-rewrite}), we get \BGE \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha^*}\Big(\frac {r}{L_1}\Big)^\alpha\le G(w,\ulin v;u) r^\alpha\le\Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*}\Big(\frac {r}{L_1}\Big)^\alpha\le \Big(\frac {r}{L_1}\Big)^\alpha. \label{G-upper}\EDE By treating $u$ as a force point of $\eta$ with force value $0$, we may apply Theorem \ref{Thm-est1} to conclude that \BGE \PP[\dist(\eta,u)\le r]\lesssim ( {L_1}/{(L_0+L_1)} )^{\alpha_*} ( {r}/{L_1} )^{\alpha},\quad\mbox{for any } r\in(0,L_1),\label{est-u}\EDE which implies that \BGE \PP[\dist(\eta,u)\le r]\lesssim ( {r}/{L_1})^{\alpha}\quad\mbox{for any } r>0.\label{est-u*}\EDE When $v_m<w^-$, i.e., $L_0>0$, by Theorem \ref{Thm-est1}, \BGE \PP[\dist(\eta,v_m)\le r]\lesssim ( {r}/{L_0} )^{\alpha_*},\quad \mbox{for any }r>0.\label{est-vm}\EDE Let $K_t,g_t,\F,W,\ulin V=(V_1,\dots,V_m)$ be as in the previous section. Let $U$ and $D_u$ be defined by (\ref{U-Du}). We have the following estimate. \begin{Lemma Let $R>r\in (0,|v_m-u|)$. Let $\tau$ be an $\F$-stopping time. Then \BGE\PP[\tau^u_r<\infty |\F_\tau,\{\tau\le \tau^u_R\}\}]\lesssim ( r/R)^{\alpha}.\label{interior-estimate-formula}\EDE \label{interior-estimate} \end{Lemma} \begin{proof} We may assume that $r<R/2$ since the conditional probability is bounded by $1$. We only need to prove (\ref{interior-estimate-formula}) with $E:=\{ \tau\le \tau^u_R\}$ replaced by $E_0:=\{ \tau<\tau^*_u\wedge \tau^u_R\}$ since the conditional probability on $E\sem E_0$ is $0$. Suppose $E_0$ occurs. By DMP of SLE$_\kappa(\ulin\rho)$, the conditional probability equals the probability that an SLE$_\kappa(\ulin\rho)$ curve $\eta^\tau$ in $\HH$ started from $W(\tau)$ with force points $\ulin V(\tau)$ visits $g_\tau(B_{\C}(u,r))$. Recall that $g_\tau$ maps $\C\sem (K_\tau^{\doub}\cup [v_m,\infty))$, which contains $u$, conformally onto $\C\sem [V_m(\tau),\infty)$. By Koebe $1/4$ theorem, $|V_m(\tau)-U(\tau)|\ge D_u(\tau) R/4$. Since $r<R/2$, by Koebe distortion theorem, $\rad_{U(\tau)} (g_\tau(B_{\C}(u,r)))\le 4 D_u(\tau)r$. So $\rad_{U(\tau)} (g_\tau(B_{\C}(u,r)))/|V_m(\tau)-U(\tau)|\le 16 r/R$. By (\ref{est-u*}), the conditional probability that $\eta^\tau$ visits $g_\tau(B_{\C}(u,r))$ is bounded by a positive constant times $(r/R)^{\alpha}$. So we have (\ref{interior-estimate-formula}). \end{proof} For $t\ge 0$, we write $\pa_{\HH}^+ K_t$ for the union of the ``right side'' of $\pa K_t$ and the real interval $[b_{K_t},\infty)$. More precisely, $\pa_{\HH}^+ K_t$ is the $g_{t}^{-1}$ image of the interval $[W(t),\infty)$ as a set of prime ends of $\HH\sem K_t$. We have the following estimate. \begin{Lemma Let $\tau$ be an $\F$-stopping time. On the event $\tau<\infty$, let $I$ be an $\F_\tau$-measurable crosscut of ${\HH\sem K_\tau}$, which disconnects $g_\tau(V_m(\tau))=v_m\wedge a_{K_\tau}$, viewed as a prime end of ${\HH\sem K_\tau}$, from $\pa^+_{\HH} K_\tau)$ in ${\HH\sem K_\tau}$. Then \BGE \PP[\eta \mbox{ visits }I|\F_\tau,\{\tau<\infty\}]\lesssim \exp (-\pi \alpha_* d_{\HH\sem K_\tau}(I,\pa^+_{\HH} K_\tau) ).\label{boundary-estimate-formula}\EDE \label{boundary-estimate} \end{Lemma} \begin{proof} By DMP the conditional probability equals the probability that an SLE$_\kappa(\ulin\rho)$ curve $\eta^\tau$ in $\HH$ started from $W(\tau)$ with force points $\ulin V(\tau)$ visits $g_\tau(I)$. By the assumption of $I$, $g_\tau(I)$ is a crosscut of $\HH$ that disconnects $V_m(\tau)$ from $[W(\tau),\infty)$. By conformal invariance of extremal length, $d_{\HH}(g_\tau(I), [W(\tau),\infty))=d_{\HH\sem K_\tau}(S,\pa^+_{\HH} K_\tau)$. By Proposition \ref{extrem-prop}, $1\wedge \frac{\rad_{V_m(\tau)} (g_\tau(I))}{|W(\tau)-V_m(\tau)|}\lesssim \exp({-\pi d_{\HH}(g_\tau(I), [W(\tau),\infty))|})$. So we get the conclusion using (\ref{est-vm}). \end{proof} The following lemma is the first two-point estimate in this paper. \begin{Lemma} Assume $v_m<w$. Then for any $r_0,r_1>0$, \BGE \PP[\dist(v_m,\eta)\le r_0,\dist(u,\eta)\le r_1]\lesssim ( {r_0}/{L_0} )^{\alpha_*} ( {r_1}/{L_1} )^{\alpha}.\label{two-pt-ineq}\EDE \label{two-pt-lem} \end{Lemma} \begin{proof} If $r_1\ge L_1/6$, then we get (\ref{two-pt-ineq}) by applying (\ref{est-vm}) to bound $\PP[\dist(v_m,\eta)\le r_0]$. If $r_0\ge (L_0\wedge L_1)/6$ and $r_1<L_1/6$, then we get (\ref{two-pt-ineq}) by applying (\ref{est-u}) to bound $\PP[\dist(u,\eta)\le r_1]$ and using the fact that $\frac{L_1}{L_0+L_1}\asymp \frac{L_0\wedge L_1}{L_0}$. So we may now assume that $r_0<(L_0\wedge L_1)/6$ and $r_1< L_1/6$. WLOG, we may assume that $r_1=e^{-N}L_1/2$ for some $N\in\N$. Let $\tau^0=\tau^{v_m}_{r_0}$ and $\tau^1_n=\tau^u_{L_1 e^{-n}/2}$ , $n\in\Z$. Then (\ref{two-pt-ineq}) is equivalent to \BGE \PP[\tau^0<\infty,\tau^1_{N}<\infty]\lesssim ( {r_0}/{L_0} )^{\alpha_*} e^{-\alpha N} .\label{transform*}\EDE By (\ref{est-vm},\ref{est-u}), \BGE \PP[\tau^0<\infty]\lesssim (r_0/L_0)^{\alpha_*};\label{tau1<infty}\EDE \BGE \PP[\tau^1_n<\infty]\lesssim ({L_1}/({L_0}+L_1))^{\alpha_*} e^{-\alpha n},\quad n\in\N.\label{tau2n}\EDE By (\ref{interior-estimate-formula}), \BGE \PP[\tau^1_N<\infty|\F_{\tau^1_n},\tau^1_{n}<\infty]\lesssim e^{-\alpha(N-n)},\quad n\le N.\label{tau2n-N}\EDE Note that, when $\tau^1_n<\tau^0$, every curve in $\HH\sem K_{\tau^1_n}$ connecting $I:=\{z\in\lin\HH:|z-v_m|= r_0\}$ with $\pa_{\HH}^+ K_{\tau^1_n}$ must cross $A_{\HH}(u,L_1 e^{-n}/2,L_1/2)$ and $A_{\HH}(v_m,r_0,L_1/2)$. By comparison principle of extremal length, $$d_{ \HH\sem K_{\tau^1_n}}(I,\pa_{\HH}^+ K_{\tau^1_n})\ge \frac 1\pi \ln \Big(\frac{L_1/2}{r_0} \Big)+\frac 1\pi \ln\Big (\frac{L_1/2}{L_1 e^{-n}/2 }\Big).$$ So by (\ref{boundary-estimate-formula}), \BGE \PP[\tau^0<\infty|\F_{\tau^1_n},\tau^1_n<\tau^0]\lesssim ({r_0}/{L_1 })^{\alpha_*} e^{-\alpha_* n},\quad n\in\N;\label{tau2n-tau1}\EDE We now use (\ref{tau1<infty}-\ref{tau2n-tau1}) to prove (\ref{transform*}). First, (\ref{tau1<infty}) and (\ref{tau2n-N}) (applied to $n=1$) together imply that \BGE \PP[ \tau^0<\tau^1_1\le\tau^1_{N}<\infty ]\lesssim ( {r_0}/{L_0} )^{\alpha_*} e^{-\alpha N}. \label{E0}\EDE Second, (\ref{tau2n},\ref{tau2n-tau1},\ref{tau2n-N}) together imply that \BGE \PP[\tau^1_n<\tau^0<\tau^1_{n+1}\le \tau^1_N<\infty]\lesssim ( {r_0}/{L_0 })^{\alpha_*} e^{-\alpha N}e^{-\alpha_* n} ,\quad 1\le n\le N-1.\label{En}\EDE Third, (\ref{tau2n}) and (\ref{tau2n-tau1}) together (applied to $n=N$) imply that \BGE \PP[\tau^1_N<\tau^0<\infty]\lesssim ( {r_0}/{L_0 })^{\alpha_*} e^{-\alpha N}e^{-\alpha_* N} .\label{EN}\EDE Summing up (\ref{E0},\ref{En},\ref{EN}), we get the desired inequality (\ref{transform*}). \end{proof} We are now able to prove Theorem \ref{main-thm1} in the case $m=1$. \begin{Theorem} Suppose $m=1$. Define $ \beta_\Sigma=\frac{\rho_\Sigma +3}{\rho_\Sigma +4}$. Then there is a constant $C_{\FF}>0 $ depending only on $\kappa,\rho_\Sigma$ such that, as $r/{L_1}\to 0^+$, \BGE \PP[\eta\cap B_{\FF}(u,r)\ne\emptyset]=C_{\FF} \Big(\frac{L_1} { L_0+L_1}\Big)^{\alpha_0} \Big(\frac r{L_1}\Big)^{\alpha}\Big(1+O\Big(\frac r{L_1}\Big)^{\beta_\Sigma}\Big),\label{one-force-point}\EDE where the implicit constants also depend only on $\kappa,\rho_\Sigma$. \label{m=1} \end{Theorem} \begin{proof} Since $m=1$, $\rho_\Sigma=\rho_1$. Throughout the proof, a constant depends only on $\kappa,\rho_1$. By translation and scaling, it suffices to derive (\ref{one-force-point}) in the case that $w=1$ and $u=0$. For $x\in(0,1]$, let $\eta_x$ be an SLE$_\kappa(\rho_1)$ curve in $\HH$ started from $w=1$ with the force point $v_1=x$. Here if $x=1$, then the force point $v_1$ is set to be $1^-$. For $s\in(0,1)$, let $p_{\FF}(x,s)$ denote the probability that $\eta_x$ hits $B_{\FF}(0, x\cdot s)$. By scaling and translation, we need to show there is a constant $C_{\FF}> 0$ such that \BGE p_{\FF}(x,s)=C_{\FF} x^{\alpha_0} s^{\alpha}(1+O(s^{\beta_\Sigma})),\quad \mbox{as }s\to 0.\label{pC0}\EDE Define $R$ and $\Phi$ on $[0,\tau_u^*)$ by $$R= \frac{V_1 -U }{W -U },\quad \Phi = -\frac \kappa2 \ln\Big(\frac{|V_1-U| }{D_u x}\Big);$$ and let $\ha R(t)= R\circ \Phi^{-1}(t)$, $0\le t<\ha \tau_u^*:=\sup \Phi[0,\tau^*_u)$. These functions differ from the $R,\Phi,\ha R$ in the proof of Theorem \ref{Thm-est1} by that $V_1$ and $v_1$ are used here in place of $V_0$ and $v_0$. An argument similar to the one used to derive (\ref{dhaR}) shows that $\ha R$ satisfies the SDE (\ref{Bessel-SDE-X}) with $\delta_+:=\frac 4\kappa (\rho_1+2)>0$ and $\delta_-:=\frac 4\kappa(\kappa-4-\rho_\Sigma)<2$. Here we use the assumption that $\rho_1>-2$ and $\rho_\Sigma>\frac\kappa 2-4$. As $t\uparrow \tau_u^*$, $\eta_x(t)$ tends to $\infty$ (when $\tau_u^*=\infty$) or some point on $(-\infty,u)$ (when $\tau_u^*<\infty$). In either case, by comparison principle, the extremal distance between $[u,v_1\wedge a_{K_t}]$ and $\pa_{\HH}^+ K_t$ in $\HH\sem K_t$ tends to $\infty$ as $t\uparrow \tau_u^*$. Thus, by conformal invariance of extremal distance and Proposition \ref{extrem-prop}, $\lim_{t\uparrow \tau^*_u}R(t)=0$, which implies that $\lim_{t\uparrow \ha \tau^*_u}\ha R(t)=0$. So we may apply Proposition \ref{Bessel-transition} (iii) and (\ref{invariant-def},\ref{transition-rate}) to conclude that $\ha R$ has a transition density $p^{\ha R}_t(x,y)$ and a quasi-invariant density $p^{\ha R}_\infty(y)=C_1 (1-y)^{\alpha_+}$ with decay rate $\beta_0$, such that \BGE p^{\ha R}_t(x,y)=C_2 x^{\alpha_-} p^{\ha R}_\infty(y) e^{-\beta_0 t}(1+O(e^{-(\beta_1-\beta_0)t})),\quad \mbox{as }t\to\infty,\label{transition-invariant}\EDE where $C_1,C_2$ are positive constants, and $$ \alpha_+=\frac{\delta_+}2-1,\quad \alpha_-=1-\frac{\delta_-}2,\quad \beta_n=\frac 12(n+1+\alpha_+)(n+\alpha_-),\quad n=0,1.$$ Let $L>0$, and $\tau_L=\inf(\{t\in [0,\tau_u^*):\Phi(t)\ge L\}\cup \{\infty\}) $. By Koebe $1/4$ theorem, if $\tau_L<\tau^*_u$, \BGE x e^{-\frac 2\kappa L}=x e^{-\frac 2\kappa \Phi(\tau_L)}=\frac{|V_1(\tau_L)-U(\tau_L)|}{D_u(\tau_L)} \le \dist(u,\eta_x[0,\tau_L])\wedge x\le 4x e^{-\frac 2\kappa L}.\label{Koebe-dist}\EDE Let $\til g_t=\frac{g_t-U(t)}{W(t)-U(t)}$, $0\le t<\tau^*_u$. Then $\til g_t$ maps $\HH\sem K_t$ conformally onto $\HH$, fixes $\infty$ and $0$, and sends $\eta_x(t)$ and $v_1\wedge a_{K_t}$ respectively to $1$ and $R(t)$. By DMP, there is a random curve $\eta^L$ in $\lin\HH$ defined on the event $\{\tau_L<\tau^*_u\}$, whose law conditionally on $\F_{\tau_L}$ and $\{\tau_L<\tau^*_u\}$ is that of $\eta_{\ha R(L)}$, i.e., an SLE$_\kappa(\rho_1)$ curve started from $1$ with the force point $ \ha R(L)$, such that $\eta_x(\tau_L+\cdot)=\til g_{\tau_L}^{-1}(\eta^L)$. Suppose $s< \frac 14 e^{-\frac 2\kappa L}$. Then $\eta_x$ hits $B_{\FF}(0,xs)$ iff $\tau_L<\tau_u^*$ and $\eta^L$ hits $\til g_{\tau_L}(B_{\FF}(0,xs))$. Since $\til g_{\tau_L}'(u)=\frac{D_u(\tau_L)}{W(\tau_L)-U(\tau_L)}=\ha R(L)e^{\frac 2\kappa L} $, by (\ref{Koebe-dist}) and Koebe distortion theorem, $$B_{\FF}\Big(0, \frac{\ha R(L) se^{\frac 2\kappa L} }{(1+4s e^{\frac 2\kappa L})^2}\Big) \subset \til g_{\tau_L}(B_{\FF}(0, xs)) \subset B_{\FF}\Big(0, \frac{\ha R(L) s e^{\frac 2\kappa L} }{(1-4s e^{\frac 2\kappa L})^2}\Big),$$ which implies that $$p_{\FF}(x,s)\gtreqless \EE\Big[{\bf 1}_{\{\tau_L<\tau_u^*\}}\cdot p_{\FF}\Big(\ha R(L), \frac{s e^{\frac 2\kappa L} }{(1\pm 4s e^{\frac 2\kappa L})^2}\Big)\Big].$$ Here when we choose $+$ (resp.\ $-$) in $\pm$, the inequality holds with $\ge$ (resp.\ $\le$). Since $\ha R$ has a transition density $p^{\ha R}_t(x,y)$, the above inequality further implies that \BGE p_{\FF}(x,s) \gtreqless \int_0^1 p^{\ha R}_L(x,y) p_{\FF} \Big(y, \frac{s e^{\frac 2\kappa L}}{(1\pm 4s e^{\frac 2\kappa L})^2}\Big) dy.\label{p(x,s)-integral}\EDE Let $p_{\FF}(s)=\int_0^1 p_{\FF}(x,s) p^{\ha R}_\infty(x) dx$ and $q_{\FF}(s)=s^{-\frac \kappa 2\beta_0 } p_{\FF}(s)$. By (\ref{invariant=}), if $0<s< \frac 14 e^{-\frac 2\kappa L}$, then $$p_{\FF}(s) \gtreqless \int_0^1\!\int_0^1 p^{\ha R}_\infty(x) p^{\ha R}_L(x,y) p_{\FF} \Big(y, \frac{se^{\frac 2\kappa L} }{(1\pm 4s e^{\frac 2\kappa L})^2}\Big) dy dx$$ $$=e^{-\beta_0L} \int_0^1 p^{\ha R}_\infty(y) p_{\FF} \Big(y, \frac{se^{\frac 2\kappa L} }{(1\pm 4s e^{\frac 2\kappa L})^2}\Big) dy =e^{-\beta_0L} p_{\FF}\Big(\frac{se^{\frac 2\kappa L} }{(1\pm 4s e^{\frac 2\kappa L})^2}\Big) $$ and so \BGE q_{\FF}(s) \gtreqless (1\pm 4s e^{\frac 2\kappa L})^{-\kappa \beta_0} q_{\FF}\Big(\frac{se^{\frac 2\kappa L} }{(1\pm 4s e^{\frac 2\kappa L})^2}\Big).\label{ineq-q}\EDE Let $\eps\in(0,1)$, $a_+=\eps$, and $x_0=\frac{a_+/4}{(1+a_+)^2}$. Choose $a_-\in(0,a_+)$ such that $a_++\frac 1{a_+}+2=a_-+\frac 1{a_-}-2$. Then $\frac{a_-/4}{(1+a_-)^2}=x_0$. Let $s\in (0,a_-/4)$. Then there are $L_+,L_->0$ such that $4s e^{\frac 2\kappa L_\pm}=a_\pm$, which implies $\frac{se^{\frac 2\kappa L_\pm} }{(1\pm 4s e^{\frac 2\kappa L_\pm})^2}=\frac{a_\pm /4}{(1\pm a_\pm)^2}=x_0$. By (\ref{ineq-q}), \BGE q_{\FF}(s)\gtreqless (1\pm a_\pm) ^{-\kappa \beta_0} q_{\FF}(x_0 ) \gtreqless (1\pm \eps)^{-\kappa \beta_0}q_{\FF}(x_0 ) ,\quad \forall s\in (0,a_-). \label{ineq-q2}\EDE This inequality implies that $\lim_{s\to 0^+} q_{\FF}(s)$ converges. Let $Q_{\FF}$ denote the limit. On the other hand, for any $x\in(0,1/16)$, there are $a_+>a_-\in(0,1)$ such that $\frac{a_\pm/4}{(1\pm a_\pm)^2}= x$, and as $x\to 0^+$, $a_\pm=O(x)$. For $s\in(0,a_-/4)$, there are $L_+>L_->0$ such that $4s e^{\frac 2\kappa L_\pm}=a_\pm$. Using (\ref{ineq-q2}), we get $q_{\FF}(s) \gtreqless (1\pm a_\pm) ^{-\kappa \beta_0} q_{\FF}(x)$. Sending $s$ to $0^+$, we get $q_{\FF}(x)=Q_{\FF} (1+O(x))$ as $x\to 0^+$. Thus, \BGE p_{\FF}(s)=Q_{\FF} s^{\frac \kappa 2\beta_0 } (1+O(s)),\quad \mbox{as } s\to 0^+.\label{p-Q}\EDE Setting $L =(\frac 2\kappa+(\beta_1-\beta_0))^{-1}\ln(1/s)>0$ and using (\ref{transition-invariant},\ref{p(x,s)-integral}), we find that as $s\to 0^+$, $$ p_{\FF}(x,s) \gtreqless \int_0^1 C_2 x^{\alpha_-} p^{\ha R}_\infty(y) e^{-\beta_0 L} (1+O(e^{-(\beta_1-\beta_0) L})) p_{\FF} \Big(y, \frac{s e^{\frac 2\kappa L}}{(1\pm 4s e^{\frac 2\kappa L})^2}\Big) dy.$$ $$=C_2 x^{\alpha_-} e^{-\beta_0 L} (1+O(e^{-(\beta_1-\beta_0) L})) p_{\FF} \Big( \frac{s e^{\frac 2\kappa L}}{(1\pm 4s e^{\frac 2\kappa L})^2}\Big)$$ $$=C_2 Q_{\FF} x^{\alpha_-} s^{\frac \kappa2 \beta_0}(1+O(e^{-(\beta_1-\beta_0)L})+O(s e^{\frac 2\kappa L}))=C_2Q_{\FF} x^{\alpha_-} s^{\frac \kappa 2 \beta_0} (1+O(s^{\frac{\beta_1-\beta_0}{\frac 2\kappa+\beta_1-\beta_0}})).$$ Since ${\frac \kappa2 \beta_0}=\alpha$, $\alpha_-=\alpha_0$, and $\frac{\beta_1-\beta_0}{\frac 2\kappa+\beta_1-\beta_0}=\beta_\Sigma$, we conclude that (\ref{pC0}) holds with $C_{\FF}=C_2 Q_{\FF}$. Finally, we check that $C_{\FF}>0$, which is equivalent to that $Q_{\FF}>0$ since $C_2>0$. Since $q_{\FF}\ge 0$, we have $Q_{\FF}\ge 0$. If $Q_{\FF}=0$, then (\ref{pC0}) implies that there is $c_0\in(0,1)$ such that $p_{\FF}(x,c_0)=0$ for all $0<x\le 1$. First, suppose $\FF=\C$. By the properties of the transition density of $\ha R$, $\PP[\tau_L<\tau^*_u]>0$ for any $L>0$. By (\ref{Koebe-dist}), $\PP[\dist(u,\eta_x)\le \eps]>0$ for any $\eps>0$, which contradicts that $p_{\C}(x,c_0)=0$. Second, suppose $\FF=\R$. Then we have $\rho_\Sigma<\frac\kappa 2-2$ by assumption. Let $\eta_{\bf x}$ be an SLE$_\kappa(\rho_\Sigma)$ curve started from $1$ with a random force point ${\bf x}\in(0,1)$, whose law is the quasi-invariant density of $\ha R$. Since $\rho_\Sigma<\frac\kappa 2-2$, $\eta_{\bf x}$ a.s.\ hits $(-\infty,0)$ (cf.\ \cite{MS1}). So there is $N<0$ such that $\PP[\eta_{\bf x}\cap (N{\bf x},0)\ne \emptyset]>0$. Choose $L>0$ such that $\frac{c_0 e^{\frac 2\kappa L}}{576}>|N|+1$. Let $\til g_{\tau_L}$ and $\eta^L$ be associated with $\eta_{\bf x}$. Recall that, conditionally on $\F_\tau$, $\eta^L$ is an SLE$_\kappa(\rho_1)$ curve started from $1$ with the force point $\ha R(L)$. Let $X=\til g_{\tau_L}(-c_0 {\bf x})<0$ and $r=\dist(0,\eta_{\bf x}[0,\tau_L])$. By (\ref{Koebe-dist}), $r\le 4{\bf x} e^{-\frac 2\kappa L}$. Since $(-\infty,-c_0 {\bf x})$ and $[0,{\bf x}\wedge a_{K_{\tau_L}})$ could be separated by $\A_{\HH}(0,r,c_0{\bf x})$ in $\HH\sem K_{\tau_L}$, by Proposition \ref{extrem-prop} and conformal invariance and comparison principle of extremal length, $\frac{\ha R(L)}{\ha R(L)-X}\le \frac{144r}{c_0{\bf x}}\le \frac{576 {\bf x} e^{-\frac 2\kappa L}}{c_0{\bf x}}<\frac 1{|N|+1}$, which implies that $X<N \ha R(L)$. Since $\ha R(L)$ is the force point for $\eta^L$, and conditionally on $\{\tau_L<\tau^*_u\}$ has the same law as ${\bf x}$ (because the law of ${\bf x}$ is the quasi-invariant density of $\ha R$), we know that $(\eta^L,\ha R(L))$ conditionally on $\{\tau_L<\tau^*_u\}$ has the same law as $(\eta_{\bf x},{\bf x})$. From $X<N \ha R(L)$ we know that $\PP[\eta^L\cap (X,0)\ne\emptyset]>0$, which implies that $\PP[\eta_{\bf x}\cap (c_0 {\bf x},0)\ne\emptyset]>0$. But this contradicts that $p_{\R}({\bf x},c_0)=0$. \end{proof} \begin{Remark} Theorem \ref{m=1} holds with slight modification if $\eta$ has two force points $v_1,u$, and the force values satisfy $\rho_1>-2$ and $\rho_1+\rho_u>\frac\kappa 2-4$. In this case, we redefine $\rho_\Sigma,\alpha,\beta$ using $\rho_\Sigma=\rho_1+\rho_u$, $\alpha=\frac{(\rho_1+2)(2\rho_\Sigma+8-\kappa)}{2\kappa}$, $\beta=\frac{\rho_1+\rho_\Sigma +6}{\rho_1+\rho_\Sigma +8}$, still define $\alpha_0$ using (\ref{alphaj}), and allow all constants to depend on $\kappa,\rho_1,\rho_\Sigma$. Then the proof works without further modification. \end{Remark} \begin{Lemma} Let $\delta=|v_1-v_m|$ and $R\ge 10\delta$. Suppose $|w-v_m|\ge R$. Let $\PP_m$ denote the law of the $m$-force-point SLE$_\kappa(\ulin\rho)$ curve $\eta$ in Theorem \ref{main-thm1}. Let $\PP_1$ denote the law of a one-force-point SLE$_\kappa(\rho_\Sigma)$ curve started from $w$ with the force point $v_m$. Then $\PP_m$ and $\PP_1$ are mutually absolutely continuous on $\F_{\tau^{v_m}_R}$, and there is a constant $C>0$ depending only on $\kappa,\rho_\Sigma,\sigma^*$ such that \BGE \Big|\ln\Big(\frac{d\PP_m|\F_{\tau^{v_m}_R}}{d\PP_1|\F_{\tau^{v_m}_R}}\Big)\Big|\le C\cdot \frac {\delta } R.\label{dP1/P2}\EDE \label{lemma-merge} \end{Lemma} \begin{proof} Throughout this proof, an explicit or implicit constant depends only on $\kappa,\rho_\Sigma,\sigma^*$. First suppose $\eta$ follows the law $\PP_m$. When $\tau^*_{v_m}<\infty$, we have $W=V_1=\cdots=V_m$ at $\tau^*_{v_m}$. The DMP asserts that, under $\PP_m$, conditionally on $\F_{\tau^*_{v_m}}$ and the event $\{\tau^*_{v_m}<\infty\}$, the law of $\eta(\tau^*_{v_m}+\cdot)$ is that of the $f_{\tau^*_{v_m}}$-image of an SLE$_\kappa(\rho_\Sigma)$ curve in $\HH$ started from $W(\tau^*_{v_m})$ with force point $W(\tau^*_{v_m})^-$. Here we use the fact that all force point processes merge together after the time $\tau^*_{v_m}$. The statement clearly also holds if $\eta$ follows the law $\PP_1$. Thus, it suffices to prove that (\ref{dP1/P2}) holds with $\tau_0:=\tau^*_{v_1}\wedge \tau^{v_m}_R$ in place of $\tau^{v_m}_R$. Since $W(0)=w>v_1=V_1(0)$, we have $W>V_1$ on $[0,\tau_{v_1}^*)$. Let $D_j(t)=g_t'(v_j)$. Then each $D_j$ satisfies the ODE $\frac{d D_j}{D_j}=\frac{-2}{(W-V_j)^2}$. Define $N_1$ and $N_m$ on $[0,\tau_{v_1}^*)$ by $$N_1={D_m^{ \frac{\rho_\Sigma(\rho_\Sigma+4-\kappa)}{4\kappa}} |W-V_m|^{ \frac{\rho_\Sigma}\kappa} },\quad N_m={\prod_{j=1}^m \Big( D_j^{\frac{\rho_j(\rho_j+4-\kappa)}{4\kappa}} |W-V_j|^{\frac{\rho_j}\kappa} \Big)\prod_{1\le j<k\le m} |V_j-V_k|^{\frac{\rho_j\rho_k}{2\kappa}}},$$ and let $N=N_m/N_1$. Here for $j\in\{1,m\}$, $N_j$ is a local martingale introduced in \cite{SW}, such that $\PP_j$ can be obtained by locally weighting a chordal SLE$_\kappa$ curve (with no force point) from $w$ to $\infty$ by $N_j/N_j(0)$. Thus, if we locally weight $\PP_1$ by $N/N(0)$, then we get $\PP_m$. More precisely, this means that, if $\tau$ is an $\F$-stopping time with $\tau\le \tau_{v_1}^*$ such that $N/N(0)$ is uniformly bounded on $[0,\tau)$, then \BGE \frac{d\PP_m|\F_{\tau}}{d\PP_1|\F_{\tau}}=\frac{N(\tau)}{N(0)}.\label{P2/P1-tau-N}\EDE Here if $\tau=\tau_{v_1}^*$, then $N(\tau)$ is understood as $\lim_{t\to \tau^-} N(t)$, which exists a.s.\ on the event $\{\tau=\tau_{v_1}^*\}$. These facts can also be checked directly using It\^o's formula and Girsanov theorem. We claim that \BGE \sup_{0\le t<\tau_0} \Big|\ln\Big(\frac{N(t)}{N(0)}\Big)\Big|\lesssim \frac {\delta } R.\label{N/N-bound}\EDE If the claim is true, then $N/N(0)$ is uniformly bounded on $[0,\tau_0)$, and $\ln( \frac{N(\tau_0)}{N(0)})\lesssim \frac {\delta } R$. So we may apply (\ref{P2/P1-tau-N}) to $\tau=\tau_0$ and get (\ref{dP1/P2}) with $\tau_0$ in place of $\tau^{v_m}_R$. Now we prove (\ref{N/N-bound}). Let $X_j = \frac{W -V_j }{W-V_m} $. Then $0\le X_1\le \cdots\le X_m=1$ on $[0,\tau_{v_1}^*)$. Define positive functions $D_{j,k}$ on $[0,\tau_{v_1}^*)$, $1\le j,k\le m+1$, such that, $$D_{j,k}(t)=\left\{\begin{array}{ll} \frac{V_j(t)-V_k(t)}{v_j-v_k},& \mbox{if }j,k\le m\mbox{ and }v_j\ne v_k;\\ D_j(t),&\mbox{if }j,k\le m\mbox{ and }v_j=v_k;\\ 1, & \mbox{if }j\mbox{ or }k=m+1. \end{array} \right.$$ Then we have \BGE N/N(0)=D_{m,m}^{-\frac{\rho_\Sigma(\rho_\Sigma+4-\kappa)}{4\kappa}} \prod_{j=1}^{m-1}( X_j/X_j(0))^{\frac{\rho_j}\kappa} \prod_{j_1=1}^m \prod_{j_2=1}^m D_{j_1,j_2}^{\frac{\rho_{j_1} \rho_{j_2}}{4\kappa}} \prod_{j=1}^m D_{j,j}^{\frac{\rho_j(4-\kappa)}{4\kappa}} \label{M-re1}.\EDE Define $Y_j$, $1\le j\le m-1$, and $E_{j_1,j_2}$, $1\le j_1,j_2\le m$, on $[0,\tau_{v_1}^*)$ by $$Y_j=\frac{X_j}{X_{j+1}},\quad E_{j_1,j_2}=\frac{D_{j_1,j_2}D_{j_1+1,j_2+1}}{D_{j_1+1,j_2}D_{j_1,j_2+1}}.$$ Then each $Y_j\in(0,1]$. By (\ref{dVU}) and the ODE for $D_j$, $E_{j_1,j_2}$ satisfies the ODE: $$\frac{d E_{j_1,j_2}}{E_{j_1,j_2}}=-2\prod_{s=1}^2 \Big(\frac 1{W-V_{j_s}}-\frac 1{W-V_{j_s+1}}\Big) dt\le 0,$$ where $\frac{1}{W-V_{m+1}}$ is understood as $0$. So $E_{j_1,j_2}$ is decreasing in $t$. From $E_{j_1,j_2}(0)=1$, we get $0<E_{j_1,j_2}\le 1$. Since $X_m=D_{k_1,m+1}=D_{m+1,k_2}=1$, we get $$X_j=\prod_{k=j}^{m-1} Y_k,\quad D_{j_1,j_2}=\prod_{k_1=j_1}^m \prod_{k_2=j_2}^m E_{k_1,k_2}.$$ Recall that $ \sum_{j=1}^k \rho_j=\sigma_k-2$, $1\le k\le m$. So $$ \prod_{j=1}^{m-1} X_j^{ {\rho_j} }= \prod_{j=1}^{m-1} \prod_{k=j}^{m-1} Y_k^{ {\rho_j} }=\prod_{k=1}^{m-1} \prod_{j=1}^k Y_k^{{\rho_j} } =\prod_{k=1}^{m-1} Y_k^{ {\sigma_k-2} };$$ $$\prod_{j_1=1}^m \prod_{j_2=1}^m D_{j_1,j_2}^{ {\rho_{j_1} \rho_{j_2}} } = \prod_{j_1=1}^m \prod_{j_2=1}^m \prod_{k_1=j_1}^m \prod_{k_2=j_2}^m E_{k_1,k_2}^{ {\rho_{j_1} \rho_{j_2}} }$$ $$=\prod_{k_1=1}^m \prod_{k_2=1}^m \prod_{j_1=1}^{k_1} \prod_{j_2=1}^{k_2} E_{k_1,k_2}^{ {\rho_{j_1} \rho_{j_2}} }=\prod_{k_1=1}^m \prod_{k_2=1}^m E_{k_1,k_2}^{ {(\sigma_{k_1}-2)(\sigma_{k_2}-2)} }$$ $$\prod_{j=1}^m D_{j,j}^{\rho_j}=\prod_{j=1}^m \prod_{k_1=j}^m \prod_{k_2=j}^m E_{k_1,k_2}^{ {\rho_{j}} }=\prod_{k_1=1}^m \prod_{k_2=1}^m \prod_{j=1}^{k_1\wedge k_2} E_{k_1,k_2}^{\rho_j} =\prod_{k_1=1}^m \prod_{k_2=1}^m E_{k_1,k_2}^{\sigma_{k_1\wedge k_2}-2}.$$ Let $S=\{(j,k)\in\N^2:1\le j,k\le m\}\sem\{(m,m)\}$. By (\ref{M-re1}) and the above formulas, $$N/N(0)= \prod_{j=1}^{m-1} (Y_j/Y_j(0)) ^{\frac{\sigma_j-2}\kappa} \prod_{(k_1,k_2)\in S} E_{k_1,k_2}^{\frac{(\sigma_{k_1}-2)(\sigma_{k_2}-2)}{4\kappa}+\frac{(4-\kappa)(\sigma_{k_1\wedge k_2}-2)}{4\kappa}}.$$ Thus, it suffices to show that (\ref{N/N-bound}) holds with $N/N(0)$ replaced respectively by $$\prod_{j=1}^{m-1} Y_j ,\quad \prod_{j=1}^{m-1} Y_j ^{ {\sigma_j } },\quad \prod_{(j,k)\in S} E_{j,k}^{ {\sigma_j\sigma_k} }, \quad \prod_{(j,k)\in S} E_{j,k}^{ {\sigma_j} },\quad \prod_{(j,k)\in S} E_{j,k}^{ {\sigma_{j\wedge k}} }, \quad \prod_{(j,k)\in S} E_{j,k} .$$ We observe that $\prod_{(j,k)\in S} E_{j,k}={D_1}/{D_m}$. Since $E_{j,k}\le 1$ and $\sigma_j\le \sigma^*\vee (\rho_\Sigma+2)$, we have $$ \prod_{(j,k)\in S} E_{j,k}^{ {\sigma_j} },\prod_{(j,k)\in S} E_{j,k}^{ {\sigma_{j\wedge k}} }, \prod_{(j,k)\in S} E_{j,k}^{ {\sigma_j\sigma_k} }\in \Big[\Big(\frac{D_1}{D_m}\Big)^{ (\sigma^*\vee (\rho_\Sigma+2))^2 },1\Big].$$ Thus, to prove the statement for $\prod_{(j,k)\in S} E_{j,k}^{ {\sigma_j\sigma_k} }$, $\prod_{(j,k)\in S} E_{j,k}^{ {\sigma_j} }$, and $\prod_{(j,k)\in S} E_{j,k}$, it suffices to show that (\ref{N/N-bound}) holds with $\frac N{N(0)}$ replaced by $\frac{D_1}{D_m}$. Since $\prod_{j=1}^{m-1} Y_j=X_1=\frac{|W-V_1|}{|W-V_m|}$, $0<Y_j\le 1$ and $\sigma_j\le \sigma^*$, to prove the statement for $\prod_{j=1}^{m-1} Y_j $ and $ \prod_{j=1}^{m-1} Y_j ^{ {\sigma_j } }$, it suffices to show that (\ref{N/N-bound}) holds with $\frac N{N(0)}$ replaced by $\frac{|W-V_1|}{|W-V_m|}$. The statements for $ \frac {D_1}{D_m} $ follow immediately from Koebe distortion theorem since $D_j(t)=g_t'(v_j)$, $j=1,m$, $g_t$ is analytic on $B_{\C}(v_m,R)$, and $|v_1-v_m|=\delta \le R/10$. The statements for $\frac{|W-V_1|}{|W-V_m|}$ follows from a combination of Koebe distortion theorem with Koebe $1/4$ theorem. To see this, let $t\in [0,\tau_0)$. Since $W(t)=g_t(\eta(t))$, and $|\eta(t)-v_m|\ge R$, by Koebe $1/4$ theorem, $|W(t)-V_m(t)|\ge D_m(t) R/4$. By Koebe distortion theorem, $|V_1(t)-V_m(t)|\le \frac {D_m(t)\delta }{(1-\delta /R)^2}$. Since $\frac{\delta } R\le \frac 1{10}$, we have $\frac{|V_1(t)-V_m(t)|}{|W(t)-V_m(t)|}\le \frac{4 \delta /R}{(1-\delta /R)^2}\le 5 \frac {\delta } R\le\frac 12$. Since $|\ln(1\pm x)|\le \ln (4)|x|$ when $|x|\le\frac 12$, we get $\ln(\frac{|W(t)-V_1(t)|}{|W(t)-V_m(t)|})\lesssim \frac {\delta } R$, as desired. \end{proof} \begin{Lemma} Let $C_{\FF}>0$ be as in Theorem \ref{m=1}. Let $\delta=|v_1-v_m|$. Let $\beta_\Sigma=\frac{\rho_\Sigma+3}{\rho_\Sigma+4}$ and $\beta_*=\frac{\alpha_*}{\alpha_*+1}$. Then as $\frac r{L_1},\frac {\delta}{L_0\wedge L_1 } \to 0^+$, \BGE \PP[\eta\cap B_{\FF}(u,r)\ne\emptyset]= C_{\FF}\Big(\frac r{L_1}\Big)^\alpha \Big(\Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_0} + O\Big (\frac r{L_1}\Big)^{\beta_\Sigma}+O\Big (\frac {\delta}{L_0}\Big)^{\beta_*} \Big)\label{v-close}\EDE \BGE =C_{\FF} G(w,\ulin v;u) r^{\alpha}+ \Big(\frac r{L_1}\Big)^\alpha\Big( O\Big (\frac r{L_1}\Big)^{\beta_\Sigma}+O\Big (\frac {\delta}{L_0 }\Big)^{\beta_*}+O\Big(\frac {\delta}{L_1 }\Big)\Big) .\label{v-close'}\EDE \label{v-close-lemma} \end{Lemma} \begin{proof} It suffices to prove (\ref{v-close}) since by (\ref{G-univ}, $G(w,\ulin v;u) = (\frac{L_1}{L_0+L_1} )^{\alpha_0}(\frac 1{L_1} )^\alpha(1+O(\frac{\delta}{L_1}))$ as $ \delta/{L_1}\to 0^+$. Suppose $ {\delta}/{L_0 }\in(0,1)$ is very small. Choose $R\in(\delta,L_0)$ to be determined later. Let $\PP_m$ and $\PP_1$ be as in Lemma \ref{lemma-merge}. Note that $\PP_m=\PP$. By Lemma \ref{two-pt-lem}, $$ |\PP_j[\tau^u_{r;\FF}<\infty]-\PP_j[\tau^u_{r;\FF}< \tau^{v_m}_R]|\lesssim ( R/{L_0} )^{\alpha_*} ( r/{L_1} )^{\alpha},\quad j=1,m. $$ By Lemma \ref{lemma-merge} and (\ref{est-u*}), if $10\delta<R$, then $$|\PP_m[\tau^u_{r;\FF}<\tau^{v_m}_R]-\PP_1[\tau^u_{r;\FF}<\tau^{v_m}_R]|\lesssim ( \delta/ R) ( r/{L_1})^{\alpha}.$$ By Theorem \ref{m=1}, $$\Big|\PP_1[\tau^u_{r;\FF}<\infty]-C_{\FF}\Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_0}\Big(\frac{r}{L_1}\Big)^\alpha \Big|\lesssim \Big(\frac{r}{L_1}\Big)^{\alpha+\beta_\Sigma}.$$ Combining the above three displayed formulas, we find that, when $\delta/R<1/10$, $$\Big|\PP_m[\tau^u_{r;\FF}<\infty]-C_{\FF}\Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_0}\Big(\frac{r}{L_1}\Big)^\alpha \Big|\lesssim \Big(\frac r{L_1}\Big)^{\alpha}\Big(\Big(\frac R{L_0}\Big)^{\alpha_*}+ \frac \delta R + \Big(\frac{r}{L_1}\Big)^{ \beta_\Sigma}\Big).$$ Setting $R=\delta^{\frac 1{\alpha_*+1}} L_0^{\frac{\alpha_*}{\alpha_*+1}}$ in the above estimate, we get (\ref{v-close}). \end{proof} \begin{proof}[Proof of Theorem \ref{main-thm1}] Let $C_{\FF}>0$ be as in Theorem \ref{m=1}. Let $r<L_1$. Fix $R\in (r,L_1)$ to be determined later. Define $M$ on $[0,\tau_u^*)$ by $$ M=D_u^\alpha G(W,\ulin V;U)=D_u^{\alpha} |W-U|^{-\alpha_0} \prod_{j=1}^m |V_j-U|^{-\alpha_j} .$$ A straightforward application of It\^o's formula shows that $M$ is a local martingale. By (\ref{G-upper}), \BGE M\le ( {|V_m-U|}/{|W-U|} )^{\alpha_*} ( {D_u}/{|V_m-U|} )^\alpha .\label{M-other}\EDE Let $\tau=\tau^u_R$. By Koebe $1/4$ theorem, $\frac{D_u}{|V_m-U|}\le \frac 4R $ on $[0,\tau\wedge \tau_u^*)$. So by (\ref{M-other}) $M\le (\frac 4R)^\alpha$ on $[0,\tau\wedge \tau_u^*)$. By a standard extremal length argument, as $t\uparrow \tau_u^*$, $\frac{|V_m(t)-U(t)|}{|W(t)-U(t)|}\to 0$. This implies by (\ref{M-other}) that, one the event $\{\tau_u^*\le \tau^u_R\}$, as $t\uparrow \tau_u^*$, $M(t)\to 0$. Thus, $M(t\wedge \tau\wedge \tau_u^*)$, $0\le t<\infty$, is a uniformly bounded martingale, where when $t\wedge \tau\wedge \tau_u^*\ge \tau_u^*$, $M(t\wedge \tau\wedge \tau_u^*)$ is set to be $0$. In particular, we have $\EE[{\bf 1}_{\{\tau<\tau_u^*\}} M(\tau)]=M(0)=G(w,\ulin v;u)$. By DMP, $\PP[\tau^u_{r;\FF}<\infty|\F_\tau,\tau<\tau^*_u]$ equals the probability that an SLE$_\kappa(\ulin\rho)$ curve $\eta^\tau$ in $\HH$ started from $W(\tau)$ with force points $\ulin V(\tau)$ hits $g_\tau(B_{\FF}(u,r))$. Suppose $\tau<\tau_u^*$. By Koebe distortion theorem, there are $r_1,r_2=D_u(\tau) r(1+O(r/R))$ such that $$ B_{\FF}(U(\tau),r_1)\subset g_\tau(B_{\FF}(u,r))\subset B_{\FF}(U(\tau),r_2).$$ By Koebe $1/4$ theorem, $|V_m(\tau)-U(\tau)|\ge D_u(\tau) R/4$. Note that the interval $[v_m\wedge a_{K_\tau},v_1\wedge a_{K_\tau}]$ can be disconnected in $\HH\sem K_\tau$ from both $(-\infty,u]$ and $\pa_{\HH}^+ K_\tau$ by $A_{\HH}(u,R,|v_m-u|)$. Thus, by Proposition \ref{extrem-prop}, $\frac{|V_1(\tau)-V_m(\tau)|}{|W(\tau)-V_m(\tau)|}$ and $\frac{|V_1(\tau)-V_m(\tau)|}{|V_m(\tau)-U(\tau)|}$ both have the order $O(\frac{R}{|v_m-u|})$. Since $M(\tau)=D_u(\tau)^\alpha G(W(\tau),\ulin V(\tau);U(\tau))$, by (\ref{v-close'}) and the above estimates, the conditional probability that $\eta^\tau$ hits $g_\tau(B_{\FF}(u,r))$ equals $$C_{\FF}M(\tau) r^{\alpha}+\Big(\frac{r}{R}\Big)^\alpha\Big( O\Big(\frac{r}{R}\Big)^{\beta_\Sigma}+ O\Big(\frac{R}{|v_m-u|}\Big)^{\beta_*}\Big).$$ Setting $R$ such that $R^{\beta_\Sigma+\beta_*}=r^{\beta_\Sigma} |v_m-u|^{\beta_*}$, and letting $\beta=\frac{\beta_*\beta_\Sigma}{\beta_*+\beta_\Sigma}$, we see that the above conditional probability equals $C_{\FF}M(\tau) r^{\alpha}+ (\frac{r}{R} )^\alpha O (\frac{r}{|v_m-u|} )^\beta$. Thus, $$\PP[\eta\cap B_{\FF}(u,r)\ne\emptyset]= \EE [{\bf 1}_{\{\tau<\tau_u^*\}}C_{\FF}M(\tau) r^{\alpha}] + \PP[\tau<\tau_u^*] \Big(\frac{r}{R}\Big)^\alpha O \Big(\frac{r}{|v_m-u|}\Big)^\beta $$ $$=C_{\FF} G(w,\ulin v;u) r^\alpha+ \Big(\frac{|v_m-u|}{|w-u|}\Big)^{\alpha_*}\cdot O\Big(\frac{r}{|v_m-u|}\Big)^{\alpha+\beta},$$ where we used $\EE[{\bf 1}_{\{\tau<\tau_u^*\}} M(\tau)]=G(w,\ulin v;u)$ and $\PP[\tau<\tau_u^*]\lesssim (\frac{|v_m-u|}{|w-u|} )^{\alpha_*} (\frac R{|v_m-u|})^\alpha$ by (\ref{est-u}). \end{proof} Combining Theorem \ref{main-thm1} with the lower bound of $G(w,\ulin v;u)$ in (\ref{G-upper}), we get the following lower bound of the hitting probability. \begin{Corollary} There is a constant $c\in(0,1)$ such that when ${r}/{L_1}\le c ( {L_1}/({L_0+L_1}))^{\frac{\alpha^*-\alpha_*}\beta}$, we have $\PP[ \tau^u_{r;\FF}<\infty]\ge C_{\FF} G(w,\ulin v;u) r^\alpha/2$. \label{Cor-lower-1pt} \end{Corollary} Define $G_{t,\FF}(u)$ for $t\in[0,\infty)$ by \BGE G_{t;\FF}(u)=\left\{\begin{array}{ll} C_{\FF} D_u(t)^\alpha G(W(t),V(t);U(t)), & \mbox{if } t<\tau_u^*;\\ 0, &\mbox{if }t\ge \tau_u^*. \end{array} \right. \label{Gt} \EDE By Koebe $1/4$ theorem and (\ref{G-upper}), \BGE G_{t;\FF}(u)\lesssim \dist(u, \eta[0,t])^{-\alpha}.\label{Gt-upper}\EDE The following lemmas will be used in the next section. \begin{Lemma} For any $\F$-stopping time $\tau$ and $R>r\in (0,|v_m-u|)$, as $r/R\to 0^+$, \BGE \PP[ \tau^u_{r;\FF}<\infty|\F_\tau, \tau<\tau^u_R ]= G_{\tau;\FF}(u) r^\alpha +O(r/R)^{\alpha+\beta}.\label{Green-r-R}\EDE \label{Cor-main-thm1} \end{Lemma} \begin{proof} If $\tau^*_u\le \tau<\tau^u_R$, the part of $\eta$ after $\tau$ is disconnected from $u$ by $\eta[0,\tau^*_u]$, which has distance greater than $R$ from $u$. Thus, $\PP[\ \tau^u_{r;\FF}<\infty|\F_\tau, \tau^*_u\le \tau<\tau^u_R ]=0$. So it suffices to prove (\ref{Green-r-R}) with $\tau<\tau^*_u\wedge \tau^u_R$ in place of $\tau<\tau^u_R$. By DMP, $\PP[ \tau^u_{r;\FF}<\infty|\F_\tau, \tau<\tau^*_u\wedge \tau^u_R ]$ equals the probability that an SLE$_\kappa(\ulin\rho)$ curve $\eta^\tau$ started from $W(\tau)$ with force points $\ulin V(\tau)$ visits $g_\tau(B_{\FF}(u,r))$. Assume that $\tau<\tau^*_u\wedge \tau^u_R$. By Koebe's $1/4$ theorem and distortion theorem, $|U(\tau)-V_m(\tau)|\ge D_u(\tau) R/4$, and there are $r_1,r_2=D_u(\tau) r(1+O(r/R))$ such that $ B_{\FF}(U(\tau),r_1)\subset g_\tau(B_{\FF}(u,r))\subset B_{\FF}(U(\tau),r_2)$. So we get the estimate of the conditional probability by applying Theorem \ref{main-thm1} and (\ref{Gt-upper}) . \end{proof} \begin{Lemma} Let $t_1<t_2\in [0,\tau_u^*)$. Suppose $I$ is a crosscut of $\HH\sem K_{t_1}$ connecting a point on the real interval $(u,v_m\wedge a_{K_{t_1}}]$ with a point on $\pa_{\HH}^+ K_{t_1}$ such that $\eta[t_1,t_2]$ lies in the closure of the bounded connected component of $(\HH\sem K_{t_1})\sem I$, and $d_{\HH\sem K_{t_1}}(I,(-\infty,u])>\ln(1440)/\pi$. Then \BGE|\ln(G_{t_2;\FF}(u))-\ln(G_{t_1;\FF}(u))|\lesssim \exp({-\pi d_{\HH\sem K_{t_1}}(I,(-\infty,u])}).\label{Gt1t2-eqn}\EDE \label{Gt1t2} \end{Lemma} \begin{proof} Let $\til I=g_{t_1}(I)$. Then $\til I$ is a crosscut in $\HH$ connecting $(U(t_1),V_m(t_1)]$ with $[W(t_1),\infty)$, and so it encloses $W(t_1),V_1(t_1),\dots,V_m(t_1)$. Moreover, since $\eta[t_1,t_2]$ does not intersect the unbounded component of $(\HH\sem K_{t_1})\sem I$, we see that $\til I$ also encloses $\til K:=g_{t_1}(K_{t_2}\sem K_{t_1})$. Let $X=d_{\HH\sem K_{t_1}}(I,(-\infty,u])$. By conformal invariance of extremal length, $d_{\HH}(\til I,(-\infty,g_t(u)])=X$. Since $X\ge \ln(1440)/\pi$, by Proposition \ref{extrem-prop}, \BGE {\rad_{W(t_1)}([V_m(t_1),W(t_1)]\cup \til K)} \le {\rad_{W(t_1)}(\til I)}\le 144 e^{-\pi X}{|W(t_1)-U(t_1)|}.\label{rad}\EDE In order to prove (\ref{Gt1t2-eqn}), it suffices to prove that (\ref{Gt1t2-eqn}) holds with $D_u({t_\iota})$, $|W(t_\iota)-U(t_\iota)|$, and $|V_j(t_\iota)-U(t_\iota)|$, $1\le j\le m$, respectively, in place of $G_{t_\iota;\FF}(u)$, $\iota=1,2$. Since $\til K=g_{t_1}(K_{t_2}\sem K_{t_1})$, we have $g_{t_2}=g_{\til K}\circ g_{t_1}$. So $U(t_2)=g_{\til K}(U(t_1))$ and $D_u(t_2)=g_{\til K}'(U(t_1))D_u(t_1)$. \begin{comment} Statement (i) is \cite[Corollary 3.44]{Law-SLE} with a translation. Now we prove (ii). For each nonempty $\HH$-hull $L$, we define $a_L=\min(\lin L\cap \R)$, $b_L=\max(\lin L\cap \R)$, $c_L=\sup g_L(-\infty, a_L)$ and $d_L=\inf g_L(b_L,\infty)$ as in \cite[Section 5.2]{LERW}. Define an $\HH$-hull $K_0=\{z\in\HH:|z-x_0|\le r\}$. Then $K\subset K_0$, $g_{K_0}(z)=z+\frac{r^2}z$, $\hcap(K_0)=r^2$, and $[c_{K_0},d_{K_0}]=[x_0-2r,x_0+2r]$. By the monotonicity of the $\HH$-capacity, $\hcap(K)\le \hcap(K_0)=r^2$. By \cite[Lemma 5.3]{LERW}, $[c_K,d_K]\subset [c_{K_0},d_{K_0}]=[x_0-2r,x_0+2r]$. Let $f_K=g_K^{-1}$. By \cite[(5.1)]{LERW}, there is a measure $\mu_K$ supported by $[c_K,d_K]$ with total mass $\hcap(K)$ such that, for any $z\in\lin\HH\sem [c_K,d_K]$, $$f_K(z)-z=\int_{c_K}^{d_K} \frac{-1}{z-x}\, \mu(dx),$$ which implies that $$|f_K'(z)-1|=\Big|\int_{c_K}^{d_K} \frac{1}{(z-x)^2}\, \mu(dx)\Big|\le \frac{\hcap(K)}{\dist(z,[c_K,d_K])^2} \le \frac{r^2}{\dist(z,[c_K,d_K])^2}.$$ Thus, if $\dist(z,[c_K,d_K])\ge 5r$, then $\ln|f_K'(z)|\le \frac{25}{24} \frac{r^2}{\dist(z,[c_K,d_K])^2}$. Now suppose $z\in\lin\HH$ and $|z-x_0|\ge 10 r$. Then $|g_K(z)-x_0|\ge |z-x_0|-3r$ by (i), and so $\dist(g_K(z),[c_K,d_K])\ge |z-x_0|-5r\ge |z-x_0|/2\ge 5r$ because $[c_K,d_K]\subset [x_0-2r,x_0+2r]$. Then we get (ii) because $|\ln(g_K'(z))|=|\ln(f_K'(g_K(z)))|$ and $\frac{r^2}{\dist(z,[c_K,d_K])^2}\le \frac{4r^2}{|z-x_0|^2}$. \end{comment} Note that $g_{t_1}(K_{t_1+s}\sem K_{t_1})$ and $g_{t_1+s}\circ g_{t_1}^{-1}$, $s\ge 0$, are chordal Loewner hulls driven by $W(t_1+s)$, $s\ge 0$, and $\til K$ is the hull in the process at the time $t_2-t_1$. Let $r= {\rad_{W(t_1)}(\til I)} $ and $R=|U(t_1)-W(t_1)|$. Then $\frac rR\le 144 e^{-\pi X}<\frac 1{10}$ by (\ref{rad}) and that $X>\ln(1440)/\pi$. By Proposition \ref{basic-chordal} (i,ii) and (\ref{rad}), $|W(t_2)-W(t_1)|\le 2 r$ and $|U(t_2)-U(t_1)|\le 3r$. Since $|U(t_1)-W(t_1)|=R>10 r$, these estimates imply $|\frac{|U(t_2)-W(t_2)|}{|U(t_1)-W(t_1)|}-1|\le \frac {5r}R\le \frac 12$, and so $$\Big|\ln\Big(\frac{|U({t_2})-W(t_2)|}{|U({t_1})-W(t_1)|}\Big)\Big|\le 2 \Big|\frac{|U({t_2})-W(t_2)|}{|U({t_1})-W(t_1)|}-1\Big|\le \frac{10 r}R\le 1440 e^{-\pi X}.$$ This means that (\ref{Gt1t2-eqn}) holds with $|U({t_\iota})-W(t_\iota)|$ in place of $G_{t_\iota;\FF}(u)$, $\iota=1,2$. Fix $j\in\{1,\dots,m\}$. By (\ref{rad}), we have $|V_j(t_1)-W(t_1)|\le 2r$. So $|U({t_1})-V_j(t_1)|\ge R-2r\ge \frac 4 5 R\ge 8r$. We have $V_j(t_2)=\lim_{x\uparrow V_j(t_1)} g_{\til K}(x)$. By Proposition \ref{basic-chordal} (i), $|V_j(t_1)-V_j(t_2)|\le 3r$. Thus, $|\frac{|U({t_2})-V_j(t_2)|}{|U({t_1})-V_j(t_1)|}-1|\le \frac {6r}{4 R/5}\le \frac 34$, which implies $$\Big|\ln\Big(\frac{|U({t_2})-V_j(t_2)|}{|U({t_1})-V_j(t_1)|}\Big)\Big|\le 4 \Big|\frac{|U({t_2})-V_j(t_2)|}{|U({t_1})-V_j(t_1)|}-1\Big|\le \frac{30 r}R\le 4320 e^{-\pi X}.$$ This means that (\ref{Gt1t2-eqn}) holds with $|U({t_\iota})-V_j(t_\iota)|$ in place of $G_{t_\iota;\FF}(u)$, $\iota=1,2$. Finally, by Proposition \ref{basic-chordal} (iii), $|\ln ( {D_u(t_2)}/{D_u(t_1)})|\le 5 (r/R)^2 \le 72 e^{-\pi X}$. So (\ref{Gt1t2-eqn}) also holds with $D_u(t_\iota)$ in place of $G_{t_\iota;\FF}(u)$, $\iota=1,2$. The proof is now complete. \end{proof} \section{Two-point Green's Function} \label{Section 5} We use the setup of the previous section. Now we fix $u_1>u_2\in (-\infty,v)$, and let $L_0=|w-v|$, $L_1=|v-u_1|$, $L_2=|u_1-u_2|$, and $ L_\wedge =L_1\wedge L_2$. We are going to prove the following theorem. \begin{Theorem} [Two-point Green's function] There is a number $G_{\FF}(w,\ulin v;u_1,u_2)\in (0,\infty)$ (depending also on $\kappa,\ulin\rho$), which satisfies \BGE G(w,\ulin v;u_1) L_2^{-\alpha} \lesssim G_{\FF}(w,\ulin v;u_1,u_2)\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} L_1^{-\alpha}L_2^{-\alpha}\label{G(u1,u2)-upper}\EDE such that, for some constants $\zeta_1,\zeta_2\in(0,1)$, as $\frac{r_1}{L_\wedge},\frac{r_2}{L_2}\to 0^+$, $$ \PP[\tau^{u_j}_{r_j;\FF}<\infty, j=1,2]=G_{\FF}(w,\ulin v; u_1,u_2) r_1^\alpha r_2^\alpha $$ \BGE+ \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^\alpha \Big(\frac{r_2}{L_2}\Big)^\alpha \Big[O \Big(\frac{r_1}{L_\wedge}\Big)^{\zeta_1}+O\Big(\frac{r_2}{L_2}\Big)^{\zeta_2}\Big]. \label{G(u1,u2)-est}\EDE In particular, we have \BGE G_{\FF}(w,\ulin v;u_1,u_2)=\lim_{r_1,r_2\to 0^+} r_1^{-\alpha}r_2^{-\alpha} \PP[\eta\cap B_{\FF}(u_j,r_j)\ne \emptyset,j=1,2].\label{Green-2pt}\EDE \label{main-thm2} \end{Theorem} We now write $\tau^j_{r;\FF}$ for $\tau^{u_j}_{r;\FF}$ and $\tau^j_r$ for $\tau^{u_j}_r=\tau^j_{r;\C}$. Let $U_j(t)=g_t(u_j)$ and $D_j(t)=g_t'(u_j)$. \begin{Lemma} Let $r_1\in (0,L_1)$ and $r_2>0$. Then \BGE \PP[\tau^j_{r_j}<\infty,j=1,2]\lesssim ( {L_1}/({L_0+L_1}) )^{{\alpha_*}} ( {r_1}/{L_1} )^{\alpha} ( {r_2}/{L_2} )^{\alpha}.\label{transform-2}\EDE \label{two-pt-lem-u1u2} \end{Lemma} \begin{Remark} This lemma is another two-point estimate. Its proof is more complicated than that of Lemma \ref{two-pt-lem}. The main reason is that the boundary exponent $\alpha_*$ in Lemma \ref{boundary-estimate} agrees with the exponent $\alpha_*$ for $v_m$ in Lemma \ref{two-pt-lem}, but may be strictly less than the exponent $\alpha$ for $u_1$ in this lemma. The idea of the proof, originated in \cite{LW}, is to exponentially bound the probability that $\eta$ travels back and forth between small discs centered at $u_1$ and $u_2$. \end{Remark} \begin{proof} If $r_2\ge L_2/6$, then we get (\ref{transform-2}) by applying (\ref{est-u}) to bound $\PP[\tau^1_{r_1}<\infty]$. Suppose $r_2<L_2/6$ and $r_1\ge L_\wedge /6$. From $\frac 1{L_1+L_2}\asymp \frac{L_\wedge}{L_1 L_2}$, we get $\frac{r_2}{L_1+L_2} \lesssim \frac{r_1r_2}{L_1L_2}$. There are two cases. Case 1. $L_1\ge L_0$. In this case, since $\frac{L_1}{L_0+L_1}\gtrsim 1$, we get (\ref{transform-2}) by applying (\ref{est-u*}) to bound $\PP[\tau^2_{r_2}<\infty]$. Case 2. $L_1\le L_0$. In this case, $ (\frac{L_1+r_1}{L_0} )^{\alpha_*}\lesssim (\frac{L_1}{L_0+L_1} )^{\alpha_*}$, and we get (\ref{transform-2}) by applying (\ref{two-pt-ineq}) to bound $\PP[\tau^{v_m}_{L_1+r_1}<\infty,\tau^{u_2}_{r_2}<\infty]$. So we always have (\ref{transform-2}) when $r_2<L_2/6$ and $r_1\ge L_\wedge/6$. From now on, we assume that $r_1 <L_\wedge /6$ and $r_2<L_2/6$. Let $L_1'=L_\wedge $ and $L_2'=L_2$. WLOG, we assume $r_j= e^{-N_j}L_j'/2$, $j=1,2$, for some $N_1,N_2\in\N$. With a slight abuse of notation, we write $\tau^j_n$ for $\tau^j_{L_j' e^{-n}/2}$. It suffices to prove \BGE \PP[\tau^j_{N_j}<\infty,j=1,2]\lesssim ( {L_1}/({L_0+L_1}) )^{\alpha_*} ( {L_\wedge}/{L_1} )^{\alpha} e^{-\alpha(N_1+N_2)}.\label{weaker}\EDE By (\ref{est-u}), \BGE \PP[\tau^1_n<\infty]\lesssim ( {L_1}/({L_0+L_1}) )^{\alpha_*} ( {L_\wedge}/{L_1} )^{\alpha} e^{-\alpha n} ;\label{tau1n'}\EDE \BGE \PP[\tau^2_n<\infty]\lesssim ( ({L_1+L_2})/({L_0+L_1+L_2}))^{\alpha_*} e^{-\alpha n}.\label{tau2n'}\EDE By (\ref{interior-estimate-formula}), \BGE \PP[\tau^j_{n_2}<\infty|\F_{\tau^j_{n_1}},\tau^j_{n_1}<\infty]\lesssim e^{-\alpha(n_2-n_1)},\quad n_2>n_1.\label{tau1-tau2}\EDE Note that if $\tau^1_{n_1-1}<\tau^2_{n_2}<\tau^1_{n_1}$, at the time $\tau^2_{n_2}$, $\{|z-u_1|=L_1' e^{1-n_1}/2\}\cap (\HH\sem K_{\tau^2_{n_2}})$ has a connected component $I$, which disconnects $v_m\wedge a_{K_{\tau^2_{n_2}}}$ from $\infty$; and in order for $\eta$ to visit $\{|z-u_1|\le L_1' e^{-n}/2\}$, it must visit $I$ after $\tau^2_{n_2}$. Since any curve in $\HH\sem K_{\tau^2_{n_2}}$ connecting $I$ and $\pa_{\HH}^+ K_{\tau^2_{n_2}}$ must cross $A_{\HH}(u_1,L_\wedge e^{1-n_1}/2,L_2/2)$ and $A_{\HH}(u_2, L_2 e^{-n_2}/2,L_2/2)$, we have $$d_{\HH\sem K_{\tau^2_{n_2}}}(I,\pa_{\HH}^+ K_{\tau^2_{n_2}})\ge \frac 1\pi \ln\Big(\frac{L_2/2}{L_\wedge e^{1-n_1}/2}\Big) +\frac 1\pi \ln\Big(\frac{L_2/2}{L_2 e^{-n_2}/2}\Big).$$ So by (\ref{boundary-estimate-formula}), \BGE \PP[\tau^1_{n_1}<\infty|\F_{\tau^2_{n_2}},\tau^1_{n_1-1}<\tau^2_{n_2}<\tau^1_{n_1}]\lesssim ( {L_\wedge}/{L_2} )^{\alpha_*} e^{-\alpha_* (n_1+n_2)}.\label{tau2-tau1}\EDE Now we prove (\ref{weaker}) using (\ref{tau1n'}-\ref{tau2-tau1}). We break the event $E:=\{\tau^j_{N_j}<\infty,j=1,2\}$ into a disjoint union of events according to the order of those hitting times $\tau^j_{n}$, $n\le N_j$, $j=1,2$. For $j=1,2$, $\tau^j_n$ is increasing in $n$. But the order between any $\tau^1_{n_1}$ and $\tau^2_{n_2}$ is uncertain. For $\ii,\oo\in\{1,2\}$ and $s\in\N$, let $\Xi^{(\ii,\oo)}_s$ be the collection of $(S_1,S_2)$ such that \begin{itemize} \item $S_j\subset \{n\in\N:n\le N_j\}$, $j=1,2$; \item $|S_{\ii}|=s-|\ii-\oo|$ and $|S_{3-\ii}|=s-1$. \end{itemize} For $\ii,\oo\in\{1,2\}$ and $\xi=(S_1,S_2)\in\Xi^{(\ii,\oo)}_s$, we order the elements in $S_j$ as $n^{(j)}_1<\cdots <n^{(j)}_{s_j}$, where $s_j=|S_j|$, $j=1,2$. We also set $n^{(j)}_0=0$ and $n^{(j)}_{s_j+1}=N_j+1$. Define $$E_{(\ii,\oo);\xi}=\Big(\bigcap_{k=1}^{s} \Big \{\tau^{\ii}_{n^{(\ii)}_{k}-1}< \tau^{3-\ii}_{n^{(3-\ii)}_{k-1}}\Big\}\Big)\bigcap\Big( \bigcap_{k=1}^{s-|\ii-\oo|}\Big \{\tau^{3-\ii}_{n^{(3-\ii)}_k-1}<\tau^{\ii}_{n^{(\ii)}_k}\Big\}\Big)\bigcap \{\tau^{\oo}_{N_{\oo}}<\infty\}$$ $$=\{\tau^{\ii}_{n^{(\ii)}_0}\le \tau^{\ii}_{n^{(\ii)}_1-1}<\tau^{3-\ii}_{n^{(3-\ii)}_0}\le \tau^{3-\ii}_{n^{(3-\ii)}_1-1}<\tau^{\ii}_{n^{(\ii)}_1}\le \cdots \le \tau^{3-\oo}_{N_{3-\oo}}<\tau^{\oo}_{n^{\oo}_{s_{\oo}}}\le \tau^{\oo}_{N_{\oo}}<\infty\}. $$ The symbols $\ii$ and $\oo$ respectively stand for ``in'' and ``out'': on the event $E_{(\ii,\oo);\xi}$, $\tau^{\ii}_0$ happens before $\tau^{3-\ii}_0$, and $\tau^{\oo}_{N_{\oo}}$ happens after $\tau^{3-\oo}_{N_{3-\oo}}$. It is straightforward to check that \BGE E=\bigcup_{\ii\in\{1,2\}} \bigcup_{\oo\in\{1,2\}} \bigcup_{s\in\N} \bigcup_{\xi\in \Xi^{(\ii,\oo)}_s} E_{(\ii,\oo);\xi} .\label{unionE}\EDE \begin{comment} \begin{Example} Suppose $N_1=15$, $N_2=12$, $i=1$, $o=1$, $s=5$. Let $S_1=\{3, 6, 9, 10, 13\}$ and $S_2=\{4, 6,7, 12\}$. Then $(S_1,S_2)\in\Xi^{(1,1)}_5$, and \begin{align*} E_{(i,o);(S_1,S_2)}=&\{\tau^1_0<\tau^1_2<\tau^2_0<\tau^2_3<\tau^1_3<\tau^1_5<\tau^2_4<\tau^2_5<\tau^1_6 <\tau^1_8<\tau^2_6\\ &<\tau^1_9<\tau^2_7<\tau^2_{11}<\tau^1_{10}<\tau^1_{12}<\tau^2_{12} <\tau^1_{13}<\tau^1_{15}<\infty.\} \end{align*} \end{Example} \end{comment} Using (\ref{tau1n'}-\ref{tau2-tau1} we find that, for some constant $C\ge 1$, \BGE \PP[E_{\ii,\oo;\xi}]\le \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big(\frac{L_\wedge}{L_1}\Big)^{\alpha} \prod_{j\in\{1,2\}} \prod_{k=1}^{s_j+1} C e^{-\alpha(n^{(j)}_k-n^{(j)}_{k-1} )} \prod_{k=1}^{s_2+|\oo-2|} C e^{-\alpha_*(n^{(2)}_k+n^{(1)}_{k-|\ii-1|} )}\label{PE-pre}\EDE \BGE \le C^{3s+3} \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big(\frac{L_\wedge}{L_1}\Big)^{\alpha} e^{-\alpha N_1} e^{-\alpha N_2} \prod_{n\in S_1} e^{-\alpha_* n} \prod_{n\in S_2} e^{-\alpha_* n}.\label{PE}\EDE To see that (\ref{PE-pre}) holds, consider two cases. Case 1. $\ii=1$. By (\ref{tau1n'},\ref{tau1-tau2},\ref{tau2-tau1}), $$\PP[\tau^1_{n^{(1)}_1}<\infty]\le C\Big (\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big (\frac{L_\wedge }{L_1} \Big)^{\alpha} e^{-n_1^{(1)}\alpha};$$ \BGE \PP[\tau^j_{n^{(j)}_{k}-1}<\infty|\F_{\tau^j_{n^{(j)}_{k-1}}},\tau^j_{n^{(j)}_{k-1}}<\infty]\le C e^{-\alpha(n^{(j)}_k-n^{(j)}_{k-1})}; \label{same1}\EDE \BGE \PP[\tau^1_{n^{(1)}_{k-|\ii-1|}}<\infty | \F_{\tau^2_{n^{(2)}_{k}-1}},\tau^2_{n^{(2)}_{k}-1}<\infty]\le C \Big (\frac {L_\wedge}{L_2}\Big)^{\alpha_*} e^{-\alpha_*(n^{(2)}_k+n^{(1)}_{k-|\ii-1|})}.\label{same2}\EDE Note that the product of upper bounds is bounded by the RHS of (\ref{PE-pre}) since $\frac{L_\wedge}{ L_2}\le 1$. Also note that $s_2+|\oo-2|=s-|\ii-\oo|$. Case 2. $\ii=2$. Then $\PP[\tau^2_{n^{(2)}_1}<\infty]\le C(\frac{L_1+L_2}{L_0+L_1+L_2})^{\alpha_*} (\frac{L_2}{L_1+L_2} )^\alpha e^{-n^{(2)}_1\alpha }$ by \ref{tau2n'}. We still have (\ref{same1},\ref{same2}). The product of these upper bounds is also bounded by the RHS of (\ref{PE-pre}) because (i) $ \frac{L_2}{L_1+L_2} \asymp \frac{L_\wedge}{L_1} $; (ii) there is at least one factor $(\frac{L_\wedge}{L_2})^{\alpha_*}$ coming from (\ref{same2}); (iii) $ \frac{L_1+L_2}{L_0+L_1+L_2} \cdot \frac {L_\wedge}{L_2} \lesssim \frac{L_1}{L_0+L_1} $; and (iv) $s_2+|o-2|=s$. To get (\ref{PE}), we note that $s_1,s_2\in \{s,s-1\}$, $\sum_{k=1}^{s_j+1} (n^{(j)}_k-n^{(j)}_{k-1})=N_j+1$, $j=1,2$, and $n^{(2)}_k$ and $n^{(1)}_{k-|\ii-1|}$, $1\le k\le s_2+|\oo-2|$, exhaust the sets $S_1$ and $S_2$ Combining (\ref{unionE},\ref{PE}), we get (\ref{weaker}) because $$ \sum_{\ii,\oo\in\{1,2\}} \sum_{s=1}^\infty \sum_{\xi=(S_1,S_2)\in \Xi^{(\ii,\oo)}_s} C^{3s+3} \prod_{k=1}^{s_1} e^{-\alpha_* n^{(1)}_k} \prod_{k=1}^{s_2} e^{-\alpha_* n^{(2)}_k}$$ $$\le \sum_{\ii,\oo\in\{1,2\}} \sum_{s=1}^\infty C^{3s+3} \sum_{n^{(1)}_1=1}^\infty e^{-\alpha_* n^{(1)}_1}\cdots \sum_{n^{(1)}_{s_1}=s_1}^\infty e^{-\alpha_* n^{(1)}_{s_1}} \sum_{n^{(2)}_1=1}^\infty e^{-\alpha_* n^{(2)}_1}\cdots \sum_{n^{(2)}_{s_2}=s_2}^\infty e^{-\alpha_* n^{(2)}_{s_2}} $$ $$= \sum_{\ii,\oo\in\{1,2\}} \sum_{s=1}^\infty C^{3s+3} \prod_{k=1}^{s_1} \frac{e^{-k\alpha_*}}{1-e^{-\alpha_*}} \prod_{k=1}^{s_2} \frac{e^{-k\alpha_*}}{1-e^{-\alpha_*}}\le \sum_{\ii,\oo\in\{1,2\}} \sum_{s=1}^\infty \frac{C^{3s+3} e^{-\alpha_*(s_1^2+s_2^2)/2} }{(1-e^{-\alpha_*})^{s_1+s_2}}=: C_*.$$ Here in the second line we used the fact that $n^{(j)}_k\ge k$. We have $C_*<\infty$ because $s_1$ and $s_2$ are determined by $s_{\ii}=s-|\ii-\oo|$ and $s_{3-\ii}=s-1$. \end{proof} The following lemma improves the two-point estimate in the case that $\eta$ first visits a small disc centered at $u_2$ and then visits a small disc centered at $u_1$. \begin{Lemma} For $r_1\in(0,L_\wedge /6)$, $R_2>r_2>0$, we have \BGE \PP[\tau^2_{R_2}<\tau^1_{r_1;\FF}<\infty]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big(\frac{r_1}{L_1}\Big)^{\alpha} \Big(\frac{R_2}{L_2}\Big)^{\alpha+\alpha_*};\label{21-ineq}\EDE \BGE \PP[\tau^2_{R_2}<\tau^1_{r_1;\FF}<\tau^2_{r_2}<\infty]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big(\frac{r_1}{L_1}\Big)^{\alpha} \Big(\frac{r_2}{L_2}\Big)^{\alpha} \Big(\frac{R_2}{L_2}\Big)^{\alpha_*}.\label{212-ineq}\EDE \label{two-pt-cor} \end{Lemma} \begin{proof} If $R_2\ge L_2/6$, these formulas clearly follow from (\ref{est-u},\ref{transform-2}). Suppose now $R_2< L_2/6$. If $r_2\ge R_2/6$, then (\ref{212-ineq}) follows from (\ref{21-ineq}). Suppose now $r_2< R_2/6$. Let $E_{r_1}$ and $E_{r_1,r_2}$ be respectively the events in (\ref{21-ineq}) and (\ref{212-ineq}), and let $\tau=\tau^2_{R_2}$. Let $E_0=\{\tau<\tau^1_{L_\wedge/3}\}$ and $E_n=\{\tau^1_{e^{1-n} L_\wedge /3}<\tau<\tau^1_{e^{-n}L_\wedge/3 }\}$, $n\in\N$. By (\ref{est-u},\ref{transform-2}), \BGE \PP[E_0]\lesssim \Big(\frac{L_1+L_2}{L_0+L_1+L_2}\Big)^{\alpha_*} \Big(\frac{R_2}{L_1+L_2}\Big)^{\alpha}\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big(\frac{L_2}{L_\wedge}\Big)^{\alpha_*} \Big(\frac{L_\wedge }{L_1}\Big)^{\alpha} \Big(\frac{R_2}{L_2}\Big)^{\alpha} ;\label{tauR20}\EDE \BGE \PP[E_n]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big(\frac{e^{-n}L_\wedge}{L_1}\Big)^{\alpha} \Big(\frac{R_2}{L_2}\Big)^{\alpha},\quad n\ge 1. \label{tauR2n}\EDE Here we used that $ {L_1L_2}\asymp (L_1+L_2)L_\wedge$ in the second inequality of (\ref{tauR20}). By DMP, $\PP[E_{r_1}|\F_\tau,\tau<\infty]$ equals the probability that an SLE$_\kappa(\ulin\rho)$ curve $\eta^\tau$ in $\HH$ started from $W(\tau)$ with force points $\ulin V(\tau)$ visits $g_\tau(B_{\FF}(u_1,r_1))$, and $\PP[E_{r_1,r_2}|\F_\tau,\tau<\infty]$ equals the probability that such $\eta^\tau$ visits $g_\tau(B_{\FF}(u_1,r_1))$ and $g_\tau(B_{\C}(u_2,r_2))$ in order. Let $L_0'=|W(\tau)-V_m(\tau)|$, $L_1'=|V_m(\tau)-U_1(\tau)|$, $L_2'=|U_1(\tau)-U_2(\tau)|$, and $r_j'= \rad_{U_2(\tau)}(g_\tau(B_{\C}(u_j,r_j)))$, $j=1,2$. Since $r_2\le R_2/6$, by Koebe's $1/4$ theorem and distortion theorem, $L_2'\ge D_2(\tau) R_2/4$ and $r_2'\le \frac {36}{25} D_2(\tau) r_2$, which imply \BGE r_2'/L_2'\le 6(r_2/R_2)<1.\label{L2'r2'}\EDE Suppose some $E_n$, $n\ge 0$, happens. Consider two cases. Case 1. $r_1< e^{-n} L_\wedge /6$. By Koebe's $1/4$ theorem and distortion theorem, $L_1'\ge D_1(\tau) e^{-n} L_\wedge /4$, $r_1'\le \frac {36}{25} D_1(\tau) r_1$, and so \BGE r_1'/L_1'\le 6 r_1/(e^{-n}L_\wedge)<1 .\label{L1'r1'0}\EDE Note that any curve in $\HH\sem K_\tau$ connecting $[u_1,v_m\wedge a_{K_\tau}]$ and $\pa_{\HH}^+ K_\tau$ crosses $A_{\HH}(u_2,R_2,L_2/2)$, and when $e^{-n}L_\wedge<L_2/2$, also crosses $ A_{\HH}(u_1,e^{-n}L_\wedge,L_2/2)$. So by Proposition \ref{extrem-prop} and the comparison principle and conformal invariance of extremal length, \BGE {L_1'}/({L_0'+L_1'})\lesssim ({e^{-n}L_\wedge} /{L_2}) \cdot ({R_2} /{L_2}) .\label{L01/L1}\EDE Combining (\ref{L2'r2'},\ref{L1'r1'0},\ref{L01/L1}) with (\ref{est-u},\ref{transform-2}) we get \BGE \EE[E_{r_1}|\F_\tau,E_n]\lesssim \Big(\frac{e^{-n}L_\wedge} {L_2}\Big)^{\alpha_*} \Big(\frac{R_2} {L_2}\Big)^{\alpha_*} \Big(\frac{r_1}{e^{-n} L_\wedge }\Big)^\alpha;\label{condition-Er1}\EDE \BGE \EE[E_{r_1,r_2}|\F_\tau,E_n]\lesssim \Big(\frac{e^{-n}L_\wedge} {L_2}\Big)^{\alpha_*} \Big(\frac{R_2} {L_2}\Big)^{\alpha_*} \Big(\frac{r_1}{e^{-n} L_\wedge }\Big)^\alpha\Big(\frac{r_2}{R_2}\Big)^\alpha. \label{condition-Er1r2}\EDE Combining these inequalities with (\ref{tauR20},\ref{tauR2n}) we get that, for all $n\ge 0$, \BGE \PP[E_{r_1}\cap E_n] \lesssim e^{-n\alpha_*} \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big(\frac{r_1}{L_1}\Big)^{\alpha} \Big(\frac{R_2}{L_2}\Big)^{\alpha+\alpha_*} ;\label{Er1En}\EDE \BGE \PP[E_{r_1,r_2}\cap E_n] \lesssim e^{-n\alpha_*} \Big(\frac{L_1}{L_0+L_1}\Big)^{\alpha_*} \Big(\frac{r_1}{L_1}\Big)^{\alpha} \Big(\frac{r_2}{L_2}\Big)^{\alpha} \Big(\frac{R_2}{L_2}\Big)^{\alpha_*} .\label{Er1r2En}\EDE Case 2. $r_1\ge e^{-n} L_\wedge /6$. Then $n\ge 1$ since it is assumed that $r_1< L_\wedge /6$. Suppose $E_n\cap \{\tau<\tau^1_{r_1;\FF}\}$ happens. Using Proposition \ref{extrem-prop} and two separation semi-annuli $A_{\HH}(u_1, e^{1-n} L_\wedge/3, L_2/2)$ and $A_{\HH}(u_2,R_2,L_2/2)$, we find that \BGE 1\wedge \frac{\rad_{V_m(\tau)}(g_\tau(B_{\FF}(u_1,r_1)\cap\lin\HH))}{L_0'}\lesssim \frac{ e^{-n}L_\wedge}{L_2}\cdot \frac{R_2}{L_2 .\label{1wedge vm}\EDE Combining (\ref{1wedge vm}) with (\ref{est-vm}), we find that $ \EE[E_{r_1}|\F_\tau,E_n]\lesssim e^{-n \alpha_*}(\frac{L_\wedge}{L_2})^{\alpha_*} (\frac{R_2} {L_2} )^{\alpha_*} $, which together with (\ref{tauR2n}) and that $e^{-n}L_\wedge \le 6r_1$ implies (\ref{Er1En}). Similarly, combining (\ref{1wedge vm}) with (\ref{L2'r2'}) and (\ref{two-pt-ineq}), we find that $ \EE[E_{r_1,r_2}|\F_\tau,E_n]\lesssim e^{-n \alpha_*} (\frac{R_2}{L_2})^{\alpha_*}(\frac{r_2}{R_2})^{\alpha}$, which together with (\ref{tauR2n}) and that $e^{-n}L_\wedge \le 6 r_1$ implies (\ref{Er1r2En}). So (\ref{Er1En}) and (\ref{Er1r2En}) hold in both cases. Finally, summing up (\ref{Er1En}) and (\ref{Er1r2En}) over $n\in\N\cup\{0\}$, we get (\ref{21-ineq}) and (\ref{212-ineq}). \end{proof} Combining Lemma \ref{two-pt-cor} with Corollary \ref{Cor-lower-1pt}, we get the following corollary. \begin{Corollary} There is a constant $c\in(0,1)$ such that when \BGE \frac{r_1}{L_\wedge}\le c \Big(\frac{L_1}{L_0+L_1}\Big)^{\frac{\alpha^*-\alpha_*}\beta}\quad \mbox{and}\quad \frac{r_2}{L_2}\le c \Big(\frac{L_1}{L_0+L_1}\Big)^{\frac{\alpha^*-\alpha_*}{\alpha+\alpha_*}}, \label{2pt-condition}\EDE we have \BGE \PP[\tau^j_{r_j;\FF}<\infty,j=1,2]\gtrsim G(w,\ulin v;u_1) L_2^{-\alpha} r_1^\alpha r_2^\alpha.\label{lower-prob}\EDE \label{Cor-lower-2pt} \end{Corollary} \begin{proof} By Corollary \ref{Cor-lower-1pt} and (\ref{21-ineq},\ref{G-upper}), there is a constant $c_1\in(0,1)$ such that when (\ref{2pt-condition}) holds with $c_1$ in place of $c$, we have \BGE\PP [\tau^1_{r_1;\FF}<\tau^2_{r_2}] \gtrsim G(w,\ulin v;u_1) r_1^\alpha.\label{lower-prob-tau}\EDE Suppose that $\tau^1_{r_1;\FF}<\tau^2_{r_2}$ and (\ref{2pt-condition}) holds with $c_1$ in place of $c$. If $\eta(\tau^1_{r_1;\FF})\not\in\R$, the union of $[v_m\wedge a_{K_{\tau^1_{r_1;\FF}}},a_{K_{\tau^1_{r_1;\FF}}}]$ and the boundary arc of $\pa K_{\tau^1_{r_1;\FF}}$ from $a_{K_{\tau^1_{r_1;\FF}}}$ to $\eta(\tau^1_{r_1;\FF})$ can be disconnected from $(-\infty,u_2]$ in $\HH\sem K_{\tau^1_{r_1;\FF}}$ by $A_{\HH}(u_1,r_1,L_2)$. By Proposition \ref{extrem-prop} and conformal invariance and comparison principle of extremal length, if $\frac{r_1}{L_2}<\frac 1{288}$, then $\frac{|W(\tau^1_{r_1;\FF})-V_m(\tau^1_{r_1;\FF})|}{|W(\tau^1_{r_1;\FF})-U_2(\tau^1_{r_1;\FF})|}\le 144 \frac{r_1}{L_2}< \frac 12$. If $\eta(\tau^1_{r_1;\FF})\in\R$, then from $r_1<L_1=|v_m-u_1|$ we see that $\eta(\tau^1_{r_1;\FF})<v_m$, which implies $W(\tau^1_{r_1;\FF})=V_m(\tau^1_{r_1;\FF})$. So in any case we have \BGE \frac{|V_m(\tau^1_{r_1;\FF})-U_2(\tau^1_{r_1;\FF})|}{|W(\tau^1_{r_1;\FF})-U_2(\tau^1_{r_1;\FF})|}=1- \frac{|W(\tau^1_{r_1;\FF})-V_m(\tau^1_{r_1;\FF})|}{|W(\tau^1_{r_1;\FF})-U_2(\tau^1_{r_1;\FF})|} > \frac 12. \label{12}\EDE By Koebe $1/4$ theorem, on the event $\{\tau^1_{r_1;\FF}<\tau^2_{r_2}\}$, \BGE g_{\tau^1_{r_1;\FF}}(B_{\FF}(u_2,r_2))\supset B_{\FF}(U_2(\tau^1_{r_1;\FF}),D_2(\tau^1_{r_1;\FF}) r_2/4) ;\label{contains}\EDE \BGE |V_m(\tau^1_{r_1;\FF})-U_2(\tau^1_{r_1;\FF})|\le D_2(\tau^1_{r_1;\FF}) \dist(u_2,\eta[0,\tau^1_{r_1;\FF}]\cup [v_m,\infty))\le D_2(\tau^1_{r_1;\FF}) L_2 .\label{Vm-U}\EDE By DMP, Corollary \ref{Cor-lower-1pt}, and (\ref{G-upper},\ref{12},\ref{contains},\ref{Vm-U}), there is a constant $c_2\in(0,1)$ such that when $\frac{r_1}{L_2}<\frac 1{288}$ and $\frac{r_2}{L_2}<c_2$, \BGE \PP[\tau^2_{r_2;\FF}<\infty|\F_{\tau^1_{r_1;\FF}},\tau^1_{r_1;\FF}<\tau^2_{r_2}]\gtrsim ( {r_2}/{L_2})^\alpha.\label{tau2r2}\EDE Thus, if (\ref{2pt-condition}) holds for $c:=c_1\wedge c_2\wedge \frac 1{288}$, then (\ref{lower-prob-tau}) and (\ref{tau2r2}) both hold, which together imply (\ref{lower-prob}). \end{proof} For $0\le t<\tau^*_u$ and $r>0$, let $I_t(u,r)$ denote the connected component of $\{|z-u|=r\}\cap (\HH\sem K_t)$ which disconnects $u$ from $\infty$. Note that one endpoint of $I_t(u,r)$ is $u-r$. \begin{Lemma} Let $S_1>R_1>r_1\in (0, L_\wedge /6)$ and $r_2\in(0,L_2/2)$. Then \BGE \PP[\tau^1_{r_1;\FF},\tau^2_{r_2}<\infty;\eta[\tau^1_{R_1;\FF},\tau^1_{r_1;\FF}]\cap I_{\tau^1_{R_1;\FF}}(u_1,S_1)\ne \emptyset]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}}\Big (\frac{R_1}{S_1}\Big)^{{\alpha_*}} \Big(\frac{r_1r_2}{L_1L_2}\Big)^{\alpha} ;\label{E-P}\EDE \BGE \PP[\tau^1_{r_1;\FF}<\infty;\eta[\tau^1_{R_1;\FF},\tau^1_{r_1;\FF}]\cap I_{\tau^1_{R_1;\FF}}(u_1,S_1)\ne \emptyset]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}}\Big (\frac{R_1}{S_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^{\alpha} ;\label{E-P'}\EDE \label{comeout-2} \end{Lemma} \begin{proof} We only prove (\ref{E-P}). The proof of (\ref{E-P'}) is simpler. Let $E$ denote the event in (\ref{E-P}). Let $\tau_*=\inf(\{t\ge \tau^1_{R_1;\FF}:\eta(t)\in {I_{\tau^1_{R_1;\FF}}(u_1,S_1)}\}\cup\{\infty\})$. Then $E=\{\tau_*<\tau^1_{r_1;\FF}<\infty;\tau^2_{r_2}<\infty\}$. We may assume $r_2<L_2/6$ for otherwise (\ref{E-P}) reduces to (\ref{E-P'}). Let $R_2=L_2/2$. Let $N_1,N_2\in\N$ be such that $e^{-N_j} R_j\le r_j<e^{1-N_j} R_j$, $j=1,2$. Define $$ E^1_{n_1}=\{\tau^1_{e^{1-n_1} R_1;\FF}\le \tau_*<\tau^1_{e^{-n_1} R_1;\FF}\},\quad 1\le n_1\le N_1;$$ $$E^2_{n_2}=\{\tau^2_{e^{1-n_2} R_2}< \tau^1_{R_1;\FF}<\tau^2_{e^{-n_2}R_2}\},\quad 1\le n_2\le N_2;$$ $$E^2_0=\{\tau^1_{R_1;\FF}<\tau^2_{R_2}\},\quad E^2_{N_2+1}=\{\tau^2_{e^{-N_2} R_2}< \tau^1_{R_1;\FF}<\infty\}.$$ When $E$ happens, we have $\tau^1_{R_1;\FF}<\infty$ and $\tau^1_{R_1;\FF}<\tau_*<\tau^1_{r_1;\FF}\le \tau^1_{e^{-N_1}R_1;\FF}$ because $r_1\ge e^{-N_1} R_1$. So we have \BGE E\subset\bigcup_{n_1=1}^{N_1} \bigcup_{n_2=0}^{N_2+1} (E^1_{n_1}\cap E^2_{n_2}).\label{E-union}\EDE Note that all $E^1_{n_1}\cap E^2_{n_2}\in\F_{\tau_*}$. By (\ref{est-u},\ref{21-ineq}), for all $n_1,n_2$, \BGE \PP[E^1_{n_1}\cap E^2_{n_2}] \lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}}\Big(\frac{e^{1-n_1}R_1}{L_1}\Big)^\alpha \Big(\frac{ e^{1-n_2}R_2 }{L_2}\Big)^{\alpha+{\alpha_*}} .\label{n2>0}\EDE Suppose for some $1\le n_1\le N_1$ and $0\le n_2\le N_2$, $E^1_{n_1}\cap E^2_{n_2}$ happens. Let $I= I_{\tau_*}(u_1,R_1 e^{1-n_1})$. Then $I$ is a crosscut of $\HH\sem K_{\tau_*}$, which disconnects $[u_1,v_m\wedge a_{K_{\tau_*}}]\cup (B_{\FF}(u_1,r_1)\cap \lin\HH)$ from $\infty$, and can be disconnected from $\pa_{\HH}^+ K_{\tau_*}$ in $\HH\sem K_{\tau_*}$ by $A_{\HH}(u, R_1 e^{1-n_1},S_1)$. By Proposition \ref{extrem-prop} and conformal invariance and comparison principle of extremal length, \BGE 1\wedge \frac{\rad_{U_1(\tau_*)}([U_1(\tau_*),V_m(\tau_*)]\cup g_{\tau_*}(B_{\FF}(u_1,r_1)\cap \lin\HH))}{|W(\tau_*)-U_1({\tau_*})|}\lesssim \frac{R_1 e^{1-n_1}}{S_1}.\label{extrem-est-WV}\EDE By Koebe $1/4$ theorem, $$|V_m(\tau_*)-U_1({\tau_*})|\ge D_1({\tau_*}) R_1 e^{-n_1}/4,\quad |U_1(\tau_*)-U_2({\tau_*})|\ge D_2({\tau_*}) R_2 e^{-n_2}/4.$$ By Koebe distortion theorem, for $j=1,2$, if $R_je^{-n_j}>6r_j$, then $$\rad_{U_j({\tau_*})} (g_{\tau_*}(B_{\C}(u_j,r_j)))\le \frac 32 D_j({\tau_*}) r_j <|V_m(\tau_*)-U_j({\tau_*})|.$$ Combining these three estimates with DMP and (\ref{transform-2}), we see that, for $1\le n_1\le N_1$ and $1\le n_2\le N_2$, when $R_j e^{-n_j}>6r_j$, $j=1,2$, \BGE \PP [E|\F_{\tau_*},E^1_{n_1}\cap E^2_{n_2}]\lesssim \Big(\frac{R_1 e^{1-n_1}}{S_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{R_1 e^{-n_1}}\Big)^\alpha \Big(\frac{r_2}{R_2 e^{-n_2}}\Big)^\alpha.\label{two-pt-bdry-temp1}\EDE This estimate also holds without assuming $R_j e^{-n_j}>6r_j$, $j=1,2$. In fact, if $R_j e^{-n_j}>6r_j$ holds for only one of $j\in\{1,2\}$, then we get (\ref{two-pt-bdry-temp1}) using (\ref{est-u}) instead of (\ref{transform-2}). If $R_j e^{-n_j}\le 6r_j$ for $j\in\{1,2\}$, then (\ref{two-pt-bdry-temp1}) follows from (\ref{est-vm},\ref{extrem-est-WV}). A similar argument shows that \BGE \EE[E|\F_{\tau_*},E^1_{n_1}\cap E^2_{N_2+1} ]\le \EE[\tau^1_{r_1;\FF}<\infty |\F_{\tau_*},E^1_{n_1}\cap E^2_{N_2+1} ]\lesssim \Big(\frac{R_1 e^{1-n_1}}{S_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{R_1 e^{-n_1}}\Big)^\alpha.\label{two-pt-bdry-temp2-N}\EDE Here when $R_1 e^{-n_1}>6r_1$ we use (\ref{est-u}); and when $R_1 e^{-n_1}\le 6 r_1$, we use (\ref{est-vm}). Combining (\ref{n2>0}) with (\ref{two-pt-bdry-temp1},\ref{two-pt-bdry-temp2-N}), we see that, for all $n_1,n_2$, $$\PP[E\cap E^1_{n_1}\cap E^2_{n_2}]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}} \Big(\frac{R_1 }{S_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^\alpha \Big(\frac{r_2}{L_2}\Big)^\alpha e^{-(n_1+n_2){\alpha_*}}.$$ Summing up these inequalities and using (\ref{E-union}), we get the desired upper bound of $\PP[E]$. \end{proof} \begin{comment} Using a similar and simpler argument, we get the following lemma. \begin{Lemma} Let $S_1>R_1>r_1\in (0, L_\wedge /2)$. Let $E$ denote the event that $\tau^{1}_{r_1}<\infty$, and $\eta[\tau^1_{R_1},\tau^1_{r_1}]$ hits ${I_{\tau^1_{R_1}}(u_1,S_1)}$. Then $$\PP[E]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}}\Big (\frac{R_1}{S_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^{\alpha} .$$ \label{comeout} \end{Lemma} \end{comment} \begin{proof}[Proof of Theorem \ref{main-thm2}] Suppose $r_1\in(0,L_\wedge/6 )$ and $r_2\in (0,L_2 /6)$. Choose $S_1>R_1\in (r_1, L_\wedge/6 )$ and $R_2\in (r_2,L_2/6 )$, whose values are to be determined later. Recall the $I_t(u,r)$ defined before Lemma \ref{comeout-2}. Let $\ha\tau^1_{R_1,S_1}=\inf(\{t\ge \tau^1_{R_1;\FF}:\eta(t)\in I_{\tau^1_{R_1}}(u_1, S_1)\}\cup\{\infty\})$. Define the events: $$ E=\{\tau^j_{r_j;\FF}<\infty,j=1,2\}; \quad \ha E_1=\{\tau^1_{r_1;\FF}<\ha \tau^1_{R_1,S_1}\};\quad E_{ 2}^{x}=\{\tau^1_{x;\FF}<\tau^2_{R_2}\},\quad x\in\{r_1,R_1\}.$$ In the following, we use $X\st{e}{\approx}Y$ to denote the approximation relation $|X-Y|=e$, and call $e$ the error term. We are going to use the following approximation relations: \BGE \begin{aligned} &\PP[E] \st{e_1}{\approx} \PP[E\cap \ha E_1]\st{e_2}{\approx} \PP[E\cap \ha E_1\cap E_2^{r_1}] =\EE[{\bf 1}_{\ha E_{1}\cap E_2^{r_1}} \PP[\tau^2_{r_2;\FF}<\infty|\F_{\tau^1_{r_1;\FF}},E_2^{r_1}]]\\ \st{e_3}{\approx}& \EE[{\bf 1}_{ \ha E_{1}\cap E_2^{r_1}} G_{\tau^1_{r_1;\FF};\FF}(u_2) ]r_2^\alpha \st{e_4}{\approx} \EE[{\bf 1}_{\ha E_{1}\cap E_2^{r_1}} G_{\tau^1_{R_1;\FF};\FF}(u_2) ]r_2^\alpha\\ \st{e_5}{\approx}& \EE[{\bf 1}_{\ha E_{1}\cap E_2^{R_1} } G_{\tau^1_{R_1;\FF};\FF}(u_2)] r_2^\alpha\st{e_6}{\approx} \EE[{\bf 1}_{\{\tau^1_{r_1;\FF}<\infty\}\cap E_2^{R_1}} G_{\tau^1_{R_1;\FF};\FF}(u_2)] r_2^\alpha\\ =&\EE[{\bf 1}_{E_2^{R_1}} G _{\tau^1_{R_1;\FF}}(u_2) \PP[\tau^1_{r_1;\FF}<\infty|\F_{\tau^1_{R_1;\FF}}]] r_2^\alpha \st{e_7}{\approx} \EE[{\bf 1}_{E_2^{R_1}} G _{\tau^1_{R_1;\FF}}(u_1) G _{\tau^1_{R_1;\FF}}(u_2) ] r_1^\alpha r_2^\alpha. \end{aligned} \label{approximation} \EDE By (\ref{est-u}), \BGE \PP[E_2^{x}]\le \PP[\tau^1_{x;\FF}<\infty] \lesssim ( {L_1}/({L_0+L_1}) )^{{\alpha_*}} ( {x}/{L_1} )^{\alpha},\quad x\in\{r_1,R_1\}.\label{Ex-upper}\EDE By Lemma \ref{Cor-main-thm1}, there is a constant $c_1\in(0,1)$ such that if $r_1/R_1<c_1$, \BGE |\PP[\tau^1_{r_1;\FF}<\infty|\F_{\tau^1_{R_1}},\tau^1_{R_1}<\infty]- G_{\tau^1_{R_1};\FF}(u_1)r_1^\alpha|\lesssim (r_1/R_1)^{\alpha+\beta};\label{G1r1}\EDE and if $r_2/R_2<c_1$, then for $x=r_1$ or $R_1$, \BGE |\PP[\tau^2_{r_2;\FF}<\infty|\F_{\tau^1_{x}},E_2^x]- G_{\tau^1_{x};\FF}(u_2)r_2^\alpha|\lesssim (r_2/R_2)^{\alpha+\beta}, \label{G2r2}\EDE and so \BGE G_{\tau^1_{x};\FF}(u_2)r_2^\alpha \lesssim \PP[\tau^2_{r_2;\FF}<\infty|\F_{\tau^1_{x}} ]+ (r_2/R_2)^{\alpha+\beta}\quad \mbox{on }E_2^{x}.\label{G<}\EDE We now bound the error terms in (\ref{approximation}). By (\ref{E-P}), $$e_1=\PP[E\sem \ha E_1]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}}\Big (\frac{R_1}{S_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^{\alpha} \Big(\frac{r_2}{L_2}\Big)^{\alpha}.$$ Since $E\sem E_2^{r_1}=\{\tau^2_{R_2}<\tau^1_{r_1}<\tau^2_{r_2}<\infty\}\cup \{\tau^2_{r_2}<\tau^1_{r_1}<\infty\}$, by (\ref{21-ineq},\ref{212-ineq}), $$e_2\le \PP[E\sem E_{2}^{r_1}]\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}} \Big(\frac{R_2}{L_2}\Big)^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^\alpha \Big(\frac{r_2}{L_2}\Big)^\alpha.$$ Combining (\ref{Ex-upper}) and (\ref{G2r2}) with $x=r_1$, we find that, when $r_2/R_2<c_1$, $$e_3\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^\alpha \Big(\frac{r_2}{R_2}\Big)^{\alpha+\beta}.$$ By (\ref{21-ineq}), $\PP[\ha E_1\cap( E_2^{R_1} \sem E_2^{r_1})]\le \PP[\tau^2_{R_2}<\tau^1_{r_1;\FF}<\infty] \lesssim (\frac{L_1}{L_0+L_1} )^{{\alpha_*}} (\frac{r_1}{L_1} )^\alpha (\frac{R_2}{L_2} )^{\alpha+{\alpha_*}}$. By (\ref{G-upper}) and Koebe's $1/4$ theorem, ${\bf 1}_{E_2^{R_1}}G_{\tau^1_{R_1;\FF}}(u_2)\lesssim R_2^{-\alpha}$. These estimates together imply that $$e_5\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}} \Big(\frac{R_2}{L_2}\Big)^{ {\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^\alpha \Big(\frac{r_2}{L_2}\Big)^{\alpha}.$$ By (\ref{G<},\ref{E-P},\ref{E-P'}), when $r_2/R_2<c_1$, $$e_6\lesssim \EE[{\bf 1}_{E_2^{R_1}\cap (\{\tau^1_{r_1;\FF}<\infty\}\sem \ha E_1)} (\PP[\tau^2_{r_2;\FF}<\infty|\F_{\tau^1_{R_1}}]+(r_2/R_2)^{\alpha+\beta})]$$ $$\le \PP[(\{\tau^1_{r_1;\FF}<\infty\}\sem \ha E_1)\cap \{\tau^2_{r_2;\FF}<\infty\}]+(r_2/R_2)^{\alpha+\beta} \PP[ \{\tau^1_{r_1;\FF}<\infty\}\sem \ha E_1]$$ $$\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}}\Big (\frac{R_1}{S_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^{\alpha}\Big[ \Big(\frac{r_2}{L_2}\Big)^{\alpha}+ \Big(\frac{r_2}{R_2}\Big)^{\alpha+\beta} \Big] .$$ By (\ref{G1r1},\ref{G<}), Koebe $1/4$ theorem, and (\ref{est-u},\ref{transform-2}), when $r_j/R_j<c_1$, $j=1,2$, $$e_7\lesssim (r_1/R_1)^{\alpha+\beta}\EE[{\bf 1}_{E_2^{R_1}} \PP[\tau^2_{r_2;\FF}<\infty|\F_{\tau^1_{R_1}}]+(r_2/R_2)^{\alpha+\beta}] $$ $$\le (r_1/R_1)^{\alpha+\beta}( \PP[\tau^1_{R_1;\FF}<\infty,\tau^2_{r_2;\FF}<\infty]+(r_2/R_2)^{\alpha+\beta} \PP[\tau^1_{R_1;\FF}<\infty])$$ $$\lesssim \Big(\frac{L_1}{L_0+L_1}\Big)^{{\alpha_*}} \Big(\frac{r_1}{R_1}\Big)^\beta \Big(\frac{r_1}{L_1}\Big)^\alpha\Big[\Big(\frac{r_2}{L_2}\Big)^\alpha+\Big(\frac{r_2}{R_2} \Big)^{\alpha+\beta}\Big].$$ When $\tau^1_{R_1}<\infty$, $I_{\tau^1_{R_1}}(u_1, S_1)$ can be disconnected in $\HH\sem K_{\tau^1_{R_1}}$ from $(-\infty,u_2]$ by $ A_{\HH}(u,S_1,L_2)$. So $d_{\HH\sem K_{\tau^1_{R_1}}}(I_{\tau^1_{R_1}}(u_1, S_1), (-\infty,u_2])\ge \frac 1\pi \ln(\frac{L_2}{S_1})$. On the event $\ha E_1$, by Lemma \ref{Gt1t2}, if $L_2/S_1> 1440$, $|G_{\tau^1_{R_1};\FF}(u_2)-G_{\tau^1_{r_1};\FF}(u_2)|\lesssim \frac{S_1}{L_2} G_{\tau^1_{r_1};\FF}(u_2) $. Combining this with (\ref{G<},\ref{est-u},\ref{transform-2}), we get, when $S_1/L_2<1/ 1440$ and $r_2/R_2<c_{1}$, $$e_4 \lesssim ({S_1}/{L_2}) ( \EE[{\bf 1}_{ E_2^{r_1}} \PP[\tau^2_{r_2;\FF}<\infty|\F_{\tau^1_{r_1}}] ]+ ( {r_2}/{R_2})^{\alpha+\beta} \PP[ E_2^{r_1}])$$ $$\le ({S_1}/{L_2}) ( \PP[\tau^j_{r_j;\FF}<\infty,j=1,2]+ ( {r_2}/{R_2} )^{\alpha+\beta} \PP[\tau^1_{r_1;\FF}<\infty] )$$ $$\lesssim \frac{S_1}{L_2} \Big(\frac{L_1}{L_0+L_1}\Big )^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^\alpha \Big[ \Big(\frac{r_2}{L_2}\Big)^\alpha+\Big(\frac{r_2}{R_2}\Big)^{\alpha+\beta}\Big].$$ Now we determine $R_1,S_1,R_2$. Let $\ha r_1\in [r_1,L_\wedge/6)$ and $\ha r_2\in [r_2,L_2/6)$. Define $$R_1=(L_\wedge /6)^{\frac{\alpha_*}{\alpha_*\beta+{\alpha_*}+\beta}} \cdot \ha r_1^{\frac{\alpha_*\beta+\beta}{\alpha_*\beta+{\alpha_*}+\beta}}, \quad S_1=(L_\wedge /6)^{\frac{ {\alpha_*}+\beta }{\alpha_*\beta+{\alpha_*}+\beta}}\cdot \ha r_1^{\frac{\alpha_*\beta }{\alpha_*\beta +{\alpha_*}+\beta}},$$ $$R_2=(L_2/6)^{\frac{\alpha+{\alpha_*}}{\alpha+{\alpha_*}+\beta}} \cdot \ha r_2^{\frac{\beta}{\alpha+{\alpha_*}+\beta}}.$$ Then $ r_1\le \ha r_1<R_1<S_2<L_\wedge /6$ and $r_2\le \ha r_2<R_2<L_2/6$. \begin{comment} Moreover, \BGE \frac{r_1}{R_1}\le \frac{\ha r_1}{R_1}\asymp \Big(\frac{\ha r_1}{L_\wedge}\Big)^{\frac{\zeta_1}{\beta}},\quad \frac{r_2}{R_2}\le \frac{\ha r_2}{R_2}\asymp \Big(\frac{\ha r_2}{L_2}\Big)^{\frac{\alpha+\alpha_*}{\alpha+\alpha_*+\beta}},\quad \frac{S_1}{L_2}\le \frac{S_1}{L_\wedge}\asymp \Big(\frac{\ha r_1}{L_\wedge}\Big)^{\zeta_1},\label{r1R1r2R2}\EDE \BGE \frac{R_1}{S_1}\asymp \Big(\frac{\ha r_1}{L_\wedge}\Big)^{\frac{\zeta_1}{\alpha_*}},\quad \frac{R_2}{L_2} \asymp \Big(\frac{\ha r_2}{L_2}\Big)^{\frac{\zeta_2}{\alpha_*}}, \label{r1R1r2R2'}\EDE \BGE \Big(\frac{r_2}{R_2}\Big)^{\alpha+\beta}\le \Big(\frac{r_2}{L_2}\Big)^\alpha \Big(\frac{L_2}{R_2}\Big)^\alpha \Big(\frac{\ha r_2}{R_2}\Big)^\beta\asymp \Big(\frac{r_2}{L_2}\Big)^\alpha \Big(\frac{\ha r_2}{L_2}\Big)^{\zeta_2}. \label{r1R1r2R2''}\EDE \end{comment} Straightforward computation shows that there is a constant $c\in(0,1)$ such that if $ \ha r_1/L_\wedge<c$ and $\ha r_2/L_2<c$, then $r_j/R_j<c_1$, $j=1,2$, and $S_1/L_2<1/1440$, and so the estimates for $e_j$, $1\le j\le 7$, all hold, and we have \begin{comment} . Summing up these estimates and using (\ref{r1R1r2R2},\ref{r1R1r2R2'},\ref{r1R1r2R2''}), we get $$\sum_{j=1}^7 e_j\lesssim \Big(\frac{L_1}{L_0+L_1}\Big )^{{\alpha_*}} \Big(\frac{r_1}{L_1}\Big)^\alpha \Big[\Big(\frac{r_2}{R_2}\Big)^{\alpha+\beta}+\Big(\frac{r_2}{L_2}\Big)^\alpha\Big[\frac{S_1}{L_2}+\Big(\frac{R_1}{S_1}\Big)^{\alpha_*}+ \Big(\frac{r_1}{R_1}\Big)^\beta+\Big(\frac{R_2}{L_2}\Big)^{\alpha_*}\Big] \Big].$$ \end{comment} $$|r_1^{-\alpha}r_2^{-\alpha} \PP[\tau^j_{r_j;\FF}<\infty,j=1,2]-\EE[{\bf 1}_{E_2^{R_1}}G _{\tau^1_{R_1}}(u_1) G _{\tau^1_{R_1}}(u_2) ]| $$ $$\le r_1^{-\alpha} r_2^{-\alpha} \sum_{j=1}^7 e_j\lesssim \Big(\frac{L_1}{L_0+L_1}\Big )^{{\alpha_*}} \Big(\frac{ 1}{L_1}\Big)^\alpha \Big(\frac{1}{L_2}\Big)^\alpha\Big[ \Big(\frac{\ha r_1}{L_\wedge}\Big)^{\zeta_1}+ \Big(\frac{\ha r_2}{L_2}\Big)^{\zeta_2}\Big],$$ where $\zeta_1:=\frac{\alpha_*\beta}{\alpha_*\beta+\alpha_*+\beta}$ and $\zeta_2:=\frac{\alpha_*\beta}{\alpha+\alpha_*+\beta}$. Now we treat $\ha r_1,\ha r_2$ as fixed. Since the RHS of the above inequality and $\EE[{\bf 1}_{E_2^{R_1}}G _{\tau^1_{R_1}}(u_1) G _{\tau^1_{R_1}}(u_2) ]$ do not depend on $r_1,r_2$, we see that the family $r_1^{-\alpha}r_2^{-\alpha} \PP[\tau^j_{r_j;\FF}<\infty,j=1,2]$ is Cauchy as $r_1,r_2\to 0^+$. So $G_{\FF}(w,v ;u_1,u_2)\in [0,\infty)$ could be well defined by (\ref{Green-2pt}), and satisfies (\ref{G(u1,u2)-est}) by the above inequality. Formula (\ref{G(u1,u2)-upper}) follows immediately from (\ref{transform-2}) and (\ref{lower-prob}), which implies that $G_{\FF}(w,v ;u_1,u_2)\in(0,\infty)$. \end{proof} \begin{Remark} With some more work, one can prove that $G_{\FF}(w,\ulin v;u_1,u_2)$ has an expression that resembles the ordered two-interior-point Green's function for chordal SLE$_\kappa$ in \cite{LW}: $$G_{\FF}(w,\ulin v;u_1,u_2)= G_{0;\FF}( u_1) \EE_*[ G_{\tau^*_{u_1};\FF}(u_2)],$$ where $\EE_*$ stands for the expectation w.r.t.\ the SLE$_\kappa(\ulin\rho,\kappa-8-2\rho_\Sigma)$ curve started from $w$ with force points $(\ulin v,u_1)$. Such curve almost surely hits $u_1$ at the time $\tau^*_{u_1}$, and can be understood as the SLE$_\kappa(\ulin\rho)$ curve $\eta$ conditioned to pass through $u_1$. \end{Remark} \begin{comment} \begin{Corollary} The process $M_{(u_1,u_2)}(t)$, $0\le t<\tau^*_{u_1}$, defined by $$M_{(u_1,u_2)}(t)= D_1(t)^\alpha D_2(t)^\alpha G_{\FF}(W(t),\ulin V(t);U_1(t),U_2(t))$$ is a positive local martingale. \label{M-u1-u1} \end{Corollary} \begin{Corollary} Define $M_{(u_1,u_2)}(t)$, $0\le t<\infty$, by $$M_{(u_1,u_2)}(t)={\bf 1}_{\{t<\tau^*_{u_1}\}}D_1(t)^\alpha D_2(t)^\alpha G_{\FF}(W(t),\ulin V(t);U_1(t),U_2(t)).$$ Then for any $\F$-stopping time $\tau$, $$\lim_{r_1,r_2\to 0^+}r_1^{-\alpha} r_2^{-\alpha} \PP[\eta\cap B_{\R}(u_j,r_j)\ne\emptyset,j=1,2|\F_\tau,\tau<\tau^j_{r_j},j=1,2] =M_{(u_1,u_2)}(\tau).$$ Thus, by Fatou lemma, $M_{(u_1,u_2)}$ is a positive supermartingale. \label{M-u1-u1} \end{Corollary} \begin{Corollary} Assume that $\rho_\Sigma\in((-2)\vee (\frac\kappa 2-4),\frac\kappa 2-2)$. For $u\in (-\infty,v_m)$, let $$M_u(t)={\bf 1}_{\{t< \tau^*_u\}} g_t'(u)^\alpha C_{\R} G(W(t),\ulin V(t);g_t(u)),\quad t\ge 0.$$ Let $\lambda$ denote the Lebesgue measure on $\R$. For any bounded measurable set $F\subset (-\infty,v_m)$ with $\dist(v_m,F)>0$, define $$M_F(t)=\int_F M_u(t) \lambda(du),\quad t\ge 0.$$ Then a.s.\ there exists a unique continuously increasing process $\theta_F$ with $\theta_F(0)=0$ such that $M_F+\theta_F$ is a uniformly integrable martingale. \end{Corollary} \begin{proof} By Lemma \ref{M1-lemma}, each $M_u$ is a positive supermartingale. Thus, by (\ref{G-upper}), for any $t\ge 0$, $$\EE[M_F(t)]=\int_F \EE[M_u(t)] \lambda(du)\le \int_F M_u(0) \lambda(du)= \int_F C_{\R} G(w,\ulin v;u) \lambda(du)$$ $$\lesssim \int_F |u-v_m|^{-\alpha} \lambda(du)\le \lambda(F) \dist(v_m,F)^{-\alpha}<\infty.$$ This shows that, for each $t\ge 0$, $M_F(t)$ is a.s.\ finite. So $M_F$ is also a positive supermartingale. Let $u_1>u_2\in(-\infty,v_m)$. From (\ref{G-upper},\ref{G(u1,u2)-upper}) we get $$C_{\R}^2G(w,\ulin v;u_1)G(w,\ulin v;u_2)\lesssim C_{\F} G(w,\ulin v;u_1) |u_1-u_2|^{-\alpha}\lesssim G_{\R}(w,\ulin v;u_1,u_2).$$ Let $M_{(u_1,u_2)}$ be the supermartingale given by Lemma \ref{M-u1-u1} with $\FF=\R$. The above inequality implies that $M_{u_1}(t)M_{u_2}(t)\lesssim M_{(u_1,u_2)}(t)$ for all $t\ge 0$. Let $\Delta_F=\{(u_1,u_2)\in F^2:u_1>u_2\}$. Then for any finite $\F$-stopping time $\tau$, $$\EE[M_{F}(\tau)^2]=\EE\Big[2\int \!\!\int_{\Delta_F} M_{u_1}(\tau) M_{u_2}(\tau) \lambda(du_1) \lambda(du_2)\Big] \lesssim \EE\Big[\int\!\!\int_{\Delta_F} M_{(u_1,u_2)}(\tau) \lambda(du_1) \lambda(du_2)\Big]$$ $$= \int\!\!\int_{\Delta_F}\EE[ M_{(u_1,u_2)}(\tau)] \lambda(du_1) \lambda(du_2) \le \int\!\!\int_{\Delta_F} G_{\R}(w,\ulin v;u_1,u_2) \lambda(du_1) \lambda(du_2)$$ $$\lesssim \dist(F,v_m)^{-\alpha} \int\!\!\int_{\Delta_F} |u_1-u_2|^{-\alpha} \lambda(du_1) \lambda(du_2)<\infty,$$ where we used (\ref{G(u1,u2)-upper}) in the second last inequality, and used $\alpha\in (0,1)$ in the last inequality. This implies that the family $(M_F(\tau))$ is $L^2$-bounded, where $\tau$ ranges over the set of all finite stopping times. So the supermartingale $M_F$ is of class (D). By Doob-Meyer decomposition, there exists an a.s.\ unique increasing right-continuous adapted process $\theta_F$ with $\theta_F(0)=0$ such that $M_F+\theta_F$ is a uniformly integrable martingale. \end{proof} \end{comment} \section{Minkowski Content Measure} \label{Section 6} Let $\eta$ be as in Theorems \ref{main-thm1} and \ref{main-thm2}. The purpose of this section is to use those theorems to construct a covariant Borel measure, called the Minkowski content measure, on $\eta\cap(-\infty,v_m]$, which is closely related to the Minkowski contents of subsets of $\eta\cap(-\infty,v_m]$. \subsection{General theory} \label{Section 6.1} In this subsection, we first review the Minkowski contents, and then define the Minkowski content measures, and derive some basic properties. Let $n\in\N$. Let $\lambda^n$ denote the Lebesgue measure on $\R^n$. Fix $d\in(0,n)$. Let $S\subset\R^n$. For $r>0$, let $B(S,r)=\{x\in\R^n: \dist(x,S)\le r\}$, and $\Cont(S;r)=r^{d-n} \lambda^n(B_{\R}(S,r))$. The upper and lower ($d$-dimensional) Minkowski content of $S$ are respectively defined by $$\lin{\Cont}(S):=\limsup_{r\to 0^+} \Cont(S;r),\quad \ulin{\Cont}(S):=\liminf_{r\to 0^+} \Cont(S;r).$$ When $\lin{\Cont}(S)=\ulin{\Cont}(S)$, the common value, denoted by $\Cont(S)$ or $\Cont_d(S)$, is called the ($d$-dimensional) Minkowski content of $S$. \begin{Remark} The definition here differs from that in \cite[3.2.37]{MC} in two aspects. First, we allow $d$ to be not an integer; second, we omit a multiplicative constant: $\frac{\Gamma(\frac{n-d}2+1)}{\pi^{\frac{n-d}2}}$ for simplicity. \end{Remark} For $S_1,S_2\subset \R^n$, from $\lambda^n(B(S_1\cup S_2,r))\le \lambda^n (B(S_1,r))+\lambda^n(B(S_2,r))$, we get \BGE \lin{\Cont}(S_1\cup S_2)\le \lin{\Cont} (S_1)+\lin{\Cont} (S_2).\label{Cont-union-upper-n}\EDE \BGE \ulin{\Cont}(S_1\cup S_2)\le \ulin{\Cont}(S_1)+\lin{\Cont}(S_2).\label{Cont-union-lower-n}\EDE \begin{Lemma} For any compact sets $S_1,S_2\subset\R^n$, \BGE \ulin\Cont(S_1)+\ulin\Cont(S_2)\le \ulin\Cont(S_1\cup S_2)+\lim_{\eps\to 0^+} \lin\Cont(S_1\cap B(S_1\cap S_2,\eps)). \label{Cont-union-bigger}\EDE \end{Lemma} \begin{proof} Fix $\eps>0$. Since $ S_1\cap B(S_2,1/k)\downarrow S_1\cap S_2$, there is $k_0\in\N$ such that $S_1\cap B(S_2,1/k_0)\subset B(S_1\cap S_2,\eps)$. Let $r\in(0,1/(2k_0))$. Suppose $x\in B(S_1,r)\cap B(S_2,r)$. Then there exist $y_j\in S_j$ such that $|x-y_j|\le r$, $j=1,2$. Then $|y_1-y_2|\le 2r\le 1/k_0$. So $y_1\in S_1\cap B(S_2,1/k_0)\subset S_1\cap B(S_1\cap S_2,\eps)$, which implies that $x\in B(S_1\cap B(S_1\cap S_2,\eps),r)$. So we get $B(S_1,r)\cap B(S_2,r)\subset B(S_1\cap B(S_1\cap S_2,\eps),r)$. Thus, $$\Cont(S_1;r)+\Cont(S_2;r)\le \Cont(S_1\cup S_2;r)+\Cont(S_1\cap B(S_1\cap S_2,\eps);r).$$ We then get (\ref{Cont-union-bigger}) by first sending $r$ to $0^+$ and then sending $\eps$ to $0^+$. \end{proof} For $T\subset S\subset\R$, we write $\pa_S T$ for the relative boundary of $T $ in $S$, i.e., $S\cap \lin { T}\cap \lin{S\sem T}$. \begin{Definition} Let ${\cal M}$ denote the family of all sets in $\R^n$ that can be expressed as the intersection of an open set with a closed set. Let $S\in\cal M$. A $d$-dimensional Minkowski content measure on $S$ is a Borel measure $\mu$ on $S$ which satisfies that, for any compact set $K\subset S$, (i) $\mu(K)<\infty$, and (ii) whenever $\mu(\pa_S K)=0$, $\Cont_d(K)$ exists and equals $\mu(K)$. \label{Def-Mink} \end{Definition} \begin{Remark} A Minkowski content measure $\mu$ is $\sigma$-finite since every set in $\cal M$ can be expressed as a countable union of compact sets. So for $1\le k\le n$, the set $E_{\mu,k}$ of points $c\in\R$ such that $\mu\{x\in\R^n:x_k=c\}>0$ is countable. Let $\cal R_\mu$ denote the family of sets $R=\prod_{k=1}^n I_k\subset \R^n$ such that each $I_k$ is a compact interval with $\pa I_k\cap E_{\mu,k}=\emptyset$. Then $\mu(\pa R)=0$ for $R\in \cal R_{\mu}$, and every compact rectangle in $\R^n$ could be approximated by elements in $\cal R_\mu$. \label{sigma-finite} \end{Remark} \begin{Lemma} The Minkowski content measure on any $S\in\cal M$ is unique. \label{unique} \end{Lemma} \begin{proof} Let $S=F\cap G\in \cal M$, where $F$ is closed and $G$ is open. Let $\mu_1,\mu_2$ be Minkowski content measures on $S$. Let ${\cal R}_{\mu_1,\mu_2}^G$ be the family of $R\in {\cal R}_{\mu_1}\cap {\cal R}_{\mu_2}$ that are contained in $G$. For any $R\in{\cal R}_{\mu_1,\mu_2}^G$, from $\mu_j(\pa R)=0$ and $\pa_S(R\cap F)\subset \pa R$, we get $\mu_j(R)=\mu_j(R\cap F)=\Cont(R\cap F)$, $j=1,2$, which implies that $\mu_1(R)=\mu_2(R)$. Since every compact rectangle in $G$ can be approximated by elements in ${\cal R}_{\mu_1,\mu_2}^G$, we get $\mu_1=\mu_2$. \end{proof} \begin{comment} \begin{Example} Let $n=1$, $p >0$, and $S=\{0\}\cup \{n^{-p}:n\in\N\}$. Let $d=\frac 1{1+p}$. Then $\Cont_d(S)=\frac{2^{1-d}}{d^d(1-d)^{1-d}}$. If $T$ is a finite subset of $S\sem \{0\}$, then $\Cont_d(T)=0$ and $\Cont_d(S\sem T)=\Cont_d(S)$. So $\mu:=\frac{2^{1-d} \delta_0}{d^d(1-d)^{1-d}}$ is the $d$-dimensional Minkowski content measure on $S$. In this example, the (minimal) support of $\mu$ is not $S$ but $\{0\}$, and it is not true that $\Cont_d(T)=\mu(T)$ for all compact $T\subset S$, even if $\Cont_d(\pa_S T)=0$. One obvious counterexample is $T=\{0\}$. \end{Example} \end{comment} \begin{Example} Let $n=1$. Let $B$ be an $1$-dimensional Brownian motion, and let $L_t$ be the usual local time of $B$ at $0$ (cf.\ \cite{RY}). Let $Z$ be the zero set $\{t:B_t=0\}$. There is a constant $C_0\in(0,\infty)$ such that for any $t_2\ge t_1\ge 0$, $\Cont_{1/2}(Z\cap [t_1-t_2])=C_0 (L_{t_2}-L_{t_1})$ (cf.\ \cite{Law-BM}). Then $C_0 dL$ is the $1/2$-dimensional Minkowski content measure on $Z$. \end{Example} \begin{Example} Let $n=2$. Let $\kappa\in(0,8)$ and $d=1+\frac \kappa 8$. Let $\eta$ be an SLE$_\kappa$ curve in $\HH$ from $0$ to $\infty$. The natural parametrization of $\eta$ is a random continuously strictly increasing function $\theta$ on $\R_+$ that satisfies that a.s.\ for any $a<b\in\R_+$, $\Cont_d(\eta[a,b])=\theta(b)-\theta(a)$ (\cite{LR}). Then $\eta_*(d\theta)$ is the $d$-dimensional Minkowski content measure on the image of $\eta$. \end{Example} \begin{Example} Let $n\ge 3$. Let $B$ be an $n$-dimensional Brownian motion. There is a constant $C_n\in(0,\infty)$ such that for any $t_2\ge t_1\ge 0$, $\Cont_{2}(B[t_1,t_2])=C_n (t_2-t_1)$ (cf.\ \cite{Law-BM}). Then $C_n B_* \lambda^1|_{[0,\infty)}$ is the $2$-dimensional Minkowski content measure on the image of $B$ \end{Example} \begin{Lemma} (i) If $\mu$ is the Minkowski content measure on $S\in\cal M$, then for any compact set $K\subset S$, $\lin\Cont(K)\le \mu(K)$. (ii) If $S\subset\R^n$ is compact, and $\mu$ is a measure on $S$ such that $\Cont(S)=\mu(S)<\infty$, and for any compact set $K\subset S$, $\lin\Cont(K)\le \mu(K)$, then $\mu$ is the Minkowski content measure on $S$. \label{Lemma-upper-cont} \end{Lemma} \begin{proof} (i) Let $S=F\cap G$, where $F$ is closed and $G$ is open. Let $K$ be a compact subset of $S$. Then for any $k>0$, $K$ can be covered by finitely many rectangles in $\cal R_\mu$ that are contained in $B(K,1/k)\cap G$. Let $K_k$ be the union of these rectangles. Then $K_k$ is a compact subset of $ B(K,1/k)\cap G$, and $\mu(\pa K_k)=0$. Now $K_k\cap S=K_k\cap F$ is a compact subset of $S$, and $\pa_S(K_k\cap S)\subset \pa K_k$. So $\mu(\pa_S (K_k\cap S))=0$. Thus, $\lin\Cont(K)\le \lin\Cont(K_k\cap S)=\mu(K_k\cap S)\le \mu(B(K,1/k))$. Sending $k$ to $\infty$, we get $\lin\Cont(K)\le \mu(K)$. (ii) Let $K$ be a compact subset of $S$ such that $\mu(\pa_S K)=0$. Let $L=\lin{S\sem K}$. Then $L$ is also a compact subset of $S$, $S=K\cup L$, and $K\cap L=\pa_S K$. From $\mu(\pa_S K)=0$ we get $\Cont(S)=\mu(S)=\mu(K)+\mu(L)$. Thus, by the assumption and (\ref{Cont-union-lower-n}), $$\mu(S)=\Cont(S)\le \ulin\Cont(K)+\lin\Cont(L)\le \lin\Cont(K)+\lin\Cont(L)\le \mu(K)+\mu(L)=\mu(S).$$ So we get $\Cont(K)=\mu(K)$, and conclude that $\mu$ is the Minkowski content measure on $S$. \end{proof} \begin{Theorem} Let $S\subset\R^n$ be compact. For $r>0$, let $\mu_r$ be the measure such that $d\mu_r=r^{d-n} {\bf 1}_{B(S,r)} d\lambda^n$. Then $\mu$ is the Minkowski content measure on $S$ iff $\mu_r\to \mu$ weakly as $r\to 0^+$. \label{weak} \end{Theorem} \begin{proof} First, suppose $\mu_r\to \mu$ weakly as $r\to 0^+$. Since $\mu_r$ is supported by $B(S,r)$, $\mu$ is supported by $S$. Since $|\mu_r|=\Cont(S;r)$ by definition, we get $|\mu|=\lim |\mu_r|=\Cont(S)$. Let $K$ be a compact subset of $S$. Fix $r_0>0$. Then for any $r\in(0,r_0)$, $\Cont(K;r)\le \mu_r(B(K;r_0))$. Sending $r$ to $0^+$, we get $\lin\Cont(K)\le \limsup_{r\to 0^+} \mu_r(B(K,r_0))\le \mu(B(K,r_0))$ since $B(K,r_0)$ is closed. Sending $r_0$ to $0^+$, we get $\lin\Cont(K)\le \mu(K)$. By Lemma \ref{Lemma-upper-cont} (ii), $\mu$ is the Minkowski content measure on $S$. Second, suppose $\mu$ is the Minkowski content measure on $S$. Then $|\mu|=\Cont(S)=\lim_{r\to 0^+} |\mu_r|$. Thus, the family $(\mu_r)_{r\in(0,1]}$ is pre-compact w.r.t.\ the weak convergence. Suppose $\mu_{r_n}\to \mu'$ weakly for some sequence $r_n\to 0$. Then $\mu'$ is supported by $S$, and $|\mu'|=\lim|\mu_{r_n}|=\lim\Cont(S;r_n)=\Cont(S)$. Let $K$ be a compact subset of $S$ such that $\mu(\pa_S K)=0$. Then $\Cont(K)=\mu(K)$. Fix $r_0\in(0,1]$. Choose $n$ big enough such that $r_n<r_0$. Then $\mu_{r_{n}}(B(K,r_0))\ge \Cont(K,r_{n})$. First sending $n$ to $\infty$ and then sending $r_0$ to $0^+$, we get $\mu'(K)\ge \Cont(K)=\mu(K)$. Now let $K$ be any compact subset of $S$. Then we could find a decreasing sequence of compact subsets $(K_n)$ of $S$ such that $K=\bigcap_n K_n$ and $\mu(\pa_S K_n)=0$ for each $n$. Then $\mu'(K)=\lim\mu'(K_n)\ge \lim\mu(K_n)=\mu(K)$. Since this holds for any compact $K\subset S$ and $|\mu'|=|\mu|$, we get $\mu'=\mu$. This means that any subsequential weak limit of $(\mu_r)$ as $r\to 0^+$ is $\mu$. Thus, $\mu_r\to \mu$ weakly as $r\to 0^+$. \end{proof} \begin{Theorem} [Restriction Property] Let $\mu$ be the Minkowski content measure on $S\in \cal M$. Let $F\subset\R^n$ be closed and $G\subset\R^n$ be open. Suppose $\mu(\pa_{S} (S\cap F))=0$. Let $S'=S\cap F\cap G$ and $\mu'=\mu|_{S'}$. Then $\mu'$ is the Minkowski content measure on $S'$. \label{restriction} \end{Theorem} \begin{proof} We have $S'\in \cal M$ since $\cal M$ is closed under finite intersections. Let $K\subset S'$ be compact. Then $K$ is also a compact subset of $S$, and so $\mu'(K)=\mu(K)<\infty$. Suppose $\mu'(\pa_{S'} K)=0$. Then $\mu(\pa_{S'} K)=0$. Let $S_F=S\cap F$. Since $S'=S_F\cap G$ is relatively open in $S_F$, and $K\subset S'$ is compact, we get $\pa_{S'} K=\pa_{S_F} K$. From $K\subset S_F\subset S$, we get $\pa_{S}K\subset \pa_{S_F} K\cup \pa_{S} S_F$. Thus, $\mu(\pa_{S} K)\le \mu(\pa_{S'} K)+\mu(\pa_{S} S_F)=0$. Since $\mu$ is the Minkowski content measure on $S_1$, we get $\Cont(K)=\mu(K)=\mu'(K)$. So $\mu'$ is the Minkowski content measure on $S'$. \end{proof} The following theorem asserts that the Minkowski content measures are preserved under covariant transforms \cite[Definition 1.1]{cov}. \begin{Theorem}[Conformal Covariance] Let $\mu$ be the Minkowski content measure on $S\in\cal M$, which is relatively closed in an open set $G\subset \R^n$. Let $\phi:G\to\R^n$ be $C^1$, injective, and conformal. Then $\nu:=\phi_*(|\phi'|^d\cdot \mu)$ is the Minkowski content measure on $\phi(S)$, where $|\phi'(x)|:=|\det D\phi(x)|^{1/n}$, $|\phi'|^d\cdot \mu$ is the measure on $S$ which is absolutely continuous w.r.t.\ $\mu$ with Radon-Nikodym derivative $|\phi'|^d$, and $\nu$ is the pushforward of $|\phi'|^d\cdot \mu$ under $\phi$. \label{phi(K)-Cont} \end{Theorem} \begin{Remark} We emphasize that this theorem holds for all $n\in\N$. The assumption that $\phi$ is conformal on $G$ means that for every $x\in G$, $D\phi(x)$ is some nonzero scalar multiple of a rotation matrix. When $n=1$, the conformal assumption is equivalent to that $\phi'\ne 0$. When $n=2$, $\phi$ is holomorphic or anti-holomorphic on each component of $G$. When $n\ge 3$, by Liouville's theorem, $\phi$ is a M\"obius transformation on each component of $G$. \end{Remark} \begin{proof}[Proof of Theorem \ref{phi(K)-Cont}] Since $S$ is relatively closed in $G$ and $\phi$ is a homeomorphism on $G$, both $\phi(G)$ and $\phi(G\sem S)$ are open. So $\phi(S)=\phi(G)\sem \phi(G\sem S)\in \cal M$. Let $K$ be any compact subset of $G$. Let $m_K=\min_{x\in K} |\phi'(x)|$ and $M_K=\max_{x\in K} |\phi'(x)|$. Let $c>1$. Then there is $\delta>0$ such that $|\phi'(x)|<cM_K$ for $x\in B(K,\delta)$, and $|((\phi)^{-1})'(y)|\le c/m_K$ for $y\in B(\phi(K),\delta)$. Thus, for $r>0$ small enough, we have $$\phi(B(K,\frac r{ cM_K}))\subset B(\phi(K),r)\subset \phi(B(K,\frac {r}{m_K/c})),$$ which then implies that $$\frac{(m_K/c)^n}{(cM_K)^{n-d}} \Cont(K;\frac {r}{cM_K})\le \Cont(\phi(K );r)\le \frac {(cM_K)^n}{(m_K/c)^{n-d}} \Cont(K ;\frac {r}{m_K/c}).$$ First sending $r$ to $0^+$ and then sending $c$ to $1^+$, we get \BGE {m_K^n}{M_K^{d-n}}\cdot \ulin\Cont(K)\le \ulin\Cont(\phi(K))\le \lin\Cont(\phi(K))\le {M_K^n}{m_K^{d-n}} \cdot \lin\Cont(K).\label{phi(K)-ineq}\EDE Now assume $K\subset S$ and $\mu(\pa_S K)=0$. Since $\Cont(K)=\mu(K)$ and $m_K\le |\phi'|\le M_K$ on $K$, we get \BGE \frac {m_k^n}{M_K^n} \int_K |\phi'(x)|^d \mu (dx)\le \ulin{\Cont}(\phi (K ))\le \lin{\Cont}(\phi(K )) \le \frac {M_K^n}{m_K^n} \int_K |\phi'(x)|^d \mu (dx). \label{phimM}\EDE Let $K_\phi\subset \phi(S)$ be compact. Let $K=\phi^{-1}(K_\phi)$. Then $K\subset S$ is compact. So $\mu(K)<\infty$. Since $|\phi'|$ is bounded on $K$, $\nu(K_\phi)<\infty$. Suppose $\nu(\pa_{\phi(S)} K_\phi)=0$. We want to show that $\Cont(K_\phi)=\nu(K_\phi)$. Since $\pa_S K=\phi^{-1}(\pa_{\phi(S)} K_\phi)$, we get $\mu(\pa_S K)=0$. Fix $c>1$. We may find $R_1,\dots,R_N\in {\cal R}_\mu$ which intersect each other only at their boundaries such that $K\subset \bigcup_{j=1}^N R_j\subset G$, and for each $1\le j\le N$, $ \max_{x\in R_k}|\phi'(x)|\le c \min_{x\in R_k}|\phi'(x)|$. Let $K_j=K\cap R_j$, $1\le j\le N$. Then $\mu(\pa_S K_j)\le \mu(\pa_S K)+\mu(\pa R_j)=0$. By (\ref{phimM}), \BGE c^{-n} \int_{K_j} |\phi'(x)|^d \mu (dx)\le \ulin{\Cont}(\phi (K_j ))\le \lin{\Cont}(\phi(K_j )) \le c^n \int_{K_j} |\phi'(x)|^d \mu (dx). \label{phimM-j}\EDE Summing up the last inequality of (\ref{phimM-j}) over $1\le j\le N$, and using (\ref{Cont-union-upper-n}) and that $\mu(K_j\cap K_k)\le \mu(R_j\cap R_k)=0$ when $j\ne k$, we get $\lin{\Cont}(K_\phi) \le c^n \int_{K} |\phi'(x)|^d \mu (dx)$. Let $1\le k<j\le N$. Since $\mu(K_j\cap K_k)=0$, we get $\lim_{\delta\to 0^+} \lin\Cont( K_j\cap B(K_j\cap K_k,\delta))=0$ by Lemma \ref{Lemma-upper-cont} (i). By (\ref{phi(K)-ineq}), $\lim_{\eps\to 0^+} \lin\Cont( \phi(K_j)\cap B(\phi(K_j)\cap \phi(K_k),\eps))=0$. By (\ref{Cont-union-upper-n}), we then get $\lim_{\eps\to 0^+} \lin\Cont( \phi(K_j)\cap B(\phi(K_j)\cap \bigcup_{k=1}^{j-1} \phi(K_k),\eps))=0$ for $2\le j\le n$. By (\ref{Cont-union-bigger}) and induction, we get $\ulin\Cont(K_\phi)\ge \sum_{j=1}^N \ulin\Cont( \phi(K_j))$, which together with the first inequality of (\ref{phimM-j}) implies that $\ulin\Cont(K_\phi)\ge c^{-n} \int_{K} |\phi'(x)|^d \mu (dx)$. Combining this inequality with $\lin{\Cont}(K_\phi) \le c^n \int_{K} |\phi'(x)|^d \mu (dx)$ and sending $c$ to $1^+$, we get $\Cont(K_\phi)=\int_{K} |\phi'(x)|^d \mu (dx)=\nu(K_\phi)$. Thus, $\nu$ is the Minkowski content measure on $\phi(S)$. \end{proof} \begin{Remark} It's easy to extend the notions of Minkowski content and Minkowski content measure to Riemannian manifolds. The propositions in this subsection will still hold. \end{Remark} \begin{Remark} Most well-known deterministic fractal sets do not have non-trivial Minkowski contents. However, we may define the average Minkowski content by $$\Cont_d^{\aaa}(S)=\lim_{t\to \infty} \frac 1t \int_0^t \Cont_d(S;e^{-r})dr,$$ and use this to define the average Minkowski content measure. Then most propositions in this subsection except for Theorem \ref{weak} still hold for average Minkowski content measures. \end{Remark} \begin{Example} Let $S\subset [0,1]$ be the $1/3$-Cantor set. Let $d=\frac{\ln 2}{\ln 3}$. Then $\Cont_d(S,e^{-t})$, $t\ge 0$, has period $\ln(3)$, and is not constant. So the Minkowski content $\Cont_d(S)$ does not exist, but the average Minkowski content $\Cont_d^{\aaa}(S)=C_0$ for some $C_0\in(0,\infty)$. Let $\mu$ be the Haar probability measure on $S$. Then $C_0 \mu$ is the average Minkowski content measure on $S$. \end{Example} \subsection{Boundary Minkowski content of SLE$_\kappa({\protect\ulin\rho})$} \label{Section 6.2} Now we come back to the SLE$_\kappa(\ulin\rho)$ curve $\eta$ in Theorem \ref{main-thm1} with the additional assumption that $\rho_\Sigma\in (\frac\kappa 2-4,\frac\kappa 2-2)$. This means that $\eta$ intersects the interval $(-\infty,v_m)$. Let $$d=1-\alpha=\frac{(\rho_\Sigma+4)(\kappa-4-2\rho_\Sigma)}{2\kappa}.$$ It is known that $d$ is the Hausdorff dimension of $\eta\cap (-\infty,v_m)$ (cf.\ \cite{MW}). We are going to prove the existence of the $d$-dimensional Mknowski content measure on $\eta\cap (-\infty,v_m)$. We write $G_1(u)$ for the function $C_{\R} G(w,\ulin v;u)$ in Theorem \ref{main-thm1}, and $G_2(u_1,u_2)$ for the function $G_{\R}(w,\ulin v;u_1,u_2)$ in Theorem \ref{main-thm2}. We also define $G_2(u_1,u_2)=G_2(u_2,u_1)$ if $u_1<u_2\in (-\infty, v_m)$. \begin{Theorem} Almost surely the (random) $d$-dimensional Minkowski content measure $\mu_\eta$ on $\eta\cap (-\infty,v_m]$ in $\R$ exists, is atomless, and satisfies the following properties. \begin{itemize} \item [(i)] If $\phi$ is a deterministic nonnegative measurable function on $(-\infty,v_m]$, then \BGE\EE\Big[\int \phi(u) \mu_\eta(du)\Big]=\int \phi(u) G_1(u) \lambda(du).\label{EC1-phi}\EDE \item [(ii)] If $\psi$ is a deterministic nonnegative measurable function on $(-\infty,v_m]^2$, then \BGE \EE\Big[\int\!\!\int \psi(u_1,u_2) \mu_\eta^2(du_1\otimes du_2)\Big]=\int \!\!\int \psi(u_1,u_2) G_2(u_1,u_2) \lambda^2(du_1\otimes du_2).\label{EC2-psi}\EDE \item [(iii)] For any deterministic compact set $S\subset (-\infty,v_m]$ with $\Cont(\pa S)=0$, almost surely \BGE \Cont(\eta\cap S)=\mu_\eta(S).\label{Cont=muL}\EDE \item [(iv)] The (minimal) support of $\mu_\eps$ (i.e., the smallest closed set $F\subset \R$ such that $\mu_\eps(F^c)=0$) is a.s.\ $\eta\cap (-\infty,v_m]$. \end{itemize} \label{Thm-Mink} \end{Theorem} \begin{Remark} We emphasize here that the a.s.\ existence of the Minkowski content measure $\mu_\eta$ means that, on an event with probability $1$, $\mu_\eta$ satisfies properties (i) and (ii) in Definition \ref{Def-Mink} simultaneously for all compact subsets of $\eta\cap[-\infty,v_m]$. \end{Remark} \begin{proof} For $r>0$, define random functions $I_r,J_r$ on $(-\infty,v_m)$ by $I_r(u)= {\bf 1}_{B_{\R}(\eta\cap \R,r)}(u)$ and $J_r(u)=r^{-\alpha} I_r(u)$. By Theorems \ref{main-thm1} and \ref{main-thm2}, for $u\in (-\infty,v_m)$, $\EE[J_r(u)]\to G_1(u)$ as $r\to 0^+$, and for $u_1\ne u_2\in (-\infty,v_m)$, $\EE[J_{r_1}(u_1)J_{r_2}(u_2)]\to G_2(u_1,u_2)$ as $r_1,r_2\to 0^+$. By (\ref{est-u*},\ref{G-upper}), for any $u\in (-\infty,v_m)$ and $r>0$, \BGE \EE[J_r(u)],G_1(u) \lesssim |v_m-u|^{-\alpha}.\label{JG1'}\EDE By (\ref{transform-2},\ref{G(u1,u2)-upper}), for $u_1\ne u_2\in (-\infty,v_m)$ and $r_1,r_2>0$, \BGE \EE[J_{r_1}(u_1)J_{r_2}(u_2)],G_2(u_1,u_2)\lesssim |v_m-u_1|^{-\alpha} |u_1-u_2|^{-\alpha}.\label{JG2'}\EDE Combining (\ref{JG2'}) with (\ref{G(u1,u2)-est}) and setting $\zeta:=\zeta_1\wedge \zeta_2\wedge \frac{1-\alpha}2\in (0,1-\alpha)$, we get \begin{align} &|\EE[J_{r_1}(u_1)J_{r_2}(u_2)]-G_2(u_1,u_2)| \nonumber \\ \le & |v_m-u_1|^{-\alpha}|u_1-u_2|^{-\alpha}\Big(1\wedge \Big(\frac{r_1\vee r_2}{|v_m-u_1|\wedge |u_1-u_2|}\Big)^\zeta \Big). \label{J-G2'} \end{align} Fix a bounded measurable set $S\subset (-\infty,v_m]$. Let $A=\sup\{|v_m-u|:u\in S\}$. Define \BGE Y_r =\int_S J_r(u) \lambda(du) =r^{-\alpha} \lambda(B_{\R}(\eta\cap\R,r)\cap S).\label{YrS'}\EDE From (\ref{JG2'}) and that $\alpha\in(0,1)$ we get $$\int_S\!\int_S G_2(u_1,u_2)\lambda(du_1)\lambda(du_2)\lesssim \int_0^A L_1^{-\alpha}\int_0^{A-L_1} L_2^{-\alpha} \lambda(dL_2)\lambda(dL_1)<\infty.$$ The finiteness of this integral together with (\ref{J-G2'}) implies that, for any $r_1,r_2>0$, \begin{align*} & \EE[(Y_{r_1}-Y_{r_2})^2]=\EE\Big[\Big(\int_S J_{r_1}(u)\lambda(du)-\int_S J_{r_2}(u)\lambda(du)\Big)^2\Big]\\ \le &2 \sum_{j=1}^2 \sum_{k=1}^2 \int\!\!\int_{\{(u_1,u_2)\in S^2:u_1>u_2\}} |\EE[J_{r_j}(u_1) J_{r_k}(u_2)]-G(u_1,u_2)| \lambda(du_1) \lambda(du_2)\\ \lesssim & \int_{0}^{A}\!\!\int_{0}^{A } L_1^{-\alpha} L_2^{-\alpha} \Big(1\wedge \Big(\frac {r_1\vee r_2}{L_1\wedge L_2}\Big)^\zeta\Big)\lambda(dL_1) \lambda(dL_2) \end{align*} where in the last line we used a change of variables: $L_1=v_m-u_1$ and $L_2=u_1-u_2$. Since $\alpha\in(0,1)$ and $\zeta\in (0,1-\alpha)$, we can easily bound the last integral and get \BGE \|Y_{r_1}-Y_{r_2}\|_{L^2}^2 \lesssim A^{2-2\alpha-\zeta} (r_1\vee r_2)^{\zeta}. \label{L2diff'}\EDE This implies that $(Y_r)$ is Cauchy in $L^2$ as $r\to 0^+$. Let $\xi_S\in L^2$ be the limit. By (\ref{L2diff'}) and Chebyshev's inequality, $$\sum_{n=1}^\infty \PP[|Y_{Ae^{-n}}-Y_{Ae^{1-n}}|\ge A^{1-\alpha} e^{-n\zeta/3}]\lesssim \sum_{n=1}^\infty e^{-n\zeta/3}<\infty.$$ By Borel-Cantelli lemma, with probability one, $|Y_{Ae^{-n}}-Y_{Ae^{1-n}}|\le A^{1-\alpha} e^{-n\zeta/3}$ for sufficiently large $n$, which implies that a.s.\ $\lim_{n\to\infty} Y_{e^{-n}}$ converges. The limit is $\xi_S$ since $Y_{Ae^{-n}}\to \xi_S$ in $L^2$. A similar argument shows that, for any $k\in\N$, a.s.\ $Y_{Ae^{-n/2^k}}\to \zeta$ as $n\to\infty$. Let $E_0$ be the event that $\lim_{n\to\infty} Y_{e^{-n/2^k}}= \zeta$ for every $k\in\N$. Then $\PP[E_0]=1$. By the definition of $I_r$ and $J_r$, for $r_1\ge r_2>0$, we have $I_{r_1}(u)\ge I_{r_2}(u)$ and $J_{r_1}(u)\ge (r_2/r_1)^\alpha J_{r_2}(u)$, which implies that $Y_{r_1}\ge (r_2/r_1)^\alpha Y_{r_2}$. Let $k\in\N$. For $r\in(0,1]$, there is a unique $n_k(r)\in\N$ such that $e^{-n_k(r)/2^k}< r\le e^{(1-n_k(r))/2^k}$. Then we get $ e^{-\alpha /2^k} Y_{e^{-n_k(r)/2^k}}\le Y_r\le e^{\alpha/2^k} Y_{e^{(1-n_k(r))/2^k}}$, which implies that $e^{-\alpha/2^k} \zeta\le \liminf_{r\to 0^+} Y_r\le \limsup_{r\to 0^+} Y_r\le e^{\alpha/2^k}\zeta$ on the event $E_0$. Since this holds for every $k\in\N$, and $\PP[E_0]=1$, we get a.s.\ $Y_r\to\xi_S$ as $r\to 0^+$. So far we have proved that, for any bounded measurable set $S\subset (-\infty,v_m]$, there is a nonnegtive random variable $\xi_S\in L^2$ such that, as $r\to 0^+$, $\int_S J_r(u) \lambda(du)\to \xi_S$ almost surely and in $L^2$. It is clear that a.s.\ $\xi_{S_1\cup S_2}=\xi_{S_1}+\xi_{S_2}$ if $S_1\cap S_2=\emptyset.$ By Fubini Theorem, $$\int_S \EE[J_r(u)]\lambda(du)\to \EE[\xi_S],\quad \int_S\!\int_S \EE[J_r(u_1)J_r(u_2)] \lambda(du_1) \lambda(du_2)\to \EE[\xi_S^2].$$ By polarization, the second limit further implies that for any bounded measurable sets $S_1,S_2\subset (-\infty,v_m]$, $$\int_{S_1}\!\int_{S_2} \EE[J_r(u_1)J_r(u_2)] \lambda(du_1) \lambda(du_2)\to \EE[\xi_{S_1}\xi_{S_2}].$$ Since $\EE[J_r(u)]\to G_1(u)$ and $\EE[J_r(u_1)J_r(u_2)]\to G_2(u_1,u_2)$ as $r\to 0^+$, by (\ref{JG1'},\ref{JG2'}) and dominated convergence theorem, for any bounded measurable sets $S,S_1,S_2\subset (-\infty,v_m]$, \BGE \EE[\xi_S]=\int_S G_1(u) \lambda(du),\quad \EE[\xi_{S_1 }\xi_{S_2}]=\int_{S_1}\!\int_{S_2} G_2(u_1,u_2) \lambda(du_1) \lambda(du_2).\label{moment-zeta}\EDE We now define $\theta(u)=\xi_{[u,v_m]}$, $u\in (-\infty,v_m]$. Then for $a<b\in (-\infty,v_m)$, by (\ref{moment-zeta},\ref{JG2'}) $$\EE[(\theta(a)-\theta(b))^2]=\EE[\xi_{[a,b)}^2]=\int_{a}^{b}\int_{a}^{b} G_2(u_1,u_2) du_1 du_2$$ $$\lesssim \int_{v_m-b}^{v_m-a} \int_0^{b-a} L_1^{-\alpha} L_2^{-\alpha} \lambda(dL_2)\lambda(dL_1)\lesssim |v_m-b|^{-\alpha} |b-a|^{ 2-\alpha}.$$ By Kolmogorov continuity theorem, for any $b\in (-\infty,v_m)\cap \Q$, $\theta$ has a locally H\"older continuous version on $(-\infty,b]$. Thus, $\theta$ has a continuous version on $(-\infty,v_m)$. Such continuous version is nonnegative and decreasing since for any $a<b\in(-\infty,v_m]$, a.s.\ $\theta(a)-\theta(b)=\xi_{[a,b)}\ge 0$. By (\ref{moment-zeta},\ref{JG1'}), $\EE[\theta(b)]\to 0$ as $b\to v_m^-$. So $\theta$ has a continuous decreasing version on $(-\infty,v_m]$ with $\theta(v_m)=0$. Such version of $\theta$ determines an atomless locally finite measure $\mu_\eta$ on $(-\infty,v_m]$. Then for any set $S\subset (-\infty,v_m]$, which is a finite disjoint union of intervals of the form $(a_j,b_j]$, we have a.s.\ $\mu_\eta(S)=\xi_S$. By monotone class theorem, for any bounded measurable set $S\subset (-\infty,v_m]$, a.s.\ $\mu_\eta(S)=\xi_S$. By (\ref{moment-zeta}), we get \BGE\EE[\mu_\eta(S)]=\int_S G_1(u) \lambda(du),\label{EC1}\EDE and if $T=S_1\times S_2$ for some bounded measurable sets $S_1,S_2\subset (-\infty,v_m]$, then \BGE \EE[\mu_\eta^2(T)]=\int \!\!\int_{T} G_2(u_1,u_2) \lambda^2(du_1\otimes du_2).\label{EC2}\EDE By monotone convergence theorem and monotone class theorem, (\ref{EC1}) holds for any measurable set $S\subset (-\infty,v_m]$, and (\ref{EC2}) holds for any measurable set $T\subset (-\infty,v_m]^2$. By monotone convergence theorem, these results further imply (\ref{EC1-phi},\ref{EC2-psi}). Let $S$ be a compact subset of $(-\infty,v_m]$ with $\Cont(\pa S)=0$. For $r>0$, since the symmetric difference between $B_{\R}(\eta\cap \R,r)\cap S$ and $B_{\R}(\eta\cap S,r)$ is contained in $B_{\R}(\pa S,r)$, by (\ref{YrS'}) we get $| \int_S J_r(u) \lambda(du)-\Cont(\eta\cap S;r)|\le \Cont(\pa S;r)$. Sending $r$ to $ 0^+$ and using that a.s.\ $\int_S J_r(u) \lambda(du)\to \xi_S=\mu_\eta(S)$, we get (\ref{Cont=muL}). Applying (\ref{Cont=muL}) to $S$ of the form $[a,b]$, we find that there is an event $E$ with $\PP[E]=1$ such that on the event $E$, for any $a<b\in (-\infty,v_m]\cap \Q$, $\Cont(\eta\cap [a,b])=\mu_\eta[a,b]=\theta(a)-\theta(b)$. By the continuity of $\theta$ and the increasingness of $\lin\Cont$ and $\ulin\Cont$, we find that the equality holds for any $a<b\in (-\infty,v_m]$ on the event $E$. Assume that $E$ happens. For any $a<b\in (-\infty,v_m]$ such that $[a,b]\cap \eta=\emptyset$, $\mu_\eta[a,b]=\Cont(\eta\cap [a,b])=\Cont(\emptyset)=0$. This implies that $\mu_\eta$ is a measure on $\eta\cap (-\infty, v_m]$. Fix $a<v_m$. Then $\mu_\eta[a,v_m]=\Cont(\eta\cap [a,v_m])$. Let $F$ be any compact subsets of $[a,v_m]$. We may find a decreasing sequence of sets $(F_n)_{n=0}^\infty$ such that $F_n\downarrow F$ and each $F_n$ is a finite union of compact intervals. For each $n$, we have $\Cont(F_n\cap \eta)=\mu_\eta(F_n)$. Since $\mu_\eta(F_n)\to \mu_\eta(F)$, by the increasingness of $\lin\Cont$, we have $\lin\Cont(F)\le \mu_\eta(F)$. By Lemma \ref{Lemma-upper-cont} (ii), $\mu_\eta|_{[a,v_m]}$ is the Minkowski content measure on $\eta\cap [a,v_m]$ on the event $E$. Since this holds for any $a<v_m$, and $\PP[E]=1$, $\mu_\eta$ is a.s.\ the Minkowski content measure on $\eta\cap (-\infty,v_m]$. To prove Part (iv), we need the lemma below. \begin{comment} Assume that $E$ happens. For any $a<b\in (-\infty,v_m]$ such that $[a,b]\cap \eta=\emptyset$, $\mu_\eta[a,b]=\Cont(\eta\cap [a,b])=\Cont(\emptyset)=0$. This implies that $\mu_\eta$ is a measure on $\eta\cap (-\infty, v_m]$. Let $F$ be a compact subset of $\eta\cap (-\infty,v_m]$. Let $m_F=\min F$ and $M_F=\max F$. Then $\mu_\eta(F)\le \mu_\eta[m_F,M_F]=\theta(m_F)-\theta(M_F)<\infty$. Assume in addition that \BGE \mu_\eta(\pa_{\eta\cap (-\infty,v_m]} F)= \mu_\eta(F\cap \lin{\eta\cap (-\infty,v_m]\sem F})=0.\label{assump}\EDE We may find a decreasing sequence of sets $(F_n)_{n=0}^\infty$ such that $F_0=[m_F,M_F]$, $\bigcap_n F_n=F$, and each $F_n$ is a finite union of compact intervals. For each $n$, we have $\Cont(F_n)=\mu_\eta(F_n)$. Since $\mu_\eta(F_n)\to \mu_\eta(F)$, by the increasingness of $\lin\Cont$, we have $\lin\Cont(F)\le \mu_\eta(F)$. Let $F'=\lin{[m_F,M_F]\sem F}$. A similar argument shows that $\lin{\Cont}(F')\le \mu_\eta(F')$. Since $ F'\subset \lin{\eta\cap (-\infty,v_m]\sem F}$, by (\ref{assump}) we get $\mu_\eta(F\cap F')=0$. So by (\ref{Cont-union-lower-n}), $$\lin\Cont(F)+\lin{\Cont}(F')\le \mu_\eta(F)+\mu_\eta(F')=\mu_\eta(F\cup F')$$ $$=\mu_\eta[m_F,M_F]=\Cont(\eta\cap [m_F,M_F])\le \ulin\Cont(F)+\lin\Cont(F').$$ This implies that $\Cont(F)=\mu_\eta(F)$. Thus, $\mu_\eta$ is the Minkowski content measure on $\eta\cap (-\infty,v_m]$ on the event $E$. To prove Part (iv), we need the lemma below. \end{comment} \begin{Lemma} Let $\tau$ be a finite $\F$-stopping time. Suppose $\eta^\tau$ is a continuous curve in $\lin\HH$, which satisfies that a.s.\ $f_\tau(\eta^\tau)=\eta(\tau+\cdot)$. Then $\mu^\tau:=(g_\tau)_*((g_\tau')^d\cdot \mu_\eta|_{(-\infty,v_m\wedge a_{K_\tau})})$ is a.s.\ the Minkowski content measure on $\eta^\tau\cap (-\infty,V_m(\tau)]$. \label{lemma-mu-tau} \end{Lemma} \begin{proof} By DMP, conditionally on $\F_\tau$, $\eta^\tau$ is an SLE$_\kappa(\ulin\rho)$ curve started from $W(\tau)$ with force points $\ulin V(\tau)$. Thus, the Minkowski content measure $\mu_{\eta^\tau}$ on $\eta^\tau\cap (-\infty,V_m(\tau)]$ a.s.\ exists. By Theorem \ref{restriction}, $\mu_\eta|_{(-\infty,v_m\wedge a_{K_\tau})}$ and $\mu_{\eta^\tau}|_{ (-\infty,V_m(\tau))}$ are respectively the Minkowski content measure on $\eta\cap (-\infty,v_m\wedge a_{K_\tau})$ and $\eta^\tau\cap (-\infty,V_m(\tau))$. Since $g_\tau$ is injective and $C^1$ on $(-\infty,v_m\wedge a_{K_\tau})$, and sends $\eta\cap (-\infty,v_m\wedge a_{K_\tau})$ to $\eta^\tau\cap (-\infty,V_m(\tau))$, by Theorem \ref{phi(K)-Cont}, $\mu_{\eta^\tau}|_{ (-\infty,V_m(\tau))}=(g_\tau)_*((g_\tau')^d\cdot \mu_\eta|_{(-\infty,v_m\wedge a_{K_\tau})})=\mu^\tau$. Since $\mu_{\eta^\tau}$ is supported by $(-\infty,V_m(\tau)]$, and is atomless, we get $\mu^\tau=\mu_{\eta^\tau}$. So we get the conclusion. \end{proof} \begin{comment} \begin{Lemma} Let $\tau$ be a finite $\F$-stopping time. Let $\eta^\tau$ be the SLE$_\kappa(\ulin\rho)$ curve started from $W(\tau)$ with force points $\ulin V(\tau)$ obtained from DMP of $\eta$, which satisfies that $f_\tau(\eta^\tau)=\eta(\tau+\cdot)$. Let $\mu^\tau$ be the Minkowski content measure on $\eta^\tau\cap (-\infty,V_m(\tau)]$. Then $$\mu^\tau=(g_\tau)_*((g_\tau')^d\cdot \mu_\eta|_{(-\infty,v_m\wedge a_{K_\tau})}).$$ \label{lemma-mu-tau} \end{Lemma} \begin{proof} By Theorem \ref{restriction}, $\mu|_{(-\infty,v_m\wedge a_{K_\tau})}$ is the Minkowski content measure on $\eta\cap (-\infty,v_m\wedge a_{K_\tau})$, and $\mu^{\tau}|_{(-\infty,V_m(\tau))}$ is the Minkowski content measure on $\eta^\tau\cap (-\infty,V_m(\tau))$. Since $g_\tau$ is injective and $C^1$ on $(-\infty,v_m\wedge a_{K_\tau})$, and sends $\eta\cap (-\infty,v_m\wedge a_{K_\tau})$ to $\eta^\tau\cap (-\infty,V_m(\tau))$, by Theorem \ref{phi(K)-Cont}, $ \mu^{\tau}|_{(-\infty,V_m(\tau))}=(g_\tau)_*((g_\tau')^d\cdot \mu|_{(-\infty,v_m\wedge a_{K_\tau})})$. Since $\mu^\tau$ is supported by $(-\infty,V_m(\tau)]$, and is atomless, $\mu^\tau= \mu^{\tau}|_{(-\infty,V_m(\tau))}$. So we get the conclusion. \end{proof} \end{comment} Now we prove Part (iv). Let $\supp(\nu)$ denote the (minimal) support of a measure $\nu$. First consider the case that $w=0$ and $v_m=0^-$. Then $\eta$ is an SLE$_\kappa(\rho_\Sigma)$ curve started from $0$ with the force point $0^-$. It satisfies the scaling property: for any $a>0$, $a\eta$ has the same law as $\eta$ up to a linear time-change. For $L\in(-\infty,0)$, let $p(L)=\PP[\mu_\eta[L,0]=0]$. Then $p(L)$ is increasing with $\lim_{L\to -\infty} p(L)= \PP[\mu_\eta=0]$ and $\lim_{L\to 0^-} p(L)=\PP[0\not\in \supp(\mu_\eta)]$. By Theorem \ref{phi(K)-Cont}, $\mu_\eta$ has the same law as $a^d \mu_\eta(\cdot/a)$, which implies that $p(L)=p(aL)$ for any $a>0$. So $p$ is constant on $(-\infty,0)$. Denote the constant value by $p_0\in[0,1]$. Then $\PP[\mu_\eta=0]=\PP[0\not\in \supp(\mu_\eta)]=p_0$. Let $\tau=\tau^*_{-1}$. Then $\tau$ is a finite $\F$-stopping time, and $\eta(\tau)\in(-\infty,-1)$. By DMP, $g_\tau\circ \eta(\tau+\cdot)-W(\tau)$ has the same law as $\eta$, and is independent of $\F_\tau$. By Lemma \ref{lemma-mu-tau}, we have $\PP[\eta|_{(-\infty,\eta(\tau)]}=0|\F_\tau]=p_0$. Since $\mu_\eta[\eta(\tau),0]=\Cont(\eta[0,\tau]\cap (-\infty,0])$ is $\F_\tau$-measurable, we have $$p_0=\PP[\mu_\eta=0]=\PP[\{\mu_\eta|_{[\eta(\tau),0]}=0\}\cap \{\mu|_{(-\infty,\eta(\tau)]}=0\}]=p_0 \PP[\mu_\eta|_{[\eta(\tau),0]}=0]\le p_0^2.$$ If $p_0=1$, then by $\PP[\mu_\eta=0]=p_0$ and (\ref{EC1-phi}), we get $0=\EE[\mu_\eta(-\infty,0]]=\int_{-\infty}^0 G_1(u) \lambda(du)>0$, which is a contradiction. So $p_0=0$, and we get $\PP[0\in\supp(\mu_\eta)]=1$. Now we consider the general case. To prove that a.s.\ $\supp(\mu_\eta)=\eta\cap (-\infty,v_m]$, it suffices to prove that, for any $a<b\in(-\infty,v_m]\cap \Q$, on the event $\eta\cap (a,b)\ne\emptyset$, a.s.\ $\mu_\eta(a,b)>0$. Fix $a<b\in(-\infty,v_m]\cap \Q$. Let $\tau=\tau^*_b$. Then $\tau$ is a finite $\F$-stopping time, and $\eta(\tau)\in (-\infty,b]$. By DMP, $g_\tau(\eta(\tau+\cdot))-W(\tau) $, denoted by $\eta^\tau$, has the law of an SLE$_\kappa(\rho_\Sigma)$ curve started from $0$ with the force point $0^-$. Let $\mu^\tau$ be the Minkowski content measure on $\eta^\tau\cap(-\infty,0]$. By the last paragraph, a.s.\ $0\in \supp(\mu^\tau)$. By Lemma \ref{lemma-mu-tau}, a.s.\ $\eta(\tau)\in \supp(\mu_\eta|_{(-\infty,\eta(\tau)]})$. On the event $\eta\cap (a,b)\ne\emptyset$, we have $\eta(\tau)\in (a,b]$, which together with a.s.\ $\eta(\tau)\in \supp(\mu_\eta|_{(-\infty,\eta(\tau)]})$ implies that a.s.\ $\mu_\eta(a,b)>0$ on this event. The proof is now complete. \end{proof} \begin{Remark} Let $S=[v_m-A,v_m]$ for some $A>0$. By (\ref{EC2-psi}) and (\ref{JG2'}), for any $\eps\in(0,d)$, $$\EE\Big[\int\!\!\int_{S^2} \frac{\mu_\eta^2 (du_1\otimes du_2)}{|u_1-u_2|^{d-\eps}}\Big] =\int\!\!\int_{S^2} |u_1-u_2|^{-c} G_2(u_1,u_2) \lambda^2(du_1\otimes du_2)$$ $$\lesssim \int_0^{A}\!\! \int_0^{A } L_1^{-\alpha} L_2^{-\alpha-(d-\eps)}\lambda(dL_1) \lambda( dL_2)\le\frac{A^{1-\alpha}}{1-\alpha}\cdot \frac{A^{\eps}}{\eps} <\infty,$$ which implies that a.s.\ $\int\!\!\int_{S^2} \frac{\mu_\eta^2 (du_1\otimes du_2)}{|u_1-u_2|^{d-\eps}}<\infty$. This shows that $\mu_\eta$ has finite $(d-\eps)$-energy. So, on the event $\{\eta\cap S\ne \emptyset\}$, $\mu_\eta|_S$ could serve as a Frostman measure on $\eta\cap S$. \label{Frostman} \end{Remark} \begin{Theorem} For $u\in (-\infty,v_m)$, let $$M_u(t)={\bf 1}_{\{t< \tau^*_u\}} g_t'(u)^\alpha C_{\R} G (W(t),\ulin V(t);g_t(u)),\quad t\ge 0.$$ Let $S\subset (-\infty,v_m)$ be compact, and define the supermartingale $M_{S}(t)=\int_{S} M_u(t) \lambda(du)$. Then we have the Doob-Meyer decomposition: $$M_{S}(t)=\EE[\mu_\eta(S)|\F_t]-\mu_\eta(\eta[0,t]\cap S),\quad t\ge 0.$$ \end{Theorem} \begin{proof} Since $ \mu_\eta(S) -\mu_\eta(\eta[0,t]\cap S)=\mu_\eta(\eta(t,\infty)\cap S) $ and $\mu_\eta(\eta[0,t]\cap S)=\Cont(\eta[0,t]\cap S)\in\F_t$, it suffices to show $\EE[\mu_\eta(\eta(t,\infty)\cap S)|\F_t]=M_S(t)$. Since $\eta(t,\infty)\subset (-\infty,a_{K_t})$ and $\mu_\eta$ is supported by $(-\infty,v_m)$, $\mu_\eta(\eta(t,\infty)\cap S)=\mu_\eta((-\infty,a_{K_t}\wedge v_m)\cap S)$. Let $\eta^{t}$ be the SLE$_\kappa(\ulin\rho)$ curve conditionally on $\F_{t}$ started from $W(t)$ with force points $\ulin V(t)$ such that $\eta(t+\cdot)=f_{t} \circ \eta^{t}$. Let $\mu^{t}$ be the the Minkowski content measure on $\eta^{t}\cap (-\infty, V_m(t)]$. By (\ref{EC1-phi}), if $\phi$ is an $\F_{t}$-measurable random nonnegative measurable function on $(-\infty,V_m(t))$\footnote{More precisely, $\phi$ is an $\F\times {\cal B}(\R)$-measurable function defined on $\{(\omega,x)\in \Omega\times \R: x<V_m(\omega,t)\}$. Here we use $V_m(\omega,t)$ to emphasize the dependence of $V_m(t)$ on $\omega\in\Omega$.}, then \BGE \EE\Big[\int \phi(x) \mu^{t}(du)\Big|\F_{t}\Big] =\int \phi(x) C_{\R} G(W(t),\ulin V(t);x) \lambda(dx).\label{eta-S*}\EDE By Lemma \ref{lemma-mu-tau}, $\mu^{t}=(g_{t})_*((g_{t}')^d\cdot \mu_\eta|_{(-\infty, a_{K_{t}}\wedge v_m)})$. Thus, \begin{align*} &\EE[\mu_\eta((-\infty,a_{K_t}\wedge v_m)\cap S)|\F_{t_0}]=\EE\Big[\int_{(-\infty,a_{K_t}\wedge v_m)\cap S} g_{t}'(g_{t}^{-1}(x))^{-d} \mu^{t}(dx)\Big|\F_{t}\Big]\\ =& \int_{(-\infty,V_m(t))\cap g_t(S)} g_{t}'(g_{t}^{-1}(x))^{-d}C_{\R} G(W(t),\ulin V(t);x) \lambda(dx)\qquad (\mbox{by }(\ref{eta-S*}))\\ =& \int_{(-\infty,a_{K_t}\wedge v_m)\cap S} g_{t}'(u)^{\alpha} C_{\R} G(W(t),\ulin V(t);g_{t}(u)) \lambda(du)\qquad (x=g_{t}(u))\\ = &\int_{(-\infty,a_{K_t}\wedge v_m)\cap S} M_u(t ) \lambda(du)= \int_{S} M_u(t ) \lambda(du)= M_{S}(t), \end{align*} where the second last ``$=$'' holds because $S\subset (-\infty,v_m]$ and $M_u(t)=0$ for $u\ge a_{K_t}$. So we get $\EE[\mu_\eta(\eta(t,\infty)\cap S)|\F_t]=M_S(t)$, as desired. \end{proof} \begin{Remark} Note that the above proof does not use the Doob-Meyer decomposition theorem. When all $\rho_j$ equal $0$, the theorem shows that, if $\eta$ is a chordal SLE$_\kappa$ curve ($\kappa\in(4,8)$) with no force point, the measure $\mu_\eta$ agrees with the covariant measure derived in \cite{cov} up to a multiplicative constant. So we proved the conjecture in \cite[Section 7.3]{cov}. \end{Remark}

proofpile-arXiv_069-12981

\section{Introduction} We examine a quasi-two-dimensional (2D) synthetic active nematic fluid, composed of biologically derived subunits~\cite{Ndlec97}, in which the rod-like microtubule subunits are driven relative to each other by the action of kinesin motor proteins. In recent years, this system has become an important prototype for active fluids, with numerous experimental and theoretical studies~\cite{Sanchez12,Henkin14,Giomi15,DeCamp15,Guillamat16,Doostmohammadi17,Guillamat17,Shendruk17,Lemma19,Tan19}. A key characteristic of this fluid is the emergence of mobile topological defects; points where the orientational order of the nematic phase breaks down. Positive and negative defects occur spontaneously as a result of the active flows, separate from each other and ultimately annihilate with other defects of their opposite charge in the fluid. In recent work, our group analyzed the motion of these defects in the context of chaotic mixing~\cite{Tan19}, revealing that the defects can be treated as virtual stirring rods, which braid around each other, driving advective flows that stretch and fold the material. One of the most striking features of 2D microtubule-based active nematics is the visually distinctive patterns of folded dark and light bands of varying intensity (Fig.~\ref{fig8}a). These folded patterns appear fractal, with finer structure apparent at smaller length scales. These fractal patterns are the subject of the current paper. The fractal analysis of images has a rich history across numerous disciplines, with many quantitative measures of fractal behavior introduced and utilized~\cite{Peitgen88,Soille96,Lopes09,Bies16,Nayak19}. Fractal dimension is a well known tool and various definitions are commonly used. Here we choose to investigate the closely related decay of the power spectrum at large wavenumbers, i.e. small length scales. We analyze experimental images of active nematics, using data sets first reported in Ref.~\onlinecite{Tan19}. We find that power, i.e. the norm-squared of the Fourier coefficients, scales as a power-law $k^{-\beta}$ in the wavenumber $k$. This power-law behavior begins at the active length scale of the system, roughly the spacing between topological defects, and continues down to smaller scales, eventually being dominated by pixel noise. In a log-log plot, power versus $k$ displays a strikingly linear behavior over a wide range of $k$ values, from which we extract $\beta$. The value of $\beta$ is roughly constant in time. Fractal patterns are well known to occur within the passive advection of material in a traditional 2D fluid~\cite{Tel05,Aguirre09,Lai10,Aref89,Muzzio92b,Pentek95}. This is evident on large planetary scales, e.g. plankton blooms in the ocean~\cite{Tel05,Sandulescu07}, or on small laboratory scales, e.g. microfluidic devices~\cite{Truesdell03,Phelan08}. However, in a confined, incompressible flow, such fractal behavior is transient; eventually the system becomes well mixed and the fractal pattern washes out~\cite{Muzzio92b}. Persistent fractal behavior can arise in an open incompressible flow, when a constant stream of impurity enters and then exits a chaotic mixing region. However, the fractal structure witnessed in the active microtubule-based nematic is both persistent and confined. Furthermore, the microtubule system is also incompressible, or area-preserving, at least when averaged over a large enough domain. On smaller length scales, laboratory videos do show local compression of microtubule bundles and the creation and expansion of small voids. (See the online supplemental video.) Despite this, most continuum-based simulations assume local incompressibility from the start~\cite{Giomi15, Doostmohammadi17,Shendruk17}. \begin{figure} \includegraphics[width = 1\columnwidth]{firstframecombo.pdf} \caption{\label{fig8} a) A fluorescence microscopy image of the microtubule system. b) The intensity across the top row of panel a.} \end{figure} We present a simple model for the creation of fractal structure that does not assume incompressibility on small length scales. Area is, however, preserved at the active length scale. Similarly, density is not constant; the patterns seen in the active nematic are, after all, due to local variations in density. Our model contains three distinct steps: 1) extension of the material in the direction of the director field, i.e. along the microtubule bundles, and compression in the perpendicular direction; 2) folding of the material back over itself in a horseshoe pattern; and 3) creation of new large-scale density fluctuations at the active length scale. A critical aspect of this model is that the compressibility in Step 1 varies with density. We find that this simple model generates $k^{-\beta}$ scaling in the power spectrum with a well defined $\beta$ value. To better understand the model's consequences, we conduct an analysis of the fractal formation to first order in the density fluctuations. Within this linear analysis, we analytically compute $\beta$ and find that it depends solely on a dimensionless parameter $r$, that characterizes the compressibility. The fractal behavior is lost if the system is incompressible. This model is not intended to be a quantitatively faithful simulation of all the physics of a microtubule-based active nematic. Rather it is a reduced model designed to highlight the key physical processes necessary to create fractal structure. The lessons learned here could be incorporated into more quantitatively accurate numerical simulations in the future. Note that our work should be distinguished from prior studies of scale-free behavior in active nematics~\cite{Giomi15,Alert20}. These studies assume uniform density, so are not sensitive to the density fluctuations considered here. Instead, they consider the decay in the energy and enstrophy spectra, which have a universal power-law behavior at small scales. This paper is organized as follows. Section~\ref{sec:experiments} presents our results on the spectral decay of experimental images of the active nematic material. Section~\ref{sec:model} explains our three-step model of fractal generation. Section~\ref{sec:linear} linearizes the model in the density fluctuations. Section~\ref{sec:incompressible} discusses the special case in which Step 1 of our model is incompressible but Step 3 is not. Conclusions are in Sect.~\ref{sec:conclusions}. \section{Experimentally measured power-spectrum decay of microtubule-based active nematics} \label{sec:experiments} The analysis here is based on the experimental data presented in Ref.~\onlinecite{Tan19}, where a complete description of the experimental technique is given. In brief, a quasi-2D layer of microtubules is suspended at an oil-water boundary. The microtubules are at high density and form bundles due to the use of a depletion agent (polyethylene glycol). Kinesin molecular motors, joined together in clusters, serve to cross-link the microtubules. ATP is added to the solution to power the molecular motors, which walk along the microtubules from their negative end to their positive end. Neighboring microtubules of opposite polarity are pushed in opposing directions by the motor clusters to produce the active stress in the material. This local stress produces large-scale dynamics in the 2D microtubule layer. The system is imaged using fluorescence microscopy. See Fig.~\ref{fig8}a and the online supplemental video. Experiments were performed with six different ATP concentrations. The higher the ATP concentration, the more activity is induced in the system and the faster the system evolves, assuming all other factors are held constant. \begin{figure} \includegraphics[width = 1\columnwidth]{ExperimentalPS.pdf} \caption{\label{fig7} a) Power spectrum (black) of Fig.~\ref{fig8}b. The blue curve is the power spectrum averaged in $\log k$ space using a Gaussian of width $\sigma = 0.4$. (See Appendix.) The red line is a linear fit to this average. The vertical dashed line denotes the $k$ value corresponding to the velocity autocorrelation length. Note that the scaling is chosen so that $k = L_0/\lambda$ for wavelength $\lambda$, where $L_0$ is the width of the image; b) The 1D power spectrum averaged over all rows of the image in Fig.~\ref{fig8}a; c) The angle-averaged 2D power spectrum of the image in Fig.~\ref{fig8}a; d) The 1D (black) and 2D (green) $\beta$ values as a function of time. The red curve is the green curve shifted down by 1. All data in these figures were taken at an ATP concentration of $50 \mu$M.} \end{figure} We analyse the power spectrum of the resulting experimental images $f(x,y)$ in two ways. In the first technique, we compute the 1D Fourier transform $\tilde{f}(k_x;y)$ of each row; here $y$ labels the row and $k_x$ is the $x$ wave vector. The power spectrum of row $y$ is then the norm-squared of the Fourier coefficients versus $k = |k_x|$, i.e. $P(k;y) = |\tilde{f}(k;y)|^2$. For example, Fig.~\ref{fig8}b shows the intensity of the top row in Fig.~\ref{fig8}a; Fig.~\ref{fig7}a shows a log-log plot of the resulting power spectrum (black). To better see the spectral decay, we smooth the power spectrum in $\log k$ as discussed in the Appendix. This average is shown as the smooth blue curve. Finally, the blue curve is fit to a linear decay, shown in red, with slope $-1.33$. The vertical dashed line marks the value $k = L_0/\ell$, where $L_0$ is the width of the image and $\ell$ is the active length scale, defined here as the velocity autocorrelation length~\cite{Lemma19,Hemingway16} of the active nematic, computed in Ref.~\onlinecite{Tan19}. Note that the linear fall off begins at $k$ values larger than approximately $L_0/\ell$. The initial (black) power spectrum in Fig.~\ref{fig7}a has large fluctuations, which we can reduce by averaging $P(k;y)$ over all the rows $y$ of the image, producing the averaged 1D power spectrum $P_1(k)$. This is shown as the black power spectrum in Fig.~\ref{fig7}b, where the fluctuations have diminished dramatically. The red fit line is obtained as in Fig.~\ref{fig7}a, i.e. the power spectrum is first smoothed in $\log k$ (not shown in Fig.~\ref{fig7}b), and then the red line is fit to this average. The resulting slope is $-1.37$, comparable to that of Fig.~\ref{fig7}a. The second technique for computing the spectral decay is to first take the 2D Fourier transform $\tilde{f}(k_x,k_y)$ of the entire image (in practice, a square subset of the image). Then $P(k_x,k_y) = |\tilde{f}(k_x,k_y)|^2$ is averaged over angle to produce the 2D power spectrum \begin{equation} P_2(k) = \frac{1}{2 \pi} \int_0^{2 \pi} P(k \cos(\phi), k \sin(\phi)) \, d\phi, \label{r21} \end{equation} which is shown in Fig.~\ref{fig7}c. Its fluctuations are roughly comparable to that of averaging over the rows (Fig.~\ref{fig7}b). The data is again smoothed in $\log k$ and subsequently fit to the red line, with slope $-2.37$. Note that these power spectra show that there is a single active length scale, which can be defined as the velocity autocorrelation length $\ell$, as done here. This is consistent with what is known from prior publications, e.g. Ref.~\onlinecite{Lemma19}. On scales smaller than the active length scale, the density distribution appears scale free. \begin{figure} \includegraphics[width = .7\columnwidth]{betavsATP.pdf} \caption{\label{fig9} The $\beta$ value averaged over all 500 frames of the active nematic video for experiments performed at six different ATP concentrations: 50, 75, 100, 250, 500, 1000 $\mu$M. The black data is computed from $P_1$ and the red data from $P_2$, shifted down by 1. Error bars are the standard deviations of $\beta$ over all frames.} \end{figure} We now investigate the time-dependence of the $\beta$'s. Figure~\ref{fig7}d shows $\beta$ computed from $P_1$ (black) and $P_2$ (green) for each of 500 experimental frames. There is modest variation in each graph. Note that the two curves differ by about 1. This is made clearer by the red curve in Fig.~\ref{fig7}d, which is the green curve shifted down by 1. The reason for this correspondence can be understood as follows. Suppose the original image $f(x,y)$ were independent of $y$, consisting only of vertical stripes. Then $\tilde{f}(k_x, k_y)$ could be written as $\tilde{f}(k_x; y) \delta_{k_y 0}$, i.e. the 2D Fourier transform depends on $k_x$ just as the 1D Fourier transform of any (equivalent) row of the image and is nonzero only when $k_y = 0$, since the image is constant in $y$. If $P(k_x;y)$ falls off like $k^{-\beta}$, then the angular average of $P(k_x, k_y)$ is one factor of $k$ smaller due to dividing by the circumference $2 \pi k$. The close agreement between the red and black curves confirms the validity of our analysis. Separate experiments were run for six different ATP concentrations. Figure~\ref{fig9} shows the $\beta$ value averaged over all 500 frames for each ATP concentration. The black data is computed from $P_1$ and the red data from $P_2$, shifted down by one. Error bars are the standard deviation of $\beta$ over the 500 frames. Note that the $P_1$ and $P_2$ data are consistent across all six experimental runs. However, there are significant variations in $\beta$ from one experiment to the next, indicating that the nature of the scale-free behavior is not universal. \section{Model of fractal generation} \label{sec:model} \subsection{Step 1: One-dimensional model of compression dynamics} We consider a model of the material in which the microtubule bundles, and hence the directors, point along the $y$ direction everywhere, and in which the density $\rho$ depends solely on $x$. See Fig.~\ref{fig10}a. We assume that this material is under a constant stress perpendicular to the directors and that the response of the material depends solely on the density $\rho(x)$. Specifically, the relative change in length $d L/L$ of any interval $[x_0, x_1]$, where $L = x_1-x_0$, over time $dt$ depends solely on the average density $\bar{\rho}$ of the interval, i.e. \begin{align} \frac{d L}{dt} & = g(\bar{\rho}) L, \label{r1} \\ \bar{\rho} &= \frac{1}{L} \int_{x_0}^{x_1} \rho(x) dx, \end{align} where $g$ is the compression rate for a given average density. The assumption that Eq.~(\ref{r1}) applies to \emph{all} intervals is equivalent to $g$ being an affine function, which we write as \begin{equation} g(\bar{\rho}) = -\alpha(1 - \bar{\rho}/\rho_m), \label{r28} \end{equation} where $\alpha$ and $\rho_m$ are constants. We take $\rho_m > 0$ and $\alpha > 0$, so that the material becomes less compressible at higher density, becoming incompressible for the given stress when the average density is $\rho_m$. Using $v(x,t)$ for the velocity of the material at position $x$ and time $t$, Eqs.~(\ref{r1})--(\ref{r28}) imply \begin{align} \frac{1}{L} \int_{x_0}^{x_1} \frac{\partial v}{\partial x} dx & = \frac{1}{L}[v(x_1) - v(x_0)] = \frac{1}{L} \frac{d L}{dt} = g(\bar{\rho}) \nonumber \\ & = -\alpha + \frac{\alpha}{\rho_m} \frac{1}{L} \int_{x_0}^{x_1} \rho(x) dx \nonumber \\ & = \frac{1}{L} \int_{x_0}^{x_1} -\alpha \left(1 - \frac{\rho(x)}{\rho_m}\right) dx, \end{align} from which we find \begin{equation} \frac{\partial v}{\partial x} = -\alpha \left( 1 - \frac{\rho(x)}{\rho_m} \right). \label{r2} \end{equation} Applying the continuity equation \begin{equation} \frac{\partial}{\partial t} \rho + \frac{\partial}{\partial x} (v \rho) = 0, \end{equation} we find \begin{equation} \frac{D}{Dt} \rho(x,t) = \frac{\partial}{\partial t} \rho + v \frac{\partial}{\partial x} \rho = -\rho \frac{\partial}{\partial x} v = \alpha \rho \left( 1 - \frac{\rho}{\rho_m}\right). \label{r3} \end{equation} Here $D/Dt$ is the advective derivative. Note that the quadratic nature of Eq.~(\ref{r3}) produces two fixed points for $\rho$ in the Lagrangian frame, one at $\rho = 0$ and one at $\rho = \rho_m$; these are linearly unstable and stable, respectively. See Fig.~\ref{fig10}b. Thus, a typical parcel of material will be driven toward the maximum density $\rho_m$ as it is compressed; a block of material already at density $\rho_m$ is incompressible. Note that the combination of Eqs.~(\ref{r2}) and (\ref{r3}) yields an integro-differential equation for the density $\rho(x,t)$, which can be solved numerically. \begin{figure} \includegraphics[width = 1\columnwidth]{DrhoDt.pdf} \caption{\label{fig10} a) The microtubules (green) and directors (black double arrow) are aligned in the $y$ direction. The microtubule density $\rho$ depends only on $x$. The microtubules are compressed in the $x$ direction. b) The time-rate-of-change of the density in the Lagrangian frame. The density is driven toward the stable fixed point $\rho_m$. c) Stretching in the $y$ direction is added to the model. d) The density is driven toward the new stable fixed point $\rho_a < \rho_m$.} \end{figure} We next incorporate into the dynamics stretching in the $y$-direction, given by the rate $\gamma>0$, which is taken to be constant in position and time and independent of density. See Fig.~\ref{fig10}c. This does not change Eq.~(\ref{r2}), but modifies Eq.~(\ref{r3}) to be \begin{align} \frac{D}{Dt} \rho(x,t) &= \frac{\partial}{\partial t} \rho + v \frac{\partial}{\partial x} \rho = \rho ( \alpha - \gamma - \alpha \frac{\rho}{\rho_m}) \nonumber \\ & = \alpha \rho \left( \frac{ \rho_a - \rho}{\rho_m} \right), \label{r4} \end{align} where \begin{equation} \rho_a = \rho_m (\alpha - \gamma)/\alpha < \rho_m. \label{r27} \end{equation} See Fig.~\ref{fig10}d. With this modification, a parcel of material is driven toward the steady state density $\rho_a$ instead of $\rho_m$. Note that the limit $\rho_m = \infty$, with $\alpha$ and $\gamma$ constant, removes the density dependence from Eqs.~(\ref{r2}) and (\ref{r4}). Density then increases at the uniform rate of $\alpha - \gamma$ everywhere. This behavior is identical to the compression-stretching step for both the baker and Lozi maps, and it is well known that both of these maps have a fractal attractor, with a singular density distribution, but only when the system is contracting, i.e. $\alpha - \gamma > 0$. This is not an accurate model for the microtubule-based system, which must be area-preserving on average. One could instead consider the incompressible limit of Eq.~(\ref{r4}) by setting $\alpha = \gamma$. Though completely area-preserving dynamics will not produce fractal structure, it is possible to produce fractal structure if Step 1 is area preserving and it is combined with density fluctuations introduced in Step 3. (See Sect.~\ref{sec:densityvariations} below.) This special case is considered in Sect.~\ref{sec:incompressible}. Until then, we take $\rho_m$ to be finite. Though area is not conserved locally in the microtubule system, we do require that area be conserved across the entirety of the material, as noted above. This means that the stretching rate must be balanced by the compression rate, i.e. $\gamma = -g(\bar{\rho}_T)$, where $\bar{\rho}_T$ is the average density of the entire block of material. Since $\gamma$ is assumed constant in time, $\bar{\rho}_T$ must also be constant in time. Equation~(\ref{r4}) then implies that \begin{equation} \bar{\rho}_T = \rho_a. \label{r29} \end{equation} The ``a'' subscript stands for ``average''. By an appropriate choice of length and time scales, we can set $\alpha = \rho_m = 1$ so that Eqs.~(\ref{r2}) and (\ref{r4}) become \begin{align} \frac{D}{Dt} \rho(x,t) & = \frac{\partial}{\partial t} \rho + v \frac{\partial}{\partial x} \rho = \rho ( r - \rho), \label{r5} \\ \frac{\partial v}{\partial x} & = \rho - 1, \label{r6} \end{align} where \begin{equation} r = \rho_a/\rho_m \le 1 \label{r26} \end{equation} is dimensionless. Equation~(\ref{r29}) for the density averaged over the entire material becomes \begin{equation} \bar{\rho}_T = r. \label{r22} \end{equation} Note that there is no incompressible limit of Eqs.~(\ref{r5}) and (\ref{r6}), since the rescaling assumes $\rho_m$ is finite. \subsection{Step 2: Folding dynamics} After the system has been compressed to one-half its original width, we introduce a fold. In the one-dimensional model, this means that the system returns to its original width $L_0$, with density \begin{equation} \rho(x,t) = \left\{ \begin{array}{ll} \rho(x,t) & 0 < x < L_0/2, \\ \rho(L_0/2-x,t) & L_0/2 < x < L_0. \end{array} \right. \label{r10} \end{equation} An alternative perspective here is to extend $\rho(x,t)$ to an $x$-periodic function with wavelength $L_0$ that is also an even function in $x$. In this case, the function $\rho(x,t)$ will naturally evolve into the form of Eq.~(\ref{r10}) once the original interval is compressed to half its length. Thus the fold step is naturally incorporated into the compression step. To implement this step, we compute the time $T$ to compress the entire block of material from length $L_0$ to an arbitrary length $L$. First, assuming without loss of generality that the position of the left edge of the block is fixed at $x=0$, the velocity at the right edge $x = L$ is computed as \begin{align} \frac{dL}{dt} & = \int_0^L \frac{\partial v}{\partial x} dx = - \int_0^L ( 1 - \rho) dx = -( 1 - r) L, \end{align} using Eqs.~(\ref{r6}) and (\ref{r22}). Hence, the time to compress the block from $L_0$ to $L$ is \begin{equation} T = \frac{ \log(L_0/L)}{1 - r} = \frac{ \log(2)}{ 1 - r }, \label{r11} \end{equation} where in the final step we have set $L = L_0/2$. \subsection{Step 3: Imposing large-scale density variations} \label{sec:densityvariations} Immediately after the folding step we impose a multiplicate density variation $R(x)$ and normalize the density so that the average density remains $r$. Explicitly, \begin{equation} \rho(x) \mapsto \frac{R(x) \rho(x)}{\langle R(x) \rho(x) \rangle} r, \label{r12} \end{equation} where the angled brackets denote the average over $x$. Consistent with the viewpoint that $\rho(x)$ is an even, $L_0$-periodic function, we similarly extend $R(x)$ to an even, $L_0$-periodic function. \subsection{Fractal generation} The sequence of compression by a factor of two, folding, and large-scale density variations generates a mapping $M$ of an original density $\rho_0(x)$ to a density $\rho(x)$; though the folding step can be ignored if $\rho(x)$ is even and $L_0$-periodic as discussed above. This mapping depends on the form $R(x)$ of the density variations imposed and the value of $r$ in the compression step. The mapping $M$ can then be repeatedly applied to the density. We numerically observe that it converges to a stationary fractal function, as seen in Fig.~\ref{fig2} for the density variation \begin{equation} R(x) = 1 + B \cos(x), \label{r18} \end{equation} with $B = 0.2$. Figure~\ref{fig3} shows the power spectrum (black) of the fractal image. The blue line is the power spectrum averaged over log(k), and the red line is a linear fit to this average. For comparison, Fig.~\ref{fig:LargeAmp} shows the stationary density distribution and power spectrum for larger amplitude density variations with $B = 0.7$. \begin{figure} \includegraphics[width = 1\columnwidth]{evolution.pdf} \caption{\label{fig2} The density $\rho$ after each iterate $n$ of the map $M$ with $r= 0.25$, $L_0 = 2 \pi$, and density variation given by Eq.~(\ref{r18}) with $B = 0.2$. More and more fine-scale structure develops at each iterate. By $n = 15$ the density has visually converged to an invariant state.} \end{figure} \begin{figure} \includegraphics[width = 0.7\columnwidth]{fractalPS.pdf} \caption{\label{fig3} The power spectrum (black) of the final graph in Fig.~\ref{fig2} ($n = 15$). The blue curve is the power spectrum smoothed according to Eq.~(\ref{r19}). The smoothed spectrum is fit to a line (red) with slope -1.71.} \end{figure} \section{Linear analysis of fractal generation} \label{sec:linear} \subsection{Linearization of compression dynamics} The constant density \begin{equation} \rho_0(x,t) = r \end{equation} and time-invariant velocity \begin{equation} v_0(x,t) = (r-1)x \label{r8} \end{equation} are steady state solutions of Eqs.~(\ref{r5}) and (\ref{r6}). Note that we have chosen a frame in which $v = 0$ at the origin $x=0$. Expanding about this solution using \begin{align} \rho & = \rho_0 + \epsilon g, \label{r14} \\ v & = v_0 + \epsilon f, \end{align} where $\langle g \rangle = 0$, we find to first order in $\epsilon$ \begin{align} \frac{\partial}{\partial t} g &= (1-r)x \frac{\partial}{\partial x} g -r g, \label{r9} \\ \frac{\partial f}{\partial x} & = g. \end{align} \begin{figure} \includegraphics[width = 1\columnwidth]{LargeAmp.pdf} \caption{\label{fig:LargeAmp} a) The stationary density for the same parameters as Fig.~\ref{fig2}, except $B = 0.7$. b) The corresponding power spectrum. The fit slope $-1.60$ differs slightly from Fig.~\ref{fig3}.} \end{figure} Note that the density perturbation $g$ decouples from $f$. Direct substitution shows that the solution to Eq.~(\ref{r9}) for a given initial condition $g(x,0)$ has the form \begin{equation} g(x,t) = h(t)g(x/\ell(t), 0), \label{r23} \end{equation} where \begin{align} \ell(t) &= \exp( (r-1) t), \\ h(t) &= \exp(-rt). \end{align} This implies that the Fourier transform $\tilde{g}(k,t)$ evolves in time as \begin{equation} \tilde{g}(k,t) = h(t) \tilde{g}(k \ell(t), 0 ). \end{equation} Thus as time evolves, the graph of the power spectrum in log-log space shifts to larger wavenumbers and smaller amplitudes by translating along a line of slope $-2r/(r-1)$. After time $T$ given by Eq.~(\ref{r11}), we find \begin{equation} \tilde{g}(k,T) = 2^{- r/(1-r) } \tilde{g}(k/2, 0). \label{r13} \end{equation} This equation is valid when the Fourier transform is smooth, but must be treated carefully when the Fourier transform is singular, as when $\rho$ is $L_0$-periodic at $t = 0$ and $t = T$. In this case $\tilde{g}(k,t)$ is zero except when $k = 2\pi n/L_0$ for $n = 0, 1, 2, 3, ...$, at which value it is a delta function whose amplitude we denote $\tilde{g}_n(t)$. These coefficients satisfy \begin{equation} \tilde{g}_n(T) = \left\{ \begin{array}{ll} 2^{- r / (1-r) } \tilde{g}_{ n/2 }(0), & n \text{ even} \\ 0, & n \text{ odd.} \end{array} \right. \label{r15} \end{equation} Thus, the spacing between nonzero values of $\tilde{g}_n$ has increased by a factor of 2 over time $T$. Thus, when smoothed over $n$, as discussed in the Appendix, $|\tilde{g}_n|^2$ scales as \begin{align} \langle |\tilde{g}_n(T)|^2 \rangle & \approx 0.5 \times 2^{- 2r/(1-r) } \langle |\tilde{g}_{\lfloor n/2 \rfloor } (0)|^2 \rangle \nonumber \\ & \approx 2^{- (1 + r)/(1-r) } \langle |\tilde{g}_{\lfloor n/2 \rfloor } (0)|^2 \rangle, \label{r17} \end{align} where the angled brackets denote the average over $n$ and $\lfloor \quad \rfloor$ denotes the floor function. From this we see that the smoothed power spectrum $\langle |\tilde{g}_n(t)|^2 \rangle$ slides down a line of slope $(1+r)/(1-r)$ in log-log space. \subsection{Linearization of new density variations} We assume the density variations introduced by Eq.~(\ref{r12}) scale as $\epsilon$, i.e. \begin{equation} R(x) = 1 + \epsilon p(x), \label{r30} \end{equation} with $\langle p(x) \rangle = 0$. Then, to first order in $\epsilon$ and using the expansion Eq.~(\ref{r14}), Eq.~(\ref{r12}) becomes \begin{equation} g(x) \mapsto g(x) + p(x), \end{equation} which in terms of the Fourier coefficients, we write as \begin{equation} \tilde{g}_n \mapsto \tilde{g}_n + \tilde{p}_n. \end{equation} \subsection{Linearized fractal generation} \label{sec:linearfractal} We denote by $A$ the linear operator that evolves $g(x,0)$ forward for time $T$ according to Eq.~(\ref{r23}). Then the total linearized dynamics is \begin{equation} g \mapsto A g+ p. \label{r24} \end{equation} Repeated application of this map, beginning with $g = 0$ and using \begin{equation} p(x) = 0.2 \cos(x), \label{r25} \end{equation} converges to the stationary function (blue) in Fig.~\ref{fig4}, which is consistent with the nonlinear stationary density from Fig.~\ref{fig2} and reproduced in Fig.~\ref{fig4} (black). It is straightforward to see that \begin{equation} P = \sum_{m = 0}^\infty A^m p, \end{equation} is the unique invariant function of Eq.~(\ref{r24}) and that any initial function will converge to it. \begin{figure} \includegraphics[width = .7\columnwidth]{fractalcompare.pdf} \caption{\label{fig4} The invariant function (blue) obtained from repeated application of the linearized dynamics Eq.~(\ref{r24}), with $r = 0.25$ and $L_0 = 2\pi$. For comparison is the invariant function (black) obtained from the nonlinear dynamics and shown previously in Fig.~\ref{fig2}.} \end{figure} It was noted to us by S.~Berman that when $p(x)$ is a cosine, as in Eq.~(\ref{r25}), the invariant function $P(x)$ is the Weierstrass function, which was the first published example of a continuous function that is nowhere differentiable. Furthermore, fixed-point equations with the form of Eq.~(\ref{r24}) were introduced by de Rham to characterize such singular functions. Such techniques have a history of applications to dynamical systems, e.g. Ref.~\onlinecite{Tasaki98}. In Fourier space, the linear operator $\tilde{A}$ that maps coefficients $\tilde{g}_n(0)$ forward to $\tilde{g}_n(T)$ is given by Eq.~(\ref{r15}), and the invariant function is \begin{equation} \tilde{P} = \sum_{m = 0}^\infty \tilde{A}^m \tilde{p}, \end{equation} with coefficients \begin{equation} \tilde{P}_n = \sum_{m = 0}^\infty 2^{- m r/(1-r)} \tilde{p}_{n/2^m}, \label{r16} \end{equation} where it is understood that $\tilde{p}_{n/2^m} = 0$ if $n/2^m$ is not an integer. \begin{figure} \includegraphics[width = .7\columnwidth]{linearfractalPS.pdf} \caption{\label{fig5} The power spectrum (black) of the blue curve in Fig.~\ref{fig4}. The blue curve above is the power spectrum smoothed according to Eq.~(\ref{r19}). The smoothed spectrum is fit to a line (red) with slope -1.67.} \end{figure} Assuming that the density variation function $p(x)$ is analytic, its Fourier coefficients $\tilde{p}_n$ decay exponentially in $n$, meaning that the sum in Eq.~(\ref{r16}) is dominated by the first nonzero term. This term will have an $m$ value satisfying $m \approx \log(n)/\log(2)$, and hence \begin{equation} \tilde{P}_n \propto n^{-r/(1-r)}, \end{equation} from which follows \begin{equation} |\tilde{P}_n|^2 \propto n^{-2r/(1-r)}, \end{equation} and finally \begin{equation} \langle |\tilde{P}_n|^2 \rangle \propto \frac{1}{n} n^{-2r/(1-r)} = n^{-(1+r)/(1-r)}, \end{equation} where the factor of $1/n$ that appears upon averaging is due to the diminishing density of nonzero coefficients, as discussed prior to Eq.~(\ref{r17}). Thus in the linearized model, we find \begin{equation} \beta = \frac{1+r}{1-r}. \label{r20} \end{equation} This is the same scaling that we previously identified for the temporal behavior in Eq.~(\ref{r17}), highlighting the origin of this scaling in the compression dynamics. To check this formula numerically, we plot in Fig.~\ref{fig5} the power spectrum of the blue curve in Fig.~\ref{fig4}, which has $r = 0.25$. The linear fit yields a slope of $-1.67$, which agrees with the value $\beta = 1.67$ obtained from Eq.~(\ref{r20}). Figure~\ref{fig6} provides a more comprehensive comparison of the nonlinear analysis, the linear analysis, and Eq.~(\ref{r20}). The black and red dots show $\beta$ versus $r$ for the nonlinear and linearized systems, respectively, using $R$ and $p$ given by Eqs.~(\ref{r18}) (with $B = 0.2$) and (\ref{r25}). The data are quite consistent with each other. The blue curve is the graph of Eq.~(\ref{r20}), and it nicely tracks the red data points. \begin{figure} \includegraphics[width = .7\columnwidth]{betavsr.pdf} \caption{\label{fig6} The variation of $\beta$ with $r$. The black dots are computed numerically from the nonlinear analysis with $B = 0.2$. The red dots are computed numerically from the linear analysis. The blue curve is the graph of Eq.~(\ref{r20}).} \end{figure} \subsection{The fluctuation injection scale} \begin{figure} \includegraphics[width = .7\columnwidth]{varyinjection.pdf} \caption{\label{fig:injection} The (smoothed) power spectra of the linearized dynamics for three different sinusoidal forms for the injected density fluctuations. The black curve uses an injection wavelength equal to the full width of the material. The red and blue curves use wavelengths that are a factor of 10 and 100 smaller, respectively. The dashed vertical lines denote the scale of the injected density fluctuations. Note that the linear behavior only occurs when $k$ is greater than the injection value.} \end{figure} The value of $\beta$ does not depend on the details of the density fluctuations introduced by $p(x)$, except that its Fourier coefficients should fall off sufficiently rapidly. However, the $k^{-\beta}$ scaling only begins at length scales below the smallest length scale of $p(x)$. Figure~\ref{fig:injection} illustrates this by plotting the smoothed power spectrum for $p(x)$ with wavelengths equal to the width of the block of material divided by 1 [$p(x) = \cos(x)$, black], 10 [$p(x) = \cos(10 x)$, blue], and 100 [$p(x) = \cos(100 x)$, red]. These wavelengths are the scales at which density fluctuations are injected into the system. The $k$ value for each wavelength is denoted by the dashed vertical line of the corresponding color. The linear behavior only begins to the right of each line, i.e. at length scales smaller than the injection scale. \section{The special case of incompressibility in Step 1} \label{sec:incompressible} Returning to Eq.~(\ref{r4}) and taking the limit $\rho_m = \infty$ and setting $\alpha = \gamma$, density is conserved in the Lagrangian frame, \begin{equation} \frac{D}{Dt} \rho(x,t) = 0, \end{equation} so that the material is incompressible, regardless of its density. Furthermore, Eq.~(\ref{r2}) reduces to \begin{equation} \frac{\partial v}{\partial x} = -\alpha, \end{equation} so that the system is uniformly compressed in the $x$ direction. Combining this with the density variations in Step 3, Eq.~(\ref{r12}), we obtain a steady state fractal density. (We ignore subtle points about the convergence of this function, which is technically a distribution.) This dynamics is admittedly physically inconsistent, as we cannot imagine a situation in which density fluctuations could arrise via folding on the active scale (Step 3) without the material also being subject to compression or expansion in Step 1. Nevertheless, it is instructive to compute the resulting value of $\beta$, which is straightforward when linearizing the density fluctuations, i.e. using Eq.~(\ref{r30}). Repeating the analysis in Sec.~\ref{sec:linearfractal}, one finds \begin{equation} \tilde{P}_n = \sum_{m = 0}^\infty \tilde{p}_{n/2^m}, \end{equation} which simply reflects the fact that as a cosine density fluctuation is squeezed in $x$, its amplitude remains constant. This is the same result as Eq.~(\ref{r16}) with $r = 0$ and therefore gives the same power spectrum decay with $\beta = 1$. Thus, even if one were to assume incompressibility in Step 1, the resulting value of $\beta$ disagrees with the experimental measurements in Fig.~\ref{fig9} (at least within the linearized analysis.) \section{Conclusions} \label{sec:conclusions} By using a simple model of fractal generation, we have highlighted the necessity both of introducing density fluctuations on large scales and of a variable compressibility in the material. At least within the linearized analysis, the details of the density fluctuations are irrelevant to the value of $\beta$. The density fluctuations can take any functional form, so long as they have a minimum length scale. The $k^{-\beta}$ scaling then manifests at scales below this smallest length scale. In the case of the microtubule-based active nematic material, the injection scale of the density fluctuations is the active length scale of the system, i.e. the average defect spacing. Physically, the injected density variations are most naturally explained by the creation of defect pairs and the associated fracturing. The $\beta$ parameter depends on the compressibility via $r$, and in the linearized analysis we derived an explicit relationship, Eq.~(\ref{r20}). It is tempting to thus translate the experimentally measured values of $\beta$ (Fig.~\ref{fig9}) into a material parameter $r$ via \begin{equation} r = \frac{\beta - 1}{\beta + 1}, \end{equation} yielding values of $r$ in the range $0.1$ -- $0.3$. But what is the physical meaning of $r$? Within the context of the simple model, we can combine Eqs.~(\ref{r27}) and (\ref{r26}), to rewrite $r$ as \begin{equation} r = 1 - \gamma/\alpha. \end{equation} Recall $\gamma$ is the (uniform and constant) extension rate and $\alpha$ can be interpreted as the transverse compression rate in the low-density limit [Eq.~(\ref{r2})]. Whether this interpretation from the simple fractal model will hold for the actual laboratory system is not clear. This issue could be addressed via a realistic 2D hydrodynamic model with a density-dependent compressibility. Specifically, the dependence of $\beta$ on the material properties and system parameters would make an interesting and informative study. Finally, it would be interesting to identify whether other physical systems exhibit a similar mechanism for fractal formation. Such systems would exhibit fractal structure not in the patterning of impurities mixed into the system, but in morphological properties such as density or surface texture. The classic example of chaos in a taffy puller comes to mind~\cite{Thiffeault18}, in which large-scale surface irregularities are introduced periodically by the folding of the taffy and then pushed down to smaller length scales by the stretching and compression dynamics. \begin{acknowledgments} The authors acknowledge generous funding from the National Science Foundation, through several awards: DMR-1808926, NSF-CREST: Center for Cellular and Biomolecular Machines at UC Merced (HRD-1547848), and from the Brandeis Biomaterials facility MRSEC-1420382, which provided materials. We would also like to thank Simon Berman for critiquing the manuscript, and in particular for pointing out the work in Ref.~\onlinecite{Tasaki98}. \end{acknowledgments} \section*{DATA AVAILABILITY} The data that support the findings of this study are available from the corresponding author upon reasonable request.

proofpile-arXiv_069-13197

\section{Introduction} Cluster algebras were first defined in \cite{clusteri}, they are generated by cluster variables. Starting with an initial seed; a quiver and a set of cluster variables, there is an operation called mutation, at any vertex, which gives a new quiver and a new cluster variable. The set of cluster variables is given by applying all possible sequences of mutations. By restricting so that we are only allowed to mutate at sinks or sources (where mutation becomes much simpler) we find cluster variables $X^i_n$ satisfying the frieze pattern: \begin{equation}\label{friezeformulaintro} X^i_nX^i_{n+1}=1+\left(\prod_{j\rightarrow i}(X^j_n)^{|b_{ji}|}\right)\left(\prod_{i\rightarrow j}(X^j_{n+1})^{|b_{ji}|}\right) \end{equation} for each vertex $i$ and $n\in\mathbb{Z}$. Here $b_{ji}=-b_{ij}$ is the number of arrows from $j$ to $i$ in $Q$. We consider (\ref{friezeformulaintro}) as an equation for determining $X^i_{n+1}$. Remarkably, if the initial quiver is of Dynkin type, then this isn't a restriction at all, as proved in \cite{clustersiv}. \begin{theorem} If the quiver $Q$ is of Dynkin type then there are finitely many cluster variables and each of them can be found on the frieze pattern. \end{theorem} If $Q$ if not Dynkin then the frieze pattern does not contain all of the cluster variables. The aim of this work is to describe what appears outside of the frieze pattern for the affine quivers $\tilde{A}$ and $\tilde{D}$. The frieze pattern was considered in \cite{kellerscherotzke} where, via the cluster category, the authors proved that linear relations exist between the frieze variables if and only if $Q$ is Dynkin or affine type. A more direct construction of these linear relations was given in \cite{fordyhone,pallisterlinear} by proving that there exists periodic quantities $J_n$ and $\tilde{J}_n$ for $\tilde{A}$ type and $J'_n$ for $\tilde{D}$ type. These are functions of the $X^i_n$ that are fixed by sending $n \mapsto n+a$ for an appropriate $a$. For example, in $\tilde{D}$ case, we have \[ J'_n:=\frac{X^1_{n+1}+X^1_{n-1}}{X^2_n}=\frac{X^1_{n+N-1}+X^1_{n+N-3}}{X^2_{n+N-2}}=J'_{n+N-2} \] where the vertices $1$ and $2$, that appear as superscripts, are given in Figure \ref{Dquiver}. The first result of this paper is that these periodic quantities are also cluster variables, but do not live on the frieze pattern. In order to find the rest of the cluster variables we rely on a method pioneered in \cite{clustersandtriangulated1} for viewing cluster variables as arcs of triangulations of particular surfaces. For $\tilde{A}_{q,p}$ the surface is an annulus with $q$ and $p$ marked points on either boundary component. An example is shown in Figure \ref{exampletriangulation} which gives an annulus by gluing along the dotted lines. For $\tilde{D}_N$ the surface is a disc with two internal marked points and $N-2$ marked points on the boundary, shown in Figure \ref{Dtypebefore}. By finding all of the possible arcs for $\tilde{A}$ and $\tilde{D}$ types, we prove that all cluster variables can be described in terms of the frieze sequence (\ref{friezeformulaintro}) and determinants: \begin{equation}\label{determinantintro} D^m_a(F_n):= \begin{vmatrix} F_n & 1 & 0 \\ 1 & F_{n+a} & 1 & 0 \\ 0 & 1 & F_{n+2a} & 1 & 0 \\ & 0 & 1 & F_{n+3a} & \ddots \\ & & 0 & \ddots & \ddots & 1 \\ & & & & 1 & F_{n+(m-1)a} \end{vmatrix} \end{equation} where $F$ is a function of the periodic quantities as described in the following two theorems. \begin{theorem}\label{Atypearcs} \begin{enumerate}[(i)] For $\tilde{A}_{q,p}$ cluster algebras the arcs on the annulus are of three types: \item The arcs that connect the two boundary components. These are in bijection with the frieze variables $X^i_n$. \item The arcs connecting the boundary component with $q$ marked points to itself, in bijection with the cluster variables $D^l_p(J_{jp})$, for $j=0,\ldots,q-1$ and $l=1,\ldots,q-1$. \item The arcs connecting the other boundary component (with $p$ marked points) to itself, in bijection with $D^l_q(\tilde{J}_{jq})$, for $j=0,\ldots,p-1$ and $l=1,\ldots,p-1$. \end{enumerate} \end{theorem} The corresponding result for $\tilde{D}$ type is: \begin{theorem} \begin{enumerate}[(i)] For $\tilde{D}_{N}$ cluster algebras we divide the arcs of the twice punctured disk into four types. \item The arcs that connect the boundary vertices such that the two punctures lie on different sides of this arc. \item The arcs connecting the boundary to the punctures. These and the arcs of (i) are in bijection with the frieze variables $X^i_n$. \item The arcs connecting the boundary component to itself, in bijection with the $D^l_1({J'}_{j})$, for $j=0,\ldots,N-3$ and $l=1,\ldots, N-3$. \item The three exceptional arcs $\Gamma_{\mathrm{except}}$: \begin{equation}\label{exceptionalarcsintro} \begin{tikzpicture}[scale=0.9, every node/.style={fill=white}] \draw (0,0) circle [radius=3.0]; \draw[fill=black] (0,1) circle [radius=0.1cm]; \draw[fill=black] (0,-1) circle [radius=0.1cm]; \draw[green, thick] plot [smooth, tension=1] coordinates { (0,1) (0,-1) }; \draw[blue, thick] plot [smooth, tension=1] coordinates { (0,1) (-0.6,0) (0,-2) (1.2,0) (0,1) }; \draw[red, thick] plot [smooth, tension=1] coordinates {(0,-1) (-1.2,0) (0,2) (0.6,0) (0,-1)}; \end{tikzpicture} \end{equation} \end{enumerate} \end{theorem} If we forget about arcs and surfaces these two theorems give a description of all of the cluster variables in $\tilde{A}$ and $\tilde{D}$ type: \begin{theorem}\label{findingallclustervarstheorem} For $\tilde{A}_{q,p}$ cluster algebras the cluster variables are \begin{equation} \left\{X^i_n\:\middle|\: \begin{aligned} &i=1,\ldots,N \\ &n\in\mathbb{Z} \end{aligned} \right\} \cup \left\{D^l_p(J_{jp})\:\middle|\: \begin{aligned} j=0,\ldots,q-1 \\ l=1,\ldots,q-1 \end{aligned} \right\} \cup \left\{D^l_q(\tilde{J}_{jq})\:\middle|\: \begin{aligned} j=0,\ldots,p-1 \\ l=1,\ldots,p-1 \end{aligned} \right\} \end{equation} For $\tilde{D}_N$ the cluster variables are \begin{equation} \left\{X^i_n\:\middle|\: \begin{aligned} &i=1,\ldots,N+1 \\ &n\in\mathbb{Z} \end{aligned} \right\} \cup \left\{D^l_1(J'_{j})\:\middle|\: \begin{aligned} j=0,\ldots,N-3 \\ l=1,\ldots,N-3 \end{aligned} \right\} \cup \Gamma_{\mathrm{except}} \end{equation} \end{theorem} Where $\Gamma_{\mathrm{except}}$ is the set of cluster variables associated with three exceptional arcs (\ref{exceptionalarcsintro}). Friezes were defined in \cite{coxeterfrieze} where they were shown to have connections with continued fractions and Farey series. Further links with diverse topics including triangulations of polygons \cite{conway1973triangulated}, Auslander-Reiten theory \cite{gabriel1980auslander} and moduli spaces of points in projective space \cite{morier20122} were later found. They were shown to be linked with cluster algebras \cite{calderochapoton} via the cluster category and a more direct link was studied in \cite{frises}. Here we prove that in $\tilde{A}$ and $\tilde{D}$ cases the off-frieze pattern cluster variables form (slightly generalised) friezes. \begin{proposition}\label{friezepropintro} In $\tilde{A}_{q,p}$ and $\tilde{D}_N$ cluster algebras the non-frieze pattern cluster variables defined by (\ref{determinantintro}) form friezes: \begin{equation}\setcounter{MaxMatrixCols}{20} \begin{matrix} \ldots & & 1 & & 1 & & 1 & & \\ & D^1_a(F_n) & & D^1_a(F_{n+a}) & & D^1_a(F_{n+2a}) & & \ldots & & \\ \ldots & & \mathcal{D}^2_{a}(F_n) & & \mathcal{D}^2_{a}(F_{n+a}) & & \mathcal{D}^2_{a}(F_{n+2a}) & & \\ & \mathcal{D}^3_{a}(F_{n-a}) & & \mathcal{D}^3_{a}(F_n) & & \mathcal{D}^3_{a}(F_{n+a}) & & \ldots \\ \ldots & & \mathcal{D}^4_{a}(F_{n-a}) & & \mathcal{D}^4_{a}(F_{n}) & & \mathcal{D}^4_{a}(F_{n+a}) & & \\ \\ & & \vdots & & \vdots && \vdots \\ \\ & \mathcal{D}^L_{a}(F_{n-3a}) & & \mathcal{D}^L_{a}(F_{n-2a}) & & \mathcal{D}^L_{a}(F_{n-a}) & & \ldots \end{matrix} \end{equation} i.e. each diamond $\begin{matrix} & \beta & \\ \alpha & & \delta \\ & \gamma & \end{matrix}$ satisfies $\alpha\delta-\beta\gamma=1$. In $\tilde{A}_{q,p}$ type there are two of these friezes, given by \[ (i) \qquad F_n=J_{jp}, \quad a=p, \quad L=q-1. \] \[ (ii) \qquad F_n=\tilde{J}_{jq}, \quad a=q, \quad L=p-1. \] In $\tilde{D}_N$ type the single frieze is given by \[ F_n=J'_{j}, \quad a=1, \quad L=N-3. \] \end{proposition} For these quivers this structure has previously been found \cite{assemdupont} in terms of the cluster category. There is a map $X_{?}$ from the objects of the cluster category to the cluster algebra, such that the frieze pattern (\ref{friezeformulaintro}) is the image of the transjective component. It was also shown that the exceptional tubes form friezes under this map, and we link our work to this construction. \begin{proposition}\label{intropropclusterfrieze} In $\tilde{A}_{q,p}$ type the frieze pattern (\ref{friezeformulaintro}) and the friezes of Proposition \ref{friezepropintro} give precisely the cluster frieze given in \cite{assemdupont}. In $\tilde{D}_N$ type the frieze pattern (\ref{friezeformulaintro}) and Proposition \ref{friezepropintro} give the cluster frieze except for the portion on the period $2$ tubes, which we have not constructed. \end{proposition} In this way we can see the $\tilde{A}$ and $\tilde{D}$ type cluster algebras as a union of friezes ($3$ friezes in the $\tilde{A}$ case and $2$ in the $\tilde{D}$ case) linked by the relation between the expression in the frieze variables defining the periodic quantities $J$, $\tilde{J}$ and $J'$. This paper is arranged as follows: In Section \ref{backgroundsection} we explain the background material we need, including the construction of the frieze pattern (\ref{friezeformulaintro}) and some properties of the periodic quantities we will use to prove the determinant formula (\ref{determinantintro}). We then describe how certain cluster algebras can be viewed as surfaces and how mutation works in this picture. Next we show that the $X^i_n$ defined by (\ref{friezeformulaintro}) form friezes on repetition quivers. We then define the cluster category and the cluster character and show that for $\tilde{A}$ and $\tilde{D}$ quivers the frieze (\ref{friezeformulaintro}) agrees with the frieze constructed on the transjective component of the cluster category via the cluster character, as given in \cite{assemdupont}. Finally we give the definition of a cluster frieze, which is a frieze given on the whole cluster category, not just the transjective component. In Section \ref{Friezeconstructionsection} we construct the friezes of Proposition \ref{friezepropintro}, while not yet proving that the frieze entries are cluster variables. In Section \ref{Atypesection} we describe the possible arcs (cluster variables) in $\tilde{A}$ type, proving the $\tilde{A}$ parts of Theorems \ref{findingallclustervarstheorem} and \ref{Atypearcs} and Proposition \ref{friezepropintro}. We then compare our friezes with those constructed in \cite{assemdupont}, proving Proposition \ref{intropropclusterfrieze} in the $\tilde{A}$ case. Section \ref{Dtypesection} is analogous to Section \ref{Atypesection}, but for $\tilde{D}$ type. \section{Review of cluster algebras, the cluster map, cluster algebras as triangulations of surfaces, friezes and the cluster category}\label{backgroundsection} Here we describe the disparate elements that combine to give our results. Firstly we give the construction of cluster mutation and cluster algebras. We then discuss how the cluster map is defined for period $1$ quivers and bipartite quivers before giving a general definition that gives the frieze pattern (\ref{friezeformulaintro}). Next we give previous results that have appeared for the frieze pattern in $\tilde{A}$ and $\tilde{D}$ types, including identifying periodic quantities for their respective cluster maps. We then show how certain cluster algebras can be described as triangulations of surfaces, since we aim to interpret these periodic quantities as arcs in these triangulations. We next define friezes and show how they relate to cluster algebras. Finally we give a very brief introduction to the cluster category and the cluster character, a map from the objects of this category to the set of cluster variables. We describe the cluster categories in $\tilde{A}$ and $\tilde{D}$ type and describe the frieze constructed in \cite{assemdupont} on the objects. \subsection{Cluster algebras} Here we give our definition of cluster algebras. In this paper we mean a cluster algebra without coefficients or frozen variables. A quiver $Q$ is a directed graph where multiple edges are allowed. We disallow loops or 2-cycles. Quiver mutation $\mu_k$ at any vertex $k$ is defined in $3$ steps: \begin{enumerate} \item For each length two path $i\rightarrow k \rightarrow j$ add a new arrow $i\rightarrow j$. \item Reverse the direction of all arrows entering or exiting $k$. \item Delete all 2-cycles that have appeared. \end{enumerate} This gives a new quiver $\mu_k(Q)$. The exchange matrix for a quiver $Q$ is the skew-symmetric matrix $B$ with entries $b_{ij}$, the number of arrows from $i$ to $j$, with $b_{ji}=-b_{ij}$. The mutation $\mu_k$ acts on the $b_{ij}$ as \begin{equation}\label{mutatedBmatrixentries} \mu_k(b_{ij})= \begin{cases} -b_{ij} & \mbox{ if } i=k \mbox{ or } j=k, \\ b_{ij}+\frac{1}{2}(|b_{ik}|b_{kj}+b_{ik}|b_{kj}|) & \mbox{ otherwise. } \end{cases} \end{equation} In addition to this, we will also attach a cluster variable $x_i$ at each vertex $i$. Cluster mutation at vertex $k$, also denoted $\mu_k$, fixes all variables $x_i$ with $i\neq k$ but transforms $x_k$ as follows: \begin{equation}\label{clustermutationformula} \mu_k(x_k):=\frac{1}{x_k}\left(\prod_{i\rightarrow k} x_i+\prod_{i\leftarrow k}x_i\right). \end{equation} Here the two products run over the arrows into and out of $k$ respectively, e.g. in the first product we have an $x_i$ for every arrow from $i$ to $k$. We consider $\mu_k$ as a mutation both of the quiver and of the cluster variables. The cluster algebra $\mathcal{A}_Q$ is the algebra over $\mathbb{Z}$ generated by the cluster variables obtained by any sequence of mutations applied to $Q$. \subsection{Periodic quivers and the cluster map}\label{dynamicalsystemssubsection} In this section we define periodic quivers and (a restricted version of) the cluster map that follows. We show how the cluster map has been extended to bipartite quivers before giving a general definition for all acyclic quivers. We then see how the frieze pattern (\ref{friezeformulaintro}) follows from this map. Period $1$ quivers were defined and classified in \cite{fordymarsh}. These are $N$ vertex quivers $Q$, labelled $0,1,\ldots,N-1$, satisfying $\mu_0(Q)=\rho(Q)$, where $\rho=(0,N,N-1,\ldots,2,1)$ is a permutation acting on the vertices of $Q$. This means that mutation at vertex $0$ is tantamount to a (specific) relabelling of the vertices of $Q$. By taking initial cluster variables $x_0,x_1,\ldots,x_{N-1}$ mutation at $0$ will give a new cluster variable, which we call $x_N$, determined by \begin{equation}\label{firstperiod1mutation} x_Nx_0=F(x_{N-1},x_{N-2},\ldots,x_2,x_1) \end{equation} for an appropriate function $F$. Next mutation at $1$ will give a new cluster variable called $x_{N+1}$ satisfying \[ x_{N+1}x_1=F(x_N,x_{N-1},\ldots,x_2) \] where, due to the quiver being period $1$, this $F$ is the same as in (\ref{firstperiod1mutation}). Continuing this process gives the recurrence \[ x_{n+N}x_n=F(x_{n+N-1},x_{n+N-2},\ldots,x_{n+2},x_{n+1}) \] for all $n$. An example of this is given by an orientation of an affine $A$ type diagram with $p$ and $q$ arrows pointing clockwise and anticlockwise, respectively, called $\tilde{A}_{q,p}$. This is given by taking the diagram shown in Figure \ref{Atypediagram}, reducing the labels modulo $N$, and orienting so that each arrow points from the lower label to the higher. \begin{lemma}\label{Atypeconstructionlemma} The $\tilde{A}_{q,p}$ quiver, with $p$ and $q$ coprime, gives the recurrence \begin{equation}\label{Atyperecurrence} x_{n+p+q}x_n=x_{n+q}x_{n+p}+1. \end{equation} \begin{figure} \centering \begin{equation*} \adjustbox{scale=0.9,center}{% \begin{tikzcd} & 0\arrow[dr, dash] \arrow[dl, dash] & \\ (p+q-1)p\arrow[d,dash] & & p\arrow[d,dash] \\ (p+q-2)p\arrow[d,dash] & & 2p\arrow[d,dash] \\ (p+q-3)p\arrow[d,dash] & & 3p\arrow[d,dash] \\ \vdots\arrow[d,dash] & & \vdots\arrow[d,dash] \\ (q+1)p\arrow[dr,dash] & & (q-1)p\arrow[dl,dash] \\ & qp & \end{tikzcd} } \end{equation*} \caption{The $\tilde{A}$ type diagram used to obtain the $\tilde{A}$ type recurrence relation.} \label{Atypediagram} \end{figure} \end{lemma} \begin{example} For $p=7$ and $q=8$ the recurrence of this lemma is given by the quiver shown in Figure \ref{EuclideanA15}. \begin{figure}[h] \centering \begin{equation*} \begin{tikzcd} 9 & 1\arrow[l]\arrow[r] & 8 & 0\arrow[l]\arrow[r] & 7\arrow[r] & 14 & \arrow[l]6\arrow[d] \\ 2\arrow[u]\arrow[d] & & & & & & 13 \\ 10 & 3\arrow[l]\arrow[rr] && 11 & 4\arrow[l]\arrow[r] & 12 & 5\arrow[l]\arrow[u] \end{tikzcd} \end{equation*}\caption{Our orientation of the $\tilde{A}_{8,7}$ quiver.} \label{EuclideanA15} \end{figure} \end{example} The map defined by these mutations \[ \varphi:(x_n,x_{n+1},\ldots,x_{n+N-1})\mapsto (x_{n+1},x_{n+2},\ldots,x_{n+N}) \] is know as the cluster map. In \cite{pallisterlinear} it was generalised to include bipartite quivers $Q$; we take $\mu_{\mathrm{sink}}$ and $\mu_{\mathrm{source}}$ to be the compositions of mutations at every sink and source in $Q$, respectively, then the product $\mu:=\mu_{\mathrm{source}}\circ\mu_{\mathrm{sink}}$ fixes the quiver but gives new cluster variables. We let $X^i_n$ be the cluster variable at vertex $i$ after $n$ applications of $\mu$ (with $n=0$ giving the initial variables) then the map \begin{equation}\label{generalisedclustermap} \varphi:(X^1_n,X^2_n,\ldots,X^{N+1}_n)\mapsto (X^1_{n+1},X^2_{n+1},\ldots,X^{N+1}_{n+1}) \end{equation} is also known as a cluster map. For a general acyclic quiver this map is obtained by first taking $X^i_0$ for each vertex $i$. We then take $X^i_n$, for $n\neq 0$, to satisfy \begin{equation}\label{friezeformula} X^i_nX^i_{n+1}=1+\left(\prod_{j\rightarrow i}(X^j_n)^{|b_{ji}|}\right)\left(\prod_{i\rightarrow j}(X^j_{n+1})^{|b_{ji}|}\right) \end{equation} where the products is taken over all arrows in $Q$. This is called a ``generalised frieze pattern" in \cite{kellerscherotzke}. It can be shown that all of the $X^i_n$ can be obtained by cluster mutation, similarly to the construction of (\ref{generalisedclustermap}). The $\tilde{A}$ type cluster variables obtained by (\ref{Atyperecurrence}) are also included here, with the identification $X^i_n\leftrightarrow x_{i-nN}$. \subsection{Periodic quantities for the cluster map} Here we examine the periodic quantities found for the cluster map where the quiver is of affine type. These immediately give linear relations between the frieze variables with periodic coefficients and, with some work, linear relations with constant coefficients can be obtained. Here we explain how this is done. In $\tilde{A}$ type, both papers \cite{fordymarsh,fordyhone} prove that the recurrence (\ref{Atyperecurrence}) has the periodic quantities \begin{equation}\label{periodicquantities} J_n:=\frac{x_{n+2p}+x_n}{x_{n+p}}, \qquad \tilde{J}_n:=\frac{x_{n+2q}+x_n}{x_{n+q}} \end{equation} with period $q$ and $p$, respectively. By this we mean that $J_{n+q}=J_n$ and $\tilde{J}_{n+p}=\tilde{J}_n$. Immediately we see that the $x_n$ satisfy the linear relations \begin{equation}\label{periodiclinearrelations} x_{n+2p}-J_nx_{n+p}+x_n=0, \qquad x_{n+2q}-\tilde{J}_nx_{n+q}+x_n=0 \end{equation} with periodic coefficients. The authors of \cite{fordyhone} then use this to construct linear relations with constant coefficients. \begin{theorem} The cluster variables $x_n$ satisfy the linear relation \[ x_{n+2qp}-\mathcal{K}x_{n+qp}+x_n=0 \] where $\mathcal{K}$ is constant. \begin{proof} We define the matrices \[ \Psi_n:= \begin{pmatrix} x_{n+p+q} & x_{n+q} \\ x_{n+p} & x_{n} \end{pmatrix} \qquad L_n:= \begin{pmatrix} J_n & 1 \\ -1 & 0 \end{pmatrix} \qquad \tilde{L}_n:= \begin{pmatrix} \tilde{J}_n & -1 \\ 1 & 0 \end{pmatrix} \] such that, by (\ref{periodiclinearrelations}), $\Psi_{n+p}=\Psi_nL_n$ and $\Psi_{n+q}=\tilde{L}_n\Psi_n$. By defining \[ M_n:=L_nL_{n+p}L_{n+2p}\ldots L_{n+(q-1)p}, \qquad \tilde{M}_n:=\tilde{L}_{n+(p-1)q}\ldots\tilde{L}_{n+2q}\tilde{L}_{n+q}\tilde{L}_n \] we have $\Psi_nM_n=\Psi_{n+qp}$ and $\tilde{M}_n\Psi_n=\Psi_{n+qp}$. We then apply the Cayley-Hamilton theorem to $M_n$: \[ M_n^2-\mathcal{K}M_n+I=0 \] where ${\mathcal{K}=\mathrm{Tr}(M_n)=\mathrm{Tr}(\tilde{M}_n)}$ is constant. Multiplying this matrix equation by $\Psi_n$ on the left gives the linear relation. \end{proof} \end{theorem} The first question we ask is about the entries of $M_n$ and $\tilde{M}_n$. We define the matrices $M^m_n$ as the product of the first $m$ of these $L$ matrices: \begin{equation}\label{matrixproduct} M^m_n:=L_nL_{n+p}L_{n+2p}\ldots L_{n+(m-1)p}. \end{equation} For example $M^3_n$ is given by \[ M^3_n= L_nL_{n+p}L_{n+2p}= \begin{pmatrix} J_{n+2p}J_{n+p}J_n-J_{n+2p}-J_n & J_{n+p}J_n-1 \\ -J_{n+2p}J_{n+p}+1 & -J_{n+p} \end{pmatrix} \] Similarly we let \[ \tilde{M}^m_n:=\tilde{L}_{n+(m-1)q}\ldots\tilde{L}_{n+2q}\tilde{L}_{n+q}\tilde{L}_n. \] We prove the following in Section \ref{Friezeconstructionsection}: \begin{proposition}\label{matrixentryclustervariables} The matrices $M^m_n$ are given by \[ M^m_n= \begin{pmatrix} A^m_n & A^{m-1}_n \\ -A^{m-1}_{n+p} & -A^{m-2}_{n+p} \end{pmatrix} \] where $A^m_n$ satisfies the recurrence in $m$: \begin{equation}\label{mainreccurence} A^m_n=J_{n+(m-1)p}A^{m-1}_n-A^{m-2}_n, \qquad A^1_n=J_n, \qquad A^0_n=1 \end{equation} for each $n$. Conversely the matrices $\tilde{M}^m_n$ are given by \[ \tilde{M}^m_n= \begin{pmatrix} \tilde{A}^m_n & -\tilde{A}^{m-1}_{n+q} \\ \tilde{A}^{m-1}_n & -\tilde{A}^{m-2}_{n+q} \end{pmatrix} \] satisfying \[ \tilde{A}^m_n=\tilde{J}_{n+(m-1)q}\tilde{A}^{m-1}_n-\tilde{A}^{m-2}_n, \qquad \tilde{A}^1_n=\tilde{J}_n, \qquad \tilde{A}^0_n=1. \] \end{proposition} In \cite{pallisterlinear} periodic quantities were found for the cluster map (\ref{generalisedclustermap}) for $\tilde{D}$ and $\tilde{E}$ quivers, as shown in Figure \ref{periodictable}. \begin{figure} \begin{center} \begin{tabular}{|c|c|c|c|} \hline Quiver & Period & Explicit Expression in \cite{pallisterlinear} \\ \hline\noalign{\smallskip} & $N-2$ & Equation 22\\ $\tilde{D}_N$ & $2$ & Lemma 4.1 \\ & $2$ & Lemma 4.1 \\ \hline\noalign{\smallskip} & $3$ & Lemma 4.9 \\ $\tilde{E}_6$ & $3$ & Lemma 4.9 \\ & $2$ & Lemma 4.12 \\ \hline\noalign{\smallskip} & $4$ & Theorem 4.20 \\ $\tilde{E}_7$ & $3$ & Theorem 4.19 \\ & $2$ & Equation 42 \\ \hline\noalign{\smallskip} & $5$ & Lemma 4.26 \\ $\tilde{E}_8$ & $3$ & Theorem 4.28 \\ & $2?$ & Conjecture 4.30 \\ \hline \end{tabular} \end{center}\caption{Periodic quantities found for the cluster map for $\tilde{D}$ and $\tilde{E}$ quivers.}\label{periodictable} \end{figure} The linear relations that follow from these were also obtained in \cite{kellerscherotzke} using the cluster category. For $\tilde{D}$ type, the period $N-2$ quantity can be written \begin{equation}\label{DtypeJintro} J'_n:=\frac{X^1_{n+1}+X^1_{n-1}}{X^2_n} \end{equation} and can also be used to give linear relations with constant coefficients, as in the $\tilde{A}$ case. \begin{proposition} In the $\tilde{D}_N$ cases, for even $N$ the cluster variables at vertex $1$ satisfy the constant coefficient linear relation: \[ X^1_{n+2N-4}-\mathcal{K}'X^1_{n+N-2}+X^1_n=0 \] and for odd $N$: \[ X^1_{n+4N-8}-\mathcal{K}'X^1_{n+2N-4}+X^1_n=0 \] where vertex $1$ is as shown in Figure \ref{Dquiver}. \end{proposition} The proof of this is similar to the $\tilde{A}$ case. For $N$ even the constant $\mathcal{K}'$ is given by the trace of \begin{equation}\label{Deven} {L'}_n{L'}_{n+1}\ldots{L'}_{n+N-3}, \qquad {L'}_n:= \begin{pmatrix} 0 & -1 \\ 1 & J'_n \end{pmatrix} \end{equation} and for odd $N$ the trace of \begin{equation}\label{Dodd} {L'}_n{L'}_{n+1}\ldots{L'}_{n+2N-5}. \end{equation} For the same ${L'}_n$. Interestingly, the same product of matrices appears as in the $\tilde{A}$ case. A similar result holds too. \begin{proposition}\label{Dtypematrixentries} The entries of the matrix products (\ref{Deven}) and (\ref{Dodd}) are determined by their upper left entry, which we call ${A'}^m_n$ in either case, which satisfies the linear recurrence \begin{equation}\label{Dtypematrixlinear} {A'}^m_n=J'_{n+m-1}{A'}^{m-1}_n-{A'}^{m-2}_n, \qquad {A'}^1_n=J'_n, \qquad {A'}^0_n=1 \end{equation} for each $n$. \end{proposition} The proof of this is similar to the one for $\tilde{A}$ type given in Section \ref{Friezeconstructionsection}. Due to this result we ask how the $A^m_n$, $\tilde{A}^m_n$ and ${A'}^m_n$ fit into their respective cluster algebras. Are they are cluster variables and, if so, what cluster variables remain outside of the $A^m_n$, $\tilde{A}^m_n$ and ${A'}^m_n$ and the variables obtained by the cluster map? To answer these we need an alternative way of viewing these cluster algebras. \subsection{Cluster algebras as triangulated surfaces} In \cite{clustersandtriangulated1} the authors describe how to view both Dynkin and affine $A$ and $D$ type cluster algebras as triangulations of surfaces with marked points. In this section we briefly explain how this correspondence works: we describe how a quiver is obtained from a triangulation and what the analogues of quiver and cluster mutation are. For $\tilde{A}_{q,p}$ cluster algebras we shall be dealing with triangulations of annuli with $q$ and $p$ marked points on opposite boundary components, an example of which can be seen in Figure \ref{exampletriangulation}, which gives an annulus by gluing along the dotted lines. In addition to the traditional triangles seen for this surface we also allow ``self-folded" triangles: \begin{equation}\label{selffolded} \scalebox{0.75}{ \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,0) circle [radius=0.15cm]; \draw[fill=gray] (0,2) circle [radius=0.15cm]; \draw [-] (0,0) to (0,2); \draw plot [smooth, tension=1] coordinates {(0,0) (-0.9,2) (0,3.4) (0.9,2) (0,0)}; \node[scale=1.2] at (0,1) {$i$}; \node[scale=1.2] at (0,3.4) {$k$}; \end{tikzpicture} } \end{equation} that will appear in the $\tilde{D}$ type triangulations, as in Figure \ref{Dtypebefore}. For $\tilde{A}$ type some possible arcs (not a full triangulation) are displayed in Figure \ref{annuluswithsomearcs}. \begin{figure} \centering \begin{tikzpicture} \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (1,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \node at (3,0) {$\ldots$}; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (0.5,2) circle [radius=0.1cm]; \draw[fill=black] (1.5,2) circle [radius=0.1cm]; \node at (2.5,2) {$\ldots$}; \draw[fill=black] (3.5,2) circle [radius=0.1cm]; \foreach \x in {-1,...,1} \draw [-] (\x,0) to (\x+1,0); \draw [-] (2,0) to (2.5,0); \draw [-] (3.5,0) to (4,0); \foreach \x in {4,...,6} \draw [-] (\x,0) to (\x+1,0); \draw [-] (7,0) to (7.5,0); \draw [-] (8.5,0) to (10,0); \foreach \x in {-1,...,1} \draw [-] (\x,2) to (\x+1,2); \foreach \x in {3,...,6} \draw [-] (\x,2) to (\x+1,2); \node at (0,-0.3) {0}; \node at (1,-0.3) {1}; \node at (2,-0.3) {2}; \node at (4,-0.3) {$q-1$}; \node at (0.5,2.3) {$q$}; \node at (1.5,2.3) {$q+1$}; \node at (3.5,2.3) {$p+q-1$}; \draw[fill=black] (5,0) circle [radius=0.1cm]; \draw[fill=black] (6,0) circle [radius=0.1cm]; \draw[fill=black] (7,0) circle [radius=0.1cm]; \node at (8,0) {$\ldots$}; \draw[fill=black] (9,0) circle [radius=0.1cm]; \draw[fill=black] (5.5,2) circle [radius=0.1cm]; \draw[fill=black] (6.5,2) circle [radius=0.1cm]; \node at (7.5,2) {$\ldots$}; \draw[fill=black] (8.5,2) circle [radius=0.1cm]; \foreach \x in {7,...,8} \draw [-] (10+\x,0) to (10+\x+1,0); \draw [-] (8,2) to (10,2); \node at (5,-0.3) {0}; \node at (6,-0.3) {1}; \node at (7,-0.3) {2}; \node at (9,-0.3) {$q-1$}; \node at (5.5,2.3) {$q$}; \node at (6.5,2.3) {$q+1$}; \node at (8.5,2.3) {$p+q-1$}; \draw[dotted] (-0.5,-1) to (-0.5,3); \draw[dotted] (4.5,-1) to (4.5,3); \draw[dotted] (9.5,-1) to (9.5,3); \draw[fill=black] (10,0) circle [radius=0.1cm]; \node at (10,-0.3) {$1$}; \draw plot [smooth, tension=1] coordinates {(3.5,2) (10,0)}; \draw plot [smooth, tension=1] coordinates {(-1.5,2) (5,0)}; \draw[fill=black] (-1.5,2) circle [radius=0.1cm]; \node at (-1.5,2.3) {$p+q-1$}; \draw [-] (-2,2) to (0,2); \draw [-] (10,0) to (10.5,0); \draw [-] (8.5,2) to (10,1.53); \draw plot [smooth, tension=1] coordinates {(0,0) (2,0.75) (4,0)}; \draw plot [smooth, tension=1] coordinates {(1,0) (2.5,0.5) (4,0)}; \draw plot [smooth, tension=1] coordinates {(6,0) (7.5,0.5) (9,0)}; \draw plot [smooth, tension=1] coordinates {(5,0) (7,0.75) (9,0)}; \draw plot [smooth, tension=1] coordinates {(0.5,2) (2,1.25) (3.5,2)}; \draw plot [smooth, tension=1] coordinates {(5.5,2) (7,1.25) (8.5,2)}; \end{tikzpicture}\caption{The annulus associated with the $\tilde{A}_{q,p}$ cluster algebra, with some possible arcs.}\label{annuluswithsomearcs} \end{figure} To return the quiver from the triangulation: \begin{enumerate}[(i)] \item Attach a vertex to every arc. \item For every pair of vertices $i$ and $j$ from step (i) that are part of the same non-self-folded triangle we draw an arrow $i\mapsto j$ if, while travelling clockwise around the triangle, $j$ comes directly after $i$. If no arcs lie on a boundary then the situation looks as follows: \[ \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (2,2) circle [radius=0.1cm]; \draw[fill=white] (1,1) circle [radius=0.15cm]; \draw[fill=white] (3,1) circle [radius=0.15cm]; \draw[fill=white] (2,0) circle [radius=0.15cm]; \draw [-] (0,0) to (2,2); \draw [-] (0,0) to (4,0); \draw [-] (2,2) to (4,0); \draw [->] (1.2,1) to (2.8,1); \draw [->] (1.8,0.2) to (1.2,0.8); \draw [->] (2.8,0.8) to (2.2,0.2) ; \end{tikzpicture} \] where the white circles are the vertices of the quiver. If an arc lies on a boundary component then we do not draw a vertex there. \item For every pair of arcs $k$ and $i$ forming a self-folded-triangle (\ref{selffolded}) and for every arrow $j\mapsto k$ we add an arrow $j\mapsto i$. Also for every arrow $k\mapsto j$ we add an arrow $i\mapsto j$. \end{enumerate} For arcs $k$ not part of self-folded triangles, quiver mutation is analogous to replacing the diagonal of a quadrilateral, $k$, with the other diagonal, $k'$: \begin{equation}\label{diagonalflip} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \draw[fill=black] (1,2) circle [radius=0.1cm]; \draw[fill=black] (3,2) circle [radius=0.1cm]; \draw [-] (0,0) to (2,0); \draw [-] (1,2) to (3,2); \draw [-] (0,0) to node[scale=0.7] {$k$} (3,2); \draw [-] (2,0) to (3,2); \draw [-] (0,0) to (1,2); \node[scale=0.7] at (2,2){a}; \node[scale=0.7] at (2.5,1){b}; \node[scale=0.7] at (1,0){c}; \node[scale=0.7] at (0.5,1){d}; \end{tikzpicture} \quad \begin{tikzpicture}[every node/.style={fill=white}] \node at (0,0) {$\;$}; \node at (0,1) {$\xrightarrow{\mu_k}$}; \end{tikzpicture} \quad \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \draw[fill=black] (1,2) circle [radius=0.1cm]; \draw[fill=black] (3,2) circle [radius=0.1cm]; \draw [-] (0,0) to (2,0); \draw [-] (1,2) to (3,2); \draw [-] (2,0) to node[scale=0.7] {$k'$} (1,2); \draw [-] (2,0) to (3,2); \draw [-] (0,0) to (1,2); \node[scale=0.7] at (2,2){a}; \node[scale=0.7] at (2.5,1){b}; \node[scale=0.7] at (1,0){c}; \node[scale=0.7] at (0.5,1){d}; \end{tikzpicture} \end{equation} known as a flip. Figures \ref{exampletriangulation} and \ref{triangulationaftermutation} show an example of this. In our work, for arcs $k$ that are the ``outside" of a self-folded triangle mutation looks like: \begin{equation}\label{selffoldedmutation} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (-1,-3) (0,-1.4) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1.5,-3.5) (3.46,-2)}; \node[scale=0.7] at (0,-3){i}; \node[scale=0.7] at (0.7,-3){k}; \node at (4,-2.7){$\xrightarrow{\mu_k}$}; \node[scale=0.7] at (1.5,-3.5){1}; \node[scale=0.7] at (2.5,-1.75){j}; \end{tikzpicture} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,-2) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (-1,-3) (0,-1.4) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1.5,-3.5) (3.46,-2)}; \node[scale=0.7] at (0,-3){i}; \node[scale=0.7] at (1.4,-2){k'}; \node[scale=0.7] at (1.5,-3.5){1}; \node[scale=0.7] at (2.5,-1.75){j}; \end{tikzpicture} \end{equation} where the arc labelled $1$ lives on the boundary. This follows by gluing certain vertices (\ref{diagonalflip}). In either case we remove the arc $k$ and replace it with the unique other arc $k'$ that forms a triangulation. For arcs inside self-folded triangles we do not allow mutation. Under these rules mutation of triangulations mimics exactly quiver mutation. \begin{remark} Due to the construction of quivers from triangulations, for any pair of arcs $i$ and $j$ forming a self-folded triangle (\ref{selffolded}) we have $b_{i,k}=b_{j,k}$ for all vertices $k$. So $\mu_i=\sigma\mu_j$ where $\sigma$ is the relabelling $i\leftrightarrow j$. Due to this, not being able to mutate at the arc inside the self-folded triangle, $i$, isn't really a restriction, we may as well just mutate at $j$. In \cite{clustersandtriangulated1} this inability to mutate inside self-folded triangles is rectified with the introduction of ``tagged arcs". \end{remark} We also assign cluster variables to each arc of a triangulation. For this we let the boundary arcs have value $1$. To each arc $i$ in an initial triangulation we give initial cluster variables $x_i$. New cluster variables are defined to satisfy \begin{equation}\label{ptolemy} x_{k}x_{k'}=x_ax_c+x_bx_d \end{equation} where the arcs are as labelled in (\ref{diagonalflip}). For quivers that are constructible as triangulated surfaces this matches the cluster mutation formula (\ref{clustermutationformula}). For the situation described in (\ref{selffoldedmutation}) we have \[ x_{k}x_{k'}=1+x_j. \] Finally in the following situation \begin{equation} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (-1.5,-3.5) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1.5,-3.5) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(-3.46,-2) (0,-0.5) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (-2,-2.5) (0,-1.3) (3.46,-2)}; \node[scale=0.7] at (0,-3){i}; \node[scale=0.7] at (0.7,-3){j}; \node at (4,-2.7){$\xrightarrow{\mu_k}$}; \node[scale=0.7] at (1.5,-3.5){1}; \node[scale=0.7] at (-1.5,-3.5){1}; \node[scale=0.7] at (0,-0.5){m}; \node[scale=0.7] at (2.5,-1.75){k}; \end{tikzpicture} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (-1.5,-3.5) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1.5,-3.5) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(-3.46,-2) (0,-0.5) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (2,-2.5) (0,-1.3) (-3.46,-2)}; \node[scale=0.7] at (0,-3){i}; \node[scale=0.7] at (0.7,-3){j}; \node[scale=0.7] at (1.5,-3.5){1}; \node[scale=0.7] at (-1.5,-3.5){1}; \node[scale=0.7] at (0,-0.5){m}; \node[scale=0.7] at (0,-1.3){k'}; \end{tikzpicture} \end{equation} we have $x_kx_{k'}=1+x_ix_jx_m$. Again, these are special cases of (\ref{ptolemy}) by gluing (\ref{diagonalflip}) in the right way. We will use this construction of cluster algebras as triangulated surfaces to ``see" the cluster variables, in particular the frieze variables $X^i_n$ from (\ref{friezeformula}). We will also use this picture to show that the $A^m_n$ of (\ref{mainreccurence}) are indeed cluster variables. The recurrence defining these also means that they form a frieze, which we now define. \subsection{Friezes} In this section we give the definition of friezes, and then of repetition quivers. We show that in $\tilde{A}$ type the repetition quiver gives rise to a frieze. We define friezes on repetition quivers and show that for any acyclic quiver the frieze variables $X^i_n$ of (\ref{friezeformulaintro}) form a frieze on its repetition quiver. Friezes were first defined in \cite{coxeterfrieze}. They were originally arrays of integers in the plane organised like brickwork and sandwiched between two rows of zeroes and ones: \[\setcounter{MaxMatrixCols}{20} \begin{matrix} & \ldots & & 0 & & 0 & & 0 & & \ldots \\ \ldots & & 1 & & 1 & & 1 & & 1 & & \ldots \\ & & \vdots & & \vdots & & \vdots & & \vdots & & \\ & \ldots & & a & & b & & c & & \ldots \\ \ldots & & d & & e & & f & & g & & \ldots \\ & \ldots & & h & & i & & j & & \ldots \\ & & \vdots & & \vdots & & \vdots & & \vdots & & \\ \ldots & & 1 & & 1 & & 1 & & 1 & & \ldots \\ & \ldots & & 0 & & 0 & & 0 & & \ldots \\ \end{matrix} \] such that each diamond $\begin{matrix} & \beta & \\ \alpha & & \delta \\ & \gamma & \end{matrix}$ satisfies $\alpha\delta-\beta\gamma=1$. For our purposes we don't demand that the entries are integers and don't require the rows of zeroes and ones. We have, for example, the friezes formed by the periodic quantities in $\tilde{A}$ and $\tilde{D}$ type shown in Proposition \ref{friezepropintro}, the proof of which we give in Section \ref{Friezeconstructionsection}. In order to see the link with the frieze patterns of Section \ref{dynamicalsystemssubsection}, we first need to make some definitions. \begin{definition}\label{repetitionquiverdef} For a quiver $Q$, the repetition $\mathbb{Z}Q$ is the quiver with vertices $(n,i)$, with $n\in \mathbb{Z}$ and $i$ a vertex of $Q$. For every arrow $i\mapsto j$ in $Q$ we have arrows $(n,i)\mapsto (n,j)$ and $(n-1,j)\mapsto (n,i)$ for every $n$. \end{definition} The repetition quiver is an example of a stable translation quiver: it has a map $\tau:(n,i)\mapsto (n-1,i)$ such that there is a $1:1$ correspondence between arrows $j\mapsto i $ and $\tau(i)\mapsto j$. \begin{definition} A frieze on a repetition quiver is an assignment $f(n,i)$ for each vertex $(n,i)$ such that \[ f(n,i)f(n-1,i)=1+\prod_{(m,j)\mapsto (n,i)} f(m,j) \] where the product is taken over all arrows into $(n,i)$. \end{definition} \begin{example}\label{repetitionexample} We draw a frieze on the repetition quiver for $\tilde{A}_{5,2}$ (the quiver $\tilde{A}_{5,2}$ can be seen in red), where we identify the top and bottom rows: \[ \adjustbox{scale=0.9,center}{% \begin{tikzcd} &\ldots&&x_{-7}&& \color{red}x_0\arrow[dl] && x_7\arrow[dl]\\ \ldots&&\ldots&&x_{-2}\arrow[dr]\arrow[ul]&& \color{red}x_5\arrow[dr]\arrow[ul, red] &&x_{12}\arrow[ul]\\ &\ldots&&x_{-4}\arrow[dr]\arrow[ur]&&\color{red}x_3\arrow[dr] \arrow[ur, red] &&x_{10}\arrow[ur]\\ \ldots&&x_{-6}\arrow[ur]&& \color{red}x_1\arrow[dl] \arrow[ur, red] &&x_{8}\arrow[dl]\arrow[ur] && \ldots\\ &\ldots&&x_{-1}\arrow[dr]\arrow[ul]&& \color{red}x_6\arrow[ll, "\tau" description, blue]\arrow[dr]\arrow[ul, red] && x_{13}\arrow[ul]\\ \ldots&&x_{-3}\arrow[dr]\arrow[ur]&&\color{red}x_4 \arrow[dr]\arrow[ur, red] &&x_{11}\arrow[ur] &&\ldots\\ &x_{-5}\arrow[dr]\arrow[ur]&& \color{red}x_2\arrow[dr] \arrow[ur, red] &&x_9\arrow[ur] &&\ldots\\ x_{-7}\arrow[ur]&&\color{red}x_0 \arrow[ur, red] &&x_7\arrow[ur] && \ldots & & \ldots \end{tikzcd} } \] here the map $\tau$ acts by shifting each vertex left, as shown in blue for $\tau (x_6)=x_{-1}$. Furthermore this is a frieze due to the $\tilde{A}$ type relation (\ref{Atyperecurrence}) \[ x_{n}x_{n+7}=1+x_{n+2}x_{n+5}. \] \end{example} To define a frieze on the repetition quiver for $\tilde{A}_{q,p}$ for general $q$ and $p$ we let $f(n,i)=x_{nN+ip}$. The $x_n$ satisfy (\ref{Atyperecurrence}) so they give the frieze: \begin{equation}\label{Atypefrieze} \begin{matrix} \ddots & & \vdots & & \vdots & & \iddots \\ &x_{n-q} & & x_{n+p} & & x_{n+N+p} & & \ldots \\ \ldots & & x_n & & x_{n+N} & & x_{n+2N} \\ &x_{n-p} && x_{n+q} & & x_{n+N+q} & & \ldots \\ \ldots &&x_{n+q-p} & & x_{n+2q} & & x_{n+N+2q} \\ & \iddots &&\vdots & & \vdots & & \ddots \end{matrix} \end{equation} where, since \[ nN+ip=(n-p)N+(i+q+p)p \] we identify $(n,i)=(n-p,i+q+p)$. \begin{remark} \begin{enumerate}[(i)] \item The initial values for the associated dynamical system (\ref{Atyperecurrence}) are highlighted in red in Example \ref{repetitionexample}. In \cite{frises} this layout of the initial values is called a frontier (after rotation so that it is horizontal) and a formula is given for the frieze entries in terms of this frontier. \item This type of construction can be found in \cite{calderochapoton} for $A$ type, with the initial cluster variables set to $1$, thus forming a traditional frieze. \end{enumerate} \end{remark} For arbitrary quivers $Q$ (including $\tilde{D}$ type) we can take $f(n,i)=X^i_n$, as defined in (\ref{friezeformula}) to give a frieze on the repetition quiver $\mathbb{Z}Q$. \subsection{Cluster algebras and representation theory} Here we give the necessary background to compare our results with \cite{assemdupont}, where friezes are constructed on the various components of the cluster category. An overview of most of the elements of this section, including the cluster category and the cluster character, can be found in the notes \cite{kelleroverview}. Throughout this section the quiver $Q$ is taken to be acyclic and to have $N$ vertices and $\mathbf{k}$ is an arbitrary algebraically closed field. \subsubsection{The cluster category} In this section we define and briefly describe the Auslander-Reiten quivers for category of modules over the path algebra $\mathbf{k}Q$ if $Q$ is Dynkin or affine. We then describe the objects of the cluster category in these cases. \begin{definition} The Auslander-Reiten quiver $\Gamma(\mathbf{k}Q)$ of the module category $\mathrm{mod}\:\mathbf{k}Q$ has isomorphism classes of finitely generated, indecomposable modules for vertices. The arrows are given by ``irreducible maps". There is an automorphism \[ \tau:\Gamma(\mathbf{k}Q)\rightarrow \Gamma(\mathbf{k}Q) \] called the Auslander-Reiten translate, such that $\tau(P)=0$ for all projectives $P$ and $\tau^{-1}(I)=0$ for all injectives $I$. All of the projective modules lie in the same component $\mathbf{p}$, which is called the preprojective component. Similarly the injectives all lie in $\mathbf{q}$, the preinjective component. The remaining components are called regular. See \cite{hugel} for an overview or \cite{auslander1974representation} for details. \end{definition} A quiver is called representation finite if, up to mutation, it has only finitely many indecomposable modules. Gabriel's theorem \cite{gabriel} says that these are precisely the Dynkin quivers. Moreover we have the following theorem, the first two points of which are from \cite{gabrielriedtmann} and paraphrased in \cite{hugel}. The third point can be found in \cite{crawleyboevey}. \begin{theorem} \begin{enumerate}[(i)] \item If $Q$ is Dynkin then $\Gamma(\mathbf{k}Q)=\mathbf{p}=\mathbf{q}$ is a full and finite subquiver of $\mathbb{N}Q^{\mathrm{op}}$. \item If $Q$ is not Dynkin then $\mathbf{p}=\mathbb{N}Q^{\mathrm{op}}$ and $\mathbf{q}=-\mathbb{N}Q^{\mathrm{op}}$ with $\mathbf{p}\cap \mathbf{q}=0$. The modules of $\mathbf{p}$ and $\mathbf{q}$ are uniquely determined by their dimension vectors. Finally $\mathbf{p}\cup \mathbf{q}\neq \Gamma(\mathbf{k}Q)$. \item If $Q$ is affine then \[ \mathbf{p}=\{\tau^{-m}P_j\mid m\geq 0,\: j=1,\ldots, N\}, \qquad \mathbf{q}=\{\tau^{m}I_j\mid m\geq 0,\: j=1,\ldots, N\} \] and the regular modules $M$ are those that satisfy $\tau^m M\neq 0$ for all $m\in\mathbb{Z}$. The regular components are parametrised by $\lambda\in\mathbb{P}^1(\mathbf{k})$. By defining the quiver $A_{\infty}$ as \[ \begin{tikzpicture}[every node/.style={fill=white}] \draw [->] (0,0) to (1.8,0); \draw [->] (2,0) to (3.8,0); \draw [->] (4,0) to (5.8,0); \node at (6.2,0){\ldots}; \node at (0,0.0){$1$}; \node at (2,0.0){$2$}; \node at (4,0.0){$3$}; \end{tikzpicture} \] we can describe the regular components as ``tubes". They are given by ${\mathbb{Z}A_{\infty}/<\tau^{p_{\lambda}}>}$ for an appropriate $p_{\lambda}\in\mathbb{Z}_{\geq 0}$, known as the width of the tube, and ${\tau:(n,i)\mapsto(n-1,i)}$ where $n\in\mathbb{Z}$ and $i$ is a vertex of $A_{\infty}$. The modules of the form $(n,1)$ for $n=1,2,\ldots,p_{\lambda}$ are called quasi-simple. If $p_{\lambda}=1$ then the tube is called homogeneous, otherwise it is called exceptional. We define \[ \mathbb{P}^{\mathcal{E}}=\{\lambda\in\mathbb{P}^1(\mathbf{k})\mid p_{\lambda}>1\} \] and let $T_{\lambda}$ be the tube of width $p_{\lambda}$. A list of the widths of the exceptional tubes can be found in \cite{crawleyboevey}. In particular $|\mathbb{P}^{\mathcal{E}}|\leq 3$. \end{enumerate} \end{theorem} The cluster category $\mathcal{C}_Q$ was first defined in \cite{tiltingcluster}. It has a suspension functor $[1]$ that coincides with the Auslander-Reiten translation $\tau$. The indecomposable objects of $\mathcal{C}_Q$ can be identified with the disjoint union of the objects of $\Gamma(\mathbf{k}Q)$ and the shifts of the projectives $P_i[1]$. The Auslander-Reiten quiver of $\mathcal{C}_Q$ is a stable translation quiver with translate $\tau$. There is a connection between the cluster category and the cluster algebra given by the cluster character $X_?:\mathrm{Ob}(\mathcal{C}_Q)\rightarrow \mathcal{A}_Q$ which has a few components we need to define. \subsubsection{The cluster character} For a $\mathbf{k}Q$ module $M$ the quiver Grassmanian of dimension $\underline{e}\in\mathbb{Z}_{\geq 0}^N$ is \[ \mathrm{Gr}_{\underline{e}}(M)=\{N\subset M \mid \underline{\mathrm{dim}}(N)=\underline{e}\} \] and $\chi(\mathrm{Gr}_{\underline{e}}(M))$ is its Euler characteristic with respect to an appropriate cohomology. We let $b_{ij}$ be the number of arrows between vertices $i$ and $j$ in $Q$, then the Euler form $<-\:,\:->$ acts on pairs $a,a'\in \mathbb{Z}^N$ by \[ <a\:,\:a'>=\sum_{i=1}^N a_ia'_i+\sum_{i,j=1}^N b_{ji}a_ia'_j. \] \begin{definition}\cite{calderochapoton} For a cluster algebra with initial cluster variables $u_i$, the cluster character is a map $X_?:\mathrm{Ob}(\mathcal{C}_Q)\rightarrow \mathcal{A}_Q$ defined by: \begin{enumerate}[(i)] \item If $M$ is an indecomposable $kQ$ module with $\underline{m}=\underline{\mathrm{dim}}(M)$ then \[ X_M=\sum_{\underline{e}}\chi(\mathrm{Gr}_{\underline{e}}(M))\prod_i u_i^{-<\underline{e}\:,\:\underline{m}>-<\alpha_i\:,\:\underline{m}-\underline{e}>} \] where $\alpha_i=\underline{\mathrm{dim}}(S_i)$, the dimension of the simple module at $i$. The sum is taken over the $\underline{e}\in\mathbb{Z}^N$ such that $\chi(\mathrm{Gr}_{\underline{e}}(M))\neq 0$ and the product is taken over all vertices $i$. \item If $M=P_i[1]$ then $X_M=u_i$. \item For any two objects $M$ and $N$ \[ X_{M\bigoplus N}=X_MX_N. \] \end{enumerate} \end{definition} \begin{theorem} \cite{calderokellerii} Theorem 4. The map $X_?$ gives a bijection between the set of isomorphism classes of rigid indecomposable modules in $\mathcal{C}_Q$ and the set of cluster variables $\mathcal{A}_Q$. \end{theorem} \begin{proposition}\cite{assemdupont} Proposition 2.2. For an acyclic quiver $Q$ the cluster character $X_?$ induces a frieze on the repetition quiver $\Gamma(\mathcal{C}_Q)$. \end{proposition} \subsubsection{Friezes on the cluster category} We are interested affine quivers, in which case the transjective component of $\Gamma(\mathcal{C}_Q)$ is isomorphic to the repetition quiver $\mathbb{Z}Q$ and has the objects $P_i[1]$ at the vertices $(0,i)$. The frieze on $\mathbb{Z}Q$ given by (\ref{friezeformula}) places the initial cluster variables $X^i_0$ at $(0,i)$, so by \cite{assemdupont}, Corollary 3.2, the frieze $X^i_n$ coincides with the frieze given by $X_?$ on the transjective component. Throughout this section the vertex $e$ is taken to be a fixed extending vertex. In the $\tilde{A}$ case the extending vertices are all vertices and in the $\tilde{D}$ case these are the vertices labelled $1,2,N$ or $N+1$ in Figure \ref{Dquiver}. When we use these results in Sections \ref{Atypeclusterfriezesection} and \ref{Dtypeclusterfriezebit} we choose $e$ explicitly. For any $\lambda\in\mathbb{P}^1(k)$ there exists a unique quasi-simple module $M_{\lambda}$ in $\mathcal{T}_{\lambda}$ such that $\mathrm{dim}\:M_{\lambda}(e)=1$. Set $N_{\lambda}=M_{\lambda}[1]$ if $e$ is a source or $N_{\lambda}=M_{\lambda}[-1]$ if $e$ is a sink. In \cite{assemdupont} it is proved that there exists transjective $B_{\lambda}$ and $B'_{\lambda}$ that are uniquely determined by the existence of non-split triangles \[ \begin{tikzcd} N_{\lambda}\arrow[r] & B_{\lambda}\arrow[r] & S_{e}\arrow[r] & N_{\lambda}[1] \end{tikzcd} \qquad \begin{tikzcd} S_e\arrow[r] & B'_{\lambda}\arrow[r] & N_{\lambda}\arrow[r] & S_e[1] \end{tikzcd} \] for a source $e$ and \[ \begin{tikzcd} N_{\lambda}\arrow[r] & B'_{\lambda}\arrow[r] & S_{e}\arrow[r] & N_{\lambda}[1] \end{tikzcd} \qquad \begin{tikzcd} S_e\arrow[r] & B_{\lambda}\arrow[r] & N_{\lambda}\arrow[r] & S_e[1] \end{tikzcd} \] if $e$ is a sink. For $\tilde{A}$ and $\tilde{D}$ type these $B_{\lambda}$ and $B'_{\lambda}$ are then constructed explicitly. These triangles are used to allow consideration of friezes on the whole of $\Gamma(\mathcal{C}_Q)$. \begin{definition} For an affine quiver $Q$, a cluster frieze on $\Gamma(\mathcal{C}_{Q})$ is a frieze $f$ on $\Gamma(\mathcal{C}_{Q})$ such that, for any $\lambda\in\mathbb{P}^{\mathcal{E}}$ we have \begin{equation}\label{clusterfrieze} f(N_{\lambda}[k])=\frac{f(B_{\lambda}[k])+f(B'_{\lambda}[k])}{f(S_{e}[k])} \end{equation} for any $k=1,2,\ldots p_{\lambda}-1$. \end{definition} We can consider this as ``gluing" the friezes on the exceptional tubes to the frieze on the transjective component. Due to (\ref{clusterfrieze}) the values of a cluster frieze on the exceptional tubes are determined by the values on the transjective component. \begin{proposition}\cite{assemdupont}, Proposition 2.2. If $f$ is a cluster frieze on $\Gamma(\mathcal{C}_Q)$ such that $f(P_i[1])=u_i$ for each $i$ then $f(M)=X_M$ for all $M\in\mathcal{C}_Q$. \end{proposition} For $\tilde{A}$ and $\tilde{D}$ quivers we have seen that the $X^i_j$ of (\ref{friezeformula}) form a frieze on their transjective components. Our goal is to define friezes on the exceptional components, in terms of the periodic quantities $J_n$ and $\tilde{J}_n$ ($\tilde{A}$ type) and $J'_n$ ($\tilde{D}$ type) satisfying the cluster frieze condition (\ref{clusterfrieze}). This is done in Sections \ref{Atypeclusterfriezesection} and \ref{Dtypeclusterfriezebit}. \section{Linear relations and friezes for the matrix entries $A^m_n$, $\tilde{A}^m_n$ and ${A'}^m_n$}\label{Friezeconstructionsection} In this section we prove Propositions \ref{matrixentryclustervariables} and \ref{Dtypematrixentries} which give linear relations between the $A^m_n$ and $\tilde{A}^m_n$ in the $\tilde{A}$ case and ${A'}^m_n$ in the $\tilde{D}$ case, the matrix entries that appear in the construction of the constant coefficient linear relations. For both we show that we can construct friezes from these matrix entries. We describe the entries of $M_n^m$ as \[ M^m_n:= \begin{pmatrix} A^m_n & B^m_n \\ C^m_n & D^m_n \end{pmatrix} \] These satisfy $M^{m+1}_n=M^m_nL_{n+mp}$ so we have \[ \begin{pmatrix} A^{m+1}_{n} & B^{m+1}_{n} \\ C^{m+1}_{n} & D^{m+1}_{n} \end{pmatrix} = \begin{pmatrix} A^m_n & B^m_n \\ C^m_n & D^m_n \end{pmatrix} \begin{pmatrix} J_{n+mp} & 1\\ -1 & 0 \end{pmatrix} = \begin{pmatrix} A^m_nJ_{n+mp}-B^m_n & A^m_n \\ C^m_nJ_{n+mp}-D_n & C^m_n \end{pmatrix} \] This gives $B^m_{n}=A^{m-1}_{n}$ and $D^m_n=C^{m-1}_{n}$, so $A^m_n$ and $C^m_n$ satisfy \[ A^{m+1}_{n}=A^m_nJ_{n+mp}-A^{m-1}_{n}, \qquad C^{m+1}_{n}=C^m_nJ_{n+mp}-C^{m-1}_{n} \] with initial values \[ A^1_n=J_n, \qquad A^0_n=1, \qquad C^1_n=-1, \qquad C^0_n=0 \] since $M^0_n=I$ and $M^1_n=L_n$. We have $C^2_n=-J_{n+p}=-A^1_{n+p}$ and $C^1_n=-1=-A^0_{n+p}$ so $C^m_n=-A^{m-1}_{n+p}$ for all $n,m$. This proves Proposition \ref{matrixentryclustervariables} for $M^m_n$ and the result for $\tilde{M}^m_n$ is proved similarly. \begin{remark} In \cite{fordyhone}, for $q=1$, the traces $\mathcal{K}^m_n=A^m_n+D^m_n$ are constructed in terms of a recursion operator \[ \mathcal{R}^{(m)}_n=J_{n+m}J_{n+m+1}\frac{\partial^2}{\partial J_n\partial J_{n+m-1}}-J_{n+m}\frac{\partial}{\partial J_n}-J_{n+m+1}\frac{\partial}{\partial J_{n+m-1}}+J_{n+m}J_{n+m+1}-1 \] satisfying $\mathcal{K}^{m+2}_n=\mathcal{R}^m_n\mathcal{K}^m_n$. \end{remark} The recurrence (\ref{mainreccurence}) appears in \cite{coxeterfrieze}, with the iterates forming a frieze with the boundaries of ones and zeroes. Coxeter gives a general solution, which still works in our case; we define the determinants \begin{equation}\label{determinant} {D}^{m}_{a}(F_n):= \begin{vmatrix} F_n & 1 & 0 \\ 1 & F_{n+a} & 1 & 0 \\ 0 & 1 & F_{n+2a} & 1 & 0 \\ & 0 & 1 & F_{n+3a} & \ddots \\ & & 0 & \ddots & \ddots & 1 \\ & & & & 1 & F_{n+(m-1)a} \end{vmatrix} \end{equation} for an arbitrary function $F_n$ and integer $a$. By expanding along the last row these satisfy \[ {D}^{m}_{a}(F_n)=F_{n+(m-1)a}{D}^{m-1}_{a}(F_n)-{D}^{m-2}_{a}(F_n) \] with ${D}^{0}_{a}(F_n)=1$ and ${D}^{1}_{a}(F_n)=F_n$, hence we can express each of $A^m_n$, $\tilde{A}^m_n$ and ${A'}^m_n$ in terms of these determinants: \[ {D}^{m}_{p}(J_n)=A^m_n, \qquad D^m_q(\tilde{J}_n)=\tilde{A}^m_n, \qquad D^m_1(J'_n)={A'}^m_n. \] Furthermore applying the Desnanot-Jacobi identity to ${D}^{m}_{a}(F_n)$ gives \[ D^{m}_a(F_n)D^{m-2}_a(F_{n+a})=D^{m-1}_a(F_n)D^{m-1}_a(F_{n+a})-1 \] so the determinants $D^m_a(F_n)$ form a frieze: \begin{equation}\label{generaldetfrieze}\setcounter{MaxMatrixCols}{20} \begin{matrix} &1 & & 1 & & 1 & & 1 & & \ldots \\ \ldots & & F_n & & F_{n+a} & & F_{n+2a} & & F_{n+3a} \\ &D^2_a(F_{n-a}) & & D^2_a(F_n) & & D^2_a(F_{n+a}) & & D^2_a(F_{n+2a}) & & \ldots \\ \ldots && D^3_a(F_{n-a}) & & D^3_a(F_{n}) & & D^3_a(F_{n+a}) & & D^3_a(F_{n+2a}) \\ &\vdots & & \vdots & & \vdots && \vdots \end{matrix} \end{equation} This gives two friezes in $\tilde{A}$ type: \begin{equation}\label{frieze} (i) \qquad \:\: F_n=J_n, \qquad a=p. \end{equation} \begin{equation}\label{tildefrieze} (ii) \qquad F_n=\tilde{J}_n, \qquad a=q. \end{equation} and one frieze in $\tilde{D}$ type, \begin{equation}\label{friezeDtype} \qquad \quad \:\: F_n=J'_n, \qquad a=1. \end{equation} \begin{remark} \begin{enumerate}[(i)] \item The determinants (\ref{determinant}) are examples of continuants. See \cite{muir}, for example, where they are shown to be related to continued fractions. \item This determinant solution for Dynkin $A$ type appears in \cite{dupont}. There all cluster variables $x_n$ come from the cluster map and they form a frieze (\ref{frieze}), with the $x_n$ replacing the $J_n$ in an appropriate way. There the $D^m_n$ are called ``generalised Chebyshev polynomials". \end{enumerate} \end{remark} \section{Triangulated surfaces and the cluster frieze in $\tilde{A}$ type}\label{Atypesection} In Section \ref{Atypetriangulatedsection} we construct $\tilde{A}_{q,p}$ cluster algebras as triangulated surfaces. We first find the cluster variables $x_n$, obtained by the cluster map, as arcs before proving that the remaining arcs are given by the determinants $D^l_p(J_{jp})$ and $D^l_q(\tilde{J}_{jq})$ of Theorem \ref{findingallclustervarstheorem}, proving the $\tilde{A}$ part of that theorem. Finally in Section \ref{Atypeclusterfriezesection} we prove that the friezes constructed in this case, (\ref{Atypefrieze}) for the $x_n$ and (\ref{frieze}) and (\ref{tildefrieze}) for the $J_n$ and $\tilde{J}_n$, are precisely the cluster friezes given in \cite{assemdupont}. \subsection{$\tilde{A}$ type cluster algebras as triangulated surfaces}\label{Atypetriangulatedsection} The $\tilde{A}_{q,p}$ cluster algebra is obtained by triangulations of an annulus with $q$ and $p$ vertices on the inner and outer boundaries, respectively. By cutting this annulus we represent the surface as a strip with $q$ vertices on the top and $p$ vertices on the bottom, as in Figure \ref{annuluswithsomearcs}, where we have displayed some possible arcs (cluster variables) but not a full triangulation. In order to present the $\tilde{A}_{q,p}$ quiver constructed in Section \ref{dynamicalsystemssubsection} (for example Figure \ref{EuclideanA15}) as a triangulation, we take a strip with vertices labelled as in Figure \ref{annuluswithsomearcs} and draw the arcs $kp$ for ${k=0,1,\ldots,p+q-2}$ from vertex $k- \lfloor{kp/N} \rfloor$ (on the bottom) to vertex $q+\left \lfloor{kp/N}\right \rfloor$ (on the top). We also have an arc labelled $(p+q-1)p$ from $0$ on the bottom to $p+q-1$ on the top. We then reduce the labels modulo $N$. \begin{example} In Figure \ref{exampletriangulation} we demonstrate this construction for the quiver from Figure \ref{EuclideanA15}. Before reducing modulo $N$ our vertex labels are $0,7,14,21,28,\ldots, 98$. The first three, $k=0,1,2$, satisfy $\lfloor{kp/N} \rfloor=0$ so they give arcs $k\mapsto q=8$ labelled $0,7$ and $14$. The next, $k=3$ has $\lfloor{kp/N} \rfloor=1$ so it gives an arc $2\mapsto q+1=9$ which we label $6$, after reducing $21$ modulo $15$. Continuing this gives the rest of the triangulation. \begin{figure} \centering \[ \begin{tikzpicture}[every node/.style={fill=white}] \foreach \x in {0,...,7} \draw[fill=black] (\x,0) circle [radius=0.1cm]; \foreach \x in {0,...,6} \draw[fill=black] (\x+1/2,2) circle [radius=0.1cm]; \foreach \x in {-2,...,8} \draw [-] (\x,0) to (\x+1,0); \foreach \x in {-2,...,8} \draw [-] (\x,2) to (\x+1,2); \foreach \x in {0,...,7} \node[scale=0.7] at (\x,-0.3) {\x}; \node[scale=0.7] at (0.5,2.3) {$8$}; \node[scale=0.7] at (1.5,2.3) {$9$}; \node[scale=0.7] at (2.5,2.3) {$10$}; \node[scale=0.7] at (3.5,2.3) {$11$}; \node[scale=0.7] at (4.5,2.3) {$12$}; \node[scale=0.7] at (5.5,2.3) {$13$}; \node[scale=0.7] at (6.5,2.3) {$14$}; \node[scale=0.7] at (8,-0.3) {$0$}; \node[scale=0.7] at (-1.5,2.3) {$14$}; \draw [-] (0,0) to node[scale=0.7] {0} (0.5,2); \draw [-] (1,0) to node[scale=0.7] {7} (0.5,2); \draw [-] (2,0) to node[scale=0.7] {14} (0.5,2); \draw [-] (2,0) to node[scale=0.7] {6} (1.5,2); \draw [-] (3,0) to node[scale=0.7] {13} (1.5,2); \draw [-] (3,0) to node[scale=0.7] {5} (2.5,2); \draw [-] (4,0) to node[scale=0.7] {12} (2.5,2); \draw [-] (4,0) to node[scale=0.7] {4} (3.5,2); \draw [-] (5,0) to node[scale=0.7] {11} (3.5,2); \draw [-] (5,0) to node[scale=0.7] {3} (4.5,2); \draw [-] (6,0) to node[scale=0.7] {10} (4.5,2); \draw [-] (6,0) to node[scale=0.7] {2} (5.5,2); \draw [-] (7,0) to node[scale=0.7] {9} (5.5,2); \draw [-] (7,0) to node[scale=0.7] {1} (6.5,2); \draw [-] (8,0) to node[scale=0.7] {8} (6.5,2); \draw [-] (0,0) to node[scale=0.7] {8} (-1.5,2); \draw[dotted] (-0.5,-1) to (-0.5,3); \draw[dotted] (7.5,-1) to (7.5,3); \draw[fill=black] (-1.5,2) circle [radius=0.1cm]; \draw[fill=black] (8,0) circle [radius=0.1cm]; \end{tikzpicture} \] \caption{The triangulation for the $\tilde{A}_{8,7}$ quiver.} \label{exampletriangulation} \end{figure} \end{example} We see that in this construction every arc connects the top boundary to the bottom. Of course, any arc will either do this or connect two vertices on the same boundary component. The following proposition tells us precisely how to obtain every arc of the first kind. \begin{proposition}\label{Atypebipartitebelt} The arcs connecting the top of the strip to the bottom are in bijection with the cluster variables $x_n$ obtained by the recurrence (\ref{Atyperecurrence}). \begin{proof} We temporarily forget the periodicity of the strip and consider the (now infinitely many) vertices on the top and bottom to be labelled by $v_i$ and $v'_i$ for $i\in\mathbb{Z}$, as in Figure \ref{alternativelabelling}. \begin{figure} \centering \begin{tikzpicture} \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (1,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \node at (3,0) {$\ldots$}; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (0.5,2) circle [radius=0.1cm]; \draw[fill=black] (1.5,2) circle [radius=0.1cm]; \node at (2.5,2) {$\ldots$}; \draw[fill=black] (3.5,2) circle [radius=0.1cm]; \foreach \x in {-1,...,1} \draw [-] (\x,0) to (\x+1,0); \draw [-] (2,0) to (2.5,0); \draw [-] (3.5,0) to (4,0); \foreach \x in {4,...,6} \draw [-] (\x,0) to (\x+1,0); \foreach \x in {-1,...,1} \draw [-] (\x,2) to (\x+1,2); \foreach \x in {3,...,6} \draw [-] (\x,2) to (\x+1,2); \draw [-] (-1.5,2) to (-0.5,2); \node at (0,-0.3) {$v_0$}; \node at (1,-0.3) {$v_1$}; \node at (2,-0.3) {$v_2$}; \node at (4,-0.3) {$v_{q-1}$}; \node at (5,-0.3) {$v_{q}$}; \node at (6,-0.3) {$v_{q+1}$}; \node at (7,-0.3) {$v_{q+2}$}; \node at (-1.5,2.3) {$v'_{-1}$}; \node at (0.5,2.3) {$v'_0$}; \node at (1.5,2.3) {$v'_{1}$}; \node at (3.5,2.3) {$v'_{p-1}$}; \node at (5.5,2.3) {$v'_{p}$}; \node at (6.5,2.3) {$v'_{p+1}$}; \draw[fill=black] (5,0) circle [radius=0.1cm]; \draw[fill=black] (6,0) circle [radius=0.1cm]; \draw[fill=black] (7,0) circle [radius=0.1cm]; \draw[fill=black] (5.5,2) circle [radius=0.1cm]; \draw[fill=black] (-1.5,2) circle [radius=0.1cm]; \draw[fill=black] (6.5,2) circle [radius=0.1cm]; \draw[dotted] (-0.5,-1) to (-0.5,3); \draw[dotted] (4.5,-1) to (4.5,3); \end{tikzpicture}\caption{Alternative labelling of the vertices on the strip.}\label{alternativelabelling} \end{figure} The mutation $\mu_{k}$ will mutate each arc $k$ while it is a diagonal of a square (\ref{DiagonalAtype}) \begin{equation}\label{DiagonalAtype} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \draw[fill=black] (1,2) circle [radius=0.1cm]; \draw[fill=black] (3,2) circle [radius=0.1cm]; \draw [-] (-1,0) to (4,0); \draw [-] (-1,2) to (4,2); \draw [-] (0,0) to node[scale=0.7] {$k$} (3,2); \draw [-] (2,0) to (3,2); \draw [-] (0,0) to (1,2); \node[scale=1] at (0,-0.5) {$v_j$}; \node[scale=1] at (2,-0.5) {$v_{j+1}$}; \node[scale=1] at (1,2.5) {$v'_i$}; \node[scale=1] at (3,2.5) {$v'_{i+1}$}; \end{tikzpicture} \end{equation} to give the arc on the other diagonal $v'_{i}\:\mbox{---}\:v_{j+1}$, for example mutating the triangulation of Figure \ref{exampletriangulation} at $0$ gives Figure \ref{triangulationaftermutation}. The whole mutation sequence ${\mu=\mu_N\mu_{N-1}\ldots\mu_1\mu_0}$ will do this to every arc. With the labelling in Figure \ref{alternativelabelling}, the initial quiver has arcs $kp$ for ${k=0,1,\ldots,p+q-1}$ from vertex $k- \lfloor{kp/N} \rfloor$ (on the bottom) to vertex $\left \lfloor{kp/N}\right \rfloor$ (on the top). To show that a general arc $v_{i}\:\mbox{---}\:v'_{j}$ appears due to the cluster map, we reinstate the periodicity on Figure \ref{alternativelabelling} and have that \[ v_{i}\:\mbox{---}\:v'_{j}=v_{i+mp}\:\mbox{---}\:v'_{j+mq} \] for all $m\in\mathbb{Z}$, so by taking the correct $m$ we can assume that $0\leq i+j\leq p+q-1$. In this case the arc $k=i+j$ that appears in the initial triangulation is from $k- \lfloor{kp/N} \rfloor$ on the bottom to $\lfloor{kp/N}\rfloor $ on the top. Applying $\mu^{\lfloor{kp/N}\rfloor-j}$ will give the arc $v_{i}\:\mbox{---}\:v'_{j}$. \end{proof} \end{proposition} \begin{figure} \centering \[ \begin{tikzpicture}[every node/.style={fill=white}] \foreach \x in {0,...,7} \draw[fill=black] (\x,0) circle [radius=0.1cm]; \foreach \x in {0,...,6} \draw[fill=black] (\x+1/2,2) circle [radius=0.1cm]; \foreach \x in {-2,...,8} \draw [-] (\x,0) to (\x+1,0); \foreach \x in {-2,...,8} \draw [-] (\x,2) to (\x+1,2); \foreach \x in {0,...,7} \node[scale=0.7] at (\x,-0.3) {\x}; \node[scale=0.7] at (0.5,2.3) {$8$}; \node[scale=0.7] at (1.5,2.3) {$9$}; \node[scale=0.7] at (2.5,2.3) {$10$}; \node[scale=0.7] at (3.5,2.3) {$11$}; \node[scale=0.7] at (4.5,2.3) {$12$}; \node[scale=0.7] at (5.5,2.3) {$13$}; \node[scale=0.7] at (6.5,2.3) {$14$}; \node[scale=0.7] at (8,-0.3) {$0$}; \node[scale=0.7] at (9,-0.3) {$1$}; \node[scale=0.7] at (-1,-0.3) {$7$}; \node[scale=0.7] at (-1.5,2.3) {$14$}; \draw [-] (1,0) to node[scale=0.7] {0} (-1.5,2); \draw [-] (1,0) to node[scale=0.7] {7} (0.5,2); \draw [-] (2,0) to node[scale=0.7] {14} (0.5,2); \draw [-] (2,0) to node[scale=0.7] {6} (1.5,2); \draw [-] (3,0) to node[scale=0.7] {13} (1.5,2); \draw [-] (3,0) to node[scale=0.7] {5} (2.5,2); \draw [-] (4,0) to node[scale=0.7] {12} (2.5,2); \draw [-] (4,0) to node[scale=0.7] {4} (3.5,2); \draw [-] (5,0) to node[scale=0.7] {11} (3.5,2); \draw [-] (5,0) to node[scale=0.7] {3} (4.5,2); \draw [-] (6,0) to node[scale=0.7] {10} (4.5,2); \draw [-] (6,0) to node[scale=0.7] {2} (5.5,2); \draw [-] (7,0) to node[scale=0.7] {9} (5.5,2); \draw [-] (7,0) to node[scale=0.7] {1} (6.5,2); \draw [-] (8,0) to node[scale=0.7] {8} (6.5,2); \draw [-] (0,0) to node[scale=0.7] {8} (-1.5,2); \draw [-] (-1,0) to node[scale=0.7] {1} (-1.5,2); \draw [-] (6.5,2) to node[scale=0.7] {0} (9,0); \draw[dotted] (-0.5,-1) to (-0.5,3); \draw[dotted] (7.5,-1) to (7.5,3); \draw[fill=black] (-1.5,2) circle [radius=0.1cm]; \draw[fill=black] (8,0) circle [radius=0.1cm]; \draw[fill=black] (9,0) circle [radius=0.1cm]; \draw[fill=black] (-1,0) circle [radius=0.1cm]; \end{tikzpicture} \] \caption{The triangulation for the $\tilde{A}_{8,7}$ quiver after mutation at $0$.} \label{triangulationaftermutation} \end{figure} Aside from the arcs discussed in the previous proposition, we only have arcs connecting the top (or bottom) of the strip to itself, for example the arc $2\:\mbox{---}\: p$ in Figure \ref{annuluswithsomearcs}. We call these $J_{jp}$ and $\tilde{J}_{-iq}$, as shown in Figure \ref{arcJ}, for $j=0,1,\ldots,q-1$ and $i=0,1,\ldots,p-1$. \begin{figure} \centering \begin{tikzpicture} \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (3,2) circle [radius=0.1cm]; \draw[fill=black] (5,2) circle [radius=0.1cm]; \draw[fill=black] (7,2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,0) (2,0.7) (4,0)}; \draw plot [smooth, tension=1] coordinates {(3,2) (5,1.3) (7,2)}; \draw plot [smooth, tension=1] coordinates {(0,0) (2,0)}; \draw plot [smooth, tension=1] coordinates {(0,2) (7,2)}; \draw plot [smooth, tension=1] coordinates {(2,0) (7,0)}; \node at (2,1.0) {$J_{jp}$}; \node at (5,1.0) {$\tilde{J}_{-iq}$}; \node at (0,-0.3) {$j$}; \node at (2,-0.3) {$j+1$}; \node at (4,-0.3) {$j+2$}; \node at (3,2.3) {$q+i-2$}; \node at (5,2.3) {$q+i-1$}; \node at (7,2.3) {$q+i$}; \end{tikzpicture}\caption{The arcs $J_{jp}$ and $\tilde{J}_{-iq}$.}\label{arcJ} \end{figure} This naming is justified by the following lemma. \begin{lemma}\label{constructingJarcslemma} The arcs $J_{jp}$ and $\tilde{J}_{-iq}$ are the arcs associated with the periodic quantities (\ref{periodicquantities}), which are cluster variables. \begin{proof} We first prove that for any arc labelled $\overline{kp}$, for some $k$, at vertex $j$ on the bottom boundary we have $\overline{kp}\equiv jp \mod q$. Here $\overline{kp}$ denotes the label after reduction modulo $N$. We prove this by induction and assume that the statement is true for the arcs labelled up to $\overline{(k-1)p}$. There are two possibilities: \begin{equation*} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (5,2) circle [radius=0.1cm]; \draw [-] (-1,0) to (6,0); \draw [-] (3,2) to (6,2); \draw [-] (4,0) to node[scale=0.7] {$\overline{kp}$} (5,2); \draw [-] (0,0) to node[scale=0.7] {$\overline{(k-1)p}$} (5,2); \node at (0,-0.4) {$j-1$}; \node at (4,-0.4) {$j$}; \end{tikzpicture} \end{equation*} or \begin{equation*} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (9,2) circle [radius=0.1cm]; \draw[fill=black] (5,2) circle [radius=0.1cm]; \draw [-] (3,0) to (7,0); \draw [-] (3,2) to (9,2); \draw [-] (4,0) to node[scale=0.7] {$\overline{(k-1)p}$} (5,2); \draw [-] (4,0) to node[scale=0.7] {$\overline{kp}$} (9,2); \node at (4,-0.4) {$j-1$}; \end{tikzpicture} \end{equation*} In the first case $\overline{kp}=\overline{(k-1)p}+p$ so \[ \overline{kp}\equiv \overline{(k-1)p}+p\equiv (j-1)p+p\equiv jp \mod q \] where we have used the induction assumption. In the second case \[ \overline{kp}=\overline{(k-1)p}+p-N=\overline{(k-1)p}-q \] so \[ \overline{kp}\equiv \overline{(k-1)p} \equiv (j-1)p \mod q. \] To construct the $J_n$ we let $\overline{kp}$ be the rightmost arc at $j$. Near $j+1$ there are two situations: \begin{equation*} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (8,0) circle [radius=0.1cm]; \draw[fill=black] (5,2) circle [radius=0.1cm]; \draw [-] (-1,0) to (9,0); \draw [-] (3,2) to (7,2); \draw [-] (4,0) to node[scale=0.7] {$\overline{kp}+p$} (5,2); \draw [-] (0,0) to node[scale=0.7] {$\overline{kp}$} (5,2); \draw [-] (8,0) to node[scale=0.7] {$\overline{kp}+2p$} (5,2); \node at (0,-0.4) {$j$}; \node at (4,-0.4) {$j+1$}; \node at (8,-0.4) {$j+2$}; \end{tikzpicture} \end{equation*} or \begin{equation*} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (8,0) circle [radius=0.1cm]; \draw[fill=black] (9,2) circle [radius=0.1cm]; \draw[fill=black] (5,2) circle [radius=0.1cm]; \draw [-] (-1,0) to (9,0); \draw [-] (3,2) to (9,2); \draw [-] (4,0) to node[scale=0.7] {$\overline{kp}+p$} (5,2); \draw [-] (0,0) to node[scale=0.7] {$\overline{kp}$} (5,2); \draw [-] (4,0) to node[scale=0.7] {$\overline{kp}+2p-N$} (9,2); \draw [-] (8,0) to node[scale=0.7] {$\overline{kp}+3p-N$} (9,2); \node at (0,-0.4) {$j$}; \node at (4,-0.4) {$j+1$}; \node at (8,-0.4) {$j+2$}; \end{tikzpicture} \end{equation*} In the first case mutation at $\overline{kp}+p$ gives $J_{\overline{kp}}$. Mutating the second case at $\overline{kp}+2p-N$ gives the first case so we can obtain $J_{\overline{kp}}$ between vertices $j$ and $j+2$ again. Since $\overline{kp}\equiv jp \mod q$ and $J_n$ is period $q$ we see that the arc obtained is indeed $J_{jp}$. For $\tilde{J}$ we look at the leftmost arc at $q+i$ on the top boundary, labelled $\overline{kp}$. The situation is this: \begin{equation} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (-2,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (8,0) circle [radius=0.1cm]; \draw[fill=black] (9,2) circle [radius=0.1cm]; \draw[fill=black] (5,2) circle [radius=0.1cm]; \draw[fill=black] (-3,2) circle [radius=0.1cm]; \draw [-] (-3,0) to (9,0); \draw [-] (-4,2) to (9,2); \draw [-] (4,0) to node[scale=0.7] {$\overline{kp}+q-p$} (5,2); \draw [-] (2,0) to (5,2); \draw [-] (-2,0) to (5,2); \draw [-] (-2,0) to (-3,2); \draw [-] (8,0) to node[scale=0.7] {$\overline{kp}+q$} (5,2); \draw [-] (8,0) to node[scale=0.7] {$\overline{kp}$} (9,2); \node at (5,2.4) {$q+i-1$}; \node at (2,0.7) {$\ldots$}; \node at (0,0) {$\ldots$}; \node at (9,2.4) {$q+i$}; \end{tikzpicture} \end{equation} We mutate at each of the arcs at $q+i-1$ from left to right, except for the arc labelled $\overline{kp}+q$ which gives, near $q+i-1$, \begin{equation} \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (8,0) circle [radius=0.1cm]; \draw[fill=black] (9,2) circle [radius=0.1cm]; \draw[fill=black] (5,2) circle [radius=0.1cm]; \draw[fill=black] (-3,2) circle [radius=0.1cm]; \draw [-] (-3,0) to (9,0); \draw [-] (-4,2) to (9,2); \draw [-] (8,0) to node[scale=0.7] {$\overline{kp}+2q$} (-3,2); \draw [-] (8,0) to node[scale=0.7] {$\overline{kp}+q$} (5,2); \draw [-] (8,0) to node[scale=0.7] {$\overline{kp}$} (9,2); \node at (5,2.4) {$q+i-1$}; \node at (-3,2.4) {$q+i-2$}; \node at (9,2.4) {$q+i$}; \node at (1,0.7) {$\ldots$}; \node at (3,0) {$\ldots$}; \end{tikzpicture} \end{equation} Mutation at $\overline{kp}+q$ will then give $\tilde{J}_{\overline{kp}}$. By a similar argument to the one at the start of this proof we have that $\tilde{J}_{\overline{kp}}=\tilde{J}_{-iq}$ \end{proof} \end{lemma} The $J$ and $\tilde{J}$ connect vertices on the same boundary by ``jumping" over one vertex. The arcs we haven't discussed so far are those jumping over more than one vertex. We define these as $J^{l-1}_{j}$, which starts at $j$ and jumps right over $l-1$ vertices to reach $j+l$, and $\tilde{J}_{i}^{m-1}$ which starts at $q+i$ and jumps left over $m-1$ vertices to reach $q+i-m$, as shown in Figure \ref{LongerJ}. Again we define these for $j=0,1,\ldots,q-1$ and $i=0,1,\ldots,p-1$. We remark that $J^{q-1}_{j}$ and $\tilde{J}^{p-1}_{i}$ are the widest arcs, as $J^{q}_{j}$ and $\tilde{J}^{p}_{i}$ self-intersect. \begin{figure} \centering \begin{tikzpicture} \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (3,2) circle [radius=0.1cm]; \draw[fill=black] (5.6,2) circle [radius=0.1cm]; \draw[fill=black] (9,2) circle [radius=0.1cm]; \draw[fill=black] (8,0) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,0) (4,0.7) (8,0)}; \draw plot [smooth, tension=1] coordinates {(3,2) (6,1.3) (9,2)}; \draw plot [smooth, tension=1] coordinates {(0,0) (2,0)}; \draw plot [smooth, tension=1] coordinates {(0,2) (6,2)}; \draw plot [smooth, tension=1] coordinates {(8,2) (10,2)}; \draw plot [smooth, tension=1] coordinates {(2,0) (4,0)}; \draw plot [smooth, tension=1] coordinates {(7,0) (9,0)}; \node at (6,0) {$\ldots$}; \node at (7,2) {$\ldots$}; \node at (2,0.9) {$J^{l-1}_{j}$}; \node at (8,1.1) {$\tilde{J}^{m-1}_{i}$}; \node at (0,-0.3) {$j$}; \node at (2,-0.3) {$j+1$}; \node at (4,-0.3) {$j+2$}; \node at (8,-0.3) {$j+l$}; \node at (3,2.3) {$q+i-m$}; \node at (5.8,2.3) {$q+i-m+1$}; \node at (9,2.3) {$q+i$}; \end{tikzpicture}\caption{The arcs $J^{l-1}_{j}$ and $\tilde{J}^{m-1}_{i}$.}\label{LongerJ} \end{figure} For small $l$ we have \[ J^0_{j}=1, \qquad J^1_{j}=J_{jp}, \qquad \tilde{J}^0_{i}=1, \qquad \tilde{J}^1_{i}=\tilde{J}_{-iq} \] and the following theorem allows us to calculate $J^l_{j}$ and $\tilde{J}^l_i$ for $l>1$. \begin{theorem} The arcs $J^l_j$ satisfy the recurrence relation \[ J^{l-1}_{j}=J^{l-2}_{j}J_{(j+l-2)p}-J^{l-3}_{j} \] for $l=3,4,\ldots,q$, with initial values $J^l_0=1$ and $J^1_{j}=J_{jp}$. Similarly, the arcs $\tilde{J}^m_i$ satisfy \[ \tilde{J}_i^{m-1}=\tilde{J}^{m-2}_i\tilde{J}_{-(i-m+2)q}-\tilde{J}^{m-3}_i \] for $m=3,4,\ldots,p$, with initial values $\tilde{J}^0_{i}=1$ and $\tilde{J}^1_{i}=\tilde{J}_{-iq}$. Hence $J^l_{j}={D}^{l}_p(J_{jp})$, as defined in (\ref{determinant}), and they form a frieze (\ref{frieze}). Similarly we have $\tilde{J}^m_i=D^m_q(\tilde{J}_{-iq})$ forming a frieze (\ref{tildefrieze}). \begin{proof} We look at the quadrilateral with diagonals $J_{(j+l-2)p}$ and $J^{l-2}_{j}$: \begin{equation}\label{Ptolemypic} \begin{tikzpicture} \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (8,0) circle [radius=0.1cm]; \draw[fill=black] (10,0) circle [radius=0.1cm]; \draw[fill=black] (12,0) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,0) (6,3) (12,0)}; \draw plot [smooth, tension=1] coordinates {(0,0) (4,0.7) (8,0)}; \draw plot [smooth, tension=1] coordinates {(0,0) (5,1.7) (10,0)}; \draw plot [smooth, tension=1] coordinates {(8,0) (10,0.7) (12,0)}; \draw plot [smooth, tension=1] coordinates {(-1,0) (4,0)}; \draw plot [smooth, tension=1] coordinates {(7,0) (13,0)}; \node at (6,0) {$\ldots$}; \node at (6,3.3) {$J^{l-1}_{j}$}; \node at (10,1.0) {$J_{(j+l-2)p}$}; \node at (4,1.0) {$J^{l-3}_{j}$}; \node at (5,2.0) {$J^{l-2}_{j}$}; \node at (0,-0.3) {$j$}; \node at (2,-0.3) {$j+1$}; \node at (4,-0.3) {$j+2$}; \node at (8,-0.3) {$j+l-2$}; \node at (10,-0.3) {$j+l-1$}; \node at (12,-0.3) {$j+l$}; \end{tikzpicture} \end{equation} In this case the Ptolemy relation (\ref{ptolemy}) gives \[ J^{l-2}_{j}J_{(j+l-2)p}=J^{l-3}_{j}+J^{l-1}_{j} \] since the boundary arcs have the value $1$. The widest arcs are $J^{q-1}_{j}$, since the next arc, $J^q_{j}$, would self-intersect. The result for the $\tilde{J}^m_i$ arcs follows from a similar construction. \end{proof} \end{theorem} This proves the $\tilde{A}$ parts of Theorem \ref{findingallclustervarstheorem} and Proposition \ref{friezepropintro}. What remains is to prove the $\tilde{A}$ part of Proposition \ref{intropropclusterfrieze}. \subsection{$\tilde{A}$ type cluster friezes}\label{Atypeclusterfriezesection} Here we intend to show that the frieze pattern (\ref{friezeformulaintro}) and the friezes of Proposition \ref{friezepropintro} give a cluster frieze on the Auslander-Reiten quiver $\Gamma(\mathcal{C}_Q)$. They are already friezes, so we just need to ensure that they are connected by the relation (\ref{clusterfrieze}). In this case there are two exceptional tubes given by $\lambda=0,1$. Firstly we need a description of the modules $B_{\lambda}$ and $B'_{\lambda}$, as given in \cite{assemdupont}. The vertex $e$ is required to be a sink so we first mutate the $\tilde{A}_{q,p}$ quiver at $0$ and take $e=0$. The portion of the quiver near $0$ now looks like \begin{equation*} \begin{tikzcd} 2q-N\arrow[r] & {q}\arrow[r] & {0} & p\arrow[l] \end{tikzcd} \end{equation*} with cluster variables \begin{equation}\label{newnearzero} \begin{tikzcd} x_{2q-N}\arrow[r] & x_{q}\arrow[r] & x_{N} & x_p\arrow[l] \end{tikzcd} \end{equation} and \cite{assemdupont} gives $B_0=P_{q}$ and $B'_0=P_{p}[1]$. By definition $X_{P_{p}[1]}=x_p$. $X_{P_0}=X_{S_0}$ has $x_N$ as its denominator, hence is given by mutating (\ref{newnearzero}) at $N$, so $X_{P_0}=x_0$. Similarly $X_{P_q}$ has denominator $x_qx_N$ so is given by performing $\mu_q\mu_{N}$ on (\ref{newnearzero}), so $X_{P_q}=x_{-p}$. Collecting this we have that (\ref{clusterfrieze}) is \[ X_{N_0}=\frac{X_{P_{q}}+X_{P_{p}[1]}}{X_{P_{N}}}=\frac{x_{-p}+x_{p}}{x_{0}}=J_{0}. \] The map $M\mapsto M[1]$ gives an automorphism of the cluster algebra, so \[ X_{N_0[j]}=\frac{X_{P_{q}[j]}+X_{P_{p}[j+1]}}{X_{P_{N}[j]}}=\frac{x_{-p+jN}+x_{p+jN}}{x_{-1+jN}}=J_{jN}=J_{jp}. \] Conversely the modules $B_1=P_{p}$ and $B'_1=P_{q}[1]$ give \[ X_{N_1[j]}=\frac{X_{P_{q}[j]}+X_{P_{p}[j+1]}}{X_{P_{N}[j]}}=\frac{x_{-q+jN}+x_{q+jN}}{x_{jN}}=\tilde{J}_{jN}=\tilde{J}_{jq}. \] This proves Proposition \ref{intropropclusterfrieze} in that $\tilde{A}$ case. \section{Triangulated surfaces and the cluster frieze in $\tilde{D}$ type}\label{Dtypesection} In this section we look at the construction of $\tilde{D}$ quivers as triangulations of discs with two punctures. We first show how to construct the arcs corresponding to the cluster map variables $X^i_n$. We then construct a periodic frieze of cluster variables, with the first row given by the periodic quantities $J'_n$, as in the $\tilde{A}$ case. We show that there are only $3$ exceptional arcs outside of these. Finally we identify the two friezes constructed here as the cluster friezes given in \cite{assemdupont}. \subsection{$\tilde{D}$ type cluster algebras as triangulated surfaces} As discussed in Subsection \ref{dynamicalsystemssubsection} we take the bipartite orientation of the $\tilde{D}$ diagram as shown in Figure \ref{Dquiver}. To ensure that this is bipartite the orientation of the arrows at the right end of the diagram depend on the parity of $N$, which we have signified with double ended arrows. \begin{figure} \centering \begin{equation*} \begin{tikzcd} X^1\arrow[dr] & & & & & & & X^N\\ & X^3 & X^4\arrow[l]\arrow[r] & X^5 & \arrow[l]\ldots\arrow[r, leftrightarrow] & X^{N-2}\arrow[r, leftrightarrow] & X^{N-1}\arrow[ur, leftrightarrow]\arrow[dr, leftrightarrow] \\ X^2\arrow[ur] & & & & & & & X^{N+1} \end{tikzcd} \end{equation*} \caption{The $\tilde{D}_N$ quiver.}\label{Dquiver} \end{figure} The cluster map \begin{equation}\label{Dtypeclustermap} \varphi: \begin{pmatrix} X^1_n, & X^2_n, &\ldots &X^{N+1}_n \end{pmatrix} \mapsto \begin{pmatrix} X^1_{n+1}, & X^2_{n+1}, &\ldots &X^{N+1}_{n+1} \end{pmatrix} \end{equation} is obtained by applying \[ \mu:=\mu_{\mathrm{source}}\circ\mu_{\mathrm{sink}} \] to $Q$, where \[ \mu_{\mathrm{source}}:=\mu_1\mu_2\mu_4\ldots, \qquad \mu_{\mathrm{sink}}:=\mu_3\mu_5\mu_7\ldots \] which are the compositions of mutations at the sources and sinks in Figure \ref{Dquiver}. As shown in \cite{clustersandtriangulated1} we can see this quiver as a triangulation of a disk with $2$ marked points inside and $N-2$ marked points on the boundary. An example of this is seen in Figure \ref{Dtypebefore}. In order to determine all of the arcs that can occur in a triangulation, we firstly note that every arc $\gamma$ connecting boundary vertices splits the disc into two connected components. There are two possibilities: either the punctures $p_1$ and $p_2$ lie in the same connected component or in different components. We denote these sets of arcs by: \[ \Gamma_{2,0}:=\{\gamma \mid \gamma \textrm{ connects boundary vertices and } p_1 \textrm{ and } p_2 \textrm{ appear on the same side of } \gamma\} \] \[ \Gamma_{1,1}:=\{\gamma \mid \gamma \textrm{ connects boundary vertices and } p_1 \textrm{ and } p_2 \textrm{ appear on different sides of } \gamma\} \] There are also arcs connecting the boundary to the punctures, which we call \[ \Gamma_{\mathrm{punc}}:=\{\gamma \mid \gamma \textrm{ connects a puncture to the boundary}\} \] Outside of these three sets there are three exceptional arcs involving only the punctures (\ref{3exceptionalarcs}), which we call $\Gamma_{\mathrm{except}}$. \begin{equation}\label{3exceptionalarcs} \begin{tikzpicture}[scale=0.9, every node/.style={fill=white}] \draw (0,0) circle [radius=3.0]; \draw[fill=black] (0,1) circle [radius=0.1cm]; \draw[fill=black] (0,-1) circle [radius=0.1cm]; \draw[green, thick] plot [smooth, tension=1] coordinates { (0,1) (0,-1) }; \draw[blue, thick] plot [smooth, tension=1] coordinates { (0,1) (-0.6,0) (0,-2) (1.2,0) (0,1) }; \draw[red, thick] plot [smooth, tension=1] coordinates {(0,-1) (-1.2,0) (0,2) (0.6,0) (0,-1)}; \end{tikzpicture} \end{equation} \begin{lemma}\label{Dsurfacepossiblearcs} The arcs of $\Gamma_{1,1}$ look like those shown in Figure \ref{just2boundaryverticesfinal}. They start from a vertex $v_i$ and encircle the line $L$ and the punctures $m\in \mathbb{Z}$ times, as shown in Figure \ref{just2boundaryvertices} in blue for $m>0$. After this the curve crosses $L$ and takes the only possible path to $v_j$ (red). For $m<0$ the curve is shown by reflecting this picture in the vertical axis. We denote these by $\gamma(v_i,v_j,m)$. The arcs of $\Gamma_{\mathrm{punc}}$ are the arcs that form self-folded triangles with a $\gamma(v_i,v_i,m)$. \begin{proof} We firstly look at an arc $\gamma\in \Gamma_{1,1}$ connecting two boundary vertices $v_i$ and $v_j$. Since this arc separates the two punctures it necessarily crosses the line joining them, which we call $L$. The portion of $\gamma$ before this crossing can only live in the annulus obtained by removing an area enclosing the two punctures and $L$, as shown in Figure \ref{just2boundaryvertices} on the left, so it simply orbits the drawn ellipse $m\in \mathbb{Z}$ times, where we take $m>0$ to mean that the blue portion of the arc travels anticlockwise, starting from $v_i$, while $m<0$ means clockwise. When $\gamma$ finally crosses $L$, as shown on the right of Figure \ref{just2boundaryvertices} (we draw this new portion of $\gamma$ in red for clarity), there is only one choice, to follow the maze back to $v_j$. The only remaining issue is that perhaps the curve crosses $L$ from the other direction, i.e. the curve of Figure \ref{just2boundaryvertices} is blue to the right of $L$ and red to the left. In this case we just need to permute $i\leftrightarrow j$ to obtain a curve of the right form. Finally we note that every arc of $\Gamma_{\mathrm{punc}}$ appears as the internal arc of a self-folded triangle. The other arc of this self-folded triangle, as we have just shown, is of the form $\gamma(v_i,v_i,m)$. \begin{figure} \centering \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,1) circle [radius=0.1cm]; \draw[fill=black] (0,-1) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \node[scale=1] at (-4,2){$v_j$}; \node[scale=1] at (0,-4.4){$v_i$}; \node[scale=0.7] at (0.2,0){$L$}; \draw[dotted] (0,-1) to (0,1); \draw (0,0) ellipse (1cm and 2cm); \draw[blue, thick] plot [smooth, tension=1] coordinates {(0,-4) (3.5,0) (0,3.5) (-3.5,0) (0,-3) (2.9,0) (0,2.9) (-2.9,0) (0,-2.5) (2.5,0) (0,2.5)}; \end{tikzpicture} \qquad \begin{tikzpicture}[every node/.style={fill=white}] \draw[fill=black] (0,1) circle [radius=0.1cm]; \draw[fill=black] (0,-1) circle [radius=0.1cm]; \node[scale=0.7] at (0.2,0.65){$L$}; \draw[dotted] (0,-1) to (0,1); \draw[blue, thick] plot [smooth, tension=1] coordinates { (-1,3.5) (-3.5,0) (0,-3) (2.9,0) (0,2.9) (-2.9,0) (0,-2.5) (2.5,0) (0,2.3) (-1.8,0.8) (0,0)}; \draw[red, thick] plot [smooth, tension=1] coordinates {(0,0) (1,-0.8) (0,-2) (-2,0)}; \end{tikzpicture} \caption{The portion of an arc (blue) before crossing the line $L$ and the behaviour just after crossing (red).} \label{just2boundaryvertices} \end{figure} \begin{figure} \centering \begin{tikzpicture}[every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,1) circle [radius=0.1cm]; \draw[fill=black] (0,-1) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \node[scale=1] at (-3.9,2){$v_j$}; \node[scale=1] at (0,-4.4){$v_i$}; \node[scale=0.7] at (0.2,0.65){$L$}; \draw[dotted] (0,-1) to (0,1); \draw[blue, thick] plot [smooth, tension=1] coordinates {(0,-4) (3.5,0) (0,3.5) (-3.5,0) (0,-3) (2.9,0) (0,2.9) (-2.9,0) (0,-2.5) (2.5,0) (0,2.3) (-1.8,0.8) (0,0)}; \draw[red, thick] plot [smooth, tension=1] coordinates {(0,0) (1,-0.8) (0,-2) (-2,0) (-1,2.4) (1,2.2) (2.6,0.7) (2.0,-1.7) (0,-2.7) (-2,-2) (-3.2,0) (-2,2.6) (0,3.2) (2,2.6) (3.2,0) (2,-2.4) (0,-3.3) (-2.1,-2.9) (-3.7,0) (-3.3,1.6) (-3.46,2) }; \end{tikzpicture} \caption{The full curve after the red portion takes the only path to reach $v_j$.} \label{just2boundaryverticesfinal} \end{figure} \end{proof} \end{lemma} Next we try to identify the arcs obtained by the cluster map. Our initial triangulation is shown in Figure \ref{Dtypebefore} for $N=8$. Its construction can be tentatively described in two steps: \begin{enumerate}[(i)] \item Draw two self folded triangles; one at $v_1$ and $p_1$ and one at $v_6$ and $p_2$. \item Draw arcs $i$ from $v_{i-2}$ to $v_{i-1}$ for $i=3,4,\ldots, N-1$. \end{enumerate} We remark that in the second step there is only one choice for each of the arcs. We need to see how this triangulation changes under $\mu$, which we perform in several steps, as shown in Figure \ref{fullDmutation}. \begin{figure} \begin{subfigure}{0.31\textwidth} \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (0,2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (-0.7,3) (0,1.6) (0.7,3) (0,4)}; \draw plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (-1,-3) (0,-1.4) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (1,3) (0,1.4) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,1) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,-2) (0,-1) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,0) (-3.46,-2)}; \node[scale=0.7] at (0,-3){1}; \node[scale=0.7] at (0.7,-3){2}; \node[scale=0.7] at (-1,-2.7){3}; \node[scale=0.7] at (0,-1){4}; \node[scale=0.7] at (0,0){5}; \node[scale=0.7] at (0,1){6}; \node[scale=0.7] at (1,2.7){7}; \node[scale=0.7] at (-0.7,3){8}; \node[scale=0.7] at (0,3){9}; \node[scale=0.7] at (-3.9,-2){$v_3$}; \node[scale=0.7] at (-3.9,2){$v_5$}; \node[scale=0.7] at (3.9,2){$v_4$}; \node[scale=0.7] at (0,-4.4){$v_1$}; \node[scale=0.7] at (0,4.4){$v_6$}; \node[scale=0.7] at (3.9,-2){$v_2$}; \node[scale=0.7] at (-0.2,-2.35){$p_1$}; \node[scale=0.7] at (0.2,2.35){$p_2$}; \end{tikzpicture} \caption{The initial triangulation for $N=8$.} \label{Dtypebefore} \end{subfigure}% \hspace{40mm} \begin{subfigure}{0.31\textwidth} \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (0,2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (-0.7,3) (0,1.6) (0.7,3) (0,4)}; \draw plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1,-3) (0,-1.4) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (-1,3) (0,1.4) (3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,1) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,-2) (0,-1) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(-3.46,2) (0,0) (3.46,-2)}; \node[scale=0.7] at (0,-3){1}; \node[scale=0.7] at (-0.7,-3){2}; \node[scale=0.7] at (1,-2.7){3}; \node[scale=0.7] at (0,-1){4}; \node[scale=0.7] at (0,0){5}; \node[scale=0.7] at (0,1){6}; \node[scale=0.7] at (-1,2.7){7}; \node[scale=0.7] at (0.7,3){8}; \node[scale=0.7] at (0,3){9}; \end{tikzpicture} \caption{After performing $\mu_7\mu_5\mu_3$.} \label{fig:1b} \end{subfigure}% \begin{subfigure}{0.31\textwidth} \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (0,2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (-0.7,3) (0,1.6) (0.7,3) (0,4)}; \draw plot [smooth, tension=1] coordinates {(0,-2) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1,-3) (0,-1.4) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (-1,3) (0,1.4) (3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,1) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,-2) (0,-1) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(-3.46,2) (0,0) (3.46,-2)}; \node[scale=0.7] at (0,-3){1}; \node[scale=0.7] at (-0.7,-2){2}; \node[scale=0.7] at (1,-2.7){3}; \node[scale=0.7] at (0,-1){4}; \node[scale=0.7] at (0,0){5}; \node[scale=0.7] at (0,1){6}; \node[scale=0.7] at (-1,2.7){7}; \node[scale=0.7] at (0.7,3){8}; \node[scale=0.7] at (0,3){9}; \end{tikzpicture} \caption{After $\mu_2$.} \label{Dtypenext} \end{subfigure} \hspace{40mm} \begin{subfigure}{0.31\textwidth} \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(-3.46,-2) (-0.5,-1.7) (0.5,-2) (-0.5,-2.3) (-3.46,-2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (0,2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (-0.7,3) (0,1.6) (0.7,3) (0,4)}; \draw plot [smooth, tension=1] coordinates {(0,-2) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1,-3) (0,-1.4) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (-1,3) (0,1.4) (3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,1) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,-2) (0,-1) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(-3.46,2) (0,0) (3.46,-2)}; \node[scale=0.7] at (0.5,-2){1}; \node[scale=0.7] at (-0.7,-2){2}; \node[scale=0.7] at (1,-2.7){3}; \node[scale=0.7] at (0,-1){4}; \node[scale=0.7] at (0,0){5}; \node[scale=0.7] at (0,1){6}; \node[scale=0.7] at (-1,2.7){7}; \node[scale=0.7] at (0.7,3){8}; \node[scale=0.7] at (0,3){9}; \end{tikzpicture} \caption{After $\mu_1$.} \end{subfigure} \begin{subfigure}{0.31\textwidth} \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(-3.46,-2) (-0.5,-1.7) (0.5,-2) (-0.5,-2.3) (-3.46,-2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (0,2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (-0.7,3) (0,1.6) (0.7,3) (0,4)}; \draw plot [smooth, tension=1] coordinates {(0,-2) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1,-3) (0,-1.4) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (-1,3) (0,1.4) (3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (-2,3) (0,0.7) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (2,-3) (0,-0.7) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(-3.46,2) (0,0) (3.46,-2)}; \node[scale=0.7] at (0.5,-2){1}; \node[scale=0.7] at (-0.7,-2){2}; \node[scale=0.7] at (1,-2.7){3}; \node[scale=0.7] at (0,-0.7){4}; \node[scale=0.7] at (0,0){5}; \node[scale=0.7] at (0,0.7){6}; \node[scale=0.7] at (-1,2.7){7}; \node[scale=0.7] at (0.7,3){8}; \node[scale=0.7] at (0,3){9}; \end{tikzpicture} \caption{After $\mu_6\mu_4$.} \label{Dtypenext2} \end{subfigure} \hspace{40mm} \begin{subfigure}{0.31\textwidth} \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(-3.46,-2) (-0.5,-1.7) (0.5,-2) (-0.5,-2.3) (-3.46,-2)}; \draw plot [smooth, tension=1] coordinates {(3.46,2) (0.5,1.7) (-0.5,2) (0.5,2.3) (3.46,2)}; \draw plot [smooth, tension=1] coordinates {(0,-2) (-3.46,-2)}; \draw plot [smooth, tension=1] coordinates {(0,2) (3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1,-3) (0,-1.4) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (-1,3) (0,1.4) (3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (-2,3) (0,0.7) (3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (2,-3) (0,-0.7) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(-3.46,2) (0,0) (3.46,-2)}; \node[scale=0.7] at (0.5,-2){1}; \node[scale=0.7] at (-0.7,-2){2}; \node[scale=0.7] at (1,-2.7){3}; \node[scale=0.7] at (0,-0.7){4}; \node[scale=0.7] at (0,0){5}; \node[scale=0.7] at (0,0.7){6}; \node[scale=0.7] at (-1,2.7){7}; \node[scale=0.7] at (0.7,2){8}; \node[scale=0.7] at (-0.5,2){9}; \node[scale=0.7] at (-3.9,-2){$v_1$}; \node[scale=0.7] at (-3.9,2){$v_3$}; \node[scale=0.7] at (3.9,2){$v_6$}; \node[scale=0.7] at (0,-4.4){$v_2$}; \node[scale=0.7] at (0,-2.35){$p_1$}; \node[scale=0.7] at (0,2.35){$p_2$}; \node[scale=0.7] at (0,4.4){$v_5$}; \node[scale=0.7] at (3.9,-2){$v_4$}; \end{tikzpicture} \caption{After $\mu_9\mu_8$.} \label{laststep} \end{subfigure} \caption{A full application of $\mu$ to the initial quiver.} \label{fullDmutation} \end{figure} In Figure \ref{laststep}, after a complete application of $\mu$, we can see that the boundary vertices of the self-folded triangles have moved clockwise. We have also rotated the boundary vertices to obtain a new labelling for the boundary of $\mu(Q)$, so now the new quiver can be described by the same two steps as before. In Figure \ref{incompleteDtype} we display the full triangulation for $\mu^2(Q)$ and some of the triangulation for $\mu^3(Q)$ (because the full picture would be too busy to be helpful) which are also constructed by the same two steps, if we relabel as before. \begin{figure} \begin{tikzpicture}[every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(-3.46,2) (0,-2)}; \draw[red, thick] plot [smooth, tension=1] coordinates {(-3.46,2) (-0.1,-1.6) (-0.3,-1.9) (-3.46,2)}; \draw[green, thick] plot [smooth, tension=1] coordinates {(-3.46,2) (0.3,-1.7) (-0.3,-2.6) (-2,-1.0) (-3.46,-2)}; \draw[green, thick] plot [smooth, tension=1] coordinates {(3.46,-2) (-0.3,1.7) (0.3,2.6) (2,1.0) (3.46,2)}; \draw[orange, thick] plot [smooth, tension=1] coordinates {(3.46,2) (-0.8,2.45) (2.4,-2.1) (0,-4)}; \draw[orange, thick] plot [smooth, tension=1] coordinates {(-3.46,-2) (0.8,-2.45) (-2.4,2.1) (0,4)}; \draw[blue, thick] plot [smooth, tension=1] coordinates {(0,-4) (2,-2) (0,0)}; \draw[blue, thick] plot[smooth, tension=1] coordinates {(0,4) (-2,2) (0,0)}; \draw plot [smooth, tension=1] coordinates {(3.46,-2) (0,2)}; \draw[red, thick] plot [smooth, tension=1] coordinates {(3.46,-2) (0.1,1.6) (0.3,1.9) (3.46,-2)}; \end{tikzpicture} \qquad \begin{tikzpicture}[every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,4) (-2.5,1) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,-4) (2.5,-1) (0,2)}; \draw[red, thick] plot [smooth, tension=1] coordinates {(0,4) (-2.3,1) (0.4,-2.1) (-0.1,-2.25) }; \draw[red, thick] plot[smooth, tension=1] coordinates {(-0.1,-2.25) (-2.6,1) (0,4)}; \draw[green, thick] plot [smooth, tension=1] coordinates {(0,4) (-2.1,1) (0.6,-1.9) (0.6,-2.7) }; \draw[green, thick] plot [smooth, tension=1] coordinates {(0.6,-2.7) (-1,-1.8) (-2.8,0) (-3.46,2)}; \draw[red, thick] plot [smooth, tension=1] coordinates {(0,-4) (2.3,-1) (-0.4,2.1) (0.1,2.25) }; \draw[red, thick] plot[smooth, tension=1] coordinates {(0.1,2.25) (2.6,-1) (0,-4)}; \draw[orange, thick] plot [smooth, tension=1] coordinates {(-3.46,2) (-2.5,-2) (0,-3.15) (1.3,-2.3) (0,0)}; \draw[orange, thick] plot [smooth, tension=1] coordinates {(3.46,2) (0.5,3.15) (-1.3,2.3) (0,0)}; \draw[blue, thick] plot [smooth, tension=1] coordinates {(3.46,2) (0.7,2.9) (-0.9,2.3) }; \end{tikzpicture} \caption{Further applications: $\mu^2(Q)$ on the left and an incomplete drawing of $\mu^3(Q)$ on the right.} \label{incompleteDtype} \end{figure} \begin{figure} \centering \begin{tikzpicture}[every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (3,1.8) (0,3)}; \draw plot [smooth, tension=1] coordinates {(0,3) (-1.3,1.2) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (-3,-1.8) (0,-3)}; \draw plot [smooth, tension=1] coordinates {(0,-3) (1.3,-1.2) (0,2)}; \node[scale=0.7] at (0,-2.3){$p_1$}; \node[scale=0.7] at (0,1.5){$p_2$}; \node[scale=0.7] at (2.5,0){1}; \node[scale=0.7] at (-2.5,0){9}; \node[scale=0.7] at (0,-4.3){$v_1$}; \node[scale=0.7] at (0,4.3){$v_6$}; \draw[dotted] (2,-4) to (0,-2); \end{tikzpicture} \caption{The arcs of $\mu^6(Q)$ connecting the punctures to the boundary.} \label{lotsmoremu} \end{figure} The self folded triangles are wrapping around each other more with each $\mu$. In Figure \ref{lotsmoremu} we show the arcs connected to the punctures after $6$ applications of $\mu$, which is enough to construct the rest of the triangulation, following step (ii). We see, however, that in both $Q$ and $\mu^6(Q)$ we have two self folded triangles, one at $v_1$ and $p_1$ and one at $v_6$ and $p_2$, so the description of step (i) is not sufficient. To amend this we draw a line from $p_1$ that cuts the circle somewhere between $v_1$ and $v_2$ (as shown in Figure \ref{boundarylabelling}) and consider how many times the arc from $v_1$ to $p_1$ crosses this line. This is the dashed line drawn Figure \ref{lotsmoremu}, where arc $1$ crosses it once. In this example we shall have one crossing for every $6$ applications of $\mu$. We update step (i) in our description of $\mu^l(Q)$ to capture this in the following lemma. \begin{lemma}\label{descriptionoftriangulations} To describe the triangulation given by applying $\mu^l$ to (\ref{Dquiver}), for any $l\in \mathbb{Z}$, we first take the boundary vertex labelling and dashed line of Figure \ref{boundarylabelling} and rotate it by $\frac{2\pi\overline{l}}{N-2}$ clockwise, where \[ 0\leq \overline{l}< N-2, \qquad \overline{l} \equiv l \mod N-2 \] while fixing $p_1$ and $p_2$. After this rotation the dashed line will still go from $p_1$ to the boundary circle between $v_1$ and $v_2$. The triangulation is then given in three steps: \begin{enumerate}[(i)] \item Draw a self folded triangle at $v_1$ and $p_1$, such that the arc inside the triangle crosses the dashed line $\left \lfloor{\frac{l}{N-2}}\right \rfloor$ times and travels anticlockwise (clockwise) from $v_1$ to $p_1$ if $l$ is positive (negative). \item Draw a self-folded triangle at $v_{N-2}$ and $p_2$. There is only one choice due to step (i). \item Draw arcs $i$ from $v_{i-2}$ to $v_{i-1}$ for $i=3,4,\ldots, N-1$. Again, there is only one choice for each of these. \end{enumerate} \begin{figure} \centering \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (-4,0) circle [radius=0.1cm]; \draw[fill=black] (2.26,-3.3) circle [radius=0.1cm]; \draw[fill=black] (-2.26,-3.3) circle [radius=0.1cm]; \node[scale=1] at (-4.0,-2){$v_5$}; \node[scale=1] at (2.36,-3.8){$v_2$}; \node[scale=1] at (2.26,3.4){$\ddots$}; \node[scale=1] at (-2.26,3.4){$\iddots$}; \node[scale=1] at (-2.36,-3.8){$v_3$}; \node[scale=1] at (-4.0,2){$v_9$}; \node[scale=1] at (4.5,0){$v_6$}; \node[scale=1] at (-4.5,0){$v_7$}; \node[scale=1] at (4.0,2){$v_8$}; \node[scale=1] at (0,-4.5){$v_1$}; \node[scale=1] at (0,4.5){$v_{N-2}$}; \node[scale=1] at (4.0,-2){$v_4$}; \node[scale=1] at (0,-2.45){$p_1$}; \node[scale=1] at (0,2.4){$p_2$}; \draw[dotted] (1.5,-4) to (0,-2); \end{tikzpicture} \caption{Labelling around the boundary.} \label{boundarylabelling} \end{figure} \end{lemma} Analogously to Proposition \ref{Atypebipartitebelt} we have the following result. \begin{proposition} The set of cluster variables obtained by the cluster map (\ref{Dtypeclustermap}), \[ \{X^i_n \mid i=1,2,\ldots,N+1, \quad n\in \mathbb{Z}\} \] is precisely the set of cluster variables associated with $\Gamma_{1,1}\cup \Gamma_{\mathrm{punc}}$. \begin{proof} For this proof we take a fixed labelling of the boundary vertices as shown in Figure \ref{boundarylabelling}. We first show that every $\gamma(v_i,v_i,m)$ in $\Gamma_{1,1}$ appears due to the cluster map. From Lemma \ref{Dsurfacepossiblearcs} each of these arcs will orbit the two punctures and the line between them $m$ times before crossing the line $L$. For $i=1$ this arc appears on the outside of the self-folded triangle at $v_1$ in the quiver $\mu^{m(N-2)}(Q)$, since this encircles $L$ once for every $N-2$ applications of $\mu$. A further $\mu$ will then give $\gamma(v_3,v_3,m)$ since $\mu$ just moves the end points of $\gamma(v_1,v_1,m)$ clockwise. More applications of $\mu$ will give us $\gamma(v_i,v_i,m)$ for $i=1,2,\ldots, N-2$ (but not necessarily in this order). To show that $\gamma(v_i,v_j,m)$ occurs for any $i,j$ we give another labelling, as shown in Figure \ref{Arooo}. We have just shown that the arc \[ \gamma(v_{\left \lfloor{\frac{i+j}{2}}\right \rfloor},v_{\left \lfloor{\frac{i+j}{2}}\right \rfloor},m) \] appears due to the cluster map. Our construction of Lemma \ref{descriptionoftriangulations} then has us draw \[ \gamma(v_{\left \lfloor{\frac{i+j}{2}}\right \rfloor},v_{\left \lfloor{\frac{i+j}{2}}\right \rfloor+1},m) \] since, away from the boundary, every arc drawn in Lemma \ref{descriptionoftriangulations} is homotopic to the one drawn before. The next two we draw are \[ \gamma(v_{\left \lfloor{\frac{i+j}{2}}\right \rfloor-1},v_{\left \lfloor{\frac{i+j}{2}}\right \rfloor+1},m), \qquad \gamma(v_{\left \lfloor{\frac{i+j}{2}}\right \rfloor-1},v_{\left \lfloor{\frac{i+j}{2}}\right \rfloor+2},m) \] continuing this will eventually give $\gamma(v_i,v_j,m)$. We also note that every arc of $\Gamma_{\mathrm{punc}}$ appears inside a $\gamma(v_i,v_i,m)$ and so will also be given by the cluster map. Finally it is clear from Lemma \ref{descriptionoftriangulations} that every $X^i_n$ is in $\Gamma_{1,1}\cup \Gamma_{\mathrm{punc}}$. \begin{figure} \centering \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (-4,0) circle [radius=0.1cm]; \draw[fill=black] (2.26,-3.3) circle [radius=0.1cm]; \draw[fill=black] (-2.26,-3.3) circle [radius=0.1cm]; \node[scale=1] at (-4.2,-2){$v_{i+2}$}; \node[scale=1] at (2.4,-3.9){$v_{j-2}$}; \node[scale=1] at (-2.26,-3.3){$\ddots$}; \node[scale=1] at (-3.95,2){$v_i$}; \node[scale=1] at (4.5,0){$v_{j}$}; \node[scale=1] at (-4.7,0){$v_{i+1}$}; \node[scale=1] at (4.2,-2){$v_{j-1}$}; \end{tikzpicture} \caption{A different labelling around the boundary.} \label{Arooo} \end{figure} \end{proof} \end{proposition} \begin{proposition}\label{Jproposition} The arcs of $\Gamma_{2,0}$ live on an annulus obtained by removing a region containing the two punctures and the line between them, so we can display them on a strip, as in Figure \ref{DtypeJ}. $\Gamma_{2,0}$ is precisely the arcs ${J'}^{l}_i$ for $i=0,1,\ldots,N-3$ and $l=1,2,\ldots,N-3$. These satisfy the linear relation \begin{equation}\label{Dtypelinear} {J'}^{l-1}_i={J'}^{l-2}_iJ'_{i+l-2}-{J'}^{l-3}_i \end{equation} where $J'_{i+l-2}$ is the period $N-2$ quantity (\ref{DtypeJintro}). Hence ${J'}^l_j=D^l_1(J'_j)$, where $D^l_1(J'_j)$ is given in (\ref{determinant}), and these cluster variables form a frieze (\ref{friezeDtype}) but with final bottom row \begin{equation*} \begin{matrix} \\ \ldots & D^{N-3}_1(J'_{-2}) & & D^{N-3}_1(J'_{-1}) & & D^{N-3}_1(J'_0) & & D^{N-3}_1(J'_1) & & D^{N-3}_1(J'_2) & \ldots \\ \end{matrix} \end{equation*} \begin{proof} These arcs divide the disc in two such that the punctures live in the same connected component after this division. Hence the arcs avoid the part of the disc containing the line $L$ between the two punctures, so they must live in the annulus obtained by removing a small region around $L$. We draw this as in Figure \ref{DtypeJ}, where we glue along the dotted lines, and define ${J'}^{l-1}_i$ as the arc starting at $i$ and ``jumping over" $l-1$ vertices to reach $i+l$. We remark that this picture is the same as in the $\tilde{A}$ case, Figure \ref{LongerJ}, but with $0$ marked points on the internal boundary of the annulus. As such, the picture is the same as (\ref{Ptolemypic}) (with $p=1$) so we have the relation \begin{equation}\label{Dtypelinearincomplete} {J'}^{l-1}_j={J'}^{l-2}_j{J'}^1_{j+l-2}-{J'}^{l-3}_j. \end{equation} The periodic quantities for the cluster map, $J'_n$, have many equivalent expressions given in \cite{pallisterlinear}. For this proof we use \[ J'_n=\frac{X^3_{n+1}+X^5_n}{X^4_n}. \] We can obtain $J'_{-1}$ by mutating our initial quiver (triangulation), $Q$, at $3$ and then at $4$, as shown in Figure \ref{DtypeJarc}. In general $J'_n$ will be given by applying $\mu_4\mu_3$ to $\mu^{n+1}(Q)$. From this we see that $J^1_{j+l-2}=J'_{j+l-2}$ so (\ref{Dtypelinearincomplete}) becomes (\ref{Dtypelinear}). This linear relation is identical to the one in the $\tilde{A}$ case, so the determinant and frieze construction are the same as in Section \ref{Friezeconstructionsection}. \begin{figure} \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (0,2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (-0.7,3) (0,1.6) (0.7,3) (0,4)}; \draw plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1,-3) (0,-1.4) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (1,3) (0,1.4) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,1) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,-2) (0,-1) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,0) (-3.46,-2)}; \node[scale=0.9] at (1.2,-2.5){$X^3_{1}$}; \node[scale=0.9] at (0,-0.95){$X^4_0$}; \node[scale=0.9] at (0,0){$X^5_0$}; \end{tikzpicture} \qquad \begin{tikzpicture}[scale=0.85, every node/.style={fill=white}] \draw (0,0) circle [radius=4.0]; \draw[fill=black] (0,4) circle [radius=0.1cm]; \draw[fill=black] (0,2) circle [radius=0.1cm]; \draw[fill=black] (0,-2) circle [radius=0.1cm]; \draw[fill=black] (0,-4) circle [radius=0.1cm]; \draw[fill=black] (3.46,2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,2) circle [radius=0.1cm]; \draw[fill=black] (3.46,-2) circle [radius=0.1cm]; \draw[fill=black] (-3.46,-2) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,-4) (0,-2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (0,2)}; \draw plot [smooth, tension=1] coordinates {(0,4) (-0.7,3) (0,1.6) (0.7,3) (0,4)}; \draw plot [smooth, tension=1] coordinates {(0,-4) (-0.7,-3) (0,-1.6) (0.7,-3) (0,-4)}; \draw plot [smooth, tension=0.9] coordinates {(0,-4) (1,-3) (0,-1.4) (-3.46,-2)}; \draw plot [smooth, tension=0.9] coordinates {(0,4) (1,3) (0,1.4) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,1) (-3.46,2)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (3,-1) (1,-3.4) (0,-4)}; \draw plot [smooth, tension=0.9] coordinates {(3.46,2) (0,0) (-3.46,-2)}; \node[scale=0.9] at (1.2,-2.3){$X^3_{1}$}; \node[scale=0.9] at (3,-1){$J'_{-1}$}; \node[scale=0.9] at (0,0){$X^5_0$}; \end{tikzpicture} \caption{Applying $\mu_3$ then $\mu_4$ to the initial triangulation gives the arcs $J'_{-1}$.} \label{DtypeJarc} \end{figure} \begin{figure} \centering \begin{tikzpicture} \draw[fill=black] (0,0) circle [radius=0.1cm]; \draw[fill=black] (2,0) circle [radius=0.1cm]; \draw[fill=black] (4,0) circle [radius=0.1cm]; \draw[fill=black] (8,0) circle [radius=0.1cm]; \draw[fill=black] (12,0) circle [radius=0.1cm]; \draw plot [smooth, tension=1] coordinates {(0,0) (4,0.7) (8,0)}; \draw plot [smooth, tension=1] coordinates {(-1,0) (2,0)}; \draw plot [smooth, tension=1] coordinates {(-1,2) (9,2)}; \draw plot [smooth, tension=1] coordinates {(11,2) (13,2)}; \draw plot [smooth, tension=1] coordinates {(11,0) (13,0)}; \draw plot [smooth, tension=1] coordinates {(2,0) (4,0)}; \draw plot [smooth, tension=1] coordinates {(7,0) (9,0)}; \node at (6,0) {$\ldots$}; \node at (2,0.9) {${J'}^{l-1}_i$}; \node at (0,-0.3) {$v_i$}; \node at (12,-0.3) {$v_i$}; \node at (2,-0.3) {$v_{i+1}$}; \node at (4,-0.3) {$v_{i+2}$}; \node at (8,-0.3) {$v_{i+l}$}; \node at (10,0) {$\ldots$}; \node at (10,2) {$\ldots$}; \draw[dotted] (-0.5,-1) to (-0.5,3); \draw[dotted] (11.5,-1) to (11.5,3); \end{tikzpicture}\caption{The arcs ${J'}^{l-1}_{i}$.}\label{DtypeJ} \end{figure} \end{proof} \end{proposition} We have now found all the arcs in the $\tilde{D}$ case, except for the three only involving the punctures, $\Gamma_{\mathrm{except}}$. We collect these results in the following theorem. \begin{theorem} The $\tilde{D}$ type cluster variables are given by \begin{equation} \left\{X^i_n\:\middle|\: \begin{aligned} &i=1,\ldots,N+1 \\ &n\in\mathbb{Z} \end{aligned} \right\} \cup \left\{D^l_1(J'_{j})\:\middle|\: \begin{aligned} j=0,\ldots,N-3 \\ l=1,\ldots,N-3 \end{aligned} \right\} \cup \Gamma_{\mathrm{except}} \end{equation} where the $X^i_n$ are obtained by the cluster map (\ref{Dtypeclustermap}) and the ${J'}^{l}_i$ are defined in Proposition \ref{Jproposition}. The three arcs of $\Gamma_{\mathrm{except}}$ are shown in (\ref{3exceptionalarcs}). \end{theorem} We have now proven the $\tilde{D}$ part of Theorem \ref{findingallclustervarstheorem} and Proposition \ref{friezepropintro}. In the next section we complete our results for $\tilde{D}$ type with a proof of Proposition \ref{intropropclusterfrieze}. \subsection{$\tilde{D}$ type cluster friezes}\label{Dtypeclusterfriezebit} In $\tilde{D}$ type there are $3$ exceptional tubes, for $\lambda=1,0,\infty$. whose periods are $N-2,2,2$ respectively. We have not constructed friezes of width two, only one of width $N-2$, so our goal is to prove that our friezes agree with the friezes of \cite{assemdupont} away from the width two tubes. To do this we prove that the relation (\ref{clusterfrieze}), holds for $\lambda=1$ Firstly we follow \cite{assemdupont} to find the cluster variables associated with $B_1$ and $B'_1$. The vertex $e$ is required to be a sink, so we first mutate Figure \ref{Dquiver} at $1$ and $2$, so the quiver near $e=2$ is \begin{equation*} \begin{tikzcd} X_{-1}^1 \\ & X_0^3\arrow[ul]\arrow[dl] & X_0^4\arrow[l] \\ X_{-1}^2 \end{tikzcd} \end{equation*} We have $B_1=P_3/P_1$ and $B'_1=P_1[1]$ with $X_{P_1[1]}=X^1_{-1}$ and $X_{S_2}=X^2_0$. Performing $\mu_3\mu_2$ gives \begin{equation*} \begin{tikzcd} X_{-1}^1\arrow[dr] \\ & (X_0^3)'\arrow[dl]\arrow[r] & X_0^4\arrow[ull] \\ X_{0}^2\arrow[uu] \end{tikzcd} \end{equation*} where $(X_0^3)'=\frac{X^2_0X^4_0+X^1_{-1}}{X^3_0}=\frac{1+X^3_0+X^2_{-1}X^1_{-1}}{X^2_{-1}X^3_0}$ is the cluster variable associated with $B_1$. We can express this as \[ (X_0^3)'=\frac{X^1_0X^2_0X^4_0+X^1_{-1}X^1_0}{X^1_0X^3_0}=\frac{X^3_1X^3_0-1+X^3_0+1}{X^1_0X^3_0}=\frac{X^3_1+1}{X^1_0}=X^1_1 \] so we have \[ X_{N_{0}}=\frac{X_{B_1}+X_{B'_1}}{X_{S_1}}=\frac{X^1_1+X^1_{-1}}{X^2_0}=J'_0 \] which gives \[ X_{N_{0}[j]}=\frac{X_{B_1[j]}+X_{B'_1[j]}}{X_{S_1[j]}}=\frac{X^1_{1-j}+X^1_{-1-j}}{X^2_{-j}}=J'_{-j} \] as desired. \clearpage \bibliographystyle{plain}

proofpile-arXiv_069-13206

\section{Introduction} \label{Sec:Intro} A thorough understanding of the impact of electrostatic fields on superconductivity is of great relevance from the fundamental point of view, as it may represent an easy-access active knob to control fundamental quantum states of matter and open the way to a number of technological applications. Electrostatic fields have been successfully employed in systems where the carrier density is low, such as thin crystalline films \cite{shi2015superconductivity,leng2012indications,shiogai2016electric,hendrickx2018gate-controlled}, band insulators \cite{ueno2008electric,ueno2011discovery,ye2012superconducting}, interfaces \cite{caviglia2008electric}, semiconducting and low-density two-dimensional materials \cite{saito2015metallic,li2016controlling,sajadi2018gate-induced,chen2019signatures}, or to tune the proximity effect \cite{morpurgo1999gate-controlled,jarillo-herrero2006quantum,tsuneta2007gate-controlled}. In most of the cases the electrostatic field controls the carrier density by shifting the active bands of the material. A low electronic density yields poor screening and a sufficiently thin structure ensure full penetration of the electric field. In contrast, metals characterized by a high carrier density screen very well the electrostatic field within few Angstroms from the surface \cite{PinesNozieres,AshcroftMermin,Mahan,GiulianiVignale} and the residual skin contribution is typically negligible at the level of carrier density and density of states. Superconductors realized in diffusive metals are typically hardly affected by electrostatic fields. Recently, a series of experiments conducted on metallic superconducting Dayem nanobridges have shown that a strong electric field, generated by a high voltage applied on a side gate, is able to switch the critical current $I_c$ of the Dayem bridge off in a reversible ambipolar way \cite{desimoni2018metallic,paolucci2018ultra,paolucci2019magnetotransport,paolucci2019field,desimoni2020niobium}. Although a certain material dependence is observed, with a pronounced effect in Nb, Va, Ti, the effect seams to be relatively general, as it occurs in Al-Cu-Al proximity Josephson junctions \cite{desimoni2019josephson} and in Al Dayem bridges \cite{bours2020unveiling}. Possible leakage currents and related overheating effects \cite{Golokolenov2020ontheorigin,alegria2020high-energy,ritter2020superconducting} have been minimized by constructing suspended nanobridges \cite{rocci2020suspended}. Switching current distribution measurements show the presence of very strong gate-induced phase fluctuations \cite{puglia2020phase-slip}. Theoretical attempts to explain the origin of the observed field effect suggested a surface orbital polarization \cite{mercaldo2020electrically,bours2020unveiling,fukaya2020orbital} and a possible Schwinger effect \cite{solinas2020schwinger}. These findings are yet to be understood in the usual framework of the BCS theory and represent a challenge from the fundamental point of view, whose solution may reveal of great technological interest. Although the observed phenomenology involves diffusive polycrystalline metals, it offer the opportunity to study the impact of an electric field in systems at the boundary between low-density crystalline materials and diffusive metals. In this work we address the problem of gate-controlling superconductivity in thin metallic clean systems, consisting in a crystal composed by $N$ layers and characterized by a large carrier density, large DoS at the Fermi level, but with still a well defined notion of discreteness. Rather then studying the effect of variation of the carrier density, we focus on the impact of a redistribution of charge in response to an applied electric field by choosing an anti-symmetric profile of the potential in a capacitor-like configuration, that guarantees only the bipolar part of the effect is described, leaving aside global carrier density modifications. We numerically solve the fully self-consistent gap equation and Poisson electrostatic equation describing simultaneous condensation of a superconducting gap and screening of the applied field within the BCS theory. The screening length is only slightly increased from its metallic counterpart, in agreement with random-phase approximation results first described by Anderson \cite{anderson1958rpa} and Thouless \cite{thouless1960perturbation}, that predict a correction of order $(\Delta/E_F)^2$ [\onlinecite{Mahan}], with $\Delta$ the superconducting gap and $E_F$ the Fermi energy \cite{virtanen2019superconducting}. In order to enhance the responsiveness of the system, we choose as the basic system a tight-binding model in the cubic lattice, but results are qualitatively confirmed with an in-plane triangular lattice. We find that for a small relative dielectric constant, relatively large system size, and strong superconducting pairing, the gap is essentially insensitive to the applied field. In turn, by reducing the pairing strength we observe an increased sensitivity to the applied field, with the gap showing sudden rises and falls as the applied voltage is increased. This behavior originates from the density of states modification induced by the screened potential. The latter adds to the confining potential and plays the role of a layer chemical potential. Although the exponentially decaying profile significantly modifies only the outermost few layers, our results show that it can result in sizable bulk effects. For a perfectly clean crystalline structure the entire density of states spanning the whole bandwidth is necessary to account for the observed behavior. In turn, the introduction of a weak energy smearing in the DoS, that emulates the effect of weak disorder, washes out the effect and the gap follows mainly the DoS at the Fermi level. Our results apply to weak coupling limit and show how an evolution from a clean to a weakly disordered system takes place. These results are expected to be significant for layered materials and thin cristalline metals and predicts a certain degree of control of superconductivity an applied electrostatic field. \section{Mean-field BCS with screening} \label{Sec:1} We consider a system composed by $N$ layers, as depicted in Fig.~\ref{Fig:1}, described each by a spin-degenerate microscopic tight-binding model. For simplicity we assume a single orbital per unit cell, with nearest neighbor hopping $t$, that results in a dispersion $\epsilon_{\bf k}$, with ${\bf k}$ in-plane momentum. Interlayer nearest neighbor hopping is described by the same hopping $t$. Electrons interact via a purely local two-body attraction described by a Hubbard term with strength $U$, that acts as a pairing interaction. In addition, we apply an external electric field ${\bf E}_g$ along the out-of-plane $z$ direction via a side gate. The latter is screened by the electron gas and the full electric field $E^z_i$ can be introduced, that is described via an electrostatic energy potential $\phi_i$, such that $E^z_{i+1}=-(\phi_{i+1}-\phi_i)/ae$, with $a$ the lattice constant and $e$ the electric charge. The full Hamiltonian then reads \begin{eqnarray}\label{FullH} H&=&\sum_{{\bf k},i,s}(\epsilon_{\bf k}-\phi_i)c^\dag_{{\bf k},i,s}c_{{\bf k},i,s}-t\sum_{i,s}c^\dag_{{\bf k},i+1,s}c_{{\bf k},i,s}+{\rm H.c.}\nonumber\\ &-&U\sum_{{\bf k},{\bf k}'}c^\dag_{{\bf k},\uparrow}c^\dag_{-{\bf k},\downarrow}c_{-{\bf k}',\downarrow}c_{{\bf k}',\uparrow}+V_C, \end{eqnarray} where $V_C$ is the Coulomb interaction. The latter is crucial to correctly describe screening of the applied field. Its continuum form in Fourier transform is given by $V_C({\bf q},q_z)=4\pi e^2/(\epsilon(q^2+q_z^2))$, with $\epsilon$ the dielectric coupling constant. The momentum transfer $({\bf q},q_z)$ is restricted to non zero value to account for the background positive charge. We separate the in-plane and out-of-plane Coulomb interaction by singling out the ${\bf q}=0$, $q_z\neq0$ term, that reads \begin{equation}\label{Eq:VCeff} V_C=\frac{1}{2}\int dzdz'n(z)V_{\rm eff}(z-z')n(z'), \end{equation} with $V_{\rm eff}(z-z')=\frac{1}{L}\sum_{q_z\neq 0}e^{iq_z(z-z')}4\pi e^2/(\epsilon q_z^2)$, $n(z_i)=\sum_{{\bf k},s}c^\dag_{{\bf k},i,s}c_{{\bf k},i,s}$, and neglect the residual interaction. Standard decoupling of Eq.~(\ref{Eq:VCeff}) gives rise to the Poisson equation. By noticing that $q_z^2$ is the eigenvalue of the Laplacian in 1D, we directly write a discrete version of the Poisson equation in 1D as \begin{equation}\label{Eq:Poisson} -\phi_{i+1}-\phi_{i-1}+2\phi_i=\frac{4\pi e^2}{\epsilon a}\left[n_i-n_0\right], \end{equation} that ensures charge conservation locally on each layer. \begin{figure}[t] \includegraphics[width=0.45\textwidth]{Fig1.pdf} \caption{a) Schematics of the system composed by $N$ layers. The electric field is applied along the out-of-plane direction and it is modeled by fixing the electrostatic potential at the $i=0$ and $i=N+1$ layers. b) Diagrammatic form of the Poisson equation and the gap equation. \label{Fig:1}} \end{figure} The rest of the Hamiltonian is decoupled in the Cooper channel at mean-field level by defining a local layer dependent gap, $\Delta_i=\langle c_{-{\bf k},i,\downarrow}c_{{\bf k},i,\uparrow}\rangle$. The Bogoliubov deGennes tight-binding Hamiltonian is written as \begin{equation}\label{Eq:BdG} H=\sum_{{\bf k},ij}h_{ij}({\bf k})c^\dag_{{\bf k},i,s}c_{{\bf k},j,s}+\Delta_ic^\dag_{{\bf k},i,\uparrow}c^\dag_{-{\bf k},i,\downarrow}+{\rm H.c.} \end{equation} with $h_{ij}({\bf k})=[\epsilon_{\bf k}-\phi_i]\delta_{ij}-t(\delta_{i+1,j}+\delta_{i,j+1})$ and the chemical potential has been absorbed in the dispersion. The gap and the electron density are then written as \begin{equation}\label{Eq:GapCharge} n_i=\frac{2}{N_k}\sum_{\bf k}v^2_i({\bf k}),\qquad \Delta_i=\frac{U}{N_k}\sum_{\bf k}u_i({\bf k})v_i({\bf k}). \end{equation} where the $u_i({\bf k})$ and $v_i({\bf k})$ are the particle and hole part of the eigenvectors at layer $i$ of the BdG Hamiltonian Eq.~(\ref{Eq:BdG}) and $N_k$ is the number of ${\bf k}$-points in the BZ. The equation for the gap and the density (\ref{Eq:GapCharge}) are solved iteratively together with the Poisson equation (\ref{Eq:Poisson}). For simplicity we assume to insert the layered system in a capacitor like structure, that fixes the value of the gate voltage to be opposite on the two sides of the system. A uniform gap $\Delta^{(0)}$ and a linear potential $\phi^{(0)}_i=(2i/N-1) \phi_g$ are assigned at the first step and the charge density and the gap are recursively updated. The potential $\phi_i$ is obtained via inversion of the Poisson equation (\ref{Eq:Poisson}) and by imposing a fixed boundary condition at the two most external layers $\phi_0=-\phi_{N+1}=\phi_g$. The mean density $n_0$ is kept fixed at half-filling and the chemical potential is updated to keep the half-filling condition at every step. Convergence is achieved within a threshold error smaller than $10^{-5}$ of a chi-squared error function for the three quantities $n_i,\Delta_i,\phi_i$. By increasing the gate in a discretized way, convergence speed highly increases using as guesses for the gap, density, and potential those self-consistently obtained at a smaller value of the gate potential. \section{Density of States and Screening} \label{Sec:2} \begin{figure}[t] \includegraphics[width=0.45\textwidth]{Fig2.png} \caption{a) Histogram of the eigenvalues multiplicity (total density of states) in absence of applied gate. b) Profile of the self-consistent potential for different values of the relative dielectric constant $\epsilon_r=10, 20, 50$. \label{Fig:2}} \end{figure} We assume the in-plane system to be described by a square lattice, so that the dispersion reads \begin{equation} \epsilon_{\bf k}=-2t(\cos(k_x)+\cos(k_y)). \end{equation} At half filling the chemical potential is $\mu=0$. For a structure composed by a few layers some concerns may arise from the half-filling van Hove singularity that characterizes the one-layer density of states. Strictly speaking, if $N$ is odd a smeared van Hove singularity still appears at half filling, whereas for $N$ even it is shifted at positive and negative energies by inter-layer tunneling. In order to avoid peaks in the DoS at the Fermi level we always choose an even number of layers. We measure the strength of the Hubbard attraction $U$ in units of the hopping energy $t$ and group the lattice spacing $a$ and the dielectric constant $\epsilon$ in the energy scale $e^2/(a\epsilon)$. Setting $a=1$~\AA, we define our Rydberg as $Ry=e^2/(\epsilon_0a)=18.05~{\rm eV}\equiv 18.05~ t$. This way, the only free parameters in the system are the number of layers $N$, the attraction strength $U$, the number $N_k=N_xN_y$ of ${\bf k}$-points in the BZ and the relative dielectric constant $\epsilon_r$. The smaller is the dielectric constant $\epsilon_r$ the strongest is the screening. In Fig.~\ref{Fig:2}a) we show a histogram of the multiplicity of the eigenvalues of a $N_x=N_y=100$ grid in momentum space, for a system constituted by $N=20$ layers. The histogram tends to the typical DoS of a 3D cubic lattice once a smearing in energy is assumed. Deviations due to finite size effects both in-plane and out-of-plane are evident and $N=20$ peaks reminiscent of the original van Hove singularities are still present. \begin{figure}[t] \includegraphics[width=0.45\textwidth]{Fig3.png} \caption{Results of the simulations for a strong pairing $U=1~t$ for a thick slab with $N=30$. a) Color plot of the local gap versus layer $i_N$ and applied gate voltage $\phi_g$. b) Section of a) at $\phi_g=24~t$ showing gap suppression in the outer most layers. c) Value of the self-consistent charge density versus the layer $i_N$ and the applied gate, showing charge depletion in the outer most layers. d) Self-consistent charge density at $\phi_g=24~t$ showing full depletion at the outermost layer. \label{Fig:S1}} \end{figure} In Fig.~\ref{Fig:2}b) we show the self consistent potential normalized to its maximum strength for values of the applied gate $0<\phi_g<5t$, for three different values of the relative dielectric constant $\epsilon_r=10, 20, 50$. The field is screened very well by the metal and an overall exponential decay is recognized. Besides, oscillations of the charge on the scale of the screening length are present, indicating a local increase/decrease of the layer chemical potential $\phi_i$ in the bulk of the slab. The charge distribution screens the external field via charge accumulation at the outermost layers. The latter overshoots the one necessary to screen the field and a series of alternating electric dipoles on the scale of the screening lengths are generated to compensate local overshooting. \section{Simulations} We now present the results of full numerical calculations. As pointed out in Sec.~\ref{Sec:Intro}, we expect no effect of the applied gate for a large system, characterized by strong screening and a relatively strong pairing, within the weak coupling limit. We confirm this expectation for a system composed by $N=30$ layers, pairing strength $U=1~t$, relative dielectric constant $\epsilon_r=10$ and $N_x=N_y=100$ points in momentum space. The results of the simulations are shown in Fig.~\ref{Fig:S1}. The self-consistent gap and potential are shown in Fig.~\ref{Fig:S1}a) and c), respectively. The gap is mostly uniform through the slab for all values of the applied electric field. Small oscillations on the scale of the lattice constant appear on top of the average value. For strong field the gap in the outermost layer approaches zero at $\phi_g=24~t$. This arise because of complete charge depletion/saturation in the outermost layer (shown in Fig.~\ref{Fig:S1}d) and the absence of available particle (hole) states locally suppresses the gap. \begin{figure}[t] \includegraphics[width=0.45\textwidth]{Fig4.pdf} \caption{Average gap $\Delta$ versus applied gate $\phi_g$ for $U=t$ varying a) the number of layers $N=10,20,30$ for $\epsilon_{r}=10$, and b) the relative dielectric constant $\epsilon_r=10, 20, 50$ for $N=20$. In-plane grid with $N_x=N_y=100$. \label{Fig:4}} \end{figure} {\bf Average gap varying $N$ and $\epsilon_{r}$.---} We now study the impact of the applied field on the gap for samples with different thickness $N=30, 20, 10$, for $U=t$. Results are shown in Fig.~\ref{Fig:4}a). The gap at zero applied gate voltage is not a monotonic function of the sample thickness and shows well know quantum oscillations. The dependence on the applied gate is very smooth for $N=20, 30$ and the gap shows robustness to the applied electric field. For $N=10$ a sizable and smooth modulation of the gap is observed. We then fix the slab thickness to $N=20$ and study the dependence of the layer-averaged gap $\Delta=\sum_i\Delta_i/N$ on the applied field for three values of the dielectric constant $\epsilon_{r} = 10, 20 , 50$. Results are shown in Fig.~\ref{Fig:4}b). By increasing the value of $\epsilon_r$ the gap shows an average smooth linear decrease with the applied gate, that seams to saturate for higher values of $\epsilon_r$, accompanied by fast irregular fluctuations on top of the linear decrease. The analysis is repeated for $U=0.8~t$ (results not shown). The average gap becomes more sensitive to the applied gate and strongly non-monotonic rises and falls appear in correspondence of the smooth variations observed for $U=t$. \begin{figure}[t] \includegraphics[width=0.45\textwidth]{Fig5.pdf} \caption{a) Normalized average gap versus applied gate voltage $\phi_g$ for different values of the pairing strength $U/t = 1, 0.8, 0.6$ for $\epsilon_r=10$. b) Gap calculated through Eq.~(\ref{Eq:GapEqBCS}) with the density of states $\nu(E)$ resulting from the self-consistent potential. Fixed parameters are $N=20$ and $N_x=N_y=100$. \label{Fig:5}} \end{figure} {\bf Average gap varying $U$.---} We now study the gap versus gate voltage curve varying $U$. The zero voltage gap clearly diminishes by decreasing $U$, so that we normalize the curves with their zero voltage value $\Delta_0=\Delta(\phi_g=0)$. In Fig.~\ref{Fig:5}a) we show the results for $\epsilon_r=10$. We see that by decreasing $U$ the effect of the gate becomes more and more pronounced. For $U=0.6$ at $\phi_g\simeq 1.5~t$ the gap drops to zero after a 50\% increase and it stays zero a part from very sharp and sudden revivals. For the chosen value $\epsilon_r=10$ the field penetrates only for few layers. The weak modulation of the gap observed for large $U$ is strongly amplified for smaller $U$. The analysis is repeated for $\epsilon_{r}=50$ (results not shown) and the dependence on the gate voltage becomes more and more frustrated. We observe again that the small fluctuations appearing in the $U=t$ curves are strongly amplified for smaller $U$, suggesting a common density of states origin. \section{Gap from DoS} The analysis so far presented shows that a strong sensitivity to the gate appears when reducing the pairing strength $U$. We point out that in all simulations a smooth convergence of the self-consistent calculation is observed at every steps, ruling out numerical instability. The BCS theory predicts that all properties of the superconducting gap are determined by the DoS of the system. The latter is typically assumed to be uniform over a large range of energies where the pairing is active and approximated with its value at the Fermi level. Clearly, this approximation fails if the DoS suffers strong modifications close to the Fermi level due to confinement, as shown in Fig.~\ref{Fig:2}a), where the DoS at the Fermi level is ill-defined. We then calculate the gap that results assuming the effect of the electric field comes solely by the screened potential $\phi_i$. This is done by taking the self-consistently screened potential for two given $\epsilon_r=10, 50$ for every value of the applied gate voltage, calculating the DoS $\nu(E)$ in the normal phase, and self-consistently solving for the gap via the BCS gap equation \begin{equation}\label{Eq:GapEqBCS} 1=U\int dE \frac{\nu(E)}{2\sqrt{E^2+\Delta^2}}, \end{equation} by varying only $U$. The result is shown in Fig.~\ref{Fig:5}b): there is a striking similarity between the curves shown in Fig.~\ref{Fig:5}a), showing how the unexpected behavior of the gap is totally understood in terms of the full DoS and its readjustment with the screened potential. Although the dependence on the field is smoother in the curve in Fig.~\ref{Fig:5}b), the rises and falls appear in the same ranges of gate voltage. \begin{figure}[t] \includegraphics[width=0.45\textwidth]{Fig6.png} \caption{a) Density of states versus energy after a smearing with broadening $10^{-5}t$ for a system with $N=20$ layers and $\epsilon_r=50$ at zero applied field. Inset: zoom on a small energy window around the Fermi level at $E=0$. b) DoS at the Fermi level as a function of the applied gate voltage for $\epsilon_r=10, 20, 50$. c) Gap resulting from solution of Eq.~(\ref{Eq:GapEqBCS}) for $\epsilon_r=50$ and $U=0.6 t$. \label{Fig:6}} \end{figure} \section{Extrapolations} It is clear that finite size effects, in particular the finite grid in momentum space, are at the origin of the observed sensitivity to the applied gate voltage for weaker value of $U$. We can mimic a denser grid in momentum space and a small amount of local disorder by smearing the DoS on a small energy window. We notice that it is sufficient to introduce a very small broadening on order $10^{-5}t$ to smear the DoS without completely erasing its discrete origin, as shown in Fig.~\ref{Fig:6}a). The DoS at the Fermi level is now a meaningful quantity and it is shown in Fig.~\ref{Fig:6}b) for three values of $\epsilon_r=10, 20, 50$. We can then calculate the self-consistent gap via numerically integrating Eq.~(\ref{Eq:GapEqBCS}) for $U=0.6 t$ and $\epsilon_r=50$. We see that the gap closely follows the behavior of the DoS at the Fermi level. This result reliably predicts a modulation of the gap in a weakly disordered metal via an external electric field that penetrates sufficiently in the system. It also shows how the sensitivity to the applied gate evolves from clean thin crystal to a weakly disordered metal. Furthermore, comparison of Fig.~\ref{Fig:6}c) with Fig.~\ref{Fig:4}b) for $\epsilon_{r}=50$ shows how the gap versus gate voltage for small $U$ and weak energy smearing is compatible with the gap dependence at large $U$ without energy smearing. This way, the results for weakly disordered weak-coupling limit are analogous to those of moderate-strong pairing in a clean system. \section{Small grains and dot} Finally, we study a smaller system that rather describes a clean dot. This is done by keeping the number of layers to $N=20$ and reducing the size of the in-plane lattice by setting $N_x=N_y=80$. This results in a reduced DoS, with consequent reduction of the gap size at zero gate voltage with respect to the cases analyzed in the previous sections. The average gap $\Delta$ versus applied gate is shown in Fig.~\ref{Fig:7} and smoothly decreases to zero, in a fashion similar to the BCS temperature dependence. Fig.~\ref{Fig:7}a) we fix $U=0.5~t$ and vary the relative dielectric constant $\epsilon_r$. The curves fall all on top of each other upon proper field rescaling (not shown). In Fig.~\ref{Fig:7}b) we increase the pairing strength and confirm that for a larger gap the result remains valid. \begin{figure}[t] \includegraphics[width=0.45\textwidth]{Fig7.pdf} \caption{Gap as a function of the applied gate voltage for a system characterized by $N=20$ and $N_x=N_y=80$, with in-plane periodic boundary conditions. a) $U=0.5~ t$ varying $\epsilon_r$. b) $\epsilon_r=10$ varying $U$. \label{Fig:7}} \end{figure} \section{Discussion} \label{Sec:3} \begin{figure}[t] \includegraphics[width=0.45\textwidth]{Fig8.pdf} \caption{Gap as a function of the applied gate potential for a in-plane triangular tight-binding model characterized by $N=20$, $\epsilon_r=10$ and $N_x=N_y=100$ for different values of the pairing $U/t=0.8, 0.6, 0.5$. \label{Fig:8}} \end{figure} The analysis presented tackles the problem of the simultaneous condensation of a superconducting gap and screening of an externally applied field in a fully self-consistent way. The Poisson equation (\ref{Eq:Poisson}) includes the part of the Coulomb interaction that describes repulsion of the average charge of each layer and results from minimization of the total energy. It does not include a finite in-plane momentum transfer. The problem is solved exactly at the mean-field level and does not account for phase fluctuations. The results presented show that the behavior of the system is totally due to modification of the density of state induced by the gate voltage, that acts as a confining potential. The observations are qualitatively confirmed by changing the in-plane lattice model. It is well known that the square lattice at half filling represents somewhat a peculiar case, with van Hove singularities close to the Fermi level. In Fig.~\ref{Fig:8} we show the results of the simulation for an in-plane triangular lattice. The electric field is screened in few lattice constants, depending on the relative dielectric constant $\epsilon_r$. Fluctuations of the gap are seen for reduced strength of the pairing $U$, confirming the results obtained for the square lattice. The exponential dependence of the gap on the density of states enhances weak variation of the latter when reducing the pairing strength $U$. Differently from the case of larger in-plane systems, the results are not confirmed for a small system, for which the gap is never suppressed. The results and methodology open the way to reliable modeling of systems where the surface physics has a non-trivial content, such as the case of a Rashba spin-orbit interaction, that is controlled by an applied electric field, or a multi-orbital character of the band structure, that enables an electric-field controlled orbital-Rashba effect. Screening in absence of pairing converges very quickly, and the joint impact of electric field, sample geometry and Rashba field can show non-trivial results on the gap. In summary, we find that in cristalline thin metals a strong sensitivity to an applied electric field appears in the weak coupling limit, with the gap showing sudden rises and falls as the applied voltage is increased. This behavior reflects the density of states modification induced by the screened potential acting mostly on the outermost few layers. For a perfectly clean crystalline structure the observed behavior can be understood only in terms of the entire density of states spanning the whole bandwidth. The introduction of a weak energy smearing in the DoS emulates the effect of weak disorder and washes out the effect, showing a gap that follows mainly the DoS at the Fermi level. Our results are expected to be significant for layered materials and thin cristalline metals allowing control of superconductivity by an externally applied electrostatic field. \section{Acknowledgments} L.C. acknowledges the European Commission for funding through the MSCA Global Fellowship grant TOPOCIRCUS-841894. T.C. acknowledges fundings from the European Commission, under the Graphene Flagship, Core 3, grant number 881603. F.G. acknowledges the EU’s Horizon 2020 research and innovation program under grant agreement No. 800923 (SUPERTED), and the European Research Council under the EU’s Horizon 2020 Grant Agreement No. 899315-TERASEC for partial financial support.

proofpile-arXiv_069-13297

\section{Introduction} Given some undirected graph $G=(V,E)$, an {\em $r$-coloring} for $G$ is a partition of the vertex set $V$ into $r$ independent sets. The smallest integer $r$ such that a graph $G$ permits an $r$-coloring is referred to as the {\em chromatic number} of $G$. Deciding whether a graph has a $3$-coloring is NP-complete. However, there are efficient solutions for the Chromatic Number problem on special graph classes, such as co-graphs \cite{CLS81}, chordal graphs \cite{Gol80}, and comparability graphs \cite{Hoa94}. For directed graphs the concept of acyclic colorings introduced by Neumann-Lara \cite{NL82} received a lot of attention in \cite{NL82,Moh03,BFJKM04} and also in recent works \cite{LM17,MSW19,SW20,GKR20c,GKR21}. Given some directed graph $G=(V,E)$, an {\em acyclic $r$-coloring} for $G$ is a partition of the vertex set $V$ into $r$ acyclic sets.\footnote{A set $V'$ of vertices of a digraph $G$ is called {\em acyclic} if the subdigraph induced by $V'$ is acyclic.} The {\em dichromatic number} of a directed graph $G$ is the smallest integer $r$ such that $G$ permits an acyclic $r$-coloring. In this paper we consider the principle of oriented colorings on oriented graphs, which has been introduced by Courcelle \cite{Cou94}. Given some oriented graph $G=(V,E)$, an {\em oriented $r$-coloring} for $G$ is a partition of the vertex set $V$ into $r$ independent sets, such that all the arcs linking two of these subsets have the same direction. The {\em oriented chromatic number} of an oriented graph $G$, denoted by $\chi_o(G)$, is the smallest integer $r$ such that $G$ has an oriented $r$-coloring. Oriented colorings have applications in scheduling models in which incompatibilities are oriented \cite{CD06}. In the Oriented Chromatic Number problem ($\OCN$ for short) there is given an oriented graph $G$ and an integer $r$ and one has to decide whether there is an oriented $r$-coloring for $G$. If $r$ is constant, i.e.\ not part of the input, the corresponding problem is denoted by $\OCN_{r}$. Even $\OCN_{4}$ is NP-complete~\cite{CD06}. So far, the definition of oriented coloring is mostly applied to undirected graphs. In this case, the maximum value $\chi_o(G')$ of all possible orientations $G'$ of an undirected graph $G$ is considered. For several special undirected graph classes the oriented chromatic number has been bounded. Among these are outerplanar graphs \cite{Sop97}, planar graphs \cite{Mar13}, and Halin graphs \cite{DS14}. See \cite{Sop16} for a survey. Oriented colorings of special classes of oriented graphs seems to be nearly uninvestigated. In this paper, we consider the Oriented Chromatic Number problem restricted to acyclic transitive digraphs, oriented co-graphs, msp-digraphs, and digraphs of bounded directed clique-width. Oriented complement reducible graphs, oriented co-graphs for short, have been studied by Lawler in \cite{Law76} and by Corneil et al.\ in \cite{CLS81} using the notation of transitive series parallel (TSP) digraphs. Oriented co-graphs can be defined from the single vertex graph by applying the disjoint union and the order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time \cite{GKR19d}. We generalize this result using the concept of perfect orderable graphs by showing that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. In order to obtain an upper bound we show that for acyclic transitive digraphs and thus also for oriented co-graphs the oriented chromatic number is at most the maximum vertex degree plus one. Minimal series-parallel digraphs, msp-digraphs for short, have been analyzed in \cite{VTL82}. By \cite[Section 11.1]{BG18} these digraphs can be used for modeling flow diagrams and dependency charts and have applications within scheduling under constraints. Msp-digraphs can be defined from the single vertex graph by using the parallel composition and series composition. We prove an upper bound of $7$ for the oriented chromatic number for msp-digraphs and we give an example to verify that this is bound best possible. We use this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs. Further, we show an upper bound of $3$ for the chromatic number of underlying undirected graphs of msp-digraphs. We also consider the parameterized complexity of the Oriented Chromatic Number problem parameterized by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure. The existence of an $\fpt$-algorithm\footnote{FPT is the class of all parameterized problems which can be solved by algorithms that are exponential only in the size of a fixed parameter while polynomial in the size of the input size \cite{DF13}.} or an $\xp$-algorithm\footnote{XP is the class of all parameterized problems which can be solved by algorithms that are polynomial if the parameter is considered as a constant \cite{DF13}.} w.r.t.\ some structural parameter allows an efficient computation of the Oriented Chromatic Number problem on graph classes of bounded parameter values. As already mentioned in \cite{GHKLOR14}, the Oriented Chromatic Number problem is not in $\xp$ when parameterized by directed tree-width, directed path-width, Kelly-width, or DAG-width, unless $\p=\np$. Better results can be achieved considering the parameter directed clique-width. By extending our solution on msp-digraphs we can show an algorithm for the Oriented Chromatic Number problem on digraphs on $n$ vertices given by a directed clique-width $k$-expression with running time in $\text{$\mathcal O$}(n\cdot k^2 \cdot 2^{r(r+k)})$. This implies that the Oriented Chromatic Number problem is in $\fpt$ when parameterized by the directed clique-width and $r$, which was already known by defineability in monadic second order logic (MSO) \cite{GHKLOR14}. Thus, for every integer $r$ it holds that $\OCN_{r}$ is in $\fpt$ when parameterized by directed clique-width and for every class of graphs of bounded directed clique-width and every integer $r$ the $r$-Oriented Chromatic Number problem can be solved in polynomial time. Ganian has shown an FPT-algorithm for $\OCN$ w.r.t.\ the parameter tree-width (of the underlying undirected graph) \cite{Gan09}. Further, he has shown that $\OCN$ is DET-hard\footnote{DET is the class of decision problems which are reducible in logarithmic space to the problem of computing the determinant of an integer valued $n\times n$-matrix.} for classes of oriented graphs such that the underlying undirected class has bounded rank-width. Beside these, we consider the standard parameter, i.e.\ the threshold value given in the instance, and the parameter ''number of vertices''. In Table \ref{fpt-sum} we summarize the known results for $\OCN$ and $\OCN_r$ parameterized by parameters. \begin{table}[ht] {\small \begin{center} \begin{tabular}{l||ll|ll|} & \multicolumn{2}{c|}{$\OCN$} & \multicolumn{2}{c|}{$\OCN_r$} \\ \hline directed tree-width & $\not\in\xp$ & Corollary \ref{cor-xp-ro}&$\not\in\xp$ & Corollary \ref{cor-xp-ro} \\ \hline directed path-width & $\not\in\xp$ & Corollary \ref{cor-xp-ro}& $\not\in\xp$ & Corollary \ref{cor-xp-ro} \\ \hline DAG-width & $\not\in\xp$ & Corollary \ref{cor-xp-ro} & $\not\in\xp$ & Corollary \ref{cor-xp-ro} \\ \hline Kelly-width & $\not\in\xp$ &Corollary \ref{cor-xp-ro} & $\not\in\xp$ & Corollary \ref{cor-xp-ro} \\ \hline tree-width of $\un(G)$ & \fpt &\cite{Gan09} & \fpt & \cite{Gan09} \\ \hline rank-width of $\un(G)$ & DET-h &\cite{Gan09} & ? & \\ \hline directed modular-width & ? & & \fpt & Corollary \ref{ocnk2} \\ \hline directed clique-width & ? & & \fpt & Corollary \ref{cor11ax} \\ \hline standard parameter $r$ & $\not\in\xp$ & Corollary \ref{cor-xp-r} & /// & \\ \hline directed clique-width + $r$& \fpt & Corollary \ref{o} & /// & \\ \hline number of vertices $n$ & \fpt & Corollary \ref{xp-n} & \fpt & Corollary \ref{xp-n} \\ \hline \end{tabular} \end{center} \caption{Complexity of $\OCN$ and $\OCN_r$ parameterized by parameters. The ''///'' entries indicate that by taking $r$ out of the instance the considered parameter makes no sense and the ''?'' entries indicate that the parameterized complexity remain open. We assume that $\p\neq \np$.}\label{fpt-sum} } \end{table} \section{Preliminaries}\label{intro} We use notations of Bang-Jensen and Gutin \cite{BG09} for graphs and digraphs. \subsection{Graphs} A {\em graph} is a pair $G=(V,E)$, where $V$ is a finite set of {\em vertices} and $E \subseteq \{ \{u,v\} \mid u,v \in V,~u \not= v\}$ is a finite set of {\em edges}. For a vertex $v\in V$, the set $N(v)=\{u\in V \mid \{v,u\}\in E\}$ is called the {\em set of all neighbors} of $v$ or {\em neighborhood} of $v$. We will use the following indexed graphs. \begin{itemize} \item By $P_n=(\{v_1,\ldots,v_n\},\{ \{v_1,v_2\},\ldots, \{v_{n-1},v_n\}\})$, $n \ge 2$, we denote the path on $n$ vertices. \item By $C_n=(\{v_1,\ldots,v_n\},\{\{v_1,v_2\},\ldots, \{v_{n-1},v_n\},\{v_1,v_n\}\})$, $n \ge 3$, we denote the cycle on $n$ vertices. \item By $K_n=(\{v_1,\ldots,v_n\},\{ \{v_i,v_j\}\mid 1\leq i<j\leq n\})$, $n \ge 1$, we denote the complete graph on $n$ vertices. \item By $K_{n,m}=(\{v_1,\ldots,v_n,w_1,\ldots,w_m\},\{ \{v_i,w_j\}\mid 1\leq i\leq n,1\leq j\leq m\})$, $n,m \ge 1$ we denote the complete bipartite graph with $n+m$ vertices. \end{itemize} \subsection{Digraphs} A {\em directed graph} or {\em digraph} is a pair $G=(V,E)$, where $V$ is a finite set of {\em vertices} and $E\subseteq \{(u,v) \mid u,v \in V,~u \not= v\}$ is a finite set of ordered pairs of distinct vertices called {\em arcs} or {\em directed edges}. For a vertex $v\in V$, the sets $N^+(v)=\{u\in V \mid (v,u)\in E\}$ and $N^-(v)=\{u\in V \mid (u,v)\in E\}$ are called the {\em set of all successors} and the {\em set of all predecessors} of $v$. The set $N(v)=N^+(v) \cup N^-(v)$ is the {\em set of all neighbors}. The {\em outdegree} of $v$, $\text{outdegree}(v)$ for short, is the number of successors of $v$ and the {\em indegree} of $v$, $\text{indegree}(v)$ for short, is the number of predecessors of $v$. The {\em maximum (vertex) degree} is defined by $\Delta(G)=\max_{v\in V} (\text{outdegree}(v)+\text{indegree}(v))$. For some given digraph $G=(V,E)$, we define its underlying undirected graph by ignoring the directions of the arcs, i.e. $\un(G)=(V,\{\{u,v\} \mid (u,v)\in E, u,v\in V\})$. For some (di)graph class $F$ we define $\free(F)$ as the set of all (di)graphs $G$ such that no induced sub(di)graph of $G$ is isomorphic to a member of $F$. A digraph $G'=(V',E')$ is a {\em subdigraph} of digraph $G=(V,E)$ if $V'\subseteq V$ and $E'\subseteq E$. If every arc of $E$ with both end vertices in $V'$ is in $E'$, we say that $G'$ is an {\em induced subdigraph} of $G$ and we write $G'=G[V']$. An {\em oriented graph} is a digraph with no loops and no opposite arcs. We will use the following indexed oriented graphs. \begin{itemize} \item By $\overrightarrow{P_n}=(\{v_1,\ldots,v_n\},\{ (v_1,v_2),\ldots, (v_{n-1},v_n)\})$, $n \ge 2$, we denote the oriented path on $n$ vertices. \item By $\overrightarrow{C_n}=(\{v_1,\ldots,v_n\},\{(v_1,v_2),\ldots, (v_{n-1},v_n),(v_n,v_1)\})$, $n \ge 2$, we denote the oriented cycle on $n$ vertices. \item By $\overrightarrow{K_{n,m}}=(\{v_1,\ldots,v_n,w_1,\ldots,w_m\},\{ (v_i,w_j)\mid 1\leq i\leq n,1\leq j\leq m\})$, $n,m \ge 1$ we denote an oriented complete bipartite digraph with $n+m$ vertices. \end{itemize} An {\em oriented forest (tree)} is an orientation of a forest (tree). An {\em out-rooted-tree} ({\em in-rooted-tree}) is an orientation of a tree with a distinguished root such that all arcs are directed away from (directed to) the root. A {\em directed acyclic graph (DAG for short)} is a digraph without any oriented cycle $\overrightarrow{C_n}$, for $n\geq 2$, as subdigraph. A {\em tournament} is a digraph in which there is exactly one edge between every two distinct vertices. A vertex $v$ is {\em reachable} from vertex $u$ in $G$, if $G$ contains an oriented path $\overrightarrow{P_n}$ as a subdigraph having start vertex $u$ and end vertex $v$. A {\em topological ordering} of a directed graph is a linear ordering of its vertices such that for every directed edge $(u,v)$, vertex $u$ is before vertex $v$ in the ordering. A digraph $G$ is bipartite if $\un(G)$ is bipartite and a digraph $G$ is planar if $\un(G)$ is planar. A digraph $G=(V,E)$ is {\em transitive} if for every pair $(u,v)\in E$ and $(v,w)\in E$ of arcs with $u\neq w$ the arc $(u,w)$ also belongs to $E$. The {\em transitive closure} $tc(G)$ of a digraph $G$ has the same vertex set as $G$ and for two distinct vertices $u,v$ there is an arc $(u,v)$ in $tc(G)$ if and only if $v$ is reachable from $u$ in $G$. \subsection{Coloring undirected graphs}\label{co-und} \begin{definition}[Graph coloring]\label{def-oricol} An \emph{$r$-coloring} of a graph $G=(V,E)$ is a mapping $c:V\to \{1,\ldots,r\}$ such that: \begin{itemize} \item $c(u)\neq c(v)$ for every $\{u,v\}\in E$.\footnote{The single condition on the mapping to be a feasible coloring will be extended for oriented colorings in Definition \ref{def-oc}.} \end{itemize} The {\em chromatic number} of $G$, denoted by $\chi(G)$, is the smallest integer $r$ such that $G$ has a $r$-coloring. \end{definition} We consider the following decision problem. \begin{desctight} \item[Name] Chromatic Number (CN) \item[Instance] A graph $G=(V,E)$ and a positive integer $r \leq |V|$. \item[Question] Is there a $r$-coloring for $G$? \end{desctight} If $r$ is a constant, i.e.\ not part of the input, the corresponding problem is denoted by $r$-Chromatic Number ($\CN_{r}$). Even on 4-regular planar graphs $\CN_{3}$ is NP-complete \cite{Dai80}. It is well known that bipartite graphs are exactly the graphs which allow a 2-coloring and that planar graphs are graphs that allow a 4-coloring. On undirected co-graphs, the Chromatic Number problem is easy to solve by the following result proven by Corneil et al.: \begin{lemma}[\cite{CLS81}]\label{colo-und} Let $G_1$ and $G_2$ be two vertex-disjoint graphs. \begin{enumerate} \item $\chi_o((\{v\},\emptyset))=1$ \item $\chi(G_1\cup G_2) = \max(\chi(G_1),\chi(G_2))$ \item $\chi(G_1\times G_2)= \chi(G_1) +\chi(G_2)$ \end{enumerate} \end{lemma} \begin{proposition} Let $G$ be a co-graph. Then, $\chi(G)$ can be computed in linear time. \end{proposition} Coloring a graph $G=(V,E)$ can be done by a greedy algorithm. For some given ordering $\pi$ of $V$, the vertices are ordered as a sequence in which each vertex is assigned to the minimum possible value that is not forbidden by the colors of its neighbors, see Algorithm \ref{algo-bip}. Obviously, different orders can lead to different numbers of colors. But there is always an ordering yielding to the minimum number of colors, which is hard to find in general. \begin{algorithm}[ht] \KwData{A graph $G=(\{v_1,\ldots,v_n\},E)$ and an ordering $\pi: v_1 < \hdots < v_n$ of its vertices.} \KwResult{An admitted vertex coloring $c:\{v_1,\ldots,v_n\} \mapsto {\mathbb N} $ of $G$.} \For{($i=1$ to $n$)}{ $c(v_i)=\infty$ } $c(v_1)=1$;\\ \For{($i=2$ to $n$)}{ $c(v_i)=\min\{{\mathbb N} \setminus \{c(v) \mid v\in N(v_i)\}\}$ \tcc*[r]{${\mathbb N}$ denotes the set of all positive integers.} } \caption{{\sc Greedy Coloring}} \label{algo-bip} \end{algorithm} The class of perfectly orderable graphs consists of those graphs for which the given greedy algorithm leads to an ordering yielding to an optimal coloring, not only for the graph itself but also for all of its induced subgraphs. \begin{definition}[Perfectly orderable graph \cite{Chv84}]\label{def-peog} Let $G=(V,E)$ be a graph. A linear ordering on $V$ is {\em perfect} if a greedy coloring algorithm with that ordering optimally colors every induced subgraph of $G$. A graph $G$ is {\em perfectly orderable} if it admits a perfect order. \end{definition} \begin{theorem}[\cite{Chv84}]\label{th-pe} A linear ordering $\pi$ of a graph $G$ is perfect if and only if there is no induced $$P_4=(\{a,b,c,d\},\{\{a,b\},\{b,c\},\{c,d\}\})$$ in $G$ such that $\pi(a)<\pi(b)$, $\pi(b)<\pi(c)$, and $\pi(d)<\pi(c)$. \end{theorem} \begin{example}\label{ex-pe} Every co-graph is perfectly orderable, since it does not have any induced $P_4$. \end{example} The Chromatic Number problem can be solved by an $\fpt$-algorithm w.r.t.\ the tree-width of the input graph \cite{Gur08c}. In contrast, this is not true for clique-width, since it has been shown in \cite{FGLS10a}, that the Chromatic Number problem is $\w[1]$-hard w.r.t.\ the clique-width of the input graph. That is, under reasonable assumptions an $\xp$-algorithm is the best one can hope for. Such algorithms are known, see \cite{EGW01a,KR01}. In order to show fixed parameter tractability for $r$-Chromatic Number w.r.t.\ the parameter clique-width one can use its defineability within monadic second order logic \cite{CMR00}. \subsection{Coloring oriented graphs} Oriented graph coloring has been introduced by Courcelle \cite{Cou94} in 1994. We consider oriented graph coloring on oriented graphs, i.e. digraphs with no loops and no opposite arcs. \begin{definition}[Oriented graph coloring]\label{def-oc} Let $G=(V,E)$ be an oriented graph. An \emph{oriented $r$-coloring} of $G$ is a mapping $c:V\to \{1,\ldots,r\}$ such that: \begin{itemize} \item $c(u)\neq c(v)$ for every $(u,v)\in E$, \item $c(u)\neq c(y)$ for every two arcs $(u,v)\in E$ and $(x,y)\in E$ with $c(v)=c(x)$. \end{itemize} The {\em oriented chromatic number} of $G$, denoted by $\chi_o(G)$, is the smallest integer $r$ such that $G$ has an oriented $r$-coloring. The vertex sets $V_i=\{v\in V\mid c(v)=i\}$, with $1\leq i\leq r$, divide $V$ into a partition of so called {\em color classes}. \end{definition} f For two oriented graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ a {\em homomorphism} from $G_1$ to $G_2$, $G_1 \to G_2$ for short, is a mapping $h: V_1 \to V_2$ such that $(u,v) \in E_1$ implies $(h(u),h(v)) \in E_2$. A homomorphism from $G_1$ to $G_2$ can be regarded as an oriented coloring of $G_1$ that uses the vertices of $G_2$ as colors classes. Therefore, digraph $G_2$ is called the {\em color graph} of $G_1$. This leads to equivalent definitions for the oriented coloring and the oriented chromatic number. There is an oriented $r$-coloring of an oriented graph $G_1$ if and only if there is a homomorphism from $G_1$ to some oriented graph $G_2$ with $r$ vertices. Thus, the oriented chromatic number of $G_1$ is the minimum number of vertices in an oriented graph $G_2$ such that there is a homomorphism from $G_1$ to $G_2$. Obviously, it is possible to choose $G_2$ as a tournament. \begin{observation}\label{oc-tournament} There is an oriented $r$-coloring of an oriented graph $G_1$ if and only if there is a homomorphism from $G_1$ to some tournament $G_2$ with $r$ vertices. Further, the oriented chromatic number of $G_1$ is the minimum number of vertices in a tournament $G_2$ such that there is a homomorphism from $G_1$ to $G_2$. \end{observation} \begin{observation}\label{obs-low} Let $G$ be an oriented graph. Then, it holds that $\chi(\un(G))\leq \chi_o(G)$. \end{observation} On the other hand it is not possible to bound the oriented chromatic number of an oriented graph $G$ by a function of the (undirected) chromatic number of $\un(G)$. This has been shown in \cite[Section 3]{Sop16} by an orientation $K'_{n,n}$ of a $K_{n,n}$ satisfying $\chi_o(K'_{n,n})=2n$ and $\chi(\un(K'_{n,n}))=2$. \begin{lemma}\label{le-col-subdigraph} Let $G$ be an oriented graph and $H$ be a subdigraph of $G$. Then, an oriented $r$-coloring of $G$ leads also an oriented $r$-coloring for $H$. \end{lemma} \begin{corollary}\label{le-isubdigraph} Let $G$ be an oriented graph and $H$ be a subdigraph of $G$. Then, it holds that $\chi_o(H)\leq \chi_o(G)$. \end{corollary} \begin{example}\label{ex-color} For oriented paths and oriented cycles we know: $\chi_o(\overrightarrow{P_2})=2$, $\chi_o(\overrightarrow{P_3})=3$, $\chi_o(\overrightarrow{C_4})=4$, $\chi_o(\overrightarrow{C_5})=5$. \end{example} An oriented graph $G=(V,E)$ is an {\em oriented clique} ({\em o-clique}) if $\chi_o(G) = |V|$. Thus all graphs given in Example~\ref{ex-color} are oriented cliques. Further by Observation \ref{obs-low} every tournament is an oriented clique. Thus, for DAGs the oriented chromatic number is unbounded. We consider the following decision problem. \begin{desctight} \item[Name] Oriented Chromatic Number ($\OCN$) \item[Instance] An oriented graph $G=(V,E)$ and a positive integer $r \leq |V|$. \item[Question] Is there an oriented $r$-coloring for $G$? \end{desctight} If $r$ is constant, i.e.\ not part of the input, the corresponding problem is denoted by $r$-Oriented Chromatic Number ($\OCN_{r}$). If $r\leq 3$, then $\OCN_{k}$ can be decided in polynomial time, while $\OCN_{4}$ is NP-complete \cite{KMG04}. $\OCN_{4}$ is even known to be NP-complete for several restricted classes of digraphs, e.g. for bounded degree DAGs \cite{CD06}, bounded degree bipartite oriented graph \cite{CD06}, graphs with K-width 1 and DAG-depth 3 \cite{GH10}, DAGs of K-width 3 and DAG-depth 5 \cite{GHKLOR14}, digraphs of DAG-width 2, K-width 1 and DAG-depth 3 \cite{GHKLOR14}, and acyclic oriented graphs whose underlying graph is connected, planar, bipartite and has maximum degree 3 \cite{CFGK16}. Up to now, the definition of oriented coloring was frequently applied to undirected graphs. For an undirected graph $G$ the maximum value $\chi_o(G')$ of all possible orientations $G'$ of $G$ is considered. In this sense, every tree has oriented chromatic number at most $3$ and every cycle $C_n$ has oriented chromatic number at most $5$. For several further graph classes there exist bounds on the oriented chromatic number. Among these are outerplanar graphs \cite{Sop97}, Halin graphs \cite{DS14}, and planar graphs \cite{Mar13}. See \cite{Sop16} for a survey. To advance research in this field, we consider oriented graph coloring on recursively defined oriented graph classes. \section{Coloring transitive acyclic digraphs}\label{sec-ta} In this section we will use the concept of perfectly orderable graphs and Theorem \ref{th-pe} in order to find oriented colorings of transitive acyclic digraphs. \begin{theorem}\label{algop2} Let $G$ be a transitive acyclic digraph. Then, every greedy coloring along a topological ordering of $G$ leads to an optimal oriented coloring of $G$ and $\chi_o(G)$ can be computed in linear time. \end{theorem} \begin{proof} Let $G$ be a transitive acyclic digraph. Since $G=(V,E)$ is acyclic there is a topological ordering $t$ for $G$. Since $G$ is transitive, it does not contain the following orientation of a $P_4$ as an induced subdigraph. $$\bullet \rightarrow \bullet \rightarrow\bullet\leftarrow \bullet$$ By Theorem \ref{th-pe} every linear ordering and thus also $t$ is perfect on the vertex set of graph $\un(G)=(V,E_u)$. Let $c:V\to \{1,\ldots,k\}$ be a coloring for $\un(G)$ obtained by the greedy algorithm (Algorithm \ref{algo-bip}) for $t$ on $V$. We show that $c$ is an oriented coloring for $G$ by verifying the two properties of Definition \ref{def-oc}. \begin{itemize} \item Property $c(u)\neq c(v)$ holds for every $(u,v)\in E$ since $c(u)\neq c(v)$ holds for every $\{u,v\}\in E_u$. \item Property $c(u)\neq c(y)$ for every two arcs $(u,v)\in E$ and $(x,y)\in E$ with $c(v)=c(x)$ holds by the following argumentation. Assume there is an arc $(v_i,v_j)\in E$ with $v_i<v_j$ in $t$ but $c(v_i)>c(v_j)$. Then, when coloring $v_i$ we would have taken $c(v_j)$ if possible, as we always take the minimum possible color value. Since this was not possible there must have been an other vertex $v_k<v_i$ which was colored before $v_i$ with $c(v_k)=c(v_j)$ and $(v_k,v_i)\in E$. But if $(v_k,v_i)\in E$ and $(v_i,v_j)\in E$, due to transitivity it must also hold that $(v_k,v_j)\in E$ and consequently, $c(v_k)=c(v_j)$ is not possible. Thus, the assumption was wrong and for every arc $(v_i,v_j)\in E$ with $v_i<v_j$ in $t$ it must hold that $c(v_i)<(c_j)$. \end{itemize} The optimality of oriented coloring $c$ follows since the lower bound of Observation \ref{obs-low} is achieved. \end{proof} In order to state the next result, let $\omega(G)$ be the number of vertices in a largest clique in the (undirected) graph $G$. \begin{corollary}\label{cor-tt-a} Let $G$ be a transitive acyclic digraph. Then, it holds that $$\chi_o(G)=\chi(\un(G))=\omega(\un(G))$$ and all three values can be computed in linear time. \end{corollary} For some oriented graph $G$ we denote by $\ell(G)$ the length of a longest oriented path in $G$. \begin{proposition}\label{pro1} Let $G$ be a transitive acyclic digraph. Then, it holds that $\chi_o(G)= \ell(G)+1$. \end{proposition} \begin{proof} The proof of Theorem \ref{algop2} leads to an optimal oriented coloring using $\ell(G)+1$ colors. \end{proof} Next, we consider oriented colorings of oriented graphs with bounded vertex degree. For every oriented graph the oriented chromatic number can be bounded (exponentially) by its maximum vertex degree $\Delta$ \cite{KSZ97}. For small vertex degrees $\Delta\leq 7$ there are better bounds in \cite{Duf19,DOPS20}. \begin{corollary}\label{cor-tt-av} Let $G$ be a transitive acyclic digraph. Then, it holds that $\chi_o(G)\leq \Delta(G)+1$. \end{corollary} \begin{proof}Let $G$ be a transitive acyclic digraph. By Proposition \ref{pro1} and the fact that the first vertex of a longest path within a transitive digraph $G$ has outdegree at least $\ell(G)$, it follows that the oriented chromatic number of $G$ can be estimated by $\chi_o(G)= \ell(G)+1 \leq \Delta(G)+1$. \end{proof} \begin{proposition}\label{pro2} Let $G$ be an acyclic digraph. Then, it holds that $\chi_o(G)\leq \ell(G)+1$. \end{proposition} \begin{proof} Let $G$ be an acyclic digraph and $G'$ its transitive closure. Using Corollary \ref{le-isubdigraph} and Proposition \ref{pro1} we know that $\chi_o(G)\leq \chi_o(G')= \ell(G')+1=\ell(G)+1$. \end{proof} Now, we consider oriented graph coloring on recursively defined oriented graph classes. \section{Coloring oriented co-graphs}\label{sec-cog} We recall operations which have been considered by Bechet et al.\ in \cite{BGR97}. Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be two vertex-disjoint digraphs. \begin{itemize} \item The {\em disjoint union} of $G_1$ and $G_2$, denoted by $G_1 \oplus G_2$, is the digraph with vertex set $V_1\cup V_2$ and arc set $E_1\cup E_2$. \item The {\em order composition} of $G_1$ and $G_2$, denoted by $G_1\oslash G_2$, is defined by their disjoint union plus all possible arcs from vertices of $G_1$ to vertices of $G_2$. \end{itemize} By omitting the series composition within the definition of directed co-graphs in \cite{CP06}, we obtain the class of all {\em oriented co-graphs}. \begin{definition}[Oriented co-graphs]\label{dcog} The class of {\em oriented complement reducible graphs}, {\em oriented co-graphs} for short, is recursively defined as follows. \begin{enumerate} \item Every digraph on a single vertex $(\{v\},\emptyset)$, denoted by $v$, is an {\em oriented co-graph}. \item If $G_1$ and $G_2$ are two vertex-disjoint oriented co-graphs, then \begin{enumerate} \item the disjoint union $G_1\oplus G_2$, and \item the order composition $G_1\oslash G_2$ are {\em oriented co-graphs}. \end{enumerate} \end{enumerate} The class of oriented co-graphs is denoted by $\OC$. \end{definition} Every expression $X$ using the operations of Definition \ref{dcog} is called a {\em di-co-expression}. Example \ref{ex-orico} illustrates these notations. \begin{example}\label{ex-orico} The following di-co-expression $X$ defines the oriented graph shown in Figure \ref{F03}. $$ X=((v_1\oslash v_3) \oslash(v_2 \oplus v_4)) \label{eq-ori-c4x} $$ \end{example} \begin{figure}[hbtp] \centering \centerline{\includegraphics[width=0.25\textwidth]{ex_oc.pdf}} \caption{Digraph defined by di-co-expression $X$ in Example \ref{ex-orico}.} \label{F03} \end{figure} Several classes of digraphs are included in the set of all oriented co-graphs. \begin{proposition} Every transitive tournament is an oriented co-graph. \end{proposition} \begin{proposition} Every oriented bipartite graph $\overrightarrow{K_{n,m}}$ is an oriented co-graph. \end{proposition} The set of all oriented co-graphs is closed under taking induced subdigraphs. Using the notations of \cite{VTL82} we denote the following orientation of a $P_4$ as the $N$ graph. $$N=\bullet \rightarrow \bullet \leftarrow\bullet\rightarrow \bullet$$ The class of oriented co-graphs can be characterized by excluding the four forbidden induced subdigraphs $\overleftrightarrow{P_2}=(\{u,v\},\{(u,v),(v,u)\})$, $\overrightarrow{P_3}$, $\overrightarrow{C_3}$, and $N$, see \cite{GKR21b}. The class of oriented co-graphs has already been analyzed by Lawler in \cite{Law76} and Corneil et al.\ in \cite{CLS81} using the notation of {\em transitive series-parallel (TSP) digraphs}. For every oriented co-graph we can define a tree structure, denoted as {\em di-co-tree}. The leaves of the di-co-tree represent the vertices of the digraph and the inner vertices of the di-co-tree correspond to the operations applied on the subexpressions defined by the subtrees. For every oriented co-graph one can construct a di-co-tree in linear time, see \cite{CP06}. Oriented co-graphs are a subclass of directed co-graphs \cite{CP06} and both classes are interesting from an algorithmic point of view since several hard graph problems can be solved in polynomial time by dynamic programming along the tree structure of the input graph, see \cite{Ret98,BM14,Gur17a,GR18c,GKR19f,GKR19d,GHKRRW20,GKRRW20}. \begin{lemma}[\cite{GKR19d}]\label{le-co-gr}Let $G_1$ and $G_2$ be two vertex-disjoint oriented co-graphs. Then, the following equations hold. \begin{enumerate} \item $\chi_o((\{v\},\emptyset)) = 1$ \item $\chi_o(G_1 \oplus G_2) = \max(\chi_o (G_1), \chi_o(G_2))$ \item $\chi_o(G_1\oslash G_2) = \chi_o(G_1) + \chi_o(G_2)$ \end{enumerate} \end{lemma} \begin{theorem}[\cite{GKR19d}]\label{algop} Let $G$ be an oriented co-graph. Then, an optimal oriented coloring for $G$ and $\chi_o(G)$ can be computed in linear time. \end{theorem} The result in \cite{GKR19d} concerning the oriented coloring on oriented co-graphs is based on a dynamic programming along a di-co-tree for the given oriented co-graph as input. Since every oriented co-graph is transitive and acyclic, Theorem \ref{algop2} leads to the next result, which re-proves Theorem \ref{algop}. \begin{corollary}\label{c-co} Let $G$ be an oriented co-graph. Then, every greedy coloring along a topological ordering of $G$ leads to an optimal oriented coloring of $G$ and $\chi_o(G)$ can be computed in linear time. \end{corollary} Theorem \ref{algop2} is more general than Corollary \ref{c-co} since it does not exclude $N$ which is a forbidden induced subdigraph for oriented co-graphs. It holds that $$\OC=\free\{\overleftrightarrow{P_2},\overrightarrow{P_3}, \overrightarrow{C_3}, N\}\subseteq \free\{\overleftrightarrow{P_2},\overrightarrow{P_3}, \overrightarrow{C_3}\}$$ and $\free\{\overleftrightarrow{P_2},\overrightarrow{P_3}, \overrightarrow{C_3}\}$ is equivalent to the set of all acyclic transitive digraphs. Since every oriented co-graph is transitive and acyclic, Corollary \ref{cor-tt-av} leads to the following bound. \begin{corollary}\label{th-degxxco2} Let $G$ be an oriented co-digraph. Then, it holds that $\chi_o(G)\leq \Delta(G)+1$. \end{corollary} There are classes of oriented co-graphs, e.g., the class of all $\overrightarrow{K_{1,n}}$, for which the oriented chromatic number is even bounded by a constant and thus smaller than the shown bound. The set of all transitive tournaments shows that the bound given in Corollary \ref{th-degxxco2} is best possible. \section{Coloring msp-digraphs} We recall the definitions from \cite{BG18} which are based on \cite{VTL82}. First, we introduce two operations for two vertex-disjoint digraphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$. Let $O_1$ be the set of vertices of outdegree $0$ (set of sinks) in $G_1$ and $I_2$ be the set of vertices of indegree $0$ (set of sources) in $G_2$. \begin{itemize} \item The {\em parallel composition} of $G_1$ and $G_2$, denoted by $G_1 \cup G_2$, is the digraph with vertex set $V_1\cup V_2$ and arc set $E_1\cup E_2$. \item The {\em series composition} of $G_1$ and $G_2$, denoted by $G_1 \times G_2$ is the digraph with vertex set $V_1\cup V_2$ and arc set $E_1\cup E_2\cup \{(v,w)\mid v\in O_1, w\in I_2\}$. \end{itemize} \begin{definition}[Msp-digraphs]\label{def-msp} The class of {\em minimal series-parallel digraphs}, {\em msp-di\-graphs} for short, is recursively defined as follows. \begin{enumerate} \item Every digraph on a single vertex $(\{v\},\emptyset)$, denoted by $v$, is a {\em minimal series-parallel digraph}. \item If $G_1$ and $G_2$ are vertex-disjoint minimal series-parallel digraphs then, \begin{enumerate} \item the parallel composition $G_1 \cup G_2$ and \item then series composition $G_1 \times G_2$ are {\em minimal series-parallel digraphs}. \end{enumerate} \end{enumerate} The class of minimal series-parallel digraphs is denoted as $\MSP$. \end{definition} Minimal (vertex) series-parallel digraphs are the line digraphs of edge series-parallel digraphs \cite{VTL82}, which are an oriented version of the well known class of series-parallel graphs. Every expression $X$ using the operations of Definition \ref{def-msp} is called an {\em msp-expression}. The digraph defined by the expression $X$ is denoted by $\g(X)$. We illustrate this by two expressions which we will refer to later. \begin{example}\label{ex-msp} The following msp-expressions $X_1$ and $X_2$ define msp-digraphs on five and six vertices shown in Figure~\ref{F06} and Figure~\ref{F03y}. $$ X_1= (v_1\times((v_2\times v_3)\cup v_4))\times v_5 $$ $$ X_2= (v_1\times(((v_2\times v_3)\times v_4)\cup v_5))\times v_6 $$ \end{example} \begin{figure}[hbtp] \centering \parbox[b]{67mm}{ \centerline{\includegraphics[width=0.4\textwidth]{ex_erbb.pdf}} \caption{$\gr(X_1)$ in Example \ref{ex-msp}. } \label{F06}} ~~~~~~~~~ \parbox[b]{67mm}{ \centerline{\includegraphics[width=0.5\textwidth]{ex_erb.pdf}} \caption{$\gr(X_2)$ in Example \ref{ex-msp}.} \label{F03y}} \end{figure} By removing vertex $v_3$ from $\g(X_2)$ in Example \ref{ex-msp}, we obtain an oriented graph which is no msp-digraph. This implies that the set of all msp-digraphs is not closed under taking induced subdigraphs. A further remarkable property of msp-digraphs is that the oriented chromatic number of the disjoint union of two msp-digraphs can be larger than the maximum oriented chromatic number of the involved digraphs. This follows by the digraphs defined by expressions $X_1$ and $X_2$ in Example \ref{ex-six2}, which both have oriented chromatic number $4$ but their disjoint union leads to a digraph with oriented chromatic number $5$. \begin{example}\label{ex-six2} In the following two msp-expressions we assume that the series composition $\times$ binds more strongly than the parallel composition $\cup$. $$X_1=v_{1} \times (v_2 \cup v_3 \times v_4) \times v_5 \times v_6$$ $$X_2=w_{1} \times (w_2 \cup w_3 \times (w_{4} \cup w_{5} \times w_{6} ))\times w_{7}$$ \end{example} Several classes of digraphs are included in the set of all msp-digraphs. \begin{proposition} Every in- and out-rooted tree is an msp-digraph. \end{proposition} \begin{proposition} Every oriented bipartite graph $\overrightarrow{K_{n,m}}$ is an msp-digraph. \end{proposition} For every msp-digraph we can define a tree structure $T$, which is denoted as {\em msp-tree}. (In \cite{VTL82}, the tree-structure for an msp-digraphs is denoted as binary decomposition tree.) The leaves of an msp-tree represent the vertices of the digraph and the inner vertices of the msp-tree correspond to the operations applied on the subexpressions defined by the subtrees. For every msp-digraph one can construct an msp-tree in linear time, see \cite{VTL82}. Next we want to give an algorithm to compute the oriented chromatic number of an msp-digraph. Considering the solutions of Sections \ref{sec-ta} and \ref{sec-cog} we conclude that a greedy coloring of $\un(G)$ along a topological ordering of $G$ does not work for computing the oriented chromatic number of an msp-digraph $G$. An oriented path would be colored by only two colors which is not an admitted oriented coloring. Further, a dynamic programming solution using similar formulas to Lemma \ref{le-co-gr} is not possible for computing the oriented chromatic number of msp-digraphs. Example \ref{ex-six2} implies that the oriented chromatic number of the disjoint union of two msp-digraphs can be larger than the maximum oriented chromatic number of the involved digraphs. In order to give an algorithm to compute the oriented chromatic number of msp-digraphs, we first show that this value can be bounded by a constant. The class of undirected series-parallel graphs was considered in \cite{Sop97} by showing that every orientation of a series-parallel graph has oriented chromatic number at most 7. This bound can not be applied to msp-digraphs, since the set of all $\overrightarrow{K_{n,m}}$ is a subset of msp-digraphs and the underlying graphs are even of unbounded tree-width \cite{Bod98} and thus, no series-parallel graphs. Nevertheless, we can show that $7$ is also an upper bound for the oriented chromatic number of msp-digraphs. Therefore we introduce recursively defined oriented graphs $M_i$. \begin{lemma}\label{le-mi} We recursively define oriented digraphs $M_i$ as follows. $M_0$ is a single vertex graph and for $i\geq 1$ we define $$M_i=M_{i-1}\cup M_{i-1}\cup (M_{i-1} \times M_{i-1}).$$ Then, every msp-digraph $G$ is a (-n induced) subdigraph of some $M_i$ such that every source in $G$ is a source in $M_i$ and every sink in $G$ is a sink in $M_i$. \end{lemma} \begin{proof} The lemma can be shown by induction over the number of vertices in some msp-digraph $G$. If $G$ has exactly one vertex, the claim holds true by choosing $M_0$. Next, assume that $G$ has $k>1$ vertices. Then, it holds that $G=G_1 \Box G_2$ for some $\Box\in\{\cup,\times\}$ and $G_1$ and $G_2$ are msp-digraphs with less than $k$ vertices. By the induction hypothesis we conclude that there are two integers $i_1$ and $i_2$ such that digraph $G_1 \Box G_2$ is a subdigraph of digraph $M_{i_1} \Box M_{i_2}$. (If $\Box=\times$, it is important that every sink in $G_1$ is a sink in $M_{i_1}$ and every source in $G_2$ is a source in $M_{i_2}$.) Thus, $G$ is a subdigraph of digraph $M_{i_1}\cup M_{i_2}\cup (M_{i_1} \times M_{i_2})$. W.l.o.g. we assume that $i_1\leq i_2$. By construction it follows that $M_{i_1}$ is a subdigraph of $M_{i_2}$. Consequently, $G$ is a subdigraph of digraph $M_{i_2}\cup M_{i_2}\cup (M_{i_2} \times M_{i_2})=M_{i_2+1}$ and every source in $G$ is a source in $M_{i_2+1}$ and every sink in $G$ is a sink in $M_{i_2+1}$. This completes the proof of the claim. \end{proof} \begin{theorem}\label{algspd} Let $G$ be an msp-digraph. Then, it holds that $\chi_o(G)\leq 7$. \end{theorem} \begin{proof} By Lemma \ref{le-col-subdigraph} we can show the theorem by coloring the digraphs $M_i$ defined in Lemma \ref{le-mi}. Further, the first two occurrences of $M_{i-1}$ in $M_i$ can be colored in the same way. Thus, we can restrict to oriented graphs $M'_i$ which are defined as follows. $M'_0$ is a single vertex graph and for $i\geq 1$ we define $$M'_i=M'_{i-1}\cup (M'_{i-1} \times M'_{i-1}).$$ We define an oriented $7$-coloring $c$ for $M'_i$ as follows. For some vertex $v$ of $M'_i$ we define by $c(v,i)$ the color of $v$ in $M'_i$. First, we color $M'_0$ by assigning color $0$ to the single vertex in $M'_0$.\footnote{Please note that using colors starting at value $0$ instead of $1$ does not contradict Definition \ref{def-oc}.} For $i\geq 1$ we define the colors for the vertices $v$ in $M'_i=M'_{i-1}\cup (M'_{i-1} \times M'_{i-1})$ according to the three copies of $M'_{i-1}$ in $M'_i$ (numbered from left to right). Therefore, we use the two functions $p(x)=(4\cdot x) \bmod 7$ and $q(x)=(4\cdot x + 1) \bmod 7$. We define \[ c(v,i)= \left\{\renewcommand{\arraystretch}{1.3}\begin{array}{ll} c(v,i-1) & \text{if } v \text{ is from the first copy,}\\ p(c(v,i-1)) & \text{if } v \text{ is from the second copy, and}\\ q(c(v,i-1)) & \text{if } v \text{ is from the third copy.}\\ \end{array}\right.~\] It remains to show that $c$ leads to an oriented coloring for $M_i$. Let $C_i=(W_i,F_i)$ with $W_i=\{0,1,2,3,4,5,6\}$ and $F_i=\{(c(u,i),c(v,i))\mid (u,v)\in E_i\}$ be the color graph of $M'_i=(V_i,E_i)$. By the definition of $M'_i$ we follow that \[ \begin{array}{lclllll} F_{i}&=&F_{i-1}&\cup &\{(p(x),p(y)) \mid (x,y)\in F_{i-1}\} \\ & & &\cup & \{(q(x),q(y)) \mid (x,y)\in F_{i-1}\} \\ & & &\cup & \{(p(c(v,i-1)),q(c(w,i-1))) \mid v \text{ sink of } M'_{i-1}, w \text{ source of } M'_{i-1} \}. \\ \end{array} \] In order to ensure an oriented coloring of $M'_i$, we verify that $C_i$ is an oriented graph. In Figure \ref{F09} the color graph $C_i$ for $i\geq 5$ is given. \begin{figure}[hbtp] \centering \centerline{\includegraphics[width=0.3\textwidth]{C7.pdf}} \caption{Color graph $C_i$ for $i\geq 5$ used in the proof of Theorem \ref{algspd}, which is also known as the Paley tournament of order 7.} \label{F09} \end{figure} Every source in $M'_i$ is colored by $0$ since $p(0)=0$. Every sink in $M'_i$ is colored by $0$, $1$, or $5$ since $q(0)=1$, $q(1)=5$, and $q(5)=0$. Consequently, the arcs of $$\{(p(c(v,i-1)),q(c(w,i-1))) \mid v \text{ sink of } M'_{i-1}, w \text{ source of } M'_{i-1}\}$$ belong to the set $$\{(p(0),q(0)),(p(1),q(0)),(p(5),q(0))\}=\{(0,1),(4,1),(6,1)\}.$$ For every $(u,v)\in\{(0,1),(4,1),(6,1)\}$ we know that \begin{equation} (v-u) \bmod 7 \in\{1,2,4\} \label{mod7} \end{equation} which implies $(u-v) \bmod 7 \not\in\{1,2,4\}$ and thus, condition (\ref{mod7}) does not hold for the reverse arcs of $\{(0,1),(4,1),(6,1)\}$. It remains to show that (\ref{mod7}) remains true for all arcs $(u,v)$ when applying $p$ and $q$ to $M'_{i-1}$: \[ \begin{array}{lclllll} (q(v)-q(u)) \bmod 7&=& (((4\cdot v +1) \bmod 7) - ((4\cdot u +1) \bmod 7)) \bmod 7 \\ & = & (((4\cdot v) \bmod 7) - ((4\cdot u) \bmod 7)) \bmod 7\\ & = & (p(v)-p(u)) \bmod 7\\ & = &(4 (v-u)) \bmod 7 \end{array} \] Since $(v-u) \bmod 7 \in\{1,2,4\}$ leads to $(4(v-u)) \bmod 7 \in\{1,2,4\}$, the result follows. \end{proof} Digraph $G$ on 27 vertices defined by expression $X$ in Example \ref{ex-six} satisfies $\chi_o(G)=7$, which was found by a computer program.\footnote{We implemented an algorithm which takes an oriented graph $G$ and an integer $k$ as an input and which decides whether $\chi_o(G)\leq k$.} This implies that the bound of Theorem \ref{algspd} is best possible. \begin{example}\label{ex-six} In the following msp-expression we assume that the series composition $\times$ binds more strongly than the parallel composition $\cup$. $$ \begin{array}{c} X=v_{1} \times (v_2 \cup v_3 \times (v_4 \cup v_5 \times v_6))\times (v_7 \cup (v_8 \cup v_9 \times v_{10})\times (v_{11} \cup v_{12} \times v_{13}))\times \\ (v_{14}\cup (v_{15} \cup (v_{16} \cup v_{17} \times v_{18})\times (v_{19} \cup v_{20} \times v_{21}))\times (v_{22} \cup(v_{23} \cup v_{24} \times v_{25})\times v_{26}) ) \times v_{27} \end{array} $$ \end{example} In oder to compute the oriented chromatic number of an msp-digraph $G$ defined by an msp-expression $X$, we recursively compute the set $F(X)$ of all triples $(H,L,R)$ such that $H$ is a color graph for $G$, where $L$ and $R$ are the sets of colors of all sinks and all sources in $G$ with respect to the coloring by $H$. The number of vertex labeled, i.e., the vertices are distinguishable from each other, oriented graphs on $n$ vertices is $3^{\nicefrac{n(n-1)}{2}}$. By Theorem \ref{algspd} we can conclude that $$|F(X)|\leq 3^{\nicefrac{7(7-1)}{2}}\cdot 2^7 \cdot 2^7\in \text{$\mathcal O$}(1)$$ which is independent of the size of $G$. For two color graphs $H_1=(V_1,E_1)$ and $H_2=(V_2,E_2)$ we define $H_1+H_2=(V_1\cup V_2,E_1\cup E_2)$. \begin{lemma}\label{le1} \begin{enumerate} \item\label{le1-1} For every $v\in V$ it holds $F(v)=\{((\{i\},\emptyset ),\{i\},\{i\}) \mid 0\leq i \leq 6\}$. \item\label{le1-2} For every two msp-expressions $X_1$ and $X_2$ we obtain set $F(X_1\cup X_2)$ from sets $F(X_1)$ and $F(X_2)$ as follows. For every $(H_1,L_1,R_1) \in F(X_1)$ and every $(H_2,L_2,R_2) \in F(X_2)$ such that graph $H_1+H_2$ is oriented, we put $(H_1 + H_2,L_1\cup L_2,R_1\cup R_2)$ into $F(X_1\cup X_2)$. \item\label{le1-3} For every two msp-expressions $X_1$ and $X_2$ we obtain set $F(X_1\times X_2)$ from sets $F(X_1)$ and $F(X_2)$ as follows. For every $(H_1,L_1,R_1) \in F(X_1)$ and every $(H_2,L_2,R_2) \in F(X_2)$ such that graph $H_1+H_2$ together with the arcs in $R_1\times L_2$ is oriented, we put $((V_1\cup V_2,E_1\cup E_2\cup R_1\times L_2),L_1,R_2)$ into $F(X_1\times X_2)$. \end{enumerate} \end{lemma} \begin{proof} \begin{enumerate} \item Set $F(v)$ includes obviously all possible solutions to color every vertex on its own with the seven given colors. \item Let $(H_1,L_1,R_1)$ be any possible solution for coloring $\g(X_1)$, which therefore is included in $F(X_1)$, as well as a possible solution $(H_2,L_2,R_2)$ for coloring $\g(X_2)$ which is included in $F(X_2)$. Let further $H_1+H_2$ be an oriented graph. Since the operation $\cup$ creates no additional edges in $\g(X_1\cup X_2)$, the vertices of $\g(X_1)$ can still be colored with $H_1$ and the vertices of $\g(X_2)$ can still be colored with $H_2$ such that all vertices from $\g(X_1\cup X_2)$ are legally colored. Further, all sinks in $\g(X_1)$ and $\g(X_2)$ are also sinks in $\g(X_1\cup X_2)$. The same holds for the sources. For an oriented digraph $H_1+H_2$ this leads to $(H_1+H_2,L_1\cup L_2, R_1\cup R_2)\in F(X_1\cup X_2)$. Let $(H,L,R)\in F(X_1\cup X_2)$, then there is an induced subdigraph $H_1$ of the color graph $H$ which colors $\g(X_1)$, an induced subdigraph of $\g(X_1\cup X_2)$. Since $H$ is oriented, $H_1$ is oriented. Let $L_1\subseteq L$ be the sources with vertices in $\g(X_1)$ and $R_1\subseteq R$ be the sinks for vertices in $\g(X_1)$. Then, it holds that $(H_1,L_1,R_1)\in F(X_1)$. The same arguments hold for $X_2$, such that $(H_2,L_2,R_2)\in F(X_2)$. \item Let $(H_1,L_1,R_1)$ be any possible solution for coloring $\g(X_1)$, which therefore is included in $F(X_1)$, as well as a possible solution $(H_2,L_2,R_2)$ for coloring $X_2$ which is included in $F(X_2)$. Further, let $H_1+H_2$ together with edges from $R_1\times L_2$ be an oriented graph. Then, $H=(V_1\cup V_2,E_1\cup E_2\cup R_1\times L_2)$ is an oriented coloring for $X=X_1\times X_2$. Since the sinks of $\g(X_1)$ are connected with the sources of $\g(X_2)$ in $\g(X)$ the sources of $L_1$ are the only sources left in $\g(X)$ as well as the sinks in $R_2$ are the only sinks left in $\g(X)$. This leads to $(H,L_1,R_2)\in F(X)$. Let $(H,L,R)\in F(X_1\times X_2)$, then there is an induced subdigraph $H_1$ of the color graph $H$ which colors $\g(X_1)$ which is an induced subdigraph of $\g(X_1\times X_2)$. Since $H$ is oriented, $H_1$ is also oriented. Since all the sources of $\g(X_1\times X_2)$ are in $\g(X_1)$ it holds that $L_1=L$ are also sources of $\g(X_1)$. Let $R_1$ be the vertices in $\g(X_1)$ which only have out-going neighbors in $\g(X_2)$ but not in $\g(X_1)$, then $R_1$ are the sinks of $\g(X_1)$. Thus, it holds that $(H_1,L_1, R_1)\in F(X_1)$. Simultaneously, there is an induced subdigraph $H_2$ of the color graph $H$ which colors $\g(X_2)$ which is an induced subdigraph of $\g(X_1\times X_2)$. Since $H$ is oriented, $H_2$ is also oriented. Since all the sinks of $\g(X_1\times X_2)$ are in $\g(X_2)$ it holds that $R_2=R$ are also sinks of $\g(X_2)$. Let $L_2$ be the vertices in $\g(X_2)$ which only have in-going neighbors in $\g(X_1)$ but not in $\g(X_2)$, then $L_2$ are the sources of $\g(X_2)$. Thus, it holds that $(H_2,L_2, R_2)\in F(X_2)$. \end{enumerate} This shows the statements of the lemma. \end{proof} Since every possible coloring of $G$ is part of the set $F(X)$, where $X$ is an msp-expression for $G$, it is possible to find a minimum coloring for $G$. \begin{corollary}\label{cor1} There is an oriented $r$-coloring for some msp-digraph $G$ which is given by some msp-expression $X$ if and only if there is some $(H,L,R)\in F(X)$ such that color graph $H$ has $r$ vertices. Therefore, $\chi_o(G)=\min\{|V| \mid ((V,E),L,R)\in F(X)\}$. \end{corollary} \begin{theorem}\label{algspdd} Let $G$ be an msp-digraph. Then, the oriented chromatic number of $G$ can be computed in linear time. \end{theorem} \begin{proof} Let $G$ be an msp-digraph on $n$ vertices and $m$ edges. Further, let $T$ be an msp-tree for $G$ with root $r$. For some vertex $u$ of $T$ we denote by $T_u$ the subtree rooted at $u$ and $X_u$ the msp-expression defined by $T_u$. In order to solve the Oriented Chromatic Number problem for some msp-digraph $G$, we traverse msp-tree $T$ in a bottom-up order. For every vertex $u$ of $T$ we compute $F(X_u)$ following the rules given in Lemma \ref{le1}. By Corollary~\ref{cor1} we can solve our problem by $F(X_r)=F(X)$. An msp-tree $T$ can be computed in $\text{$\mathcal O$}(n+m)$ time from msp-digraph $G$, see \cite{VTL82}. Our rules given in Lemma~\ref{le1} show the following running times. \begin{itemize} \item For every vertex $v\in V$ set $F(v)$ is computable in $\text{$\mathcal O$}(1)$ time. \item For every two msp-expressions $X_1$ and $X_2$ set $F(X_1 \cup X_2)$ can be computed in $\text{$\mathcal O$}(1)$ time from $F(X_1)$ and $F(X_2)$. \item For every two msp-expressions $X_1$ and $X_2$ set $F(X_1 \times X_2)$ can be computed in $\text{$\mathcal O$}(1)$ time from $F(X_1)$ and $F(X_2)$. \end{itemize} Since we have $n$ leaves and $n-1$ inner vertices in msp-tree $T$, the running time is in $\text{$\mathcal O$}(n+m)$. \end{proof} Due Corollary \ref{cor-tt-a} we know that for every oriented co-graph $G$ it holds that $\chi_o(G)=\chi(\un(G))$. This equality does not hold for msp-digraphs by Example \ref{ex-six} and the next result. \begin{proposition}\label{le-msp-uG} Let $G$ be an msp-digraph. Then, it holds that $\chi(\un(G))\leq 3$. \end{proposition} \begin{proof} We show the result by giving a 3-coloring $c$ of $\un(M'_i)$ for the oriented graphs $M'_i$ defined in Lemma \ref{le-mi}. For some vertex $v$ of $\un(M'_i)$ we define by $c(v,i)$ the color of $v$ in $\un(M'_i)$. First we color $\un(M'_0)$ by assigning color $0$ to the single vertex in $\un(M'_0)$. For $i\geq 1$ we define the colors for the vertices $v$ in $\un(M'_i)=\un(M'_{i-1})\cup (\un(M'_{i-1} \times M'_{i-1}))$ according to the three copies of $\un(M'_{i-1})$ in $\un(M'_i)$ (numbered from left to right). Therefore, we use the two functions $p(x)=(2\cdot x) \bmod 3$ and $q(x)=(2\cdot x +1) \bmod 3$. We define \[ c(v,i)= \left\{\renewcommand{\arraystretch}{1.3}\begin{array}{ll} c(v,i-1) & \text{if } v \text{ is from the first copy,}\\ p(c(v,i-1)) & \text{if } v \text{ is from the second copy, and}\\ q(c(v,i-1)) & \text{if } v \text{ is from the third copy.}\\ \end{array}\right. \] In order to show the correctness of the given coloring for $\un(M'_i)$ by Definition \ref{def-oricol} it suffices to verify that the end vertices of every edge in $\un(M'_i)$ are colored differently. This follows since differently labeled endvertices remain labeled differently when applying the permutations by $p$ and $q$. Further, every edge $\{u,v\}$ in $\un(M'_i)$ arises from some arc $(u,v)$ in $M'_i$ defined by the series composition and satisfies $c(u,i)=0$ and $c(v,i)=1$ or $c(u,i)=2$ and $c(v,i)=1$. This holds true since before applying the permutations by $p$ and $q$ all sources $w$ are labeled by $c(w,i-1)=0$ and all sinks $v$ are labeled by $c(v,i-1)=0$ or $c(v,i-1)=1$. \end{proof} For expression $X_1$ given in Example \ref{ex-msp} we obtain by $\un(\g(X_1))$ a cycle on five vertices $C_5$ of chromatic number $3$, which implies that the bound of Proposition \ref{le-msp-uG} is sharp. In \cite{GKL21} we introduced the concept of $g$-oriented $r$-colorings which generalizes both oriented colorings and colorings of the underlying undirected graph. \section{Parameterized Results} A {\em parameterized problem} is a pair $(\Pi,\kappa)$, where $\Pi$ is a decision problem, ${\mathcal I}$ the set of all instances of $\Pi$ and $\kappa: \mathcal{I} \to \mathbb{N}$ a so-called {\em parameterization}. The idea of parameterized algorithms is to restrict the combinatorial explosion to a parameter $\kappa(I)$ that is expected to be small for all inputs $I\in \mathcal{I}$. An algorithm $A$ is an {\em FPT-algorithm with respect to $\kappa$}, if there is a computable function $f: \mathbb{N} \to \mathbb{N}$ such that for every instance $I\in {\mathcal I}$ the running time of $A$ on $I$ is at most $f(\kappa(I))\cdot |I|^{\text{$\mathcal O$}(1)}$ or equivalently at most $f(\kappa(I))+ |I|^{\text{$\mathcal O$}(1)}$. If there is an fpt-algorithm with respect to $\kappa$ that decides $\Pi$ then $\Pi$ is called {\em fixed-parameter tractable}. FPT is the class of all parameterized problems which can be solved by an FPT-algorithm. An algorithm $A$ is an {\em XP-algorithm with respect to $\kappa$}, if there are two computable functions $f,g: \mathbb{N} \to \mathbb{N}$ such that for every instance $I\in {\mathcal I}$ the running time of $A$ on $I$ is at most $f(\kappa(I))\cdot |I|^{g(\kappa(I))}$. If there is an xp-algorithm with respect to $\kappa$ which decides $\Pi$ then $\Pi$ is called {\em slicewise polynomial}. XP is the class of all parameterized problems which can be solved by an XP-algorithm. In order to show fixed-parameter intractability, it is useful to show the hardness with respect to one of the classes $\w[t]$ for some $t\geq 1$, which were introduced by Downey and Fellows \cite{DF95} in terms of weighted satisfiability problems on classes of circuits. The relations $$\fpt \subseteq \w[1] \subseteq \w[2] \subseteq \ldots \subseteq \xp$$ are called {\em $\w$-hierarchy} and all inclusions are assumed to be strict. In the case of hardness with respect to some parameter $\kappa$ a natural question is whether the problem remains hard for {\em combined} parameters, i.e.\ parameters $(\kappa_1,\ldots,\kappa_r)$ that consists of $r\geq 2$ parts of the input. The given notations can be carried over to combined parameters, e.g.\ an FPT-algorithm with respect to $(\kappa_1,\ldots,\kappa_r)$ is an algorithm of running time $f(\kappa_1(I),\ldots,\kappa_r(I))\cdot |I|^{\text{$\mathcal O$}(1)}$ for some computable function $f: \mathbb{N}^r \to \mathbb{N}$ depending only on $\kappa_1,\ldots,\kappa_r$. For several problems the {\em standard parameter}, i.e.\ the threshold value given in the instance, is very useful. Unfortunately, for the Oriented Chromatic Number problem the standard parameter is the number of necessary colors and does even not allow an $\xp$-algorithm, since $\OCN_{4}$ is NP-complete \cite{KMG04}. \begin{corollary}\label{cor-xp-r} The Oriented Chromatic Number problem is not in $\xp$ when parameterized by $r$, unless $\p=\np$. \end{corollary} \medskip From an algorithmic point of view so called so-called {\em structural parameters}, which are measuring the difficulty of decomposing a graph into a special tree-structure, are interesting. For undirected graphs the clique-width \cite{CO00} and tree-width \cite{RS86} are the most important structural parameters. Clique-width is more general than tree-width since graphs of bounded tree-width have also bounded clique-width~\cite{CR05}. Conversely, the tree-width can only be bounded by the clique-width under certain conditions \cite{GW00}. A lot of NP-hard graph problems admit poly\-nomial-time solutions when restricted to graphs of bounded tree-width or graphs of bounded clique-width \cite{EGW01a}. For directed graphs there are several attempts to generalize tree-width such as directed tree-width, directed path-width, DAG-width, or Kelly-width, which are representative for what people are working on, see the surveys \cite{GHKLOR14,GHKMORS16}. Unfortunately, none of these attempts allows polynomial-time algorithms for a large class of problems on digraphs of bounded width. This also holds for $\OCN_r$ and $\OCN$ by the following section. \subsection{Parameterization by directed tree-width and related parameters} As mentioned above, for $r\geq 4$ the $r$-Oriented Chromatic Number problem is hard on DAGs \cite{CD06}. An $\xp$-algorithm with respect to parameters directed tree-width, directed path-width, Kelly-width, and DAG-width of the input digraph would imply a polynomial time algorithm for every fixed parameter, but even for parameter values $0$ or $1$ the problems are NP-hard, since DAGs have width $0$ or $1$ for these parameters. \begin{corollary}\label{cor-xp-ro} The Oriented Chromatic Number problem and for every positive integer $r\geq 4$ the $r$-Oriented Chromatic Number problem is not in $\xp$ when parameterized by directed tree-width, directed path-width, Kelly-width, or DAG-width, unless $\p=\np$. \end{corollary} \subsection{Parameterization by directed clique-width} By \cite{GHKLOR14}, directed clique-width performs much better than directed path-width, directed tree-width, DAG-width, and Kelly-width from the parameterized complexity point of view. Hence, we consider the parameterized complexity of $\OCN$ parameterized by directed clique-width. The directed clique-width of digraphs has been defined by Courcelle and Olariu \cite{CO00} as follows. \begin{definition}[Directed clique-width]\label{D4} The {\em directed clique-width} of a digraph $G$, $\dcws(G)$ for short, is the minimum number of labels needed to define $G$ using the following four operations: \begin{enumerate} \item Creation of a new vertex $v$ with label $a$ (denoted by $a(v)$). \item Disjoint union of two labeled digraphs $G$ and $H$ (denoted by $G\oplus H$). \item Inserting an arc from every vertex with label $a$ to every vertex with label $b$ ($a\neq b$, denoted by $\alpha_{a,b}$). \item Change label $a$ into label $b$ (denoted by $\rho_{a\to b}$). \end{enumerate} An expression $X$ built with the operations defined above using $k$ labels is called a {\em directed clique-width $k$-expression}. Let $\g(X)$ be the digraph defined by $k$-expression $X$. \end{definition} A class of graphs $\mathcal{L}$ has {\em bounded directed clique-width} if there is some integer $k$ such that every graph in $\mathcal {L}$ has directed clique-width at most $k$. \begin{proposition}\label{prop-cw-oc} Every oriented co-graph has directed clique-width at most 2. \end{proposition} \begin{proof} We next show how to give a directed clique-width $2$-expression for some oriented co-graph $G$ given by some di-co-expression. The vertices of $G$ are labeled by $a$ and within an order composition we relabel the vertices of one involved subgraph to $b$. The following three expressions recursively allow to give a directed clique-width $2$-expression $X$ for $G$. \begin{itemize} \item If $G=(\{v\},\emptyset)$, then $X=a(v)$. \item If $G=G_1 \oplus G_2$, such that $G_1$ and $G_2$ are defined by a directed clique-width $2$-expression $X_1$ and $X_2$, respectively, then we obtain a directed clique-width $2$-expression for $G$ by $X=X_1 \oplus X_2$. \item If $G=G_1 \oslash G_2$, such that $G_1$ and $G_2$ are defined by a directed clique-width $2$-expression $X_1$ and $X_2$, respectively, then we obtain a directed clique-width $2$-expression for $G$ by $$X= \rho_{b\to a}(\alpha_{a,b}(X_1 \oplus \rho_{a\to b}(X_2))).$$ \end{itemize} This shows the statements of the proposition. \end{proof} In \cite{GKR21b} the set of oriented co-graphs is characterized by excluding cycles of length 2, i.e.\ $\overleftrightarrow{P_2}=(\{u,v\},\{(u,v),(v,u)\})$ as a proper subset of the set of all graphs of directed clique-width 2, while for the undirected versions both classes are equal~\cite{CO00}. \begin{proposition}\label{prop-cw-msp} Every msp-digraph has directed clique-width at most $7$. \end{proposition} \begin{proof} We next show how to give a directed clique-width $7$-expression for some msp-digraph $G$ given by some msp-expression. Therefore we use the following four labels for the vertices of $G$: \begin{itemize} \item All vertices which are source and sink in $G$ are labeled by $a$. \item All vertices which are sink (and no source) in $G$ are labeled by $b$. \item All vertices which are source (and no sink) in $G$ are labeled by $c$. \item All vertices which are no sink and no source in $G$ are labeled by $d$. \end{itemize} Furthermore, we use three auxiliary labels $a',b',c'$ to distinguish the sinks and sources of the two combined graphs within a series composition. The following three expressions recursively allow to give a directed clique-width $7$-expression $X$ for $G$. \begin{itemize} \item If $G=(\{v\},\emptyset)$, then $X=a(v)$. \item If $G=G_1 \cup G_2$, such that $G_1$ and $G_2$ are defined by a directed clique-width $7$-expression $X_1$ and $X_2$, respectively, then we obtain a directed clique-width $7$-expression for $G$ by $X=X_1 \oplus X_2$. \item If $G=G_1 \times G_2$, such that $G_1$ and $G_2$ are defined by a directed clique-width $7$-expression $X_1$ and $X_2$, respectively, then we obtain a directed clique-width $7$-expression for $G$ by $$X= \rho''(\alpha'(X_1 \oplus \rho'(X_2))),$$ where $\rho'(X')=\rho_{c\to c'}(\rho_{b\to b'}(\rho_{a\to a'}(X')))$ relabels all sources and sinks in $\g(X')$ to the corresponding auxiliary labels, $\rho''(X')=\rho_{c'\to c}(\rho_{b'\to b}(\rho_{a'\to a}(X')))$ relabels all sources and sinks in $\g(X')$ to the corresponding original labels, and $$\alpha'(X'\oplus X'')=\alpha_{a,a'}(\alpha_{a,b'}(\alpha_{c,a'}(\alpha_{c,b'}(X' \oplus X''))))$$ inserts the edges of the series composition between $\g(X')$ and $\g(X'')$. \end{itemize} This shows the statements of the proposition. \end{proof} By the given definition every graph of directed clique-width at most $k$ can be represented by a tree structure, denoted as {\em $k$-expression-tree}. The leaves of the $k$-expression-tree represent the vertices of the digraph and the inner nodes of the $k$-expression-tree correspond to the operations applied to the subexpressions defined by the subtrees. Using the $k$-expression-tree many hard problems have been shown to be solvable in polynomial time when restricted to graphs of bounded directed clique-width \cite{GWY16,GHKLOR14}. In order to show fixed parameter tractability for $\OCN_{r}$ w.r.t.\ the parameter directed clique-width one can use its defineability within monadic second order logic (MSO). We restrict to $\text{MSO}_1$-logic, which allows propositional logic, variables for vertices and vertex sets of digraphs, the predicate $\arc(u,v)$ for arcs of digraphs, and quantifications over vertices and vertex sets \cite{CE12}. For defining optimization problems we use the $\text{LinEMSO}_1$ framework given in~\cite{CMR00}. The following theorem is from \cite[Theorem 4.2]{GHKLOR14}. \begin{theorem}[\cite{GHKLOR14}]\label{th-ghk} For every integer $k$ and $\text{MSO}_1$ formula $\psi$, every $\psi$-$\text{LinEMSO}_1$ optimization problem is fixed-parameter tractable on digraphs of clique-width $k$, with the parameters $k$ and $|\psi|$. \end{theorem} In \cite[Proposition 4.19]{GHKLOR14} the following monadic second order logic formula $\psi$ for $\OCN_{r}$ is given. \begin{remark}Let $G=(V,E)$ be an oriented graph. We can define $\OCN_{r}$ by the $\text{MSO}_1$ formula $$\begin{array}{lcl} \psi&=&\exists V_1,\ldots, V_r: \\ &&\displaystyle\left( \bigwedge_{i=1,\ldots,r} \forall x,y\in V_i \left(\neg \arc(x,y)\right) \wedge \bigwedge_{i,j=1,\ldots,r} \forall x,y \in X_i, z,t\in X_j \left(\arc(x,z) \rightarrow \neg \arc(t,y)\right) \right), \end{array} $$ where the vertices of set $V_i$ are mapped onto vertex $i$ of an orientation of a complete graph $K_r$. \end{remark} Since for the length of the given formula $\psi$ it holds $|\psi|\in \text{$\mathcal O$}(r)$ by Theorem \ref{th-ghk} we obtain the following result. \begin{corollary}\label{o} The Oriented Chromatic Number problem is fixed parameter tractable on digraphs of clique-width $k$, with the parameters $k$ and $r$. \end{corollary} By Corollary \ref{o} we know the existence of an fpt-algorithm for the Oriented Chromatic Number problem w.r.t.\ the combined parameter of directed clique-width and standard parameter. Next we give such an algorithm and estimate its running time. \medskip In order to compute the oriented chromatic number of some oriented graph $G$ of oriented chromatic number at most $r$ which is defined by some directed $k$-clique-width expression $X$ we extend our solution given for msp-digraphs in Section \ref{sec-cog}. Therefore, we recursively compute the set $F(X)$ of all labeled color graphs $H=(V,E)$ on $V=\{0,\ldots,r-1\}$. Every vertex $v\in V$ is labeled by set $L\subseteq \{1,\ldots,k\}$ consisting of all clique-width labels of the vertices in $G$ with color $c$. The number of vertex labeled, i.e., the vertices are distinguishable from each other, oriented graphs on $n$ vertices is $3^{\nicefrac{n(n-1)}{2}}$. Since for every $v\in V$ there at most $2^{k}-1$ label sets $L$, we can conclude $$|F(X)|\leq 3^{\nicefrac{r(r-1)}{2}} \cdot (2^{k}-1)^r \leq 3^{\nicefrac{r^2}{2}} \cdot 2^{k\cdot r} \leq 2^{r^2} \cdot 2^{k\cdot r} = 2^{r(r+k)} $$ which is independent on the size of $G$. For two color graphs $H_1=(V_1,E_1)$ and $H_2=(V_2,E_2)$ we define $H_1+H_2=(V,E)$ as follows. Vertex set $V$ is obtained from $V_1\cup V_2$ by merging the label sets of vertices for the same color. Formally, for every $0\leq c \leq r-1$ we consider all $(c,L')\in V_1\cup V_2$ and replace them by $(c,\cup_{(c,L')\in V_1\cup V_2}L' )$. Edge set $E$ is obtained by $E_1\cup E_2$. \begin{lemma}\label{le1cw} \begin{enumerate} \item For every $v\in V$ it holds $F(a(v))=\{(\{(i,\{a\})\},\emptyset ) \mid 0\leq i \leq r-1\}$. \item For every two $k$-expressions $X_1$ and $X_2$ we obtain set $F(X_1\oplus X_2)$ from sets $F(X_1)$ and $F(X_2)$ as follows. For every $H_1 \in F(X_1)$ and every $H_2 \in F(X_2)$ such that graph $H_1+H_2$ is oriented we put $H_1 + H_2$ into $F(X_1\oplus X_2)$. \item Set $F(\alpha_{a,b}(X))$ can be obtained from $F(X)$ as follows. First we remove from $F(X)$ all color graphs $(V,E)$ such that there is some $(c_1,L_1)\in V$ and some $(c_2,L_2)\in V$ such that $((c_2,L_2),(c_1,L_1))\in E$ and $a\in L_1$ and $b\in L_2$. Afterwards we modify every color graph $H=(V,E)$ in $F(X)$ as follows. If there is some $(c_1,L_1)\in V$ and some $(c_2,L_2)\in V$ such that $a\in L_1$ and $b\in L_2$ we insert $((c_1,L_1),(c_2,L_2))$ into $E$. The resulting set is $F(\alpha_{a,b}(X))$. \item It holds that $F(\rho_{a \to b}(X)) = \{\rho_{a \to b}((V,E)) \mid (V,E)\in F(X)\}$. Here we apply $\rho_{a \to b}((V,E))=(\rho_{a \to b}(V),\rho_{a \to b}(E))$, $\rho_{a \to b}(V)=\{(c,\rho_{a \to b}(L))\mid(c,L)\in V\}$, $\rho_{a \to b}(L)=\{\rho_{a \to b}(x)\mid x\in L \}$, $\rho_{a \to b}(x)=b$, if $x=a$ and $\rho_{a \to b}(x)=x$, if $x\neq a$, $\rho_{a \to b}(E)=\{((c_1,\rho_{a \to b}(L_1)),(c_2,\rho_{a \to b}(L_2))) \mid ((c_1,L_1),(c_2,L_2))\}$. \end{enumerate} \end{lemma} \begin{proof} \begin{enumerate} \item Set $F(a(v))$ includes obviously all possible solutions to color every vertex on its own with the $r$ given colors. \item Similar to (\ref{le1-2}.) of Lemma \ref{le1}. \item Let $F'(X)$ be the set obtained from $F(X)$ using the modifications given in the lemma. We next show that $F'(X)=F(\alpha_{a,b}(X))$. Let $H=(V,E)\in F(X)$ be some color graph for $\g(X)$. The modifications of $H$ given in the lemma either remove $H$ or lead to a color graph for $\g(\alpha_{a,b}(X))$. Thus, it holds that $F'(X)\subseteq F(\alpha_{a,b}(X))$. Next, let $H=(V,E)\in F(\alpha_{a,b}(X))$ be some color graph for $\g(\alpha_{a,b}(X))$. Then we can assume, that there is some $(c_1,L_1)\in V$ and some $(c_2,L_2)\in V$ such that $a\in L_1$ and $b\in L_2$ and $((c_1,L_1),(c_2,L_2))\in E$. Since the edge insertion $\alpha_{a,b}$ does not change the vertices or clique-width labels there is some color graph $H'=(V,E')\in F(X)$ for $\g(X)$ such that $H'$ is a subgraph of $H$ in which the edge $((c_1,L_1),(c_2,L_2))$ can be missing. The modifications of $H'$ given in the lemma insert the edge $((c_1,L_1),(c_2,L_2))$ into $H'$ which implies that $H$ is in $F'(X)$. \item The relabeling $\alpha_{a,b}(X)$ can change the labels but not the structure of $\g(X)$. Thus, for every color graph $H=(V,E)$ in $F(X)$ for every vertex $(c,L)\in V$ we have to change the set $L\subseteq \{1,\ldots,k\}$ of clique-width labels according to the performed relabeling operation $\rho_{a \to b}$, which is done by the rules given in the lemma. \end{enumerate} This shows the statements of the lemma. \end{proof} Since every possible coloring of $G$ is part of the set $F(X)$, where $X$ is a directed $k$-clique-width expression for $G$, it is possible to find a minimum coloring for $G$. \begin{corollary} \label{cor3} Let $G=(V,E)$ be an oriented graph given by a directed clique-width $k$-expression $X$. There is an oriented $r$-coloring for $G$ if and only if there is some $H \in F(X)$ which has $r$ vertices. Therefore, $\chi_o(G)=\min\{|V| \mid (V,E)\in F(X)\}$. \end{corollary} \begin{theorem}\label{xp-dca} The Oriented Chromatic Number problem on digraphs on $n$ vertices given by a directed clique-width $k$-expression can be solved in $\text{$\mathcal O$}(n\cdot k^2 \cdot 2^{r(r+k)})$ time. \end{theorem} \begin{proof} Let $G=(V,E)$ be a digraph of directed clique-width at most $k$ and $T$ be a $k$-expression-tree for $G$ with root $w$. For some vertex $u$ of $T$ we denote by $T_u$ the subtree rooted at $u$ and $X_u$ the $k$-expression defined by $T_u$. In order to solve the Oriented Chromatic Number problem for $G$, we traverse $k$-expression-tree $T$ in a bottom-up order. For every vertex $u$ of $T$ we compute $F(X_u)$ following the rules given in Lemma \ref{le1cw}. By Corollary \ref{cor3} we can solve our problem by $F(X_w)=F(X)$. Our rules given Lemma \ref{le1cw} show the following running times. For every $v\in V$ and $a\in\{1,\ldots,k\}$ set $F(a(v))$ can be computed in $\text{$\mathcal O$}(1)$. The set $F(X \oplus Y)$ can be computed in time $\text{$\mathcal O$}(2^{r(r+k)})$ from $F(X)$ and $F(Y)$. The sets $F(\alpha_{a,b}(X))$ and $F(\rho_{a \to b}(X))$ can be computed in time $\text{$\mathcal O$}(2^{r(r+k)})$ from $F(X)$. In order to bound the number and order of operations within directed clique-width expressions, we can use the normal form for clique-width expressions defined in \cite{EGW03}. The proof of Theorem 4.2 in \cite{EGW03} shows that also for directed clique-width expression $X$, we can assume that for every subexpression, after a disjoint union operation first there is a sequence of edge insertion operations followed by a sequence of relabeling operations, i.e. between two disjoint union operations there is no relabeling before an edge insertion. Since there are $n$ leaves in $T$, we have $n-1$ disjoint union operations, at most $(n-1)\cdot (k-1)$ relabeling operations, and at most $(n-1)\cdot \nicefrac{k(k-1)}{2}$ edge insertion operations. This leads to an overall running time of $\text{$\mathcal O$}(n\cdot k^2 \cdot 2^{r(r+k)})$. \end{proof} Up to now there are only very few digraph classes for which we can compute a directed clique-width expression in polynomial time. This holds for directed co-graphs, digraphs of bounded directed modular width, orientations of trees, and directed cactus forests. For such classes we can apply the result of Theorem \ref{xp-dca}. In order to find directed clique-width expressions for general digraphs one can use results on the related parameter bi-rank-width \cite{KR13}. By \cite[Lemma 9.9.12]{BG18} we can use approximate directed clique-width expressions obtained from rank-decomposition with the drawback of a single-exponential blow-up on the parameter. If we restrict to some constant value of $r$ the running time of Theorem \ref{xp-dca} leads to the following result, which reproves a result shown in \cite{GHKLOR14} using Theorem \ref{th-ghk}. \begin{corollary}\label{cor11ax} For every integer $r$ the $r$-Oriented Chromatic Number problem is in $\fpt$ when parameterized by directed clique-width. \end{corollary} \begin{corollary}\label{ocnk2bb} For every class of graphs of bounded directed clique-width and every positive integer $r$ the $r$-Oriented Chromatic Number problem can be solved in polynomial time. \end{corollary} By Propositions \ref{prop-cw-oc} and \ref{prop-cw-msp} we know that on oriented co-graphs and msp-digraphs for every positive integer $r$ the $r$-Oriented Chromatic Number problem can be solved in polynomial time. Comparing this to Corollary \ref{c-co} and Theorem \ref{algspdd} we have shown more general solutions before. If we restrict to some constant value of $k$ the running time of Theorem \ref{xp-dca} leads to the following result. \begin{corollary}\label{cor11ay} For every class of graphs of bounded directed clique-width the Oriented Chromatic Number problem is in $\fpt$ when parameterized by $r$. \end{corollary} \subsection{Parameterization by directed modular-width} Next, we discuss parameterization of $\OCN$ and $\OCN_r$ w.r.t. the parameter directed modular-width ($\dmws$). Directed modular-width was introduced and applied to the Acyclic Chromatic Number problem in \cite{SW19,SW20}. A class of graphs $\mathcal{L}$ has {\em bounded directed modular-width} if there is some integer $k$ such that every graph in $\mathcal {L}$ has directed modular-width at most $k$. \begin{proposition}[\cite{SW20}]\label{prop-co-mw} Every (directed and thus every) oriented co-graph has directed modular width $2$. \end{proposition} The relation of directed clique-width and directed modular width is as follows. \begin{lemma}[\cite{SW20}]\label{le-mw-cw} For every digraph $G$ it holds that $\dcws(G)\leq \dmws(G)$. \end{lemma} On the other hand, there exist several classes of digraphs of bounded directed clique-width and unbounded directed modular width, e.g. the set of all oriented paths $\{\overrightarrow{P_n} \mid n\geq 1\}$, the set of all oriented cycles $\{\overrightarrow{C_n} \mid n\geq 1\}$, and the set of all msp-digraphs. An advantage of directed modular width is that it can be computed efficiently \cite{SW19,SW20}. Lemma \ref{le-mw-cw} and Corollary \ref{cor11ax} lead to the following result. \begin{corollary}\label{ocnk2} For every positive integer $r$ the $r$-Oriented Chromatic Number problem is fixed parameter tractable w.r.t.\ the parameter directed modular-width. \end{corollary} \begin{corollary}\label{ocnk2a} For every class of graphs of bounded directed modular-width and every positive integer $r$ the $r$-Oriented Chromatic Number problem can be solved in polynomial time. \end{corollary} By Proposition \ref{prop-co-mw} we conclude that on oriented co-graphs for every positive integer $r$ the $r$-Oriented Chromatic Number problem can be solved in polynomial time. Comparing this to Corollary \ref{c-co} we have shown a more general solution before. \subsection{Parameterization by the width of the underlying undirected graph} In \cite{Gan09}, Ganian has shown an $\fpt$-algorithm for $\OCN$ w.r.t.\ the parameter tree-width (of the underlying undirected graph). Further, he has shown that $\OCN$ is DET-hard for classes of oriented graphs, such that the underlying undirected class has bounded rank-width. \subsection{Parameterization by number of vertices} A positive result can be obtained for parameter ''number of vertices'' $n$. Since integer linear programming is fixed-parameter tractable for the parameter ''number of variables'' \cite{Len83} the following binary integer program for $\OCN$ using $n^2+n$ variables implies an $\fpt$-algorithm for parameter $n$. \begin{remark}\label{rem-bip} To formulate the Oriented Chromatic Number problem for some oriented graph $G=(V,E)$ as a binary integer program, we introduce a binary variable $y_j\in\{0,1\}$, $j\in \{1,\ldots, n\}$, such that $y_j=1$ if and only if color $j$ is used. Further we use $n^2$ variables $x_{i,j}\in\{0,1\}$, $i,j\in \{1,\ldots,n\}$, such that $x_{i,j}=1$ if and only if vertex $v_i$ receives color $j$. The main idea is to ensure the two conditions of Definition \ref{def-oc} within (\ref{01-p3-cn0x}) and (\ref{01-px}). \begin{eqnarray} \text{minimize} \sum_{i=1}^{n} y_i \label{01-p1-cn0a} \end{eqnarray} subject to \begin{eqnarray} \sum_{j=1}^{n} x_{i,j} & = &1 \text{ for every } i \in \{1,\ldots,n\} \label{01-p2-cn0a} \\ x_{i,j} + x_{i',j}& \leq &y_j \text{ for every } (v_i,v_{i'})\in E, j \in \{1,\ldots,n\} \label{01-p3-cn0x} \\ \bigvee_{j=1}^n x_{i,j} \wedge x_{i''',j} & \leq & 1- \bigvee_{j=1}^n x_{i',j} \wedge x_{i'',j} \text{ for every } (v_i,v_{i'}),(v_{i''},v_{i'''})\in E \label{01-px} \\ y_j &\in & \{0,1\} \text{ for every } j \in \{1,\ldots,n\} \label{01-p4-cna0a} \\ x_{i.j} &\in & \{0,1\} \text{ for every } i,j \in \{1,\ldots,n\} \label{01-p5-cn0a} \end{eqnarray} Equations in (\ref{01-px}) are not in propositional logic. In order to reformulate them for binary integer programming, one can use the results of \cite{Gur14}. \end{remark} \begin{corollary}\label{xp-n} The Oriented Chromatic Number problem is in $\fpt$ when parameterized by the number of vertices $n$. \end{corollary} \section{Conclusions and outlook} In this paper we considered oriented colorings of recursively defined digraphs. We used the concept of perfect orderable graphs in order to show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring, which generalizes a known dynamic programming solution for the Oriented Chromatic Number problem on oriented co-graphs in \cite{GKR19d}. Further, we showed that every msp-digraph has oriented chromatic number at most $7$, which is best possible. We applied this bound together with the recursive structure of msp-digraphs to give a linear time solution for computing the oriented chromatic number of msp-digraphs. In Figure \ref{grcl} we summarize the relation of special graph classes considered in this work. Among these are directed acyclic graphs (DAGs), transitive directed acyclic graphs (transitive DAGs), series-parallel digraphs ($\SPD$), minimal series-parallel digraphs ($\MSP$), series-parallel order graphs ($\SPO$), and oriented co-graphs ($\OC$). The directed edges represent the existing relations between the graph classes, which follow by their definitions. For the relations to further graph classes we refer to \cite[Figure 11.1]{BG18}. \begin{figure}[ht] \begin{center} \centerline{\includegraphics[width=0.37\textwidth]{classes1.pdf}} \caption{The figure shows the inclusions of special graph classes. A directed edge from class $A$ to class $B$ indicates that $B\subseteq A$. Two classes $A$ and $B$ are incomparable if there is neither a directed path from $A$ to $B$, nor a directed path from $B$ to $A$.}\label{grcl} \end{center} \end{figure} Furthermore we considered the parameterized complexity of the Oriented Chromatic Number problem by structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure. The parameterized results of Corollary \ref{o} and Corollary \ref{cor11ax} also hold for any parameter which is larger or equal than directed clique-width directed linear clique-width \cite{GR19c}. Furthermore, restricted to semicomplete digraphs the shown parameterized solutions also hold for directed path-width \cite[Lemma 2.14]{FP19}. For future work it could be interesting to extend our solutions to further classes such as edge series-parallel digraphs \cite{VTL82}. The parameterized complexity of $\OCN$ w.r.t.\ the parameter directed clique-width remains open. Since the directed clique-width of a digraph is always greater or equal the undirected clique-width of the corresponding underlying undirected graph \cite{GWY16}, the result of Ganian \cite{Gan09} does not imply a hardness result. Furthermore, the parameterized complexity of $\OCN$ w.r.t.\ structural parameter directed modular-width and of $\OCN_r$ w.r.t.\ parameter rank-width of $\un(G)$ remains open (cf.\ Table \ref{fpt-sum}). \section*{Acknowledgements} \label{sec-a} The work of the second and third author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- 388221852 \newcommand{\etalchar}[1]{$^{#1}$}

proofpile-arXiv_069-13314

\section{Introduction} According to the current most reliable knowledge, the final evolutionary state of main-sequence stars leads to ultra-dense stellar remnants which are commonly referred to as compact objects. Depending on the star's initial mass, the stellar object can evolve under a self-gravitational collapse process into a white dwarf, a neutron star, or a black hole. As black holes provide no direct electromagnetic radiation (besides their shadows \cite{ab1,aki}), they are rather difficult objects to be used in order to test and compare gravitational theories, where only gravitational waves provide reliable information. On the contrary, neutron stars, can be observed not only via gravitational waves \cite{ab2} but also through electromagnetic radiation \cite{hew1,hew2}, allowing to constrain models of the nuclear matter at ultra-high densities as well as our models of the gravitational interaction. A typical neutron star has a mass of $1.5$ solar masses, and a radius around $10$ kilometers, making it the smallest and densest stars known so far. Their structural properties can be roughly understood by assuming that they are composed of neutrons supported by their degeneracy pressure (Pauli exclusion principle), which makes them particularly suitable to test our theories of gravity and matter at the highest densities achievable in the universe. Indeed, the actual matter composition at densities above the saturation density $\sim2.8\times10^{17}\frac{\text{kg}}{\text{m}^3}$ at the star's center is still unknown, which is a crucial element to modeling such objects. The recent observation of gravitational waves and electromagnetic radiation of a neutron star merger, and its ability to rule out certain families of modified gravity theories \cite{ez,bor,san,ab3,jana} is a clear example of their relevance for current research. Despite this success, there is still a long way to go in order to extract useful information about the internal structure, dynamics and composition of neutron stars. This fact introduces a fundamental degeneracy, since we cannot distinguish whether the differences in the mass-radius profiles are caused either by the equation of state assumed or from the theory of gravity from which the hydrostatic equilibrium equation was obtained. White dwarf stars \cite{shap,koe} are also the final state of the stellar evolution but for main sequence stars whose masses lie below $8-10$ solar masses. When the stage of hydrogen-fusing of a main-sequence stars ends, it leads into an expansion to a red giant star. Subsequently, helium is fused to carbon and oxygen, which are building up at the star's center. However, the core temperature is not enough to further fuse carbon (the temperature depends on the red giant's mass) and thus the outer layers, composed of lighter elements, will be ejected under the form of a planetary nebula, leaving behind a carbon-oxygen core. Since there is no further fusion reaction in a new-born white dwarf providing energy to support the star against gravitational collapse, the star is supported by electron degeneracy pressure only. After the planetary nebula is formed, the mass of a white dwarf becomes around $1$ solar mass packed into an Earth-size volume, making up densities around $10^9 \frac{\text{kg}}{\text{m}^3}$ , and including it into the family of compact stars. The same intense radiation that makes the planetary nebula form will cool down the white dwarf causing the core material to eventually crystallize. The cooling process of such an object is very long. In order to become a cold black dwarf (a star which does not emit any heat or light and therefore would be undetectable by optical observations), the white dwarf would need more time than the current age of the Universe. All these elements make white dwarfs important objects to study since the cooling process provides information on the star's formation in the Milky Way. There are two main problems related to the physics of white dwarfs: their age (related to the very low-mass) and super-Chandrasekhar masses, which are hard to explain by current astrophysical models. It indicates that we are far from understanding these objects; it may also mean that there are processes in the stellar evolution which we do not take into account. The very curious case is a discovery of a very low mass white dwarf with 0.2 solar masses in the binary system KIC 8145411 \cite{masuda}. According to the commonly accepted models \cite{la}, such a white dwarf would have to evolve from a very small progenitor star which had had to be older than the Universe ($13.6$ billion years) in order to run out of its fuel and then to follow the known scenario of turning into a white dwarf. The age of an observed white dwarf is roughly speaking the sum of the lifetime of the main sequence progenitor star and of the white dwarf cooling time. The lifetime of the main sequence star is proportional to $\tau\sim M^{-3}$ , where $M$ is a main sequence star's mass. Thus, the less massive the star, the longer time needed to turn it into a white dwarf. Usually such low-mass white dwarfs are observed in binary systems, in which their companions steal the outer layers, speeding the process of the exposition of the core. But in the considered binary KIC 8145411 the orbit of the second companion, being a Sun-like star, is far wider than the required one to produce such a low-mass white dwarf via their interactions. There could have existed a third star which had changed the orbits of the binary, making the cannibalism process possible, when the white dwarf progenitor was a red giant - but then the current orbit would have not been nearly circular as observed. Another explanation of its low mass could be the existence of large planets in the past which could have siphoned the gas from the star's surface. However, none of those scenarios is satisfactory, especially in the light of other single low-mass white dwarfs whose companions have not been yet detected, assuming that they exist. Another ambiguity related to the age of these compact objects are discoveries of white dwarfs being older than their host cluster, assuming that the cluster's stars had come into being at the same time. This way, there are white dwarfs such as for example WD 0346+246 \cite{ham}, SDSS J091709.55+463821.8 \cite{kilic} (which is a binary system of two low-mass white dwarfs \cite{kilic2}) that are very old - around 11-12 billion years. Usually they are very faint (as in the case of one of the members of J091709.55+463821.8, which was not detected optically neither via X-ray and radiopulsar observations \cite{agu}, which is the reason why the companion should be a white dwarf) and cold, with the effective temperatures very close to the limit $T=3000$K. This is a minimal temperature that this type of star can have in order not to be older than the Milky Way, assuming that those stars are already crystallized. It is accepted that the crystallization has to happen during the cooling period of a white dwarf - this is a process which appears as the fluid in the star cools and as a result, turns into a solid state. However, when fluid crystallizes it releases latent heat which must be also added in the total energy supply, being further radiated. This increases the cooling time. Although it was shown that the crystallization phenomena can shorten the cooling time \cite{mestel}, this process is still a source of uncertainty in the age of cold white dwarfs. The lifetime of a main sequence star as well as the cooling time of white dwarfs (taking into account the crystallization process) are quantities which depend on the luminosity and mass of the stars \cite{her}. It was shown \cite{sak1,sak2,gonzalo,aneta4,aneta5,davis,benito} that in the case of some gravitational theories both mass and luminosity change because they are sensitive to modifications of Einstein's gravity. Indeed, although white dwarfs can be treated by the Newtonian hydrostatic equilibrium equation, which is used in the computations of the lifetime of the main sequence stars and cooling processes, many theories of gravity modify that equation (see e.g. \cite{reva} and references therein). Moreover, it was also demonstrated that modified gravity has also an impact on the early stars' evolution, such us for example Hayashi tracks and radiative core development \cite{aneta4,chow,chang}, which clearly indicates that re-examination of the evolutionary processes in the framework of modified gravity can provide answers to the white dwarfs' age problem. Another important comment related to the star's age is that lithium abundance in stellar atmosphere depends on gravity model, too, introducing an additional uncertainty to age determination techniques of young stars and globular clusters since some of them rely on the lithium depletion. It was shown that in the case of Palatini $f(\mathcal{R})$ gravity the age of an individual star is significantly shorter than the age in the model provided by GR \cite{aneta5}. Therefore, when the total star's evolution is re-analyzed, it will very probably happen that the age of the KIC 8145411 white dwarf lies in the reasonable range. The intriguing issue also associated to the physics of white dwarfs is super luminous SNIas, e.g. \cite{howell,scalzo,hicken,yaman,tau,silv}. This suggests that the star progenitors of such events could be super-Chandrasekhar white dwarfs which exceeded the Chandrasekhar mass limit (in the Chandrasekhar's model, a white dwarf's mass cannot be more than $1.44M_\odot$ \cite{chandra}). A white dwarf, whose mass crosses this limit by siphoning a nearby star's mass, turns into a type Ia supernovea explosion. Even more curious, including the relativistic effects to the stellar equations turns out to reduce the maximum mass white dwarfs with magnetic fields \cite{bera,deh}, thus other explanations, as for example provided by modified gravity, are necessary. White dwarfs have already been studied in modified gravity models. In \cite{das1,das2} the authors were studying the $f(R)$ gravity effects on limiting mass (super-Chandrasekhar and sub-Chandrasekhar unified by the model) and its not-uniqueness, while in \cite{carv} the $f(R,T)$ model provided the WD macroscopic properties and well as lower energy density in comparison to other models. White dwarfs are also used to constrain theories of gravity, such as for example in \cite{gar1,nn1,alt,gar,cor,ben,biesiada,ben2,is,jain,sal,babi,kalita,cris,nn2,biesiada2,eslam}. Re-examined cooling and heating processes in modified gravity and dark matter models were studied for instance in \cite{is2,pano,hor}. In this work we will briefly introduce Palatini $f(\mathcal{R})$ gravity with the stellar equations needed to study the Chandrasekhar's model of white dwarfs. We will solve modified Lane-Emden equation for the polytropic parameter $n=3$ in order to discuss the Chandrasekhar mass in this framework. We use the $(-+++)$ metric signature while $\kappa^2=\frac{8\pi G}{c^4}$. \section{Palatini $f(\mathcal{R})$ gravity} Palatini $f(\mathcal{R})$ gravity is one of the simplest generalization of the General Relativity: the last one can be also considered in Palatini approach. It means that the assumption on the metricity of the connection is waived and therefore one deals with two independent objects, metric $g$ and connection $\hat\Gamma$. However, in the case of the linear functional $f(\mathcal{R})$, the field equations taken with respect to the independent connection $\hat\Gamma$ provide that the connection is the Levi-Civita connection of the metric $g$. Considering more general Lagrangians this is not true anymore: as a result of the field equations the connection turns out to be a Levi-Civita connection of a certain metric which happens to be conformally related to the metric $g$. To demonstrate it, we start with the action \begin{equation} S=S_{\text{g}}+S_{\text{m}}=\frac{1}{2\kappa^2}\int \sqrt{-g}f(\mathcal{R}) d^4 x+S_{\text{m}}[g_{\mu\nu},\psi_m],\label{action} \end{equation} where the curvature scalar defined as $\mathcal{R}=\mathcal{R}^{\mu\nu}g_{\mu\nu}$ is constructed with the metric $g$ and Ricci tensor $\mathcal{R}_{\mu\nu}$ built of the independent connection $\hat\Gamma$. The variation of (\ref{action}) with respect to the metric $g_{\mu\nu}$ gives \begin{equation} f'(\mathcal{R})\mathcal{R}_{\mu\nu}-\frac{1}{2}f(\mathcal{R})g_{\mu\nu}=\kappa^2 T_{\mu\nu},\label{structural} \end{equation} where $T_{\mu\nu}$ denotes the energy momentum tensor of the matter field, defined as usually by $T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g_{\mu\nu}}$. For now, the primes denote derivatives with respect to the function's argument, that is, $f'(\mathcal{R})=\frac{df(\mathcal{R})}{d\mathcal{R}}$. On the other hand, the variation with respect to the independent connection $\hat\Gamma$ provides \begin{equation} \hat{\nabla}_\beta(\sqrt{-g}f'(\mathcal{R})g^{\mu\nu})=0.\label{con} \end{equation} From above we immediately notice that $\hat{\nabla}_\beta$ is the covariant derivative calculated with respect to $\hat\Gamma$, that is, it is the Levi-Civita connection of the conformal metric \begin{equation}\label{met} h_{\mu\nu}=f'(\mathcal{R})g_{\mu\nu}. \end{equation} The structural equation is obtained from the trace of (\ref{structural}) taken with respect to the metric $g_{\mu\nu}$ \begin{equation} f'(\mathcal{R})\mathcal{R}-2 f(\mathcal{R})=\kappa^2 T,\label{struc} \end{equation} where $T=g_{\mu\nu}T^{\mu\nu}$ is the trace of the energy-momentum tensor. For particular functionals $f(\mathcal R)$ one may find a solution of the structural equation (\ref{struc}) in the form of the relation $\mathcal{R}=\mathcal{R}(T)$, allowing to express $f(\mathcal{R})$ as a function of the trace of the energy momentum tensor $T$. One may also express the fields equations as dynamical equations for the conformal metric $ h_{\mu\nu}$ \cite{DeFelice:2010aj,BSS,SSB} and the scalar field, which is algebraically related to the trace of the energy momentum tensor, defined as $\Phi=f'(\mathcal{R})$: \begin{subequations} \begin{align} \label{EOM_P1} \bar R_{\mu\nu} - \frac{1}{2} h_{\mu\nu} \bar R & =\kappa^2 \bar T_{\mu\nu}-{1\over 2} h_{\mu\nu} \bar U(\Phi) \end{align} \begin{align} \label{EOM_scalar_field_P1} \Phi\bar R & - (\Phi^2\,\bar U^(\Phi))^\prime = \end{align} \end{subequations} where the potential $\bar U(\Phi)=\frac{\mathcal{R}\Phi-f(\mathcal{R})}{\Phi^2}$ was introduced together with an appropriate energy momentum tensor $\bar T_{\mu\nu}=\Phi^{-1}T_{\mu\nu}$. This approach to the Palatini gravity significantly simplify studies on some particular physical problems, as shown in, e.g. \cite{menchon, afonso3, afonso4, sporea, aneta, aneta2, artur, gonzalo, afonso5}. It is also worth to comment that in the vacuum the Palatini gravity turns out to be Einstein vacuum solution with the cosmological constant. It does not depend on the $f(\mathcal{R})$ form - it can be easily demonstrated from the structural equation (\ref{struc}). Besides, in the case of analytic functionals $f(\mathcal{R})$ the center-of-mass orbits are the same as in GR \cite{junior} while the modifications of energy and momentum which appear in Euler equation are not sensitive to the experiments performed for the solar system orbits. That can change when atomic level experiments are available \cite{sch,ol1,ol2}. Let us stress that Palatini $f(\mathcal{R})$ gravity differes significantly from the metric approach in which the field equations result as the $4$th order differencial equations for the metric tensor, providing different cosmological and astrophysical scenarios (see eg. \cite{cap0}). However, the extra degree of freedom related to the curvature scalar is present when the equations are transformed to the scalar-tensor representation of the metric theory, providing the 2nd order differential equations for the metric tensor. It clearly demonstrates the difference between the metric and Palatini approach since in the last one there is no extra degree of freedom. \subsection{Non-relativistic stars for quadratic Palatini gravity} Before going to the stellar equations, let us briefly comment that the stellar structure in the metric-affine theory (for detailed review see \cite{reva}) was studied mainly for spherical-symmetric solutions and mass-radius relation \cite{mr1,mr2,mr3, mr4, mr5,mr7,mr8,mr9,mr10}, the last one provided by the modified Tolman-Oppenheimer-Volkoff equations (TOV). Some shortcomings and their solutions were considered further in \cite{ba1,ba2,ba3,pani,ba4,ba5,gonzalo2,kim,b6,gd} while problems addressed to stability can be viewed in \cite{aneta,stab1,stab2,stab3,stab4,stab5}. Non-relativistic stars, which are here our main concern, were discussed in \cite{aneta2,gonzalo,artur,nn1,nn2,aneta3,aneta4,aneta5,benito}. The relativistic hydrostatic equilibrium equations (TOV) for arbitrary Palatini $f(\mathcal{R})$ gravity are given in \cite{aneta} in the form \begin{eqnarray} \frac{d}{d\tilde r}\left(\frac{\Pi}{\Phi({\tilde r})^2}\right)&=&-\frac{GA\mathcal{M}}{\tilde r^2}\left(\frac{Q+\Pi}{\Phi({\tilde r})^2}\right) \left(1+\frac{4\pi \tilde r^3\frac{\Pi}{\Phi({\tilde r})^2}}{\mathcal{M}}\right)\label{tov_kon}\\ A&=&1-\frac{2G \mathcal{M}(\tilde r)}{\tilde r} \\ \mathcal{M}(\tilde r)&=& \int^{\tilde r}_0 4\pi \tilde{x}^2\frac{Q(\tilde{x})}{\Phi(\tilde{x})^2} d\tilde{x} \ ,\label{mass} \end{eqnarray} where the tilde indicates the quantities in conformal frame, and the generalized energy density $Q$ and pressure $\Pi$ are defined as \begin{subequations}\label{defq} \begin{equation} \bar{Q}=\bar{\rho}-\frac{\bar{U}}{2\kappa^2 c^2}=\frac{\rho}{\Phi^2}-\frac{U}{2\kappa^2 c^2\Phi^2}=\frac{Q}{\Phi^2} \ , \end{equation} \begin{equation} \bar{\Pi}=\bar{p}+\frac{\bar{U}}{2\kappa^2}=\frac{p}{\Phi^2}+\frac{U}{2\kappa^2\Phi^2}=\frac{\Pi}{\Phi^2} \ . \end{equation} \end{subequations} Let us recall that $\bar{U}$ and $\Phi$ depend on the choice of the gravitational model one is interested in. It is readily seen that, in the GR limit, $\Phi=1,\bar{U}=0$ and one recovers the standard TOV equations. We will now consider the quadratic model \begin{equation}\label{quad} f(\mathcal{R})=\mathcal{R}+\beta \mathcal{R}^2; \end{equation} then $\Phi=f'(\mathcal{R})=1+2\beta \mathcal{R}$. Using (\ref{structural}), one easily finds $\mathcal{R}=-\kappa^2 T$ such that $\Phi=1-2\kappa^2\beta T$. Then, the above equations in the Newtonian limit, that is, $p<<\rho$ together with $4\pi r^3 p<< \mathcal{M}$ and $\frac{2G\mathcal{M}}{r}<<1$, in the Einstein frame are \cite{aneta2} \begin{align} \frac{dp}{d\tilde r}&=-\frac{G\mathcal{M}\rho}{\Phi\tilde r^2} \ ,\\ \mathcal{M}&=\int_0^{\tilde r}4\pi x^2\rho(x)dx \ , \end{align} where $\tilde r^2=\Phi r^2$ and $\Phi=1+2\kappa^2c^2\beta\rho$. Coming back to the Jordan frame we have \begin{align} \frac{\Phi^\frac{1}{2}}{1+\frac{1}{2}r\frac{\Phi'}{\Phi}}\frac{dp}{dr}&=-\frac{G\mathcal{M}(r)\rho}{\Phi^2 r^2} \label{equi} ,\\ \frac{\Phi^\frac{1}{2}}{1+\frac{1}{2}r\frac{\Phi'}{\Phi}}\frac{d\mathcal{M}(r)}{dr}&=4\pi \Phi r^2\rho(r) \ \end{align} which are the modified Newtonian hydrostatic equilibrium equations used to described the Newtonian stars. \subsection{Lane-Emden equation for quadratic Palatini gravity and Chandrasekhar mass} We will now obtain the Chandrasekhar mass as a function of solutions of the modified Lane-Emden equation. The Lane-Emden equation given in \cite{aneta2} was obtained from (\ref{equi}) after applying the polytropic equation of state \begin{equation} p=K\rho^\Gamma, \end{equation} where $K$ and $\Gamma=1+\frac{1}{n}$ are polytropic constants. For quadratic Palatini gravity (\ref{quad}) it has the following form: \begin{equation}\label{LEeq} \xi^2\theta^n\Phi+\displaystyle\frac{\Phi^{-1/2}}{1+\frac12 \xi\Phi_{\xi}/\Phi}\displaystyle\frac{d}{d\xi} \left(\displaystyle\frac{\xi^2\Phi^{3/2}}{1+\frac12 \xi\Phi_{\xi}/\Phi}\displaystyle\frac{d\theta}{d\xi} \right)=0 \end{equation} which for $n=3$ simplifies to \begin{align} 0=\frac{\theta^2\Big(2\alpha\theta(2+3\xi^2\theta^2)\theta'+6\alpha\xi\theta'^2-\xi\theta\Big)}{\xi} -\frac{2\theta'}{\xi}+(2\alpha\theta^3-1)\theta'' \end{align} Let us recall that for $n=3$ the mass and radius of a white dwarf can be expressed as functions of the solutions and their first zeros of the Lane-Emden equation: \begin{align} \mathcal{M}&=-4\pi\left(\frac{K}{\pi G}\right)^{\frac{3}{2}}\xi_R^2\theta'(\xi_R),\\ R&=\left(\frac{K}{\pi G}\right)^\frac{1}{2}\rho_c^{-\frac{1}{3}}\xi_R, \end{align} where $\xi_R$ is the first zero of $\theta(\xi)$. The function $\theta(\xi)$ is a solution of the Lane-Emden equation, $\rho_c$ the central density, $G$ the gravitational constant while $K=1.2435\times10^{15}\mu_e^{-4/3}$ cgs for $n=3$. Since the eventual differences in masses and in radii with respect to a theory of gravity are carried by $\xi_R$ and $\theta'(\xi_R)$, we may write down \begin{align} M&=0.7212475 M_\odot \left(\frac{2}{\mu_e}\right)^2 \xi^2_R |\theta'(\xi_R)|,\\ R&=0.48515\times10^4\left( \frac{10^6\text{g/cm}^{3} }{\rho_c} \right)^{1/3} \left(\frac{2}{\mu_e}\right)^{2/3}\xi_R \text{ km} \end{align} We will consider here the Chandrasekhar model, and thus the mean molecular weight per electron is $\mu_e=2$. In the table \ref{tab} we have obtained the radii and masses for a few values of the parameter $\alpha=\kappa^2 c^2 \beta\rho_c$, were $\beta$ is the Starobinsky parameter. Let us notice that $\alpha$ depends on the central density $\rho_c$ which is a feature of the metric-affine (Palatini) gravity. Because of the choice of the Starobisnky model, the conformal function for the polytropic EoS is expressed as $\Phi=1-2\alpha\theta^3$ in the Lane-Emden variables and as discussed in \cite{aneta2}, the function is singular for $\alpha=1/2$. Therefore, the range of the values for the parameter is $\alpha\in(-\infty;0.5)$ because above the value $\alpha=0.5$ one obtains non-physical values. It should be also recalled that in \cite{aneta3} the stability of a non-relativistic polytrope was discussed for the model we have focused on. Thus, it was shown that for $n=3$ one deals with a stationary point if the parameter $\alpha$ is negative. This is also visible from the table: for positive values of $\alpha$ the radius' value increases while the mass decreases, decreasing density of the star. Let us just comment the main difference between the Newtonian limit and Lane-Emden equation given for metric $f(R)$ gravity in \cite{cap1,cap2,cap3} - the equation is much complicated and it does not recover the well-known solutions in the GR limit which is however not a case in Palatini $f(\mathcal{R})$ gravity. Further works in this area, as it was presented for the metric $f(R)$ gravity in \cite{nn1,kal}, will allow to constrain Palatini gravity. \begin{table} \centering \caption{Numerical values of the first zeros $\xi_R$ and the functions $\xi_R^2|\theta'(\xi_R)|$, together with the radii $R$ $\Big(\times10^4\left( \frac{10^6\text{g/cm}^{3} }{\rho_c} \right)^{1/3}\text{ km}\Big)$ and masses $M=\mathcal{M}/M_\odot$ for various values of $\alpha$.} \label{tab} \begin{tabular}{|c||c c|c c|} \hline $\alpha$ & $\xi_R$ & $\xi_R^2|\theta'(\xi_R)|$ & $R$ & $M$ \\ \hline\hline 0.49 & 10.32 & 1.396 & 5.008 & 1.0072 \\ 0.4 & 9.523 & 1.407 & 4.62 & 1.01471 \\ 0.3 & 8.486 & 1.468 & 4.12 & 1.05912 \\ 0.2 & 7.651 & 1.595 & 3.712& 1.15068 \\ 0.15 & 7.354 & 1.682 & 3.568 & 1.2128 \\ 0.05 & 6.985 & 1.894 & 3.389 & 1.36579 \\ \hline 0 & 6.897 & 2.018 & 3.347 & 1.457 \\ \hline -0.05 & 6.863 & 2.155 & 3.3 & 1.55413 \\ -0.1 & 6.875 & 2.30 & 3.34 & 1.66139 \\ -0.15 & 6.932 & 2.465 & 3.363 & 1.7778 \\ -0.2 & 7.027 & 2.64 & 3.41 & 1.90393 \\-0.3 & 7.322 & 3.035 & 3.55 & 2.18932 \\-0.4 & 7.748 & 3.505 & 3.759 & 2.52798 \\-0.5 & 8.30119 & 4.07 & 4.03 & 2.93518 \\ \hline \end{tabular} \end{table} \section{Conclusions} The aim of this letter was to introduce a few aspects related to white dwarfs which lie in a field of interest to the author. Thus, we have briefly discussed the age of such objects and super-Chandrasekhar mass. The first problem is strongly linked to the early evolution of white dwarf's progenitor star and cooling process of the star's compact remnant. It was already demonstrated in various works that both phenomena are strongly affected by modifications of the gravitational counterpart of the stellar equations. On the other hand, the second issue was broadly examined in modified gravity confirming that indeed super luminous SNIas could be caused by white dwarfs which exceed the (General Relativity) Chandrasekhar limit since the limit significantly differs in many theories of gravity. Therefore studying those extraordinary objects in different gravitational frameworks may not only answer the posted questions but also shed light on the dynamical effects introduced by modified gravity because of new effective interactions in the matter sector which may end up altering the effective pressure generated by the matter sources. The high densities and pressures achievable by white dwarfs turn them into excellent laboratories to confront the standard and modified theories with observations. In the last part of this paper we obtained Chandrasekhar limit for quadratic Palatini $f(\mathcal R)$ gravity which has not been yet studied in this theory of gravity. We have shown that for the negative parameter $\alpha$ one deals with limited masses bigger than GR equivalent and thus, after more detailed investigations and realistic description, Palatini gravity could also contribute to explanation the mentioned problems related to white dwarfs and stars' modeling. \section*{Acknowledgments} The work is supported by the European Union throughthe ERDF CoE grant TK133.

proofpile-arXiv_069-13343

\section{Introduction} Presenting quantum mechanics as a statistical theory on a classical phase space has attracted a great deal of attention since the very early days of this discipline. This framework gives an alternative point of view that provides more insight and understanding~\cite{Hillery:1984aa,Lee:1995aa,Schroek:1996aa,Ozorio:1998aa,Schleich:2001aa,QMPS:2005aa}, and, in addition, avoids the operator formalism, thereby freeing quantization of the burden of the Hilbert space~\cite{Ali:2005aa}. The main ingredient for any successful phase-space method is a \emph{bona fide} mapping that relates operators with functions defined on a smooth manifold $\mathcal{M}$, the phase space of the system, endowed with a very precise mathematical structure~\cite{Kirillov:2004aa}. This mapping, first suggested by Weyl~\cite{Weyl:1928aa} and later put on solid grounds by Stratonovitch~\cite{Stratonovich:1956aa}, is not unique. In fact, a whole family of $s$-parametrized functions can be assigned to each operator and the choice of a particular element of the family depends on its convenience for each problem. In particular, the time-honored quasiprobability distributions are the functions connected with the density operator. The most common choices of $s$ are $+1$, 0, and $-1$, which correspond to the $P$ (Glauber-Sudarshan)~\cite{Glauber:1963aa,Sudarshan:1963aa}, $W$~(Wigner)~\cite{Wigner:1932uq}, and $Q$~(Husimi)~\cite{Husimi:1940aa} functions, respectively. For continuous variables (such as Cartesian position and momentum), the quintessential example that fuelled the interest for this field, the parameter $s$ defines different orderings of the basic variables. These quasiprobability distributions and their corresponding equations of motion give the right tools for the representation of quantum dynamics entirely in the language of phase-space variables. Actually, a substantial step to addressing this question came from the work of Groenewold~\cite{Groenewold:1946aa} and Moyal~\cite{Moyal:1949aa}, who showed that the evolution equation can be written as \begin{equation} \partial_{t}W_{\varrho}^{(s)}(\Omega, t) = \ML W_{\varrho}^{(s)}(\Omega, t), W_{H}^{(s)}(\Omega ) \MR \, , \label{ee1} \end{equation} where $\Omega \in \mathcal{M}$ are points in phase space, $W_{H}^{(s)}(\Omega )$ is the Weyl symbol of the Hamiltonian and the Moyal bracket $\ML \cdot, \cdot \MR$ is the image of the commutator [times $(i \hbar)^{-1}$] under the Weyl-Stratonovitch map. Therefore, (\ref{ee1}) is formally identical with its quantum version for the density operator, if one replaces the commutator with the Moyal bracket. This equation contains, in general, higher-order derivatives, which reflects the analytical properties of the map. This complicates, in general, getting any analytical solution~\cite{Polkovnikov:2010aa}. It turns out that the dynamics in (\ref{ee1}) can be rewritten as a continuity equation \begin{equation} \partial_{t}W_{\varrho}^{(s)}(\Omega, t)= - \nabla \cdot \mathbf{J}^{(s)}(\Omega , t) \, , \label{flow} \end{equation} as it was recently discussed for one-dimensional systems~\cite{Bauke:2011aa,Steuernagel:2013aa,Kakofengitis:2017aa,Kakofengitis:2017ab,Oliva:2019aa,Oliva:2019ab}. Nonetheless, there is a substantial difference between the classical and quantum regimes. In classical statistical mechanics, the conservative evolution of the (positive) phase-space distribution function $f(\Omega, t)$ is given by the famous Liouville equation~\cite{Pathria:2011aa} \begin{equation} \label{eq:liouv} \partial_{t} f(\Omega,t ) = \{ f(\Omega, t), H_{\mathrm{cl}} \} \, , \end{equation} with $\{ \cdot , \cdot \}$ being the Poisson brackets on $\mathcal{M}$. This can also be expressed as a continuity equation \begin{equation} \label{eq:contclass} \partial_{t}f (\Omega, t)= - \nabla \cdot \mathbf{J}_{\mathrm{cl}}(\Omega , t) \, , \end{equation} but now the classical current is \begin{equation} \label{eq:Jcl} \mathbf{J}_{\mathrm{cl}} = \mathbf{v} (\Omega) \, f(\Omega, t) \ \end{equation} where $\mathbf{v}(\Omega )\propto \nabla H_{\mathrm{cl}}$ is related to the velocity field generated by the Hamiltonian $H_{\mathrm{cl}}$ on $\mathcal{M}$. In other words, $\mathbf{v}(\Omega )$ induces a Hamiltonian propagation \begin{equation} f(\Omega ,0)\overset{\mathbf{v}}{\longmapsto } f(\Omega ,t)=f(\Omega_{\mathrm{cl}}(t),0) \, , \label{ft} \end{equation} so that the distribution function is preserved along every classical trajectory $\Omega _{\mathrm{cl}} (t)$. The stagnation points $\{\Omega _{j}\}$, where $\mathbf{J}_{\mathrm{cl}}(\Omega_{j},t)=0$, coincide either with the zeros of $\mathbf{v}(\Omega)$\ or those of $f(\Omega , 0)$. For the quantum dynamics, $\mathbf{J}^{(s)}(\Omega ,t)$ can be conveniently represented as a (differential) operator acting on the quasiprobability distribution; i.e., \begin{equation} \mathbf{J}^{(s)}(\Omega ,t) = \hat{\mathbf{J}}^{(s)}(\Omega )\; W_{\varrho}^{(s)}(\Omega ,t)\,. \label{Jsin} \end{equation} In the semiclassical limit, when the dynamics of distribution resembles the motion of an incompressible fluid, $\hat{\mathbf{J}}^{(s)}(\Omega ) \rightarrow \mathbf{v} (\Omega )$. In contradistinction, in the full quantum regime trajectories do not exist globally for systems whose phase-space distributions can develop areas with nonpositive values and the velocity is not a well-defined function~\cite{Oliva:2018aa}. Consequently, (\ref{Jsin}) not only describes the propagation and deformations of $W_{\varrho }^{(s)}(\Omega ,t)$, as in (\ref{ft}), but also the emergence of interference patterns (in particular, the evolution of $W_{\varrho }^{(s)}(\Omega )$ into a zero-amplitude function at the nonclassical stagnation points). The currents $J^{(s)}(\Omega ,t)$ bear several unexpected features from a classical viewpoint~\cite{Steuernagel:2013aa,Kakofengitis:2017aa,Kakofengitis:2017ab,Oliva:2019aa,Oliva:2019ab}. The distribution and character of the stagnation points can be used for detecting the quantumness of the evolution, both from the Wigner and the Husimi currents~\cite{Veronez:2016aa}. Interestingly, the quasiprobability currents can be properly defined not only in the unitary case, but also in the presence of dissipation~\cite{Kossakowski:1972aa,Lindblad:1976aa,Gorini:1976aa}. The analysis of these dissipative currents can provide interesting insights into the decoherence dynamics~\cite{Braasch:2019aa}, especially in the presence of purely quantum effects, such as self-interference and tunnelling. Recently, it has been shown that the description of the evolution in terms of currents can be extended in natural ways to spinlike systems, where the classical phase space is the unit sphere~\cite{Yang:2019aa}. Actually, in the simplest yet relevant case of a nonlinear Kerr medium, the spatial distribution of the phase-space Wigner current and in, particular, of the stagnation lines (one of the components of the Kerr current is always zero) allows one to distinguish quantum from classical dynamics, even for short times. In the present paper we introduce the $s$-ordered currents for spinlike systems whose evolution is governed by quadratic Hamiltonians and apply it to the analysis of the dynamics of the Kerr~\cite{Kitagawa:1993aa, Agarwal:1997aa} and Lipkin-Meshkov-Glick~\cite{Lipkin:1965aa,Meshkov:1965aa,Glick:1965aa} models, both in the unitary and the dissipative case. The most representative examples of spin dynamics, stable, unstable and tunnelling (evolution in the classically forbidden regions), will be analyzed on the basis of spatial distributions of the quantum currents. Although we shall get analytical expressions for any $s$-ordered distribution, our numerical analysis will be focused on the Wigner propagation, for it reveals in the most conspicuous manner the quantum dynamical behavior in case of nonlinear evolution. \section{Quasiprobability distributions on the sphere} We consider a system whose dynamical symmetry group is SU(2). The corresponding Lie algebra $\mathfrak{su} (2)$ is spanned by the operators $\{ \hat{S}_{x}, \hat{S}_{y}, \hat{S}_{z} \}$ satisfying the standard commutation relations $ [\hat{S}_{x}, \hat{S}_{y} ] = i \hat{S}_{z}$ and cyclic permutations (in units $\hbar =1$, which will be used throughout). The Casimir operator is $\hat{\mathbf{S}}^{2} = \hat{S}_{x}^{2} + \hat{S}_{y}^{2}+ \hat{S}_{z}^{2} = S (S+1) \openone$, so the eigenvalue $S$ (which is a nonnegative integer or half integer) labels the irreducible representations (irreps). We take a fixed irrep of spin $S$, with a $(2S+1)$-dimensional carrier space $\mathcal{H}_{S}$ spanned by the standard angular momentum basis $\{ |S, m\rangle \mid m= -S, \ldots, S\}$, whose elements are simultaneous eigenstates of $\hat{\mathbf{S}}^{2}$ and $ \hat{S}_{z}$: \begin{equation} \hat{\mathbf{S}}^{2} |S, m \rangle = S (S+1) |S, m \rangle \, , \qquad \hat{S}_{z} |S, m\rangle = m |S, m \rangle \, . \end{equation} The highest weight state is $|S,S\rangle$ and it is annihilated by the ladder operator $\hat{S}_{+}$ (with $\hat{S}_{\pm} = \hat{S}_{x} \pm i \hat{S}_{y}$). The isotropy subgroup (i.e., the largest subgroup that leaves the highest weight state invariant) consists of all the elements of the form $\exp( i \chi \hat{S}_{z})$, so it is isomorphic to U(1). The coset space is then SU(2)$/$U(1), which is just the unit sphere $\mathcal{S}_{2}$ (the so-called Bloch sphere) and it is the classical phase space, the natural arena to describe the dynamics. The $s$-parametrized Weyl-Stratanovich map \begin{equation} \hat{A} \mapsto W_{A}^{(s)} (\Omega ) = \Tr [ \hat{A} \, \hat{w}^{(s)}(\Omega ) ] , \label{map} \end{equation} where $\Omega =(\theta ,\phi )\in \mathcal{S}_{2}$, puts in one-to-one correspondence each operator $\hat{A}$ invariantly acting on $\mathcal{H}_{S}$ with a function on the sphere $\mathcal{S}_{2}$. The corresponding kernels $\hat{w}^{(s)}$ are defined as~\cite{Berezin:1975mw,Agarwal:1981bd,Varilly:1989ud} \begin{equation} \hat{w}^{(s)}(\Omega )= \sqrt{\frac{4\pi}{2S+1}} \sum_{K=0}^{2S} \sum_{q=-K}^{K} ( C_{SS,K0}^{SS} )^{-s} \, Y_{Kq}^{\ast}(\Omega) \hat{T}_{Kq}^{S}\,, \label{ker} \end{equation} where $Y_{Kq}(\Omega )$ are the spherical harmonics, $C_{S_{1}m_{1},S_{2}m_{2}}^{Sm}$ the Clebsch-Gordan coefficients and $\hat{T}_{Kq}^{S}$ the irreducible tensor operators~\cite{Fano:1959ly,Varshalovich:1988ct} \begin{equation} \hat{T}_{Kq}^{S} = \sqrt{\frac{2K+1}{2S+1}} \sum_{m,m^{\prime}=-S}^{S} C_{Sm,Kq}^{Sm^{\prime}} \, |S,m^{\prime}\rangle \langle S,m| \, . \end{equation} As expected, they are properly normalized \begin{equation} \Tr [ \hat{w}^{(s)} ( \Omega ) ]=1 \, , \qquad \frac{2S+1} {4\pi}\int_{\mathcal{S}_{2}} d\Omega \; \hat{w}^{(s)} (\Omega )=\openone \,, \end{equation} with $d\Omega =\sin \theta\ d\theta\ d\phi $ the invariant measure on the sphere. Consequently, the symbol of $\hat{A}$ can be concisely expressed as \begin{equation} W_{A}^{(s)} ( \Omega ) = \sqrt{\frac{4\pi}{2S+1}} \sum_{K=0}^{2S} \sum_{q=-K}^{K} ( C_{SS,K0}^{SS} )^{-s} A_{Kq} \; Y_{LM}^{\ast} (\Omega ) \, , \label{eq:symb} \end{equation} where $A_{Kq}=\Tr ( \hat{A} \hat{T}_{Kq}^{S \dagger} )$. As some relevant examples we shall need in what follows we quote \begin{equation} \begin{array}{rcl} \label{eq:symbols} \hat{S}_{i} & \mapsto & W^{(s)}_{S_{i}} = \displaystyle \left ( \frac{S}{S+1} \right )^{-s/2} \sqrt{S(S+1)} \, n_{i} \, , \\ & & \\ \{\hat{S}_{i},\hat{S}_{j}\}_{+} & \mapsto & W^{(s)}_{\{S_{i},S_{j}\}_{+}} = \mathcal{C}_{ij}^{(s)} \; n_{i}n_{j} + \frac{1}{3}\delta_{ij} [ 2S(S+1) - \mathcal{C}_{ij}^{(s)} ] , \end{array} \end{equation} where $\mathbf{n} $ is a unit vector in the direction of $\Omega \in \mathcal{S}_{2}$, $\{ \cdot, \cdot \}_{+}$ stands for the anticommutator and $\mathcal{C}_{ij}^{(s)}=(1-\frac{1}{2}\delta_{ij})[S(2S-1)]^{(1-s)/2}[(2S+3)(S+1)]^{(1+s)/2}$. The traditional SU(2) quasiprobability distributions are just the $s$-symbols of the density operator $\hat{\varrho}$. The value $s=0$ corresponds to the standard Wigner function, whereas $s=\pm 1$ leads to $P$ and $Q$ functions respectively, defined as dual coefficients in the basis of spin coherent states~\cite{Arecchi:1972aa,Perelomov:1986ly} \begin{equation} |\Omega \rangle = \exp [ \case{1}{2} \theta (\hat{S}_{+}e^{-i\phi}- \hat{S}_{-}e^{i\phi} ) ] |S,S\rangle \, , \label{CS} \end{equation} according to \begin{equation} Q(\Omega ) = \langle \Omega |\hat{\varrho}|\Omega \rangle \, , \qquad \hat{\varrho} =\frac{2S+1}{4\pi}\int_{\mathcal{S}_{2}}d\Omega \; P(\Omega) \; |\Omega \rangle \langle \Omega | \, . \end{equation} The symbols $W_{A}^{(s)} (\Omega )$ are covariant under SU(2) transformations and provide the overlap relation \begin{equation} \Tr (\hat{\varrho} \hat{A}) = \frac{2S+1}{4\pi} \int_{\mathcal{S}_{2}} d\Omega \, W_{\varrho}^{(s)}(\Omega ) \,W_{A}^{(-s)}(\Omega ) \, . \end{equation} A number of alternative generalized quasiprobability distributions can be found using the method of Cohen~\cite{Cohen:1966aa} (see also Ref.\cite{Liu:2011ab}). In this representation, the Moyal equation (\ref{ee1}), as indicated in the Introduction, involves higher-order derivatives. However, it admits an expansion on the parameter $\varepsilon =(2S+1)^{-1}$. When $\varepsilon \ll 1$ we are in the semiclassical limit and one can show that~\cite{Gilmore:1975aa,Amiet:1991aa,Benedict:1999aa,Klimov:2005aa} \begin{equation} \partial_{t}W_{\varrho}^{(s)}(\Omega, t ) \simeq 2 \varepsilon \{W_{\varrho}^{(s)} (\Omega, t ), W_{H}^{(s)}(\Omega )\} + s \mathcal{O}(\varepsilon^{2})+ \mathcal{O}(\varepsilon^{3}), \end{equation} where $\{\cdot, \cdot \}$ are the Poisson brackets on the sphere $\mathcal{S}_{2}$ \begin{equation} \{f (\Omega ),g(\Omega )\} = \frac{1}{\sin \theta} ( \partial_{\phi}f \; \partial_{\theta}g - \partial_{\theta}f \; \partial_{\phi}g ) . \label{PB} \end{equation} Therefore, the first-order corrections to the classical evolution $\sim \mathcal{O}(\varepsilon^{2})$ vanish for the Wigner function and consist of second-order derivatives of $W_{\varrho}^{(\pm 1)}(\Omega )$. The lowest order approximation, known as the Truncated Wigner Approximation (TWA)~\cite{Heller:1976aa,Davis:1984aa,Bagrov:1992aa,Drummond:1993aa,Polkovnikov:2003aa,Drummond:2017aa,Valtierra:2017aa}, describes propagation of every point of the initial distribution along the corresponding classical trajectories $\Omega (t)$, which are solutions of the Hamilton equations; viz, \begin{equation} W_{\varrho}^{(s)}(\Omega, t) \simeq W_{\varrho}^{(s)}(\Omega(-t), t=0) \, . \label{TWA} \end{equation} This semiclassical evolution allows one to predict the short-time behavior~\cite{Valtierra:2016aa,Sundar:2019aa}. The positive and negative parts of $W_{\varrho }^{(s)}(\Omega ,t=0)$ are deformed according to (\ref{TWA}), so that their volumes are preserved during the validity of the TWA. \section{Hamiltonian dynamics and currents on the sphere} The exact evolution equation for $W_{\varrho}^{(s)}(\Omega )$ has been derived in \cite{Klimov:2002cr} (see also \cite{Zueco:2007aa} and \cite{Klimov:2017aa}, where the corresponding star-product for the map is discussed). In most physical applications only Hamiltonians quadratic in the spin generators play an important role. Typical examples include second-harmonic generation, homogeneous spin-spin interactions, spin-orbit splitting, and atom-field interactions in the dipole approximation~\cite{Cohen-Tannoudji:1989aa}. The generic second-order Hamiltonian reads \begin{equation} \hat{H} = \sum_{i}a_{i} \, \hat{S}_{i}+\sum_{jk}b_{jk} \, \{ \hat{S}_{j}, \hat{S}_{k}\}_{+} \equiv \hat{H}_{L}+ \hat{H}_{NL} \ \label{H} \end{equation} where $\hat{H}_{L}$ and $\hat{H}_{NL}$ refer to the linear and nonlinear parts of the Hamiltonian. The evolution equation for the Hamiltonian (\ref{H}) can be rewritten in terms of the Poisson brackets (\ref{PB}) as follows: \begin{eqnarray} \partial _{t}W_{\varrho }^{(s)} & = & \left( \frac{S}{S+1}\right)^{-s/2} \sum_{i}a_{i}\{W_{\varrho }^{(s)},n_{i}\} \nonumber \\ & + & \frac{1}{2\varepsilon } \sum_{j,k}b_{jk}[\{\hat{G}_{k}W_{\varrho }^{(s)},n_{j}\}+\{\hat{G} _{j}W_{\varrho }^{(s)},n_{k}\}]\,, \label{EE2} \end{eqnarray} where \begin{eqnarray} \hat{G}_{k}^{(\pm 1)} &=&n_{k}(1\pm \varepsilon )\,\pm i \varepsilon (\mathbf{n}\times \hat{\mathbf{L}})_{k}\,, \nonumber \\ && \\ \hat{G}_{k}^{(0)} &=&\frac{1}{2}n_{k}\Phi (\mathcal{L}^{2})-\frac{1}{2} \varepsilon ^{2}[n_{k}+ 2 i (\mathbf{n}\times \hat{\mathbf{L}})_{k}]\Phi^{-1}(\mathcal{L}^{2})\,. \nonumber \end{eqnarray} Here, $\hat{\mathbf{L}}=(\hat{L}_{x},\hat{L}_{y},\hat{L}_{z})$ are a differential realization of the angular momentum operators; viz, \begin{eqnarray} \hat{L}_{x}& = & i (\sin \phi \partial _{\theta }+ \cot \theta \cos \phi \partial _{\phi }) \,, \nonumber \\ \hat{L}_{y}& = & i (-\cos \phi \partial _{\theta}+ \cot \theta \sin \phi \partial _{\phi })\,,\\ \hat{L}_{z} & = & -i \partial_{\phi }\,, \nonumber \end{eqnarray} and $\hat{\mathcal{L}}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}$ is the realization of the Casimir operator on the sphere, namely \begin{equation} \hat{\mathcal{L}}^{2} =-\left( \partial _{\theta \theta }+\cot \theta \,\partial _{\theta }+ \frac{1}{\sin ^{2}\theta }\partial _{\phi \phi}\right) \;, \label{L2} \end{equation} so that $\hat{\mathcal{L}}^{2}Y_{Lm}(\Omega )=L(L+1)Y_{Lm}(\Omega )$ (note that, except for a sign, it is the Laplacian operator on the sphere). Finally, the function $\Phi $ reads \begin{equation} \Phi (x)=\left[ 2-\varepsilon ^{2}(2x^{2}+1) + 2 \sqrt{1-\varepsilon^{2}(2x^{2}+1)+\varepsilon ^{4}x^{4}}\right] ^{1/2}\,. \end{equation} The equation of motion (\ref{EE2}) can be immediately recast in the form (\ref{flow}). The currents (\ref{Jsin}), $\mathbf{J}^{(s)} = (J_{\theta}^{(s)},J_{\phi}^{(s)})$, can be conveniently represented as the actions of differential operators on the corresponding quasiprobability distributions: \begin{equation} \mathbf{J}^{(s)}(\Omega , t) = \hat{\mathbf{J}}^{(s)} \; W_{\varrho}^{(s)}(\Omega , t) \, , \label{Js} \end{equation} with \begin{eqnarray} \hat{J}_{\theta}^{(s)} & = & \left ( \frac{1-\varepsilon}{1+\varepsilon} \right)^{-s/2} \frac{1}{\sin \theta} \sum_{i}a_{i}\partial_{\phi}n_{i} \nonumber \\ & + & \frac{1}{2\varepsilon \sin \theta} \sum_{jk}b_{jk} [\partial_{\phi}n_{j} \hat{G}_{k}^{(s)} + \partial_{\phi}n_{k}\hat{G}_{j}^{(s)}] \, , \nonumber \\ & & \label{Jops} \\ \hat{J}_{\phi}^{(s)} & = & - \left( \frac{1-\varepsilon}{1+\varepsilon} \right)^{-s/2} \sum_{i}a_{i}\,\partial_{\theta}n_{i} \nonumber \\ & - & \frac{1}{2\varepsilon} \sum_{jk}b_{jk} [\partial_{\theta}n_{j}\hat{G}_{k}^{(s)} + \partial_{\theta}n_{k}\hat{G}_{j}^{(s)}] \, . \nonumber \end{eqnarray} It is worth noticing that $\mathbf{\hat{J}}^{(\pm 1)} $ are first-order operators (observe that $\mathbf{n}\times \hat{\mathbf{L}} $ is just the angular part of the gradient operator in spherical coordinates~\cite{Varshalovich:1988ct}), whereas $\mathbf{\hat{J}}^{(0)}$ contains higher-order derivatives because of the to dependence on the Casimir operator~$\hat{\mathcal{L}}^{2}$. The current associated to the linear Hamiltonian $\hat{H}_{L}$ generates a rigid rotation of the initial distribution; i.e., the quantum and classical currents coincide in this case. In the general case, the quantum dynamics is described by current operators that do not reduce to a multiplication by some phase-space function, as in the classical case. This leads to a nontrivial evolution of the stagnation points $\Omega _{j}$, wherein $J^{(s)}(\Omega _{j},t)=0$, as we shall see in Sec.~\ref{sec:ex}. The properties of the vector field $J^{(s)}(\Omega ,t)$ in the vicinity of the stagnation points can be studied, e.g., with the winding number~\cite{Steuernagel:2013aa,Veronez:2016aa} $ I(\Omega _{j}) = \frac{1}{2\pi } \oint_{L} d\varphi$, where $\varphi$ is the angle between the flow and some fixed reference axis in the loop $L$. This number takes the values $I(\Omega _{j})=\pm 1$ for vortices and saddle points, correspondingly. Nontrivial stagnation points [that do not coincide with zeros of the initial distribution and the gradient field $\nabla W_{H}^{(s)}(\Omega )$] dynamically emerge/disappear only by pairs (topological dipoles~\cite{Veronez:2016aa}) according to Poincar\'{e}-Hopf theorem. It is worth noting that the evolution (\ref{EE2}) can be rewritten in terms solely of the Poisson brackets with the Weyl symbol of the Hamiltonian (\ref{H}) in two instances: for the linear case $\hat{H}_{L}$ and when the Hamiltonian is the square of an element of the su(2) algebra; i.e., up to an SU(2) rotation, it has the form \begin{equation} \hat{H}_{NL} = b_{z} \, \hat{S}_{z}^{2}, \qquad \qquad \partial _{t} W_{\varrho }^{(s)} = \{\hat{\Gamma}_{z}^{(s)}W_{\varrho}^{(s)}, W_{H_{NL}}^{(s)}\}\,, \label{flow2} \end{equation} where \begin{eqnarray} \hat{\Gamma}_{z}^{(\pm )} & = & \frac{(1\pm \varepsilon )} {\varepsilon }\,\pm i \frac{(\mathbf{n}\times \hat{\mathbf{L}})_{z}\,}{n_{z}}\, , \nonumber\\ & & \\ \hat{\Gamma}_{z}^{(0)} & = & \frac{1}{2\varepsilon} \Phi (\mathcal{L}^{2}) - \frac{1}{2}\varepsilon \left [ 1+2i\frac{(\mathbf{n}\times \hat{\mathbf{L}})_{z}}{n_{z}} \right ] \, \Phi ^{-1}(\mathcal{L}^{2}) \,, \nonumber \end{eqnarray} and here $i(n\times \hat{\mathbf{L}})_{z}/n_{z} =\tan \theta \ \partial _{\theta }$ (in general, an arbitrary direction can be chosen instead of the $z$ component). The equation of motion in the form (\ref{flow2}) appears as the so-called second-kind continuity equation~\cite{Liu:2011aa}. \section{Dissipative quasiprobability currents on the sphere} \subsection{Dissipative currents} Models of dissipation address the interaction of a system with an environment, whose characteristics are encoded in its spectral density~\cite{Breuer:2003aa}. Here, we assume that the spin system is coupled to a thermal bath at temperature $T$. The resulting effective dynamics is appropriately described by the Lindblad equation~\cite{Kossakowski:1972aa,Lindblad:1976aa,Gorini:1976aa} \begin{equation} \partial_{t}{\hat{\varrho}}=-i [ \hat{H},\hat{\varrho} ] + \case{1}{2} \gamma \, ( \overline{n}+1) \hat{\Lambda}_{1}(\hat{\varrho}) + \case{1}{2} \gamma \, \overline{n} \hat{\Lambda}_{2}(\hat{\varrho}) \, , \label{LE} \end{equation} where $\hat{\Lambda}_{1,2}$ are the superoperators \begin{eqnarray} \hat{\Lambda}_{1}(\hat{\varrho}) & = & 2\hat{S}_{-}\hat{\varrho}\hat{S}_{+}- \hat{S}_{+}\hat{S}_{-}\hat{\varrho} - \hat{\varrho}\hat{S}_{+}\hat{S}_{-} \, ,\nonumber \\ & & \\ \hat{\Lambda}_{2}(\hat{\varrho}) & = & 2\hat{S}_{+}\hat{\varrho}\hat{S}_{-} - \hat{S}_{-}\hat{S}_{+}\hat{\varrho} - \hat{\varrho} \hat{S}_{-}\hat{S}_{+} \, ,\nonumber \end{eqnarray} and $\overline{n}=[\exp (\hbar \omega_{0}/kT)-1]^{-1}$ is the average number of excitations in the bath. In the phase-space picture, the action of the superoperators $\hat{\Lambda}_{1,2}(\hat{\varrho})$ is represented by the following differential operators \begin{widetext} \begin{eqnarray} s& = & \pm 1 \quad \left \{ \begin{array}{lcl} \hat{\Lambda}_{1} (\hat{\varrho}) & \mapsto & [ -\hat{\mathcal{L}}^{2} (1 \pm \cos \theta ) + \hat{L}_{z}^{2} - \frac{1}{\varepsilon} (2\cos \theta + \sin \theta \partial_{\theta}) ] W_{\varrho}^{(\pm 1)}(\Omega ), \\ \\ \hat{\Lambda}_{2}(\hat{\varrho}) & \mapsto & [ - \hat{\mathcal{L}}^{2} (1 \mp \cos \theta ) + \hat{L}_{z}^{2} + \frac{1}{\varepsilon} (2\cos \theta +\sin \theta \partial_{\theta}) ] W_{\varrho}^{(\pm 1)}(\Omega )\, , \end{array} \right . \nonumber \\ \\ s& = & 0 \quad \; \; \; \left \{ \begin{array}{lcl} \hat{\Lambda}_{1}(\hat{\varrho}) & \mapsto & [ -\hat{\mathcal{L}}^{2}+ \hat{L}_{z}^{2} - \frac{1}{\varepsilon} ( \cos \theta + \frac{1}{2}\sin \theta \partial_{\theta} ) \Phi (\mathcal{L}^{2}) +\varepsilon ( \hat{\mathcal{L}}^{2}- \cos \theta - \frac{1}{2}\sin \theta \partial_{\theta} ) \Phi^{-1} (\hat{\mathcal{L}}^{2}) ] W_{\varrho}^{(0)}(\Omega ), \\ \\ \hat{\Lambda}_{2}(\hat{\varrho}) & \mapsto & [ - \hat{\mathcal{L}}^{2} + \hat{L}_{z}^{2} + \frac{1}{\varepsilon} ( \cos \theta + \frac{1}{2}\sin \theta \partial_{\theta} ) \Phi (\hat{\mathcal{L}}^{2}) -\varepsilon ( \hat{\mathcal{L}}^{2} - \cos \theta -\frac{1}{2}\sin \theta \partial_{\theta} ) \Phi^{-1}(\hat{\mathcal{L}}^{2})] W_{\varrho}^{(0)}(\Omega ) \, . \end{array} \right . \nonumber \end{eqnarray} \end{widetext} The dissipative phase-space dynamics of these spinlike systems has been investigated by a number of authors~\cite{Perelomov:1986ly,Carmichael:1999aa,Kalmykov:2016aa}. Interestingly, the Lindblad evolution can also be recast as a continuity equation with current operators given by \pagebreak \begin{eqnarray} \label{Jtpm} \hat{J}_{\theta}^{(\pm 1)} & = & \case{1}{2} \gamma \left[ \frac{1}{\varepsilon} \sin \theta - \partial_{\theta} ( 1 + 2 \overline{n} \pm \cos \theta ) \right] , \nonumber \\ \hat{J}_{\phi}^{(\pm 1)} &=&-\case{1}{2} \gamma \left[ (\pm \tan \theta + ( 2 \overline{n} + 1 ) \frac{\cos \theta}{\tan \theta}\right] \partial_{\phi}; \nonumber \\ & & \\ \hat{J}_{\theta}^{(0)} & = & \case{1}{2} \gamma \left[ \frac{1}{2\varepsilon} \sin \theta \, \Phi (\hat{\mathcal{L}}^{2} ) - ( 2 \overline{n} + 1) \partial_{\theta}\right . \nonumber \\ & + & \left . \varepsilon \left( \partial_{\theta} + \case{1}{2}\sin \theta \right) \ \Phi^{-1}(\hat{\mathcal{L}}^{2})\right] , \nonumber \\ \hat{J}_{\phi}^{(0)} &=&-\case{1}{2} \gamma \left [ (2\overline{n} +1) \frac{\cos \theta}{\tan \theta} - \frac{\varepsilon}{\sin \theta}\ \Phi^{-1}(\hat{\mathcal{L}}^{2} ) \right] \partial_{\phi}. \nonumber \end{eqnarray} In the high-temperature regime ($\overline{n}\gg 1$), all the maps ($s=\pm 1,0$) lead to the same equation of motion; viz, \begin{equation} \hat{\Lambda}(\hat{\varrho}) \simeq \case{1}{2} \gamma \, \overline{n} \left( \hat{\Lambda}_{1}+ \hat{\Lambda}_{2}\right) (\hat{\varrho}) \mapsto - \case{1}{2} \gamma \,\overline{n} \left( \hat{\mathcal{L}}^{2} + \partial_{\phi \phi} \right ) W_{\varrho}^{(s)}(\Omega ), \end{equation} and the currents take the simple form \begin{equation} \label{Jdn} \hat{J}_{\theta}^{(s)} \simeq -\gamma \, \overline{n}\, \partial_{\theta}, \qquad \hat{J}_{\phi}^{(s)} \simeq -\gamma \, \overline{n} \, \frac{\cos \theta}{\tan \theta} \, \partial_{\phi} \,. \end{equation} In the presence of dissipation, the stagnation points may become sinks and sources. For instance, (\ref{Jdn}) generate a single source of a free-evolving vector field at the point $( \theta =\pi /2,\phi =0 ) $ for an initial spin coherent state centered on the $X$ axis $|\Omega = (\pi /2, 0) \rangle$. In the opposite limit of $\overline{n} =0$ in addition to that source, several sinks appear. \subsection{Classical limit} In the large spin limit, $\varepsilon \rightarrow 0$, only the Wigner current tends to the classical form. Indeed, taking into account that $\Phi (\hat{\mathcal{L}}^{2}) \simeq 2+ \mathcal{O}(\varepsilon^{2})$, the operators $\mathbf{\hat{J}}^{(0)}$ reduce to $c$-numbers, that is, $ \mathbf{\hat{J}}^{(0)} \simeq \mathbf{v} + \mathcal{O} (\varepsilon)$, with \begin{equation} \mathbf{v} = 2 \varepsilon \left ( \begin{array}{c} \displaystyle \frac{1}{\sin \theta} \partial_{\phi} W_{H}^{(0)}(\Omega ) \\ - \partial_{\theta}W_{H}^{(0)}(\Omega ) \end{array} \right) = 2 \varepsilon \nabla W_{H}^{(0)} (\Omega ) \, . \end{equation} Here, $W_{H}^{(0)} (\Omega)$ is the corresponding symbol of the Hamiltonian, which, using (\ref{eq:symbols}), is \begin{equation} W_{H}^{(0)} (\Omega ) = \frac{1}{2\varepsilon} \sum_{i}a_{i}\, n_{i}+\frac{1}{4\varepsilon^{2}} \sum_{jk}b_{jk}\, n_{k}n_{j} \, . \label{WH} \end{equation} According to (\ref{Js}), in this limit the points of the distribution $W_{\varrho}^{(0)}(\Omega )$ just follow the flow generated by $\mathbf{J}_{ \mathrm{cl}}$. In contradistinction, the $Q$ and $P$ currents have nonvanishing corrections in this semiclassical limit. In the same limit, the dominant term in the dissipative currents (\ref{Jtpm}) is $\hat{J}_{\theta}^{(s)}$ and this yields Fokker-Planck equations. This also indicates that the main direction of the dissipative motion is towards the South pole of the Bloch sphere, representing the ground state of the system. \section{Examples: Kerr and Lipkin-Meshkov-Glick models} \label{sec:ex} \begin{figure}[t] \centering \includegraphics[width=\columnwidth]{figure1} \caption{The Wigner symbol $W^{(0)}_{H}(\Omega )$ for (a) the Kerr and (b) the LMG Hamiltonians.} \label{fig:potenciales} \end{figure} \subsection{Kerr model} The simplest quadratic Hamiltonian corresponds to the so-called Kerr medium, which is described by \begin{equation} \hat{H}_{\mathrm{Kerr}}= \chi \, \hat{S}_{z}^{2} \, . \label{kerr} \end{equation} This leads to a remarkable non-Gaussian operation that has set off a lot of interest due to possible applications in a variety of fields, such as quantum nondemolition measurements~\cite{Braginskii:1968aa,Unruh:1979aa,Milburn:1983aa,Imoto:1985aa,Sanders:1989aa}, generation of quantum superpositions~\cite{Milburn:1986aa,Yurke:1986aa,Tombesi:1987aa,Gantsog:1991aa,Tara:1993aa,Luis:1995aa,Chumakov:1999aa,Rigas:2013aa}, quantumteleportation~\cite{Vitali:2000aa,Zhu:2011aa}, and quantum logic~\cite{Turchette:1995aa,Semiao:2005aa,Matsuda:2007aa,You:2012aa}. \begin{figure*}[t] \centering {\includegraphics[width=1.75\columnwidth]{figure2}} \caption{Snapshots of the Wigner function corresponding to the Kerr Hamiltonian (\ref{kerr}) and the initial spin coherent state $|\Omega = (\pi /2,0) \rangle $ at the best squeezing time $\chi t=0.32$ and two-component Schr\"{o}dinger cat time $ \chi t=\pi /2$. Left panel, unitary evolution; right panel, dissipative evolution with $\gamma =0.015$. Red and white lines are stagnation lines for $J_{\theta }$ and $J_{\phi }$, respectively. The pseudocolor encodes the Wigner function and the black arrows indicate the direction and strength of the flow. We have used $S=10$.} \label{fig:kerr} \end{figure*} As shown in Fig.~\ref{fig:potenciales}, the Wigner symbol of $\hat{H}_{\mathrm{Kerr}}$ has the aspect of a valley centered at $\theta =\pi /2$. The unitary currents have only one nonzero component, ($\hat{J}_{\theta }^{(s)}=0$), and they are \begin{eqnarray} \hat{J}_{\phi }^{(0)} &=&\sin \theta \left[ \tfrac{1}{2 \varepsilon}\cos \theta \;\Phi (\hat{\mathcal{L}}^{2}) - \tfrac{1}{2}\varepsilon (\cos \theta + 2\sin \theta \partial _{\theta })\Phi ^{-1}(\hat{\mathcal{L}}^{2})\right] \,, \nonumber \label{Jt0} \\ \hat{J}_{\phi }^{(1)} &=&\sin \theta \left[ 2(S+1)\cos \theta +\sin \theta \partial _{\theta }\right] \,, \\ \hat{J}_{\phi }^{(-1)} &=&\sin \theta \left[ 2S\cos \theta -\sin \theta \partial _{\theta }\right] . \nonumber \end{eqnarray} A peculiarity of these currents is the existence of stagnation lines, where $J_{\phi}^{(s)}(\Omega )=0$ (see also the discussion in \cite{Yang:2019aa}). When dissipation is included, both components are present. In Fig.~\ref{fig:kerr} we plot two snapshots of the Wigner function for an initial coherent state (\ref{CS}) located at the equator $\Omega = (\pi /2,0) $ both for unitary and dissipative dynamics. The first one is at the best squeezing time ($\chi t\sim S^{-2/3}$), where the maximum value of the spin squeezing, evaluated as the normalized minimum fluctuation of spin components $\Delta^{2} \mathbf{s}(t)$ on the tangent plane, orthogonal to the initial mean spin vector $\mathbf{n} =\langle \hat{\mathbf{S}}(t)\rangle$ ($\mathbf{s\cdot n}=0$) is achieved. The second one is at the two-component cat time ($\chi t=\pi /2$)~\cite{Agarwal:1997aa}, when the state becomes a superposition of two spin coherent states ($|\Omega =(\pi /2,0)\rangle + |\Omega =(\pi /2,\pi )\rangle $). The currents (\ref{Jt0}) apparently generate a fast motion of initial distributions to the south pole of the Bloch sphere; i. e., a decay into the ground state $|S,-S\rangle $. Nonetheless, distributions localized inside the potential valley move quite slowly into the south pole, so that even some quantum interference effects like residual Schr\"{o}dinger cat states can be observed. In Fig.~\ref{fig:kerr} we see that the interference pattern is partially destroyed for times $\chi t\sim 1$, as expected, but the distribution is still mainly concentrated inside the valley, in spite of the strong dissipation. This can be understood by taking into account that for $S=10$ at the minimum of the valley $W_{H}^{(0)}(\theta_{\min },\phi )\simeq 0.125$ and the energy fluctuation in the state $|\Omega = (\pi /2,0) \rangle $ is $\Delta H\simeq 6.8$, while at the south pole $W_{H}(\theta =\pi ,\phi )\simeq 109.75$. Thus, the distribution should overcome a significant potential barrier in order to reach the south pole, which slows down the decay of equatorially localized distributions to the ground state. It is interesting to stress that, in contrast to the previous behavior, when the distributions are localized below the equator they rapidly slide toward the south pole. This can be readably observed in Fig.~\ref{fig:kerrc}, where the Wigner function of an initial spin coherent state with $\Omega = (3\pi /4,0)$ at the two-component cat time, $\chi t=\pi /2$, is shown. \begin{figure}[b] \centering {\includegraphics[width=.90\columnwidth]{figure3}} \caption{Pseudocolor plot of the Wigner function for the initial state $|\Omega = (3\pi /4,0)\rangle $ at the two-component cat time in case of dissipative evolution ($\gamma =0.015$) with $S=10$.} \label{fig:kerrc} \end{figure} \subsection{Lipkin-Meshkov-Glick model} The Lipkin-Meshkov-Glick (LMG) model~\cite{Lipkin:1965aa,Meshkov:1965aa,Glick:1965aa} was originally proposed to deal with phase transitions in the nuclei. The model captures well the physics of two-mode Bose-Einstein condensates~\cite{Ulyanov:1992aa} and Josephon junctions~\cite{zibold:2010aa,Julia:2012aa}. In the language of spin operators the LMG Hamiltonian can be written as \begin{equation} \hat{H}_{\mathrm{LMG}} = - h \, \hat{S}_{x}+\frac{\lambda}{2(2S+1)} ( \hat{S}_{z}^{2}-S_{y}^{2} ) \, . \label{lipkin} \end{equation} For different values of the parameter $\lambda $, the associated classical symbol has either one or two minima. Here, we take $\lambda \sim S$, which corresponds to a double-well potential, as shown in Fig.~\ref{fig:potenciales}. The minima are located at $(\theta_{\min}=\pi /2, \phi_{\min}=1.459)$ and $(\theta_{\min}=\pi /2, \phi_{\min}=-1.459 )$, separated by a local maximum with a saddle point at $(\theta_{s}=\pi /2,\phi_{s}=0)$. The LMG current operators are significantly more involved than (\ref{Jt0}), and have the form in accordance with the general expressions~(\ref{Jops}). We consider two dynamical regimes: \subsubsection{Stable motion} \begin{figure*}[t] \centering {\includegraphics[width= 1.45\columnwidth]{figure4}} \caption{Snapshots of the Wigner function corresponding to the LMG Hamiltonian (\ref{lipkin}) for the stable case. The initial spin coherent state is $|\Omega = (\pi /2,-1.459) \rangle $ at the times $h t/T=0.1$ (back), $h t/T= 0.5$ (middle) and $h t/T=1$ (front), where $T$ here is the period of oscillation between the two wells. Left panel, unitary evolution; right panel, dissipative evolution with $\gamma =1\times 10^{-7}$. Red and white lines are stagnation lines for $J_{\theta}$ and $J_{\phi}$, respectively. In both cases the total spin is $S=10$.} \label{fig:stable} \end{figure*} \begin{figure}[b] \centering \includegraphics[width=0.80 \columnwidth]{figure5} \caption{Zoom of the Wigner function and the distribution of currents at $h t/T=0.1$ in the vicinity of the saddle point for the LMG Hamiltonian for the stable unitary evolution. The initial stat is the same as in Fig.~\ref{fig:stable}.} \label{fig:esfera} \end{figure} We take an initial spin coherent state with $\Omega= (\pi /2,1.459 )$ located inside one of the potential wells (actually, the right one). The classical stable motion corresponds to oscillations inside that well. Yet, due to tunnelling, the state is slowly transfered to the other well. In Fig.~\ref{fig:stable} we plot snapshots of the Wigner current at some representative times. The dynamics of the tunnelling as well as the formation of the corresponding interference patterns can be clearly appreciated. The current directions and its intensity distribution provide nontrivial information about tunnelling dynamics indicating the main paths of the state transfer. In particular, phase-space currents explicitly show the spatial distribution of the quasiprobability flow in classically forbidden areas, where the standard density current $\mathbf{j}\propto \I (\psi ^{\ast}\mathbf{\nabla }\psi )$ is zero. The red/white lines correspond to zero lines of $J_{\phi }^{(0)}/J_{\theta }^{(0)}$, respectively, and their intersections reveals the position of the stagnation points. In Fig.~\ref{fig:esfera} we plot a magnification of the vicinity of the saddle point, $\phi_{s}=0$ [at the same time as the first plot in Fig.~\ref{fig:stable}]. The tunnelling flow in both directions is manifest. \begin{figure}[b] \centering \par \includegraphics[width=0.80 \columnwidth]{figure6} \caption{The flow $\mathcal{I} ( \phi =0, t)$ (\ref{I}) for the same unitary (blue line) and dissipative (red line) dynamics as in Fig.~\ref{fig:stable} in terms of the dimensionless time $ht/T$.} \label{fig:intphi} \end{figure} \begin{figure*}[t] \centering {\includegraphics[width=1.65\columnwidth]{figure7}} \caption{Snapshots of the Wigner function corresponding to the LMG Hamiltonian for the unstable case. The initial spin coherent state is $|\Omega = (\pi /2,0) \rangle $ and the dimensionless times $h t=0.279$ (back) and $h t= 0.837$ (front). Left panel, unitary evolution; right panel, dissipative evolution with $\gamma =0.05$. Red and white lines are stagnation lines for $J_{\theta}$ and $J_{\phi}$, respectively. Again, the total spin is $S=10.$} \label{fig:unstable} \end{figure*} The counterclockwise vector field in the vicinity of the stable point $ (\theta_{\min}=\pi /2,\phi_{\min}=-1.459)$ extends into the classically forbidden region, generating a left-to-right flow with the highest intensity in the region $\theta <\pi /2$ (above the equator). The tunnelling flow freely crosses the stagnation (white) line $J_{\theta}^{(0)}(\Omega )=0$. The flow in the opposite direction is mainly below the equator, as one can see in Fig.~\ref{fig:esfera}. When a moderate dissipation is present, the tunnelling becomes slower, in agreement with general considerations~\cite{Caldeira:1981aa}. The interference pattern is largely destroyed. Nonetheless, in spite of the very long transfer times from one well into another, the distribution does not show any fingerprint of decay to the ground state at the \emph{half-period} of motion (when the distribution has passed to another minimum). The reason of such a behavior is the same as in the Kerr medium: to reach the south pole following the lines of the dissipative current, a distribution initially localized in the minimum of the potential should overcome a potential barrier, which is significantly higher than the local maximum: $ W_{H}^{(0)}(\theta_{\min},\phi_{\min})=-47.567$; the energy fluctuation in the state $|\Omega = (\pi /2,1.459) \rangle $ is $\Delta H \sim 2.996$, $W_{H}(\theta _{s},\phi_{s})=-10.488$, while $W_{H}(\theta =\pi ,\phi )=46.982$, for our case of $S=10 $. The dissipative evolution for larger times drastically differs from the unitary one: while the Hamiltonian evolution is quasiperiodic and the distribution \emph{oscillates} between the potential wells, the dissipation does not allow to the transfer of the whole distribution to the other well, and the \emph{inverse tunnelling} back to the original well is significantly suppressed in comparison with the unitary case. Actually, the decoherence in a two-well tunnelling acts as a viscous medium, in the sense that it leads to a phase-space equilibration at long-times, when the initial quasiprobability becomes equally distributed between the wells in the form of an incoherent superposition. Useful information about the tunnelling is provided by the integral flow at the line $\phi =0$ (which separates the potential wells), \begin{equation} \mathcal{I}(\phi =0,t) = \int d\theta \sin \theta \; J_{\phi}^{(0)}(\theta ,\phi =0, t) \, , \label{I} \end{equation} which typifies the dominant direction of the propagation at a given time. In Fig.~\ref{fig:intphi} we plot the flow (\ref{I}) for one \emph{period} of the tunnelling oscillation. During this time the initial distribution is transferred to the other potential minimum and returns back. One can observe that the direction of the Hamiltonian evolution changes from right-to-left to left-to-right when the distribution is completely transferred from the right to the left potential well. The flow in the presence of dissipation is significantly smaller in the second half-period of motion. \subsubsection{Unstable motion.} Next, we consider the initial spin coherent state $|\Omega = (\pi /2,0) \rangle $ centered at the saddle point (the classical separatrix). The directions of the current clearly indicate the hyperbolic nature of the stagnation point $(\theta_{s}=\pi /2,\phi_{s}=0)$. Quantum instability is reflected in a separation of the initial distribution into two symmetric pieces moving toward the minima, according to the current direction, with a subsequent formation of a complex interference picture. This can be seen in Fig.~\ref{fig:unstable}, where the snapshots of the Wigner function along with the corresponding current lines are plotted. As the state is initially at the local maxima, the decay to the ground state is quite fast in the presence of dissipation. The distribution clearly tends to the south pole along the current lines at times approximately corresponding to the half-period of oscillations in the Hamiltonian case, in sharp contrast with the stable situation. \section{Conclusions} In summary, we expect to have provided compelling evidence demonstrating that the quantum currents $\mathbf{J}^{(s)}(\Omega |t)$are a useful tool for the analysis of the evolution in phase space. Indeed, the spatial distribution of the quantum current allows one to visualize the main directions of propagation of the distribution. Our analytic current operators explicitly underlines the strong differences between the Wigner and the $Q$ and $P$-currents for SU(2) quadratic Hamiltonians: while for $Q$ and $P$ the quantum effects are generated by the gradient operator, the Wigner current also involves action of the Laplace operator, which leads to a significantly more involved phase-space interference patterns. The effect of dissipation is twofold: it destroys the interference and generates a flow towards the south pole. Nevertheless, as we have seen in the example of the stable LMG evolution, the impact of the decoherence on a given Hamiltonian dynamics depends essentially on the location of the initial distribution. It is worth noting that in multi-spin case, when the classical phase space is a direct product of several two-spheres, the geometrical representation of the currents (\ref{Js}) would not be directly possible, but the analytical properties of the stagnation points (e.g. the winding numbers) still can provide a useful information about the character of nonlinear evolution. \section*{Acknowledgments} This work is partially supported by the Grant 254127 of CONACyT (Mexico). L.L.S.S acknowledges the support of the Spanish MINECO (Grants FIS2015-67963-P and PGC2018-099183-B-I00).

proofpile-arXiv_069-13493

\section{Conclusion}\label{Conclusion} Smart contracts are immutable, verifiable, and autonomous software, hosted and ran on blockchains, like Ethereum. Due to their immutable nature, smart contracts cannot be changed, once they are deployed on the blockchain. Therefore, it is important for developers to fully analyze their smart contracts before deployment. In this research, we analyzed a comprehensive list of known bugs in Solidity smart contracts, and proposed 10 classes of mutation operators. Our experiments show that our operators can regenerate the real faults for 10 smart contracts out of 15 famous buggy ones. We have also extended the universal mutator tool with our mutation operators to automatically generate mutants for smart contracts. We believe that our designed operators can help Solidity developers (and testers) to better develop bug-free and safe smart contracts. \section{Discussion} \section{Evaluation}\label{Evaluation} Mutation testing is a well-known technique in software testing, and different mutation operators have been proposed for several languages. There are a few methods to evaluate the effectiveness of the designed mutation operators~\cite{jia2010analysis,gopinath2014mutations}. One way is to analyze the real faults already detected in software developed in the studied language, and find the number of faulty codes that can be regenerated by applying one of the mutation operators to the corresponding corrected code. High percentage in this evaluation can indicate the resemblance of the mutation operators to the real faults, and can be an indication to the power of the mutants in selecting effective test cases (the ones that can find the bugs in the code). To evaluate the effectiveness of our proposed mutation operators using this technique, we first gathered a comprehensive set of real world known bugs from a number of available lists including~\cite{atzei2016survey, not-so-smart-contracts,dasp,awesome-buggy-erc20,solidity-known-attacks, luu2016making}. Then, we applied our mutation operators to the source code of the fixed version of these smart contracts to see if we can generate the previously detected bugs. Table~\ref{table:evaluation} summarizes our evaluation results for these real world case studies, which shows that our mutation operators can regenerate 10 out of 15 real world bugs. Here, we list all the designed mutation operators related to each real world bug, and the operators that could regenerate the bug have been marked with ``*''. \begin{table*} \centering \scriptsize \begin{tabular}{|p{2cm}|l|p{2.4cm}|l|l|c|} \hline \textbf{Bug Class} & \textbf{The Bug} & \textbf{Related Operators} & \textbf{Value} & \textbf{Date} & \textbf{Reproducable?} \\\hline Re-Entrancy & The DAO \cite{dao-report-slockit} & SCL*, SCR & 3M \textit{Ether} ($\sim$ \$50M) & June 2016 & \OK \\\hline \multirow{2}{2cm}{Delegated Call} & Parity: ``MultiSig" \cite{parity-multisig-bug} & IPuR*, IPrR, IER & 153K \textit{Ether} ($>$ \$30M) & July 2017 & \OK \\ & Parity: ``I Accidentally Killed It!" \cite{parity-accidentally-kill} & RSF, ASF* & 500K \textit{Ether} ($\sim$ \$300M) & Nov 2017 & \OK \\\hline \multirow{3}{2cm}{Arithmetic} & PoWN Overflow \cite{powh-underflow} & IST, DST, DReq* & 886 \textit{Ether} ($\sim$ \$800K) & Feb 2018 & \OK \\ & BeautyChain Batch Overflow Bug \cite{bec-batch-overflow}& IST, DST, DA* & $10^{58}$ \textit{BEC} & April 2018 & \OK \\ & ICON Disable All Transactions \cite{icon-fatal} & LCR* & Potentially \$800M & Jun 2016 & \OK \\\hline \multirow{2}{2cm}{Constructor Name} & Rubixi \cite{rubixi} & CCN*, CFC & Potentially 108 \textit{Ether} & April 2016 & \OK \\ & Morph Case Sensitive Constructor \cite{morph-token-constructor} & CCN*, CFC & $\sim$ 400 \textit{Ether} & June 2018 & \OK \\\hline \multirow{2}{2cm}{Timestamp-dependence} & The Run \cite{why-smart-contracts-fail} & & & & \NOK \\ & GovernMental: ``Timestamp" \cite{atzei2016survey} & & & & \NOK \\\hline \multirow{3}{2cm}{Not Checked Return Values} & KoET Return Check \cite{koet-postmortem} & CSR*, SCR, DReq* & 98.5 \textit{Ether} & Feb 2016 & \OK \\ & Etherpot Blockhash Bug \cite{etherbot-blockhash} & & & Sept 2016 & \NOK \\ & SmartBillions Randomness \cite{smart-billions} & & $\sim$ 400 \textit{Ether} & Oct 2017 & \NOK \\\hline Dynamic Libraries & Re-Entrancy Honey Pot \cite{reentrancy-honeypot} & CAA* & 1 \textit{Ether} & Feb 2018 & \OK \\\hline Denial of Service & Governmental: ``Too Much Gas" \cite{governmental-too-much-gas} & & 1100 \textit{Ether} & Apr 2016 & \NOK \\\hline \end{tabular} \caption{Evaluation Results}\label{table:evaluation} \end{table*} In the following, we discuss more details on some of these faults in the real world examples, especially the ones that cannot be regenerated by our mutation operators. ``Re-Entrancy'' and ``Delegated Call'' bugs have been the most costly faults in the Ethereum history. Our mutation operators can successfully regenerate these classes of faults in smart contracts. Arithmetic issues are also the other most repeated faults in smart contracts. To detect these arithmetic issues, we have proposed Solidity-specific mutation operators, in addition to the classical operators for arithmetic. These operators help us to cover more real faults in the arithmetic class of bugs in Solidity language. Another class of known real world bugs are Timestamp-dependence faults, which are very common in lottery ERC20 tokens (e.g. \cite{atzei2016survey, why-smart-contracts-fail}). This bug happens when a smart contracts uses the block timestamp values as a random seed, ignoring the fact that a malicious miner can manipulate the timestamp value within 900 seconds to her benefits~\cite{luu2016making}. Similarly, some smart contracts use the \lstinline|blockhash| functions to compute the hash values of blocks. This function can only compute the hash values for the last 256 Ethereum blocks and in other cases, it will return zero. There are many faulty lottery smart contracts that use \lstinline|blockhash| function to generate random numbers~\cite{etherbot-blockhash, smart-billions}. To avoid these two faults, we suggest developers to use 3rd party libraries, which provide secure random generators~\cite{chatterjee2019probabilistic, rando}. Since the solutions (corrections) for the bugs related to generating random numbers are complicated mechanisms, we could not design any mutation operator that can regenerate this bug from the corrected code. To evaluate our mutation operators in practice, we have also extended a well-know tool for generating mutants, called \textit{Universal Mutator}~\cite{universal}, and added our Solidity-related mutation operators to it. The enhanced tool accepts a Solidity smart contract and generates all its possible mutants based on our proposed operators. One of the great features about this tool is detecting invalid (the ones that cannot be compiled) and redundant mutants. The source code for this tool is available online on Github~\cite{universal-mutator}, and can be used by other researchers and smart contract developers. We used this tool to generate the mutants for a number of well-known smart contracts. The number of valid, invalid, and redundant mutants generated for each smart contract can be seen in Table~\ref{table:Mutants generated by Universal Mutator}. \begin{table}[] \centering \begin{tabular}{|l|l|l|l|l|} \hline Smart Contract & Valid & Invalid & Redundant & Total \\ \hline Smartex & 60 & 57 & 25 & 142 \\ \hline ethBank & 154 & 22 & 47 & 223 \\ \hline NEST\_LoanContract & 129 & 169 & 89 & 387 \\ \hline BITNOMO & 71 & 136 & 65 & 272 \\ \hline WOR & 45 & 83 & 42 & 170 \\ \hline \end{tabular} \caption{Mutants generated by the extended universal mutator} \label{table:Mutants generated by Universal Mutator} \end{table} \section{Introduction}\label{sec:introduction} Blockchain is an immutable distributed ledger, where users can trigger transactions by creating a wallet~\cite{nakamoto2008bitcoin}. In blockchain, there is no central database, and the full history of transactions is stored by all network nodes. Adding new transactions is only devoted to {\em miners}, who try to produce new blocks out of received transactions to earn rewards. In late 2013, Buterin et al. published the Ethereum white paper~\cite{buterin2014ethereum}, where they introduced Ethereum as a global, open-source platform for decentralized applications (known as {\em smart contracts}) based on blockchain. Smart contracts are immutable pieces of code, stored and executed autonomously by the Ethereum miners. They can hold \textit{Ether} or new defined assets (ERC20 tokens), and include rules for sending (or dispensing) these assets. Several smart contracts have been published on the Ethereum network for different applications, including security tokens, voting, gambling and lottery, property ownership, stocktaking, etc. Millions of dollars are carried by smart contracts, and hence, any security vulnerability in these smart contracts can lead to huge monetary losses. As an example, we can mention the famous DAO attack~\cite{daoattack}, which resulted in loss of 150 million dollars. Due to the immutable nature of smart contracts, their correctness is significant. There are more than 34000 vulnerable smart contracts on the Ethereum blockchain, carrying about 4905 \textit{Ether}~\cite{nikolic2018finding}, which shows the need for effective analysis techniques for them. Therefore, guaranteeing the correctness of smart contracts have recently attracted researchers in software engineering and distributed computing. There have been several attempts to analyze the correctness of smart contracts. Most of these works focus on static analysis of smart contracts based on the known bug patterns~\cite{luu2016making,kalra2018zeus}. Static analysis is a strong tool for evaluating software quality, however, it has its own limitations. First of all this technique can only find the bugs which are matched with the known patterns, and any other bug cannot be detected. Moreover, since the code is not executed in this technique, there may be many false positives in the reported bugs. In other words, some reported bugs may correspond to the paths of the code which are not possible to execute. Therefore, static analysis techniques need to be strengthened with dynamic analysis methods including testing. One of the main challenges in testing is test design. More specifically, from the large set of possible scenarios, the question is how to select a sufficiently small subset of test cases that are more likely to find the potential bugs. Several techniques have been introduced in the test community for designing effective test cases. It is crucial to have a test design technique, considering the specific features of smart contracts, and the bugs resulting from the distributed nature of their running environment. Our idea is to use mutation testing approach for smart contract testing. Mutation testing is known as the strongest technique for test design. There are several mutation operators designed for different programming languages. In this paper, our goal is to propose a set of mutation operators specifically designed for the Solidity programming language, considering the known real bugs made by smart contract developers. There is one paper with similar idea~\cite{wu2019mutation}, where 15 mutation operators are proposed {\em based on Solidity documentation}. Note that the power of mutation testing is very much dependent on its mutation operators, and we believe the operators that can mimic the real bugs can select more effective test cases. We did an extensive study to identify the known bugs in the scope of smart contracts using many academic papers, open-source smart contracts (and their open or closed issues) on Github~\cite{github}, articles in the websites like Medium~\cite{medium}, and Q\&A websites like Stack-Exchange~\cite{exchange}. We used this list of real world known bugs to design a set of mutation operators. We also proposed a set of mutation operators for Solidity specific features, including constructor, modifiers, methods for transaction call, and also self-destruct. \\ Our contributions in this work are as follows: \begin{itemize} \item We propose a set of mutation operators for Solidity inspired by the real faults in smart contracts written in this language. \item We evaluated the effectiveness of our mutation operators by applying them to the fixed version of famous buggy smart contracts, and finding the percentage of real bugs generated by our operators. \item We extended the {\em Universal Mutator}~\cite{universal} tool by our mutation rules to generate mutants for smart contracts automatically. \end{itemize} This paper is organized as follows. In Section~\ref{ssec:Preliminaries}, we briefly discuss the preliminary concepts including smart contracts and mutation testing. In Section~\ref{sec: Mutation Testing of Smart Contracts}, we present our mutation operators with details about the intuition behind them. In Section \ref{Evaluation}, we discuss our techniques for evaluating the effectiveness of our mutation operators. We briefly mention the related works in Section~\ref{ssec:Related Work}, and concluding remarks are presented in Section~\ref{Conclusion}. \section{Mutation Testing of Smart Contracts}\label{sec: Mutation Testing of Smart Contracts} In this research, our goal is to propose a systematic approach for selecting effective test cases for smart contracts based on mutation. The strength of this method is very much dependent to its mutation operators. Mutation operators are usually inspired by the common faults in a language or domain. In other words, they usually mimic the common bugs of the programs written in a language. Different mutation operators may be applicable for many programming languages, while some of them may be language-specific, and depending on the specific features of that language. There are different languages for writing smart contracts in Ethereum, among which, in this paper we just focus on the Solidity programming language. Solidity is very similar to JavaScript, but it includes more functions to support writing smart contracts. To design effective mutation operators for Solidity smart contracts, we first conducted an extensive study on the known real world bugs in the scope of smart contracts. Our sources of investigation are academic papers, open-source smart contracts repositories including open or closed issues on Github, articles in the websites like Medium and Q\&A websites like Stack-Exchange~\cite{exchange}. We ended up with a collection of most repeated bugs that may happen in the implementation of a smart contract in Solidity programming language. Studying the bugs in smart contracts, we categorize them to two groups: \begin{enumerate} \item {\em Classic Bugs:} These bugs occur in almost any programming language, from which we can mention arithmetic issues or logical bugs (inside conditions). \item {\em Solidity Bugs:} These faults are mostly related to the Solidity programming languages, and the distributed nature of blockchain and smart contracts. \end{enumerate} Hence, we conclude that classical mutation operators designed for general-purpose programming languages, e.g. JavaScript, are not sufficient for the Ethereum platform, and we need to design other mutation operators to mimic the Solidity specific bugs. So, we divide our mutation operators into two groups: \textit{(i)} Classic Mutation Operators, and \textit{(ii)} Solidity Mutation Operators. \subsection{Classic Mutation Operators (CMO)}\label{ssec:Designing of CMO} Classic mutation operators include a set of mutation operators, which can be used for almost any programming languages (with maybe some minor differences due to the difference in syntax). Among these operators, we can mention arithmetic and logical operators. They are designed to target a group of bugs that are common in most programming languages. Effective classic mutation operators have been designed for different languages, which also can be used for Solidity smart contracts~\cite{ma2006evaluation}. These operators mostly contain insertion, deletion, and replacement of arithmetic and logical operators, some of which are depicted in Table~\ref{table:classic mutation operators}. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline ABS & ABSolute value insertion \\ \hline AOR & Arithmetic Operator Replacement \\ \hline CRP & Constants RePlacement \\ \hline CRR & Constants for Reference Replacement \\ \hline LCR & Logical Connector Replacement \\ \hline ROR & Relational Operator Replacement \\ \hline RCR & Reference for Constant Replacement \\ \hline FDL & Formula DeLetion \\ \hline FRC & Formula Replacement with Constant \\ \hline RFR & Reference Replacement \\ \hline UOI & Unary Operator Insertion \\ \hline SCL & Swap Code Lines \\ \hline \end{tabular} \caption{Classic mutation operators} \label{table:classic mutation operators} \end{table} \subsection{Solidity Mutation Operators (SMO)}\label{ssec:Designing of SMO} Considering our list of known real-world bugs, we have designed 9 classes of Solidity-specific mutation operators to mimic the common security and software bugs in smart contracts. In this section, we discuss different classes of Solidity-specific bugs and our designed mutation operators for them. \subsubsection{Overflow-Underflow}\label{sssub:Overflow-Underflow} Overflow and underflow bugs are common in many programming languages, but in Solidity smart contracts, an underflow or overflow breach can cause huge monetary losses. Overflow happens when a number exceeds the upper bound of its type. For example, assume a \lstinline $uint32$ variable that has the maximum value of $2^{32}-1$. If this variable increases in an operation beyond its maximum value, it will return to its minimum value of $0$. More specifically, consider this scenario of a smart contract, where it is needed to increase someone's balance, but because of overflow, the balance value will be set to $0$. A developer should recheck each statement using some guard functions, to avoid overflow or underflow related bugs. In Listing~\ref{lst:overflow}, you can see an example for a smart contract containing an overflow bug, and a method to secure it. \begin{lstlisting}[caption={Overflow bug},label={lst:overflow},numbers=none] mapping (address => uint32) public balanceOf; // INSECURE function transfer(address to, uint32 value) { //Check if sender has balance require(balanceOf[msg.sender] >= value); //Add and subtract new balances balanceOf[msg.sender] -= value; balanceOf[to] += value; } // SECURE function transfer(address to, uint32 value) { //Check if sender has balance and for overflows */ require( balanceOf[msg.sender] >= value && balanceOf[to] + value >= balanceOf[to] ); //Add and subtract new balances balanceOf[msg.sender] -= value; balanceOf[to] += value; } \end{lstlisting} Similar bug happens with underflow, when one tries to decrease a \lstinline|uint32| variable, for example, below its minimum value (i.e. $0$), and instead of getting a negative value, the variable gets its maximum value ($2^{32}-1$). For example, consider someone tries to withdraw all of her balance from a contract, but instead of setting her balance to $0$, she ends up with the maximum value of the variable type, because of a simple underflow. Listing~\ref{lst:underflow} shows an example source code containing an underflow bug: \begin{lstlisting}[caption={Underflow bug},label={lst:underflow},numbers=none] mapping (address => uint32) public balanceOf; // INSECURE function transfer(address to, uint32 value) { //Check if sender has balance require(balanceOf[msg.sender] >= value); //Add and subtract new balances balanceOf[msg.sender] -= value; balanceOf[_to] += value; } // SECURE function transfer(address to, uint32 value) { //Check if sender has balance and for overflows require( balanceOf[msg.sender] >= value && balanceOf[to] - value <= balanceOf[to] ); //Add and subtract new balances balanceOf[msg.sender] -= value; balanceOf[to] += value; } \end{lstlisting} In Listings~\ref{lst:overflow} and~\ref{lst:underflow}, assuming that balances are real assets like \textit{Ether}, simple overflow or underflow can cost millions of dollars. To detect these bugs, we propose Overflow-Underflow mutation operators, which are depicted in Table~\ref{table:overflow-underflow}. In IST and DST operators, the size of a type is increased/decreased respectively. For example, the type \lstinline $int32$ is replaced by \lstinline $int8$, or \lstinline $int64$. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline USP & Unsigned to Signed Replacement \\ \hline PSU & Signed to Unsigned Replacement \\ \hline IST & Increase Size of a Type \\ \hline DST & Decrease Size of a Type \\ \hline \end{tabular} \caption{Mutation operators for Overflow-Underflow} \label{table:overflow-underflow} \end{table} \subsubsection{Access Control}\label{sssec:Acess Control} It is a must for every developer to understand how she can restrict parts of her code from other parts. In smart contracts, the access controls is very important and the corresponding bugs can be very dangerous and potentially leads to big loss of assets. In Solidity, there are four levels of access control, namely Public, External, Private, and Internal~\cite{soldoc}. The difference between these types can be seen in Table~\ref{table:AClevels}. \begin{table}[] \centering \begin{tabular}{|l|p{5.7cm}|} \hline Access Control & Definition \\ \hline Public & All can access \\ \hline External & Cannot be accessed internally, only externally \\ \hline Internal & Only this contract and contracts deriving from it can access \\ \hline Private & Can be accessed only from this contract \\ \hline \end{tabular} \caption{Access control levels in Solidity} \label{table:AClevels} \end{table} Listing~\ref{lst:access control} shows an example for access control bug in a smart contract. This contract has a public function called \lstinline {helpCharity} that everyone can call to transfer some assets to the contract address from her balance. Also, there is another function named \lstinline $transfer$, which should be called from \lstinline {helpCharity} to transfer asset to the charity's address. As this function is mistakenly defined as public, an attacker can call \lstinline $transfer$ function and transfer all assets to her address, instead of the charity's address. \begin{lstlisting}[caption={Access control bug},label={lst:access control},numbers=none] mapping (address => uint256) public balanceOf; //Public function so everyone can help the charity function helpCharity(uint256 value) public { let charityAddress = 0xe0f5206bbd039e7b0592d8918820024e2a7437b9; transfer(to: charityAddress, value: 1); balanceOf[msg.sender] -= value; } // transfer mistakenly defined as a public instead of private function transfer(address to, uint256 value) public { balanceOf[to] += value; } \end{lstlisting} To detect the access control bugs, we have designed a set of mutation operators, which are listed in Table~\ref{table:access control}. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline PuPrR & Public to Private Replacement \\ \hline PuIR & Public to Internal Replacement \\ \hline PuER & Public to External Replacement \\ \hline PrPuR & Private to Public Replacement \\ \hline PrIR & Private to Internal Replacement \\ \hline PrER & Private to External Replacement \\ \hline IPuR & Internal to Public Replacement \\ \hline IPrR & Internal to Private Replacement \\ \hline IER & Internal to External Replacement \\ \hline EPuR & External to Public Replacement \\ \hline EPrR & External to Private Replacement \\ \hline EIR & External to Internal Replacement \\ \hline \end{tabular} \caption{Mutation operators for access control} \label{table:access control} \end{table} \subsubsection{Transaction Call Mechanism}\label{sssec: Transaction Call Mechanism} In Ethereum, we can use transactions to send an asset from one account to another account or trigger another contract's code. There are three methods for creating a transaction in Solidity smart contracts. These methods are different in terms of gas usage and their return value, and hence, each one should be used in appropriate situations. These three methods are as follows: \begin{enumerate} \item \lstinline $send$: It only includes 21000 \textit{wei} (the smallest denomination of \textit{Ether}) into the transaction. That means the execution process of the called function must not exceed 21000 \textit{wei}, otherwise the transaction will be incomplete, and the \lstinline|send| returns \lstinline|false|. \item \lstinline $transfer$: \lstinline $transfer$ also includes only 21000 \textit{wei} into the transaction. However, if the execution of a function exceeds 21000 \textit{wei} of gas, it will throw an exception, rather than returning \lstinline|false|. \item \lstinline $call$: Using this method can be dangerous, as it transfers all the remaining gas for the transaction to the called function. \end{enumerate} An example of a bug caused by inappropriate usage of transaction call is {\em reentrancy}. This attack occurs when a function from the first smart contract makes an external call to another one using \lstinline $call$ function. Listing~\ref{lst:reentrancy example} shows an example of this bug, where an attacker can deploy a contract and define its fallback function to rerun the \lstinline|withdraw| function. After deploying the contract, she can call \lstinline $Victim$'s \lstinline|withdraw| function, and \lstinline $msg.sender.call.value(amount)()$ is going to send some assets to \lstinline $Attacker$'s contract, which results in calling the fallback function of the \lstinline|Attacker| that will call the \lstinline|withdraw| function again, and so on. Since \lstinline $Victim$ uses \lstinline $call$ as a mechanism to send asset, \lstinline $call$ is going to send all the remaining gas to the attacker's contract and the attacker can take control of the transaction flow and recursively call \lstinline $withdraw$ function, till all the gas burns. Therefore, in this scenario an attacker can steal all the contract's assets before the \lstinline|Victim| contract sets her balance to zero. But, if \lstinline $Victim$ uses \lstinline $send$ function instead of \lstinline $call$, it would only use 21000 \textit{wei}, and \lstinline $Attacker$'s fallback function would throw ``OutofGas" exception because of lack of gas. \begin{lstlisting}[caption={Reentrancy Attack},label={lst:reentrancy example},numbers=none] //Victim function function withdraw() external { uint256 amount = balances[msg.sender]; require(msg.sender.call.value(amount)()); balances[msg.sender] = 0; } //Attacker fallback function function() external payable { Victim v; v.withdraw(); } \end{lstlisting} We have designed 6 mutation operators to take care of transaction call mechanism bugs that can be seen in Table~\ref{table:transaction call mechanism mutant operators}. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline TCR & Transfer to Call Replacement \\ \hline TSR & Transfer to Send Replacement \\ \hline SCR & Send to Call Replacement \\ \hline STR & Send to Transfer Replacement \\ \hline CSR & Call to Send Replacement \\ \hline CTR & Call to Transfer Replacement \\ \hline \end{tabular} \caption{Mutation operators for transaction call mechanism} \label{table:transaction call mechanism mutant operators} \end{table} \subsubsection{Guard Mechanism}\label{sssub:Guard Mechanism} In Solidity, it is possible to use {\em guards} as a way of handling errors. Guards are some state-reverting exceptions which will undo all changes in the EVM if a condition (a Boolean expression) is not met. There are three guard functions available in Solidity programming language, which are \lstinline $assert$, \lstinline $require$, and \lstinline{revert}. Each one has its own behavior, and a developer should be careful about using the appropriate one. Careless usage can result in freezing the contract, or making it vulnerable to attacks. When we use \lstinline $assert$ as a guard function, this operator will use all the associated gas, but \lstinline $require$ and \lstinline $revert$ will send back the remaining gas. As an example, in Listing~\ref{lst:guard mechanism}, if the developer forgets to use \lstinline $require$ in the \lstinline $justOwner$ function, anyone can call \lstinline $justOwner$ which will lead to abnormal and unexpected behavior. \begin{lstlisting}[caption={Guard mechanism bug},label={lst:guard mechanism},numbers=none] address owner; function justOwner() { require(msg.sender == owner) //if someone rather than owner call this function something bad will happen. } \end{lstlisting} Table~\ref{table:guard mechanism mutant operators} shows our mutation operators for the guard mechanism. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline ARevR & Assert to Revert Replacement \\ \hline AReqR & Assert to Require Replacement \\ \hline RevReqR & Revert to Require Replacement \\ \hline RevAR & Revert to Assert Replacement \\ \hline ReqAR & Require to Assert Replacement \\ \hline ReqRevR & Require to Revert Replacement \\ \hline AReq & Add Require \\ \hline AA & Add Assert \\ \hline ARev & Add Revert\\ \hline DReq & Delete Require \\ \hline DRev & Delete Revert \\ \hline DA & Delete Assert \\ \hline \end{tabular} \caption{Mutation operators for guard mechanism} \label{table:guard mechanism mutant operators} \end{table} \subsubsection{Transaction Origin}\label{sssub: Tx Origin Attacks} There are a number of fields in the \lstinline|msg| object that provide more information for the receiver of the message (transaction). For example, \lstinline|msg.sender| identifies who has triggered the transaction, \lstinline|msg.value| contains the value of attached assets, and \lstinline|msg.data| returns the attached date to the transaction, if any. Also, \lstinline|tx.origin| can be used to find out the original first user who has triggered the transaction. A transaction can be the result of a chain of other transactions, which have called each other, and in these cases, \lstinline|tx.origin| can identify who is the root of this call chain. Inappropriate usage of \lstinline $tx.origin$ and \lstinline $msg.sender$ can put a smart contract into the danger of an attack. As an example, in Listing~\ref{lst:tx origin attacks}, there are two smart contracts, called \lstinline $Victim$ and \lstinline $Attacker$. In \lstinline $Victim$, only the owner of the smart contract can call the \lstinline $transfer$ function and send assets to an address. The developer has mistakenly used \lstinline $tx.origin$ to see whether the caller is the owner or not. If the owner wants to transfer some assets to the \lstinline $Attacker$ contact, after triggering the transaction, assets will be transferred to the \lstinline $Attacker$ contract, resulting in running its fallback function. The statements in the fallback function causes to transfer any number of funds to the \lstinline $Attacker$, because the very first origin of the transaction is the \lstinline|Victim|'s owner. To prevent this, the developer should check \lstinline|msg.sender|, instead of \lstinline|tx.origin|. \begin{lstlisting}[caption={A transaction origin atack},label={lst:tx origin attacks},numbers=none] //VICTIM contract Victim { address owner; constructor() public { owner = msg.sender; } function transfer(address to, uint64 amount) public { require(tx.origin == owner); to.call.value(amount)(); } } //ATTACKER interface Victim { function transfer(address to, uint amount); } contract Attacker { address owner; constructor() public { owner = msg.sender; } function() payable public { let randomValue = 64; Victim(msg.sender). transfer(owner, randomValue); } } \end{lstlisting} Table~\ref{table:Tx origin mutation operators} shows our mutation operators, designed for Transaction Origin attacks. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline TMR & Tx.origin to Msg.sender Replacement \\ \hline MTR & Msg.sender to Tx.origin Replacement \\ \hline \end{tabular} \caption{Mutation operators for Transaction Origin} \label{table:Tx origin mutation operators} \end{table} \subsubsection{\lstinline $selfdestruct$ Operator}\label{sssub: Manipulate selfdestruct} The \lstinline|selfdestruct| function (previously known as \lstinline $suicide$), helps developers to destruct a smart contract. This function takes an address as argument to transfer all the stored assets in the smart contract to that, before destructing the smart contract. For example, calling \lstinline $selfdestruct(addr)$ sends all the contract's current balance to the address \lstinline $addr$. This operator is useful, when we are done with a smart contract, as it transfers assets to a given address with consuming less gas amount than simple transfer using \lstinline $addr.send(this.balance)$. In fact, because of freeing up some space in the EVM, \lstinline|selfdestruct| actually uses negative gas. Thus, using this operator wisely can be helpful, however, if a developer mistakenly uses this function, it will lead to destroying the smart contract and it may cause loss of stored assets. In order to deal with this bug, we have designed two mutation operators to remove \lstinline $selfdestruct$ from the code, and swap its location to adjacent lines (Table~\ref{table:Manipulate selfdestruct}). \\ \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline RSF & Remove selfdestruct from a Function \\ \hline SSL & Swap selfdestruct's location to adjacent lines \\ \hline \end{tabular} \caption{Mutation operators for \lstinline $selfdestruct$} \label{table:Manipulate selfdestruct} \end{table} \\ \subsubsection{Constant Properties}\label{sssub:Manipulate constant properties} Each smart contract can use many constant values, including smart contract or user addresses, number of tokens (e.g., the number of tokens to issue), gas amount on each transaction call, etc. Setting wrong constant values can lead to incorrect behavior of a smart contract. For example, using wrong constant address can cause sending money to a wrong user, or running wrong function from a malicious contract. Settings wrong gas amounts may fail a transaction, or wrong value of issued token can cause huge monetary losses. Therefore, developers should set the constant values carefully in smart contracts before deploying them on EVM. Table~\ref{table: manipulate constants properties mutant operators} depicts our mutation operator designed for constants values used in a smart contract. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline CAA & Change Address to another Address \\ \hline CDG & Increase and Decrease in Gas amount \\ \hline CCV & Change in any Constant Value \\ \hline \end{tabular} \caption{Mutation operators for manipulating constants} \label{table: manipulate constants properties mutant operators} \end{table} \subsubsection{Function Modifiers}\label{sssub: Manipulate Modifiers} Function modifiers can be defined as Boolean conditions added to a function declaration. When a function with modifier is called, first the modifiers are validated, and only if all its modifiers are satisfied, the function begins to execute. The function modifiers are widely used to check the authority of Ethereum users for calling a smart contract's function. Hence, wrong or buggy modifier can cause unauthorized access to important functions of a smart contract. As an example, in Listing~\ref{lst:manipulate modifiers}, there is a function, called \lstinline $justOwner$ with the \lstinline $onlyOwner$ modifier. This modifier restricts calling this function to the contract owner. Using this modifier can be very important, as the function may include transferring the contract assets. You can see that a small bug in the modifier, such as changing \lstinline $==$ to \lstinline $!=$, can cause unauthorized access of an attacker. \begin{lstlisting}[caption={An example of using modifier},label={lst:manipulate modifiers},numbers=none] address owner; modifier onlyOwner { require(msg.sender == owner); } function justOwner() onlyOwner { // transferring the contract assets } \end{lstlisting} Table~\ref{table:manipulate modifiers mutant operators} shows our proposed mutation operators for manipulating modifiers. These mutation operators change each modifier to the constant values of \lstinline|true| or \lstinline|false|. Selected test cases by these operators can better check the correctness of the implemented modifiers. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline CMT & Change Modifier to \lstinline|true| \\ \hline CMF & Change Modifier to \lstinline|false| \\ \hline \end{tabular} \caption{Mutation operators for manipulating modifiers} \label{table:manipulate modifiers mutant operators} \end{table} \subsubsection{Constructor Name}\label{sssub: Manipulate Smart Contract's Constructor} In Solidity, a smart contract can be initialized by a constructor, where the required initialization statements of a smart contract take place before deploying it to the EVM. For example, initialization of the owner property of the smart contract can be performed in its constructor. Correct spelling of the constructor name is very important, as a mistake in its spelling will cause Solidity not recognizing the function as a constructor. On the other hand, if a function is mistakenly named as the constructor's name, it is going to be behaved like a constructor, and can cause unexpected behaviors. Listing~\ref{lst:Constructor} is an example, where misspelling of a constructor function can cause damage to the contract. In this example, a typo in the name of the constructor made it a simple regular function named \lstinline $Exampl$. Thus, any user can call this method and introduce herself as the contract owner. \begin{lstlisting}[caption={An example of a bug caused by constructor name},label={lst:Constructor},numbers=none] address owner; contract Example { // Correct Behaviour function Example(address add) public { owner = add } // Wrong Behaviour (typo on contract name) function Exampl(address add) public { owner = add } } \end{lstlisting} Table~\ref{table:Manipulate Smart Contract's Constructor Name} shows our mutation operators for manipulating the smart contract's constructor name. \begin{table}[] \centering \begin{tabular}{|l|l|} \hline Operators & Description \\ \hline CCN & Change Constructor Name to something else \\ \hline CFC & Change a Function's name to Constructor \\ \hline \end{tabular} \caption{Mutation operators for manipulating constructor name} \label{table:Manipulate Smart Contract's Constructor Name} \end{table} \section{Preliminaries}\label{ssec:Preliminaries} In this section, we briefly discuss the preliminary concepts used in this work, including the notion of smart contract, as well as mutation testing concepts. \subsection{Smart Contracts}\label{ssec:Smart Contracts} As we discussed earlier, Ethereum provides a Turing-complete programming language named Solidity for developing smart contracts. Each smart contract is an Ethereum account that holds a piece of code (i.e. the source code for the smart contract), has some private storage, and also can hold some money in \textit{Ethers} (the Ethereum currency). Thus, there are two types of accounts in Ethereum: \begin{itemize} \item External accounts, which are owned by Ethereum users and are like simple bank accounts, and \item Smart contract accounts \end{itemize} Both users and smart contracts can initiate transactions for transferring currency or running a smart contract, which will change the state of the transaction-based Ethereum Virtual Machine ({\em EVM}). The execution of smart contracts and verifying the validity of transactions are devoted to a set of special network nodes, called {\em miners}, which get fee from the transactions' initiators for doing the computations. In the Ethereum network, smart contracts can be written it different languages such as JavaScript, C++, and Solidity. To deploy a smart contract on Ethereum, it should be compiled to EVM byte-code and stored on the Ethereum network. Then, the EVM allocates a limited storage for the compiled contract and assigns an address to both compiled smart contract and the allocated storage. Any Ethereum user can later use the assigned address to trigger any function from the compiled contract. To submit a transaction, either for transferring money or triggering a function from a smart contract, a user should pay some fee in {\em Ether}, known as {\em gas}. Paying the gas prevents the Ethereum network from being attacked by wasteful tasks (or getting stuck in an infinite loop), and also is an incentive for the miners to run the smart contract or verify the transaction. In the Ethereum network, the gas amount is related to the complexity of the code, meaning that an Ethereum user needs to pay more gas to run complex smart contracts. In a case that a user do not pay sufficient gas value, the EVM raises an exception and all changes caused by the transaction will revert. \subsection{Mutation Testing}\label{ssec:Mutation Testing} Mutation Testing is one of the strongest techniques for test design. The idea behind mutation testing is changing the source code of a program with tiny syntactic changes, called \textit{mutation operators}. The mutation operators are language specific changes that are designed considering the features of the language, or the common faults made by software developers. For example, a mutation operator may change the $+$ mathematical operator to $-$, or remove negation from a logical condition. To test a program, we should apply all mutation operators to the source code of the program (one after another), which results in a set of modified programs, called \textit{mutants}. To select a set of test cases from a pool of randomly generated scenarios, they are run on the program and the mutants. If the output of running the test case on the program is different from the output of running it on a mutant, the mutant is said to be {\em killed}, and that test case is selected. {\em Mutation score} for a set of test cases corresponds to the percentage of mutants killed by these scenarios, and is a metric for evaluating the effectiveness of test cases. The philosophy of mutation testing is based on two hypotheses: the competent programmer hypothesis (CPH) \cite{budd1979mutation},\cite{demillo1978hints}, and the coupling effect hypothesis \cite{demillo1978hints}. CPH states that the competition among the developers makes them develop a code that is really close to the final correct program. This indicates that although there may be faults in a program, but most of them are minor, and can be fixed by minor changes in the program. Therefore, mutation operators are designed to add tiny changes in the program. The Coupling Effect states that a test case identifying the difference between a small changing mutant and the original programs, can identify more complicated faults as well. \section{Related Work}\label{ssec:Related Work} In this section, we discuss the related works in two areas of analyzing smart contracts and mutation testing. \subsection{Mutation Testing}\label{ssec:Mutation Testing in Software Engineering} Mutation Testing was first proposed by Lipton in~\cite{lipton1971fault}. The technique can be used in different levels of testing, including unit, integration, and specification. Mutation can be used to identify effective test suite that developers can use to test their application~\cite{geist1992estimation}. Mutation operators are often designed to mimic the common faults in a programming language. The technique has been widely used for test design in different programming languages, including Fortan~\cite{budd1979mutation}, Ada~\cite{bowser1988reference}, C~\cite{agrawal1989design}, and Java~\cite{epstein1963physiological}. It has even been used to test non-functional properties of a software, such as energy testing of Android applications~\cite{jabbarvand2017mudroid}. \subsection{Analysis of Smart Contracts}\label{ssec:Mutation Testing in Smart Contracts} There are several works on analyzing smart contracts for bugs and vulnerabilities. They can be categorized into three groups; static analysis techniques, formal verification, and testing. Luu et al. in~\cite{luu2016making} introduced OYENTE, which is a tool to statically analyze smart contracts using symbolic execution . Symbolic execution can be used to identify the inputs needed to reach to specific parts of a code. Bhargavan et al. in~\cite{bhargavan2016formal} introduced a framework to convert the bytecode of a smart contract to $F^*$, a functional programming language, and then verify a set of specifications for it. The limitation of this framework is due to the complexity of verification, and as reported in the paper, only 49 smart contracts out of a list of 396 contracts could be verified by this framework. Kalra et al. in~\cite{kalra2018zeus} introduced Zeus that uses abstract interpretation and symbolic model checking to formally check smart contracts. Zeus has less false positives compared to similar frameworks. However, it does not support some Solidity operators, like \lstinline $selfdestruct$ or \lstinline $throw$. Brent et al. in~\cite{brent2018vandal} introduced Vandal as a security analysis tool for Ethereum smart contracts. Vandal uses logic-driven program analysis to analyze smart contracts and detect the security bugs and vulnerabilities. Finally, Jiang et al. developed ContractFuzzer which is a fuzzer to test smart contracts for vulnerabilities and security drawbacks~\cite{jiang2018contractfuzzer}. There is one recent work on testing smart contracts using mutation testing approach~\cite{wu2019mutation}. In this work, Wu et al. proposed 15 mutation operators based on Solidity documentations. In comparison, we have surveyed a comprehensive list of reported real bugs in Ethereum smart contracts, and designed our mutation operators based on these bugs. We have also included a set of mutation operators for Solidity-specific features. Using this approach, we have a more complete list of operators, from which we can mention the operators for overflow-underflow, transaction call mechanisms, \lstinline|selfdestruct|, and modifiers. We have also suggested including more classical operators such as SCL, ABS, etc to better detect possible bugs in Solidity smart contracts. Considering the evaluation, we have evaluated our mutation operators by checking their power in regenerating the real faults. We have also extended the \textit{Universal Mutator} with our rules to automatically generate mutants for Solidity smart contracts.

proofpile-arXiv_069-13557

\section{Introduction} In \cite{SS04}, Seidel and Smith defined a singly graded link invariant \textit{symplectic Khovanov cohomology} $Kh^*_{symp}(L)$. It is the Lagrangian intersection Floer cohomology of two Lagrangians in a symplectic manifold $\mathscr{Y}_n$. The manifold $\mathscr{Y}_n$ is built through taking a fiber of the restriction of the adjoint quotient map $\chi:\mathfrak{sl}_{2n}(\mathbb{C})\to Conf^0_{2n}(\mathbb{C})$ to a nilpotent slice $\mathcal{S}_n$. A given link $L$ in $S^3$ can be realized as a braid closure of $\beta_L\in Br_{n}$ for some $n$ depending on $L$. $\beta_L\times id \in Br_{2n}$ gives a path in the configuration space $Conf^0_{2n}(\mathbb{C})$. The parallel transport induces a symplectomorphism of $\mathscr{Y}_n$ to itself, (precisely speaking, an arbitrarily large compact subspace of $\mathscr{Y}_n$). There is a distinguished Lagrangian submanifold $\mathcal{K}$ given by iterated vanishing cycles and let $(\beta_L\times id)(\mathcal{K})$ be its image under the parallel transport. $Kh^*_{symp}(L)$ is defined to be the Floer cohomology group \begin{equation}Kh^*_{symp}(L)=HF^{*+n+w}(\mathcal{K},(\beta_L\times id)(\mathcal{K}))\end{equation}, where $w$ is the writhe of $\beta_L$. There is a conjectural relation between $Kh^*_{symp}$ and the combinatorial Khovanov homology $Kh^{*,*}$: \begin{conjecture}[Seidel-Smith, \cite{SS04}]\label{conj:SS} For any link $L\subset S^3$, $Kh^k_{symp}(L)\cong \bigoplus_{i-j=k}Kh^{i,j}(L)$. \end{conjecture} This conjecture is true over any characteristic zero field, proved by Abouzaid and Smith in \cite{AS19}. For the cases of non-characteristic zero fields, only a few examples have been computed such that the theories are isomorphic, see \cite[Proposition 55]{SS04} for the case of trefoi . Our results rely on the theorem of Abouzaid and Smith that Conjecture~\ref{conj:SS} is true over any characteristic zero field, so we assume the characteristic of the base field $\textbf{k}$ to be zero unless noted otherwise. It is also worth noting that we will be working with cohomology theories throughout the paper. Even if the Seidel-Smith invariant is called symplectic Khovanov homology in some contexts, it is of cohomological type, i.e. the differential raises the homological grading by $1$. Correspondingly, Khovanov homology is also a cohomology theory. We will work with the framework of Manolescu's Hilbert scheme reformulation. The space $\mathscr{Y}_n$ can be embedded symplectically as an open subscheme into $Hilb^n(A_{2n-1})$, the $n$-th Hilbert scheme of the Milnor fibre of $A_{2n-1}$-surface singularity. One of the advantages of Manolescu's reformulation is that we can work with \textit{bridge diagrams} instead, which are decorated link diagrams obtained by breaking the link diagrams into $n$ pairwise disjoint $\alpha$-arcs and $n$ pairwise disjoint $\beta$-arcs such that $\beta$-arcs surpass $\alpha$-arcs at any intersection. These arcs give two Lagrangians $\mathcal{K}_\alpha$ and $\mathcal{K}_\beta$ in $\mathscr{Y}_n\subset Hilb^n(A_{2n-1})$. It is proved in \cite{M04} that $Kh^*_{symp}(L)=HF^{*+n+w}(\mathcal{K}_\alpha,\mathcal{K}_\beta)$ for a specific type of diagram, called a \textit{flattened braid diagram}, where $n$ is the number of strands and $w$ is the writhe of the corresponding braid. All braids can give rise to flattened braid diagrams but not all bridge diagrams are isotopic to flattened braid diagrams. Attempts were made to generalize symplectic Khovanov cohomology to arbitrary bridge diagrams, an $\mathbb{F}_2$-coefficients version by Hendricks-Lipshitz-Sarkar in \cite[Section 7]{HLS15} and a relatively graded version by Waldron in \cite[Section 6]{W09}. In this paper, we give an absolute grading to Waldron's construction as follow: \begin{theorem}\label{thm:bridge} For any oriented bridge diagram, let $w$ be the writhe of the diagram i.e. the number of positive minus the number of negative crossings, $rot$ be the rotation number of the diagram i.e. the number of counterclockwise minus the number of clockwise Seifert circles and $x_0$ be the generator whose coordinates are the starting point of each $\beta$-arcs. Then the Floer cohomology groups \begin{equation} Kh^*_{symp}(L)=HF^{*+gr(x_0)+w+rot}(\mathcal{K}_\alpha,\mathcal{K}_\beta) \end{equation} are link invariants. \end{theorem} As Waldron proved in the relative case, see \cite[Theorem 1.1]{W09}, the absolutely graded invariant defined with bridge diagram is also canonical, i.e. we Floer groups from two equivalent bridge diagrams are related through a canonical isomorphism. The orientation of the diagram, especially for a link diagram, is crucial in computing the correction terms and locating $x_0$ (that there exactly are two generators whose coordinates are all endpoints and the choice depends on the orientation). Throughout this paper, we always assume that our bridge diagrams are oriented. Abouzaid and Smith constructed an endomorphism $\phi$ of $CF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)$, which induces a generalized eigenspace decomposition of $HF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)$, see also \cite{AS16}. The eigenvalues give an additional grading on $HF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)$, called the \textit{weight grading}. We will only use a relative version of the weight grading because an absolute grading relies on choices of auxiliary data in symplectic geometry, called \textit{equivariant structures}, but we will not specify our choices of equivariant structures in this paper. We prove that for any bridge diagram, the relative weight grading recovers the Jones grading (or quantum grading in some contexts) of Khovanov cohomology \begin{theorem}\label{thm:main} Symplectic Khovanov cohomology and Khovanov homology are isomorphic as bigraded vector spaces over any characteristic zero field, where the gradings are related by $k=i-j$ and $wt=-j+c$, where $k$ is the homological grading, $wt$ is the weight grading and $c$ is a correction term of the relative weight grading. \end{theorem} By definition, the weight grading lives in $\bar{\textbf{k}}$, the algebraic closure of the base field. The theorem implies that the weight grading is integral. At the writing of this paper, the author does not know the correction term $c$ to define an absolute weight grading. Precisely speaking, we need to make a specific choice on the equivariant structures on the Lagrangians depending on the writhe, crossing number, and other properties of the bridge diagram. To prove Theorem~\ref{thm:main}, we show that Abouzaid-Smith long exact sequence of symplectic Khovanov cohomology groups, see \cite[Equation 7.9]{AS19} decomposes with respect to the weight grading. In other words, if we fix a weight grading $wt_1$ of the first group, the only non-trivial map can happen between a single weight grading $wt_2$ of the second group and $wt_3$ of the third group. \begin{equation}\ldots\to HF^{*,wt_1}(L_+)\to HF^{*,wt_2}(L_0)\to HF^{*+2,wt_3}(L_\infty)\to HF^{*+1,wt_1}(L_+)\to \ldots \end{equation} In the singly-graded case, the isomorphism of Abouzaid and Smith between symplectic Khovanov cohomology and Khovanov homology can be used in building a commutative diagram between the exact triangle above and the exact triangle of unoriented skein relation in Khovanov homology. We show the maps between the exact triangles given by the isomorphism of Abouzaid-Smith are bigraded by induction on the number of crossings. Abouzaid-Smith purity result \cite[Theorem 1.1]{AS16} leads to a computation for crossingless diagram of an unlink: \begin{proposition}[Abouzaid-Smith, \cite{AS16}] If $L\subset S^3$ is an unlink represented by a crossingless diagram, there exists a choice of equivariant structures on Lagrangians such that for any element $x\in Kh^k_{symp}(L)$, $wt(x)=k$. \end{proposition} This finishes the proof of Theorem~\ref{thm:main}. Our argument so far is diagrammatic, we have not proved that the relative weight grading is independent of bridge diagrams yet. Now that we know the relative weight grading recovers the Jones grading up to an overall grading shift for any diagram and the fact that Jones grading is independent of link diagrams, we prove a conjecture of Abouzaid-Smith, \begin{theorem} \label{thm:relative} The relative weight grading on $Kh^*_{symp}(L)$ is independent of the choice of link diagram. \end{theorem} It is worth noting that the proof of Theorem~\ref{thm:relative} is not internal to symplectic geometry, and the invariance of the relative weight grading relies on the well-definition of the Jones grading in combinatorial Khovanov homology. \textbf{Organization:} The paper is organized as follows: In Section~\ref{ch2}, we review the definition of symplectic Khovanov cohomology and construct an absolute grading on symplectic Khovanov cohomology of bridge diagrams. In Section~\ref{ch3}, we give a precise definition of the weight grading and construct a bigraded unoriented skein exact triangle of symplectic Khovanov cohomology. In Section~\ref{ch5}, we prove the main theorem by showing Abouzaid-Smith's isomorphism between symplectic Khovanov cohomology and combinatorial Khovanov homology preserves the second grading. \textbf{Acknowledgement:} The author would like to thank his thesis advisor Mohammed Abouzaid for his guidance throughout this project. He also thanks Kristen Hendricks, Mikhail Khovanov, Francesco Lin, Robert Lipshitz, and Ivan Smith for many helpful discussions on the subject. Finally, he thanks Columbia University for its supportive studying environment during the early stage of writing the paper and Institut Mittag-Leffler for its hospitality during the later stage of preparation of this paper. The author was partially supported by his thesis advisor's NSF grants DMS-1609148, and DMS-1564172, and by Institut Mittag-Leffler Junior Fellowship. \section{A review of symplectic Khovanov cohomology $Kh^{*}_{symp}(L)$}\label{ch2} We will briefly review the original definition of symplectic Khovanov cohomology and give a formal definition of symplectic Khovanov cohomology of a bridge diagram in Section~\ref{3.2}. We will discuss the homological grading in Section~\ref{3.3}. The construction of this section is not restricted to characteristic zero fields. \subsection{Symplectic Khovanov cohomology for bridge diagrams}\label{3.2} In \cite{SS04}, the link invariant $Kh^*_{symp}(L)$ is first introduced by Seidel and Smith as the Lagrangian intersection Floer cohomology of two Lagrangians in $\mathscr{Y}_n$, constructed as a nilpotent slice in $\mathfrak{sl}_{2n}(\mathbb{C})$. But it is defined through a link diagram presented as a braid closure and the Floer cohomology is computed through the braid action on $\mathscr{Y}_n$. Manolescu introduced a reformulation using Hilbert schemes in \cite{M04}, which is easier to visualize and more similar to other low dimensional invariants, such as Heegaard Floer homology. In this subsection, Floer homology groups are relatively graded. Let us start with Hilbert schemes of points on surfaces. We choose our algebraically closed field to be $\textbf{k}=\mathbb{C}$ and $X$ to be a complex variety. The Hilbert scheme of $n$ points on $X$, $Hilb^n(X)$ is defined to be closed 0-dimensional subschemes of $X$ of length $n$. An important part of this variety is a subvariety consisting of $n$ distinct points but its diagonal where points collide together is really complicated. However, we have the following \textit{Hilbert-Chow morphism} from \cite{N99}: \begin{proposition} The Hilbert-Chow morphism $\pi$ is a natural morphism from the Hilbert scheme of $n$ points on $X$ to the $n$-fold symmetric product of $X$ such that \begin{equation} \pi(Z)=\sum_{x\in X} length(Z_x)[x] \end{equation} Moreover, if $X$ is complex 1-dimensional, then $\pi$ is an isomorphism. If $X$ is complex 2-dimensional, then $\pi$ is a resolution of singularities and $Hilb^n(X)$ is smooth. \end{proposition} Now we specify our complex surface. Consider the following complex surface \begin{equation}S=\{(u,v,z)\in \mathbb{C}^3|u^2+v^2+p(z)=0\}\in \mathbb{C}^3\end{equation}, where $p(z)=(z-p_1)\ldots(z-p_{2n})$. This is isomorphic to a fibre of nilpotent slice of block size $(1,2n-1)$ with eigenvalues $p_i$ in Seidel-Smith construction. Seidel-Smith invariant is defined in the nilpotent slice $\mathscr{Y}_n$ of block size $(n,n)$, which is proved to be an open subscheme of $Hilb^n(S)$, by Manolescu. Denote its complement \begin{equation} D_r=Hilb^n(S)\backslash \mathscr{Y}_n \end{equation} which is a complex co-dimension 1 subvariety and, in fact, a relative Hilbert subscheme given by all elements with length less than $n$. To fully characterize $\mathscr{Y}_n$, consider a projection $i:S\to\mathbb{C}$ such that $i(u,v,z)=z$, then \begin{equation}\mathscr{Y}_n=\{I\in Hilb^n(S)|i(I)\text{ has length }n\} \end{equation} Manolescu proved in \cite[Proposition 2.7]{M04} that $\mathscr{Y}_n$ is biholomorphic to the space $\mathscr{Y}_{n,\tau}$ obtained by a fibre of nilpotent slice by Seidel-Smith in \cite{SS04}. We need to construct Lagrangians from a link diagram. A \textit{bridge diagram} $D$ for a link $L$ is a triple $(\vec{\alpha},\vec{\beta},\vec{p})$, where $\vec{p}=(p_1, p_2,\ldots, p_{2n})$ are $2n$ distinct points in $\mathbb{R}^2$, $\vec{\alpha}=(\alpha_1, \alpha_2,\ldots, \alpha_n)$ are $n$ pairwise disjoint embedded arcs and $\vec{\beta}=(\beta_1, \beta_2,\ldots, \beta_n)$ are also pairwise disjoint embedded arcs such that $\partial(\cup \alpha_i)=\partial(\cup \beta_i)=\{p_1,\ldots,p_{2n}\}$ and if we let the $\beta$ arcs surpass the $\alpha$ arcs at the intersections in $\mathbb{R}^3$, we get $L$. For each arc $\alpha_i$ or $\beta_i$, we can associate a Lagrangian sphere $\Sigma_{\alpha_i}$ or $\Sigma_{\beta_i}$ in S through the following equation: \begin{equation}\Sigma_{\alpha_i}=\{(u,v,z)\in S| z\in \alpha_i, u, v \in\sqrt{-p(z)}\mathbb{R}\}\end{equation} \begin{equation}\Sigma_{\beta_i}=\{(u,v,z)\in S| z\in \beta_i, u, v \in\sqrt{-p(z)}\mathbb{R}\}\end{equation} For each interior point of the arc, we have an $S^1$, while each end point gives a point. So $\Sigma_{\alpha_i}$ and $\Sigma_{\beta_i}$ are copies of $S^2$ and it is easy to see that they are Lagrangians for an appropriate choice of K\"ahler form, see \cite[Section 4]{M04}. These spheres enable us to build two Lagrangians in $Hilb^n(S)$ and, in fact, in $\mathscr{Y}_n$ by \begin{equation}\mathcal{K}_\alpha=\Sigma_{\alpha_1}\times\Sigma_{\alpha_2}\times\ldots\times\Sigma_{\alpha_n} \end{equation} \begin{equation}\mathcal{K}_\beta=\Sigma_{\beta_1}\times\Sigma_{\beta_2}\times\ldots\times\Sigma_{\beta_n} \end{equation} It is worth pointing out that $\mathcal{K}_\alpha$ and $\mathcal{K}_\beta$ do not intersect transversely. The intersections of these spheres are points in the fibers of the endpoints of the $\alpha$ arcs and $\beta$ arcs and circles $S^1$ in the fibers of interior intersections of those arcs. Thus, we could have some tori as intersections. To deal with this, we should perturb one of the Lagrangians, see \cite[section 6.1]{M04}, or use Floer theory with clean intersections, like \cite{P99} Because $\mathcal{K}_\beta$ is a product $\Sigma_{\beta_1}\times\Sigma_{\beta_2}\times\ldots\times\Sigma_{\beta_n}$, it will be easier to perturb each $\Sigma_\beta$ in the complex surface $S$. Let $N=\Sigma_\alpha\cap\Sigma_\beta$ be the portion of the intersection consisting of copies of $S^1$ intersections in $S$ and $V$ be a neighborhood of $N$. Then according to Weinstein \cite{W73}, we use the standard height function on $N$ as a Morse-Smale function that is required by the context. Then $\Sigma_\beta$ can be isotopied into $\Sigma_\beta'$ such that they are identical outside of V and intersects $\Sigma_\beta\cap V$ exactly at the maximum and the minimum of our height function. The resulting Floer cochain $CF^*(\Sigma_\alpha,\Sigma_\beta')$ will be quasi-isomorphic to $CF^*(\Sigma_\alpha,\Sigma_\beta)$. From now on, we always treat $\mathcal{K}_\beta$ as the original Lagrangian perturbed whenever we consider Lagrangian intersections, so it will intersect transversely with $\mathcal{K}_\alpha$ at isolated points. So each intersection at the interior of arcs $\alpha_i$ and $\beta_j$ now gives the intersection of $\Sigma_{\alpha_i}$ and $\Sigma_{\beta_j}$ at two points instead of a circle. \begin{proposition}\label{prop:rela} $($\cite[Theorem 1.2]{M04}, \cite[Theorem 1.1 and Theorem 4.12]{W09}$)$ For any bridge diagram $D$, The Floer cohomology $HF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)$ in $\mathscr{Y}_n=Hilb^n(S)\backslash D_r$ is canonically isomorphic to Seidel-Smith symplectic Khovanov homology $Kh^*_{symp}(L_D)$, where $L_D$ is the link represented by the bridge diagram. \end{proposition} With Proposition~\ref{prop:rela} in mind, we finally define: \begin{definition}\label{def:symp} The symplectic Khovanov cohomology $Kh^*_{symp}(D)$ of a bridge diagram $D$ is defined to be $HF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)$ in $\mathscr{Y}_n=Hilb^n(S)\backslash D_r$. \end{definition} \begin{remark} We have not yet given an absolute grading for this definition yet, and in fact, Proposition~\ref{prop:rela} is proved in the relatively graded case. Manolescu has an absolute grading in \cite{M04} but it only works with flattened braid diagrams, where explicit choices can be made on Lagrangians to construct an absolute Maslov grading. Waldron's construction in \cite{W09} works for any bridge diagram and his isomorphism is canonical but only works in the relatively graded case. \end{remark} In the rest of the paper, whenever we mention $Kh^*_{symp}(L)$, we will always be working with Definition~\ref{def:symp}, unless noted otherwise. \subsection{An absolute grading for bridge diagrams}\label{3.3} In this subsection, we prove Theorem~\ref{thm:bridge} by showing that the homological grading shifted by $gr(x_0)+rot+w$ is invariant under isotopy, handleslide and stabilization. We reiterate our conventions here that are crucial in defining the absolute grading. Our link is oriented. At each intersection, the $\beta$-arc surpasses the $\alpha$-arc. The distinguished generator has all the coordinates at the starting points of every $\beta$-arc. The idea of this correction term is from Droz-Wagner in \cite{DW09}, where they worked with grid diagrams obtained from deforming flattened braid diagrams. A flattened braid diagram of a braid $b\in{Br_n}$ is a special bridge diagram associated to the braid $b$ whose marked points $\mu_i$ are placed on the real line with an order by their indices, $\alpha$-arcs are segments on the real line connecting $\mu_{2i-1}$ and $\mu_{2i}$, $\beta$-arcs are the images of $\alpha$-arcs after applying the braid $b\times id\in Br_{2n}$ action on the plane with $2n$ punctures at the marked points. \begin{remark}An interesting example is that if we take a bridge diagram that represents a braid closure, the rotation number is exactly the number of strands (because all Seifert circles are going counterclockwise,) and the writhe of the diagram matches the writhe of the braid. The distinguished generator has homological grading $0$ in any flattened braid diagram. Thus the correction term $gr(x_0)+rot+w$ for the braid closure diagram agrees with Manolescu's correction term $n+w$ of the flattened braid diagram, where these two diagrams are isotopic as link diagrams. \end{remark} Geometrically, such an absolute grading requires us to grade our Lagrangians by choosing some sections of the canonical bundle. In the braid closure setup, two Lagrangians are related by some fibered Dehn twists and thus the choice on the second Lagrangian can be induced from the first one canonically. In the general bridge diagram case (not a flattened braid diagram), there is no easy way to assign a choice to the second Lagrangian, so instead, we assign the generator $x_0$ to have the grading $0$. Manolescu pointed out a combinatorial method to compute the relative homological grading in \cite[Subsection 6.2]{M04}. In short, we replace each $\alpha$-arc $\alpha_i$ with an oriented (the orientation does not matter) figure eight $\gamma_i$ in a small neighborhood. We arrange our diagram such that each $\beta$-arc is horizontal and each figure eight is vertical wherever they intersect. Each intersection of Lagrangian spheres $\Sigma_{\alpha_i}$ and $\Sigma_{\beta_j}$ corresponds to an intersection of the arc $\gamma_i$ and the figure eight $\beta_j$. We travel along with the figure eight $\gamma_i$ and mark the points that have horizontal tangent lines. We assign $+1$ to those marked points if it is locally oriented counterclockwise and $-1$ if clockwise. We start moving along the oriented figure eight, at the starting point of the $\beta$-arc with $0$ (that it is essentially still relative but we are going to remove the grading of this generator in the correction term anyway) and change this number by the number on the marked point whenever we reach one. In this way, each intersection of a $\beta$-arc and the figure eight $\gamma_i$ will be labeled with a number, see Figure~\ref{fig:Isotopy1} for an idea of the computation. Each of the generators of the Floer cohomology group corresponds to $n$ intersection points and the sum of the labeled numbers will be its grading. \begin{proposition}$HF^{*+gr(x_0)+rot+w}(\mathcal{K}_\alpha,\mathcal{K}_\beta)$ is invariant under bridge diagram isotopy. \end{proposition} \begin{proof} An isotopy of a bridge diagram induces Hamiltonian isotopic Lagrangians and thus keeps the relatively graded group $HF^{*}(\mathcal{K}_\alpha,\mathcal{K}_\beta)$ unchanged. If no crossing is introduced or removed, it is easy to see that all three components in the correction term remain the same. If crossings are introduced, it must come from one of the following cases, as shown in Figure~\ref{fig:Isotopy}: \begin{figure} \includegraphics[width=6cm]{figure1.png} \centering \caption{Two possible bridge diagram isotopies corresponding to Reidemeister I and II moves. } \label{fig:Isotopy} \end{figure} These two isotopies correspond to Reidemeister move I and II, (whereas Reidemeister move III is from a combination of (de-)stabilizations and handleslides.) In the first case, there are two subcases--the $\beta$-arc oriented to the left and to the right. The easier subcase is the left-going $\beta$-curve. The crossing is a negative crossing and the bigon region gives an additional counterclockwise Seifert circle. The distinguished generator $x_0$ has no coordinate in this region and thus its grading does not change. The total change of $gr(x_0)+rot+w$ is $0$. \begin{figure} \includegraphics[width=6cm]{figure2.png} \centering \caption{Isotopy corresponding to Reidemeister I move with intersections marked with gradings and horizontal tangent points marked with signs.} \label{fig:Isotopy1} \end{figure} The other subcase is more subtle. The crossing is still negative but the Seifert circle is now clockwise, which changes the correction term by $-2$. The distinguished generator $x_0$ indeed takes a coordinate at the black dot. With the help of figure eights in Figure~\ref{fig:Isotopy1}, we can visualize that the relative grading is changed by $2$, which cancels the contribution of the other two terms. In fact, we can explicitly describe the change of the cochain. The bigon region in the complex plane lifts to some holomorphic discs in the complex surface $S$, connecting one of the two generators at the interior intersection to the endpoint. One of the moduli spaces has indeed dimension $1$ that contributes to differentials cancelling pair of generators with identical coordinates except one coordinate in the picture. At chain level, the isotopy created three copies of the complexes that have a coordinate at the black dot, but two of them are canceled on the homology level. The surviving copy should have the same grading as the original chain. It is not hard to see the other moduli space of the bigon region has dimension $2$, and thus locally changing coordinate from the lower grading coordinate at the intersection to the black dot increases the grading by $2$. So in the new diagram, the grading of the distinguished generator increases by $2$ relatively to other generators, which cancels the contribution by the sum of writhe and rotation number. In the second case which corresponds to Reidemeister II move, regardless of the orientation, two new crossings always have different signs so the writhe remains unchanged. If the two components are oriented in the same direction, the Seifert surface remains unchanged and thus the rotation number is also the same. If the two components are oriented differently, there will be two subcases, whether the two strands are from the same component or not. In the first subcase, locally one Seifert circle breaks into three circles, but the middle one is of a different direction. In the second subcase, there are two Seifert circles of the same direction in each picture. In either case, the rotation number remains the same as well. Apart from the two new intersections in the new diagram, all other generators are from the diagram before the isotopy and it is not hard to see that the gradings of such generators remain unchanged. Thus $gr(x_0)$ remains unchanged as well relative to other generators. \begin{figure} \includegraphics[width=6cm]{figure3.png} \centering \caption{Two cases for Reidemeister II isotopy when two components are oriented differently.} \label{fig:Isotopy2} \end{figure} \end{proof} \begin{proposition}$HF^{*+gr(x_0)+rot+w}(\mathcal{K}_\alpha,\mathcal{K}_\beta)$ is invariant under stabilization. \end{proposition} \begin{proof} Stabilizations will not change the writhe or the rotation number of the diagram. It suffices to show that the grading of the homology remains relatively the same to the distinguished generator $x_0$. It is true because Waldron's proof of invariance is based on chain level that the isomorphism is in fact on chain level so $gr(x_0)$ is not changed relative to cohomology, see \cite[Lemma 5.19]{W09}. \end{proof} The handleslide invariance is the hardest among the three. Before the proof, we make some topological observations. With the isotopy invariance, we should arrange our diagram to be as simple as possible. We explicitly describe the process of the handleslide of arc $\alpha_2$ over $\alpha_1$. First, we choose the boundary circle of slightly flattened $\alpha_1$, such that each midpoint intersection of any $\beta$ arc locally intersect the circle exactly twice, and the $\beta$ arcs connecting the endpoints of $\alpha_1$ locally intersect the circle exactly once. Then pick any path $\gamma$ from $\alpha_2$ to the circle that does not intersect any $\alpha$ arc in its interior, and perform a connect sum such that each intersection between the path and the $\beta$ arc creates exactly two intersections on the connected sum. \begin{proposition}$HF^{*+gr(x_0)+rot+w}(\mathcal{K}_\alpha,\mathcal{K}_\beta)$ is invariant under handleslide. \end{proposition} \begin{proof} It is clear that new intersections are created in pairs, and each pair contains exactly one negative and one positive crossing. Thus the writhe of the diagram remains unchanged. Next, we show that the rotation number remains unchanged as well. This breaks into two parts. First, we study the intersections created around the path $\gamma$. Locally, there are a series of parallel $\beta$-arcs intersecting two $\alpha$-arcs. The two $\alpha$-arcs are parallel to the path $\gamma$ and always oriented in different directions. Let us call a $\beta$-arc with positive sign if it goes from left to right, and negative otherwise. We study the local picture with only two adjacent $\beta$-arcs. There are four cases in total shown in Figure~\ref{fig:HS1} In the case $++$, the first Seifert circle remains the same. In the case $--$, the second circle remains the same. The other Seifert circle is connected to another section. \begin{figure} \includegraphics[width=6cm]{figure4.png} \centering \caption{All four possible Seifert circles at the intersections near the path $\gamma$, depending on the orientation of $\beta$-arcs.} \label{fig:HS1} \end{figure} In the case $+-$, there are two subcases, whether the two $\beta$-arcs belong to the same strand or different strands, similar to the Reidemeister II move argument. It will either break a clockwise circle into two, or merging two counterclockwise circles into one. In either case, the rotation number is reduced by $1$. In the case $-+$, one counterclockwise circle is created. The other two Seifert circles are studied already in other cases. Thus the rotation number is increased by $1$. To sum this up, each time the sign change from $+$ to $-$, the rotation number decreases by $1$, and each time the sign change from $-$ to $+$, the rotation number increases by $1$. At the intersection of the path $\gamma$ and $\alpha_1$, $\alpha_1$ is oriented to the right (to match the orientation of the $\alpha$-arcs in the handleslide picture,) and the circle will remain the same if the first $\beta$ is positive, and decreases the rotation number by $1$ if the first $\beta$ is negative. Thus, if the last $\beta$-arc is positive, the rotation number remains the same, if the last one is negative, the rotation number is reduced by $1$. The last $\beta$-arc is also related to the next part of the argument. The other part we need to compute is the contribution of new crossings created at the circle near $\alpha_2$. Without loss of generality, let us assume $\alpha_2$ is oriented from left to right. The part in the dashed box of Figure~\ref{fig:HS2} is exactly the same as the picture before the handleslide except for the last endpoint intersection, (and, in fact, the last intersection correspond to the intersection on the top-right if we keep going from the $\alpha$-arc.) The rest of the region is two parallel $\alpha$-arcs oriented differently, intersecting a series of parallel $\beta$-arcs. The study of these new intersections is similar to the first half of the argument and the conclusion is similar, except now the sign depends on the first $\beta$-arc. We claim that the rotation number remains the same if the first crossing is positive, and increases by $1$ if the first crossing is negative. \begin{figure} \includegraphics[width=8cm]{figure5.png} \centering \caption{All four possible Seifert circles at the intersections near the path $\gamma$, depending on the orientation of $\beta$-arcs.} \label{fig:HS2} \end{figure} Now we only need to check the last $\beta$ of the first part and the first $\beta$ of the last part. If both are of the same sign, the Seifert circle does not change. If the $\beta$ from the first half if $+$ and the $\beta$ from the second half is $-$, then it created a counterclockwise Seifert circle, rotation number increases by $1$. In the symmetric case, the rotation number decreases by $1$. In any of the cases above, the total change of rotation number if $0$. Lastly, we need to check the relative grading of the homology remains the same relative to the distinguished generator $x_0$. First of all, the isomorphism of two homology groups is induced by a continuation map on the cochain level. It is also easy to see that there is an injection from the generators of the original cochain to the ones of the cochain after the handleslide. It is easy to see that the homological gradings of the corresponding generators are not changed from the definition of figure-eights, because we only changed one of the arcs and we essentially only applied an isotopy to one of the $\alpha$ arcs if we allow the isotopy to move across other marked points and $\alpha$ arcs. Moreover, the distinguished generator is obviously one of the generators that are preserved in the correspondence, and thus the homological grading relative to the distinguished generator $x_0$ remains the same. \end{proof} Combining the propositions in this subsection, we complete a proof for Theorem~\ref{thm:bridge} that we obtain an absolutely graded version of symplectic Khovanov cohomology for any bridge diagram. \section{Weight grading on $Kh^*_{symp}(L)$}\label{ch3} \subsection{Construction of weight grading}\label{3.4} In this subsection, we define the weight grading $wt$ on $Kh^*_{symp}$, following the idea of Abouzaid-Smith, see \cite[Section 3]{AS16} for more details. The idea of constructing such a grading is building an automorphism, more precisely, the linear term of a \textit{non-commutative vector field}, or a \textit{nc vector field} \begin{equation}\phi:CF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)\to CF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)\end{equation} which preserves the homological grading and commutes with the differential. Then it will induce an automorphism $\Phi$ on cohomology. If $x$ is an eigenvector of eigenvalue $\lambda$, then we define $wt(x)=\lambda$. Most results of this section work for any characteristic field but we will restrict our discussion to characteristic zero ones, see Remark~\ref{re:field} for more detail. The idea of defining the map $\phi$ is to study certain moduli spaces of holomorphic maps in a partially compactified space $\bar{\mathscr{Y}}_n$ of $\mathscr{Y}_n$: \begin{itemize} \item Let $Z=\bar{\bar{A}}_{2n-1}$ be the blow up of $\mathbb{P}^1\times\mathbb{P}^1$ at $2n$ points. It admits a Lefschetz fibration structure over $\mathbb{P}^1$ and let $F_\infty$ be its fiber at $\infty$ and $s_0$, $s_\infty$ be two sections. The original surface is simply \begin{equation}A_{2n-1}=\bar{\bar{A}}_{2n-1}\backslash(F_\infty\cup s_0\cup s_\infty)\end{equation} \item Let the intermediate space be \begin{equation}\bar{A}_{2n-1}=\bar{\bar{A}}_{2n-1}\backslash(F_\infty)\end{equation}. \item Through taking Hilbert scheme, the projective variety is $\bar{\bar{M}}=Hilb^n(Z)$. \item $D_0$ is the divisor of subschemes whose support meets $s_0\cup s_\infty$. \item $D_\infty$ is the divisor of subschemes whose support meets $F_\infty$. \item $D_r$ is the relative Hilbert scheme of the projection $Z\to \mathbb{P}^1$, which is a divisor supported on a compactification of the complement of $\mathscr{Y}_n$. \item $\bar{\mathscr{Y}}_n=\bar{\bar{M}}\backslash D_\infty=Hilb^n(\bar{A}_{2n-1})$. \end{itemize} The moduli space was originally used by Seidel and Solomon in \cite{SS12} for the q-intersection number and later used by Abouzaid and Smith in \cite{AS16} for the weight grading. Let $\mathscr{R}^{k+1}_{(0,1)}$ be the moduli space of holomorphic classes of closed unit discs with the following additional data: \begin{itemize} \item two marked points $z_0=0$ and $z_1\in (0,1)$. \item $k+1$ boundary punctures at $p_0=1$ and $k$ others $p_1,\ldots p_k$ placed counterclockwise. \end{itemize} We define \begin{equation}\mathscr{R}^{k+1}_{(0,1)}(x_0;x_k,\ldots,x_1)\end{equation} to be the moduli space of finite energy holomorphic maps $u:\mathbb{D}\to \bar{\mathscr{Y}}_n$ such that \begin{itemize} \item $u^{-1}(D_r)=\emptyset$ \item $u^{-1}(D_0)=z_0$ \item $u^{-1}(D'_0)=z_1$, where $D'_0$ is a divisor linearly equivalent to $D_0$ but shares no irreducible component with $D_0$. \item $u(p_i)=x_i$ \end{itemize} Now we are ready to introduce our automorphism $\Phi$. In the case of $k=1$, the virtual dimension of $\mathscr{R}^{2}_{(0,1)}(x,y)$ is the difference of the homological grading $gr(x)-gr(y)$, see \cite[Lemma 3.16]{AS16}. It makes sense to consider a map of degree $0$ that counts holomorphic discs in $\mathscr{R}^{2}_{(0,1)}(x,y)$: \begin{equation} b^1(x)=\sum_{y|gr(y)=gr(x)}\#\mathscr{R}^{2}_{(0,1)}(x,y)y \end{equation} \begin{remark} We stated the definition above without orientation. But in fact, the moduli spaces $\mathscr{R}^{k+1}_{(0,1)}(x_0;x_k,\ldots,x_1)$ admit natural orientations relative to the orientation lines $o_{x_i}$ and the moduli spaces of curves $\mathscr{R}^{k+1}_{(0,1)}$, by fixing orientation for $\mathscr{R}^{k+1}_{(0,1)}$ by identifying the interior points of this moduli space with an open subset of $(0,1)\times (\partial\Delta)^{k+1}$ and twist the overall orientation by $\sum_{i=1}^k i gr(x_i)$. \end{remark} $\tilde{b}^1$ is the linear part of a Hochschild cochain $\tilde{b}\in CC^*(\mathcal{F}(M),\mathcal{F}(M))$ if we allow multiple inputs instead of one single input $y$. But this cochain is not closed. Two additional string maps \begin{equation}CO(gw_1)\text{, where $gw_1$ bounds the Gromov-Witten invariant $GW_1$, }\end{equation} \begin{equation} co(\beta_0)\text{, where $\beta_0$ bounds the intersection of $D_0$ and $D_0'$,}\end{equation} are needed to make a closed cochain and moreover a $nc$ vector field. By adding these two terms, we obtain a Hochschild cocycle $b$ and its linear part $b^1$. By definition, given an (exact) Lagrangian $L$, the obstruction to the existence of an equivariant structure $c$ is given by the first term of the $nc$ vector field $b^0|_L\in HF^1(L,L)\cong H^1(L)$, and the set of choices when this vanishes is an affine space equivalent to $H^0(L)$. In our case, the Lagrangians are a product of $S^2$, thus $H^1(L)\cong\{0\}$ and $H^0(L)\cong \textbf{k}$. Since we do not assume that $b^0$ vanishes, for a cocycle $b\in CC^1(\mathcal{F}(M),\mathcal{F}(M))$ and equivariant objects $(\mathcal{K}_\alpha,c_\alpha)$ and $(\mathcal{K}_\beta,c_\beta)$, the linear term $b^1$ is not always a chain map. However, we can define a chain map \begin{equation} \phi(x)=b^1(x)-\mu^2(c_\alpha,x)+\mu^2(x,c_\beta) \end{equation}. It induces an endomorphism $\Phi$ on $HF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)$. If we consider the (generalized) eigenspace decomposition, the eigenvalue of the generalized eigenvector $x$ will be its weight grading denoted as $wt(x)$. \begin{remark}\label{re:field} The construction above works in any field $\mathbf{k}$. The weight grading is a priori indexed by elements of the algebraic closure $\bar{\mathbf{k}}$. Working with characteristic zero field will not only enable us to use the result of Abouzaid-Smith that identifies symplectic Khovanov cohomology and Khovanov homology, but also will make it possible to make the weight grading integral, in contrast to finite fields. If the weight grading lives in the algebraic closure of a finite field, it is impossible to make it integral using conventional methods. \end{remark} We learn from \cite[Lemma 2.12]{AS16} that since $HF^0(\mathcal{K},\mathcal{K})\cong\textbf{k}$, changing equivariant structures shifts the overall weight by a constant \begin{equation} \Phi_{(\mathcal{K}_\alpha,c_\alpha),(\mathcal{K}_\beta,c_\beta)}= \Phi_{(\mathcal{K}_\alpha,c_\alpha+s_\alpha),(\mathcal{K}_\beta,c_\beta+s_\beta)}+(s_\alpha-s_\beta)id \end{equation} So we have the following definition of the relative weight grading: \begin{definition} Let $\Phi$ be the endomorphism constructed above on $HF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)$ and $x$ be an eigenvector of $\Phi$. The relative weight grading $wt(x)$ is defined to be the eigenvalue of $x$. This construction relies on auxiliary choices of equivariant structures on $\mathcal{K}_\alpha$ and $\mathcal{K}_\beta$, but different choices of such structures will only change all gradings by a fixed number and thus $wt(x)$ as a relative grading is independent of choices of equivariant structures. \end{definition} \begin{remark} With given equivariant structures on $\mathcal{K}_\alpha$ and $\mathcal{K}_\beta$, we can compute an absolute weight grading. But at the time of writing this paper, the author does not know the choices that would give a correction term independent of the link diagram. \end{remark} Then we can rephrase Abouzaid-Smith's purity result as follow: \begin{proposition}\cite[Proposition 6.11]{AS16}\label{prop:nocrossing} Let $D$ be a bridge diagram without any crossings. Then we can choose $c_\alpha$ on $\mathcal{K}_\alpha$ and $c_\beta$ on $\mathcal{K}_\beta$ such that for any $x\in HF^*(\mathcal{K}_\alpha,\mathcal{K}_\beta)$, we have $wt(x)=gr(x)$. \end{proposition} If we recall the Khovanov homology of an unlink of $k$-component $U_k$ is \begin{equation} Kh^{*,*}(U_k)=\bigotimes^k(\textbf{k}_{(0,1)}\oplus\textbf{k}_{(0,-1)}). \end{equation} With the choice of the equivariant structures by Abouzaid and Smith, we have \begin{equation} Kh^{*,*}_{symp}(U_k)=\bigotimes^k(\textbf{k}_{(1,1)}\oplus\textbf{k}_{(-1,-1)}) \end{equation} This proposition is a special case of our main theorem with crossing number equals to 0, if we relate gradings $(gr,wt)$ on symplectic Khovanov cohomology and $(i,j)$ on Khovanov homology with the formula: \begin{equation} i=gr-wt \end{equation} \begin{equation} j=-wt \end{equation} \subsection{Floer product and weight grading}\label{2.4} In \cite[Section 3]{AS16}, Abouzaid and Smith pointed out an important fact that the weight grading is compatible with Floer products, without actually phrasing and proving the precise statement. Also in \cite[Equation 4.9]{SS12}, Seidel and Solomon discussed the derivation property in a similar setup. We prove the following proposition: \begin{proposition}\label{prop:product} Let $\mathcal{K}_0$, $\mathcal{K}_1$, $\mathcal{K}_2$ be compact Lagrangians given by crossingless matchings in $\mathscr{Y}_n$. For any eigenvector $\alpha\in HF^*(\mathcal{K}_1,\mathcal{K}_2)$ and $\beta\in HF^*(\mathcal{K}_0,\mathcal{K}_1)$, we have \begin{equation}wt(\mu^2(\alpha,\beta))=wt(\alpha)+wt(\beta)\end{equation} \end{proposition} \begin{proof} Consider the boundary strata of the moduli space $\bar{\mathscr{R}}^3_{(0,1)}(x_0;x_1,x_2)$, where we have three boundary marked points and two interior marked points. We can exclude sphere bubbles through a correct choice of bounding cycle. There are still 6 kinds of degeneration (see Figure~\ref{fig:product}) of this moduli space. Three degenerations shown in the first row compute \begin{equation}\phi(\mu^2(x_1,x_2))\end{equation}\begin{equation}\mu^2(x_1,\phi(x_2))\end{equation}\begin{equation}\mu^2(\phi(x_1),x_2)\end{equation} The three degenerations shown in the second row compute \begin{equation}\mu^1\phi^2(x_1,x_2)\end{equation}\begin{equation}\phi^2(\mu^1(x_1),x_2)\end{equation}\begin{equation}\phi^2(x_1,\mu^1(x_2))\end{equation} If we pass to homology, those terms with $\mu^1$ will vanish and thus we have the following relation by counting all the boundary components of $1$ dimensional moduli space $\bar{\mathscr{R}}^2_{(0,1)}(x_0;x_1,x_2)$ \begin{equation}\phi(\mu^2(x_1,x_2))=\mu^2(x_1,\phi(x_2))+\mu^2(\phi(x_1),x_2)\end{equation} This is equivalent to \begin{equation}wt(\mu^2(x_1,x_2))\mu^2(x_1,x_2)=\mu^2(x_1,wt(x_2)x_2)+\mu^2(wt(x_1)x_1,x_2)=(wt(x_1)+wt(x_2))\mu^2(x_1,x_2).\end{equation}, which proves the result. \end{proof} \begin{figure} \includegraphics[width=6cm]{figure6.png} \centering \caption[6 possible breaks in the boundary]{6 possible breaks in the boundary. The incoming arrows are inputs and the outgoing ones are outputs} \label{fig:product} \end{figure} \begin{remark}\label{rmk:product} By studying the similar set up with more boundary marked points, we can generalize the result above to higher Floer products. Specifically, when $n=3$, we have $wt(\mu^3(x_1,x_2,x_3))=wt(x_1)+wt(x_2)+wt(x_3)$. \end{remark} \subsection{A long exact sequence of $Kh^{*,*}_{symp}(L)$}\label{ch4} Abouzaid and Smith constructed a long exact sequence from an exact triangle of bimodules over the Fukaya category of $\mathscr{Y}_n$, see \cite[Equation 7.9]{AS19}: \begin{equation}\ldots\to Kh_{symp}^{*}(L_+)\to Kh_{symp}^{*}(L_0)\to Kh_{symp}^{*+2}(L_\infty)\to \ldots \end{equation} where $L_+$ is a link diagram with a positive crossing and $L_0$ and $L_\infty$ are diagrams given by $0$ or $\infty$ resolutions at the positive crossing. The goal of this chapter is to give an explicit construction of such a long exact sequence with the framework of bridge diagrams that preserves the weight grading, just like the combinatorial Khovanov homology. We use the following local diagrams for the computation: we name the blue curves $\beta$, green curves $\gamma$ and yellow curves $\delta$, respectively, in Figure~\ref{fig:exact}, and they share all the other components (and to make it easier to discuss in the future, we move $\gamma$ and $\delta$ slightly away from $\beta$ in all the other components so that they don't intersect in the interior of the arcs). Pairing $\alpha$ with $\beta$ gives $L_+$, $\alpha$ with $\gamma$ gives $L_0$ and $\alpha$ with $\delta$ gives $L_-$. We have the following exact sequence: \begin{proposition}\cite[Proposition 7.4]{AS19} If we have $\alpha$, $\beta$, $\gamma$ and $\delta$ curves presented like Figure~\ref{fig:exact} locally and $\beta$, $\gamma$ and $\delta$ are the same apart from this region, then we have the following exact sequence \begin{equation}\label{eqt:exact2}\ldots\xrightarrow{c_1} HF^{*}(\mathcal{K}_\alpha,\mathcal{K}_\beta)\xrightarrow{c_2} HF^{*}(\mathcal{K}_\alpha,\mathcal{K}_\gamma)\xrightarrow{c_3} HF^{*+2}(\mathcal{K}_\alpha,\mathcal{K}_\delta)\xrightarrow{c_1} \ldots \end{equation} \begin{figure} \includegraphics[width=6cm]{figure7.png} \centering \caption[Exact triangle of Lagrangians]{Exact triangle of Lagrangians. The red, blue, green and yellow are $\alpha$, $\beta$, $\gamma$ and $\delta$ curves} \label{fig:exact} \end{figure} In particular, there are elements $c_1 \in CF^{*}(\mathcal{K}_\beta,\mathcal{K}_\delta)$, $c_2 \in CF^{*}(\mathcal{K}_\gamma,\mathcal{K}_\beta)$ and $c_3 \in CF^{*}(\mathcal{K}_\delta,\mathcal{K}_\gamma)$ such that the maps above are Floer products with the corresponding elements. \end{proposition} \begin{proof} Abouzaid and Smith proved that there is an exact triangle of bimodules of the Fukaya category of $\mathscr{Y}_n$ among the identity bimodule, the cup-cap bimodule and the bimodule representing an half-twist $\tau$. Evaluating these bimodules at $\mathcal{K}_\gamma$ as the second object, we have an exact triangle of one-sided modules between $\mathcal{K}_\gamma$, $\mathcal{K}_\delta$ and an one-sided module that is equivalent to $HF^*(\bullet,\mathcal{K}_\beta)$, such that the maps connecting those terms are Floer products with some elements $c_1 \in CF^{*}(\mathcal{K}_\beta,\mathcal{K}_\delta)$, $c_2 \in CF^{*}(\mathcal{K}_\gamma,\mathcal{K}_\beta)$ and $c_3 \in CF^{*}(\mathcal{K}_\delta,\mathcal{K}_\gamma)$. Evaluating these one-sided modules at $\mathcal{K}_\alpha$, we have the desired long exact sequence. \end{proof} Now we need to show that this long exact sequence preserves the relative weight grading in the sense that each element summing to $c_i$ has the same weight grading so that $c_i$ has a well-defined weight grading, and moreover the weight gradings of $c_1$, $c_2$ and $c_3$ sum to $0$. With a closer look into the diagram, we have the following observation: \begin{lemma} Each pair of $\beta$, $\gamma$ and $\delta$ forms a bridge diagram for an unlink of $(n-1)$-components without any crossing. \end{lemma} \begin{proof} In the region shown in Figure~\ref{fig:exact}, each two of $\beta$, $\gamma$ and $\delta$ form a crossingless unknot. In the other regions not shown in the figure, each pair of arcs forms a crossingless unknot component as well. Thus we have $1$ crossingless unknot component in Figure~\ref{fig:exact} and crossingless $(n-2)$ unknot components outside that figure. Together, each two of $\beta$, $\gamma$ and $\delta$ form a bridge diagram for an unlink of $(n-1)$ components without any crossing. \end{proof} \begin{remark} Each of the Floer groups is computed with the second Lagrangian perturbed so that it intersects transversely with the first one. In this subsection, $\mathcal{K}_\beta$, $\mathcal{K}_\gamma$, $\mathcal{K}_\delta$ are all perturbed so they intersect transversely with $\mathcal{K}_\alpha$. We also know that $\beta$, $\gamma$, and $\delta$ only intersect at endpoints, so their corresponding Lagrangians have pairwise transverse intersections. We can now assume the perturbations we apply to $\mathcal{K}_\beta$, $\mathcal{K}_\gamma$, and $\mathcal{K}_\delta$ are small enough that they still pairwise intersect transversely, while they still intersect transversely with $\mathcal{K}_\alpha$. We will keep abusing the notation of Lagrangians $\mathcal{K}_\beta$, $\mathcal{K}_\gamma$, $\mathcal{K}_\delta$ and the Lagrangians after perturbations when pairing with $\mathcal{K}_\alpha$. \end{remark} From the computations of \cite{AS19}, we have \begin{corollary}\label{cor:unlink1} With some grading shifts \begin{equation}CF^*(\mathcal{K}_\beta, \mathcal{K}_\gamma)\cong CF^*(\mathcal{K}_\gamma, \mathcal{K}_\delta)\cong CF^*(\mathcal{K}_\delta, \mathcal{K}_\beta)\cong \bigotimes^{n-1} H^*(S^2)\end{equation} and thus all generators are cocycles. \end{corollary} \begin{proposition}\label{prop:es} Fix a choice of equivariant structures on Lagrangians, there is a well-defined weight grading for $c_1$, $c_2$ and $c_3$. \end{proposition} \begin{proof}From Corollary~\ref{cor:unlink1}, we know each of the Floer cochains and cohomologies above can be made such that weight grading equals homological grading. Together with the observation that choices of equivariant structures will only apply overall grading shifts to all weight gradings, thus we know the weight grading of any element in $CF^n$ must be the same. If we fix a set of equivariant structures on $\mathcal{K}_\beta$, $\mathcal{K}_\gamma$ and $\mathcal{K}_\delta$, we have a well-defined weight grading for each $c_i$ from its homological grading plus the effect a grading shift from changing the equivariant structure from the standard one. \end{proof} Before we prove that $wt(c_1)+wt(c_2)+wt(c_3)=0$. Recall the following lemma from Seidel: \begin{lemma}\cite[Lemma 3.7]{Sei08} A triple $\mathcal{K}_\beta$, $\mathcal{K}_\gamma$ and $\mathcal{K}_\delta$ form an exact triangle of Lagrangians if and only if there exist $c_1 \in CF^{1}(\mathcal{K}_\beta,\mathcal{K}_\delta)$, $c_2 \in CF^{0}(\mathcal{K}_\gamma,\mathcal{K}_\beta)$, $c_3 \in CF^{0}(\mathcal{K}_\delta,\mathcal{K}_\gamma)$, $h_1\in HF^0(\mathcal{K}_\gamma,\mathcal{K}_\beta)$, $h_2\in HF^0(\mathcal{K}_\beta,\mathcal{K}_\delta)$ and $k\in HF^{-1}(\mathcal{K}_\beta,\mathcal{K}_\beta)$ such that \begin{equation}\label{eqt:6.6} \mu^1(h_1)=\mu^2(c_3,c_2) \end{equation} \begin{equation}\label{eqt:6.7} \mu^1(h_2)=-\mu^2(c_1,c_3) \end{equation} \begin{equation} \mu^1(k)=-\mu^2(c_1,h_1)+\mu^2(h_2,c_2)+\mu^3(c_1,c_3,c_2)-e_{\mathcal{K}_\beta} \end{equation} \end{lemma} \begin{lemma} For any choice of equivariant structures on Lagrangians, the sum of weight gradings $wt(c_1)+wt(c_2)+wt(c_3)=0$. \end{lemma} \begin{proof} We know $\mu^1$ vanishes on all the Floer groups above. If we know that $h_1=0$ and $h_2=0$, then the equation $(5.6)$ becomes $\mu^3(c_1,c_3,c_2)=e_{\mathcal{K}_\beta}$. Together with the fact weight grading is compatible with Floer product, see Remark~\ref{rmk:product}, we know that $wt(c_1)+wt(c_2)+wt(c_3)=wt(e_{\mathcal{K}_\beta})$. From \cite[Lemma 4.10]{AS16}, no matter which equivariant structure we choose on $\mathcal{K}_\beta$, the identity always has the weight grading $0$. To see $h_1=0$ and $h_2=0$, we need to look into the absolute grading. We claim that $HF^*(\mathcal{K}_\gamma,\mathcal{K}_\beta)$ and $HF^*(\mathcal{K}_\beta,\mathcal{K}_\delta)$ are supported in odd degrees. This is because pairing $\gamma$ with $\beta$ gives a flattened braid diagram for braid $\sigma$ with writhe $1$. The number of strands is even and thus this group is supported in odd degrees. Similarly, pairing $\beta$ with $\delta$ gives $\sigma^{-1}$, this also gives an odd degree supported group. Notice that the third map $c_3$ should have degree $2$ in the absolute grading case, thus the exact triangle is actually between $\mathcal{K}_\beta$, $\mathcal{K}_\gamma$ and $\mathcal{K}_\delta[2]$. But shifting the grading of Floer groups by $2$ does not change the fact that $h_1$ and $h_2$ have even degrees. Thus, they must be $0$. \end{proof} Thus we conclude the following, \begin{proposition} The long exact sequence Equation~\ref{eqt:exact2} \begin{equation}\ldots\xrightarrow{c_1} HF^{*,wt_1}(\mathcal{K}_\alpha,\mathcal{K}_\beta)\xrightarrow{c_2} HF^{*,wt_2}(\mathcal{K}_\alpha,\mathcal{K}_\gamma)\xrightarrow{c_3} HF^{*+2,wt_3}(\mathcal{K}_\alpha,\mathcal{K}_\delta)\xrightarrow{c_1} \ldots \end{equation} decomposes with respect to weight gradings. \end{proposition} \begin{proof} From Proposition~\ref{prop:es}, we know that there is a well-defined weight grading for $c_1$, $c_2$ and $c_3$. If we start with the first group $ HF^{*,wt_1}(\mathcal{K}_\alpha,\mathcal{K}_\beta)$, the only non-trivial map will be at weight grading $wt_1+wt(c_2)$ because the weight grading is compatible with Floer products. The same goes for $wt_3=wt_1+wt(c_3)+wt(c_2)$. The next weight grading should be $wt_1+wt(c_2)+wt(c_3)+wt(c_1)=wt_1$, which is exactly where we started with the first group. \end{proof} \section{Proof of the main theorem via a bigraded isomorphism}\label{ch5} Now we have enough to prove our main theorems, Theorem~\ref{thm:main} \begin{theorem}\label{thm:bigrading} Symplectic Khovanov cohomology, graded by $(gr,wt)$, and Khovanov homology, graded by $(i,j)$, are isomorphic as bigraded vector spaces over any characteristic zero field, where the gradings are related by $gr=i-j$ and $wt=-j+c$, with an ambiguity of a grading shift $c$ on relative weight grading. \end{theorem} Our isomorphism is a graded refinement of the isomorphism in \cite{AS19}. \begin{proposition}\cite[Theorem 7.5]{AS19} For any bridge diagram $L$, we have an isomorphism $H$ such that \begin{equation} H: Kh^*_{symp}(L)\to Kh^*(L) \end{equation} \end{proposition} In \cite{AS19}, we can only conclude from the original argument of Abouzaid and Smith that $H$ is canonical for braid closures. With Proposition~\ref{prop:rela} that symplectic Khovanov cohomology is defined canonically (and the same for combinatorial Khovanov homology), we can claim that $H$ is canonical for any link diagram. This is an isomorphism with only information on the homological grading. But the result of Abouzaid and Smith implies that the long exact sequence in Equation~\ref{eqt:exact2} commutes with the corresponding long exact sequence in Khovanov homology. \begin{lemma} Fix a link diagram $L_+$ and its unoriented resolutions $L_0$ and $L_\infty$ at one of the crossings. We represent their corresponding bridge diagrams with $(\alpha,\beta)$, $(\alpha,\gamma)$ and $(\alpha,\delta)$-arcs respectively such that $\alpha$, $\beta$, $\gamma$ and $\delta$ are locally shown in Figure~\ref{fig:exact}. The isomorphism $H$ is compatible with the exact sequence Equation~\ref{eqt:exact2}, i.e. the following diagram commutes \[\begin{tikzcd} \hspace{-1cm}HF^{*}(\mathcal{K}_\alpha,\mathcal{K}_\gamma) \arrow{r} \arrow[swap]{d}{H} & HF^{*+2}(\mathcal{K}_\alpha,\mathcal{K}_\delta) \arrow{r} \arrow[swap]{d}{H}& HF^{*+1}(\mathcal{K}_\alpha,\mathcal{K}_\beta) \arrow{r} \arrow[swap]{d}{H}& HF^{*+1}(\mathcal{K}_\alpha,\mathcal{K}_\gamma) \arrow{r} \arrow[swap]{d}{H}&HF^{*+3}(\mathcal{K}_\alpha,\mathcal{K}_\delta) \arrow{d}{H} \\ \hspace{-1cm}Kh^*(L_0) \arrow{r} & Kh^{*+2}(L_\infty) \arrow{r} &Kh^{*+1}(L_+) \arrow{r} &Kh^{*+1}(L_0) \arrow{r} &Kh^{*+3}(L_\infty) \end{tikzcd} \] where the first line is the exact sequence of Equation~\ref{eqt:exact2} and the second line is the exact sequence for combinatorial Khovanov homology with grading $i-j$. \end{lemma} \begin{proof} In the proof of the Abouzaid-Smith isomorphism, they show that the cup functors of Khovanov and symplectic Khovanov are identified with the isomorphism in the arc algebra. Cap functors in both cases are adjoint to the corresponding cup functors. The horizontal maps in the first squares are given by applying the same cap-cup functor in each case, and thus they commute with the isomorphisms in the first square. The diagram naturally commutes if we replace the third group of each row with the mapping cone of the other two. The third group of each row is isomorphic to a mapping cone of the other two, see \cite[Proposition 7.4]{AS19}. Moreover, in the proof of \cite[Theorem 7.6]{AS19}, the isomorphism $H$ and the long exact sequences are constructed via the mapping cones and thus factor through the cones of horizontal maps of the first square. As a result, the second and third squares are also commutative. \end{proof} \begin{corollary} The diagram involving the exact sequences of resolving a negative crossing is also commutative. \end{corollary} We compared the bigradings of two theories at the end of Section~\ref{3.4} for unlinks: \begin{proposition}\cite[Proposition 6.11]{AS16} \label{prop:unlink} The isomorphism $H$ preserves the weight grading if $D$ is crossingless diagram, with grading correspondence $gr=i-j$ and $wt=-j$. \end{proposition} \begin{proof}[Proof of Theorem~\ref{thm:main}] We only need to prove that for any fixed Jones grading $j_0$, $H$ is also an isomorphism with $wt=-j_0+c$ with some grading shift $c$, \begin{equation} H: Kh^{*,-j_0}_{symp}(L)\to Kh^{*,j_0}(L) \end{equation} We prove that this statement is true for any link (bridge) diagram by induction on the number of crossings. The base case for unlinks is proved with Proposition~\ref{prop:unlink}. Now we assume that $L_+$ is a link diagram with n crossings, whereas its resolutions $L_0$ and $L_\infty$ have $(n-1)$ crossings. Let us assume we are doing resolutions at a positive crossing, if all crossings are negative, a similar argument can be applied for the exact sequences induced by doing resolution at a negative crossing. From the inductive assumption, $H$ are bigraded isomorphisms for $L_0$ and $L_\infty$. Let us fix a Jones grading $j_1$ on $Kh(L_0)$. The maps in the long exact sequence will be trivial unless the Jones grading $j_2$ on $Kh(L_\infty)$ is $j_1-3v-2$, where $v$ is the signed count of crossings between this arc and other components, and $j_3$ on $Kh(L_+)$ to be $j_1+1$. Thus we can decompose our commutative diagram with respect to the Jones grading: \small\[\begin{tikzcd} \hspace{-2.5cm}HF^{*,-j_1}(\mathcal{K}_\alpha,\mathcal{K}_\gamma) \arrow{r}{c_2} \arrow[swap]{d}{H} & HF^{*+2,-j_2}(\mathcal{K}_\alpha,\mathcal{K}_\delta) \arrow{r}{c_3} \arrow[swap]{d}{H}& HF^{*+1,j'_3}(\mathcal{K}_\alpha,\mathcal{K}_\beta) \arrow{r}{c_1} \arrow[swap]{d}{H}& HF^{*+1,-j_1}(\mathcal{K}_\alpha,\mathcal{K}_\gamma) \arrow{r}{c_2} \arrow[swap]{d}{H}&HF^{*+3,-j_2}(\mathcal{K}_\alpha,\mathcal{K}_\delta) \arrow{d}{H} \\ \hspace{-1.5cm}Kh^{*,j_1}(L_0) \arrow{r} & Kh^{*+2,j_2}(L_\infty) \arrow{r} &Kh^{*+1,j_3}(L_+) \arrow{r} &Kh^{*+1,j_1}(L_0) \arrow{r} &Kh^{*+3,j_2}(L_\infty) \end{tikzcd} \] \normalsize where the weight grading $j'_3$ is given by $-j_2+wt(c_3)$. The first, second, fourth, and fifth columns are all isomorphisms, so by the five lemma, we conclude that the third column is also an isomorphism and thus we know $H$ is also a bigraded isomorphism for $L_+$. As for the grading correspondence, because the $c_i$ all have fixed weight grading after specifying the choice of equivariant structures, if we change $j_1$ by any number $k$, we change $j_3$ also by $k$. As for the first row, if $-j_1$ is changed into $-j_1-k$, this will result in $j'_3$ shifting by $-k$ as well. This is enough to show that as a relative grading, the weight grading recovers the Jones grading as a relative grading. \end{proof} As a corollary of the theorem above, we can also conclude Theorem~\ref{thm:relative} that the relative weight grading is independent of the choice of link diagrams. \begin{proof}[Proof of Theorem~\ref{thm:relative}] For any two bridge diagrams $D$ and $D'$ representing a link $L$, relative weight grading $wt$ and $wt'$ can be defined on $D$, and respectively $D'$. Theorem~\ref{thm:main} indicates that both $wt$ and $wt'$ coincide with $-j$ with as relative gradings. Thus $wt$ and $wt'$ are the same as relative gradings. \end{proof} At the writing of this paper, the author can not provide a pure symplectic proof of Theorem~\ref{thm:relative} without referring to the invariance of the bigrading on Khovanov homology. Such a proof consists of the invariance of weight grading under isotopy, handeslide and stabilization. Isotopy and handleslide invariance could be potentially proved via Proposition~\ref{prop:product}, if we look close enough into the bridge diagrams and the Floer products. Isotopy and handleslide invariance can also be deduced from Hamiltonian isotopy invariance. But general Hamiltonian isotopy invariance will require virtual perturbations because some of the transversality assumptions will not be preserved under general isotopies, say, $L$ might bound some Maslov zero discs after isotopy. Stabilization invariance is harder than the other two. It requires some degeneration arguments in the Hilbert scheme setup, relating holomorphic discs in $\mathscr{Y}_n$ and discs in $\mathscr{Y}_{n-1}$ so that differentials in the stabilized diagram can be identified with differentials in the original diagram. However, degeneration arguments are only used in the original nilpotent slice definition of Seidel and Smith, while the weight grading is defined using Hilbert schemes. \bibliographystyle{unsrt}

proofpile-arXiv_069-13704

\section{Old proof for Theorem \ref{odd_unicyclic}} \begin{proof} For the proof we consider the structure of $G$. Recall that the length of the cycle is $2k+1$. The proof is based on a case analysis. \begin{enumerate} \item \textbf{For every $u \in C$, $T_u$ is a rooted tree.} \\ By Lemma~\ref{zero}, there is an edge $e$ such that $Z(G-e) \leq Z(G)$. Since the $G-e$ is a tree, Corollary~\ref{unicycle} ensures that $\dim(G-e) \leq Z(G-e)$. By Lemma~\ref{tree}, $\dim(G) \leq \dim(G-e)$. The combination of these inequalities gives $\dim(G) \leq Z(G)$. \item \textbf{For every $u \in C$, $T_u$ is a rooted path.} \\ We claim that any pair of vertices $\alpha$ and $\beta$ of $C$ at distance $k$ is a resolving set. Let ${\alpha,\beta}$ be such a pair of vertices. Let us prove that for every $x \in G$, $d(x,\alpha)+d(x,\beta) \in \{k,k+1\}$ if and only if $x \in C$. Indeed as $d(\alpha,\beta)=k$ for any vertex $x$, $d(x,\alpha)+d(x,\beta) \geq k$. If $x \in C$ either $x$ is on the path between $\alpha$ and $\beta$ and $d(x,\alpha)+d(x,\beta) =k$ or $x$ is in the other part of the cycle and $d(x,\alpha)+d(x,\beta) = k+1$. If $x \notin C$ then let $y$ be the vertex of $C$ such that $x \in T_y$. Then, $d(x,\alpha)+d(x,\beta)=d(y,\alpha)+d(y,\beta)+2d(x,y) \geq 2 +d(y,\alpha)+d(y,\beta) \geq k+2$. This implies that a vertex on the cycle cannot have the same code as a vertex not on the cycle. \\ As $\alpha$ and $\beta$ are vertices of $C$, Lemma~\ref{cycle} applies. Then two vertices on $C$ cannot have the same code. Two vertices not in $C$ cannot have the same code then one of the previous case must apply. Assume by contradiction that two vertices $x$ and $y$ are not resolved by $\{\alpha, \beta\}$ with $x \in T_u$ and $y \in T_v$ for $u$ and $v$ in $C$. If $u=v$ since $T_u$ is a rooted path $\alpha$ resolves the pair $(x,y)$ immediately. Assume from now $u \neq v$. Assume by symmetry $d(x,u) \leq d(y,v)$ and let $z \in T_v$ be the vertex on $T_v$ such that $d(v,z)=d(x,u)$. Then, the pair $\{\alpha,\beta\}$ does not resolve the pair $(z,u)$ but $u \in C$ and the previous cases gives that any pair of vertices with one vertex on $C$ is resolved by $\{\alpha,\beta\}$. We get a contradiction and then pair $\{\alpha,\beta\}$ resolves $G$. So $\dim(G)=2$ and as $G$ is not a path $Z(G) \geq 2$. \item \textbf{There exists a unique $u \in C$ such that $T_u$ is a rooted tree}.\\ Let us prove that $\dim(G) \leq \dim(T_u)+1$ and $Z(G) \geq Z(T_u)+1$. Let $S$ be a basis for $T_u$ and $v \in C$ a vertex at distance $k$ from $u$. Let us show that $S \cup \{v\}$ is a resolving set of $G$. Let $\alpha \in S \cap T_u$, then two vertices in $G \setminus T_u$ cannot have the same distance to $\alpha$ and $v$. Indeed otherwise these two vertices would have the same distance to $u$ and $v$ which is impossible since in $G \setminus T_u$, for every $v \in C$ $T_v$ is a rooted path and by point (2) $\{u,v\}$ is a resolving set. Two vertices on $T_u$ cannot have the same code since $S$ is a basis for $T_u$ and $u$ is a cutvertex. Let $x \in T_w$ for some $w \in C$ and $y \in T_u$. By the triangular inequality $d(\alpha,y) \leq d(y,u)+d(u,\alpha)$. If $d(\alpha,y)=d(\alpha,x)$ then $d(x,u) \leq d(y,u)$ as $d(\alpha,x)=d(\alpha,u)+d(u,x)$. If $d(x,v)=d(y,v)$ then as $d(y,v)=d(y,u)+d(u,v) \geq d(x,u)+d(u,v)$. We get $d(x,v) \geq d(x,u)+d(u,v)$. Removing $d(x,w)$ on both side gives $d(w,v) \geq d(w,u)+d(u,v)$ which is impossible as $d(u,v)=k$ and $d(w,v) \leq k$. So $x$ and $y$ have different codes and $\dim(G) \leq \dim(T_u)+1$. \\ Let $Z$ be a minimal zero forcing set of $G$. If $Z$ contains $u$, then it should contain at least another vertex in $G \setminus T_u$. Since the restriction of $Z$ to $T_u$ is a forcing set for $T_u$ we have $Z(G) \geq Z(T_u)+1$. So we can assume that $u \notin Z$. Consider a sequence of color change rule that turns $v$ into black. Either $u$ is forced by a vertex in $G \setminus T_u$ since $(G \setminus T_u) \cup \{u\})$ contains a cycle there are at least two vertices in $Z \cap (G \setminus T_u)$ and $Z \cap T_u \cup \{u\}$ is a forcing set of $T_u$. So $Z(G) \geq Z(T_u)+1$. Otherwise $u$ is forced by a vertex of $T_u$ then there is at least one vertex in $Z \cap (G \setminus T_u)$ and $Z \cap T_u $ is a forcing set of $T_u$ so $Z(G) \geq Z(T_u)+1$. So in both cases we have $\dim(G) \leq \dim(T_u)+1 \leq Z(T_u)+1 \leq Z(G)$. \item \textbf{There exists $w \in C$ such that $T_w$ is a rooted path and there exist two vertices $u$ and $v$ in $C$ such that $T_u$ and $T_v$ are rooted trees.} \\ We assume by contradiction $\dim(G) > Z(G)$. The proof consists in finding $e \in E(C)$ such that $\dim(G-e) < Z(G-e)$ and $Z(G-e) \leq Z(G)$. We use Lemma~\ref{deg1} to find such an edge and Lemma~\ref{treecomp} to get a contradiction. \begin{enumerate} \item \textbf{There exists a component $T_u$ with $\ell_u =0$.} \begin{enumerate} \item \textbf{There exists $x \in C$ with $\ell_x \geq 1$ and $ux \notin E(C)$.} \\ Let $e$ be an edge incident to $x$ satisfying the conclusion of Lemma~\ref{deg1}. By Lemma~\ref{deg1} $Z(G-e) \leq Z(G)$. And $\dim(G-e) < Z(G-e)$ by Lemma~\ref{treecomp} since $u$ has degree at least three but is not a major vertex with terminal degree greater than 2. So $\dim(G-e) \leq Z(G)+1$ and, by Lemma~\ref{vardim}, $\dim(G) \leq \dim(G-e)+1$ which give the conclusion. \item \textbf{Now assume that $\ell_v=0$.} \\ Let $e$ be an edge incident to $w$ satisfying the conclusion of Lemma~\ref{deg1}. The edge $e$ is not incident to $u$ or $v$, assume by symmetry $e$ is not incident to $u$. Then $Z(G-e) \leq Z(G)$ and $\dim(G-e) < Z(G-e)$ by Lemma~\ref{treecomp} since $u$ has degree at least three but is not a major vertex with terminal degree at least $2$. So $\dim(G-e) \leq Z(G)+1$ and by Lemma~\ref{vardim} $\dim(G) \leq \dim(G-e)+1$ which give the conclusion. \item \textbf{So we can assume that $\ell_v \geq 1$ with $v$ and $x$ incident to $u$ and $T_u,T_v,T_x$ are the only rooted trees of $C$.} \\ Let $e$ be an edge incident to $v$ satisfying the conclusion of Lemma~\ref{deg1}. Then, $Z(G-e) \leq Z(G)$. If $e=uv$ then $x$ has degree three in $G-e$ but is not a major vertex with terminal degree at least 2 so $\dim(G-e) < Z(G-e)$ by Lemma~\ref{treecomp}. And if $e$ is the other edge of $E(C)$ incident to $v$ then $u$ has degree at least three in $G-e$ but is not a major vertex with terminal degree at least $2$ so $\dim(G-e) < Z(G-e)$ by Lemma~\ref{treecomp}. \end{enumerate} \item \textbf{Not all the rooted trees are consecutive.} \\ Let $w \in C$ be the root of a path. Then, there exist $u,w,v,z$ appearing in this order in the cycle such that $T_u$ and $T_v$ are rooted trees and $T_z$ root of a path. Let $e$ be an edge incident to $w$ satisfying the conclusion of Lemma~\ref{deg1}. Then $Z(G-e) \leq Z(G)$. And $\dim(G-e) < Z(G-e)$ by Lemma~\ref{treecomp} since $z$ is neither a terminal vertex, a major vertex or an interior degree two vertex in $G-e$. So $\dim(G-e) \leq Z(G)+1$ and by Lemma~\ref{vardim} $\dim(G) \leq \dim(G-e)+1$ which give the conclusion. \\From now we can assume the rooted trees are consecutive. \item \textbf{There are at least three consecutive rooted trees $T_u,T_v,T_w$.} \\ Without loss of generality assume $u$ is adjacent to $v$ and $w$. Let $e$ be an edge incident to $u$ satisfying the conclusion of Lemma~\ref{deg1}. Without loss of generality assume $e=uv$. Then $Z(G-e) \leq Z(G)$. By Lemma~\ref{treecomp} $\dim(G-e) < Z(G-e)$ since the vertex in $C$ incident to $w$ which is not $u$ have degree three in $G-e$ but is not a major vertex with terminal degree at least 2. So $\dim(G-e) \leq Z(G)+1$ and by Lemma~\ref{vardim} $\dim(G) \leq \dim(G-e)+1$ which gives the conclusion. \item \textbf{There is exactly two rooted trees $T_u$ and $T_v$} \\ Recall that $u$ and $v$ are supposed adjacent. \begin{enumerate} \item \textbf{Assume first $\ell_u=1$.} \\ Let $e$ be an edge incident to $u$ satisfying the conclusion of Lemma~\ref{deg1}. Then $Z(G-e) \leq Z(G)$. If $e=uv$ then by Lemma~\ref{treecomp} $\dim(G-e) < Z(G-e)$ since of $u$ which has degree at least three but is not a major vertex with terminal degree greater than 2 in $G-e$. If $e=vw$ where $w$ is the other vertex adjacent to $v$ and $\dim(G-e) = Z(G-e)$ then $u$ is a major vertex in $G-e$ so $v$ is also a major vertex. So $v$ is connected to two leaves in $G-e$. As $\ell_v=1$ the deletion of $e$ gives a new terminal vertex to $u$ which can only be the terminal vertex of the path rooted in $w$. That mean there is only one vertex which is root of a path. Let $e'$ be an edge incident to $u$ satisfying the conclusion of Lemma~\ref{deg1} in $G-e'$. with $Z(G-e') \leq Z(G)$. If $e'=uv$ then by Lemma~\ref{treecomp} $\dim(G-e') < Z(G-e')$since $w$ has degree 3 in $G-e'$ and $\ell_w=1$ ($w$ is connected to $u$ and $v$ which are not root of a rooted path). If $e'$ is the other edge incident to $u$ then by Lemma~\ref{treecomp} $\dim(G-e') < Z(G-e')$ because of $u$ which is not a major vertex with terminal degree at least 2 in $G-e'$. \item \textbf{$\ell_u \geq 2$ and $l_v \geq 2$.}\\ Let $e=uv$ we have $Z(G-e) \leq Z(G)$. Let $Z$ be a minimum forcing set of $G$ then in $Z \cap T_u$ there is a vertex in a path from $u$ to a terminal vertex. We can assume this vertex is the terminal vertex and force all the path until $u$. Same for $v$ so $e$ is not used to forced any vertex. By Lemma~\ref{treecomp} $\dim(G-e) < Z(G-e)$ because of one vertex root of a path which is not a major vertex with terminal degree greater than 2 in $G-e$. \qedhere \end{enumerate} \end{enumerate} \end{enumerate} \end{proof} \section{General bound}\ The goal of this section is the prove the following theorem: \begin{theorem}\label{main} Let $G=(V,E)$ be a graph of cycle rank at least one. Then, $\dim(G) \leq Z(G) + \min(5c(G)+\tau(G)-5,3c(G)+5\tau(G)-5)$. \end{theorem} Since $\tau(G)\leq c(G)$, we obtain the following linear bound: \begin{cor}\label{bornec} For any graph $G$ of cycle rank at least one, $\dim(G) \leq Z(G)+ 6c(G)-5$. \end{cor} \subsection{Outline of the proof of Theorem \ref{main}} The result is known if $c(G) \leq 1$, hence in the rest of the proof we will consider a graph $G$ with $c(G)\geq 2$. The main idea of the proof of Theorem \ref{main} is to remove a small number of vertices in order to guarantee that the resulting connected components $T$ are trees with nice properties. We then use the inequality $\dim(T) \leq Z(T)$ on these trees. We finally prove that the size of a resolving set cannot increase too much and the size of a zero forcing set cannot reduce too much when we return to $G$. Slightly more formally, we will do the following. \begin{enumerate} \item Find a (small) set of vertices $M$ such that: (1) $M$ contains a feedback vertex set $X$, (2) $G \setminus M$ is a forest and, (3) each connected component $H$ of $G\setminus M$ is connected to at most two vertices of $M$ (Lemma~\ref{structure}). \item Define a (small) set of vertices $P$ that identify in which connected component $H$ of $G \setminus M$ a vertex belongs to (Lemma~\ref{use_of_p}). \item For each connected component $G_i$ of $G \setminus X$, let $S_{G_i}$ be a metric basis of $G_i$ and prove that $P \cup M \cup \bigcup_i S_{G_i}$ is a resolving set of $G$ (Lemmas~\ref{toconclude},~\ref{structurecompM} and~\ref{iscode}). \item Bound the size of $M$ and $P$ as functions of $\tau$ and $c(G)$ (Lemmas ~\ref{boundN}, ~\ref{boundM},~\ref{boundMalt} and~\ref{boundP}). \item Prove the inequality $|\bigcup_i S_{G_i}| - \tau \leq Z(G)$ and conclude (Lemma~\ref{boundZ}). \end{enumerate} \subsection{Proof of Theorem~\ref{main}} Let $J$ be the graph obtained from $G$ by an iterative deletion of vertices of degree $1$. Since $c \ge 2$, $J$ is not a cycle. Let $X \subseteq V$ be a feedback vertex set of $J$ such that each vertex of $X$ has at least degree three in $J$. \begin{lemme} Let $G$ be a connected graph with no vertex of degree $1$ that is not a cycle. There exists a feedback vertex set $X$ of size $\tau(G)$ with all vertices of $X$ of degree at least $3$. \end{lemme} \begin{proof} Let $X$ be a minimum feedback vertex set with the minimum number of vertices of degree less than $3$. Assume by contradiction that $X$ contains a degree $2$ vertex $x$. Let $P$ be the maximal path of vertices of degree $2$ containing $x$. Since $G$ is not a cycle, $P$ does not contain the whole graph. Let $y$ be an endpoint of $P$. Let $X'=X \setminus \{x\} \cup \{y\}$. The set $X'$ is a feedback vertex set, a contradiction with the definition of $X$. \end{proof} From now on we consider a minimum feedback vertex set $X$ of $G$ only containing vertices of degree at least $3$. Let $G_1,G_2,...,G_k$ be the connected components of $G \setminus X$. Note that each $G_i$ induces a tree. For each $G_i$, let $X_i \subseteq X$ be the set of vertices of $X$ connected to at least one vertex of $G_i$. Let $N_i \subseteq V(G_i)$ be the set of vertices in $G_i$ adjacent to (at least) one vertex of $X_i$ and $E_i$ be the set of edges between $N_i$ and $X_i$. Let $T_i$ be the minimal subtree of $G_i$ containing the vertices of $N_i$. In other words, $T_i$ is the subtree of $G_i$ restricted to the union of the paths between $a$ and $b$ for any pair $a,b \in N_i$. Let $T'_i$ be the tree built from $T_i$ by adding to each vertex $u$ in $N_i$ as many leaves as $u$ have neighbors in $X$ in $G$. We define $M_i$ as the set of vertices in $T'_i$ of degree at least 3 in $T'_i$. Let $M=X \cup (\bigcup_{i=1}^k M_i)$. \begin{lemme}\label{structure} For each connected component $H$ of $G \setminus M$ there are at most two edges in $G$ between $H$ and $G \setminus H$. \end{lemme} \begin{proof} Let $U_H$ be the minimal subtree of $H$ containing all the vertices incident to an edge between $H$ and $G \setminus H$. Then, for each edge between a vertex $v \in H$ and a vertex in $G \setminus H$, add one new vertex in $U_H$ adjacent to $v$. Let us prove that if a vertex have degree at least 3 in $U_H$ then this vertex is in $M$. \\ Assume by contradiction $v \in U_H$ has degree 3 in $H$. Let $i$ be such that $H$ is a subgraph of $G_i$ and consider $T'_i$. The vertex $v$ is on a minimal path between two vertices of $M$ (otherwise $v \notin U_H$) so $v$ is also in $T'_i$ because the path in $G_i$ between these vertices is unique as $G_i$ is a tree. So the degree of $v$ is at least 3 in $T'_i$ and $v \in M$, a contradiction. \end{proof} Lemma~\ref{structure} implies the following. \begin{cor} Every connected component of $G \setminus M$ is connected to at most two vertices of $M$. \end{cor} A connected component of $G\setminus M$ can be attached to $M$ in three different ways, called {\em types}, that are illustrated in Figure~\ref{figexh}. A connected component has Type $A$ (respectively type $B$) if there are exactly two edges between $H$ and $M$ with distinct endpoints in $H$ and such that their endpoints in $M$ are distinct (resp. the same). A component $H$ has Type $C$ if all the edges of $G$ between $H$ and $M$ have the same endpoint in $H$ (but possibly distinct endpoints in $M$). Let $H$ be a connected component of $G \setminus M$ of type $A$ or $B$ and let $x$ and $y$ be the two endpoints in $M$ of the edges between $H$ and $M$. Let $\rho_H$ be one vertex on the path in $H$ between $x$ and $y$ such that $|d_H(x,\rho_H)-d_H(y,\rho_H)| \leq 1$. In other words, $\rho_H$ is one of the vertices in the middle of the path between $x$ and $y$ in $H$. Let $P$ be the union of the vertices $\rho_H$ for all the connected component $H$ of type $A$ and $B$. \begin{figure}[ht] \centering \includegraphics[scale=0.6]{exh.pdf} \caption{The three types of components of $G\setminus M$. } \label{figexh} \end{figure} \begin{lemme}\label{use_of_p} Let $H$ be a connected component of $G \setminus M$ of type $A$ or $B$. Let $\rho$ be the vertex of $P \cap H$. Then, for any vertex $v \in H$, there is no shortest path between $z$ and $\rho$ using vertices in $G \setminus H$ and for $z \in \partial H$ there is a shortest path using only vertices in $H \cup \{z\}$. \end{lemme} \begin{proof} Let $x$ and $y$ be the two vertices in $H$ adjacent to a vertex of $M$, $\alpha$ be the vertex of $M$ adjacent to $x$ and $\beta$ the vertex in $M$ adjacent to $y$. By definition of types $A$ or $B$, $x \neq y$ (but $\alpha$ and $\beta$ could be the same vertex if the type is $B$). By definition of $\rho$, we have $|d_H(x,\rho)-d_H(y,\rho)| \leq 1$. Let $v \in V(H)$. Assume by contradiction that a shortest path between $v$ and $\rho$ in $G$ passes through vertices in $G \setminus H$. So this shortest path between $\rho$ and $v$ passes through $x$ and $y$. By symmetry we can assume $d_G(v,\rho)=d_H(v,x)+d(x,\alpha)+d(\alpha,\beta)+d(\beta,y)+d_H(y,\rho)$. In other words, the path from $v$ to $\rho$ passes through $x$ and then $y$. Assume by contradiction that $d_H(z,x)+d(x,\alpha)+d(\alpha,\beta)+d(\beta,y)+d_H(y,\rho) \leq d_G(v,\rho)$. As $x\alpha$ and $y\beta$ are edges, we have $d_H(v,x)+d(\alpha,\beta)+d_H(y,\rho) +2 \leq d_G(v,\rho)$. By triangular inequality, $d_G(v,\rho) \leq d_H(v,\rho) \leq d_H(v,x)+d_H(x,\rho)$. So $d_H(x,\rho) \geq d_H(y,\rho)+d(\alpha,\beta)+2$ which contradicts $|d_H(x,\rho)-d_H(y,\rho)| \leq 1$. \\ If $z$ is in $\partial H$ and $\alpha=\beta$ then $z=\alpha=\beta$ and the result is immediate. Else assume $z=\alpha$ then we want to contradict $d(\alpha,\beta)+d(\beta,\rho) < d(\alpha,\rho)$. As $\alpha$ is adjacent to $x$ and $\beta$ adjacent to $y$ the inequality is equivalent to $d(\alpha,\beta) + d(y,\rho) < d(x,\rho)$. As $\alpha \neq \beta$, $d(\alpha,\beta) \geq 1$. So $d(y,\rho) +2 \leq d(x,\rho)$ which contradicts $|d_H(x,\rho)-d_H(y,\rho)| \leq 1$. \end{proof} We will now construct a resolving set of $G$. For every $i$, we define a metric basis $S_i$ of $G_i$ as follows. If $G_i$ is a path then $S_i$ is the extremity of the path the farthest to $X$. If $G_i$ is a branching tree then $S_i$ contains all but one terminal vertex for each major vertex on $G_i$. The set $S_i$ is a metric basis of $G[H_i]$ by Lemma~\ref{dimtree}. We will prove that $S_i$ resolves in $G$ the vertices inside the components of $G\setminus M$. Let us first define the projection of a vertex. \begin{defn}{} Let $H$ be a connected component of $G \setminus M$ of type $A$ or $B$ and let $x$ and $y$ be the two vertices in $H$ adjacent to $M$. Let $u$ be any vertex of $H$. The \emph{projection $z_u$ of $u$ (on the path between $x$ and $y$)} is the unique vertex in the path between $x$ and $y$ in $H$ at minimum distance to $u$. \end{defn} \begin{lemme}\label{toconclude} Let $H$ be a connected component of $G \setminus M$ of type $A$ or $B$. Let $u$ and $v$ be two vertices in $H$ and assume that a vertex $\gamma$ resolves the pair $(u,v)$ in $G_i$, the connected component of $G\setminus X$ that contains $H$. If $z_u=z_v$ then $\gamma$ resolves the pair $(u,v)$ in $G$. \end{lemme} \begin{proof} We consider two cases $z_{\gamma}=z_u$ and $z_{\gamma} \neq z_u$. If $z_{\gamma}=z_u=z_v$ then the distances to $\gamma$ are the same in $G_i$ and $G$: $d_{G_i}(\gamma,u)=d_G(\gamma,u)$ and $d_{G_i}(\gamma,v)=d_G(\gamma,v)$ so $d_G(\gamma,u) \neq d_G(\gamma,v)$ as $d_{G_i}(\gamma,u) \neq d_{G_i}(\gamma,v)$. If $z_{\gamma} \neq z_u$ then $d_{G_i}(\gamma,u)=d_{G_i}(u,z_u)+d_{G_i}(z_u,\gamma)$ and $d_{G_i}(\gamma,v)=d_{G_i}(v,z_v)+d_{G_i}(z_v,\gamma)$. As $z_u=z_v$ and $d_{G_i}(\gamma,u) \neq d_{G_i}(\gamma,v)$, we have $d_{G_i}(u,z_u) \neq d_{G_i}(v,z_v)$. Since $d_{G_i}(u,z_u)=d_G(u,z_u)$ and $d_{G_i}(v,z_v)=d_G(v,z_v)$ we obtain $d_G(u,\gamma) \neq d_G(v,\gamma)$ since $d_G(u,\gamma)=d_G(u,z_u)+d_G(z_u,\gamma)$ and $d_G(v,\gamma)=d_G(v,z_v)+d_G(z_v,\gamma)$. \end{proof} \begin{lemme}\label{structurecompM} Let $m$ be a vertex of $M$. Let $S= M \cup P \cup (\bigcup_i S_i)$. There is at most one connected component $H$ of $G \setminus M$ adjacent to $m$ and such that $H \cap S =\emptyset$. Moreover if such a component exists, it is a path with one extremity connected to $m$. \end{lemme} \begin{proof} By definition of $P$, any connected component of type $A$ or $B$ contains an element of $P$. So the only connected components $H$ which can satisfy $H \cap S =\emptyset$ are connected components $H$ of type $C$. Assume by contradiction that two connected components $H$ and $H'$ of $G \setminus M$ are connected to $m$ and that $H \cap S =\emptyset$ and $H' \cap S =\emptyset$. If $m \in X$ then $H=G_i$ for some $i$. Since $S_i$ is a resolving set of $H$, $S_i$ is non empty and $H \cap S \neq \emptyset$. So we can assume that $m \in M \setminus X$. Thus $H$ and $H'$ are in the same connected component $G_i$ for some $i \in \{1,,\ldots,k\}$ since $m \notin X$ is connected to at least one vertex of $H$ and $H'$. If $S_i$ does not contain any vertex in $H \cup H'$ then the two vertices in $H \cup H'$ adjacent to $m$ are not resolving by $S_i$ in $G_i$ by Lemma~\ref{dimbranch}, a contradiction with the definition of $S_i$. Let us now prove the second part of the statement. Let $H$ be a connected component of $G \setminus M$ connected to $w$ and such that $H \cap S = \emptyset$. We have $H \subseteq G_i$ for some $i \in [1,k]$. Let $u$ be a terminal vertex in $H$ and consider the path between $u$ and $w$. Assume by contradiction this path contains vertex $v$ of degree $3$. Then, there exists a vertex $w$ adjacent to $v$ which is not on the path between $v$ and $m$ and a vertex $x$ adjacent to $u$ on the path between $u$ and $v$. The vertices $w$ and $x$ are not resolved by any vertex out of $H$ in $G_i$ so if $S_i \cap H= \emptyset$ there is a contradiction as $S_i$ is a resolving set of $G_i$. So the path between $v$ and $m$ contains only degree-2 vertices for any terminal vertex $v$ meaning $H$ is a path and $v$ the only terminal vertex of $H$. \end{proof} \begin{lemme} \label{iscode} The set $S= M \cup P \cup (\bigcup_i S_i) $ is a resolving set of $G$. \end{lemme} \begin{proof} Assume by contradiction that $u$ and $v$ are different vertices at same distance to any vertex in $S$. Then, $u \notin M$ and $v \notin M$ as $M \subseteq S$. So there exist $H_u$ and $H_v$ connected components in $G \setminus M$ such that $u \in H_u$ and $v \in H_v$. \\ \begin{itemize} \item Assume $H_u$ is of type $A$ and let $x\alpha$ and $y\beta$ be the two edges connecting $H_u$ to $M$ (with $x,y \in H_u$). Let $\rho$ be the vertex in $P \cap H_u$. Assume $v \notin H_u$. The shortest path between $v$ and $\rho$ passes through $\alpha$ or $\beta$. Assume by symmetry $\alpha$ is on the shortest path between $v$ and $\rho$ so $d(v,\rho)=d(v,\alpha)+d(\alpha,\rho)$. As $\alpha$ and $\rho$ are in $S$ we also have $d(u,\rho)=d(u,\alpha)+d(\alpha,\rho)$ which contradicts Lemma~\ref{use_of_p}. So $H_u=H_v$, denote it by $H$ from now and let $G_i$ be the component of $G\setminus X$ that contains $H$. \\ The graph $H$ is a tree with a path between the two vertices $x$ and $y$. We want to prove that $z_u=z_v$ by contradiction with $z_u$ and $z_v$ are defined in Lemma~\ref{use_of_p}. Assume it is not the case, without loss of generality we can assume $z_u \neq \rho$. We have $d(u,\alpha)=d(u,z_u)+d(z_u,\alpha)$, $d(v,\alpha)=d(v,z_v)+d(z_v,\alpha)$, $d(u,\beta)=d(u,z_u)+d(z_u,\beta)$ and $d(v,\beta)=d(v,z_v)+d(z_v,\beta)$. As $d(u,\alpha)=d(v,\alpha)$ and $d(u,\beta)=d(v,\beta)$ we get \[d(z_u,\alpha)+d(z_v,\beta)=d(z_u,\beta)+d(z_v,\alpha)\] The vertices $z_u$ and $\rho$ are distinct and both between $\alpha$ and $\beta$ so $z_u$ is between $\alpha$ and $\rho$ or $\beta$ and $\rho$. Assume $z_u$ is between $\alpha$ and $\rho$ then $d(z_u,\alpha) \leq d(z_u,\beta)$ so $d(z_v,\alpha) \leq d(z_v,\beta)$ meaning $z_v$ is also between $\alpha$ and $\rho$. The shortest path between $\alpha$ and $\rho$ passes through $z_u$ and $z_v$ by Lemma~\ref{use_of_p}. Assume $z_u$ is closer than $z_v$ to $\rho$ then $d(\alpha,\rho)=d(\alpha,z_v)+d(z_v,z_u)+d(z_u,\rho)$ gives \[d(\alpha,z_u)+d(z_v,\rho)=d(\alpha,z_v)+d(z_u,\rho)+2d(z_u,z_v)\] Use now the paths to $\rho$: $d(u,\rho)=d(u,z_u)+d(z_u,\rho)$ as $z_u \neq \rho$ and $d(v,\rho) \leq d(v,z_v)+d(z_v,\rho)$ and reuse $d(u,\alpha)=d(u,z_u)+d(z_u,\alpha)$, $d(v,\alpha)=d(v,z_v)+d(z_v,\alpha)$ give \[d(z_v,\alpha)+d(z_u,\rho) \leq d(z_u,\alpha)+d(z_v,\rho)\] Combine with the previous equality give $d(z_u,z_v) \leq 0$ so $z_u=z_v$. The set $S$ contains $S_i$ which is a resolving set of $G_i$. So there exists $\gamma$ which resolves the pair $(u,v)$ in $G_i$ and thus resolves the pair $(u,v)$ in $G$ by Lemma~\ref{toconclude}. \item Assume now that $H_u$ is of type $B$ and let $x\alpha$ and $y\beta$ be the two edges between $H_u$ and $M$. Let $\rho$ the vertex in $P \cap H_u$. If $v \notin H_u$ then $\alpha$ is on the path between $v$ and $\rho$ and $d(v,\rho)=d(v,\alpha)+d(\alpha,\rho)$ so $d(u,\rho)=d(u,\alpha)+d(\alpha,\rho)$ which contradicts Lemma~\ref{use_of_p}, so $H_u=H_v$. We denote $H_u=H_v$ as $H$ from now. \\ We have $\alpha \in X$ as $H \cup \{\alpha\}$ contains a cycle. So $H=G_i$ for some $i \in [1,k]$. As $S$ contains $S_i$ which is a metric basis for $G_i$ if $u \neq v$ there exists $\gamma \in H$ such that $d_H(u,\gamma) \neq d_H(v,\gamma)$ and $d_G(u,\gamma) = d_G(v,\gamma)$. \\ Assume first $z_{\gamma} = \rho$ and prove $\gamma$ resolves the pair $(u,v)$. Indeed if $z_{\gamma}=\rho$ we prove that $d_G(u,\gamma)=d_H(u,\gamma)$ and $d_G(v,\gamma)=d_H(v,\gamma)$. If $z_u=z_{\gamma}$ then $d_G(u,\gamma)=d_G(u,z_u)+d_G(z_u,\gamma)=d_H(u,z_u)+d_H(z_u,\gamma)=d_H(u,\gamma)$. If $z_u \neq z_{\gamma}$ then \[d_G(u,\gamma)=d_G(u,z_u)+d_G(z_u,\rho)+d_G(\rho,\gamma).\] By Lemma~\ref{use_of_p}, $d_G(z_u,\rho)=d_H(z_u,\rho)$. Thus $d_G(u,z_u)=d_H(u,z_u)$ because the path between these two vertices is unique. Similarly $d_G(\rho,\gamma)=d_H(\rho,\gamma)$. So $d_G(u,\gamma)=d_H(u,\gamma)$. In the same way, $d_G(v,\gamma)=d_H(v,\gamma)$ and $\gamma$ still resolves the pair $(u,v)$. \\ Assume now $z_{\gamma} \neq \rho$. We have $d(u,\rho)=d(u,z_u)+d(z_u,\rho)$, $d(v,\rho)=d(v,z_v)+d(z_v,\rho)$, $d(u,\alpha)=d(u,z_u)+d(z_u,\alpha)$, $d(v,\alpha)=d(v,z_v)+d(z_v,\alpha)$, $d(u,\alpha)=d(v,\alpha)$ and $d(u,\rho)=d(v,\rho)$. Combine give \[d(z_u,\rho)+d(z_v,\alpha)=d(z_v,\rho)+d(z_u,\alpha)\] Moreover as $\rho$ and $\alpha$ are nearly opposed on the smallest cycle containing these two vertices we also have \[d(z_u,\rho)+d(z_u,\alpha)=d(z_v,\rho)+d(z_v,\alpha)+ \epsilon\] with $\epsilon \in \{-1,0,1\}$. Summing the two equalities give $2d(z_u,\rho)=2d(z_v,\rho)+ \epsilon$. So $\epsilon=0$, $d(z_u,\rho)=d(z_v,\rho)$ and $d(z_u,\alpha)=d(z_v,\alpha)$. Since $d(u,\alpha)=d(v,\alpha)$, we obtain $d(u,z_u)=d(v,z_v)$. If $z_u \neq z_v$ then $z_u$ and $z_v$ are distinct and at the same distance to three points on a cycle ($\rho$, $\alpha$ and $z_\gamma$) which contradicts Lemma~\ref{cycle3}. If $z_u=z_v$ then Lemma~\ref{toconclude} allow use to conclude. \item Assume now that $H_u$ and $H_v$ are of type $C$. Let $\alpha$ and $\beta$ the vertices of $M$ connected to $H_u$ and $H_v$ respectively. If $\alpha \neq \beta$ then the shortest path between $u$ and $v$ contains two distinct vertices of $S$ so by Lemma~\ref{technical} $u$ and $v$ are resolved. Hence we assume that $\alpha=\beta$. \\ We prove that $H_u=H_v$ by contradiction. If $H_u \neq H_v$, by Lemma~\ref{structurecompM}, without loss of generality, there must exists $\gamma \in S \cap H_u$. If $u$ is on the path between $\gamma$ and $v$ then $d(\gamma,u) < d(\gamma,v)$. Otherwise let $m_u$ in $H_u$ be at the intersection of the path between $u$ and $\gamma$ and between $\alpha$ and $\gamma$. Then, vertices $m_u$ and $\alpha$ are on the shortest path between $u$ and $v$. By Lemma~\ref{technical}, one of them must resolve $u$ and $v$. By assumption, it is not $\alpha$. If it is $m_u$ then we would have $d_G(u,\gamma) \neq d_G(v,\gamma)$ since the shortest paths between $u$, $v$ and $\gamma$ go through $m_u$, a contradiction. \\ Thus we can assume that $H_u=H_v$ and let $H=H_u$. Let $H\subseteq G_i$ for some $i$. The set $S_i$ resolves $u$ and $v$ in $G_i$. Let $\gamma\in S_i$ that resolves $u$ and $v$ in $G_i$. If $\gamma$ is in $H_u$, the distances between $u$, $v$ and $\gamma$ in $G$ and $H$ are the same so $\gamma$ resolves $u$ and $v$. Otherwise, $d_H(u,\gamma)=d_H(u,x)+d_H(x,\gamma)$ where $x$ is the unique vertex of $H$ connected to the rest of the graph. Similarly, $d_H(v,\gamma)=d_H(v,x)+d(x,\gamma)$. Hence $d_H(u,x)\neq d_H(v,x)$ and thus since $d_G(u,\alpha)=d_H(u,x)+1$ and $d_G(v,\alpha)=d_H(u,x)+1$, $\alpha$ resolves $u$ and $v$. \end{itemize} \end{proof} We now bound the size of the resolving set $S$. Recall that $E_i$ is the number of edges between $G_i$ and $X$. We use the following result on minors to get the bound. Let $G$ be a multigraph. The graph $H$ is a {\em minor} of $G$ if $H$ can be obtained from $G$ via a sequence of edge deletions, vertex deletions and edge contractions (the edge contraction operation can create parallel edges between two vertices or loops). One can easily check that the minor operation can only decrease the cycle rank.\\ \begin{lemme} \label{boundN} $\sum_{i=1}^k (|E_i|-1) \leq c+\tau-1 $. \end{lemme} \begin{proof} Let $G'$ be a multigraph obtained from $G$ by contracting all the connected components $G_i$ into one vertex $v_i$. Since $G'$ is a minor of $G$, $c(G') \leq c(G)$ and $G'$ is still connected since we have not deleted edges. The graph $G'$ contains $\tau+k$ vertices so $G'$ contains at most $c(G)+\tau+k-1$ edges. So $\sum_{i=1}^k |E_i| \leq c+\tau+k-1 $ and $\sum_{i=1}^k (|E_i|-1) \leq c+\tau-1 $. \end{proof} In the next two lemmas, we give two bounds on the size of $M$ (one depending only of $c$ and the other of $\tau$ and $c$). . Recall that $T_i$ is the minimal subtree of $G_i$ containing the vertices of $N_i$ and $T'_i$ is the tree built from $T_i$ by adding to each vertex $u$ in $N_i$ as many leaves as $u$ have neighbors in $X$ in $G$. \begin{lemme}\label{boundM} $|M| \leq c+2\tau-2$. \end{lemme} \begin{proof} Consider a subgraph $G_i$ of $G$ such that $|M \cap G_i| \neq \emptyset$ , we want to prove that $|M \cap G_i| \leq |E_i|-2$ and use Lemma~\ref{boundN}. If $|E_i| \leq 2$ then $M \cap G_i = \emptyset$ since $T_i'$ is reduced to a path and then has no degree-$3$ vertex. So we can assume that $|E_i| \geq 3$. By construction, $T_i'$ contains $|E_i|$ leaves and then at most $|E_i|-2 $ vertices of degree at least 3. Indeed let $T$ be any tree on $n$ vertices with $k_1$ leaves, $k_2$ degree-$2$ vertices, and $k_3$ vertices with degree at least $3$. The sum of the degree of all the vertices is $2n - 2$ ($T$ being connected). Thus $k_1 + 2k_2 + 3k_3 \leq 2n - 2$ and then, $k_1 + 3k_3 \leq 2n - 2 - 2k_2 = 2(k_1+k_2+k_3) -2 -2k_2 $. So $k_1 + 3k_3 \leq 2(k_1 + k_3) - 2$. That gives $k_3 \leq k_1 - 2$ which is the target. Since the vertices in $M \cap G_i$ are the vertices of degree at least $3$ in $T_i'$, we have $|M \cap G_i| \leq |E_i|-2$. Since $M=X \cup (\bigcup_{i=1}^k M_i)$, summing on all the $G_i$ with $M \cap G_i \neq \emptyset$ gives $|M| \leq \tau + \sum_{G_i, M \cap G_i \neq \emptyset} (|E_i|-2) $. So we have: \[|M| \leq \tau+ \sum_{G_i, M \cap G_i \neq \emptyset} (|E_i|-2) \leq \tau +\sum_{G_i, M \cap G_i \neq \emptyset} (|E_i|-1) - |\{i, M \cap G_i \neq \emptyset\}|\] By Lemma~\ref{boundN}, $\sum_{G_i, M \cap G_i \neq \emptyset} (|E_i|-1) \leq \sum_{G_i} (|E_i|-1) \leq c + \tau -1$. Since $|\{i, M \cap G_i \neq \emptyset\}| \ge 1$ (otherwise $|M|=0$ and the conclusion holds) gives $|M| \leq c+2\tau-2$. \end{proof} \begin{lemme}\label{boundMalt} $|M| \leq 2c-2$. \end{lemme} \begin{proof} If $|M|\leq 2$ then the inequality holds since $c \geq 2$. So we can assume that $|M| \ge 3$. Let $K$ be the multigraph (with possible loops) with vertex set $M$ where, for each connected component $H$ of $G \setminus M$ of type $A$ or $B$ with endpoints $x$ and $y$ in $M$ (that might be identical), we create an edge between $x$ and $y$ in $K$. Note that $K$ is a minor of $G$ as it can be obtained from $G$ by contracting edges in components $H$ of $G \setminus M$ into a single edge. Every vertex in $K$ has degree at least $3$. The vertices of $M\setminus X$ has degree at least $3$ by definition of $M_i$ for every $i$. By construction of $X$, $x \in X$ has degree at least three in the graph starting from $G$ and removing the degree one vertices. The three adjacent edges all contribute to the degree of $x$ in $K$ so $\deg_{K}(x) \geq 3$. We have $3 |V(K)| \le \sum_{v \in V(K)} \deg(v) = 2 |E(K)|$. So $3|M| \leq 2 |E|$. And the cycle rank of $K$ is at most $c$, so $|E| \leq c + |M| - 1 $. A combination of these inequalities gives $|M| \leq 2c-2$. \end{proof} \begin{lemme}\label{boundP} $|P| \leq c + |M|-1$. \end{lemme} \begin{proof} Let $K'$ be the multigraph (with loops) with vertex set $M$ and an edge between two vertices $x$ and $y$ if and only if there exists in $G \setminus M$ a connected component $H$ adjacent to $x$ and $y$. The graph $K'$ is a minor of $G$ since $K'$ can be obtained from $G$ by contracting edges with exactly one endpoint in $M$ until no such edge exists. One can easily notice that $K'$ there is an edge $xy$ with multiplicity $k$ if and only if in $G \setminus M$ there are $k$ connected components attached to $x$ and $y$. Since $K'$ is a minor of $G$, $c(K') \leq c(G)$. As $K'$ contains $|M|$ vertices it contains at most $c+|M|-1$ edges. Since $P$ contains one vertex in each component of type A or B, we have $|P| \leq c+|M|-1$. \end{proof} \begin{lemme} \label{boundZ} $|\bigcup_i S_i| - \tau \leq Z(G)$. \end{lemme} \begin{proof} Let $Z$ be a minimal zero forcing set for $G$. Let $\mathcal{S}$ be a sequence of forcings that turns $G$ into black starting from $Z$. Let $Z'$ be the set of vertices turned black in $\mathcal{S}$ because of a vertex of $X$. Note that $|Z'| \le \tau$. Let $G_i$ be a connected component of $G[V \setminus X]$. We claim that $(Z \cup Z') \cap V(G_i)$ is a zero-forcing set of $G_i$. If we restrict the sequence of colorings that turn $G$ to black from $Z$, then this sequence also works in $G_i'$. A vertex always has at most as many white neighbors as in $G$. So the only problem that can appear is that a vertex $v$ would have to be turned black because of $u$ in $G$ and $u \notin V(G_i)$. But the only vertices not in $V(G_i)$ that can turn black vertices of $G_i$ are the vertices of $X$. Therefore vertices turned black because of $X$ are in $Z' \cap V(G_i)$ and then are already black. So, by Lemma~\ref{treecomp}, we have $|S_{G_i}| \leq |Z' \cap G_i|$. When we consider the union over all the components, it gives $ |Z \cup Z'| \ge \cup_i |S_{G_i}|$. Since $Z'$ has size at most $\tau$, we have $ Z(G) \geq |\bigcup_i S_{G_i}| - \tau $. \end{proof} We can now conclude. \begin{proof}[Proof of Theorem~\ref{main}] We know that $S= M \cup P \cup (\bigcup_i S_{G_i}) $ is a resolving set of $G$ so $\dim(G) \leq |S|$. Moreover $ Z(G) \geq |\bigcup_i S_{G_i}| - \tau $. That give $\dim(G) \leq |M|+|P|+\tau+Z(G) $. So we get $\dim(G) \leq 2|M|+c+\tau-1+ Z(G)$. We bound $|M|$ by $c+2\tau-2$ and by $2c-2$ so $\dim(G) \leq Z(G) + \min(5c+\tau-5,3c+5\tau-5)$. \end{proof} \section{Introduction} \setcounter{page}{1} A \emph{resolving set} of a graph $G$ is a subset of vertices of $G$ such that any vertex in the graph is identified by its distances to the vertices of the resolving set. In the example of Figure~\ref{figexdim} the set $ \{u,v\}$ is a resolving set of the graph because all the vertices have a different \emph{distance vector} to $\{u,v\}$ so the knowledge of the distance to $u$ and $v$ uniquely identifies a vertex. This notion has been introduced in 1975 by Slater \cite{slater1975trees} for trees and by Harary and Melter \cite{harray1975} for graphs. Determining the minimum size of a resolving set, called \emph{metric dimension} and denoted by $\dim(G)$, is an NP-complete problem~\cite{garey1979} even restricted to planar graphs~\cite{Diaz2012}. Applications of metric dimension goes from piloting a sonar~\cite{harray1975} to the navigation of a robot in an Euclidean space~\cite{KHULLER1996}. \begin{figure}[ht] \centering \includegraphics[scale=1.7]{exdim.pdf} \caption{The black vertices form a resolving set. For each vertex $x$ the vector next to $x$, $(d(u,x),d(v,x))$ is unique.} \label{figexdim} \end{figure} One of the main issues to compute the metric dimension of a graph comes from the fact that it is unstable when the graph is modified. When a vertex is added, the metric dimension can be drastically modified. Indeed, while a path admits a constant size resolving set, a path plus a universal vertex only admits resolving sets of linear size (in the number of vertices). However, in this example even if only one vertex was added, a linear number of edges were also added which had permitted to put all the vertices at distance $2$. One can then wonder if the situation is better when only one edge is modified in the graph. Unfortunately again, the metric dimension of a graph can be drastically modified by the modification of a single edge. In Figure \ref{fig_delete_edge} (first proposed in~\cite{eroh2015effect}), the metric dimension of the left graph is $2k$ where $k$ is the number of layers in the graph while the right graph has metric dimension $k+1$. So the addition of one edge can modify the metric dimension by $\Omega(n)$. \begin{figure}[ht] \centering \includegraphics[scale=0.7]{exdeleteedge.pdf} \caption{\raggedright{On the left graph, all the pairs $(c_i,d_i)$ are twins (both vertices have the same neighbourhood) so for all $i \leq k$ any resolving set should contain $c_i$ or $d_i$. Then, for $i \leq k$, the pair $(x_i,y_i)$ can only be resolved by a vertex of $E_i$ so any resolving set should contain a vertex of $E_i$ for all $i \leq k$. So any resolving set contains at least $2k$ vertices. The set of black vertices is a resolving set so the dimension is $2k$. On the right graph the pairs $(c_i,d_i)$ are twins so for all $i \leq k$ any resolving set should contain $c_i$ or $d_i$. One can easily check that a set containing only vertices on $\{c_i;d_i, 1 \leq i \leq k\}$ is not resolving so the dimension is higher than $k$. The black vertices form a resolving set so the metric dimension is $k+1$. }} \label{fig_delete_edge} \end{figure} \paragraph{Metric dimension and cycle rank.} Eroh et al.~\cite{eroh2017comparison} proved that, if $G$ is a unicyclic graph, then the metric dimension of $G$ is at most the metric dimension of any spanning tree plus one. Moreover, all the existing examples where the metric dimension is drastically modified with an edge modification already contain many cycles. One can then wonder if the metric dimension of a graph which does not contain many cycles is close to the metric dimension of any of its spanning trees. The goal of this paper is to answer this question positively. One can easily remark that in the star $K_{1,n}$, any resolving set contains at least $n-1$ vertices. Indeed otherwise, two leaves are not in the resolving set and then these two vertices cannot be identified. This argument can be generalized to any graph as follows: if $r$ pending paths are attached to a vertex $v$, any resolving set contains a vertex in at least $r-1$ of them. Let $L(G)$ be the sum over all the vertices $v$ on which there are attached pending paths, of the number of paths attached to $v$ minus one. Chartrand et al.~\cite{CHARTRAND2000} remarked that, for every connected graph $G$, $\dim(G) \geq L(G)$ with equality for trees. However, this bound has no reason to be closed from the optimal value (a graph with no leaf can have an arbitrarily large metric dimension). A wheel (induced cycle plus a universal vertex), for instance, has no leaf and its metric dimension is linear in $n$ (see Figure \ref{figwheel}). The \emph{cycle rank} of a connected graph, denoted by $c(G)$ is the number of edges that has to be removed from $G$ to obtain a spanning tree. We prove that the following holds: \begin{restatable}{theorem}{mainbisstate} \label{thm:mainbis} For any graph $G$, $L(G) \leq \dim(G) \leq L(G) + 6c(G) $. \end{restatable} Since the value of $L(G)$ cannot decrease by removing edges in $G$ that are not bridges, it implies the following: \begin{restatable}{cor}{corST} \label{cor:ST} Let $G$ be a graph and $T$ be any spanning tree of $G$. We have \[ \dim(G) \le \dim(T)+6c(G) \] \end{restatable} Informally, Corollary~\ref{cor:ST} ensures that, even if the metric dimension can be widely modified when we add a single edge, the "amortized cost" of an edge addition is at most $6$ with respect to any spanning tree of $G$. As far as we know it is the first bound of the metric dimension in terms of the natural lower bound $L(G)$ or of the metric dimension of a spanning tree of $G$. Before explaining briefly the outline of the proof, let us discuss a bit the tightness of these results. Let $k \in \mathbb{N}$. Consider the graph $G_k$ which is a collection of $k$ $C_4$ glued on the central vertex of a path of length $3$ like in Figure \ref{fig2}. The metric dimension of $G_k$ is equal to $2k+1$ (we need to select exactly two vertices per $C_4$ distinct from the center and one extremity of the path), $L(G_k)=1$, and $c=k$. Since $L(G_k)=1$ for every $k$, there exist graphs $G$ such that $\dim(G) = L(G)+2c(G)$. We ask the following question: \begin{question}\label{ques:L+c} Is it true that for any graph $G$, $\dim(G) \leq L(G) + 2c(G) $ ? \end{question} Note that Sedlar and Skrekovski independently ask the same question in \cite{sedlar2021vertex}. The same authors also prove in~\cite{SEDLAR2021} that Question~\ref{ques:L+c} is true for cacti (and is tight since $G_k$ is a cactus). Let us now discuss the tightness of Corollary~\ref{cor:ST}. If, in the graph $G_k$, we remove one edge of each $C_4$ incident to the central vertex, the resulting spanning tree has metric dimension of order $k+1$ as long as $k \ge 2$. So there exist graphs $G$ for which there exists a spanning tree $T$ satisfying $\dim(G) = \dim(T)+c$. We actually ask the following question: \begin{question}\label{ques:dimGT} Is it true that for any graph $G$ and for every spanning tree $T$ of $G$, we have $\dim(G) \le \dim(T)+c$? \end{question} One can then wonder what happens if we select the best possible spanning tree to start with, i.e. the spanning that maximizes the metric dimension. In $G_k$, one can note that if we break the edge of the $C_4$s that are not incident to the center and denote by $T_k$ the resulting tree, then $\dim(G_k)=\dim(T_k)+1$. Surprisingly, we did not find any graph $G$ where the metric dimension is a function of $c$ larger than \textit{any} spanning tree of $G$. We left the existence of such a graph as an open problem. \medskip Let us now briefly discuss the main ingredients of the proof of Theorem~\ref{thm:mainbis}. First, it consists in finding a small feedback vertex set $X$ of the graph such that every connected component is attached to at most two vertices of $X$. We then prove that, if we add to a resolving set of every connected component of $G \setminus X$ few vertices (in terms of $c$) we can "detect" shortcuts passing through the rest of the graph and then obtain a resolving set of the whole graph $G$. The proof of Theorem~\ref{thm:mainbis} is given in Section~\ref{section3}. \medskip The second part of the paper consists in applying this result in order to prove a weak version of a conjecture linking metric dimension and zero forcing sets in graphs. \paragraph{Zero forcing sets.} A {\em zero forcing set} is a subset of vertices colored in black which colors the whole vertex set in black when we apply the following rule: \emph{A vertex is colored black if it is the unique non-black neighbor of a black vertex}. See Figure~\ref{figexz} where the initial set contains three vertices. The \emph{zero forcing number} of a graph is the minimal size of a zero forcing set, denoted by $Z(G)$. The zero forcing number has been introduced in 2008 to bound the rank of some families of adjacency matrices \cite{AIM2008}. Deciding if the zero forcing number of a graph is at most $k$ is NP-complete~\cite{TREFOIS2015}. \begin{figure}[ht] \centering \includegraphics[scale=1]{exz.pdf} \caption{Iterations of the change color rule. On the graph on the left, the three black vertices form a zero forcing set.} \label{figexz} \end{figure} In general, the gap between metric dimension and zero-forcing number can be arbitrarily large. But for some restricted sparse graph classes like paths or cycles, both the optimal parameters and optimal sets are the same. Eroh, Kang and Yi then started a systematic comparison between them~\cite{eroh2017comparison}. They proved that $\dim(G)\leq Z(G)$ when $G$ is a tree and that $\dim(G)\leq Z(G)+1$ when $G$ has one cycle (in other words $G$ is a tree plus an edge). On the other hand, $\dim(G)$ can be arbitrarily larger than the zero forcing number when the number of cycles increases. They conjectured the following: \begin{conj}[Cycle-rank conjecture \cite{eroh2017comparison}]\label{conj} For a connected graph $\dim(G) \leq Z(G) +c(G)$. \end{conj} Conjecture~\ref{conj} is tight for an infinite family of graphs: The graph $G_k$ contains a path of $3$ vertices and $k$ cycles of size $4$ with the central vertex of the path in common. Figure~\ref{fig2} shows the graph $G_3$ with $c(G_3)=3$. \begin{figure}[ht] \centering \includegraphics[scale=1.2]{ex2.pdf} \caption{Tightness of Conjecture~\ref{conj}} \label{fig2} \end{figure} Eroh et al. proved in~\cite{eroh2017comparison} that $\dim(G)\leq Z(G) + 2c(G)$ if $G$ contains no even induced cycles. Our main contribution to this question is to prove a weaker version of Conjecture~\ref{conj} in Section~\ref{sec:consZF}, whose proof is mainly based on an application of Theorem~\ref{thm:mainbis}. \begin{restatable}{theorem}{thmZFdim} \label{thm:main} For every graph $G$, we have \[\dim(G)\leq Z(G)+6 c(G). \] \end{restatable} As far as we know, it is the first upper bound of $\dim(G)$ of the form $Z(G)+f(c(G))$. Note that the dependency on $c(G)$ cannot be removed, i.e. $\dim(G)$ cannot be upper bounded by a function of $Z(G)$ only. For the wheel of $n$ vertices (a cycle plus a universal vertex, see Figure \ref{figwheel}), the zero forcing number is $3$ for any $n \geq 4$ but the metric dimension is a linear function in $n$. \begin{figure} \centering \includegraphics[scale=1]{wheel.pdf} \caption{The zero forcing number of a wheel is $3$ (while $n \geq 4$) but its metric dimension is linear in $n$.} \label{figwheel} \end{figure} We also prove Conjecture~\ref{conj} in several particular cases. We first focus on unicyclic graphs. We give an alternative proof of Conjecture~\ref{conj} for unicyclic graphs with a much shorter and simpler proof than the one of~\cite{eroh2017comparison}. We then extend our results to prove Conjecture~\ref{conj} for cactus graphs\footnote{This result is proved independently in \cite{sedlar2021vertex} with a different method.} (graphs with edge-disjoint cycles). It generalizes the result on unicyclic graphs and is based on a very simple induction whose base case is the case of unicyclic graphs. Since cactus graphs contain the class of graphs with no even cycles, it improves the result of~\cite{eroh2017comparison} on even cycle-free graphs. We finally show that $\dim(G) \leq Z(G)$ when the unique cycle of $G$ has odd length. This result is tight and cannot be extended to unicyclic graphs with an even cycle as shown in Figure \ref{figzdim}. All the results related to zero forcing sets are proved in Section~\ref{section4}. \begin{figure}[ht] \centering \includegraphics[scale=1.2]{zdimpair.pdf} \caption{Black vertices form respectively a metric basis and a minimal zero forcing set.} \label{figzdim} \end{figure} \section{Preliminaries} \label{section2} \subsection{Definitions and notations} Unless otherwise stated, all the graphs considered in this paper are undirected, simple, finite and connected. For standard terminology and notations on graphs, we refer the reader to \cite{bookgraph}. Let $G=(V,E)$ be a graph. The \emph{distance} between two vertices $u,v \in V$, denoted by $d_G(u,v)$ (or simply $d(u,v)$ when $G$ is clear from context), is the length of a shortest path from $u$ to $v$ in $G$. When no such path exists, we state $d_G(u,v)=+\infty$. For $v \in V$, let $N(v)$ be the (open) neighborhood of $v$ defined as $N(v) = \{u \in V, \; uv \in E\}$. We say that two vertices $v$ and $w$ are \emph{twins} if $N(v)\setminus \{w\} =N(w)\setminus \{v\}$. For $X \subseteq V$, let $G[X]$ be the subgraph of $G$ induced by $X$. In other words, $G[X]$ is the graph with vertex set $X$ where $xy$ is an edge if and only if it is an edge of $G$. We denote by $G \setminus X$ the subgraph of $G$ induced by $V \setminus X$. The \emph{border of $X$}, denoted by $\partial X$, is $\{ u \in G \setminus X | \; \exists v\in X, uv \in E \}$. A vertex $w \in V$ \emph{resolves} a pair of vertices $(u,v)$ if $d(w,u) \neq d(w,v)$. Let $S \subseteq V$. The set $S$ \emph{resolves} the pair $(u,v)$ if at least one vertex in $S$ resolves the pair $(u,v)$ and $S$ resolves a set $W \subseteq V$ if $S$ resolves all the pairs of $W$. A set $S$ is a \emph{resolving set} of $G$ if $S$ resolves $V$. \emph{The metric dimension} $\dim(G)$ of $G$ is the minimum cardinality of a resolving set in $G$. A resolving set of minimum size is called a \emph{metric basis}. Let $Z\subseteq V$ be a set of vertices. The vertices in $Z$ are colored in black whereas the other vertices are white. The \emph{color change rule} converts a white vertex $u$ into a black vertex if $u$ is the only white neighbor of a black vertex. The set $Z$ is a \emph{zero forcing set} of $G$ if all the vertices of $G$ can be turned black after finitely many applications of the color change rule. For $u$ and $v$ two vertices in $V$ and a sequence of color change rule, we say that $u$ \emph{forces} $v$ if at some step $u$ is turned black with the color change rule because of $v$. We say that the edge $uv$ is used to force $u$. \emph{The zero forcing number} $Z(G)$ of $G$ is the minimal cardinality of a zero forcing set in $G$. The \emph{cycle rank} of $G$ (or \emph{feedback edge set}), denoted by $c(G)$ (or $c$ if the context is clear enough), is the minimum number of edges that should be deleted to $G$ to get a forest. Note that we have $c(G)=|E|-|V|+cc(G)$ where $cc(G)$ is the number of connected components of $G$. A graph $G$ is \emph{unicyclic} if $G$ is connected with $c(G)=1$. A {\em feedback vertex set} of $G$ is a subset of vertices such that $G \setminus X$ is a forest. We denote by $\tau(G)$ (or $\tau$ if the context is clear enough) the minimum size of a feedback vertex set of $G$. Note that if $X$ has minimum size, then $\tau(G)\leq c(G)$. \subsection{Resolving sets and zero forcing sets on trees} Chartrand et al. \cite{CHARTRAND2000} introduced the following terminology to study resolving sets in trees. We extend this terminology to general graphs (see Figure~\ref{figtreename} for an illustration).\\ A vertex of degree $1$ is called a \emph{terminal vertex}.\\ A vertex of degree at least 3 is a \emph{major vertex}. A terminal vertex $u$ is called \emph{a terminal vertex of a major vertex $v$} if $d(u,v)<d(u,w)$ for every other major vertex $w$. In other words $u$ and $v$ are linked by a path of degree $2$ vertices. The \emph{terminal degree} of a major vertex $v$ is the number of terminal vertices of $v$, denoted by $\ter(v)$. A major vertex is \emph{exterior} if its terminal degree is positive, and \emph{interior} otherwise.\\ A degree-$2$ vertex is an \emph{exterior degree-2 vertex} if it lies on a path between a terminal vertex and its major vertex. It is an \emph{interior degree-2 vertex} otherwise. \begin{figure}[ht] \centering \includegraphics[scale=0.9]{treename.pdf} \caption{Vertices denomination in a graph} \label{figtreename} \end{figure} Let $\sigma(G)$ be the sum of the terminal degrees over all the major vertices in $G$ and $ex(G)$ be the number of exterior major vertices in $G$. Let $L(G)=\sigma(G)-ex(G)$. We can bound $\dim(G)$ and $Z(G)$ with this parameter: \begin{lemme}\cite{eroh2017comparison} \label{lowerbound} For any connected graph $G$, $\dim(G) \geq \sigma(G)-ex(G)$ and $Z(G) \geq L(G)$. \end{lemme} \begin{lemme}\cite{eroh2015effect} \label{dimtree} Let $T$ be a tree which is not a path. Then, $dim(T) =L(T)$. Moreover, any set containing all but exactly one terminal vertices of every major vertex is a resolving set of~$T$. \end{lemme} There is no similar result to compute the zero forcing number of a tree. The gap between the zero forcing number and the metric dimension can be arbitrarily large on trees. Lemmas~\ref{lowerbound} and~\ref{dimtree} imply that trees satisfy Conjecture~\ref{conj}. Moreover, the equality case can be characterized: \begin{lemme}\cite{eroh2017comparison}\label{treecomp} For every tree $T$, $\dim(T) \leq Z(T)$. The equality holds if and only if $T$ has no interior degree-2 vertices and each major vertex has terminal degree at least two. \end{lemme} \subsection{Elementary results on metric dimension} This section is devoted to some elementary results about metric dimension. \begin{lemme} \label{technical} Let $G$ be a graph and $u$, $v$ be two vertices of $G$. Let $s$ and $t$ be two different vertices on a shortest path between $u$ and $v$. Then, $d(u,s) \neq d(v,s)$ or $d(u,t) \neq d(v,t)$. \end{lemme} \begin{proof} Let $P$ be a shortest path from $u$ to $v$ containing $s$ and $t$. Up to symmetry, we can assume that $u,s,t,v$ appear in that order in $P$. Since $P$ is a shortest path, $d(s,v)=d(s,t)+d(t,v)>d(t,v)$ and $d(t,u)=d(s,u)+d(s,t)>d(s,u)$. Assume that $d(u,s)=d(v,s)$. Then, $d(u,t)=d(u,s)+d(t,s)$ and $d(v,s)=d(v,t)+d(t,s)$. So, $d(u,t)=d(v,t)+2d(t,s) \neq d(v,t)$ as $t \neq s$. \end{proof} \begin{lemme} \label{cycle1} Let $G$ be a unicyclic graph with a cycle $C$ of odd length. Then, every pair of vertices of $C$ resolves $C$. \end{lemme} \begin{proof} Let $u$ and $v$ be two vertices of $C$. There are two paths between $u$ and $v$ on $C$, one of odd length and the other of even length. There exists a unique vertex $w$ of $C$ at the same distance from $u$ and $v$ in $C$ and then in $G$ since $G$ is unicyclic which is the middle of the path of even length. The vertex $w$ is the unique vertex of $C$ which does not resolve the pair $(u,v)$. So any pair of vertices of $C$ resolves $C$. \end{proof} \begin{lemme} \label{cycle3} Let $G=(V,E)$ be a graph and $C$ be a cycle of $G$. If $V(C)=\{v_0,v_1,...v_k\} $ such that for any $i\leq j$ $d(v_i,v_j)=\min(j-i,k-j+i+1)$ \footnote{This condition ensures that there is no shortcut between the vertices of $C$.}, then, for any set $S \subseteq C$ of size at least $3$, $S$ resolves $C$. \end{lemme} \begin{proof} Let $S=\{v_a,v_b,v_c\}$ be any set of three vertices of $C$ and $v_x \ne v_y$ be two vertices of $C$. Assume by contradiction that $S$ does not resolve the pair $(v_x,v_y)$. Note that neither $v_x$ nor $v_y$ belongs to $ \{v_a,v_b,v_c \}$ since otherwise $(v_x,v_y)$ would be resolved. Without loss of generality, we can assume that $v_x=v_0$ and $a <b < c$. Assume first that $y < a$. The shortest path on $G$ between $v_0$ and $v_a$ cannot contain $v_y$ otherwise $d(v_x,v_a) > d(v_y,v_a)$. Thus, $d(v_0,v_a)=k-a+1$ and similarly $d(v_0,v_b)=k-b+1$ so in particular $d(v_0,b) < d(v_0,a)$. Consider now the path between $v_y$ and $v_b$. If this path passes through $v_a$, then $d(v_y,a)<d(v_y,b)$ and if this path passes through $v_0$ then $d(v_y,b)>d(v_0,b)$. Both cases give a contradiction with the assumption that $S$ does not resolve the pair $(v_0,v_y)$. Assume now that $a < y <b $. If the shortest path between $v_0$ and $v_b$ passes through $v_y$ then $d(v_0,v_b)>d(v_y,v_b)$ gives a contradiction. Thus, $d(v_0,v_b)=k-b+1$ and so $d(v_0,v_c)=k-c+1$. Similarly if the path between $v_y$ and $v_b$ passes through $v_0$ then $d(v_y,v_b)>d(v_0,v_b)$. So $d(v_y,v_b)=b-y$ and $d(v_y,v_c)=c-y$. We get $b-y=k-b+1$ and $c-y=k-c+1$ which is impossible since $b\neq c$. The two last cases, $b < y <c $ and $y>c$, are respectively symmetric to the cases $a < y <b $ and $y<a$. \end{proof} The following result has been stated in \cite{eroh2017comparison} but the proof contains a flaw. We provide a corrected version of the proof in Appendix~\ref{appendix}. It bounds the variation of the metric dimension when an edge is deleted in some conditions. \begin{restatable}{lemme}{lemmevardim} \label{lemme:vardim} Let $G=(V,E)$ be a graph and $C$ be a cycle of $G$. Let $V(C)=\{v_0,v_1,...v_k\}$ be the vertices of $C$. Denote by $G_i=(V_i,E_i)$ the connected components of the vertex $u_i$ in $G\setminus E(C)$. If, for every $i \ne j$, $V_i \cap V_j = \emptyset$, then, for any $e \in E(C)$, $\dim(G) \leq \dim(G-e)+1$. \end{restatable} The following lemma is a well-known fact about twins and resolving sets. \begin{lemme}\label{twins} Let $u$ and $v$ be two twins of a graph $G$. Any resolving set $S$ of~$G$ verifies $S \cap \{u,v\} \neq \emptyset$. \end{lemme} Lemma~\ref{dimbranch} is a crucial observation for studying resolving sets in particular in trees. \begin{lemme}\label{dimbranch} Let $G=(V,E)$ be a connected graph, $u$ be a vertex of $G$ and $S$ be a resolving set of~$G$. At most one connected component of $G \setminus \{u\}$ does not contain any vertex of $S$. \end{lemme} \begin{proof} Assume by contradiction that two connected components $G_i$ and $G_j$ of $G \setminus \{x\}$ do not contain any vertex of $S$. Let $v \in V$ and $w \in V(G_j)$ be two vertices incident to $u$. Then, no vertex in $S$ resolves the pair $(v,w)$ since, for every $s \in S$, $d(s,v)=d(s,w)=d(s,u)+1$. \end{proof} \section{Bounds for metric dimension}\label{section3} \begin{defn}\label{defboundmin} Let $G$ be a graph. Recall that $L(G)=\sigma(G)-ex(G)$. If $G$ is a path $P_n$ for some $n \geq 1$, let $L(G)=1$ (so $L(T)=\dim(T)$ for all trees). \end{defn} The goal is to prove Theorem~\ref{thm:mainbis} we recall here: \mainbisstate* This result implies the following one: \corST* \begin{proof} For any graph $G=(V,E)$ that is not a tree and any edge $e \in E$ such that $G-e$ is connected, $L(G) \leq L(G-e)$. Indeed if a major vertex $v$ has terminal degree $d \geq 2$ in $G$, then $v$ is still a major vertex in $G-e$ of terminal degree at least $d$. So $L(G) \leq L(G-e)$ and then, for $T$ a spanning tree of $G$, $L(G) \leq L(T)$. As $\dim(G) \leq L(G) + 6c(G) $ and $\dim(T)=L(T)$ we get $\dim(G) \leq \dim(T)+6c(G)$. \end{proof} The rest of the section is devoted to prove Theorem~\ref{thm:mainbis}. \subsection{Construction of the resolving set} If $c(G)=0$ then $G$ is a tree and $\dim(G)=L(G)$ by Lemma \ref{dimtree}. If $c(G)=1$ let us prove a stronger result. \begin{lemme}\label{lem:1cycle} Let $G=(V,E)$ be a connected unicyclic graph. Then $\dim(G) \leq L(G)+3$. \end{lemme} \begin{proof} Let $uv$ be an edge of the cycle. Let $T=G-e$, then $\dim(T)=L(T)$ by Lemma \ref{dimtree} and $\dim(G) \leq \dim(T)+1$ by Lemma \ref{lemme:vardim}. As $L(T) \leq L(G)+2$ we get the inequality. \end{proof} We now focus on the case $c(G) \geq 2$. The first part of the proof will consist in defining a subset $S$ of vertices. We then prove in the second part of the proof that it is, indeed, a resolving set. In order to build this set $S$, we first find a small subset of vertices $M$ such that $G \setminus M$ is a forest such that each connected component has at most two edges incident to $M$. We then construct the set $S$. Let us start with a simple lemma. \begin{lemme} Let $G$ be a connected graph with no vertex of degree $1$ that is not an induced cycle. There exists a feedback vertex set $X$ of size $\tau(G)$ only containing vertices of degree at least $3$. \end{lemme} \begin{proof} Let $X$ be a minimum feedback vertex set with the minimum number of vertices of degree less than $3$. Note that $X$ does not contain vertices of degree $1$. Assume by contradiction that $X$ contains a vertex $x$ of degree $2$. Let $P$ be the maximal path of vertices of degree $2$ containing $x$. Since $G$ is not a cycle (and is not acyclic otherwise $X$ would be empty), $P$ does not contain the whole graph. Let $y$ be an endpoint of $P$ adjacent to a vertex $z$ of $V \setminus P$. Let $X'=X \setminus \{x\} \cup \{z\}$. The set $X'$ is still a feedback vertex set, a contradiction with the minimality of $X$. \end{proof} Let $X$ be a feedback vertex set of $G$ only containing vertices of degree at least $3$ in the graph where all the vertices of degree one have been iteratively removed\footnote{Note that we can assume that the resulting graph is not a cycle since otherwise the graph is unicyclic and the conclusion follows by Lemma~\ref{lem:1cycle}.}. Let $G_1,G_2,...,G_k$ be the connected components of $G \setminus X$. Note that each $G_i$ induces a tree. For each $G_i$, let $X_i \subseteq X$ be the set of vertices of $X$ connected to at least one vertex of $G_i$. Let $N_i \subseteq V(G_i)$ be the set of vertices in $G_i$ adjacent to (at least) one vertex of $X_i$. Let $T_i$ be the minimal subtree of $G_i$ containing the vertices of $N_i$. In other words, $T_i$ is the subtree of $G_i$ restricted to the union of the paths between $a$ and $b$ for any pair $a,b \in N_i$. Let $T'_i$ be the tree built from $T_i$ by adding to each vertex $u \in N_i$, $|N(u)\cap X|$ pending leaves. Let $M_i$ be the set of vertices in $T'_i$ of degree at least $3$ and $M:=X \cup (\bigcup_{i=1}^k M_i)$. Figure \ref{fig:notation} illustrates these notations. \begin{figure}[ht] \centering \includegraphics[scale=1]{definitionnotation.pdf} \caption{Illustration of the notations $X,N_i,M_i,T_i$ and $G_i$.} \label{fig:notation} \end{figure} \begin{lemme}\label{structure2} For each connected component $H$ of $G \setminus M$ there are at most two edges in $G$ between $H$ and $G \setminus H$. \end{lemme} \begin{proof} Let $U_H$ be the minimal subtree of $H$ containing all the vertices incident to an edge between $H$ and $G \setminus H$. Then, for each edge between a vertex $v \in H$ and a vertex in $G \setminus H$, add one new vertex in $U_H$ adjacent to $v$. Let us still denote by $U_H$ the resulting graph. Note that $U_H$ has as many leaves as edges leaving $H$. So, there are at most two edges with exactly one endpoint in $H$ if and only if $U_H$ has no vertex of degree three. Let us prove by contradiction that if a vertex $v$ has degree at least $3$ in $U_H$ then this vertex should have been added in $M$. Let $i$ be such that $H$ is a subgraph of $G_i$ and consider $T'_i$. Let $P_1,P_2,P_3$ be the three paths from $v$ to $M$ in $U_H$ which are internally disjoint, i.e. only $v$ and possibly the endpoint in $M$ are common. We claim that every vertex of $M_i$ can appear at most once since otherwise $G_i$ would contain a cycle. So for every path $P_i$, we can complete $P_i$ into a path $P_i'$ $u_1,u_2,u_3$ of $H_i$ from $v$ to $X$ such that the paths are internally disjoint. So $v$ has degree at least $3$ in $T_i'$, a contradiction. \end{proof} Lemma~\ref{structure2} indeed implies the following. \begin{cor} Every connected component of $G \setminus M$ is connected to at most two vertices of $M$. \end{cor} A connected component of $G\setminus M$ can be attached to $M$ in three different ways, called {\em Types}, illustrated in Figure~\ref{figexh2}. A connected component of $G \setminus M$ has \emph{Type $A$} (respectively \emph{Type $B$}) if there are exactly two edges between $H$ and $M$ with distinct endpoints in $H$ and such that their endpoints in $M$ are distinct (resp. the same). A component $H$ has \emph{Type $C$} if all the edges of $G$ between $H$ and $M$ have the same endpoint in $H$ (but possibly distinct endpoints in $M$). Let $H$ be a connected component of $G \setminus M$ of Type $A$ or $B$ and let $x$ and $y$ be the two endpoints in $M$ of the edges between $H$ and $M$. Let $\rho_H$ be one vertex on the path in $H$ between $x$ and $y$ such that $|d_H(x,\rho_H)-d_H(y,\rho_H)| \leq 1$. In other words, $\rho_H$ is one of the vertices in the middle of the path between $x$ and $y$ in $H$. Let $P$ be the union of the vertices $\rho_H$ for all the connected components $H$ of \emph{Type} $A$ and $B$. \begin{figure}[ht] \centering \includegraphics[scale=0.6]{exh.pdf} \caption{The three \emph{Types} of connected components of $G\setminus M$. } \label{figexh2} \end{figure} \begin{lemme}\label{use_of_p2} Let $H$ be a connected component of $G \setminus M$ of Type $A$ or $B$. Let $\rho$ be the vertex of $P \cap H$. Then, for any vertex $v \in H$, there is no shortest path between $z$ and $\rho$ using vertices in $G \setminus H$. Moreover, for every $z \in \partial H$, there is a shortest path using only vertices in $H \cup \{z\}$. \end{lemme} \begin{proof} Let $x$ and $y$ be the two vertices in $H$ adjacent to a vertex of $M$, $\alpha$ be the vertex of $M$ adjacent to $x$ and $\beta$ the vertex in $M$ adjacent to $y$. By definition of \emph{Types} $A$ or $B$, $x \neq y$ (but $\alpha$ and $\beta$ could be the same vertex if the \emph{Type} is $B$). By definition of $\rho$, we have $|d_H(x,\rho)-d_H(y,\rho)| \leq 1$. Let $v \in V(H)$. Assume by contradiction that a shortest path between $v$ and $\rho$ in $G$ passes through vertices in $G \setminus H$. So this shortest path between $\rho$ and $v$ passes through $x$ and $y$. By symmetry we can assume $d_G(v,\rho)=d_H(v,x)+d(x,\alpha)+d(\alpha,\beta)+d(\beta,y)+d_H(y,\rho)$. In other words, the path from $v$ to $\rho$ passes through $x$ and then $y$. Assume by contradiction that $d_H(z,x)+d(x,\alpha)+d(\alpha,\beta)+d(\beta,y)+d_H(y,\rho) \leq d_G(v,\rho)$. As $x\alpha$ and $y\beta$ are edges, we have $d_H(v,x)+d(\alpha,\beta)+d_H(y,\rho) +2 \leq d_G(v,\rho)$. By triangular inequality, $d_G(v,\rho) \leq d_H(v,\rho) \leq d_H(v,x)+d_H(x,\rho)$. So $d_H(x,\rho) \geq d_H(y,\rho)+d(\alpha,\beta)+2$ which contradicts $|d_H(x,\rho)-d_H(y,\rho)| \leq 1$. \smallskip If $z$ is in $\partial H$ and $\alpha=\beta$ then $z=\alpha=\beta$ and the result is immediate. Otherwise assume that $z=\alpha$. We want to contradict $d(\alpha,\beta)+d(\beta,\rho) < d(\alpha,\rho)$. As $\alpha$ is adjacent to $x$ and $\beta$ adjacent to $y$ the inequality is equivalent to $d(\alpha,\beta) + d(y,\rho) < d(x,\rho)$. As $\alpha \neq \beta$, $d(\alpha,\beta) \geq 1$. So $d(y,\rho) +2 \leq d(x,\rho)$ which contradicts $|d_H(x,\rho)-d_H(y,\rho)| \leq 1$. \end{proof} We now have all the ingredients to define the set $S$ that will be a resolving set based on resolving sets of each connected component $H$ of $G \setminus M$. The union of the resolving sets of the different graphs is not a resolving set for $G$, so we will need to add a few vertices. Moreover, the size of the union of the resolving set of the connected components of $G \setminus M$, is not bounded by $L(G)+c$. Some vertices have to be removed of these resolving sets. Let $H$ be a connected component of $G \setminus M$. Let $S_H'$ be a metric basis for $H$ such that, for each major vertex of terminal degree at least $2$ in $H$, all but one of its leaves are in $S_H'$. To get the announced bound we divided the component of \emph{Type} $B$ in two parts. A component has \emph{Type $B_1$} if it is a component of \emph{Type} $B$ and the two endpoints in $H$ are the roots of a path in $H$ and the major vertex of the leaf of the rooted path has terminal degree at least $2$. A component has \emph{Type $B_2$} if it is a component of \emph{Type} $B$ and not a component of \emph{Type} $B_1$. We define the set $S_H$ which is equal to $S_H'$ but with the following slight modifications: \begin{itemize} \item $H$ has \emph{Type} $A$. For every $x \in H$ adjacent to $M$, if $x$ is a leaf and its terminal vertex $v$ in $H$ has degree at least $2$, then, if $x \in S_H'$, remove $x$ from $S_H'$, otherwise remove from $S_H'$ another leaf associated to the terminal vertex of $x$. \item $H$ has \emph{Type} $B_1$, we can assume that $S_H'$ contains the two leaves $x$ and $y$ of $H$ such that $z_x$ and $z_y$ are adjacent to $M$. \item $H$ has \emph{Type} $C$, let $x$ be the unique vertex of $H$ adjacent to $M$. If $x$ is a leaf and its terminal vertex $v$ in $H$ has degree at least $2$, then, if $x \in S_H'$, remove $x$ from $S_H'$, otherwise remove from $S_H'$ another leaf associated to the terminal vertex of $x$. \item $H$ has \emph{Type} $C$ and $H$ is a path with one extremity adjacent to $M$. Let $w$ be the vertex of $M$ adjacent to $H$. If there is only one component of \emph{Type} C attached to $w$ which is a path connected to $w$ by an endpoint of the path, let $S_H= \emptyset$. If there are several such components, then let $S_H = \emptyset$ for one of these components and $S_K$ be the extremity of the path not connected to $w$ for all the other such components $K$ (or the unique vertex of $H$ if $H$ is reduced to a single vertex). \end{itemize} The set $S$ is defined as $S= M \cup P \cup (\bigcup_i S_i)$. The goal will consist in proving that $S$ is a resolving $S$. \subsection{The set $S$ is a resolving set} Before proving the main result of this section, we start with a definition. \begin{defn}\label{defz} Let $H$ be a connected component of $G \setminus M$ of Type $A$ or $B$ and let $x$ and $y$ be the two vertices in $H$ adjacent to $M$. Let $u$ be any vertex of $H$. The \emph{projection $z_u$ of $u$ (on the path between $x$ and $y$)} is the unique vertex in the path between $x$ and $y$ in $H$ at minimum distance to $u$. \end{defn} We will prove several lemmas that restrict the components where pairs of unresolved vertices can belong to. Let us first prove that they must belong to the same connected component of $G \setminus M$. \begin{lemme} \label{sameh} Let $u$ and $v$ be two vertices of $G$. If $u,v$ are not resolved by $S$ then there exists a connected component of $H \in G \setminus M$ such that both $u,v$ belong to $H$. \end{lemme} \begin{proof} First note that since $M \subseteq S$, $u,v \notin M$. So there exist $H_u$ and $H_v$, connected components in $G \setminus M$, such that $u \in H_u$ and $v \in H_v$. Assume by contradiction that $H_u \ne H_v$. \begin{itemize} \item Assume $H_u$ is of \emph{Type} $A$ or $B$ and let $x\alpha$ and $y\beta$ be the two edges connecting $H_u$ to $M$ (with $x,y \in H_u$). Let $\rho$ be the vertex in $P \cap H_u$. Since $v \notin H_u$, the shortest path between $v$ and $\rho$ passes through $\alpha$ or $\beta$. Up to symmetry, $\alpha$ is on the shortest path between $v$ and $\rho$ so $d(v,\rho)=d(v,\alpha)+d(\alpha,\rho)$. As $\alpha$ and $\rho$ are in $S$ we also have $d(u,\rho)=d(u,\alpha)+d(\alpha,\rho)$, a contradiction with Lemma~\ref{use_of_p2}. \item Assume now that both $H_u$ and $H_v$ are of \emph{Type} $C$. Let $\alpha$ and $\beta$ the vertices of $M$ connected to $H_u$ and $H_v$ respectively. If $\alpha \neq \beta$, then the shortest path between $u$ and $v$ contains two distinct vertices of $S$. Hence, by Lemma~\ref{technical} $u$ and $v$ are resolved. We assume now that $\alpha=\beta$. Since all the components $H$ of \emph{Type} C attached to $\alpha$ but at most one contain a vertex of $S_H$, by construction, $H_u$ or $H_v$ contains a vertex of $S$. Without loss of generality, there exists $\gamma \in S \cap H_u$. If $u$ is on the path between $\gamma$ and $v$ then $d(\gamma,u) < d(\gamma,v)$. Otherwise, let $m_u$ in $H_u$ be at the intersection of the path between $u$ and $\gamma$ and between $\alpha$ and $\gamma$. Then, vertices $m_u$ and $\alpha$ are on the shortest path between $u$ and $v$. By Lemma~\ref{technical}, one of them must resolve $u$ and $v$. By assumption, it is not $\alpha$. If it is $m_u$, then we would have $d_G(u,\gamma) \neq d_G(v,\gamma)$. Since the shortest paths between $u$, $v$ and $\gamma$ go through $m_u$, we obtain a contradiction. \end{itemize} \end{proof} We now prove that, if two vertices are in the same connected component of $G \setminus M$, then they are resolved by $S$. We start with connected components of \emph{Type} A. \begin{lemme}\label{codez2} Let $H$ be a connected component of $G \setminus M$ of Type A. Let $u$ and $v$ be two vertices of $H$ such that,for all $ s$ in $S$, $d(u,s)=d(v,s)$. Then, $z_u=z_v$. \end{lemme} \begin{proof} Let $u\alpha$ and $v\beta$ be the two edges between $H$ and $M$ with $u \in H$ and $v \in H$. The graph $H$ is a tree with a path between the two vertices $u$ and $v$. Assume $z_u \neq z_v$ and, without loss of generality, we can suppose that $z_u \neq \rho$ with $\rho$ the vertex of $P \cap H$. We have $d(u,\alpha)=d(u,z_u)+d(z_u,\alpha)$, $d(v,\alpha)=d(v,z_v)+d(z_v,\alpha)$, $d(u,\beta)=d(u,z_u)+d(z_u,\beta)$ and $d(v,\beta)=d(v,z_v)+d(z_v,\beta)$. As $d(u,\alpha)=d(v,\alpha)$ and $d(u,\beta)=d(v,\beta)$ we get \[d(z_u,\alpha)+d(z_v,\beta)=d(z_u,\beta)+d(z_v,\alpha).\] The vertices $z_u$ and $\rho$ are distinct and both between $\alpha$ and $\beta$. So $z_u$ is between $\alpha$ and $\rho$ or $\beta$ and $\rho$. Assume $z_u$ is between $\alpha$ and $\rho$. Then $d(z_u,\alpha) \leq d(z_u,\beta)$, so $d(z_v,\alpha) \leq d(z_v,\beta)$ meaning $z_v$ is also between $\alpha$ and $\rho$. The shortest path between $\alpha$ and $\rho$ passes through $z_u$ and $z_v$ by Lemma~\ref{use_of_p2}. Assume $z_u$ is closer than $z_v$ to $\rho$. Then $d(\alpha,\rho)=d(\alpha,z_v)+d(z_v,z_u)+d(z_u,\rho)$ gives \[d(\alpha,z_u)+d(z_v,\rho)=d(\alpha,z_v)+d(z_u,\rho)+2d(z_u,z_v).\] Use now the paths to $\rho$: $d(u,\rho)=d(u,z_u)+d(z_u,\rho)$ as $z_u \neq \rho$ and $d(v,\rho) \leq d(v,z_v)+d(z_v,\rho)$. Then $d(u,\alpha)=d(u,z_u)+d(z_u,\alpha)$, $d(v,\alpha)=d(v,z_v)+d(z_v,\alpha)$ gives \[d(z_v,\alpha)+d(z_u,\rho) \leq d(z_u,\alpha)+d(z_v,\rho).\] A combination of the previous equality gives $d(z_u,z_v) \leq 0$ so $z_u=z_v$. \end{proof} \begin{lemme} \label{typea} Let $H$ be a connected component of $G \setminus M$ of Type A. Let $u$ and $v$ be two vertices of $H$ such that, for all $s$ in $S$, $d(u,s)=d(v,s)$. Then $u=v$. \end{lemme} \begin{proof} Assume by contradiction $u \neq v$. Let $\alpha$ and $\beta$ be the two vertices of $M$ adjacent to $H$. By construction of $S_H$, $S_H \cup \{\alpha,\beta\}$ is a resolving set of $H \cup \{\alpha,\beta\}$. Let $\gamma$ which resolves the pair $(u,v)$ in $H \cup \{\alpha,\beta\}$. By Lemma \ref{codez2}, $z_u=z_v$ so $\gamma$ still resolves $(u,v)$ in $G$. Indeed if $z_\gamma=z_u$ then the distances are the same in $G$ and in $H \cup \{\alpha,\beta\}$. If $z_\gamma \neq z_u$ then $d_G(u,\gamma)=d_H(u,z_u)+d_G(z_u,\gamma)$ and $d_G(v,\gamma)=d_H(v,z_v)+d_G(z_v,\gamma)$. As $d_H(u,\gamma) \neq d_H(v,\gamma)$ with $d_H(u,\gamma)=d_H(u,z_u)+d_H(z_u,\gamma)$ and $d_H(v,\gamma)=d_H(v,z_u)+d_H(z_u,\gamma)$ we get $d_H(u,z_u) \neq d_H(v,z_u)$ so $d_G(u, \gamma) \neq d_G(v,\gamma)$. So $\gamma$ resolves $(u,v)$ in $G$, a contradiction. \end{proof} \begin{lemme}\label{codez} Let $H$ be a connected component of $G \setminus M$ of Type B. If $u,v \in H$ are not resolved by $S$, then $z_u=z_v$. \end{lemme} \begin{proof} Let us prove it by contradiction. Let $\alpha$ be the vertex of $M$ connected to $H$. Let $x,y$ be the two vertices of $H$ connected to $\alpha$. \smallskip \noindent \textbf{Case 1: $H$ has \emph{Type} $B_1$} \\ By construction, $\{\alpha,\rho,y\} \subseteq S$ with $y$ such that $z_y$ is connected to $\alpha$. Assume by contradiction $z_u \neq z_v$. We first show that $(z_u,z_v)$ is resolved by $\{\alpha,\rho,y\}$. Indeed, $d(y,z_u)=d(y,z_y)+d(z_y,z_u)$ and $d(y,z_v)=d(y,z_y)+d(z_y,z_v)$. Lemma \ref{cycle3} ensures that $(z_u,z_v)$ is resolved by a vertex of $\{\alpha,\rho,z_y\}$ and if $z_y$ resolves $(z_u,z_v)$, then $y$ resolves $(z_u,z_v)$. So $(z_u,z_v)$ is resolved by a vertex of $\{\alpha,\rho,y\}$, let $\gamma$ be such a vertex. If $z_v=\rho$, then $d(\gamma,u)=d(\gamma,z_u)+d(z_u,u)$ and $d(\gamma,v)=d(\gamma,\rho)+d(\rho,v)$, so $d(z_u,u) \neq d(\rho,v)$. As $\rho \in S$, $d(u,\rho)=d(v,\rho)$ so $d(z_u,u)<d(\rho,v)$. We exploit now the equalities $d(\alpha,u)=d(\alpha,v)$ and $d(\alpha,u)=d(\alpha,z_u)+d(z_u,u)$. By definition of $\rho$, $d(\alpha,z_u) \leq d(\alpha,\rho)$ and $d(z_u,u)<d(\rho,v)$. So $d(\alpha,v)=d(\alpha,\rho)+d(\rho,v)>d(\alpha,u)$, a contradiction. If $z_v \neq \rho$, then $d(\gamma,u)=d(\gamma,z_u)+d(u,z_u)$ and $d(\gamma,v)=d(\gamma,z_v)+d(v,z_v)$. By hypothesis $d(\gamma,u)=d(\gamma,v)$, so $d(u,z_u) \neq d(v,z_v).$ We can assume by symmetry $d(z_u,u)<d(z_v,v)$. Let $\beta \in \{\alpha,\rho\}$, such that $d(z_u,\beta) \leq d(z_v,\beta)$. Such a vertex exists since the distances $d(\alpha,z_u)+d(z_u,\rho)$ and $d(\alpha,z_v)+d(z_v,\rho)$ are the same if $z_u,z_v$ are both on the same side of the $xy$-path with respect to $\rho$ and differ by at most one otherwise. Then, $d(\beta,u)=d(\beta,v)$ and $d(\beta,u)=d(\beta,z_u)+d(z_u,u)$. But $d(\beta,z_u) \leq d(\beta,z_v)$ and $d(z_u,u)<d(z_v,v)$. So $d(\beta,v)=d(\beta,z_v)+d(z_v,v)>d(\beta,u)$, a contradiction. \smallskip \noindent \textbf{Case 2: $H$ has \emph{Type} $B_2$} \\ As $S$ contains $S_H$ which is a resolving set of $H$, there exists $\gamma \in H$ such that $d_H(u,\gamma) \neq d_H(v,\gamma)$. By hypothesis $d_G(u,\gamma) = d_G(v,\gamma)$. Assume first $z_{\gamma} = \rho$. Let us prove that $d_G(u,\gamma)=d_H(u,\gamma)$ and $d_G(v,\gamma)=d_H(v,\gamma)$, which gives a contradiction. By symmetry it is enough to prove that $d_G(u,\gamma)=d_H(u,\gamma)$. If $z_u=z_{\gamma}=\rho$ then $d_G(u,\gamma)=d_G(u,\rho)+d_G(\rho,\gamma)=d_H(u,\rho)+d_H(\rho,\gamma)=d_H(u,\gamma)$ and the conclusion follows. If $z_u \neq z_{\gamma}$, then \[d_G(u,\gamma)=d_G(u,z_u)+d_G(z_u,\rho)+d_G(\rho,\gamma).\] By Lemma~\ref{use_of_p2}, $d_G(z_u,\rho)=d_H(z_u,\rho)$. We have $d_G(u,z_u)=d_H(u,z_u)$ and $d_G(\rho,\gamma)=d_H(\rho,\gamma)$ since the paths between these vertices are unique. So $d_G(u,\gamma)=d_H(u,\gamma)$. So, from now on, we can assume that $z_{\gamma} \neq \rho$. Since $\{\rho,\alpha\}$ does not resolve $(u,v)$, we have $d(u,z_u)+d(z_u,\rho)=d(v,z_v)+d(z_v,\rho)$, and $d(u,z_u)+d(z_u,\alpha)=d(v,z_v)+d(z_v,\alpha)$. Thus, \[d(z_u,\rho)+d(z_v,\alpha)=d(z_v,\rho)+d(z_u,\alpha).\] Since $\rho$ and $\alpha$ are almost opposed on the smallest cycle containing them, we also have \[d(z_u,\rho)+d(z_u,\alpha)=d(z_v,\rho)+d(z_v,\alpha)+ \epsilon\] with $\epsilon \in \{-1,0,1\}$. Summing the two equalities gives $2d(z_u,\rho)=2d(z_v,\rho)+ \epsilon$. So by parity $\epsilon=0$. Then, $d(z_u,\rho)=d(z_v,\rho)$ and finally $d(z_u,\alpha)=d(z_v,\alpha)$. Since $d(u,\alpha)=d(v,\alpha)$, we obtain $d(u,z_u)=d(v,z_v)$. If $z_\gamma \notin \{z_u,z_v\}$, then $z_u$ and $z_v$ are at the same distance to $\alpha$, $\rho$ and $z_\gamma$, so, by Lemma \ref{cycle3}, $z_u=z_v$ . If $z_\gamma \in \{z_u,z_v\}$, up to symmetry we can assume $z_\gamma = z_u$. Then $d(u,\gamma) \leq d(u,z_u)+d(z_u,\gamma)$ and $d(v,\gamma)=d(v,z_v)+d(z_v,z_u)+d(z_u,\gamma)$. As $d(u,\gamma)=d(v,\gamma)$ and $d(u,z_u)=d(v,z_v)$ we get $d(z_v,z_u) \leq 0$ so $z_u=z_v$. \end{proof} \begin{lemme} \label{typeb} Let $H$ be a connected component of $G \setminus M$ of Type B. The set $S$ resolves all the pairs of vertices of $H$. \end{lemme} \begin{proof} Let $u,v$ be two vertices of $H$ which are not resolved by $S$. By Lemma \ref{codez}, $z_u=z_v$. Let $z =z_u=z_v$, if $\deg(z)=2$ then $u=v=z$ and the result is proven. We can assume from now on that $\deg(z) \geq 3$. \smallskip \noindent \textbf{Case 1: There exists $\gamma \in S_H$ which resolves the pair $(u,v)$ in $H$.} If $z_{\gamma}=z_u=z_v$ then the distances between $u$ (resp. $v$) and $\gamma$ are the same in $H$ and $G$, a contradiction. So we can assume that $z_{\gamma} \neq z_u$. We have $d_{H}(\gamma,u)=d_{H}(u,z)+d_{H}(z,\gamma)$ and $d_{H}(\gamma,v)=d_{H}(v,z)+d_{H}(z,\gamma)$. Since $d_{H}(\gamma,u) \neq d_{H}(\gamma,v)$, we have $d_{H}(u,z) \neq d_{H}(v,z)$. Now, since by Lemma~\ref{use_of_p2}, for $w \in \{ u,v \}$, $d_H(w,\rho)=d_G(w,\rho)$ and $d(w,\rho)=d(w,z)+d(z,\rho)$, $\rho$ resolves $(u,v)$, a contradiction. \smallskip \noindent \textbf{Case 2: The pair $(u,v)$ is not resolved by $S_H$ in $H$.} This case can only happen if $H$ has \emph{Type} $B_1$ (since otherwise no vertex of $S_H'$ is removed). Then there exists a vertex $x$ such that $z_x$ is adjacent to $\alpha$ which resolves the pair $(u,v)$ in $H$. If $z=z_x$, then, $d(z,u)=d(z,x)-d(u,x)$ and $d(z,v)=d(z,x)-d(v,x)$ so $d(z,u) \neq d(z,v)$. If $z \neq x$ then $d(x,u)=d(x,z)+d(z,u)$ and $d(x,v)=d(x,z)+d(z,v)$. So $d(z,u) \neq d(z,v)$ in both cases. Hence $\alpha$ resolves the pair $(u,v)$ in $G$. As $z \neq \alpha$, $d(\alpha,u)=d(\alpha,z)+d(z,u) \neq d(\alpha,z)+d(z,v) = d(\alpha,v)$. \end{proof} \begin{lemme} \label{typec} Let $H$ be a connected component of $G \setminus M$ of Type C. Then $S$ resolves any pair of vertices in $H$. \end{lemme} \begin{proof} Assume by contradiction two vertices $u,v \in H$ with $u \neq v$ are not resolved by $S$. Let $x$ be the unique vertex of $H$ adjacent to $M$ and $m$ be a vertex of $M$ adjacent to $x$. Let $H'$ be the subgraph of $G$ with vertex set $V(H) \cup \{m\}$. If $S$ contains a resolving set of $H$, then, since $x$ is a cut-vertex, $S$ resolves the pair $(u,v)$. So we can assume that at least one vertex of $S_{H}'$ has been removed during the construction of $S$. Note that since $m$ does not resolve $(u,v)$, $d(u,x)=d(v,x)$ in $H$. So in particular $H$ cannot be a path with endpoint $x$. So by construction of $S$, we can assume that $x$ is a leaf in $H$ and its major vertex has terminal degree at least two in $H$. By construction of $S_H$, $S \cup \{x\}$ is a resolving set for $H$. Then $S \cup \{m\}$ is a resolving set for $H'$. Let $\gamma \in S_H \cup \{m\}$ that resolves $u$ and $v$ in $H'$. The distances between $u$, $v$ and $\gamma$ in $G$ and $H'$ are the same so $\gamma$ resolves $u$ and $v$, a contradiction. \end{proof} \begin{lemme}\label{resolving} The set $S$ is a resolving set of $G$. \end{lemme} \begin{proof} Let $(u,v)$ be a pair of vertices that is not resolved by $S$. Assume by contradiction $u \neq v$. By Lemma \ref{sameh}, there exists a connected component $H$ of $G \setminus M$ such that $u \in H$ and $v \in H$. Then, if $H$ has \emph{Type} A, by Lemma \ref{typea}, $u=v$. If $H$ has \emph{Type} B, by Lemma \ref{typeb}, $u=v$. If $H$ has \emph{Type} C then, by Lemma \ref{typec}, $u=v$. \end{proof} \subsection{Upper bound on the size of $S$} Lemma \ref{resolving} ensures that $\dim(G) \leq |S|$. So Theorem~\ref{thm:mainbis} holds if $|S| \le L(G)+6c$. The set $S$ is a union of three sets that we will bound the size separately. We use the following result on minors to get the bounds. Let $G$ be a multigraph. The graph $H$ is a {\em minor} of $G$ if $H$ can be obtained from $G$ via a sequence of edge deletions, vertex deletions and edge contractions (the edge contraction operation can create parallel edges between two vertices or loops). One can easily check that the minor operation can only decrease the cycle rank. \begin{lemme}\label{boundMalt2} $|M| \leq 2c(G)-2$. \end{lemme} \begin{proof} If $|M|\leq 2$ then the inequality holds since we can assume $c \geq 2$. So we can assume that $|M| \ge 3$. Let $K$ be the multigraph (with possible loops) with vertex set $M$ where, for each connected component $H$ of $G \setminus M$ of \emph{Type} $A$ or $B$ with endpoints $x$ and $y$ in $M$ (that might be identical), we create an edge between $x$ and $y$ in $K$. Note that $K$ is a minor of $G$ as it can be obtained from $G$ by contracting edges in components $H$ of $G \setminus M$ into a single edge. Every vertex in $K$ has degree at least $3$. The vertices of $M\setminus X$ have degree at least $3$ by definition of $M_i$ for every $i$. By construction of $X$, $x \in X$ has degree at least $3$ in the graph starting from $G$ and removing the degree one vertices. Three adjacent edges belong to cycles so contribute to the degree of $x$ in $K$ so $\deg_{K}(x) \geq 3$. We have $3 |V(K)| \le \sum_{v \in V(K)} \deg(v) = 2 |E(K)|$. So $3|M| \leq 2 |E|$. Since the cycle rank of $K$ is at most $c$, so $|E| \leq c + |M| - 1 $. A combination of these inequalities gives $|M| \leq 2c(G)-2$. \end{proof} \begin{lemme}\label{boundP2} $|P| \leq c + |M|-1$. \end{lemme} \begin{proof} Let $K'$ be the multigraph (with loops) with vertex set $M$ and an edge between two vertices $x$ and $y$ if and only if there exists in $G \setminus M$ a connected component $H$ adjacent to $x$ and $y$. The graph $K'$ is a minor of $G$ since $K'$ can be obtained from $G$ by contracting edges with exactly one endpoint in $M$ until no such edge exists. One can easily notice that in $K'$ there is an edge $xy$ with multiplicity $k$ if and only if in $G \setminus M$ there are $k$ connected components attached to $x$ and $y$. Since $K'$ is a minor of $G$, $c(K') \leq c(G)$. As $K'$ contains $|M|$ vertices, $K'$ has at most $c+|M|-1$ edges. Thus, $G \setminus M$ has at most $c+|M|-1$ components of \emph{Type} A or B. Since $P$ contains one vertex in each component of \emph{Type} A or B, we have $|P| \leq c+|M|-1$. \end{proof} \begin{lemme}\label{sizecomp} \[ \big| \bigcup_{H \text{con. comp. of } G \setminus M} S_H \big| \leq L(G) +c. \] \end{lemme} \begin{proof} For every connected component $H$ of $G \setminus M$, let $\ell_H=\sum_{x \in H} (\ter(x)-1)$ over all the major vertices $x$ in $H$. We consider the three types of components. Let $H$ be a connected component of \emph{Type} $A$. We claim that $|S_H|=\ell_H$. Indeed let $\epsilon \in \{0,1,2\}$ be the number of vertices of $H$ adjacent to $M$ which are leaf connected to a major vertex of degree at least $2$. By construction $L(H)-\epsilon=|S_H|=\dim(H)- \epsilon$. Let $H$ be a connected component of \emph{Type} $B_1$. Then $L(H)=|S_H|+1$ and $\ell_H=L(H)-2$ by construction so $|S_H| \leq \ell_H +1$. \\ Let $H$ be a connected component of \emph{Type} $B$ not $B_1$. Then, $L(H)=|S_H|$ and $\ell_H \geq L(H)-1$ because $H$ has not \emph{Type} $B_1$ so $|S_H| \leq \ell_H +1$. \\ Let $H$ be a component of \emph{Type} $C$ not a path. By construction of $S_H$, $|S_H|=\ell_H$. Indeed, let $\epsilon \in \{0,1\}$ be the number of vertices of $H$ adjacent to $M$ which are leaves connected to a major vertex of degree at least $2$. By construction $|S_H|=\dim(H)- \epsilon=L(H)-\epsilon$ and $\ell_H = L(H)-\epsilon$. \\ Let $H$ be a component of \emph{Type} $C$ with $H$ a path. If $H$ is connected to $M$ by a vertex $x$ which is not an extremity of $H$ then $m$ is a major vertex of terminal degree 2 in $G$. So $|S_H|=\ell_H=1$. If $H$ is a path connected to a vertex $m \in M$ by an extremity, let $k \in \mathbb{N}$ be the number of such components connected to $m$. Denote them by $H_1,H_2,...,H_k$. If $k\geq 2$ then $m$ is a major vertex in $G$ with terminal degree $k$ and $|\cup_{i \leq k} S_{H_i}| =k-1=|L(G) \cap \{m\} \cup (\cup_{i \leq k} H_i)|$. If $k=1$ then $S_H= \emptyset$ so $\ell_H =0$. \\ There are at most $c(G)$ components of \emph{Type} $B$: for each component $H$ of \emph{Type} $B$ we can found a cycle in $G$ by adding the vertex of $M$ adjacent to $H$. By definition of $c(G)$, this gives at most $c(G)$ components of \emph{Type} $B$. Summing the inequalities gives the result: \[ | \cup S_H | = \sum_{\text{\emph{Type} $A$}} |S_H| + \sum_{\text{\emph{Type} $B$}} |S_H| +\sum_{\text{\emph{Type} $C$}} |S_H| \] \[ \leq \sum_{\text{\emph{Type} $A$}} \ell_H + \sum_{\text{\emph{Type} $B$}} (\ell_H+1) +\sum_{\text{\emph{Type} $C$}} \ell_H \leq L(G)+c(G) \] \end{proof} Finally, we can prove Theorem~\ref{thm:mainbis}: \begin{proof} The set $S$ is a resolving set so $\dim(G) \leq |S|$. By definition $S= M \cup P \cup (\bigcup_H S_H)$. By Lemma \ref{boundMalt2}, $|M| \leq 2c(G)$, by Lemma \ref{boundP2}, $|P| \leq 3c$ and by Lemma \ref{sizecomp},$| \cup S_H | \leq L(G) +c(G)$. Summing the inequalities give $\dim(G) \leq L(G) + 6c(G)$. \end{proof} \begin{figure}[ht] \centering \includegraphics[scale=1.1]{exemple_tight_M.pdf} \caption{Tightness of Lemmas \ref{boundMalt2}, \ref{boundP2} and \ref{sizecomp}.} \label{figtightlemme} \end{figure} One can naturally ask if this upper bound is optimal. Figure \ref{figtightlemme} gives an example of graph where Lemmas~\ref{boundMalt2},~\ref{boundP2} and~\ref{sizecomp} are tight. It ensures that our analysis of the construction is optimal but not necessarily the construction itself. Indeed, the metric dimension of the graph of Figure~\ref{figtightlemme} is $8$ and the square vertices form a metric basis. \section{Metric dimension and zero-forcing sets}\label{section4} In this section, we study how the metric dimension and the zero forcing number can be modified when an edge is added to a graph. Then we prove a weakening of Conjecture \ref{conj}, as a consequence of Theorem~\ref{thm:mainbis}. We then give a short proof of Conjecture~\ref{conj} for unicyclic graphs. We will then generalize this result to prove the conjecture for cactus graphs. We finally prove a strengthening of Conjecture~\ref{conj} when the graph is unicyclic and the unique cycle has odd length. \subsection{Edge modifications and consequences for Conjecture~\ref{conj}}\label{sec:consZF} The following lemma ensures that the variations of the zero forcing number when an edge is added or deleted an edge are small \cite{EDHOLM2012}. \begin{lemme}\cite{EDHOLM2012}\label{varZ} Let $G=(V,E)$ be a graph and $e \in E(G)$, then $Z(G)-1 \leq Z(G-e) \leq Z(G)+1$. \end{lemme} We have a more precise result if $Z(G+e) < Z(G)$ which will be useful later. \begin{lemme} \label{rev} Let $G=(V,E)$ be a graph and $u$ and $v$ two vertices of $G$ such that $e=uv \notin E$. If $Z(G+e) < Z(G)$, then for any minimum zero forcing set of $G+e$, at some step $u$ forces $v$ or $v$ forces $u$. \end{lemme} \begin{proof} By contradiction if a zero forcing set of minimal size for $G+e$ does not use the edge $e$ then it is a zero forcing set of $G$ so $Z(G) \leq Z(G+e)$. \end{proof} A similar statement does not hold for metric dimension. However, Lemma \ref{lemme:vardim} gives some conditions where a similar result holds. Using these results we can get inequalities between the metric dimension and the zero-forcing number for some classes of graphs. \begin{cor}\label{ineq2} Let $G$ be a connected unicyclic graph and $e$ be an edge such that $T=G-e$ is a tree. We have $\dim(G) \leq Z(G)+2$. \end{cor} \begin{proof} By Lemma~\ref{treecomp}, $\dim(T) \leq Z(T)$. Lemmas~\ref{lemme:vardim} and~\ref{varZ} ensure that $\dim(G) \leq Z(G)+2$. \end{proof} Eroh et al. \cite{eroh2017comparison} proved Conjecture~\ref{conj} for unicyclic graphs via a very long case analysis. They start from a tree $T$ achieving $\dim(T)=Z(T)$ and make a complete study of all the places where an edge could be added. We drastically simplify their proof by starting from a unicyclic graph $G$ and delete a well-chosen edge. \begin{lemme} \label{zero} Let $G=(V,E)$ be a graph which is not a tree and $C \subseteq V$ a cycle of $G$. Then, there exists an edge $e \in E(C)$ such that $Z(G-e) \leq Z(G)$. \end{lemme} \begin{proof} Let $Z \subseteq V$ be a minimum zero forcing set of $G$. Let $F \subseteq E$ be the \emph{forcing edges} in a sequence starting from $Z$, i.e. $uv \in F$ if and only if at some stage $u$ forces $v$ or $v$ forces $u$. We claim that at least one edge of $C$ is not in $F$. Indeed, if $u$ forces $v$ then $u$ is turned black before $v$. So, the first vertex $w$ of the cycle that is turned black cannot be turned black because of an edge of $C$ (such a vertex can already be black at the beginning of the proceed). Let $w_1,w_2$ be the two neighbors of $w$ on $C$. The vertex $w$ can force at most one of its two neighbors. So, without loss of generality, $w_2$ is not forced by $w$ and is turned black after $w$. So, if we remove the edge $e=ww_2$, $Z$ is still a forcing set of $G-e$ with the same sequence of applications of the color change rule that turned $G$ into black. Therefore $Z(G-e) \leq Z(G)$. \end{proof} We obtain as a corollary the main result of~\cite{eroh2017comparison}. \begin{cor}\label{unicycle} Let $G$ be a unicyclic graph. Then, $\dim(G) \leq Z(G)+1$. \end{cor} \begin{proof} Let $e$ be an edge of $C$ such that $Z(G-e) \leq Z(G)$. Such an edge exists by Lemma~\ref{zero}. By Lemma~\ref{lemme:vardim}, $\dim(G) \leq \dim(G-e)+1$. Moreover, by Lemma~\ref{treecomp}, $\dim(G-e) \leq Z(G-e)$ since $G-e$ is a tree. The combination of these three inequalities gives $\dim(G) \leq Z(G)+1$. \end{proof} All these results together with Theorem~\ref{thm:mainbis} permits to prove a weakening of Conjecture~\ref{conj} which we restate here: \thmZFdim* \begin{proof} By induction with Lemma~\ref{zero}, there exists a spanning tree $T$ such that $Z(T) \leq Z(G)$. By Lemma \ref{treecomp}, $\dim(T) \leq Z(T)$ and thus, $\dim(G) \leq Z(G)+6c(G)$ by Theorem \ref{thm:mainbis}. \end{proof} We generalize the proof of Conjecture~\ref{conj} for unicyclic graphs to cactus graphs. Almost the same techniques can be applied. First we define the cactus graphs class. \begin{defn} Any graph $G$ is a cactus graph if any edge $e \in E$ is part of at most one cycle of~$G$. \end{defn} \begin{theorem}\label{edgedisjoint} Let $G=(V,E)$ be a cactus graph. Then, $\dim(G) \leq c(G) +Z(G)$. \end{theorem} \begin{proof} We prove Theorem~\ref{edgedisjoint} by induction on $c(G)$. If $c(G)=0$ then $G$ is a tree and $\dim(G) \leq Z(G)$. If $c(G) >0$ let $C$ be a cycle of $G$. By Lemma~\ref{zero} there exists an edge $e \in C$ such that $Z(G-e) \leq Z(G)$. By induction $\dim(G-e) \leq Z(G-e) +c(G-e) \leq Z(G) + c(G-e) = Z(G) + c(G)-1$ since $c(G-e)=c(G)-1$ for any $e \in C$. \\ To conclude, let us prove that Lemma~\ref{lemme:vardim} can be applied. Let $v_1,v_2...,v_k$ be the vertices of $C$. Let $G_i$ be the connected component of $v_i$ in $G \setminus C$. Assume by contradiction that two subgraphs $G_i$ and $G_j$ with $1 \leq i < j \leq k$ are not disjoint. Then, there exists a path $P$ between the vertices $v_i$ and $v_j$ in $G \setminus C$. Then $G$ contains two cycles with common edges: $C$ and a cycle containing $P$ and a path in $C$ between $v_i$ and $v_j$, a contradiction. So, by Lemma~\ref{lemme:vardim}, $\dim(G) \leq \dim(G-e)+1$ and then $\dim(G) \leq c(G) +Z(G)$. \end{proof} As a corollary, we obtain that cactus graphs satisfy Conjecture~\ref{conj}. It improves a result of~\cite{eroh2017comparison} that ensures that $Z(G) \le \dim(G)+2c(G)$ if $G$ has no even cycles. If $G$ has no even cycles, all its cycles are edge disjoint. Indeed, if two odd cycles share at least one edge then $G$ contains an even cycle. \subsection{Unicyclic graphs with an odd cycle} In this section, we consider the case where $G$ is unicyclic and its cycle has odd length. In this case, we will improve the inequality of Corollary~\ref{unicycle} to get $\dim(G) \leq Z(G)$. Such a result cannot be extended to $G$ with an even cycle, see Figure~\ref{figzdim} for an example. The intuitive reason why there is a difference between odd and even cycles is that, by Lemma~\ref{cycle1}, any pair of vertices resolves an odd cycle while it is false for even cycles. Before proving the main result of this section, we need some technical lemmas. Let $k \geq 1$ and let $G=(V,E)$ be a graph containing a unique cycle $C$ of length $2k+1$. For $u\in C$, let $T_u$ be the connected component of $u$ in $G'=(V,E\setminus E(C))$ rooted in $u$. Note that $T_u$ is a tree. We call $u$ the root of $T_u$. We say that $T_u$ is \emph{trivial} if $T_u=\{u\}$, is a \emph{rooted path} if $T_u$ is a path with $u$ at one extremity and is a \emph{rooted tree} otherwise. Note that a rooted path can be trivial (otherwise specified). For $u \in C$, if $T_u$ is not trivial we denote by $\ell_u$ the terminal degree of $u$ in $G$. Else we let $\ell_u=0$. \begin{lemme} \label{deg1} Let $G$ be an odd unicyclic graph. If there exists $u \in C$ such that $\ell_u \geq 1$ then there exists $e \in E(C)$ incident to $u$ such that $Z(G-e) \leq Z(G)$. \end{lemme} \begin{proof} By Lemma~\ref{rev}, it suffices to find a minimum zero forcing set for $G$ which does not use one of the two edges in $E(C)$ incident to $u$. Let $Z$ be a minimum zero forcing set of $G$. If $u\in Z$ then $u$ can force only one vertex and the result is proved since at most one edge incident to $u$ is used. We can assume $u \notin Z$. Let $P$ be an internal degree-two path between $u$ and a terminal vertex $l$ of $G$ which exists since $\ell_u \geq 1$. If there is a vertex $x$ in $P \cap Z$, then $(Z\setminus \{x\}) \cup \{l\}$ is still a minimum forcing set of $G$. Then, $l$ iteratively forces the vertices of $P$ until $u$ and we are back to the previous case. Finally, if $P$ and $u$ are initially white, then $u$ is the first vertex of $P \cup \{u\}$ which is turned black (eventually by one edge in $E(C)$). It then turns in black $P$. We cannot use the second edge of $E(C)$ since every vertex forces at most one vertex. \end{proof} \begin{lemme} \label{cycle} Let $G$ be an odd unicyclic graph. Let $S\subseteq V$ be such that for any $u$ on the cycle, $S$ is not a subset of $T_u$. Then, $S$ resolves $C$. \end{lemme} \begin{proof} Let $\alpha$ and $\beta$ in $S$ such that $\alpha \in T_u$ and $\beta \in T_v$ with $u \neq v$. Assume by contradiction that $x$ and $y$ in $C$ satisfies $d(x,\alpha)=d(y,\alpha)$ and $d(x,\beta)=d(y,\beta)$. Then, since $d(x, \alpha) = d(x,u) + d(u,\alpha)$ and $d(y,\alpha) = d(y,u) + d(u,\alpha)$, we have $d(x,u)=d(y,u)$. Similarly $d(x,v)=d(y,v)$, a contradiction with Lemma~\ref{cycle1}. \end{proof} \begin{lemme} \label{tree} Let $G$ be an odd unicyclic graph. If, for any $u \in C$, $T_u$ is a rooted tree then for all $e \in E(C)$, $\dim(G) \leq \dim(G-e)$. \end{lemme} \begin{proof} Let $e \in E(C)$ and $S$ be a metric basis of $G-e$. We will prove that $S$ is still a resolving set of $G$. Let us first prove that for every $u \in C$ since $T_u$ is a rooted tree $S \cap T_u \neq \emptyset$.\\ By definition of rooted tree, $T_u$ contains a vertex of degree $3$ in $G$. Let $r$ be such a vertex. By Lemma \ref{dimbranch}, at most one connected component of $G-e\setminus \{r\}$ does not contain element of $S$. The tree $T_u$ contains at least two connected components of $G-e \setminus \{r\}$, so $S \cap T_u \neq \emptyset$. Let $(x,y)$ be any pair of vertices. We prove that $S$ resolves $(x,y)$ in $G$. \begin{enumerate} \item Assume first that $x$ and $y$ are in the same component $T_u$ for some $u \in C$. Let $\alpha \in S$ that resolves $(x,y)$ in $G-e$. If $\alpha \in T_u$ then $d_{G}(\alpha,x) = d_{G-e}(\alpha,x) \neq d_{G-e}(\alpha,y) = d_{G}(\alpha,y)$. We can assume that $\alpha \notin T_u$. Since $e \notin T_u$ and $u$ is a cut-vertex of $G$ and $G-e$, we have $d_{G-e}(u,w)=d_{G}(u,w)$ for every $w \in T_u$. For every $w \in T_u$, \[d_{G}(\alpha,w)=d_{G}(\alpha,u)+d_{G}(u,w)=d_{G}(\alpha,u)+d_{G-e}(u,w).\] Since $d_{G-e}(\alpha,w)=d_{G-e}(\alpha,u)+d_{G-e}(u,w)$ and $d_{G-e}(\alpha,x) \ne d_{G-e}(\alpha,y)$, we have $d_{G-e}(u,x) \ne d_{G-e}(u,y)$. Thus, $d_{G}(\alpha,x) \ne d_{G}(\alpha,y)$ and then $\alpha$ resolves $(x,y)$ in $G$. \item Assume now $x$ and $y$ are in different components, respectively $ T_u$ and $ T_v$. Then, there exist $\alpha \in S \cap T_u$ and $\beta \in S \cap T_v$. Assume $d_G(\alpha,x)=d_G(\alpha,y)$ and $d_G(\beta,x)=d_G(\beta,y)$. Then, $d_G(\alpha,x) \leq d_G(\alpha,u)+d_G(u,x)$ and $d_G(\alpha,y)=d(\alpha,u)+d_G(u,v)+d_G(v,y)$ so $ d_G(u,v)+d_G(v,y) \leq d_G(u,x)$. The symmetric relation is $ d_G(v,u)+d_G(u,x)\leq d_G(v,y)$. Summing the two gives $2d_G(u,v) \leq 0$ which is a contradiction since $u \neq v$.\qedhere \end{enumerate} \end{proof} We will prove by case distinction the following result. \begin{theorem}\label{odd_unicyclic} Any odd unicyclic graph $G$ satisfies $\dim(G) \leq Z(G)$. \end{theorem} \begin{proof} We make a case analysis on the structure of $G$. \\ \noindent \textbf{Case 1: For every $u \in C$, $T_u$ is a rooted tree.} \\ By Lemma~\ref{zero}, there is an edge $e$ such that $Z(G-e) \leq Z(G)$. Since the $G-e$ is a tree, Corollary~\ref{treecomp} ensures that $\dim(G-e) \leq Z(G-e)$. By Lemma~\ref{tree}, $\dim(G) \leq \dim(G-e)$. The combination of these inequalities gives $\dim(G) \leq Z(G)$. \\ \noindent \textbf{Case 2: For every $u \in C$, $T_u$ is a rooted path.} \\ We prove that any pair of vertices $\alpha$ and $\beta$ of $C$ at distance $k$ is a resolving set of $G$. \\ Let ${\alpha,\beta}$ be such a pair of vertices. Let us first prove that for every $x \in G$, $d(x,\alpha)+d(x,\beta) \in \{k,k+1\}$ if and only if $x \in C$. Indeed as $d(\alpha,\beta)=k$ for any vertex $x$, $d(x,\alpha)+d(x,\beta) \geq k$. If $x \in C$, then either $x$ is on the path between $\alpha$ and $\beta$, and $d(x,\alpha)+d(x,\beta) =k$. Or $x$ is in the other part of the cycle and $d(x,\alpha)+d(x,\beta) = k+1$. If $x \notin C$, then let $y$ be the vertex of $C$ such that $x \in T_y$. Then, $d(x,\alpha)+d(x,\beta)=d(y,\alpha)+d(y,\beta)+2d(x,y) \geq 2 +d(y,\alpha)+d(y,\beta) \geq k+2$. This implies that a vertex on $C$ cannot have the same distance vector as a vertex in $V \setminus C$. \\ By Lemma~\ref{cycle}, the set $\{\alpha,\beta\}$ resolves $C$. To conclude we have to show that $\{ \alpha,\beta\}$ resolves $V \setminus C$. Let us prove that two vertices not in $C$ are resolved. Assume by contradiction that two vertices $x$ and $y$ are not resolved by $\{\alpha, \beta\}$, with $x \in T_u$ and $y \in T_v$ for $u$ and $v$ in $C$. If $u=v$, since $T_u$ is a rooted path, $\alpha$ resolves the pair $(x,y)$. From now on, we can assume that $u \neq v$. Assume by symmetry $d(x,u) \leq d(y,v)$, and let $z \in T_v$ be the vertex on $T_v$ such that $d(v,z)=d(x,u)$. Then, the pair $\{\alpha,\beta\}$ does not resolve the pair $(z,u)$. But $u \in C$ and the previous cases ensures that any pair of vertices with one vertex on $C$ is resolved by $\{\alpha,\beta\}$, a contradiction. So $\dim(G)=2$ and $G$ is not a path, so $Z(G) \geq 2$. \\ \noindent \textbf{Case 3: There exists a unique $u \in C$ such that $T_u$ is a rooted tree}.\\ Let us prove that $\dim(G) \leq \dim(T_u)+1$ and $Z(G) \geq Z(T_u)+1$. Let $S$ be a metric basis for $T_u$ and $v \in C$ a vertex at distance $k$ from $u$.\\ Let us show that $S \cup \{v\}$ is a resolving set of $G$. Let $\alpha \in S \cap T_u$, then $\{ \alpha,v\}$ resolves $G \setminus T_u$. Indeed otherwise, these two vertices would have the same distances to $u$ and $v$ which is impossible since in $G \setminus T_u$, for every $v \in C$, $T_v$ is a rooted path and by the claim in Case 2, $\{u,v\}$ is a resolving set for $G \setminus (T_u \setminus \{u\})$. Two vertices on $T_u$ are resolved since $S$ is a metric basis for $T_u$ and $u$ is a cut-vertex. Let $x \in T_w$ for some $w \in C$, and $y \in T_u$. By triangular inequality, $d(\alpha,y) \leq d(y,u)+d(u,\alpha)$. If $d(\alpha,y)=d(\alpha,x)$, then $d(x,u) \leq d(y,u)$ as $d(\alpha,x)=d(\alpha,u)+d(u,x)$. If $d(x,v)=d(y,v)$, then as $d(y,v)=d(y,u)+d(u,v) \geq d(x,u)+d(u,v)$. We get $d(x,v) \geq d(x,u)+d(u,v)$. Removing $d(x,w)$ on both side gives $d(w,v) \geq d(w,u)+d(u,v)$ which is impossible since $d(u,v)=k$ and $d(w,v) \leq k$. So $x$ and $y$ have different codes and $\dim(G) \leq \dim(T_u)+1$. \\ Let $Z$ be a minimal zero forcing set of $G$. If $Z$ contains $u$, then it should contain at least another vertex in $G \setminus T_u$. Since the restriction of $Z$ to $T_u$ is a forcing set for $T_u$, we have $Z(G) \geq Z(T_u)+1$. So we can assume that $u \notin Z$. Consider a sequence of color change rule that turns $u$ into black. Either $u$ is forced by a vertex in $G \setminus T_u$. Since $(G \setminus T_u) \cup \{u\}$ contains a cycle, there are at least two vertices in $Z \cap (G \setminus T_u)$ and $Z \cap T_u \cup \{u\}$ is a forcing set of $T_u$. So $Z(G) \geq Z(T_u)+1$. Otherwise $u$ is forced by a vertex of $T_u$. Then there is at least one vertex in $Z \cap (G \setminus T_u)$ and $Z \cap T_u $ is a forcing set of $T_u$ so $Z(G) \geq Z(T_u)+1$. So in both cases we have $\dim(G) \leq \dim(T_u)+1 \leq Z(T_u)+1 \leq Z(G)$.\\ For the other cases we use the following process: we exhibit an edge $e$ such that $Z(G-e) \leq Z(G)$ (by Lemma \ref{zero} or \ref{deg1}). Then, find a vertex $z$ in $G-e$ such that $z$ is an interior degree-2 vertex or a major vertex with terminal degree $0$ or $1$. By Lemma \ref{dimtree}, $\dim(G-e)< Z(G-e)$ and by Lemma \ref{lemme:vardim}, $\dim(G) \leq \dim(G-e)+1$, so $\dim(G) \leq Z(G)$. We just give the construction of $e$ and $z$. \\ \noindent \textbf{Case 4: There exists $u,v,w,x \in C$ in this order (not necessarily adjacent) such that $T_u$ and $T_w$ are rooted trees and $T_v$ and $T_x$ are rooted paths.} \\ Let $e \in E(C)$ such that $Z(G-e) \leq Z(G)$. Such an edge exists by Lemma \ref{zero}. In $G-e$, either the vertex $v$ or $x$ is on the path between $u$ and $w$. Let $z$ be this vertex, $z$ is an interior degree-2 vertex or a major vertex with terminal degree $0$ or $1$ in $G-e$. \item \textbf{Case 5: There exist $u$ and $v$ adjacent with $T_u$ and $T_v$ rooted trees and $w$ with $T_w$ a rooted path.} \begin{itemize} \item \textbf{$l_u=0$.} Let $e \in E(C)$ such that $Z(G-e) \leq Z(G)$. Such an edge exists by Lemma \ref{zero}. In $G-e$, $u$ is an interior degree-2 vertex or a major vertex with terminal degree $0$ or $1$ so $z=u$. \item \textbf{$l_u=1$.} Let $e \in E(C)$ adjacent to $u$ such that $Z(G-e) \leq Z(G)$. Such an edge exists by Lemma \ref{deg1}. If $e=uv$ then $w$ is an interior degree-2 vertex or a major vertex with terminal degree $0$ or $1$ in $G-e$ so $z=w$. If $e$ in the other edge in $E(C)$ adjacent to $u$ then $u$ is an interior degree-2 vertex or a major vertex with terminal degree $0$ or $1$ in $G-e$ so $z=u$. \item \textbf{$l_u \geq 2$ and $l_v \geq 2$.} Let $e=uv$, then $Z(G-e) \leq Z(G)$. Indeed a minimal zero-forcing set of $G$ is also a zero-forcing set of $G-e$. Let $Z$ be a zero-forcing set of $G$. The set $Z$ contains at least one leaf of $u$ and one leaf of $v$. The leaves can turn black $u$ and $v$ so there is a sequence of forces for $G$ such that the edge $e$ is not a forcing edge. So $Z(G-e) \leq Z(G)$ by Lemma $\ref{rev}$. Then $w$ is an interior degree-2 vertex or a major vertex with terminal degree $0$ or $1$ in $G-e$ so $z=w$. \end{itemize} \end{proof} \printbibliography \newpage \section{Introduction} \setcounter{page}{1} A \emph{resolving set} of a graph $G$ is a subset of vertices of $G$ such that any vertex in the graph is identified by its distances to the vertices of the resolving set. In the example of Figure~\ref{figexdim} the set $ \{A,B\}$ is a resolving set of the graph because all the vertices have a different \emph{distance vector} to $\{A,B\}$ so the knowledge of the distance to $A$ and $B$ identifies uniquely a vertex. This notion has been introduced in 1975 by Slater \cite{slater1975trees} for trees and by Harary and Melter \cite{harray1975} for graphs. Since its introduction the notion became an important notion in graph theory and has been widely studied. Harary and Melter have introduced the locating set with the problem to pilot a sonar but it is also valid for the navigation of a robot in an Euclidean space~\cite{KHULLER1996}. Determining the minimum size of a resolving set, called \emph{metric dimension} and denoted by $\dim(G)$, is an NP-complete problem~\cite{garey1979} even restricted to planar graphs~\cite{Diaz2012}. \begin{figure}[ht] \centering \includegraphics[scale=1.7]{exdim.pdf} \caption{The black vertices form a resolving set. For each vertex $x$ the vector next to $x$, $(d(A,x),d(B,x))$ is unique.} \label{figexdim} \end{figure} A {\em zero forcing set} is a subset of vertices colored in black which colors the whole vertex set in black when we apply the following rule: \emph{A vertex is colored black if it is the unique non black neighbor of a black vertex}. See Figure~\ref{figexz} where the initial set contains three vertices. The \emph{zero forcing number} of a graph is the minimal size of a zero forcing set, denoted by $Z(G)$. The zero forcing number has been introduced in 2008 to bound the rank of some families of adjacency matrices \cite{AIM2008}. Deciding if the zero forcing number of a graph is at most $k$ is also NP-complete~\cite{TREFOIS2015}. \begin{figure}[ht] \centering \includegraphics[scale=1]{exz.pdf} \caption{Iterations of the change color rule. On the graph on the left, the three black vertices form a zero forcing set.} \label{figexz} \end{figure} In general, the gap between metric dimension and zero-forcing number can be arbitrarily large. But for some restricted sparse graph classes like paths or cycles, both the optimal parameters and optimal sets are the same. Eroh, Kang and Yi then started a systematic comparison between them~\cite{eroh2017comparison}. They proved that $\dim(G)\leq Z(G)$ when $G$ is a tree and that $\dim(G)\leq Z(G)+1$ when $G$ has one cycle (in other words $G$ is a forest plus an edge). On the other hand, $\dim(G)$ can be arbitrarily larger than the zero forcing number when the number of cycles increases. They have then stated the following conjecture: \begin{conj}[Cycle-rank conjecture \cite{eroh2017comparison}]\label{conj} For a connected graph $\dim(G) \leq Z(G) +c(G)$ where $c(G)$ is the minimal number of edges that have to be removed from $G$ to obtain a tree. \end{conj} Conjecture~\ref{conj} is tight for an infinite family of graphs : The graph $G_k$ contains a path of $3$ vertices and $k$ cycles of size $4$ with the central vertex of the path in common. Figure~\ref{fig2} shows the graph $G_3$ with $c(G_3)=3$. \begin{figure}[ht] \centering \includegraphics[scale=1.2]{ex2.pdf} \caption{Tightness of Conjecture~\ref{conj}} \label{fig2} \end{figure} Towards this conjecture, Eroh et al. proved in~\cite{eroh2017comparison} that $\dim(G)\leq Z(G) + 2c(G)$ if $G$ contains no even induced cycles. In this paper, we prove Conjecture~\ref{conj} in several particular cases and prove a weaker version of the conjecture in general. We first focus on unicyclic graphs (that is graphs for which the deletion of exactly one edge gives a tree). In Section~\ref{section3}, we give a proof of Conjecture~\ref{conj} for unicyclic graphs which is much shorter and simpler than the one of~\cite{eroh2017comparison}. We also show that $\dim(G) \leq Z(G)$ when the unique cycle of $G$ has odd length. This result is tight and cannot be extended to unicyclic graphs with an even cycle as shown in Figure \ref{figzdim}. \begin{figure}[ht] \centering \includegraphics[scale=1.2]{zdimpair.pdf} \caption{Black vertices form respectively a metric basis and a minimal zero forcing set.} \label{figzdim} \end{figure} In Section~\ref{section4}, we prove Conjecture~\ref{conj} for cactus graphs (graphs with edge-disjoint cycles). It generalizes the result on unicyclic graphs and is based on a very simple induction whose base case is the case of unicyclic graphs. Since cactus graphs contain the class of graphs with no even cycles, it improves the results of~\cite{eroh2017comparison}. We finally prove a weaker version of Conjecture~\ref{conj} which is our main result : \begin{theorem}\label{thm:main} For every graph $G$, we have \[\dim(G)\leq Z(G)+6\cdot c(G). \] \end{theorem} As far as we know, it is the first upper bound of $\dim(G)$ of the form $Z(G)+f(c(G))$. Note that one can wonder if the dependency on $c(G)$ can be removed and if $\dim(G)$ can be upper bounded by a function of $Z(G)$ only. The answer is negative : for $G_n$ the complete bipartite graph $K_{1,n}$, $Z(G)=2$ for any $n \geq 2$ but $\dim(G_n)$ is a linear function in $n$. In our proof of Theorem~\ref{thm:main}, feedback vertex sets are playing an important role. Actually, our result is even stronger since we prove that $\dim(G)\leq Z(G) + \min\{5\cdot c(G)+ |\tau(G)|,3\cdot c(G)+5\tau(G)\}-5$ where $\tau(G)$ is the size of a minimum feedback vertex set in $G$. We ask the following question, which is a weakening of Conjecture~\ref{conj}: \begin{conj} There exists a function $f$ such that, for any connected graph $G$, $\dim(G) \leq Z(G) +c(G)+f(\tau(G))$ where $c(G)$ is the minimal number of edges that have to be removed from $G$ to obtain a tree and $\tau(G)$ is a minimum size of a feedback vertex set of $G$. \end{conj}

proofpile-arXiv_069-13756

\section{\@startsection {section}{1}{\z@} {-30pt \@plus -1ex \@minus -.2ex} {2.3ex \@plus.2ex} {\normalfont\normalsize\bfseries\boldmath}} \renewcommand\subsection{\@startsection{subsection}{2}{\z@} {-3.25ex\@plus -1ex \@minus -.2ex} {1.5ex \@plus .2ex} {\normalfont\normalsize\bfseries\boldmath}} \renewcommand{\@seccntformat}[1]{\csname the#1\endcsname. } \makeatother \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \newtheorem{corollary}{Corollary} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem{conjecture}{Conjecture} \newtheorem{remark}{Remark} \newcommand{\ssum}[4][1.8em]% {\mathop{\sum_{#2}^{#3}}_{\makebox[#1][l]{$\scriptstyle{#4}$}}} \newcommand{\heading}[1]{\smallskip\noindent% \textbf{#1.}\hspace*{0.5em}\ignorespaces} \newcommand{^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}}{^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}} \newcommand{\tr}[1]{\mathord{\mathrm{tr}}{\left( #1 \right)}} \DeclareMathAlphabet{\columnfont}{OT1}{cmr}{bx}{it} \DeclareMathAlphabet{\matrixfont}{OT1}{cmr}{bx}{sl} \newcommand{\col}[1]{\mathord{\columnfont{#1}}} \newcommand{\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}}{\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}} \newcommand{\M}[1]{\mathord{\matrixfont{#1}}} \newcommand{\M{1}_{n}}{\M{1}_{n}} \newcommand{\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}}{\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}} \begin{document} \begin{center} \uppercase{\bfseries Multiplicative functions arising from the study % of mutually unbiased bases} \vskip 20pt {\bfseries Heng Huat Chan}\\ {\smallit Department of Mathematics, % National University of Singapore, % Singapore, 119076, Singapore}\\ {\ttfamily matchh@nus.edu.sg}\\ \vskip 10pt {\bfseries Berthold-Georg Englert}\\ {\smallit Centre for Quantum Technologies, % 3 Science Drive 2, Singapore 117543, Singapore; % Department of Physics, National University of Singapore, % 2 Science Drive 3, Singapore 117551, Singapore; % MajuLab, CNRS-UCE-SU-NUS-NTU, International Joint Research Unit, CNRS, Singapore 117543, Singapore}\\ {\ttfamily englert@u.nus.edu}\\ \end{center} \vskip 30pt \centerline{\bfseries Abstract} \noindent% We embed the somewhat unusual multiplicative function, which was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics, into two families of multiplicative functions that we construct as generalizations of that particular example. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum. \pagestyle{myheadings} \markboth{Heng Huat Chan and Berthold-Georg Englert}% {Multiplicative functions arising from mutually unbiased bases} \thispagestyle{empty} \baselineskip=12.875pt \vskip 30pt \section{Multiplicative functions} We begin by recalling a few basic definitions and results. A real or complex valued function defined on the positive integers is called an arithmetic function or a number-theoretic function. For any two positive integers $m$ and $n$, we use $(m,n)$ to denote the greatest common divisor of $m$ and $n$. An arithmetic function $f$ is called multiplicative if $f$ is not identically zero and if \begin{equation*} f(mn)=f(m)f(n)\,\,\text{whenever $(m,n)=1$}. \end{equation*} It follows that $f(1)=1$ if $f$ is multiplicative. In general, given an arithmetic function $f$, it is impossible to determine $f(n)$ for a large positive integer $n$. However, if $f$ is a multiplicative function, then we can evaluate $f(n)$ provided that the factorization of $n$ into prime powers and the formulas for the multiplicative function $f$ at prime powers are known. Euler's totient function defined by \begin{equation*} \varphi(n) =\ssum{k=1}{n}{(k,n)=1} 1, \end{equation*} which is the count of positive integers that are less than $n$ and coprime with $n$, is an important example of a multiplicative function. For a proof of the multiplicative property of $\varphi(n)$ using the Chinese Remainder Theorem, see \cite[Section 5.5]{Hardy-Wright}. It is known that \begin{equation*} \varphi(p^\alpha) = p^\alpha-p^{\alpha-1} \quad \text{for}\ \alpha\geq1 \end{equation*} and consequently, if \begin{equation*} n=\prod_{j=1}^J p_j^{\alpha_j} \end{equation*} is the known factorization of $n$ into powers of distinct primes, then \begin{equation*} \varphi(n)=\prod_{j=1}^J (p_j^{\alpha_j}-p_j^{\alpha_j-1}), \end{equation*} and the values of $\varphi(n)$ can be computed in this way. If $f$ and $g$ are two arithmetic functions, we define their Dirichlet product $f*g$ to be the arithmetic function given by \begin{equation*} (f*g)(n)= \sum_{d|n} f(d)g(n/d), \end{equation*} where $\displaystyle\sum_{d|n}f(d)$ denotes the sum of $f(d)$ over all positive divisors of $n$, and analogously for the union over divisors denoted by \raisebox{0pt}[0pt][0pt]{$\displaystyle\bigcup_{d|n}$} below. The following result is well known: \begin{theorem}\label{Dprod} If $f$ and $g$ are multiplicative, then $f*g$ is multiplicative. \end{theorem} \noindent For a proof, see \cite[Theorem 2.14]{Apostol}. As an application of Theorem \ref{Dprod}, we have the following corollary, which appears as an exercise in \cite[p. 49]{Apostol}: \begin{corollary}\label{gcdsum} Let $f$ be a multiplicative function. Then the function \begin{equation*} \xi_f(n)=\sum_{k=1}^n f\bigl((k,n)\bigr) \end{equation*} is multiplicative. \end{corollary} \begin{proof} Note that \begin{equation*} \{k|1\leq k\leq n\} =\bigcup_{d|n}\{k| 1\leq k\leq n\ \text{and}\ (k,n)=d\} \end{equation*} and, therefore, \begin{eqnarray*} \xi_f(n)&=&\sum_{k=1}^n f\bigl((k,n)\bigl)\; =\sum_{d|n} \ssum{k=1}{n}{(k,n)=d} f\bigl((k,n)\bigr) \\&=&\sum_{d|n}f(d)\varphi\big(n/d\big)=(f*\varphi)(n). \end{eqnarray*} Since $f$ and $\varphi$ are both multiplicative, we deduce by Theorem \ref{Dprod} that $\xi_f=f*\varphi$ is multiplicative, too. \end{proof} \section{A curious multiplicative function} In \cite[Appendix C]{DEBZ}, T. Durt, B.-G. Englert, I. Bengtsson, and K. \.{Z}yczkowski observed, during their study of the properties of mutually unbiased bases, the following interesting identity associated with the Gauss sum: \begin{theorem}\label{Gauss-sum} For any two integers $m$ and $n$ with ${0<m<n}$, and $\zeta_n=e^{2\pi i/n}$, one has \begin{equation*} \frac{1}{\sqrt{n}}\left|\sum_{l=0}^{n-1}\zeta_{2n}^{(n-l)l m}\right| =\begin{cases} 0\quad\text{if $\,n\,$ is even with both $\,n/(m,n)\,$ and $\,m/(m,n)\,$ odd,} \\[1ex] \sqrt{(m,n)}\quad\text{otherwise.} \end{cases} \end{equation*} \end{theorem} \noindent They arrived at this result by linear-algebra arguments and without relying on the properties of Jacobi symbols and contour integrals that are usually employed when evaluating Gauss sums. This connection between linear algebra and number theory is noteworthy and, therefore, we revisit the matter in Section~\ref{sec:MUB}. In \cite{DEBZ}, they also indicated that \begin{equation}\label{DEBZ_h} h(n) =\frac{1}{\sqrt{n}}\sum_{m=1}^{n-1} \left|\sum_{l=0}^{n-1} \zeta_{2n}^{(n-l)l m}\right| +\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{2}\sqrt{n} & \text{if $n$ is even}\\[2ex] \sqrt{n} & \text{if $n$ is odd} \end{array}\right\} \end{equation} is a multiplicative function. It is possible to simplify the expression of $h(n)$. We first introduce $v_p(n)$, the count of prime number factors $p$ contained in the positive integer $n$, defined by \begin{equation*} v_p(n) = \alpha \,\,\text{whenever $p^\alpha|n$ but $p^{\alpha+1}\nmid n$.} \end{equation*} Then we can, after applying Theorem \ref{Gauss-sum}, rewrite $h(n)$ as \begin{equation}\label{DEBZ-h} h(n)=\begin{cases} \displaystyle \sum_{k=1}^n \sqrt{(k,n)}-\frac{1}{2}\sqrt{n} -\ssum{k=1}{n-1}{v_2(k)=v_2(n)}\sqrt{(k,n)} &\text{if $n$ is even,}\\ \displaystyle\sum_{k=1}^n \sqrt{(k,n)}&\text{if $n$ is odd.} \end{cases} \end{equation} Note that by Corollary \ref{gcdsum}, \raisebox{0pt}[0pt][0pt]{$\displaystyle\xi_s(n)=\sum_{k=1}^n \sqrt{(k,n)}$} is multiplicative because $s(n)=\sqrt{n}$ has this property. The function $h(n)$ appears to be a new multiplicative function. One question we can ask is whether we can construct multiplicative functions when the prime $2$ that is privileged in the definition of $h(n)$ is replaced by any odd prime $p$ and whether the function $s(n)$ can be replaced by other arithmetic functions. It turns out that this is possible. \section{First generalization} As we shall confirm in Section \ref{sec:confirm}, one generalization of $h(n)$ is the following: \begin{theorem} \label{DEBZ-general-2} Choose a multiplicative function $f$ and a prime number $p$, as well as a sequence of complex numbers $\kappa_0=0$, $\kappa_1$, $\kappa_2$, \dots, and a sequence of positive integers $a_1$, $a_2$, \dots\ with $a_{\alpha}\leq\alpha$. Then the function \begin{equation*} h^{(1)}_{f,p}(n)=\xi_f(n)-\kappa_{v_p(n)}\ssum{k=1}{n}{v_p(k)=v_p(n)-a_{v_p(n)}} f\bigl((k,n)\bigr) \end{equation*} is multiplicative. \end{theorem} \begin{proof} If $(m,n)=1$ and $(mn,p)=1$, then by Corollary \ref{gcdsum}, \begin{equation*} h^{(1)}_{f,p}(mn)=\xi_f(mn)=\xi_f(m)\xi_f(n)=h^{(1)}_{f,p}(m)h^{(1)}_{f,p}(n). \end{equation*} In order to complete the proof, we need to show that if $\alpha>0$ and $(\nu,p)=1$, then \begin{equation* h^{(1)}_{f,p}(p^\alpha\nu) = h^{(1)}_{f,p}(p^\alpha)h^{(1)}_{f,p}(\nu). \end{equation*} We evaluate the sum in \begin{equation*} h^{(1)}_{f,p}(p^\alpha \nu) =\xi_f(p^\alpha \nu) -\kappa_\alpha \ssum{k=1}{p^\alpha\nu}{v_p(k)=\alpha-a_\alpha} f\bigl((k,p^\alpha\nu)\bigr) \end{equation*} in a few steps, proceeding from \begin{align*} \ssum{k=1}{p^\alpha\nu}{v_p(k)=\alpha-a_\alpha} f\bigl((k,p^\alpha\nu)\bigr) &=f(p^{\alpha-a_{\alpha}}) \ssum{k=1}{p^{a_{\alpha}}\nu}{(k,p)=1}f\bigl((k,p^{a_{\alpha}}\nu)\bigr) =f(p^{\alpha-a_{\alpha}}) \ssum{k=1}{p^{a_{\alpha}}\nu}{(k,p)=1}f\bigl((k,\nu)\bigr)\\ &=f(p^{\alpha-a_{\alpha}})\sum_{k=1}^{p^{a_{\alpha}}\nu} f\bigl((k,\nu)\bigr) \biggl\lfloor\frac{1}{(k,p)}\biggr\rfloor, \end{align*} where $\lfloor x\rfloor$ denotes the floor of $x$, the largest integer that does not exceed $x$. Now, it is known that \begin{equation*} \biggl\lfloor\frac{1}{n}\biggr\rfloor=\sum_{d|n}\mu(d) =\begin{cases} 1 & \text{if $n=1$}\\ 0 & \text{if $n>1$} \end{cases} \end{equation*} with the M\"obius function \begin{equation*} \mu(n) = \begin{cases} 1 \quad\text{if $n=1,$} \\ (-1)^j\quad\text{if $n=p_1\cdots p_j,$ where the $p_i$s are distinct primes,}\\ 0 \quad\text{otherwise,}\end{cases} \end{equation*} which we exploit in \begin{equation*} \sum_{k=1}^{p^{a_{\alpha}}\nu} f\bigl((k,\nu)\bigr) \biggl\lfloor\frac{1}{(k,p)}\biggr\rfloor =\sum_{k=1}^{p^{a_{\alpha}}\nu} f\bigl((k,\nu)\bigr)\sum_{\substack{d|k \\ d|p}} \mu(d) =\sum_{d|p}\mu(d) \sum_{\substack{k=1\\d|k}}^{p^{a_{\alpha}}\nu}f\bigl((k,\nu)\bigr). \end{equation*} Only $d=1$ and $d=p$ contribute to the sum, so that \begin{align*} \sum_{k=1}^{p^{a_{\alpha}}\nu} f\bigl((k,\nu)\bigr) \biggl\lfloor\frac{1}{(k,p)}\biggr\rfloor &=\sum_{k=1}^{p^{a_{\alpha}}\nu}f\bigl((k,\nu)\bigr) -\sum_{k=1}^{p^{a_{\alpha}-1}\nu}f\bigl((k,\nu)\bigr)\\ &=\bigl(p^{a_{\alpha}}-p^{a_{\alpha}-1}\bigr)\xi_f(\nu) =\varphi(p^{a_{\alpha}})\xi_f(\nu), \end{align*} where it is crucial that $a_{\alpha}\geq1$, and we arrive at \begin{align*} h^{(1)}_{f,p}(p^\alpha \nu) &=\xi_f(p^\alpha \nu) -\kappa_\alpha f(p^{\alpha-a_{\alpha}})\varphi(p^{a_{\alpha}})\xi_f(\nu) \\&=\bigl[\xi_f(p^{\alpha}) -\kappa_\alpha f(p^{\alpha-a_{\alpha}})\varphi(p^{a_{\alpha}})\bigr]\xi_f(\nu) = h^{(1)}_{f,p}(p^{\alpha}) h^{(1)}_{f,p}(\nu)\,, \end{align*} where the last step recognizes that $h^{(1)}_{f,p}(p^{\alpha})=\xi_f(p^{\alpha}) -\kappa_\alpha f(p^{\alpha-a_{\alpha}})\varphi(p^{a_{\alpha}})$ as a consequence of $\xi_f(1)=1$. This completes the proof of Theorem~\ref{DEBZ-general-2}. \end{proof} \noindent% Note that ${h^{(1)}_{f,p}(p^\alpha) % =\xi_f(p^{\alpha})-\kappa_\alpha\varphi(p^{\alpha})}$ when ${a_{\alpha}=\alpha}$. \section{Second generalization.} Here is another generalization of $h(n)$, also confirmed in Section \ref{sec:confirm}: \begin{theorem} \label{DEBZ-general-3} Choose a multiplicative function $f$ and a prime number $p$, as well as a sequence of complex numbers $\kappa_0=0$, $\kappa_1$, $\kappa_2$, \dots, and a sequence of nonnegative integers $a_1$, $a_2$, \dots\ with $a_{\alpha}\leq\alpha$. Then the function \begin{equation*} h^{(2)}_{f,p}(n) =\xi_f(n)-\kappa_{v_p(n)}\ssum{k=1}{n}{v_p(k)\geq v_p(n)-a_{v_p(n)}} f\bigl((k,n)\bigr) \end{equation*} is multiplicative. \end{theorem}\enlargethispage{1.0\baselineskip \begin{proof} Since $h^{(2)}_{f,p}(n)=\xi_f(n)$ when $(n,p)=1$, we need to show that \begin{equation}\label{X} h^{(2)}_{f,p}(p^\alpha \nu) = h^{(2)}_{f,p}(p^\alpha)\xi_f(\nu) \end{equation} for $\alpha>0$ and $(\nu,p)=1$. We have \begin{align*} &\hphantom{=\ } \ssum{k=1}{p^{\alpha}\nu}{v_p(k)\geq \alpha-a_{\alpha}} f\bigl((k,p^{\alpha}\nu)\bigr) =\sum_{k=1}^{p^{a_{\alpha}}\nu}f\bigl((p^{\alpha-a_{\alpha}}k,p^{\alpha}\nu)\bigr) \\ &={\left[\enskip\ssum{k=1}{p^{a_{\alpha}}\nu}{v_p(k)=0}\quad +\ssum{k=1}{p^{a_{\alpha}}\nu}{v_p(k)=1}+\cdots +\ssum{k=1}{p^{a_{\alpha}}\nu}{v_p(k)=a_{\alpha}-1}\qquad +\ssum{k=1}{p^{a_{\alpha}}\nu}{v_p(k)\geq a_{\alpha}}\quad\enskip \right]}f\bigl((p^{\alpha-a_{\alpha}}k,p^{\alpha}\nu)\bigr)\\ &=\sum_{b=1}^{a_{\alpha}} \ssum{k=1}{p^{b}\nu}{v_p(k)=0}f\bigl((p^{\alpha-b}k,p^{\alpha}\nu)\bigr) +\sum_{k=1}^{\nu}f\bigl((p^{\alpha}k,p^{\alpha}\nu)\bigr)\\ &=\sum_{b=1}^{a_{\alpha}}f\bigl(p^{\alpha-b}\bigr) \ssum{k=1}{p^{b}\nu}{(k,p)=1}f\bigl((k,\nu)\bigr) +f(p^{\alpha})\sum_{k=1}^{\nu}f\bigl((k,\nu)\bigr)\\ &=\sum_{b=1}^{a_{\alpha}}f\bigl(p^{\alpha-b}\bigr)\varphi(p^b)\xi_f(\nu) +f(p^{\alpha})\xi_f(\nu), \end{align*} where we recall, from the proof of Theorem \ref{DEBZ-general-2}, that \begin{equation*} \ssum{k=1}{p^b\nu}{(k,p)=1}f\bigl((k,\nu)\bigr)=\varphi(p^b)\xi_f(\nu) \quad\text{for $b>0$}. \end{equation*} It follows that \eqref{X} holds with $h^{(2)}_{f,p}(p^\alpha)=\xi_f(p^{\alpha})-\kappa_{\alpha} \raisebox{0pt}[0pt][0pt]{$\displaystyle\sum_{b=0}^{a_{\alpha}}$} f\bigl(p^{\alpha-b}\bigr)\varphi(p^b)$, which concludes the proof. \end{proof} \noindent% Note that ${h^{(2)}_{f,p}(p^\alpha)=(1-\kappa_{\alpha})\xi_f(p^{\alpha})}$ when ${a_{\alpha}=\alpha}$. \section{Generalizations confirmed}\label{sec:confirm} We now confirm that $h^{(1)}_{f,p}(n)$ and $h^{(2)}_{f,p}(n)$, introduced in Theorems \ref{DEBZ-general-2} and \ref{DEBZ-general-3}, are generalizations of $h(n)$ in \eqref{DEBZ-h}. In view of the important role played by the privileged prime $p$, we write $n=p^{v_p(n)}\bigl(p^{-v_p(n)}n\bigr)$ and note that \begin{align*} h(n)&=h\bigl(2^{v_2(n)}\bigr)\xi_s\bigl(2^{-v_2(n)}n\bigr),\\ h^{(1)}_{f,p}(n)&=h^{(1)}_{f,p}\bigl(p^{v_p(n)}\bigr) \xi_f\bigl(p^{-v_p(n)}n\bigr),\\ h^{(2)}_{f,p}(n)&=h^{(2)}_{f,p}\bigl(p^{v_p(n)}\bigr) \xi_f\bigl(p^{-v_p(n)}n\bigr), \end{align*} with \begin{eqnarray}\label{R1}\notag h\bigl(2^{\alpha}\bigr)&=&\xi_s\bigl(2^{\alpha}\bigr)-2^{\alpha/2-1} \quad\text{for $\alpha\geq1$},\\ h^{(1)}_{f,p}\bigl(p^{\alpha}\bigr)&=&\xi_f\bigl(p^{\alpha}\bigr) -\kappa_\alpha f\bigl(p^{\alpha-a_{\alpha}}\bigr) \varphi\bigl(p^{a_{\alpha}}\bigr), \notag\\ h^{(2)}_{f,p}(p^\alpha)&=&\xi_f(p^{\alpha})-\kappa_{\alpha} \sum_{b=0}^{a_{\alpha}}f\bigl(p^{\alpha-b}\bigr)\varphi(p^b). \end{eqnarray} Therefore, we obtain ${h_{f,p}^{(1)}(n)=h(n)}$ and ${h_{f,p}^{(2)}(n)=h(n)}$ for $f(n)=\sqrt{n}=s(n)$ and $p=2$, if we choose $\kappa_{\alpha}$ in accordance with \begin{align*} \kappa_{\alpha}&=2^{-a_{\alpha}/2}\quad\text{for $a_{\alpha}\geq1$ in}\ h^{(1)}_{s,2}(n),\\ \kappa_{\alpha}&=\Bigl(2^{1+a_{\alpha}/2}+2^{(1+a_{\alpha})/2}-2^{1/2}\Bigr)^{-1} \quad\text{for $a_{\alpha}\geq0$ in}\ h^{(2)}_{s,2}(n), \end{align*} for all $\alpha\geq0$. Since these assignments work for all permissible choices for the $a_{\alpha}$s, we have multiple generalizations of $h(n)$ from both $h_{f,p}^{(1)}(n)$ and $h_{f,p}^{(2)}(n)$, although the conditions ${v_p(k)=v_p(n)-a_{v_p(n)}}$ in Theorem \ref{DEBZ-general-2} and ${v_p(k)\geq v_p(n)-a_{v_p(n)}}$ in Theorem \ref{DEBZ-general-3} do not really match the condition $v_2(k)=v_2(n)$ in \eqref{DEBZ-h} because $a_{\alpha}=0$ is not allowed in Theorem \ref{DEBZ-general-2}. Clearly, then, \eqref{DEBZ-h} is just one of many ways of rewriting $h(n)$ of \eqref{DEBZ_h}. \section{Each \lowercase{$h^{(1)}_{f,p}$} is a \lowercase{$h^{(2)}_{f,p}$} and % vice versa} The mappings ${(f,p)\mapsto h^{(1)}_{f,p}}$ and ${(f,p)\mapsto h^{(2)}_{f,p}}$ are both characterized by a sequence of $\kappa_{\alpha}$s and a sequence of $a_{\alpha}$s, and---for given $f$ and~$p$---one can always adjust the sequences of one of them to the sequences of the other such that the right-hand sides in \eqref{R1} are the same, ${h^{(1)}_{f,p}\bigl(p^{\alpha}\bigr)=h^{(2)}_{f,p}\bigl(p^{\alpha}\bigr)}$. In this sense, all the multiplicative functions in the $h^{(1)}_{f,p}$ family are also contained in the $h^{(2)}_{f,p}$ family, and vice versa, although the two mappings are really different. To justify this remark, we shall write $\kappa^{(1)}_{\alpha}$ and $a^{(1)}_{\alpha}$ for the parameters that specify $h^{(1)}_{f,p}$ and $\kappa^{(2)}_{\alpha}$ and $a^{(2)}_{\alpha}$ for those of $h^{(2)}_{f,p}$. Then, for a particular choice of the $\kappa^{(1)}_{\alpha}$s and $a^{(1)}_{\alpha}$s, we put $\kappa^{(2)}_{\alpha}=0$ if $\kappa^{(1)}_\alpha f\bigl(p^{\alpha-a^{(1)}_{\alpha}}\bigr)=0$; otherwise we choose either $a^{(2)}_{\alpha}=a^{(1)}_{\alpha}$ or $a^{(2)}_{\alpha}=a^{(1)}_{\alpha}-1\geq0$ such that $F=\raisebox{0pt}[\height][10pt]% {$\displaystyle\sum_{b=0}^{a^{(2)}_{\alpha}}$}% f\bigl(p^{\alpha-b}\bigr)\varphi(p^b)\neq0$ and put $\kappa^{(2)}_{\alpha}=F^{-1}\kappa^{(1)}_\alpha % f\bigl(p^{\alpha-a^{(1)}_{\alpha}}\bigr)\varphi\bigl(p^{a^{(1)}_{\alpha}}\bigr)$. Conversely, for a particular choice of the $\kappa^{(2)}_{\alpha}$s and $a^{(2)}_{\alpha}$s, we put $a^{(1)}_{\alpha}=\alpha$ and $\kappa^{(1)}_{\alpha}=\varphi\bigl(p^{\alpha}\bigr)^{-1}\kappa^{(2)}_{\alpha}% \raisebox{0pt}[\height][0pt]{$\displaystyle\sum_{b=0}^{a^{(2)}_{\alpha}}$}% f\bigl(p^{\alpha-b}\bigr)\varphi(p^b)$. These assignments ensure that ${h^{(1)}_{f,p}\bigl(p^{\alpha}\bigr)=h^{(2)}_{f,p}\bigl(p^{\alpha}\bigr)}$. \section{A linear-algebra proof of Theorem \ref{Gauss-sum}}\label{sec:MUB} We revisit here the linear-algebra proof of Theorem \ref{Gauss-sum}. While we follow the reasoning in \cite{DEBZ}, where Theorem~\ref{Gauss-sum} is a side issue and the ingredients are widely scattered, the presentation here is self-contained and adopts somewhat simpler conventions, in particular for the phase factor in the definition of $\MC_m$ in \eqref{def-Cm} below. \heading{Getting started: Columns and rows, matrices, eigenvector bases} We consider column vectors with $n$ complex entries (${n\geq2}$), their adjoint row vectors, and the $n\times n$ matrices that implement linear mappings of columns to columns and rows to rows. For any two columns $\col{x}$ and $\col{y}$, we denote the adjoint rows by $\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}$ and $\col{y}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}$; a row-times-column product such as $\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\col{y}$ is a complex number that can be understood as the inner product of the columns $\col{x}$ and $\col{y}$, or of the rows $\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}$ and $\col{y}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}$, whereby $\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\col{x}\geq0$ with ``$=$'' only for $\col{x}=\col{0}$. The row-times-column products such as $\col{y}\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}$ are $n\times n$ matrices with $\tr{\col{y}\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}}=\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\col{y}$ for the matrix trace. We recall that $(\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}})^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}=\col{x}$, $(\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\col{y})^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}=\col{y}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\col{x}$, and $(\col{y}\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}})^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}=\col{x}\col{y}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}$. Following H.~Weyl \cite[Sec.~IV.D.14]{Weyl} and J.~Schwinger \cite[Sec.~1.14]{Schwinger}, our basic ingredients are two related unitary $n\times n$ matrices $\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}$ and $\MB$ of period $n$, that is \begin{equation*} \M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}^k=\M{1}_{n},\quad \MB^k=\M{1}_{n}\quad \text{if $k\equiv0\ (\text{mod}\ n)$ and only then}, \end{equation*} where $\M{1}_{n}$ is the $n\times n$ unit matrix. The eigenvalues of $\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}$ and $\MB$ are the powers of $\zeta_n$, the basic $n$th root of unity that appears in Theorem \ref{Gauss-sum}. We denote the $j$th eigencolumn of $\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}$ by $\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j$ and the $k$th eigencolumn of $\MB$ by $\cb_k$, \begin{equation*} \M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j=\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j\,\zeta_n^j,\quad \MB\cb_k=\cb_k\,\zeta_n^k,\qquad \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}=\zeta_n^j\,\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}},\quad\cb_k^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\MB=\zeta_n^k\,\cb_k^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}, \end{equation*} where we regard the labels as modulo-$n$ integers, so that $\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_{j+n}=\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j$ and $\cb_{k+n}=\cb_k$. The sets of eigencolumns are orthonormal and complete, \begin{align*} \text{orthonormality:}&\quad \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_k\phadj=\delta^{(n)}_{j,k} =\left\{ \begin{array}{@{}l@{\enskip}l@{}} 1 & \text{if}\ \zeta_n^j=\zeta_n^k \\[1ex] 0 & \text{if}\ \zeta_n^j\neq\zeta_n^k \end{array}\right\} =\frac{1}{n}\sum_{l=0}^{n-1}\zeta_n^{(j-k)l}=\cb_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_k\phadj,\\ \text{completeness:}&\quad \sum_{j=0}^{n-1}\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j\phadj \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}=\M{1}_{n} =\sum_{k=0}^{n-1}\cb_k\phadj \cb_k^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}, \end{align*} where $\delta^{(n)}_{j,k}$ is the modulo-$n$ version of the Kronecker delta symbol, and the projection matrices associated with the eigenvectors are \begin{equation}\label{A1} \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j\phadj \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}=\frac{1}{n}\sum_{k=0}^{n-1} \Bigl(\zeta_n^{-j}\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}\Bigr)^k,\quad \cb_k\phadj \cb_k^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}=\frac{1}{n}\sum_{l=0}^{n-1}\Bigl(\zeta_n^{-k}\MB\Bigr)^l. \end{equation} The unitary matrices $\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}$ and $\MB$ are related to each other by the discrete Fourier transform that turns one set of eigenvectors into the other, \begin{equation*} \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_k\phadj=\frac{1}{\sqrt{n}}\zeta_n^{jk}, \quad \cb_k\phadj=\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j\zeta_n^{jk}, \quad \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}=\frac{1}{\sqrt{n}}\sum_{k=0}^{n-1}\zeta_n^{jk}\cb_k^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}. \end{equation*} As a consequence, we have the following identities: \begin{eqnarray}\label{A2} & \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \MB= \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_{j+1}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}},\quad \M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}\cb_k=\cb_{k+1},\quad \tr{\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}^j\MB^k}=n\,\delta^{(n)}_{j,0}\,\delta^{(n)}_{k,0},&\nonumber\\ &\zeta_n^{jk}\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}^j\MB^k=\MB^k\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}^j,\quad (\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}^j\MB^k)^l=\zeta_{2n}^{jk(l-1)l}\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}^{jl}\MB^{kl};& \end{eqnarray} we leave their verification to the reader as an exercise. \heading{More unitary matrices and their eigenstate bases} For $m=1,2,\dots,n-1$, we define \begin{equation*}\label{def-Cm} \MC_m=\zeta_{2n}^{-(n-1)m}\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}\MB^m , \end{equation*} which are unitary matrices of period $n$, \begin{equation}\label{A3} \MC_m^k=\zeta_{2n}^{-(n-1)mk}(\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}\MB^m)^k=\zeta_{2n}^{-(n-k)mk}\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}^k\MB^{mk} \xrightarrow{k=n}\M{1}_{n}. \end{equation} Upon denoting the $j$th eigencolumn of $\MC_m$ by $\cc_{m,j}=\cc_{m,j+n}$, we have $\MC_m\cc_{m,j}=\cc_{m,j}\zeta_n^j$ and $\cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\MC_m=\zeta_n^j\cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}$ as well as \begin{equation}\label{A4} \cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cc_{m,k}\phadj=\delta^{(n)}_{j,k},\quad \sum_{j=0}^{n-1} \cc_{m,j}\phadj \cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}=\M{1}_{n},\quad \cc_{m,j}\phadj \cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} =\frac{1}{n}\sum_{k=0}^{n-1}\Bigl(\zeta_n^{-j}\MC_m\Bigr)^k. \end{equation} To establish how the $\cc_{m_j}$s are related to the $\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j$s and $\cb_k$s, we first infer $\cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_k\phadj$ from the recurrence relation \begin{equation*} \cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_{k+1}\phadj= \cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C} \cb_k\phadj = \cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\zeta_{2n}^{(n-1)m}\MC_m\MB^{-m}\cb_k\phadj = \cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_k\phadj \,\zeta_{2n}^{(n-1)m}\,\zeta_n^{j-mk}, \end{equation*} which yields \begin{equation*} \cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_k\phadj =\cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_0\phadj \,\zeta_{2n}^{(n-1)mk}\,\zeta_n^{jk-mk(k-1)/2} = \frac{1}{\sqrt{n}}\,\zeta_n^{jk}\zeta_{2n}^{(n-k)km}, \end{equation*} where we adopt $\cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_0\phadj=1/\sqrt{n}$ by convention. Then we exploit the completeness of the $\cb_k$s in \begin{equation}\label{A-Gauss} \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cc_{m,k}\phadj =\sum_{l=0}^{n-1}\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_l\phadj \cb_l^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cc_{m,k}\phadj =\frac{1}{n}\sum_{l=0}^{n-1}\zeta_n^{(j-k)l}\zeta_{2n}^{-(n-l)lm}. \end{equation} This Gauss sum is our first ingredient. Next we utilize ${\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\col{y}=\tr{\col{y}\col{x}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}}}$ and the linearity of the trace as well as statements in \eqref{A1}--\eqref{A4} in \begin{align}\label{A6}\notag \bigl| \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cc_{m,k}\phadj\bigr|^2 &= \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cc_{m,k}\phadj \cc_{m,k}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j\phadj = \tr{ \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j\phadj \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\; \cc_{m,k}\phadj \cc_{m,k}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}}\\ \notag &= \frac{1}{n^2}\sum_{l,l'=0}^{n-1} \tr{ \Bigl(\zeta_n^{-j}\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}\Bigr)^{l'}\Bigl(\zeta_n^{-k}\MC_m\Bigr)^l}\\ \notag &= \frac{1}{n^2}\sum_{l,l'=0}^{n-1} \zeta_n^{-jl'-kl}\zeta_{2n}^{-(n-l)lm} \tr{\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}^{l'+l}\MB^{ml}}\\ &= \frac{1}{n}\sum_{l=0}^{n-1} \zeta_n^{(j-k)l}\zeta_{2n}^{-(n-l)lm}\delta^{(n)}_{ml,0}. \end{align} Now writing $d=(m,n)$, $n=d\nu$, $m=d\mu$ with $(\mu,\nu)=1$, we have \begin{equation*} \delta^{(n)}_{ml,0}=\delta^{(\nu)}_{l,0} \end{equation*} because ${ml\equiv0\ (\text{mod}\ n)}$ requires ${l\equiv0\ (\text{mod}\ \nu)}$. Therefore, only the terms with $l=0,\nu,2\nu,\ldots,(d-1)\nu$ contribute to the final sum in \eqref{A6}, and we arrive at \begin{align*} n \bigl| \col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cc_{m,k}\phadj\bigr|^2 & =\sum_{l=0}^{n-1}\zeta_n^{(j-k)l}\zeta_{2n}^{-(n-l)lm}\delta^{(\nu)}_{l,0} =\sum_{l=0}^{d-1}\zeta_n^{(j-k)l\nu}\zeta_{2n}^{-(n-l\nu)l\nu m}\\ & =\sum_{l=0}^{d-1}\zeta_d^{(j-k)l}(-1)^{(d-1)\mu\nu l} = \left\{\begin{array}{@{}l@{\quad}l@{}} d\,\delta^{(d)}_{j,k+d/2} & \text{if $(d-1)\mu\nu$ is odd,}\\[1ex] d\,\delta^{(d)}_{j,k} & \text{if $(d-1)\mu\nu$ is even.} \end{array}\right. \end{align*} Upon combining this second ingredient with the first in \eqref{A-Gauss}, we conclude that \begin{equation*} \frac{1}{\sqrt{n}}\left|\sum_{l=0}^{n-1} \zeta_n^{(j-k)l}\zeta_{2n}^{(n-l)lm}\right| =\left\{\begin{array}{@{}l@{\enskip}l@{}} \sqrt{(m,n)}\,\delta^{((m,n))}_{j,k+(m,n)/2} & \text{if $v_2(n)=v_2(m)\geq1$}\\[1ex] \sqrt{(m,n)}\,\delta^{((m,n))}_{j,k} & \text{otherwise.} \end{array}\right.\quad \end{equation*} For $j=k$, this is the statement in Theorem \ref{Gauss-sum}. \clearpage In passing, we found the following identity between the absolute value of a Gauss sum and a particular partial sum: \begin{corollary}\label{Gauss-sum-identity} For all integers $k,m,n$ with $n\geq1$, we have \begin{equation*} \left|\frac{1}{n}\sum_{l=0}^{n-1} \zeta_n^{kl}\zeta_{2n}^{(n-l)lm} \right|^2 =\frac{1}{n}\sum_{l=0}^{n-1} \zeta_n^{kl}\zeta_{2n}^{(n-l)lm}\delta^{(n)}_{ml,0}. \end{equation*} \end{corollary} \begin{proof} For $m$ values in the range ${0<m<n}$, this follows from comparing \eqref{A-Gauss} and \eqref{A6}, and the case of ${m=0}$ is immediate. Then, the observation that $\zeta_n^{kl}\zeta_{2n}^{(n-l)lm}$ does not change if we replace $m$ by $m\pm n$, in conjunction with replacing $k$ by $k+\frac{1}{2}n$ if $n$ is even, extends the permissible $m$ values to all integers. The identity can, of course, also be verified directly. We note that $\zeta_n^{kl}\zeta_{2n}^{(n-l)lm}$ does not change when $l$ is replaced by $l\pm n$ and, therefore, the summation over $l$ can cover any range of $n$ successive integers. Accordingly, we can replace $l$ by $l+a$ for any integer $a$ without changing the value of the sum. We exploit this when writing the left-hand side as a double sum and then processing it, \begin{align*} &\;\frac{1}{n}\sum_{l'=0}^{n-1} \zeta_n^{-kl'}\zeta_{2n}^{-(n-l')l'm} \frac{1}{n}\sum_{l=0}^{n-1} \zeta_n^{kl}\zeta_{2n}^{(n-l)lm} \\=&\;\frac{1}{n^2}\sum_{l,l'=0}^{n-1} \zeta_n^{-kl'}\zeta_{2n}^{-(n-l')l'm} \zeta_n^{k(l+l')}\zeta_{2n}^{(n-l-l')(l+l')m} \\=&\;\frac{1}{n}\sum_{l=0}^{n-1}\zeta_n^{kl}\zeta_{2n}^{(n-l)lm} \frac{1}{n}\sum_{l'=0}^{n-1}\zeta_n^{-ll'm} =\frac{1}{n}\sum_{l=0}^{n-1} \zeta_n^{kl}\zeta_{2n}^{(n-l)lm}\delta^{(n)}_{ml,0}, \end{align*} thereby arriving at the right-hand side. \end{proof} \heading{Remark: Unbiased bases} The eigenvector bases for the matrices $\M{A}}\newcommand{\MB}{\M{B}}\newcommand{\MC}{\M{C}$ and $\MB$ are such that $\bigl|\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}} \cb_k\phadj\bigr|$ has the same value for all rows $\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}$ and columns $\cb_k$, which is the defining property of a pair of \emph{unbiased} bases. Further, for each $m$, the basis $\mathcal{C}_m=\mathop{\{\cc_{m,k}\}}_{k=1}^{n}$ is unbiased with the basis $\mathcal{B}=\mathop{\{\cb_k\}}_{k=1}^n$. When $n$ is prime, each $\mathcal{C}_m$ is also unbiased with the basis $\mathcal{A}=\mathop{\{\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j\}}_{j=1}^n$. When $n$ is not prime, however, then some of the $\mathcal{C}_m$s are unbiased with $\mathcal{A}$, namely those with $(m,n)=1$, and the others are not. Further, two bases $\mathcal{C}_m$ and $\mathcal{C}_{m'}$ with $m'>m$ are unbiased if $(m'-m,n)=1$, and only then, because $\cc_{m,j}^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\cc_{m',k}\phadj =\col{a}}\newcommand{\cb}{\col{b}}\newcommand{\cc}{\col{c}_j^{\dagger}}\newcommand{\phadj}{^{\vphantom{\dagger}}\cc_{m'-m,k}\phadj$. We refer the reader to \cite{DEBZ} for a detailed discussion of unbiased bases. \enlargethispage{1.0\baselineskip} \section{Yet another multiplicative function} Here we report one more multiplicative function that is suggested by $h(n)$ in \eqref{DEBZ-h}, but is not a generalization in the spirit of $h^{(1)}_{f,p}(n)$ and $h^{(2)}_{f,p}(n)$ in Theorems \ref{DEBZ-general-2} and \ref{DEBZ-general-3}. In preparation, and for the record, we note that the Gauss sum in Theorem~\ref{Gauss-sum}, \begin{equation}\label{Smn} S(m,n)=\frac{1}{\sqrt{n}} \sum_{l=0}^{n-1} \zeta_{2n}^{(n-l)l m}, \end{equation} can be evaluated. We write $\nu=n/(m,n)$ and $\mu=m/(m,n)$ as above and distinguish three cases: \begin{enumerate} \item[(a)] If $v_2(n)=0$ or $v_2(m)>v_2(n)>0$, then $\nu$ is odd and $(n-1)\mu$ is even, and we have \begin{equation*} S(m,n) =\begin{cases} \ \bigl(\frac{1}{2}(n-1)\mu|\nu\bigr)\sqrt{(m,n)} & \text{if $i^{\nu}=i$,}\\[1ex] i\bigl(\frac{1}{2}(n-1)\mu|\nu\bigr)\sqrt{(m,n)} & \text{if $i^{\nu}=-i$.} \end{cases} \end{equation*} \item[(b)] If $v_2(n)>v_2(m)$, then $(n-1)\mu$ is odd, and we have \begin{equation*} S(m,n)=e^{\pm i\pi/4}\bigl(2\nu|(n-1)\mu\bigr)\sqrt{(m,n)} \quad\text{for}\ i^{(n-1)\mu}=\pm i. \end{equation*} \item[(c)] If $v_2(n)=v_2(m)\geq 1$, we have \begin{equation*} S(m,n) =0. \end{equation*} \end{enumerate} Here $(j|k)$ is the familiar Jacobi symbol \cite[Sec.~9.7]{Apostol}; all $(j|k)$ appearing here are equal to $+1$ or $-1$ because the arguments are co-prime in each case. We leave the proof to the reader as an exercise in the evaluation of Gauss sums, and the sum in Corollary~\ref{Gauss-sum-identity} for $k\neq0$ is another exercise. The facts required to complete these proofs can be found in \cite[Sec.~1.5]{Berndt-Evans-Williams}. The linear-algebra argument in Section~\ref{sec:MUB} establishes the absolute value of the right-hand side, namely $\sqrt{(m,n)}$ in (a) and (b) and $0$ in (c), but not its complex phase. Now, harking back to $h(n)$ in \eqref{DEBZ-h}, we observe that \begin{equation*} h(n) = \sum_{m=1}^{n} \sqrt{(m, n)} = \sum_{m=1}^{n} |S(m, n)| \quad\text{for $n$ odd,} \end{equation*} and there is an extra term when $n$ is even. Since $\displaystyle\sum_{m=1}^{n} \sqrt{(m,n)}$ is a multiplicative function of $n$ for all positive $n$, odd or even, it is natural for us to ask if \begin{equation*} s(n)=\sum_{m=1}^{n} S(m,n) \end{equation*} is a multiplicative function of $n$, too. While this turns out to be false (see for example the case of $n=6=2\times3$), it \emph{is} true for odd $n$, for which $s(n)$ is real. For even $n$, an extra term is required in addition to discarding the imaginary part of $s(n)$, reminiscent of the even-$n$ modification in \eqref{DEBZ-h}: \begin{theorem}\label{h-sharp} With the Gauss sum $S(m,n)$ in \eqref{Smn}, the function \begin{equation*} h^{\#}(n)= \sum_{m=1}^{n} \mathrm{Re}\bigl(S(m,n)\bigr) +\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{2}\sqrt{n} & \text{if $n$ is even}\\[2ex] 0 & \text{if $n$ is odd} \end{array}\right\} \end{equation*} is multiplicative. \end{theorem} \begin{proof} We evaluate the sum over $m$ in $s(n)$ by expressing $S(m,n)$ in terms of the sum over $l$ in \eqref{Smn} and carrying out the $m$ summation first, \begin{align*} \sqrt{n}\,s(n) &=\sum_{m=1}^{n}\sum_{l=0}^{n-1} \zeta_{2n}^{(n-l)l m}\\ &=\sum_{l=0}^{n-1}\sum_{m=1}^{n} \Bigl[\delta^{(2n)}_{(n-l)l,0}+\Bigl(1-\delta^{(2n)}_{(n-l)l,0}\Bigr)\Bigr] \zeta_{2n}^{(n-l)l m}\\ &=\sum_{l=0}^{n-1}\Biggl[\delta^{(2n)}_{(n-l)l,0}\,n -\Bigl(1-\delta^{(2n)}_{(n-l)l,0}\Bigr)\frac{1-\zeta_{2n}^{(n-l)ln}} {1-\zeta_{2n}^{-(n-l)l}}\Biggr], \end{align*} where we separate the terms with $\zeta_{2n}^{(n-l)l}=1$ from the others. Next, we note that \begin{equation*} \frac{1}{2}\Bigl(1-\delta^{(2n)}_{(n-l)l,0}\Bigr) \Bigl(1-\zeta_{2n}^{(n-l)ln}\Bigr) =\frac{1}{2}\Bigl(1-\delta^{(2n)}_{(n-l)l,0}\Bigr) \Bigl(1-(-1)^{(n-1)l}\Bigr) =\delta^{(2)}_{n,0}\,\delta^{(2)}_{l,1} \end{equation*} and \begin{equation*} \mathrm{Re}\left(\frac{2}{1-\zeta_{2n}^{-(n-l)l}}\right)=1. \end{equation*} Therefore, \begin{equation*} \sqrt{n}\,\mathrm{Re}\bigl(s(n)\bigr) = n\sum_{l=0}^{n-1}\delta^{(2n)}_{(n-l)l,0}-\dfrac{n}{2} \delta^{(2)}_{n,0}, \end{equation*} and we deduce that \begin{equation*} h^{\#}(n)=\mathrm{Re}\bigl(s(n)\bigr) +\frac{1}{2}\sqrt{n}\,\delta^{(2)}_{n,0} =\sqrt{n}\sum_{l=0}^{n-1}\delta^{(2n)}_{(n-l)l,0} =\sqrt{n}\sum_{l=1}^{n}\delta^{(2n)}_{(n-l)l,0}. \end{equation*} We now proceed to show that $h^{\#}(n)$ is multiplicative. We first consider the case of $n$ odd. Since \begin{equation*} (n-l)l\equiv (1-l)l\equiv 0 \pmod{2} \end{equation*} holds when $n$ is odd, and \begin{equation*} (n-l)l\equiv -l^2\pmod{n} \end{equation*} for all $n$, we have \begin{equation}\label{alt-d} \delta_{(n-l)l,0}^{(2n)}=\delta_{(n-l)l,0}^{(n)}=\delta_{l^2,0}^{(n)}. \end{equation} Now write $n=\lambda\nu^2$, where $\lambda$ is squarefree, or equivalently that $|\mu(\lambda)|=1$. The fact that $n|l^2$ implies that $\nu^2|l^2$ or $\nu|l$. Therefore, if we write $l=\nu s$ then $n|l^2$ implies that $\lambda|s^2$. But if $p|\lambda$ then $p|s^2$ and this implies that $p|s$. Since $\lambda$ is squarefree, we conclude that $\lambda|s$ and consequently, $l=\nu \lambda \omega$ for some positive integer $\omega$. Since $l\leq n$, we conclude that $\omega\leq \nu$. This implies that \begin{equation}\label{alt-id} \sum_{l=1}^{n}\delta^{(n)}_{l^2,0} = \nu. \end{equation} We remark that \eqref{alt-id} is true for any positive integer $n$. Combining \eqref{alt-id} and \eqref{alt-d}, we deduce that \begin{equation*} h^{\#}(n) = \sqrt{n}\sum_{l=1}^{n}\delta^{(2n)}_{(n-l)l,0} = \sqrt{n}\sum_{l=1}^{n}\delta^{(n)}_{l^2,0} = \sqrt{n}\,\nu. \end{equation*} Turning to $n$ even now, we write $n=2m$ with $m>0$ and observe that \begin{align*} \frac{1}{\sqrt{2m}}h^{\#}(2m) &=\sum_{l=1}^{2m}\delta^{(4m)}_{(2m-l)l,0} =\ssum{l=1}{2m}{l\ \text{even}}\delta^{(4m)}_{(2m-l)l,0}\notag\\ &=\ssum{k=1}{m}{l=2k}\delta^{(4m)}_{(2m-l)l,0} =\sum_{k=1}^{m}\delta^{(m)}_{(m-k)k,0} =\sum_{k=1}^{m}\delta^{(m)}_{k^2,0}. \end{align*} By \eqref{alt-id}, we conclude that \begin{equation*} \sum_{k=1}^m \delta^{(m)}_{k^2,0}=\eta, \end{equation*} where $\eta$ is given by $m=\dfrac{n}{2}=\kappa \eta^2$, with squarefree integer $\kappa$. This implies that \begin{equation*} h^{\#}(n) = h^{\#}(2m) = \sqrt{n}\,\eta. \end{equation*} Now, let $a$ and $b$ be two positive integers with $(a,b)=1$. If $a$ and $b$ are both odd, we may write $a=\lambda_a^{\ }\nu_a^2$ and $b=\lambda_b^{\ }\nu_b^2$ with squarefree integers $\lambda_a$ and $\lambda_b$. Then \begin{equation*} h^{\#}(ab) = h^{\#}\bigl(\lambda_a\lambda_b(\nu_a\nu_b)^2\bigr) =\sqrt{ab}\,\nu_a\nu_b = h^{\#}(a)h^{\#}(b). \end{equation*} Next suppose either $a$ or $b$ is even. We may assume that $a$ is even and $b$ is odd. Write $a/2= \lambda_a^{\ }\nu_a^2$ and $b=\lambda_b^{\ }\nu_b^2$ with squarefree integers $\lambda_a$ and $\lambda_b$; then \begin{equation*} h^{\#}(ab) = \sqrt{ab}\,\nu_a\nu_b. \end{equation*} Now $h^{\#}(a) = \sqrt{a}\,\nu_a$ and $h^{\#}(b) =\sqrt{b}\,\nu_b$ and hence \begin{equation*} h^{\#}(ab) =h^{\#}(a)h^{\#}(b), \end{equation*} and this completes the proof that $h^{\#}(n)$ is multiplicative. \end{proof} \heading{Three remarks} (i) Although the multiplicative function $h^{\#}$ is constructed differently, and is \emph{another} multiplicative function in this sense, it is also contained in the families $h^{(1)}_{f,p}$ and $h^{(2)}_{f,p}$ of Theorems \ref{DEBZ-general-2} and \ref{DEBZ-general-3}, because every multiplicative function is in these families for a corresponding $f$. This follows from the fact that the mapping $f\to\xi_f=f*\varphi$ is invertible, and we can choose $\kappa_{\alpha}=0$ for all $\alpha$. In particular, we have $h^{\#}=\xi_{f^{\#}}$ with the multiplicative function $f^{\#}(n)$ specified by its prime-power values, that is \begin{equation*} h^{\#}\bigl(2^{1+\alpha}\bigr)=2^{(1+\alpha)/2+\lfloor\alpha/2\rfloor}\,, \quad f^{\#}\bigl(2^{1+\alpha}\bigr)=\frac{2^{1/2}-1}{3} \bigl[1-(-2)^{1+\alpha}\bigr] \end{equation*} for powers of $2$, and \begin{equation*} h^{\#}\bigl(p^{\alpha}\bigr)=p^{\alpha/2+\lfloor\alpha/2\rfloor}\,, \quad f^{\#}\bigl(p^{\alpha}\bigr) =1+\frac{p-p^{1/2}}{p+1}\bigl[(-p)^{\alpha}-1\bigr] \end{equation*} for powers of odd primes, where $\alpha\geq0$.\newline (ii) It is striking that the $m$-sums of both the absolute value and the real part of $S(m,n)$ yield multiplicative functions after a suitable modification for even $n$. This makes us wonder if there are other functions of the pair $m,n$ with this property.\newline (iii) Although, right now, we do not know truly useful applications of any particular multiplicative functions in the families $h^{(1)}_{f,p}$ and $h^{(2)}_{f,p}$, it is worth recalling that $h(n)$ of \eqref{DEBZ_h} is closely related to a prime-distinguishing function \cite{DEBZ}. Similarly, the value of $h^{\#}(n)$ tells us the squarefree factor $\lambda$ in $n=\lambda\nu^2$, \begin{equation*} \lambda=\left\{ \begin{array}{cl} \displaystyle\biggl(\frac{n}{2h^{\#}(n)}\biggr)^2 & \mbox{if both $n$ and $v_2(n)$ are even,}\\[2.5ex] \displaystyle\biggl(\frac{n}{h^{\#}(n)}\biggr)^2 & \mbox{otherwise,} \end{array}\right. \end{equation*} which is a corollary to the proof of Theorem~\ref{h-sharp}. However, it is only the current lack of an efficient algorithm for the evaluation of the Gauss sum in \eqref{Smn} that prevents $h(n)$ and $h^{\#}(n)$ from being practical tools. \section{Summary} Inspired by the peculiar multiplicative function $h(n)$ in \eqref{DEBZ-h}, we found two mappings that turn a given multiplicative function into other multiplicative functions, with each image function specified by a privileged prime number, a sequence of complex numbers, and a sequence of nonnegative integers. In addition, we reported one more multiplicative function, of a different kind, also suggested by the structure of $h(n)$. \bigskip\noindent\textbf{Acknowledgments.} We sincerely thank Si Min Chan for her role in getting our collaboration started. We are grateful to Ron Evans for indicating the existence of explicit formulas for $S(m,n)$ and his encouragement. B.-G.~E. is grateful for the encouragement by Thomas Durt, Ingemar Bengtsson, and Karol \.Zyczkowski, the co-authors of the 2010 review article~\cite{DEBZ}. The Centre for Quantum Technologies is a Research Centre of Excellence funded by the Ministry of Education and the National Research Foundation of Singapore. \bigskip\noindent\textbf{Dedication.} It is a pleasure to dedicate this article to Professor Bruce Berndt on the occasion of his 80th birthday.

proofpile-arXiv_069-13774

\section{Introduction} \begin{figure}[!ht] \resizebox{\linewidth}{!} { \includegraphics{issues.pdf} } \caption{Compared with the current state-of-the-arts method Pix2Vox++~\cite{Pix2Vox++} and classic method 3D-R2N2~\cite{DBLP:conf/eccv/ChoyXGCS16}, the proposed method are more robust in reconstructing the 3D shape of an object from a single image that contains occlusion or noisy backgrounds.} \label{fig:issues} \end{figure} Reconstructing object 3D shape from a single-view RGB image is a vital but challenging computer vision task in robotics, CAD, and virtual and augmented reality applications. Humans can easily infer the 3D shape of an object from a single image due to sufficient prior knowledge and an innate ability for visual understanding and reasoning. However, this is an extremely difficult and ill-posed problem for a machine vision systems because a single-view image can not provide sufficient information for the object to be reconstructed. Most of the existing learning-based methods for single-view 3D reconstruction extract features from a single RGB image, then transform it into a 3D representation. These methods achieve promising results on the synthetic datasets (ShapeNet~\cite{DBLP:journals/corr/ChangFGHHLSSSSX15}). However, as shown in Figure~\ref{fig:issues}, they usually have trouble reconstructing the 3D shape of an object from real-world images. The performance gap between the real-world and synthetic datasets are caused by the quality of image features. The features extracted from images with noisy backgrounds or heavy occlusions usually contain insufficient useful information for 3D reconstruction. Humans can infer a reasonable 3D shape of an object from a single image, even with incomplete or noisy visual cues. This is due to the fact that humans retrieve similar shapes from their memories and apply these shape priors to recover the shape of hidden and noisy parts of the object. Motivated by human vision, we propose a novel memory-based framework for 3D reconstruction, Mem3D, which consists of four components: image encoder, memory network, LSTM shape encoder, and shape decoder. The proposed Mem3D explicitly constructs the shape priors in the memory network that help complete the missing image features to recover the 3D shape of an object that is heavy occluded or in a complex environment. To construct the shape priors, we design a writing strategy to store the ``image-voxel'' pairs into the memory network in a key-value fashion during training. To retrieve the precise 3D shapes that are highly related to the input image from the memory network, we propose a voxel triplet loss function that guarantees that images with similar 3D shapes are closer in the feature space. To better leverage the retrieved shapes, the LSTM-based shape encoder transforms the useful knowledge of these shapes into a shape prior vector. To employ both information from image and shape priors, the input image features and the output of the LSTM-based shape encoder are concatenated and are forwarded to a decoder to predict the 3D shape of the object. The main contributions are summarized as follows: \begin{itemize} \item We propose a memory-based framework for single-view 3D object reconstruction, named Mem3D. It innovatively retrieves similar 3D shapes from the constructed shape priors, and shows a powerful ability to reconstruct the 3D shape of objects that are heavily occluded or in a complex environment. \item We present a memory network that stores shape priors in the form of ``image-voxel'' pairs. To better organize the shape priors and ensure accurate retrieval, we design novel reading and writing strategies, as well as introducing a voxel triplet loss function. \item Experimental results demonstrate that the proposed Mem3D significantly improves the reconstruction quality and performs favorably against state-of-the-art methods on the ShapeNet and Pix3D datasets. \end{itemize} \section{Related Work} \noindent \textbf{Single-image 3D Reconstruction.} Recently, 3D object reconstruction from a single-view image has attracted increasing attention because of its wide applications in the real world. Recovering object shape from a single-view image is an ill-posed problem due to the limitation of visual clues. Existing works use the representation of silhouettes~\cite{Dibra_2017_CVPR}, shading~\cite{Richter_2015_CVPR}, and texture~\cite{DBLP:journals/ai/Witkin81} to recover 3D shape. With the success of deep learning, especially generative adversarial networks~\cite{DBLP:conf/nips/GoodfellowPMXWOCB14} and variational autoencoders~\cite{DBLP:journals/corr/KingmaW13}, the deep neural network based encoder-decoder has become the main-stream architecture, such as 3D-VAE-GAN~\cite{DBLP:conf/nips/0001ZXFT16}. PSGN~\cite{DBLP:conf/cvpr/FanSG17} and 3DLMNet~\cite{DBLP:conf/bmvc/MandikalLAR18} generate point representations from single-view images. 3D-R2N2~\cite{DBLP:conf/eccv/ChoyXGCS16} is a unified framework for single- and multi-view 3D reconstruction which employs a 3D convolutional LSTM to fuse the image features. To solve the permutation variance issue, Pix2Vox~\cite{Xie_2019_ICCV} employs a context-aware fusion module to adaptively select high-quality reconstructions from single-view reconstructions. However, these works that utilize shape priors implicitly are venerable to noisy backgrounds and heavy occlusions. To reconstruct the 3D shape of an object from real-world images, MarrNet~\cite{DBLP:conf/nips/0001WXSFT17} and its variants~\cite{shapehd,DBLP:conf/nips/ZhangZZTF018} reconstructs 3D objects by estimating depth, surface normals, and silhouettes. Both 3D-RCNN~\cite{DBLP:conf/cvpr/KunduLR18} and FroDO~\cite{FroDo} introduce an object detector to remove noisy backgrounds. \noindent \textbf{Memory Network.} The Memory Network was first proposed in~\cite{weston2014memory}, which augmented neural networks with an external memory module that enables the neural network to store long-term memory. Later works~\cite{DBLP:conf/icml/KumarIOIBGZPS16,DBLP:conf/nips/SukhbaatarSWF15} improve the Memory Network so it can be trained in an end-to-end manner. Hierarchical Memory Networks~\cite{ch2016hierarchical} was proposed to allow a read controller to efficiently access large scale memories. Key-Value Memory Networks~\cite{DBLP:journals/corr/MillerFDKBW16} store prior knowledge in a key-value structured memory, where keys are used to address relevant memories whose corresponding values are returned. \section{Method} \begin{figure*} \resizebox{\linewidth}{!} { \includegraphics{flow.pdf} } \caption{The proposed Mem3D reconstruct the 3D shape of an object from a single input image. The Memory Network learns to retrieve 3D volumes that are highly related to the input image. The LSTM Shape Encoder is proposed to contextually encode multiple 3D volumes into a shape prior vector, which provides the information that helps to recover the 3D shape of the object's hidden and noisy parts.} \label{fig:flow} \end{figure*} In existing single-view 3D reconstruction methods~\cite{Pix2Vox++,wang2018pixel2mesh,AttSets,DBLP:conf/eccv/ChoyXGCS16}, the shape priors are learnt into model parameters, which leads to low quality reconstructions for images containing heavy occlusion and noisy backgrounds. To alleviate this issue, the proposed Mem3D explicitly constructs the shape priors using a Key-Value Memory Network~\cite{DBLP:conf/emnlp/MillerFDKBW16}. Specifically, the image encoder extracts features from the input image. During training, the extracted features and the corresponding 3D shape are then stored in the memory network in a key-value fashion. For both training and testing, the 3D shapes whose corresponding keys have high similarities are forwarded to the LSTM shape encoder. After that, the LSTM shape encoder generates a shape prior vector. Finally, the decoder takes the both image features and the shape prior vector to reconstruct the 3D shape of the object. \subsection{Memory Network} \label{memory} The memory network aims to explicitly construct the shape priors by storing the ``image-voxel'' pairs, which memorize the correspondence between the image features and the corresponding 3D shapes.The memory items are constructed as: [key, value, age], which is denoted as $M = \left\{(\mathbf{K}_i, \mathbf{V}_i, A_i)_{i=1}^m\right\}$, where $m$ denotes the size of the memory. The ``key'' and ``value'' memory slots store the image features and the corresponding 3D volume, respectively. The ``key'' $\mathbf{K}_i \in \mathbb{R}^{n_k}$ is used to compute the cosine similarities with the input image features. The ``value'' $\mathbf{V}_i \in \mathbb{R}^{n_v}$ is returned if the similarity score between the query and the keys of memory exceeds a threshold. The $n_k$ and $n_v$ are dimension of the memory ``key'' and memory ``value'', respectively. The ``age'' $\mathbf{A}_i \in \mathbb{N}$ represents the alive time of the pair, which is to set to zero when the pair is matched by the input image features. The memory network overwrites the ``oldest'' pair when writing new pairs. \subsubsection{Memory Writer} The memory writer is presented to construct the shape priors in the memory network. We designed a writing strategy to determine how to update the memory slots when given the image features $\mathbf{F} \in \mathbb{R}^{n_k}$ and its corresponding volume $\mathcal{V}$. The memory writing only works at training because it takes the ground truth 3D volumes as input, which are not available during testing. In the memory network, the key similarity between the input image features $\mathbf{F}$ and the memory key $\mathbf{K}_i$ is defined as following \begin{equation}\label{eq:key_sim} S_k(\mathbf{F}, \mathbf{K}_i) = { \frac{\mathbf{F} \cdot \mathbf{K}_{i}}{\lVert \mathbf{F} \lVert \lVert \mathbf{K}_{i} \lVert} } \end{equation} Similarly, the value similarity between the corresponding 3D volumes $\mathcal{V}$ and the value $\mathbf{V}_i$ can be defined as \begin{equation}\label{eq:value_sim} S_v(\mathcal{V}, \mathbf{V}_i) = 1 - \frac{1}{r_v^3}\sum_{j=1}^{r_v^3}(\mathbf{V}_i^j-\mathcal{V}^j)^2 \end{equation} where $r_v$ indicates the resolution of the 3D volume. The writing strategy works for the two cases according to whether the similarity satisfies $S_v(\mathcal{V}, \mathbf{V}_{n_1}) > \delta$, where $\delta$ is the similarity threshold and $n_1$ is determined by \begin{equation} n_1=\arg \max_i S_k(\mathbf{F}, \mathbf{K}_i) \end{equation} \noindent \textbf{Strategy for Similar Examples ($S_v(\mathcal{V}, \mathbf{V}_{n_1}) \ge \delta$).} For a similar example, the value $\mathbf{V}_{n_1}$ kept unchanged, while the age $A_{n_1} = 0$ and the key $\mathbf{K}_{n_1}$ is updated as follows: \begin{equation} \mathbf{K}_{n_1} = \frac{\mathbf{F} + \mathbf{K}_{n_1}}{\lVert \mathbf{F} + \mathbf{K}_{n_1} \lVert} \end{equation} After the memory update, the ages are adjusted as $A_i = A_i + 1$ ($i \neq n_1$). \noindent \textbf{Strategy for New Examples ($S_v(\mathcal{V}, \mathbf{V}_{n_1}) < \delta$).} For a new example, the memory writer stores it to the memory network as following \begin{align} \mathbf{K}_{n_o} = \mathbf{F} \\ \mathbf{V}_{n_o} = \mathcal{V} \\ A_{n_o} = 0 \end{align} where $n_o$ is determined by \begin{equation} n_o = \arg \max_i(A_i) \end{equation} if there are no empty slots in the memory network. Otherwise, $n_o$ can be the index of any empty memory slots. After the memory update, the ages are adjusted as $A_i = A_i + 1$ ($i \neq n_o$). \subsubsection{Memory Reader} The memory reader is used for reading the values from the memory network and outputs a value sequence containing 3D volumes that are highly related to the input image features. For different input image features, there are different numbers of highly similar shapes in the memory. Therefore, retrieving a fixed number of shapes from the memory network would not be suitable for all inputs and may introduce irrelevant shapes. To solve this problem, we construct the retrieved value sequence by concatenating all values whose key satisfies $S_k(\mathbf{F}, \mathbf{K}_{n_i}) > \beta $, which can be formulated as \begin{equation} \mathbb{V} = \left[V_{n_i} | S_k(\mathbf{F}, \mathbf{K}_{n_i}) > \beta \right] \label{eq:retrieve} \end{equation} where $\beta$ is threshold and $[\cdot]$ denotes the concatenation. \subsection{LSTM Shape Encoder} The value sequence $\mathbb{V}$ retrieved by the memory reader contains 3D shapes that are similar to the object in the input image. The value sequence from the memory reader is length-variant and has been ordered by the similarities. Intuitively, different parts of different shapes in the value sequence may have a different importance in reconstructing the 3D shape from the current image. To contextually consider and incorporate knowledge useful for current reconstruction from the value sequence into the image feature to supplement the occluded or noisy parts, we leverage LSTM~\cite{lstm} to encode the value sequence $\mathbb{V}$ in a sequential manner. The LSTM shape encoder takes the length-variant value sequence as input and outputs a fixed-length ``shape prior vector''. The ``shape prior vector'' is then concatenated with the input image feature to provide extra useful information for the shape decoder. \subsection{Network Architecture} \noindent\textbf{Image Encoder.} The image encoder contains the first three convolutional blocks of ResNet-50~\cite{resnet} to extract a $512 \times 28^2$ feature map from a $224 \times 224 \times 3$ image. Then the ResNet is followed by three sets of 2D convolutional layers, batch normalization layers and ReLU layers. The kernel sizes of the three convolutional layers are $3^2$, with a padding of 1. There is a max pooling layer with a kernel size of $2^2$ after the second and third ReLU layers. The output channels of the three convolutional layers are 512, 256, and 256, respectively. \noindent\textbf{LSTM Shape Encoder.} The shape encoder is an LSTM~\cite{lstm} network with 1 hidden layer. The hidden size is set to 2,048 which indicates that the output shape prior vector is a 2,048 dimensional vector. \noindent\textbf{Shape Decoder.} The decoder contains five 3D transposed convolutional layers. The first four transposed convolutional layers are of kernel sizes $4^3$, with strides of 2 and paddings of 1. The next transposed convolutional layer has a bank of $1^3$ filter. Each of the first four transposed convolutional layers is followed by a batch normalization layer and a ReLU, and the last transposed convolutional layer is followed by a sigmoid function. The output channel numbers of the five transposed convolutional layers are 512, 128, 32, 8, and 1, respectively. The final output of decoder is a $32^3$ voxelized shape. \subsection{Loss Functions} \noindent \textbf{Voxel Triplet Loss.} We propose a voxel triplet loss that helps to retrieve precise values from the memory network by guaranteeing that images with similar 3D shapes are closer in the feature space. In the memory network, $n_p$ and $n_b$ are the memory slots of the positive and negative samples, respectively. For a positive sample, the similarity between its value $\mathbf{V}_{n_p}$ and the corresponding 3D volume of the input image $\mathcal{V}$ satisfies \begin{equation} S_v(\mathcal{V}, \mathbf{V}_{n_p}) \ge \delta \label{eq:similarity} \end{equation} Similarly, for a negative sample, the similarity satisfies \begin{equation} S_v(\mathcal{V}, \mathbf{V}_{n_b}) < \delta \end{equation} where $\mathbf{V}_{n_b}$ represents the value of the ``image-voxel'' pair for $n_b$. Therefore, the voxel triplet loss can be defined as \begin{equation} \ell_t(S_{kb}, S_{kp}, \alpha) = \max \left( S_{kb} - S_{kp} + \alpha, 0 \right) \label{eq:triplet-loss} \end{equation} where $\alpha$ is the margin in the triplet loss~\cite{TripletLoss}. $S_{kb}$ and $S_{kp}$ are the similarities between the input image features and the keys of the postive/negative sample, which are defined as $S_{kb} = S_k(\mathbf{F},{\mathbf{K}_{n_b})}$ and $S_{kp} = S_k(\mathbf{F},{\mathbf{K}_{n_p})}$, respectively. The proposed voxel triplet loss can minimize the distance among image features with similar 3D volumes and maximize the distance among image features with different 3D volumes. \noindent \textbf{Binary Cross Entropy Loss.} For the reconstruction network, we adopt the Binary Cross Entropy Loss, which is defined as the mean value of the voxel-wise binary cross entropies between the reconstructed object and the ground truth. More formally, it can be defined as \begin{equation} \ell_r(p, gt) = \frac{1}{r_v^3}\sum_{i=1}^{r_v^3} \left[gt_i\log(p_i) + (1 - gt_i)\log(1-p_i)\right] \end{equation} where $p$ and $gt$ denote the predicted 3D volume and the corresponding ground truth, respectively. The Mem3D is trained end-to-end by the combination of the voxel triplet loss and the reconstruction loss: \begin{equation} \ell_{total} = \ell_t+\ell_r \end{equation} \section{Experiments} \begin{table*}[!th] \caption{Comparison of single-view 3D object reconstruction on ShapeNet. We report the per-category and overall IoU at $32^3$ resolution. The best results are highlighted in bold.} \resizebox{\linewidth}{!} { \begin{tabular}{ccccccccc} \toprule Category & 3D-R2N2~\cite{DBLP:conf/eccv/ChoyXGCS16} & OGN~\cite{Tatarchenko_2017_ICCV} & DRC~\cite{DRC} & Pixel2Mesh~\cite{wang2018pixel2mesh}& IM-Net~\cite{IMnet}& AttSets~\cite{AttSets} & Pix2Vox~\cite{Xie_2019_ICCV} & Mem3D\\ \midrule Airplane & 0.513 & 0.587 & 0.571 &0.508 & 0.702&0.594 & 0.674 & $\textbf{0.767}$ \\ Bench & 0.421 & 0.481 & 0.453 &0.379 &0.564 &0.552 & 0.608 & $\textbf{0.651}$ \\ Cabinet & 0.716 & 0.729 & 0.635 & 0.732 &0.680 &0.783 & 0.799 & $\textbf{0.840}$ \\ Car & 0.798 & 0.828 & 0.755 & 0.670 &0.756 &0.844 & 0.858 & $\textbf{0.877}$ \\ Chair & 0.466 & 0.483 & 0.469 & 0.484 & 0.644&0.559 & 0.581 & $\textbf{0.712}$ \\ Display & 0.468 & 0.502 & 0.419 & 0.582 & 0.585& 0.565& 0.548 & $\textbf{0.631}$ \\ Lamp & 0.381 & 0.398 & 0.415 & 0.399 & 0.433& 0.445& 0.457 & $\textbf{0.535}$ \\ Speaker & 0.662 & 0.637 & 0.609 & 0.672 & 0.683&0.721 & 0.721 & $\textbf{0.778}$ \\ Rifle & 0.544 & 0.593 & 0.608 & 0.468 &0.723 & 0.601& 0.617 & $\textbf{0.746}$ \\ Sofa & 0.628 & 0.646 & 0.606 &0.622 & 0.694& 0.703& 0.725 & $\textbf{0.753}$ \\ Table & 0.513 & 0.536 & 0.424 & 0.536 &0.621 &0.590 & 0.620 & $\textbf{0.685}$ \\ Cellphone & 0.661 & 0.702 & 0.413 & 0.762 & 0.762& 0.743& 0.809 & $\textbf{0.823}$ \\ Watercraft & 0.513 & 0.632 & 0.556 & 0.471&0.607 & 0.601& 0.603 & $\textbf{0.684}$ \\ \midrule overall & 0.560 & 0.596 & 0.545 & 0.552 &0.659 & 0.642& 0.670 & $\textbf{0.729}$ \\ \bottomrule \end{tabular} } \label{table:shapenet_IOU} \end{table*} \begin{table*} \caption{Comparison of single-view 3D object reconstruction on ShapeNet. We report the per-category and overall F-Score@1\%. For voxel reconstruction methods, the points are sampled from triangular meshes generated by the marching cube algorithm. The best results are highlighted in bold.} \resizebox{\linewidth}{!} { \begin{tabular}{ccccccccc} \toprule Category & 3D-R2N2~\cite{DBLP:conf/eccv/ChoyXGCS16} & OGN~\cite{Tatarchenko_2017_ICCV} & OccNet~\cite{OccNet} & Pixel2Mesh~\cite{wang2018pixel2mesh}& IM-Net~\cite{IMnet}& AttSets~\cite{AttSets} & Pix2Vox++~\cite{Pix2Vox++} & Mem3D\\ \midrule Airplane & 0.412 & 0.487 & 0.494 & 0.376 & 0.598 & 0.489 & 0.583 & $\textbf{0.671}$ \\ Bench & 0.345 & 0.364 & 0.318 & 0.313 & 0.361 & 0.406 & 0.478 & $\textbf{0.525}$ \\ Cabinet & 0.327 & 0.316 & 0.449 & 0.450 & 0.345 & 0.367 & 0.408 & $\textbf{0.517}$ \\ Car & 0.481 & 0.514 & 0.315 & 0.486 & 0.304 & 0.497 & 0.564 & $\textbf{0.590}$ \\ Chair & 0.238 & 0.226 & 0.365 & 0.386 & 0.442 & 0.334 & 0.309 & $\textbf{0.503}$ \\ Display & 0.227 & 0.215 & 0.468 & 0.319 & 0.466 & 0.310 & 0.296 & $\textbf{0.498}$ \\ Lamp & 0.267 & 0.249 & 0.361 & 0.219 & 0.371 & 0.315 & 0.315 & $\textbf{0.403}$ \\ Speaker & 0.231 & 0.225 & 0.249 & 0.190 & 0.200 & 0.211 & 0.152 & $\textbf{0.262}$ \\ Rifle & 0.521 & 0.541 & 0.219 & 0.340 & 0.407 & 0.524 & 0.574 & $\textbf{0.626}$ \\ Sofa & 0.274 & 0.290 & 0.324 & 0.343 & 0.354 & 0.334 & 0.377 & $\textbf{0.434}$ \\ Table & 0.340 & 0.352 & 0.549 & 0.502 & 0.461 & 0.419 & 0.406 & $\textbf{0.569}$ \\ Cellphone & 0.504 & 0.528 & 0.273 & 0.485 & 0.423 & 0.469 & 0.633 & $\textbf{0.674}$ \\ Watercraft& 0.305 & 0.328 & 0.347 & 0.266 & 0.369 & 0.315 & 0.390 & $\textbf{0.461}$ \\ \midrule Overall & 0.351 & 0.368 & 0.393 & 0.398 & 0.405 & 0.395 & 0.436 & $\textbf{0.517}$ \\ \bottomrule \end{tabular} } \label{table:shapenet_fscore} \end{table*} \subsection{Datasets} \noindent\textbf{ShapeNet.} The ShapeNet dataset~\cite{DBLP:journals/corr/ChangFGHHLSSSSX15} is composed of synthetic images and corresponding 3D volumes. We use a subset of the ShapeNet dataset consisting of 44K models and 13 major categories following~\cite{DBLP:conf/eccv/ChoyXGCS16}. Specifically, we use renderings provided by 3D-R2N2 which contains 24 random views of size $137\times137$ for each 3D model. We also apply random background augmentation~\cite{Pix2Vox++,RenderForCNN} to the image during training. Note that only the ShapeNet dataset is used for training Mem3D. \noindent\textbf{Pix3D.} The Pix3D~\cite{DBLP:conf/cvpr/Sun0ZZZXTF18} dataset contains 395 3D models of nine classes. Each model is associated with a set of real images, capturing the exact object in diverse environments. The most significant category in this dataset is chairs. The Pix3D dataset is used only for evaluation. \subsection{Evaluation Metrics} We apply the intersection over union (IoU) and F-score evaluation metrics widely used by existing works. The IoU is formulated as \begin{equation} \text{IoU} = \frac{\sum_{i,j,k}\mathbb{I}(p(i,j,k)>t)\mathbb{I}(gt(i,j,k))}{\sum_{i,j,k}\mathbb{I}[\mathbb{I}(p(i,j,k)>t)+\mathbb{I}(gt(i,j,k))]} \end{equation} where $p(i,j,k)$ and $gt(i,j,k)$ indicate predicted occupancy probability and ground-truth at \emph{(i,j,k)}, respectively. $\mathbb{I}$ is the indication function which will equal to one when the requirements are satisfied. The $t$ denotes a threshold, $t=0.3$ in our experiments. Following Tatarchenko et al. \cite{Oracle}, we also take F-Score as an extra metric to evaluate the performance of 3D reconstruction results, which can be defined as \begin{equation} \textnormal{F-Score}(d) = \frac{2P(d)R(d)}{P(d) + R(d)} \end{equation} where $P(d)$ and $R(d)$ denote the precision and recall with a distance threshold $d$, respectively. $P(d)$ and $R(d)$ are computed as \begin{equation} P(d) = \frac{1}{n_{\mathcal{R}}} \sum_{r \in \mathcal{R}} \left[\min_{g \in \mathcal{G}} ||g - r|| < d \right] \end{equation} \begin{equation} R(d) = \frac{1}{n_{\mathcal{G}}} \sum_{g \in \mathcal{G}} \left[\min_{r \in \mathcal{R}} ||g - r|| < d \right] \end{equation} where $\mathcal{R}$ and $\mathcal{G}$ represent the predicted and ground truth point clouds, respectively. $n_\mathcal{R}$ and $n_\mathcal{G}$ are the number of points in $\mathcal{R}$ and $\mathcal{G}$, respectively. To adapt the F-Score to voxel models, like existing works~\cite{Pix2Vox++}, we apply the marching cube algorithm~\cite{DBLP:conf/siggraph/LorensenC87} to generate the object surface, then 8,192 points are sampled from the surface to compute F-Score between predicted and ground truth voxels. A higher IoU and F-Score indicates better reconstruction results. \subsection{Implementation Details} We used $224\times224$ RGB images as input to train the Mem3D with a batch size of 32. The whole network is trained end-to-end with the Adam optimizer with a $\beta_1$ of 0.9 and a $\beta_2$ of 0.999. The initial learning rate is set to 0.001 and decayed by 2 after 150 epochs. In the memory network, the size is $m = 4000$. The margin $\alpha$ in Equation \eqref{eq:triplet-loss} is set to $0.1$. The thresholds $\beta$ and $\delta$ in Equations \eqref{eq:retrieve} and \eqref{eq:similarity} are set to $0.85$ and $0.90$, respectively. The source code will be publicly available. \subsection{Object Reconstruction on ShapeNet} We compare the performance with other state-of-the-art methods on the ShapeNet testing set. Tables~\ref{table:shapenet_IOU} and~\ref{table:shapenet_fscore} show the IoU and F-Score@1\% of all methods, respectively, which indicates that Mem3D outperforms all other competitive methods with a large margin in terms of both IoU and F-Score@1\%. Our Mem3D benefits from the memory network which explicitly constructs shape priors and applies them according to an object's individual needs to improve reconstruction quality. \subsection{Object Reconstruction on Pix3D} Pix3D is a more challenging benchmark which contains diverse real-world images and corresponding shapes. In Pix3D, the `chair' category contains 3,839 images and the corresponding 3D models, which are the largest category of the dataset. Due to the complicated environment in images, the objects are frequently occluded by surroundings or themselves. Therefore, most of the previous works~\cite{Xie_2019_ICCV,DBLP:conf/cvpr/Sun0ZZZXTF18,Wu_2018_ECCV} evaluate their approaches using the hand-selected 2,894 untruncated and unoccluded `chair' images to guarantee their models can capture enough information from the images. However, although this avoids the occlusion problem to some extent by selecting unoccluded testing samples, the previous reconstruction models still perform imperfectly because of the complicated background. To show the superior ability to reconstruct objects with the occlusion and background issues, we evaluate Mem3D on the 2,894 chair images with less occlusions but complicated backgrounds and 945 chair images (the complementary set) with heavy occlusions. Note that the Mem3D is trained on the ShapeNet ``chair'' training set and evaluate on the Pix3D chair set. Since the memory network only writes ``image-voxel'' pairs during training, the memory network only contains shape priors extracted from the ShapeNet dataset. \subsubsection{Reconstruction with Complicated Backgrounds} \label{Sec:Reconstruction on Complicated Backgrounds} Table~\ref{pix3d1} shows the evaluation performance of Mem3D and other works on the 2,894 untruncated and unoccluded `chair' images. Note that these methods use different types of extra information. For instance, MarrNet~\cite{DBLP:conf/nips/0001WXSFT17}, DRC~\cite{DRC} and ShapeHD~\cite{Wu_2018_ECCV} use extra depth, surface normals and silhouettes information. The proposed Mem3D outperforms the state-of-the-art methods by a large margin in terms of both IoU and F-Score@1\%. The reconstruction results of our Mem3D and previous state-of-the-art works `Pix2Vox++'~\cite{Pix2Vox++} and `Pix3D'~\cite{DBLP:conf/cvpr/Sun0ZZZXTF18} are shown in Figure~\ref{fig:pix3dv1}. Compared to `Pix2Vox++'~\cite{Pix2Vox++} which employs an encoder-decoder structure, Mem3D can produce more clean and complete reconstruction results. The reconstructions from Mem3D also provide more details compared to other models. The memory network in Mem3D can explicitly store and utilize 3D volumes thus providing the reconstruction network with more detailed information about the object and eliminating the background noise. \begin{figure}[!t] \centering \includegraphics[width=1\linewidth]{pix3dv1.pdf} \caption{Reconstruction result on 5 of the 2,894 untruncated and unoccluded `chairs' in Pix3D. GT indicates the ground-truth.} \label{fig:pix3dv1} \end{figure} \begin{table} \caption{Comparison of single-view 3D object reconstruction on 2,894 untruncated and unoccluded `chair' images in Pix3D. We report IoU and F-Score@1\%. The best performance is highlighted in bold.}\label{pix3d1} \vspace{1mm} \begin{tabularx}{\linewidth}{l|YY} \toprule Method & IoU & F-Score@1\%\\ \midrule 3D-R2N2\cite{DBLP:conf/eccv/ChoyXGCS16} & 0.136 & 0.018 \\ 3D-VAE-GAN~\cite{DBLP:conf/nips/0001ZXFT16} & 0.171 & - \\ MarrNet~\cite{DBLP:conf/nips/0001WXSFT17} & 0.231 & 0.026\\ DRC~\cite{DRC} & 0.265 & 0.038\\ ShapeHD~\cite{Wu_2018_ECCV} & 0.284 & 0.046\\ DAREC~\cite{Pinheiro_2019_ICCV} & 0.241 & - \\ Pix3D~\cite{DBLP:conf/cvpr/Sun0ZZZXTF18} & 0.282 & 0.041 \\ Pix2Vox++~\cite{Pix2Vox++} & 0.292 & 0.068\\ FroDo~\cite{FroDo} & 0.325 & - \\ Mem3D & \textbf{0.387} & \textbf{0.143}\\ \bottomrule \end{tabularx} \vspace{-5mm} \end{table} \subsubsection{Reconstruction with Heavy Occlusions} The occlusion issue is another key difficulty for single-view object reconstruction. Table~\ref{pix3d2} shows the evaluation performance of Mem3D and other works on the 945 chair images with heavy occlusions. The performance of all other methods drop significantly compared to Section~\ref{Sec:Reconstruction on Complicated Backgrounds}. While Mem3D shows a favorable ability to handle the extremely cases. Figure~\ref{fig:pix3dv2} shows some reconstruction results of our Mem3D, `Pix2Vox++'~\cite{Pix2Vox++}, and 3D-R2N2~\cite{DBLP:conf/eccv/ChoyXGCS16}. It can be observed that Pix2Vox++ can reconstruct the perfectly presented parts of object in the image, but failed to reconstruct the occluded parts. Our Mem3D can provide reasonable reconstruction even for object parts that are hidden in the image. This is because Mem3D not only captures object information from images, but also obtains complete and clean shape information from the shapes read from memory. The retrieved shapes can provide detailed and complete shape information for the reconstruction network. \begin{figure}[!t] \centering \includegraphics[width=1\linewidth]{pix3dv2.pdf} \caption{Reconstruction result on 5 of the 945 `chair' images with heavy occlusions in Pix3D. GT indicates the ground-truth.} \label{fig:pix3dv2} \end{figure} \begin{table} \caption{Comparison of single-view 3D object reconstruction on 945 `chairs' with heavy occlusions in Pix3D. We report IoU and F-Score@1\%. The best performance is highlighted in bold.} \vspace{1mm} \begin{tabularx}{\linewidth}{l|YY} \toprule Method & IoU & F-Score@1\%\\ \midrule 3D-R2N2\cite{DBLP:conf/eccv/ChoyXGCS16} & 0.055 & 0.011 \\ MarrNet~\cite{DBLP:conf/nips/0001WXSFT17} & 0.138 & 0.019\\ DRC~\cite{DRC} & 0.151 & 0.025\\ ShapeHD~\cite{Wu_2018_ECCV} & 0.183 & 0.037\\ Pix2Vox++~\cite{Pix2Vox++} & 0.215 & 0.041\\ Mem3D & \textbf{0.336} & \textbf{0.105}\\ \bottomrule \end{tabularx} \vspace{-5mm} \label{pix3d2} \end{table} \begin{figure*}[!th] \centering \includegraphics[width=1\linewidth]{retrieved.pdf} \caption{An illustration of the retrieved 3D volumes and the corresponding reconstructions. ``w/o Memory'‘ indicates the reconstruction results are generated without the memory network. We only show the top-4 high-relative 3D volumes retrieved from the memory network. GT indicates the ground-truth.} \label{fig:retrieved} \vspace{-1mm} \end{figure*} \subsection{Ablation Study} In this section, we evaluate the importance of individual components by ablation studies. \noindent\textbf{Memory Network.} To quantitatively evaluate the memory network, we remove the memory network and directly employ the image encoder and decoder as the baseline model. To further prove the effectiveness of the proposed voxel triplet loss $\ell_t$ in \eqref{eq:triplet-loss}, which pulls image features with similar 3D shapes closer, we remove the voxel triplet loss $\ell_t$ from Mem3D training stage. The comparison results are shown in Table~\ref{ablation1}, which demonstrates the memory network and the proposed voxel triplet loss contribute significant improvement. We also show the reconstruction results of the baseline model (without the memory network) and our Mem3D as well as the retrieved shapes in Figure~\ref{fig:retrieved}. The baseline model which reconstruct the 3D shape of an object from a single image captured in complicated environments are vulnerable to noisy backgrounds and occlusions. Our proposed Mem3D not only obtains object shapes from images, but also can access complete and clean relevant shapes during reconstruction. The proposed Mem3D makes it possible to reconstruct the hidden parts of the object in the image and significantly improve the reconstruction quality. \begin{table} \caption{The effect of the memory network and the voxel triplet loss. The best results are highlighted in bold. `m' indicates the memory size and $\ell_t$ indicates the voxel triplet loss in Equation~\eqref{eq:triplet-loss}.} \vspace{1mm} \begin{tabularx}{\linewidth}{l|YY} \toprule Method & IoU & F-Score@1\%\\ \midrule w/o memory network & 0.273 & 0.042 \\ \midrule m = 1000 & 0.366 & 0.113\\ m = 2000 & 0.372 & 0.135\\ m = 4000 w/o $\ell_t$ &0.359 &0.111 \\ m = 4000 & \textbf{0.387} & \textbf{0.143}\\ \bottomrule \end{tabularx} \vspace{-4mm} \label{ablation1} \end{table} \begin{table} \caption{Different ways of leveraging retrieved shapes. `Top-1' indicates directly treating the first retrieved shape as the reconstruction result. The best results are highlighted in bold.} \vspace{1 mm} \begin{tabularx}{\linewidth}{l|YY} \toprule Method & IoU & F-Score@1\%\\ \midrule Top-1 & 0.287 & 0.051 \\ Average Fusion & 0.363 & 0.125 \\ LSTM Shape Encoder & \textbf{0.387}& \textbf{0.143} \\ \bottomrule \end{tabularx} \vspace{-5mm} \label{ablation2} \end{table} \noindent\textbf{LSTM Shape Encoder.} With the memory network in hand, we have different choices to leverage the retrieved shapes. For instance, we can directly use the Top-1 retrieved shape as reconstruction result, which is similar to retrieval-based reconstruction~\cite{Oracle}. We can also use average fusion or the LSTM~\cite{lstm} network to encode useful knowledge from retrieved shapes into a fixed-length vector to condition the decoder. Table~\ref{ablation2} shows the reconstruction performance when using different ways to leverage the retrieved shapes. We can also observe the retrieved shapes in Figure~\ref{fig:retrieved}. The Top-1 retrieved shape has similar overall appearance compared to the ground-truth object shape, but the details are very different. The proposed Mem3D uses the image and the retrieved shapes together to provide high-quality reconstructions, which contains unique details that are distinct from the retrieved 3D volumes. \vspace{-3mm} \section{Conclusion} \vspace{-2mm} In this paper, we propose a novel framework for 3D object reconstruction, named Mem3D. Compared to the existing methods for single-view 3D object reconstruction that directly learn to transform image features into 3D representations, Mem3D constructs shape priors that are helpful to complete the missing image features to recover the 3D shape of an object that is heavy occluded or in a complex environment. Experimental results demonstrate that Mem3D significantly improves the reconstruction quality and performs favorably against state-of-the-art methods on the ShapeNet and Pix3D datasets. {\small \bibliographystyle{ieee_fullname}

proofpile-arXiv_069-13840

\section{Introduction} Ever since the discovery of the muon and finding an empirical ``muon number'' that is separate from the electron number, the issue of lepton number violation has been pursued. Extending to the third generation of leptons, the B factory era closed with the bound~\cite{PDG}, \begin{align} {\cal B}(\tau \to \mu\gamma) < 4.4 \times 10^{-8}, \quad\quad ({\rm PDG18}) \label{taumugam_PDG} \end{align} which is from the BaBar experiment~\cite{Aubert:2009ag} and based on $\sim 0.96 \times 10^9$ $\tau$ decays. The Belle experiment has an earlier result~\cite{Hayasaka:2007vc} at $4.5 \times 10^{-8}$, based on $\sim 0.48 \times 10^9$ $\tau$ decays, but somehow has not updated. The Belle~II experiment, which has commenced B physics running, aims at improving the bound by a factor of 100, which we take conservatively as $10^{-9}$~\cite{Kou:2018nap}. Thus, there is potential for discovery in the coming decade. \begin{figure*}[t] \center \includegraphics[width=0.25 \textwidth]{tau_2_mu_gamma_1Loop.pdf} \hskip0.35cm \includegraphics[width=0.25 \textwidth]{tau_2_mu_gamma_2Loop.pdf} \hskip0.35cm \includegraphics[width=0.25 \textwidth]{tau_2_mu_gamma_2Loop-W.pdf} \caption{ One-loop, two-loop fermion and two-loop $W$ diagrams for $\tau\to \mu\gamma$.} \label{feyndiag} \end{figure*} The discovery of the 125 GeV scalar boson $h$~\cite{h125_discovery} completes the last piece of the Standard Model (SM), and is a triumph of the Large Hadron Collider (LHC). With LHC Run 1 data at 8 TeV collision energy, the CMS experiment found~\cite{Khachatryan:2015kon} an intriguing 2$\sigma$ hint for the $\tau$ lepton flavor violating ($\tau$LFV) $h \to \tau\mu$ process, which subsequently disappeared~\cite{Sirunyan:2017xzt} with 13 TeV data at Run~2, \begin{align} {\cal B}(h \to \tau\mu) < 0.25 \%. \quad\quad ({\rm CMS18}) \label{h-taumugamCMS} \end{align} Recently, with similar amount of data at $\sim 36$ fb$^{-1}$, the ATLAS experiment reported~\cite{Aad:2019ugc} a consistent bound of 0.28\%. As this is still less than 1/3 of the full Run~2 data at hand for each experiment, updates are expected. Furthermore, the scheduled Run~3 for 2021--2024 would likely add twice more data than Run~2. Thus, scaling naively by statistics, and assuming that ATLAS and CMS would make a combined analysis before the start of High Luminosity LHC (HL-LHC) targeted for 2028 --- especially if there is some hint! --- the limit could reach 0.05\%, with corresponding discovery potential. Thus, there is much to look forward to in the coming decade on the $\tau$LFV front. This paper aims at elucidating the relevant contributions and parameters of importance, enhancing what has been discussed already. To have $\tau\mu h$ couplings, the framework is a two Higgs doublet model (2HDM)~\cite{Branco:2011iw} without a $Z_2$ symmetry to forbid flavor changing neutral Higgs (FCNH) couplings, which was dubbed ``Model III''~\cite{Hou:1991un} (distinct from Models I \& II under $Z_2$ symmetry) of 2HDM a long time ago. There is a vast amount of theory work on $\tau$LFV that we cannot possibly do justice to, and we refer to the recent mini-reviw of Vicente~\cite{Vicente:2019ykr}. Instead, let us trace some major steps in the phenomenological development. The template for discussing $\tau \to \mu\gamma$ decay can be traced to the work of Chang, Hou and Keung~\cite{Chang:1993kw}, which studied the $\mu \to e\gamma$ transition in the context of 2HDM~III. The paper stressed that the top contribution to the two-loop Bjorken-Weinberg (or Barr-Zee) mechanism, by bringing in the intrinsically larger extra top Yukawa coupling, can be much larger than the one-loop effect (middle and left diagrams of Fig.~\ref{feyndiag}). One just changes the formulas from $\mu \to e$ labels to $\tau\to \mu$, which was followed by all subsequent workers. The $h \to \tau\mu$ process was proposed by Han and Marfatia~\cite{Han:2000jz} at the start of Tevatron Run II, also in the context of 2HDM III. As the Tevatron era was coming to an end, and at the dawn of the LHC, Davidson and Grenier~\cite{Davidson:2010xv} took interest in $h \to \tau\mu$ at colliders, and emphasized the link with $\tau \to \mu\gamma$ bound from B factories as an important constraint. The work, however, was oriented towards the lepton perspective. Extending from earlier and more general work~\cite{Davidson:2005cw}, the authors defined $\tan\beta_\tau = \rho_{\tau\tau}/\lambda_\tau$, where $\rho_{\tau\tau}$ is the extra diagonal $\tau$ Yukawa coupling, and $\lambda_\tau = \sqrt2 m_\tau/v$ ($v \cong 246$ GeV) is the $\tau$ Yukawa coupling of SM, and used $\tan\beta_\tau$ in place of the familiar $\tan\beta$ of 2HDM with $Z_2$ (e.g. the well known 2HDM II). Knowing that, without a $Z_2$ symmetry, $\tan\beta$ as the ratio of v.e.v.'s of the two Higgs doublets is not a physical parameter, the authors sought substitute in language and usage, but it should be clear that the ratio of Yukawa couplings is quite a different thing. The authors further extended $\tan\beta_\tau$ into the quark sector, which is a strong assumption. Adopting this, the early work of Aristizabal Sierra and Vicente~\cite{Sierra:2014nqa} in addressing the CMS hint of $h \to \tau\mu$ excess~\cite{Khachatryan:2015kon} allowed $\tan\beta_\tau = \rho_{\tau\tau}/\lambda_\tau$ to be as large as 40, i.e. the extra $\tau$ Yukawa coupling could be almost half the strength of the top Yukawa coupling. We will not take this lepton-biased view, and let extra top Yukawa couplings be independent parameters. The CMS study that showed excess~\cite{Khachatryan:2015kon} was in fact inspired by the work of Harnik, Kopp and Zupan~\cite{Harnik:2012pb}. While using the formulas of Ref.~\cite{Chang:1993kw} as usual to study the $\tau \to \mu\gamma$ constraint on the $\tau\mu h$ coupling, they showed that a direct search for $h \to \tau\mu$ at the LHC would quickly become more sensitive. The paper, however, used the language of Cheng and Sher~\cite{Cheng:1987rs}, which was adopted also in the CMS papers. While capturing the mass-mixing hierarchy suppression (Model III~\cite{Hou:1991un}) of FCNH for low energy processes, the Cheng-Sher ansatz missed one element, that the FCNH couplings are associated with the exotic (non-mass-giving) Higgs doublet, and would enter the coupling of the SM-like $h$ to e.g. $\tau\mu$ by the $h$--$H$ mixing angle between the two $CP$-even scalars. Thus, the $\tau\mu h$ coupling reads as $\rho_{\tau\mu}\cos(\beta-\alpha)$, where for the time being we retain the familiar notation of 2HDM~II. The latter approach was adopted by Omura, Senaha and Tobe~\cite{Omura:2015xcg} in correlating $h \to \tau\mu$ excess with predictions for $\tau\to \mu\gamma$, where they entertained $\rho_{\tau\mu}$, $\rho_{\tau\tau}$ up to $10\lambda_\tau$ for $c_{\beta-\alpha} \equiv \cos(\beta-\alpha) \simeq 0.1$. The point is, when the CMS excess disappeared with more data, it could just be due to the smallness of $c_{\beta-\alpha}$ (the phenomenon of {\it alignment}), rather than demanding $\rho_{\tau\mu}$ to be small. Turning this around, the proposed search~\cite{Hou:2019grj} for $H,\, A \to \tau\mu$ (where $A$ is the pseudoscalar) is not suppressed by alignment, or small $c_{\beta-\alpha}$. The process has now already been searched for by CMS~\cite{Sirunyan:2019shc}, setting bounds. We have mentioned quite a few parameters in our retracing of the development of $h \to \tau\mu$ and $\tau\to \mu\gamma$ decay studies. The main goal of this paper is to elucidate the relevant vs less relevant parameters, as the coming decade unfolds for the search of these two important $\tau$LFV processes, to clarify the landscape. Another motivation arose from the recent $H, A \to \tau\mu$ study~\cite{Hou:2019grj}, where constraints on $\rho_{tt}$ (extra top Yukawa coupling that enters $\tau \to \mu\gamma$ at two-loop) and $\rho_{\tau\mu}$ from e.g. $\tau \to \mu\gamma$ was extracted by assuming $\rho_{tt}$ to be real, ``for simplicity''. While this is a common, prevailing assumption, but just a couple of years prior, and before the hint for $h\to \tau\mu$ evaporated, it was pointed out~\cite{Fuyuto:2017ewj} that the complexity of $\rho_{tt}$ {\it could drive} the Baryon Asymmetry of the Universe (BAU). With such big issues at stake, this paper explores the possible effect of $\varphi_{tt} = {\rm arg}\, \rho_{tt}$, which has not been explored before. We shall call 2HDM III, or 2HDM without $Z_2$ symmetry and where extra Yukawa couplings are allowed, the general 2HDM (g2HDM). \section{ Parameters and Formulas in the General 2HDM} In this paper we will take the masses of the physical $CP$-even scalars $h$, $H$, $CP$-odd scalar $A$, and charged scalar $H^+$ as given, and would not be concerned with details of the Higgs potential, which can be found e.g. in Ref.~\cite{Hou:2017hiw}. The Yukawa couplings are~\cite{Davidson:2005cw,Hou:2017hiw} \begin{align} \mathcal{L} = - & \frac{1}{\sqrt{2}} \sum_{f = u, d, \ell} \bar f_{i} \Big[\big(\lambda^f_i \delta_{ij} s_\gamma + \rho^f_{ij} c_\gamma\big) h \nonumber\\ & + \big(\lambda^f_i \delta_{ij} c_\gamma - \rho^f_{ij} s_\gamma\big)H - i\,{\rm sgn}(Q_f) \rho^f_{ij} A\Big] R\, f_{j} \nonumber\\ & - \bar{u}_i\left[(V\rho^d)_{ij} R-(\rho^{u\dagger}V)_{ij} L\right]d_j H^+ \nonumber\\ &- \bar{\nu}_i\rho^\ell_{ij} R \, \ell_j H^+ +{h.c.}, \label{eff} \end{align} where $i$, $j$ are generation indices that are summed over, $L, R = (1\mp\gamma_5)/2$ are projection operators, and $V$ is the Cabibbo-Kobayashi-Maskawa matrix. Due to the very near degeneracy of the neutrinos for our processes, the corresponding matrix in lepton sector is taken as unity. The shorthand notation of $c_\gamma \equiv \cos\gamma$ (and $s_\gamma \equiv \sin\gamma$) is the $h$--$H$ mixing angle, which corresponds to the usual $\cos(\beta - \alpha)$ in 2HDM II nomenclature. The emergent {\it alignment} phenomenon, that $h$ so closely resembles the SM Higgs boson~\cite{Khachatryan:2016vau}, implies that $c_\gamma$ is rather small. But we do not quite know its value, which is especially true in g2HDM, where more parameters exist compared with 2HDM II. In the alignment limit of $c_\gamma \to 0$, the couplings of $h$, including to vector bosons, do approach SM. But as shown in Ref.~\cite{Hou:2017hiw}, small $c_\gamma$ need not imply small Higgs quartic couplings. Thus, the prerequisite~\cite{Fuyuto:2017ewj} of ${\cal O}(1)$ Higgs quartics for sake of first order electroweak phase transition for generating BAU, can be sustained. The off-diagonal coupling $\rho_{\tau\mu}$ (and $\rho_{\mu\tau}$) enters the $\tau\to \mu\gamma$ and $h\to \tau\mu$ processes of interest. Note that the first FCNH parameter studied directly at the LHC is $\rho_{tc}$ via $t \to ch$ decay~\cite{PDG}, which was pointed out already in Ref.~\cite{Hou:1991un} and reemphasized~\cite{Chen:2013qta} shortly after the $h(125)$ discovery. Whether $h\to \tau\mu$ or $t \to ch$, the SM-like $h$ boson picks up the FCNH coupling via a factor of $c_\gamma$, or $h$-$H$ mixing. From hindsight, as discussed in Ref.~\cite{Hou:2017hiw}, the alignment phenomenon that emerged with full Run~1 data can account for the absence so far of $t\to ch$ and $h \to \tau\mu$, without the need of overly suppressing extra FCNH Yukawa couplings $\rho_{tc}$ or $\rho_{\tau\mu}$. But since \begin{align} {\cal B}(h \to \tau\mu) = \frac{m_h c_\gamma^2}{16\pi\Gamma_h} (|\rho_{\tau\mu}|^2 + |\rho_{\mu\tau}|^2), \end{align} the bound of Eq.~(\ref{h-taumugamCMS}) places the constraint of \begin{align} |\rho_{\tau\mu}\, c_\gamma| \lesssim 0.0014 \simeq 0.14\lambda_\tau, \label{B_h-taumu} \end{align} where we have taken $|\rho_{\mu\tau}| = |\rho_{\tau\mu}|$ to simplify. The two chiral couplings do not interfere. As elucidated by Davidson and Grenier~\cite{Davidson:2010xv} (from the template of Ref.~\cite{Chang:1993kw} for $\mu \to e\gamma$), there are three distinct types of diagrams contributing to $\tau\to \mu\gamma$: the one-loop diagram that pairs the necessary FCNH $\rho_{\tau\mu}$ coupling with a diagonal $\tau$ Yukawa coupling, be it the $\lambda_\tau$ of SM, or the extra $\rho_{\tau\tau}$; the two-loop Bjorken-Weinberg/Barr-Zee type of diagrams with top Yukawa, be it $\lambda_t$, or $\rho_{tt}$; and the two-loop $W$ diagram. The three type of diagrams are illustrated in Fig.~\ref{feyndiag}. The $H^+$ effect is unimportant. In these diagrams, we have labeled the vertices with the compact notation of Ref.~\cite{Omura:2015xcg} (similar to Davidson and Grenier), $-y_{\phi ij}^f \bar f_iRf_j\phi$ (h.c. implied) for $f = u, d, \ell$ and $\phi = h, H, A$, where $y_{\phi ij}^f$ can be read off from Eq.~(\ref{eff}). One can now see the two-loop mechanism constitutes an insertion of $\phi \to \gamma\gamma$ (we shall neglect the $Z$ contribution), which is similar to the $gg$ fusion production of $\phi$, hence connecting with the $h \to \tau\mu$ and $H, A \to \tau\mu$ searches. The branching fraction for $\tau\to\mu\gamma$ can be written as \begin{equation} \frac{{\cal B}(\tau\to\mu\gamma)}{{\cal B}(\tau\to\mu\nu\bar\nu)} = \frac{48\pi^3\alpha}{G_F^2}\left(|A_L|^2+|A_R|^2\right), \end{equation} where ${\cal B}(\tau\to\mu\nu\bar\nu)= 17.39\%$ \cite{PDG}, and the chiral amplitudes $A_L$ and $A_R$, which do not interfere, contribute equally under our simplifying assumption of $\rho_{\tau\mu} = \rho_{\mu\tau}$. The $A_L$ amplitude corresponding to the three type of diagrams in Fig.~\ref{feyndiag} are (three separate sums) \begin{widetext} \begin{equation}\begin{aligned} A_{L} \simeq & \sum_{\phi=h, H, A } \frac{\hat y_{\phi \tau \mu}^{\ast} \hat y_{\phi \tau \tau}^{\ast }}{8 \pi^{2}v^2} x_{\tau\phi} \bigl(\log x_{\phi\tau}- {3}/{2}\bigr) - \sum_{\phi=h, H, A } \frac{N_{C} Q_{f}^2 \alpha}{4 \pi^{3} v^2} {\hat y_{\phi\tau\mu}^{\ell \,\ast}} \Bigl[\operatorname{Re} (\hat y_{\phi tt}) F_{H}(x_{t\phi}) - i\operatorname{Im}(\hat y_{\phi tt}) F_{A}(x_{t \phi})\Bigr] \\ & + \sum_{\phi=h, H} \frac{ \alpha\,\tilde g_{\phi W W}\, \hat y_{\phi\tau\mu}^{\ast}}{32\sqrt2 \,\pi^3 v^2} \Bigl\{12 F_{H}(x_{W\phi})+{23} F_{A}(x_{W \phi})+{3} G(x_{W_{\phi}})+2x_{\phi W}\bigl[F_{H}(x_{W\phi})-F_{A}(x_{W\phi})\bigr]\Bigr\}\,, \end{aligned}\end{equation} \end{widetext} where $\hat y_{\phi \tau j} = y_{\phi \tau j}/\lambda_\tau$ (and likewise $\hat y_{\phi tt} = y_{\phi tt}/\lambda_t$), $x_{a b} = m_a^2/m_b^2$, $N_c$ is the number of colors, $\tilde{g}_{h WW} = s_\gamma$, $\tilde{g}_{H WW} = c_\gamma$, and the loop functions $F_H$, $F_A$ and $G$ can be found in, e.g. Ref.~\cite{Chang:1993kw}. We include only the $\phi\gamma\gamma$ vertex contributions and neglect $\phi Z\gamma$ vertex terms, as these are suppressed by $(1-4 \sin^2\,\theta_W)$, which amounts to $\sim 10\%$ variation in our results. The $b$ and $\tau$ contributions in the second sum are suppressed by loop functions, as $x_{b\phi}$ and $x_{\tau\phi}$ are rather small. \begin{figure*}[t] \center \includegraphics[width=0.425 \textwidth]{comparison_plot.pdf} \includegraphics[width=0.453 \textwidth]{comparison_plot_zoomed.pdf} \caption{ Comparison of the three benchmark scenarios (see text for details). Black curves are for the degenerate case of $m_H = m_A$. Red (blue) curves show variation in $m_H \,(m_A)$, with the other scalar mass $m_A\,(m_H)$ heavier by $ 10, 100, 200 $ GeV. } \label{bench} \end{figure*} We find that the extra Yukawa couplings can always be normalized against the Yukawa couplings in SM, namely \begin{align} \hat\rho_{3j}^f = \rho_{3j}^f/\lambda_3^f, \label{rho-hat} \end{align} and perhaps Nature hints at such ``normalization''. After all, the extra Yukawa matrix {\boldmath $\rho$}$^f$ can be viewed as the orthogonal combination of two Yukawa matrices with respect to the mass matrix. Along this thread, we shall take throughout this work \begin{align} \rho_{32}^f, \rho_{33}^f = {\cal O}(\lambda_3^f), \label{rho-order} \end{align} as our working assumption, which is the most reasonable one without tuning, given that $\lambda_3^f$ and {\boldmath $\rho$}$^f$ emerge from the procedure of diagonalizing the mass matrix. The two Yukawa matrices should share the mass-mixing hierarchy structure~\cite{Hou:2017hiw}. For this reason, we illustrate with $\rho_{\tau\mu}$ and $\rho_{\tau\tau}$ values not exceeding $3\lambda_\tau$. We remark that $\hat \rho_{\tau\tau}$ is precisely $\tan\beta_\tau$ as defined in Ref.~\cite{Davidson:2010xv}, up to a sign. But Eq.~(\ref{rho-order}) should make clear that, while $\hat \rho_{\tau\tau}$ and $\hat \rho_{tt}$ are both ${\cal O}(1)$, their actual values could differ by an order of magnitude, and should be determined by experiment. Thus, besides scalar masses, the parameters that enter are: $\rho_{\tau\mu}$ (overall and factorized), $\rho_{\tau\tau}$ (one-loop), $\rho_{tt}$ (two-loop), and the $h$--$H$ mixing parameter $c_\gamma$. Although $c_\gamma$ is expected small, its uncertain value is relevant in bringing in the extra Yukawa couplings of $h$ that can interfere with the leading two-loop top effect, as we now elucidate. We conservatively take $c_\gamma = 0.2$ as its maximal value. \section{(Less) Relevant Contributions} Having clarified the natural setting of $\rho_{\tau\mu} = {\cal O}(\lambda_\tau)$ and $|c_\gamma| \lesssim 0.2$, we see that $h \to \tau\mu$ search at the LHC would continue to probe this space. Still, as there are multiple parameters that enter $\tau \to \mu\gamma$, one needs to discern relevant from less relevant parameters and processes. It is well known~\cite{Chang:1993kw} that, so long that $\rho_{tt} \sim \lambda_t \cong 1$ (Eq.~(\ref{rho-order})), the two-loop mechanism is by far the leading effect. But what about the other two type of diagrams in Fig.~\ref{feyndiag}. We propose two ``benchmarks'' to elucidate the leading and subleading effects, which then clarifies that, in contrast with the much larger $\rho_{\tau\mu}, \rho_{\tau\tau}$ values taken in the past, the one-loop diagram cannot really be probed by Belle~II under the rule of thumb of Eq.~(\ref{rho-order}). We define the ``BSM-benchmark'' as setting $c_\gamma = 0$ in the two-loop mechanism to decouple $h$, and take $\rho_{tt} = 1$, as larger values tend to run into flavor constraints~\cite{Hou:2019grj,Altunkaynak:2015twa}, which we shall not explore in detail here. This benchmark captures the BSM effect from extra top Yukawa couplings of $H, A$, and would stand alone in the alignment limit, when the $\rho_{tt}$ phase no longer matters. We plot the $\sqrt{{\cal B}(\tau \to\mu\gamma)}$ in Fig.~\ref{bench}, where we set $\rho_{\tau\mu} = \rho_{\mu\tau} = 0.7\lambda_\tau$ (reason clarified below), which is conservative. The current bound on $\tau \to\mu\gamma$ is the shaded region, while the (conservatively) projected Belle~II limit of $10^{-9}$ is the horizontal solid line. It is interesting that, even for the conservative value of $\rho_{\tau\mu} = 0.7\lambda_\tau$, this BSM-benchmark can itself be readily probed by Belle~II. Conversely, if we set $\rho_{tt}$ to zero, then the leading two-loop effect vanishes, but the two-loop top still has an amplitude proportional to $\lambda_t\, c_\gamma$ coming from the $h$ boson, and similarly through the two-loop $W$ diagram, also with $c_\gamma$ dependence. Combining these $m_h$-dependent effects and calling it the ``$h$-benchmark, its $\sqrt{{\cal B}(\tau \to\mu\gamma)}$ is also plotted in Fig.~\ref{bench} as the dotted line, taking the conservative maximal value of $c_\gamma = 0.2$, which implies $\rho_{\tau\mu} = 0.7\lambda_\tau$ as maximally allowed by Eq.~(\ref{B_h-taumu}). We see from Fig.~\ref{bench} that, if stand-alone, this $h$-benchmark is out of Belle~II reach. This line actually does not depend on detailed values of $c_\gamma$ or $\rho_{\tau\mu}$, but depends only on the bound of Eq.~(\ref{B_h-taumu}), which follows from Eq.~(\ref{h-taumugamCMS}), the current bound~\cite{Sirunyan:2017xzt} on $h \to \tau\mu$. This is because the $h$-benchmark is also proportional to $|\rho_{\tau\mu}\,c_\gamma|^2$. Thus, the CMS bound on $h \to \tau\mu$ excludes the possibility of observing the two-loop effect without the participation of the extra top Yukawa coupling, $\rho_{tt}$! We enlarge this branching ratio region and display in Fig.~~\ref{bench} (right), which can be used to understand our numerical discussion in the next Section. We see from Fig.~\ref{bench} that, if one has relatively light extra neutral scalars ($\lesssim 300$ GeV), then the effect from ``BSM-benchmark'' tends to predominate. However, as the extra scalar mass increases, say beyond 500--600 GeV, on one hand it would require a larger fraction of full Belle~II data to probe, on the other hand, the interference between the BSM-benchmark and $h$-benchmark becomes important. As the latter is real in amplitude, the phase $\varphi_{tt} = \arg \rho_{tt}$ matters, along with the value of $|\rho_{tt}|$, which affects the extra Higgs two-loop effect, and the value of $c_\gamma$, which controls the effect of $h$. \begin{figure*}[t] \center \includegraphics[width=0.3 \textwidth]{300_07.pdf} \includegraphics[width=0.3 \textwidth]{300_14.pdf} \includegraphics[width=0.3 \textwidth]{300_28.pdf} \vskip0.1cm \includegraphics[width=0.3 \textwidth]{500_07.pdf} \includegraphics[width=0.3 \textwidth]{500_14.pdf} \includegraphics[width=0.3 \textwidth]{500_28.pdf} \caption{ For $m_{H, A} = 300$ (500) GeV, the upper (lower) plots are for the 3, 10 and 50 ab$^{-1}$ Belle~II data reach, plotted in the $|\rho_{tt}|$--$\varphi_{tt}$ plane. For the lower $\rho_{\tau\mu} = 0.7\lambda_\tau$ value, three curves for allowed $c_\gamma$ values are illustrated, which reduces to just one low $c_\gamma$ value for the larger $\rho_{\tau\mu} = 2.8\lambda_\tau$. The shaded region is excluded by Eq.~(1). See text for further discussion.} \label{rhottconstraint} \end{figure*} Finally, we exhibit the $\sqrt{{\cal B}(\tau \to\mu\gamma)}$ of the one-loop $\tau$ effect in Fig.~\ref{bench}, where $\rho_{\tau\tau}$ can also carry a phase, and we take the nominally largest value of $|\rho_{\tau\tau}| = 3\lambda_\tau$ that satisfies Eq.~(\ref{rho-order}). It is known that the effect of $H$ and $A$ strongly cancel each other when degenerate (black dashed curve), but the cancellation weakens when degeneracy is lifted. We give three sets of dashed curves, where red (blue) corresponds to $m_H$ ($m_A$) on real axis, with the other neutral scalar heavier by 10, 100, 200 GeV (this is done also for the two-loop BSM-benchmark, where effect is minor). For a given scalar $m_H$ ($m_A$) mass, the one-loop effect varies by more than one order of magnitude as the splitting increases. In general, the amplitude is far below even the $h$-benchmark, except for rather light scalars ($\lesssim 300$~GeV). Thus, we see that Belle~II would not have the ability to probe the one-loop $\tau$ contribution, that it is more than a nuisance effect. In the next section, we neglect the one-loop effect in our illustrations, as it just smears the projections at small $|\rho_{tt}|$, but cannot be discerned by Belle II. \section{\boldmath Interplay of $h \to \tau\mu$ and $\tau \to \mu\gamma$} We have exhibited in Fig.~\ref{bench} the BSM-benchmark, which illustrates the two-loop effect from $H$ and $A$ with near maximal $|\rho_{tt}| = 1$, and the $h$-benchmark, which illustrates the two-loop effect of $h$ with near maximal $c_\gamma = 0.2$. The strength $|\rho_{tt}|$ --- and phase $\varphi_{tt}$ --- and value of $c_\gamma$ (proximity to alignment limit) together determine the strength of interference between the leading and subleading effects. We have shown that the one-loop $\tau$ effect is less than subleading, which we shall ignore in the following numerical illustration. Of course, the strength of $\rho_{\tau\mu}$ determines the overall scale for the branching fraction, as it factorizes and one cannot probe its phase. Together with $c_\gamma$, $|\rho_{\tau\mu}|$ is constrained by the bound on ${\cal B}(h \to \tau\mu)$, Eq.~(\ref{B_h-taumu}). For instance, our near maximal value of $c_\gamma = 0.2$ for the $h$-benchmark allows only $\rho_{\tau\mu} \lesssim 0.7\lambda_\tau$, while $c_\gamma = 0.1, 0.05$ can allow the larger ranges of $\rho_{\tau\mu} \lesssim 1.4\lambda_\tau, 2.8\lambda_\tau$, respectively. For the alignment limit case of $c_\gamma = 0$, one recovers the BSM-benchmark, which scales with $|\rho_{\tau\mu}\rho_{tt}|^2$, and can be read off from Fig.~\ref{bench}. To illustrate the interference effect between the leading $H, A$ with subleading $h$ contributions and the role played by $\varphi_{tt}$, we plot in Fig.~\ref{rhottconstraint} the future reach of Belle~II data at 3, 10 and 50 ab$^{-1}$ in the $|\rho_{tt}|$--$\varphi_{tt}$ plane, for the three values of $\rho_{\tau\mu} = 0.7\lambda_\tau, 1.4\lambda_\tau, 2.8\lambda_\tau$, respectively. As seen from Fig.~\ref{bench}, the BSM-benchmark does not depend strongly on $m_H$--$m_A$ splitting, so we will use a common $m_{H, A}$ mass value, taken as 300 and 500 GeV. It is illustrated e.g. in Ref.~\cite{Ghosh:2019exx} that large parameter space in Higgs potential is allowed by the electroweak precision $T$-parameter and other considerations. Let us start with the upper left plot in Fig.~\ref{rhottconstraint}, which is for the conservative value of $\rho_{\tau\mu} = 0.7\lambda_\tau$ and relatively light $m_H, m_A \simeq 300$ GeV. From Fig.~\ref{bench} one can easily understand that the current bound on $\tau \to \mu\gamma$, Eq.~(1), does not put a constraint on the displayed parameter space, but can be probed as data accumulates at Belle~II, where the three sets of curves correspond to 3, 10 and 50 ab$^{-1}$. Each set of curves is further illustrated with three curves that correspond to $c_\gamma = 0.05, 0.1, 0.2$ allowed by Eq.~(\ref{B_h-taumu}), i.e. the bound from Eq.~(2). The curves are all of similar shape, and the dependence on $\varphi$ illustrate the interference of $H, A$ with the $h$ effects, which is richer than the real value of $\rho_{tt}$ assumed in Ref.~\cite{Hou:2019grj}. For the larger value of $\rho_{\tau\mu} = 1.4\lambda_\tau$, $c_\gamma = 0.2$ becomes excluded, so we illustrate with two curves for each projected data value. The smaller $c_\gamma$ means the $h$ effect is reduced, hence the interference weakens, while the current bound of Eq.~(1) starts to cut into the $|\rho_{tt}|$ parameter space as an effect through the ``BSM-benchmark''. For the near maximal $\rho_{\tau\mu} = 2.8\lambda_\tau$, only the small $c_\gamma = 0.05$ is allowed, hence we show only one curve for each data value in the right figure, and the current bound of Eq.~(1) now cuts deeper into $|\rho_{tt}|$ parameter space. The lower plots of Fig.~\ref{rhottconstraint} are for heavier $m_H, m_A = 500$ GeV, hence the contribution from the ``BSM-benchmark" is weakened, resulting in stronger interference due to the relative importance of the ``$h$-benchmark'' contribution. For $\rho_{\tau\mu}$ at the conservative $0.7\lambda_\tau$, $\tau\to \mu\gamma$ does not yet start to probe the $|\rho_{tt}|$ parameter space even with 3~ab$^{-1}$. For $\rho_{\tau\mu} = 1.4\lambda_\tau$, the current bound is still ineffective, but 3 ab$^{-1}$ would cut into $|\rho_{tt}|$ parameter space, while for the relatively large $\rho_{\tau\mu} = 2.8\lambda_\tau$, even the current bound excludes some $|\rho_{tt}|$ parameter space. Our figures project the discovery potential of $\tau \to \mu\gamma$ by Belle II, as constrained by $h \to \tau\mu$ under our working assumption of $\rho_{\tau\mu} = {\cal O}(\lambda_\tau) \sim 0.01$. The parameter space is substantial, so long that $\rho_{\tau\mu}/\lambda_\tau$ is not far below 1, and the extra Higgs mass scale does not approach decoupling. \section{Discussion and Summary} The constraint of Eq.~(\ref{B_h-taumu}), which arises from $h \to \tau\mu$ search at the LHC, should improve in the next couple of years when the full Run~2 data is analyzed. It would likely drop further, which would imply that our ``$h$-benchmark'' line in Fig.~2 would drop. This would mean the interference effect as exhibited in Fig.~3 would shrink further, and one has less access to the phase $\varphi_{tt}$. However, it is not impossible that a hint emerges for $h \to \tau\mu$, which would suggest that neither $\rho_{\tau\mu}$ nor $c_\gamma$ vanish, and would heighten the interest in $\tau \to \mu\gamma$ search at Belle~II. Assuming no hint for signal, combining the full Run 2+3 dataset of ATLAS and CMS and scaling naively by statistics, one can probe down to $0.05\%$, compared with 0.25\% in Eq.~(\ref{B_h-taumu}). One would then be close to the ``BSM-benchmark'' scenario. If we happen to be rather close to the alignment limit, then the constraint on $\rho_{\tau\mu}$ is alleviated, with $\tau \to \mu\gamma$ probing $|\rho_{\tau\mu}\rho_{tt}|^2$, and Belle~II would still have wide discovery potential. It should be noted that exotic Higgs bosons as light as 300 GeV is {\it not} ruled out~\cite{Hou:2018zmg}. There is in fact a mild hint for a pseudoscalar $A$ around 400 GeV~\cite{Sirunyan:2019wph}, interfering with the $gg \to t\bar t$ QCD background. It could be the $gg$ fusion production and decay of $A$ via $\rho_{tt}$, or even $H$ that is produced via a purely imaginary $\rho_{tt}$ coupling~\cite{Hou:2019gpn}. The exotic Higgs spectrum for g2HDM is largely unknown, but 300 to 600~GeV is a preferred target zone, if~\cite{Hou:2017hiw} the inertial mass scale of the second (non-mass-giving) doublet is not far above the weak scale, which would be the tuned case of {\it decoupling}. Besides $gg \to H, A \to t\bar t$~\cite{Hou:2019gpn}, $t\bar c$~\cite{Altunkaynak:2015twa}, proposed searches such as $cg \to t\,H/A \to tt\bar c$, $tt\bar t$~\cite{Kohda:2017fkn} and the recently proposed $cg \to bH^+$~\cite{Ghosh:2019exx} process, give rise to signatures of same-sign top with jets, triple-top, and single top with two $b$-jets. Especially if the mass scale is below 400 GeV, we should have good hope of learning the mass spectrum in the coming years. Note that the three signatures above all require sizable $\rho_{tc}$ for production, which is in line with our working assumption of Eq.~(\ref{rho-order}). Furthermore, $\rho_{tc}$ at ${\cal O}(1)$ can {\it also drive} BAU~\cite{Fuyuto:2017ewj}. Thus, the program is well motivated. We have illustrated that the discovery potential at Belle~II does not actually depend on whether a hint for $h\to \tau\mu$ emerges at the LHC, which is in part regulated by the strength of $c_\gamma$. The actual value of $c_\gamma$, however, may be hard to extract. Although ATLAS and CMS have fitted for $\cos(\beta-\alpha)$ in the context of 2HDM~II~\cite{Khachatryan:2016vau, Aad:2019mbh}, with many more parameters in g2HDM, such a fit may not be feasible until we know more about some parameters related to the Extra Higgs, such as mass spectrum. We have conservatively taken the maximal value of 0.2 for $c_\gamma$, but we do not view $c_\gamma = 0.3$ as ruled out in g2HDM. Processes that do not depend on $c_\gamma$, such as electroweak baryogenesis (EWBG), i.e. generating BAU, are therefore of interest. Back on Earth, we note that $H^+$ and $A$ couplings do not depend on $c_\gamma$. Thus, $B^- \to \mu^- \bar\nu$ where the flavor of $\bar\nu$ is not detected, probes the product of $\rho_{tu}\rho_{\tau\mu}$~\cite{Hou:2019uxa}. Although we do not advocate that $\rho_{tu}$ should also satisfy some relation similar to Eq.~(\ref{rho-order}), we have rather poor knowledge of its value. Ref.~\cite{Hou:2019uxa} suggests that the ratio of ${\cal B}(B \to \mu\bar\nu)/{\cal B}(B \to \tau\bar\nu)$ in g2HDM may deviate from the SM expectation of 0.0045, a value that is shared by 2HDM~II. If such a result is found, which could emerge relatively early with Belle~II, it would imply nonvanishing $\rho_{\tau\mu}$, hence would also heighten the interest in $\tau\to \mu\gamma$ (as well as pursuit of the $tuh$ coupling). One could also probe $\rho_{\tau\mu}$ via searching for heavy $H, A \to \tau\mu$~\cite{Hou:2019grj}. While such search is clearly worthy~\cite{Sirunyan:2019shc}, it runs again branching ratio suppression due to the likely dominance of $t\bar t$ and $t\bar c$ decay modes in g2HDM. Although we do not think that Belle~II could effectively probe $\rho_{\tau\tau}$ through the one-loop $\tau \to \mu\gamma$ effect, $\rho_{\tau\tau}$ can be probed at the LHC in principle, both via deviations from SM rate for $h \to \tau\tau$ by $h$-$H$ mixing, or by search for heavy $H, A \to \tau\tau$~\cite{Aaboud:2017sjh, Sirunyan:2018zut}, but it might not be better than the $\tau\mu$ final state. In summary, we analyze the outlook for $\tau$LFV search via the $h \to \tau\mu$ and $\tau \to \mu\gamma$ processes, which appears quite promising in the general 2HDM. The $h \to \tau\mu$ process probes the product $\rho_{\tau\mu}c_\gamma$, where $\rho_{\tau\mu}$ is the extra flavor changing neutral Higgs coupling, and $c_\gamma$ is the $CP$-even Higgs mixing angle, which is expected to be small by the phenomenon of alignment. But whether or not a hint emerges with Run 2+3 data, our working assumption that $\rho_{\tau\mu} = {\cal O}(\lambda_\tau)$ and $\rho_{tt} = {\cal O}(\lambda_t)$ makes $\tau \to \mu\gamma$ very interesting at Belle~II, with broad parameter range for discovery. If Nature provides a finite $c_\gamma$ that is on the larger side, on one hand it increases the likelihood that $h \to \tau\mu$ may emerge, on the other hand, the interference of $h$ with $H, A$ effects in $\tau\to \mu\gamma$ decay in principle probes the phase of $\rho_{tt}$. We look forward to the unfolding of these two search modes in the coming decade. \vskip0.2cm \noindent{\bf Acknowledgments} \ We thank K.-F. Chen, S. Davidson, M. Kohda and M. Nakao for discussions. This research is supported by MOST 106-2112-M-002-015-MY3, 108-2811-M-002-626, and NTU 109L104019.

proofpile-arXiv_069-13893

\section{Introduction} Quantum gravity research usually focusses on the ``relativistic regime'', namely it considers scenarios that at low energies/large distances reduce either to general or special relativity, with finite speed of light. This is for example the case of noncommutative spacetime models \cite{Doplicher:1994zv, MR1994, Madore:2000en, Balachandran:2005eb}. They provide a formalization of the expected fuzzy behaviour of spacetime close to the Planck length $L_{p}\simeq 10^{-35} \text{m}$ \cite{AmelinoCamelia:2008qg,Garay:1994en, Hossenfelder:2012jw}, where quantum-gravitational effects break the usual description of spacetime in terms of a smooth pseudo-Riemannian manifold. These models describe a ``special relativistic regime'' of quantum gravity, since when the noncommutativity parameter (related to the Planck length) vanishes, one recovers the usual description of Minkowski spacetime. Over the last two decades, motivated by the need to make contact with phenomenological studies in the astrophysical and cosmological setting \cite{Rosati:2015pga, Barcaroli:2015eqe}, such models were extended to include a non-vanishing spatial curvature, thus developing noncommutative versions of the (Anti-)de Sitter ((A)dS) models (see~\cite{Brunoc,Stein,BM,VH, Ivetic:2013yga,BRHAdvances,Zoupanos} and references therein). As the understanding of these models advanced, it became increasingly clear that the interplay between the curvature parameter and the Planck-scale noncommutativity parameter is non-trivial \cite{Marciano:2010gq}. For example, some kinds of deformations of a particle dispersion relation might only appear when the two parameters are both non-zero \cite{Barcaroli:2015eqe,Marciano:2010gq}, and the rotation sector might be deformed \cite{Ballesteros:2017pdw}. Having realised that the introduction of a curvature parameter might have non-trivial consequences in quantum spacetime models, it is reasonable to wonder what role does the speed of light parameter play in this context. In particular, the question of what residual quantum-gravity effects might survive in the non-relativistic ($c\to \infty$) and ultra-relativistic ($c\to 0$) limits should be investigated.\footnote{Despite the possibly misleading terminology that is commonly used, both the non-relativistic and the ultra-relativistic limits of Poincar\'e invariant models retain invariance under some group of kinematical symmetries, the Galilei and the Carroll group, respectively. This is also true when the same limits are applied to systems with (Anti-)de Sitter invariance. In this case, the two limits produce models that are invariant under the so-called Newton--Hooke and curved Carroll groups of symmetries, respectively (each of them can in turn have positive or negative cosmological constant).} These are issues with possible phenomenological consequences: the $c\to \infty$ limit is relevant for the search of quantum-gravity signatures in systems where the typical velocities are small compared to the speed of light, such as in atom interferometry \cite{Garay:1999cy, Ellis:1983jz}, while the $c\to 0$ limit can be related to quantum effects in a strong gravity regime \cite{Dautcourt:1997hb, Bergshoeff:2017btm, PintoNeto:1998mz}. In previous work \cite{SnyderG}, some of us started to uncover that indeed the non-relativistic and ultra-relativistic limits of quantum spacetime models are non-trivial. This was done by studying these limits in the context of Snyder noncommutativity with zero spacetime curvature. We found that, contrary to what commonly assumed, the separation of space and time that one would usually expect in the non-relativistic and ultra-relativistic limits is not realised, because of a residual noncommutativity between time and space coordinates. Moreover, the limits are realised in a non-trivial way, since they are somewhat exchanged when looking at the corresponding momentum space geometry of each model. In this paper we further investigate these issues in the context of one of the most studied models of quantum spacetime, characterized by the so-called $\kappa$-deformation. The flat case, known as the $\kappa$-Minkowski spacetime with associated $\kappa$-Poincar\'e symmetry, has been widely investigated in the context of quantum gravity research, uncovering a rich structure and a variety of possible phenomenological consequences \cite{KowalskiGlikman:2002we, Bruno:2001mw, Gubitosi:2013rna, AmelinoCamelia:2011nt, AmelinoCamelia:1999pm, Gubitosi:2019ymi}. In more recent times, the model has been extended to include non-vanishing curvature, with the development of the $\kappa$-(A)dS spacetime~\cite{BGH2019spacetime} and its associated $\kappa$-(A)dS symmetries~\cite{BHMN2017kappa3+1}. It was indeed by using these models that the interplay between spacetime curvature and noncommutativity was most precisely characterized. Here we show that studying the non-relativistic and ultra-relativistic limits of these models allows us to investigate the possible interplay between the noncommutativity parameter, the cosmological constant and the speed of light all at once. We compute for the first time the Poisson version of the $\kappa$-deformed Newtonian and Carrollian quantum algebras, including their extension with non-vanishing curvature. These can be obtained as the $c\to \infty$ and $c\to 0$ contractions of the $\kappa$-(A)dS algebra, respectively.\footnote{The zero-curvature case is trivially obtained by sending the cosmological constant parameter to zero.} Furthermore, we obtain the noncommutative spacetimes that are invariant under such symmetries, which we call the $\kappa$-Newtonian and $\kappa$-Carrollian spacetimes (under these names we include both the vanishing and non-vanishing cosmological constant cases), showing that they are the $c\to \infty$ and $c\to 0$ limits of the $\kappa$-(A)dS spacetime. We find that, similarly to what happens for the Snyder model, also with the $\kappa$-deformation space and time maintain a residual noncommutativity in both the non-relativistic and the ultra-relativistic limits. Moreover, the form of the Casimir of the algebra of symmetries in the non-relativistic limit suggests a residual interplay between spatial momenta and energy (this is not the case in the ultra-relativistic limit). The study of the $\kappa$-deformation with non-zero spatial curvature lets us also appreciate the interplay of all of the three parameters for the model: while the non-relativistic limits preserves the deformation of the algebra of rotations that is induced by the curvature parameter in the $\kappa$-(A)dS algebra, in the ultra-relativistic limit the standard algebra of rotations is recovered, even when the curvature is non-vanishing. The structure of the paper is as follows. In Section \ref{sec2} we review the classical (A)dS algebra and its non-relativistic and ultra-relativistic limits. These are obtained after an appropriate $c$-dependent rescaling of the symmetry generators. In Section \ref{sec3} we review the contraction procedure that can be applied to quantum algebras to obtain the non-relativistic and ultra-relativistic limits, and use it to compute the $\kappa$-Newtonian algebra with non-vanishing cosmological constant and its $\kappa$-Galilei algebra limit. In Section \ref{sec4} we apply the same procedure to obtain the $\kappa$-Carrollian algebras, with vanishing and non-vanishing cosmological constant. In Section \ref{sec5} we construct the corresponding quantum spacetimes that are invariant under the $\kappa$-Newtonian and $\kappa$-Carrollian symmetries, respectively. For these spacetimes we provide both the Poisson brackets structure and their quantization. The final Section \ref{sec6} provides some concluding remarks and outlook. \section{The non-relativistic and ultra-relativistic limits of (A)dS}\label{sec2} We start by reviewing the limiting procedures that bring us from the classical (A)dS algebra to its non-relativistic ($c\to \infty$) and ultra-relativistic ($c\to 0$) limits. We show that in order to obtain consistent results, these limits need to be taken after an appropriate rescaling of the symmetry generators. The procedure we expose in this section will be adapted to the $\kappa$-deformed algebra and spacetime in the following sections. Let us consider the kinematical algebras of isometries of the (3+1)D (A)dS and Minkowskian spacetimes in the basis spanned by the time translation generator $P_0$, space translations $\>P=(P_1,P_2,P_3)$, boosts $\>K=(K_1,K_2,K_3)$ and rotations $\>J=(J_1,J_2,J_3)$. These three Lorentzian Lie algebras can collectively be described in terms of the cosmological constant $\ka$ as a one-parameter family of Lie algebras denoted by AdS$_\ka$ whose commutation relations read \bea \begin{array}{lll} [J_a,J_b]=\epsilon_{abc}J_c ,& \quad [J_a,P_b]=\epsilon_{abc}P_c , &\quad [J_a,K_b]=\epsilon_{abc}K_c , \\[2pt] \displaystyle{ [K_a,P_0]=P_a } , &\quad\displaystyle{[K_a,P_b]=\delta_{ab} P_0} , &\quad\displaystyle{[K_a,K_b]=-\epsilon_{abc} J_c} , \\[2pt][P_0,P_a]=-\ka \,K_a , &\quad [P_a,P_b]=\ka \,\epsilon_{abc}J_c , &\quad[P_0,J_a]=0 , \end{array} \label{aa} \eea where from now on sum over repeated indices will be understood and $a,b,c=1,2,3$. The family AdS$_\ka$ is endowed with two Casimir operators; one of them is the quadratic Casimir coming from the Killing--Cartan form, namely, \be {\cal C} = P_0^2-\>P^2 +\ka \left( \>K^2- \>J^2\right) , \label{ab} \ee while the other one is fourth-order, and is related to the Pauli--Lubanski 4-vector. Its explicit form can be found in~\cite{Herranz:1997us,Herranz:2006un}. Hence AdS$_\ka$ is the AdS algebra $\mathfrak{so}(3,2)$ for $\ka<0$, the dS algebra $\mathfrak{so}(4,1)$ when $\ka>0$, and the Poincar\'e algebra $\mathfrak{iso}(3,1)$ for $\ka=0$. This latter case is just the spacetime contraction of the (A)dS algebras which is associated with the composition of the parity $\mathcal {P}$ and time-reversal $\mathcal{T}$ involutive automorphisms~\cite{BLL}, which acts on the generators as follows: \be \mathcal{PT}( P_0,\>P,\>K,\>J)= (-P_0,-\>P,\>K,\>J). \label{ac} \ee The three (3+1)D Lorentzian spacetimes of constant curvature are obtained as the coset spaces that we will denote as \be \>{AdS}^{3+1}_\ka= G/H ,\qquad H= {\rm SO}(3,1)=\langle \>K,\>J\rangle , \label{ad} \ee where $G$ is the Lie group with Lie algebra AdS$_\ka$ and $H$ is the isotropy subgroup, namely the Lorentz subgroup generated by boosts and rotations. When $\ka<0$ we obtain the AdS space, when $\ka<0$ the dS one and the $\ka=0$ coset space is just the Minkowski spacetime. Hence $\>{AdS}^{3+1}_\ka$ encompasses the family of symmetrical homogeneous Lorentzian spacetimes (with involution (\ref{ac})) of constant sectional curvature $\kk=-\ka$. \subsection{The non-relativistic limit: Newtonian algebras and spacetimes} In order to apply the non-relativistic limit within the family AdS$_\ka$ we first introduce explicitly the speed of light $c$ via the map \be \>P\to \frac 1 c\, \>P,\qquad \>K\to \frac 1 c\, \>K. \label{add} \ee Applying this to the commutation rules (\ref{aa}) and taking the limit $c\to \infty$ yields \bea \begin{array}{lll} [J_a,J_b]=\epsilon_{abc}J_c ,& \quad [J_a,P_b]=\epsilon_{abc}P_c , &\quad [J_a,K_b]=\epsilon_{abc}K_c , \\[2pt] \displaystyle{ [K_a,P_0]=P_a } , &\quad\displaystyle{[K_a,P_b]=0} , &\quad\displaystyle{[K_a,K_b]=0} , \\[2pt][P_0,P_a]=-\ka \,K_a , &\quad [P_a,P_b]=0 , &\quad[P_0,J_a]=0 . \end{array} \label{ae} \eea The non-relativistic limit of the second-order Casimir is obtained by transforming (\ref{ab}) through the map (\ref{add}) and then taking the limit $\lim_{c\to\infty}(-{\cal C}/c^2)$, leading to \be {\cal C} = \>P^2 -\ka\, \>K^2 . \label{af} \ee We recall that this procedure is just an In\"on\"u--Wigner contraction~\cite{Inonu:1953sp} which corresponds to the speed-space contraction associated with the parity $\mathcal{P}$ automorphism~\cite{BLL} \be \mathcal{P}( P_0,\>P,\>K,\>J)= (P_0,-\>P,-\>K,\>J). \label{ag} \ee In this way we have obtained the family of non-relativistic or Newtonian Lie algebras, that we will denote as $\nh_\ka$: the expanding Newton--Hooke (NH) algebra $\nh_+$ for $\ka>0$, the oscillating NH algebra $\nh_-$ for $\ka< 0$ and the Galilei algebra $\nh_0$ for $\ka=0$~\cite{SnyderG,Herranz:1997us,Herranz:2006un,BLL,BHOS1994global,Aldrovandi,Duval:2014uoa,Figueroa-OFarrill:2017ycu,Gomis:2019}. These algebras have the following structure (the notation $\dsum$ stands for the semidirect sum): \be \begin{array}{llll} \nh_+= \mathbb{R}^6\dsum\bigr( \mathfrak{so}(1,1) \oplus \mathfrak{so}(3) \bigl),&\ \mathbb{R}^6=\langle \>P,\>K\rangle ,&\ \mathfrak{so}(1,1)=\langle P_0\rangle,&\ \mathfrak{so}(3)= \langle \>J \rangle. \\[4pt] \nh_-= \mathbb{R}^6\dsum\bigr( \mathfrak{so}(2) \oplus \mathfrak{so}(3) \bigl),&\ \mathbb{R}^6=\langle \>P,\>K\rangle ,&\ \mathfrak{so}(2)=\langle P_0\rangle,&\ \mathfrak{so}(3)= \langle \>J \rangle. \\[4pt] \nh_0= \mathbb{R}^4\dsum\bigr( \mathbb{R}^3 \dsum \mathfrak{so}(3) \bigl)\equiv \mathfrak{iiso}(3) ,&\ \mathbb{R}^4=\langle P_0, \>P\rangle ,&\ \mathbb{R}^3=\langle \>K \rangle,&\ \mathfrak{so}(3)= \langle \>J \rangle. \end{array} \label{ah} \ee The three corresponding (3+1)D Newtonian spacetimes of constant curvature are constructed as the coset spaces \be \>{N}^{3+1}_\ka= {\rm N}_\ka/H ,\qquad H= {\rm ISO}(3)=\langle \>K,\>J\rangle , \label{ai} \ee where $ {\rm N}_\ka$ is the Lie group with Lie algebra $\nh_\ka$ and $H$ is the isotropy subgroup of rotations and (commuting) Newtonian boosts, which is isomorphic to ${\rm ISO}(3)$. These three non-relativistic spacetimes have the same constant sectional curvature $\kk=-\ka$ as their Lorentzian counterparts. We point out that $\kk$ is the curvature of the ``main" metric (which is degenerate and provides the ``absolute-time" description), but there exists an additional invariant foliation under the Newtonian group action with a ``subsidiary" 3D non-degenerate Euclidean spatial metric restricted to each leaf of the foliation (see, e.g.~\cite{SnyderG,conf} for explicit metric models). \subsection{The ultra-relativistic limit: Carrollian algebras and spacetimes} The ultra-relativistic limit of the family AdS$_\ka$ can be performed by introducing the speed of light $c$ via the map~\cite{BLL,LevyLeblondCarroll} \be P_0\to c\, P_0,\qquad \>K\to c\, \>K. \label{aj} \ee Applying this to the commutation relations (\ref{aa}) and taking the limit $c\to 0$ generates the family of Lie algebras \bea \begin{array}{lll} [J_a,J_b]=\epsilon_{abc}J_c ,& \quad [J_a,P_b]=\epsilon_{abc}P_c , &\quad [J_a,K_b]=\epsilon_{abc}K_c , \\[2pt] \displaystyle{ [K_a,P_0]=0 } , &\quad\displaystyle{[K_a,P_b]=\delta_{ab} P_0} , &\quad\displaystyle{[K_a,K_b]= 0} , \\[2pt][P_0,P_a]=-\ka \,K_a , &\quad [P_a,P_b]=\ka \,\epsilon_{abc}J_c , &\quad[P_0,J_a]=0 . \end{array} \label{ak} \eea The ultra-relativistic limit of the second-order AdS$_\ka$ Casimir is obtained by transforming (\ref{ab}) under the map (\ref{aj}) and taking the limit $\lim_{c\to0}c^2\,{\cal C}$, thus yielding \be {\cal C} = P_0^2 +\ka\, \>K^2 . \label{al} \ee This process defines an In\"on\"u--Wigner contraction~\cite{Inonu:1953sp} which is interpreted as the speed-time contraction associated with the time-reversal $\mathcal{T}$ involution~\cite{BLL} \be \mathcal{T}( P_0,\>P,\>K,\>J)= (-P_0,\>P,-\>K,\>J). \label{am} \ee Thus we have obtained the family of Carrollian Lie algebras, denoted $\ca_\ka$, which comprises the Para-Euclidean $\ca_+$, Para-Poincar\'e $\ca_-$ and the (proper) Carroll $\ca_0$ algebras~\cite{Bergshoeff:2017btm,SnyderG,BLL,BHOS1994global,Duval:2014uoa, Figueroa-OFarrill:2017ycu,Gomis:2019,LevyLeblondCarroll,Gomis:2014, KowalskiGlikman:2014,Hartong:2015xda,Cardona:2016ytk, Daszkiewicz2019}. Their internal structure can be described as follows~\cite{SnyderG}: \be \begin{array}{lll} \ca_+\equiv \mathfrak{i'so}(4) = \mathbb{R}'^4 \dsum \mathfrak{so}(4) ,&\ \mathbb{R}'^4=\langle P_0,\>K\rangle ,&\ \mathfrak{so}(4)=\langle \>P,\>J\rangle . \\[4pt] \ca_- \equiv \mathfrak{i'so}(3,1) = \mathbb{R}'^4 \dsum \mathfrak{so}(3,1) ,&\ \mathbb{R}'^4=\langle P_0,\>K\rangle ,&\ \mathfrak{so}(3,1)=\langle \>P,\>J\rangle . \\[4pt] \ca_0\equiv \mathfrak{i'iso}(3) = \mathbb{R}'^4 \dsum \bigr( \mathbb{R}^3 \dsum \mathfrak{so}(3) \bigl) ,&\ \mathbb{R}'^4=\langle P_0, \>K\rangle ,&\ \mathbb{R}^3=\langle \>P \rangle, \quad\ \mathfrak{so}(3)= \langle \>J \rangle . \end{array} \label{an} \ee We remark that $\mathfrak{i'so}(4)$ is isomorphic to the Euclidean algebra $\mathfrak{iso}(4)$ and $\mathfrak{i'so}(3,1) $ to the Poincar\'e algebra $\mathfrak{iso}(3,1)$, although they are physically different algebras. In fact, the $'$ notation for the Para-Poincar\'e algebra $ \mathfrak{i'so}(3,1) $ means that $ \mathfrak{so}(3,1)=\langle \>P,\>J\rangle$ acts on $ \mathbb{R}'^4=\langle P_0,\>K\rangle$ through the contragredient of the vector representation, while in the Poincar\'e algebra $ \mathfrak{iso}(3,1) $ the Lorentz subalgebra $ \mathfrak{so}(3,1)=\langle \>K,\>J\rangle $ acts on $ \mathbb{R}^4=\langle P_0,\>P\rangle$ through the vector representation, and similarly for the two remaining Carrollian algebras (see~\cite{Azcarraga} for details). The three (3+1)D Carrollian spacetimes of constant curvature are identified with the coset spaces \be \>{C}^{3+1}_\ka= {\rm C}_\ka/H ,\qquad H= {\rm ISO}(3)=\langle \>K,\>J\rangle , \label{a0} \ee where $ {\rm C}_\ka$ is the Lie group with Lie algebra $\ca_\ka$ and $H$ is again the isotropy subgroup ${\rm ISO}(3)$ spanned by rotations and (commuting) Carrollian boosts. We stress that now such Carrollian spacetimes have sectional curvature $\kk=+\ka$, instead of $\kk=-\ka$ as in the Lorentzian and Newtonian spacetimes. In particular, the ``main" metric for Carrollian spacetimes is again degenerate and provides an ``absolute-space" geometry with curvature $\kk=+\ka$. But there does also exist an invariant foliation preserved by the Carrollian group action which is characterized by a ``subsidiary" 1D time metric that is restricted to each leaf of the foliation~\cite{SnyderG}. For the sake of clarity, in Table~\ref{table1} we summarize the nine kinematical algebras (Lorentzian, Newtonian and Carrollian) and the corresponding spacetimes. \begin{table}[t] {\footnotesize \caption{\small Lorentzian, Newtonian and Carrollian algebras together with their corresponding (3+1)D homogeneous spacetimes with constant sectional curvature $\kk$ according to the value of $\ka$.} \label{table1} \begin{center} \noindent \begin{tabular}{l l l } \hline \\[-0.2cm] \multicolumn{3}{c}{Lorentzian algebras and spacetimes} \\[0.2cm] \hline \\[-0.2cm] $\bullet$ AdS & $\bullet$ Poincar\'e & $\bullet$ dS \\[0.2cm] $\mathfrak{so}(3,2)$: $\ka<0$, $\kk=-\ka>0$ & $\mathfrak{iso}(3,1)$: $\ka=\kk=0$ & $\mathfrak{so}(4,1)$: $\ka>0$, $\kk=-\ka<0$ \\[0.2cm] $\>{AdS}^{3+1}={\rm SO}(3,2)/{\rm SO}(3,1)$ & $\>{M}^{3+1}={\rm ISO}(3,1)/{\rm SO}(3,1)$ & $\>{dS}^{3+1}={\rm SO}(4,1)/{\rm SO}(3,1)$ \\[0.2cm] \hline \\[-0.2cm] \multicolumn{3}{c}{Newtonian algebras and spacetimes} \\[0.2cm] \hline \\[-0.2cm] $\bullet$ Oscillating NH& $\bullet$ Galilei & $\bullet$ Expanding NH \\[0.2cm] $\nh_- =\mathbb{R}^6\dsum\bigr( \mathfrak{so}(2) \oplus \mathfrak{so}(3) \bigl)$ & $\nh_0=\mathfrak{iiso}(3)$ & $\nh_+=\mathbb{R}^6\dsum\bigr( \mathfrak{so}(1,1) \oplus \mathfrak{so}(3) \bigl)$ \\[0.2cm] $\ka<0$, $\kk=-\ka>0$ & $\ka=\kk=0$ & $\ka>0$, $\kk=-\ka<0$ \\[0.2cm] $\>{N}_-^{3+1}={\rm N}_-/{\rm ISO}(3)$ & $\>{N}_0^{3+1}\equiv \>{G}^{3+1}={\rm IISO}(3)/{\rm ISO}(3)$ &$\>{N}_+^{3+1}={\rm N}_+/{\rm ISO}(3)$ \\[0.2cm] \hline \\[-0.2cm] \multicolumn{3}{c}{Carrollian algebras and spacetimes} \\[0.2cm] \hline \\[-0.2cm] $\bullet$ Para-Poincar\'e & $\bullet$ Carroll& $\bullet$ Para-Euclidean \\[0.2cm] $\ca_- = \mathfrak{i'so}(3,1) $: $\ka=\kk<0$ & $\ca_0= \mathfrak{i'iso}(3) $: $\ka=\kk=0$ & $\ca_+= \mathfrak{i'so}(4)$: $\ka=\kk>0$ \\[0.2cm] $\>{C}_-^{3+1}={\rm I'SO(3,1)}/{\rm ISO}(3)$ & $\>{C}_0^{3+1}\equiv \>{C}^{3+1}={\rm I'ISO}(3)/{\rm ISO}(3)$ & $\>{C}_+^{3+1}={\rm I'SO(4)}/{\rm ISO}(3)$ \\[0.2cm] \hline \end{tabular} \end{center} } \end{table} \section{The $\kappa$-deformation of the Newtonian algebras} \label{sec3} In the previous section we have seen how to perform the non-relativistic limit of the AdS$_\ka$ algebra with commutation relations (\ref{aa}): one needs to introduce the speed of light parameter via a rescaling of the symmetry generators, eq. \eqref{add}, and then perform the limit $c\to \infty$. When dealing with a quantum algebra, in our case the $\kappa$-deformation of the AdS$_\ka$ algebra, one needs to identify the appropriate rescaling of the quantum deformation parameter $\kappa$ that, along with the map \eqref{add}, allows us to obtain meaningful expressions in the $c\to \infty$ limit (the idea that in quantum group contractions the deformation parameter has to be transformed was introduced for the first time in~\cite{CGST1991heisemberg, CGST1992}). In order to do so we start by recalling the fundamental structures that underlie the $\kappa$-AdS$_\ka$ algebra. A more detailed discussion can be found in \cite{BGH2019spacetime,BHMN2017kappa3+1}. The $\kappa$-deformation of the AdS$_\ka$ algebra can be generated by the classical $r$-matrix given by~\cite{BGH2019spacetime,BHMN2017kappa3+1} \be r_\ka=\frac{1}{\kappa}( K_1 \wedge P_1 + K_2 \wedge P_2 + K_3 \wedge P_3 + \ro\, J_1 \wedge J_2), \label{ba} \ee where the parameter $\ro$ is related to the cosmological constant via \be \ro^2:= {-\ka} . \label{bb} \ee The parameter $\ro$ can be either real or pure imaginary number for AdS ($r_-$) and dS ($r_+$), respectively. When $\ro=\ka =0$ we recover the $\kappa$-Poincar\'e $r$-matrix~\cite{Maslanka1993,Zakrzewski1994poincare} \be r_0=\frac{1}{\kappa}( K_1 \wedge P_1 + K_2 \wedge P_2 + K_3 \wedge P_3 ) , \label{bc} \ee underlying the well known $\kappa$-Poincar\'e algebra and group~\cite{MR1994,LRNT1991,LNR1991realforms,GKMMK1992,LNR1992fieldtheory}. The $\kappa$-AdS$_\ka$ $r$-matrix (\ref{ba}) is a solution of the modified classical Yang--Baxter equation and leads to a quasi-triangular Lie bialgebra through the commutator \be \delta(X) = [X \otimes 1 + 1 \otimes X,r_\ka], \qquad \forall X \in \mbox{AdS$_\ka$}, \label{bd} \ee namely~\cite{BGH2019spacetime,BHMN2017kappa3+1}, \begin{align} \begin{split} & \delta(P_0)=\delta(J_3)= 0, \qquad \delta(J_1)=\frac \ro\kappa \, J_1 \wedge J_3, \qquad \delta(J_2)= \frac \ro\kappa\, J_2 \wedge J_3 ,\\ & \delta(P_1)= \frac{1}{\kappa} (P_1 \wedge P_0 - \ro\, P_3 \wedge J_1 - \ro^2 K_2 \wedge J_3 + \ro^2 K_3 \wedge J_2) ,\\ & \delta(P_2)= \frac{1}{\kappa} (P_2 \wedge P_0 - \ro\, P_3 \wedge J_2 + \ro^2 K_1 \wedge J_3 - \ro^2 K_3 \wedge J_1), \\ & \delta(P_3)= \frac{1}{\kappa} (P_3 \wedge P_0 + \ro\, P_1 \wedge J_1 + \ro\, P_2 \wedge J_2 - \ro^2 K_1 \wedge J_2 + \ro^2 K_2 \wedge J_1), \\ & \delta(K_1)= \frac{1}{\kappa} (K_1 \wedge P_0 + P_2 \wedge J_3 - P_3 \wedge J_2 - \ro\, K_3 \wedge J_1) ,\\ & \delta(K_2)= \frac{1}{\kappa} ( K_2 \wedge P_0 - P_1 \wedge J_3 + P_3 \wedge J_1 - \ro\, K_3 \wedge J_2) ,\\ & \delta(K_3)= \frac{1}{\kappa} ( K_3 \wedge P_0 + P_1 \wedge J_2 - P_2 \wedge J_1 + \ro\, K_1 \wedge J_1 + \ro\, K_2 \wedge J_2). \label{be} \end{split} \end{align} Starting from these cocommutators, the complete Poisson analogue of the $\kappa$-AdS$_\ka$ quantum algebra (including fully explicit expressions for both coproducts and deformed Poisson brackets) was constructed in~\cite{BHMN2017kappa3+1}, and written in terms of the curvature $\kk\equiv \ro^2$ and quantum deformation parameter $z=\kappa^{-1}$. Recall that the term $\ro \, J_1 \wedge J_2$ in $r_\ka$ (\ref{ba}) gives rise to a sub-Lie bialgebra structure within (\ref{be}) coming from the Lie subalgebra $\mathfrak{su}(2)\simeq \mathfrak{so}(3)$ spanned by the rotation generators, which disappears in the Poincar\'e case with $\ro=0$. So one of the peculiar effects of the interplay between the curvature and quantum deformation parameters is that the algebra of rotations gets deformed, see also the discussion in \cite{Ballesteros:2017pdw}. The non-relativistic limit of the $\kappa$-AdS$_\ka$ symmetry can be obtained by applying the Lie bialgebra contraction (LBC) approach introduced in~\cite{BGHOS1995quasiorthogonal}. This completely general procedure starts from a given Lie algebra contraction and studies which transformation of the quantum deformation parameter has to be considered in order to obtain a well-defined and non-trivial Lie bialgebra structure after the contraction is performed. The initial Lie algebra contraction together with the transformation law of the quantum deformation parameter in terms of the contraction parameter defines a specific LBC, which suffices in order to define the appropriate contraction of the full quantum algebra and, through suitable consistency conditions arising from duality relations, the associated contraction of the Poisson--Lie and quantum groups (see~\cite{BGHOS1995quasiorthogonal} for details). It is important to stress that in this approach the $r$-matrix and its associated cocommutator $\delta$ given by~\eqref{bd} can behave differently under a given LBC, and the former could diverge while the latter is well-defined. This is quite natural since for non-semisimple Lie algebras (like the Newtonian and Carrollian ones) there do exist non-coboundary Lie bialgebra structures $\delta$ for which no $r$-matrix can be found, and for some of these cases they can be obtained as a LBC of a coboundary Lie bialgebra. The LBC that guarantees the existence of a non-vanishing cocommutator under contraction is called a fundamental LBC, while the one that guarantees the existence of a non-vanishing $r$-matrix is called a coboundary one. Indeed, there could also exist some LBC which is simultaneously fundamental and coboundary, and in that case the contracted $\delta$ can consistently be obtained from the contracted $r$ through the coboundary relation (\ref{bd}). As we will see in the sequel, this analysis will be essential for the obtention of Newtonian and Carrollian Lie bialgebras as contractions from AdS$_\ka$. As a first step in the computation of the non-relativistic limit of the $\kappa$-AdS$_\ka$ Lie bialgebra, we check whether there exists a LBC that is coboundary. To this aim, we transform the $r$-matrix~(\ref{ba}) through the map (\ref{add}), finding \be r_\ka=\frac{c^2}{\kappa}\left( K_1 \wedge P_1 + K_2 \wedge P_2 + K_3 \wedge P_3 + \frac 1{c^2}\,\ro\, J_1 \wedge J_2 \right). \label{bf} \ee Now, the existence of a convergent and non-trivial $c\to \infty $ limit of $r_\ka$ in (\ref{bf}) implies that the unique solution for the transformation law of the deformation parameter is \be \kappa \to c^{-2}\kappa \, . \label{bg} \ee Thus the LBC defined by the Lie algebra contraction~\eqref{add} together with the transformation law~\eqref{bg} gives rise to \be r_\ka=\frac{1}{\kappa} ( K_1 \wedge P_1 + K_2 \wedge P_2 + K_3 \wedge P_3 ), \label{bh} \ee which is a common $r$-matrix for the three Newtonian algebras $\nh_\ka$ with commutation rules (\ref{ae}). Nevertheless, if we compute the associated cocommutator $\delta$ through the coboundary relation (\ref{bd}) we get a trivial result, that is, $\delta(X)=0,$ $\forall X\in \nh_\ka$. Indeed, the same vanishing cocommutator is obtained if the same maps~\eqref{add} and~\eqref{bg} are applied onto the cocommutator (\ref{be}) and the limit $c\to \infty$ is computed. Therefore, we conclude that the coboundary LBC associated to the non-relativistic limit of the $\kappa$-deformation does exists but leads to a trivial structure. As it can be straightforwardly checked, when this LBC is applied to the full $\kappa$-AdS$_\ka$ Poisson--Hopf algebra we get a primitive coproduct and non-deformed commutation rules. Moreover, the Sklyanin bracket generated by the $r$-matrix~\eqref{bh} onto the three Newtonian groups vanishes identically, and therefore the spacetime is commutative. As we mentioned above, one can also look for a fundamental LBC which is not coboundary and which gives rise to a non-vanishing cocommutator in the Newtonian limit. In case this exists, it should be different from the previous coboundary LBC. By introducing the map (\ref{add}) within the cocommutator (\ref{be}) it is straightforward to check that the transformed $\delta$ does not contain $c$ and, therefore, there is no need to transform the deformation parameter under contraction in order to obtain a non-trivial Newtonian Lie bialgebra. Hence the non-relativistic contracted cocommutator coincides with the $\kappa$-AdS$_\ka$ one given by (\ref{be}). Note that this new fundamental LBC in which the deformation parameter does not change leads to a divergence for the $c\to\infty$ limit of the $r$-matrix~\eqref{bf}, which is consistent with the fact that the Newtonian Lie bialgebra defined by the cocommutator (\ref{be}) together with the commutation rules~\eqref{ae} is not a coboundary Lie bialgebra. Summarizing, a non-trivial $\kappa$-deformation of the Newtonian algebras can be obtained as a fundamental (and non-coboundary) LBC in which the deformation parameter is not affected by the non-relativistic limit. Therefore, if we apply the map (\ref{add}) to the complete $\kappa$-AdS$_\ka$ Poisson--Hopf algebra given in~\cite{BHMN2017kappa3+1} and perform the limit $c\to \infty$ we get fully convergent expressions for the three $\kappa$-Newtonian algebras. The coproduct $\Delta$ given in~\cite{BHMN2017kappa3+1} is found to be invariant under such contraction, so we omit it for the sake of brevity, while the contracted Poisson brackets turn out to be \be \begin{array}{lll} \multicolumn{3}{l} {\displaystyle {\left\{ J_1,J_2 \right\}= \frac{ {\rm e}^{2\ro J_3/\kappa}-1}{2 \ro/\kappa } - \frac{ \ro}{2\kappa} \left(J_1^2+J_2^2\right) ,\qquad \left\{ J_1,J_3 \right\}=-J_2 ,\qquad \left\{ J_2,J_3 \right\}=J_1 , } } \\[10pt] \left\{ J_1,P_1 \right\}=\frac \ro\kappa J_1 P_2, & \quad \left\{ J_1,P_2 \right\}= P_3-\frac \ro\kappa J_1 P_1 , & \quad \left\{J_1, P_3 \right\}= -P_2 , \\[4pt] \left\{ J_2,P_1 \right\}=-P_3+\frac \ro\kappa J_2 P_2, & \quad \left\{ J_2,P_2 \right\}= - \frac \ro\kappa J_2 P_1 , & \quad \left\{ J_2,P_3 \right\}=P_1 , \\[4pt] \left\{J_3, P_1 \right\}= P_2, & \quad \left\{ J_3,P_2 \right\}=-P_1 , & \quad \left\{ J_3,P_3 \right\}= 0 , \\[4pt] \left\{ J_1,K_1 \right\}=\frac \ro\kappa J_1 K_2 , & \quad \left\{ J_1,K_2 \right\}= K_3 -\frac \ro\kappa J_1 K_1 , & \quad \left\{J_1, K_3 \right\}= -K_2 , \\[4pt] \left\{ J_2,K_1 \right\}= -K_3+\frac \ro\kappa J_2 K_2, & \quad \left\{ J_2,K_2 \right\}= - \frac \ro\kappa J_2 K_1 , & \quad \left\{ J_2,K_3 \right\}=K_1 , \\[4pt] \left\{J_3, K_1 \right\}= K_2 , & \quad \left\{ J_3,K_2 \right\}=-K_1 , & \quad \left\{ J_3,K_3 \right\}= 0 , \\[4pt] \left\{ K_{a}, P_0 \right\}=P_a , & \quad \left\{ P_0, P_a \right\}=\ro^2 K_a , & \quad \left\{ P_0,J_a \right\}= 0 , \label{bi} \end{array} \ee \begin{eqnarray*} \left\{ K_1,P_1 \right\} \!\!\!&=&\!\!\! \frac{1}{2\kappa} \left( P_2^2+P_3^2-P_1^2\right) +\frac{\ro^2}{2\kappa} \left( K_2^2+K_3^2-K_1^2 \right) ,\\ \left\{ K_2,P_2 \right\}\!\!\!&=&\!\!\! \frac{1}{2\kappa} \left(P_1^2+P_3^2 -P_2^2\right) +\frac{\ro^2}{2\kappa} \left( K_1^2+K_3^2 -K_2^2 \right) ,\\ \left\{ K_3,P_3 \right\}\!\!\!&=&\!\!\! \frac{1}{2\kappa} \left[(P_1+ \ro K_2)^2 +(P_2- \ro K_1)^2-P_3^2 -\ro^2 K_3^2 \right] , \end{eqnarray*} \begin{eqnarray*} \left\{ P_1, K_2 \right\} \!\!\!&=&\!\!\! \frac{1}{\kappa} \left( P_1 P_2+\ro^2 K_1 K_2 - \ro P_3 K_3 \right) , \\ \left\{ P_2, K_1 \right\}\!\!\!&=&\!\!\! \frac{1}{\kappa} \left(P_1 P_2 + \ro^2 K_1 K_2+ \ro P_3 K_3 \right) ,\\ \left\{ P_1, K_3 \right\}\!\!\!&=&\!\!\! \frac{1}{\kappa} \left( P_1 P_3 +\ro^2 K_1 K_3+ \ro K_2 P_3\right) ,\\ \left\{ P_3, K_1 \right\}\!\!\!&=&\!\!\! \frac{1}{\kappa} \left(P_1 P_3+\ro^2 K_1 K_3 - \ro P_2 K_3 \right) ,\\ \left\{ P_2, K_3 \right\}\!\!\!&=&\!\!\! \frac{1}{\kappa} \left(P_2 P_3+\ro^2 K_2 K_3 -\ro K_1 P_3 \right) , \\ \left\{ P_3, K_2 \right\}\!\!\!&=&\!\!\! \frac{1}{\kappa} \left(P_2 P_3+\ro^2 K_2 K_3 + \ro P_1 K_3 \right) ,\\ \left\{ K_a,K_b \right\} \!\!\!&=&\!\!\! -\frac{\ro}{\kappa}\, \epsilon_{abc} K_cK_3 ,\qquad \left\{ P_a,P_b \right\} =-\frac{\ro}{\kappa} \, \epsilon_{abc} P_cP_3 . \end{eqnarray*} As usual, the limit $\kappa\to \infty$ corresponds to the non-deformed (``classical") limit with commutation rules (\ref{ae}), expressed as Poisson brackets together with a primitive coproduct. Note that this $c\to\infty$ limit preserves the non-trivial properties of the rotation sector that were already present in the $\kappa$-AdS$_\ka$ case and is due to the interplay between the curvature and quantum deformation parameters. The deformed version of the second-order Newtonian Casimir (\ref{af}) is obtained as the limit $\lim_{c\to\infty}{(-\cal C_\kappa}/c^2)$ after the $\kappa$-AdS$_\ka$ Casimir ${\cal C_\kappa}$ has been transformed under the automorphism (\ref{add}), yielding \bea {\cal C}_\kappa \!\!\!&=&\!\!\! {\rm e}^{P_0/\kappa} \left( \mathbf{P}^2 +\ro^2 \mathbf{K}^2 \right) \left[ \cosh(\ro J_3/\kappa)+ \frac { \ro^2}{2\kappa^2} (J_1^2+J_2^2){\rm e}^{-\ro J_3/\kappa} \right]\label{bj}\\ &&-2\ro^2 {\rm e}^{P_0/\kappa} \left[ \frac{\sinh(\ro J_3/\kappa)}{ \ro} \, R_3+ \frac 1\kappa \left( J_1 R_1 +J_2 R_2+ \frac { \ro}{2\kappa} (J_1^2+J_2^2) R_3 \right) e^{- \ro J_3/\kappa} \right], \nonumber \eea where $ R_a=\epsilon_{abc} K_b P_c$. By comparison with the classical Newtonian Casimir, we see that the presence of the time translation generator is a purely quantum effect, while the presence of the rotation generators is due to the interplay between the curvature and quantum deformation parameters. This can be confirmed by looking at the $\kappa$-Galilei Casimir that is written explicitly below, eq.~\eqref{bm}. The Newtonian limit of the deformed fourth-order ``Pauli--Lubanski'' Casimir presented in~\cite{BHMN2017kappa3+1} can be obtained in a similar manner and leads to a very involved expression. It is worth stressing that the above results provide a $\kappa$-Newtonian Poisson--Hopf algebra whose quantization should be further performed in order to obtain the noncommutative quantum algebra. In the Newton-Hooke cases (i.e. when $\eta\neq 0$) this task implies taking into account all ordering ambiguities appearing in both the coproduct and the Poisson brackets and we omit the final expressions for the sake of brevity, since the essential structure of the deformation can be fully appreciated at the Poisson bracket level. However, for the Galilean case (i.e. with $\ro \to 0$) no ordering ambiguities arise and we get the $\kappa$-Galilei quantum algebra: \begin{eqnarray} && \Delta ( P_0 ) = P_0 \otimes 1+1 \otimes P_0 ,\nonumber\\ && \Delta ( J_a ) = J_a \otimes 1+1 \otimes J_a , \nonumber\\ && \Delta ( P_a ) = P_a \otimes 1+{\rm e}^{- P_0/\kappa} \otimes P_a , \label{bk} \\ && \Delta ( K_a ) = K_a \otimes 1+{\rm e}^{- P_0/\kappa} \otimes K_a+\frac 1\kappa \, \epsilon_{abc} P_b \otimes J_c , \nonumber \end{eqnarray} \be \begin{array}{lll} [ J_a,J_b] =\epsilon_{abc}J_c ,& \quad [ J_a,P_b] =\epsilon_{abc}P_c , &\quad [ J_a,K_b] =\epsilon_{abc}K_c , \\[2pt] \displaystyle{ [ K_a,P_0] =P_a } , &\quad\displaystyle{ [ K_a,K_b] =0 } , &\quad {\displaystyle {[ K_a, P_b ] = \delta_{ab} \, \frac{1}{2\kappa}\, \>P^2 - \frac 1\kappa\, P_a P_b }} , \\[4pt] [ P_0,P_a] = 0 , &\quad [ P_a,P_b] =0 , &\quad \displaystyle{ [ P_0,J_a] =0 } , \end{array} \label{bl} \ee \be {\cal C}_\kappa= {\rm e}^{P_0/\kappa} \, \mathbf{P}^2 . \label{bm} \ee Note that the resulting Hopf algebra is written in a bicrossproduct basis~\cite{MR1994,Majida,Majidb,Azcarragab} with respect to the four commuting translations $ \mathbb{R}^4=\langle P_0,\>P\rangle$. We recall that the $\kappa$-Galilei algebra was firstly obtained in~\cite{GKMMK1992} through contraction from the $\kappa$-Poincar\'e algebra (but without an LBC analysis) and was expressed in a ``symmetrical" basis which is related to the bicrossproduct one here considered through a nonlinear map. To the best of our knowledge, the $\kappa$-deformation of the oscillating and expanding Newton-Hooke algebras $(\ro\ne 0)$ is a completely new result. \section{The $\kappa$-deformation of the Carrollian algebras} \label{sec4} The ultra-relativistic limit $c\to 0$ of the $\kappa$-AdS$_\ka$ algebra can be performed by following the same LBC approach~\cite{BGHOS1995quasiorthogonal} described in the previous section. In this case the Lie algebra contraction map is given by (\ref{aj}), which in the classical case leads to the Carrollian Lie algebra $\ca_\ka$ (\ref{ak}). As done in the previous section, we first check whether there is a coboundary LBC. To this aim we apply the contraction map (\ref{aj}) to the $r$-matrix (\ref{ba}), obtaining \be r_\ka=\frac{1}{c\,\kappa}( K_1 \wedge P_1 + K_2 \wedge P_2 + K_3 \wedge P_3 + c\,\ro\, J_1 \wedge J_2). \label{ca} \ee Asking that this $r$-matrix has a well-defined and non-trivial limit for $c\to 0$ implies that the deformation parameter must be transformed as \be \kappa\to c\,\kappa . \label{cb} \ee Then the $c\to 0$ limit for the $r$-matrix reads \be r_\ka=\frac{1}{ \kappa}( K_1 \wedge P_1 + K_2 \wedge P_2 + K_3 \wedge P_3 ), \label{cc} \ee which again coincides with the $\kappa$-Poincar\'e $r$-matrix (\ref{bc}) for any value of the cosmological constant parameter $\ro$. The remarkable point now is that, in contrast with the Newtonian case, the cocommutator obtained through the relation (\ref{bd}) is a non-trivial one, namely \begin{align} \begin{split} & \delta(P_0)=\delta(J_a)= 0, \qquad \delta(K_a)=\frac 1\kappa\, K_a\wedge P_0 ,\\ & \delta(P_a)= \frac{1}{\kappa} (P_a \wedge P_0 - \ro^2\, \epsilon_{abc}K_b \wedge J_c) . \label{cd} \end{split} \end{align} Furthermore, if we apply the LBC to the AdS$_\ka$ commutator (\ref{be}) given by the map~\eqref{aj} we find that the very same transformation (\ref{cb}) is the one ensuring the convergence of the cocommutator and gives rise to the same contracted expressions (\ref{cd}). Therefore the LBC defined by the maps (\ref{aj}) and (\ref{cb}) together with the limit $c\to 0$ is both a coboundary and a fundamental one. From it, the ultra-relativistic limit of the coproduct and Poisson brackets can directly be computed from the $\kappa$-AdS$_\ka$ Poisson algebra~\cite{BHMN2017kappa3+1} thus giving rise to the $\kappa$-Carrollian Poisson--Hopf algebras: \begin{eqnarray} && \Delta ( P_0 ) = P_0 \otimes 1+1 \otimes P_0 ,\qquad \Delta ( J_a ) = J_a \otimes 1+1 \otimes J_a , \nonumber\\ && \Delta ( P_a ) = P_a \otimes 1+{\rm e}^{- P_0/\kappa} \otimes P_a-\frac {\ro^2}\kappa \, \epsilon_{abc} K_b \otimes J_c , \label{ce} \\ && \Delta ( K_a ) = K_a \otimes 1+{\rm e}^{- P_0/\kappa} \otimes K_a , \nonumber \end{eqnarray} \bea \begin{array}{lll} \left\{J_a,J_b\right\}=\epsilon_{abc}J_c ,& \qquad \left\{J_a,P_b\right\}=\epsilon_{abc}P_c , &\quad \left\{J_a,K_b\right\}=\epsilon_{abc}K_c , \\[3pt] \displaystyle{ \left\{K_a,P_0\right\}=0 } , &\qquad \displaystyle{\left\{K_a,K_b\right\}= 0} , &\quad \left\{P_0,J_a\right\}=0 , \\[3pt] \left\{P_0,P_a\right\}=\ro^2 K_a , &\qquad \left\{P_a,P_b\right\}=-\ro^2 \epsilon_{abc}J_c , &\quad \\[6pt] \multicolumn{3}{l} { \displaystyle{ \left\{K_a,P_b\right\}=\delta_{ab}\left( \frac{1-{\rm e}^{-2 P_0/\kappa}}{2/\kappa} +\frac{\ro^2}{2\kappa} \,\>K^2 \right) -\frac{\ro^2}{\kappa}\, K_a K_b . }} \end{array} \label{cf} \eea Finally, the ultra-relativistic limit of the $\kappa$-AdS$_\ka$ deformed quadratic Casimir ${\cal C_\kappa}$ is obtained by applying the LBC map onto ${\cal C_\kappa}$ and then computing $\lim_{c\to 0}c^2\,{\cal C_\kappa}$. The final result is \be {\cal C_\kappa}=2\kappa^2\left( \cosh(P_0/\kappa)-1\right) -\ro^2 {\rm e}^{P_0/\kappa}\,\>K^2. \label{cg} \ee By inspection of the expressions above, we notice that now the rotation sector is trivial, so that the ultra-relativistic limit erases this kind of effect of the interplay between the curvature and quantum deformations, contrary to what happens in the non-relativistic limit. Moreover, the only residual mixing between the two parameters is of the form $\eta^{2}/\kappa$ (to the first-order in $1/\kappa$), while in the non-relativistic case we had effects already at the first-order in $\eta$, going as $\eta/\kappa$. This gives a first characterization of the fact that also the speed of light parameter $c$ has a non-trivial interplay with the curvature and quantum deformation parameters. The proper $\kappa$-Carroll algebra arises under the limit $\ro\to 0$ and is characterized by the following coproducts, Poisson brackets and Casimir functions: \begin{eqnarray} && \Delta ( P_0 ) = P_0 \otimes 1+1 \otimes P_0 ,\qquad \Delta ( J_a ) = J_a \otimes 1+1 \otimes J_a , \nonumber\\ && \Delta ( P_a ) = P_a \otimes 1+{\rm e}^{- P_0/\kappa} \otimes P_a , \qquad \Delta ( K_a ) = K_a \otimes 1+{\rm e}^{- P_0/\kappa} \otimes K_a ,\label{ch} \nonumber \end{eqnarray} \bea \begin{array}{lll} \left\{J_a,J_b\right\}=\epsilon_{abc}J_c ,& \quad \left\{J_a,P_b\right\}=\epsilon_{abc}P_c , &\quad \left\{J_a,K_b\right\}=\epsilon_{abc}K_c , \\[3pt] \displaystyle{ \left\{K_a,P_0\right\}=0 } , &\quad { \displaystyle{ \left\{K_a,P_b\right\}=\delta_{ab} \,\frac{1-{\rm e}^{-2 P_0/\kappa}}{2/\kappa} ,}} &\quad \displaystyle{\left\{K_a,K_b\right\}= 0} , \\[3pt] \left\{P_0,P_a\right\}=0 , &\quad \left\{P_a,P_b\right\}=0 , &\quad \left\{P_0,J_a\right\}=0 ,\end{array} \label{ci} \eea \be {\cal C_\kappa}=2\kappa^2\left( \cosh(P_0/\kappa)-1\right) . \label{cj} \ee We stress that the full quantization providing the three $\kappa$-Carrollian quantum algebras is immediate since the time translation $P_0$ and the Carrollian boosts do commute among themselves for any value of $\ro$, so no ordering problems appear and the Poisson brackets can be replaced by commutators. We also remark that the resulting quantum algebras are endowed with a bicrossproduct structure~\cite{MR1994,Majida,Majidb,Azcarragab} but now with respect to $ \mathbb{R}'^4=\langle P_0,\>K\rangle$ (see (\ref{an})). Finally, we recall that twist deformations of the Carroll algebra have recently been constructed in~\cite{Daszkiewicz2019}. They are different from the deformations constructed in this work since here the time translation generator $P_0$ plays a prominent role. \section{The $\kappa$-Newtonian and $\kappa$-Carrollian noncommutative\\ spacetimes} \label{sec5} In the previous sections we have obtained the $\kappa$-Newtonian and $\kappa$-Carrollian algebras. Now we will deduce their associated noncommutative spacetimes as the corresponding non-relativistic and ultra-relativistic contractions of the noncommutative $\kappa$-AdS$_\ka$ spacetime, which has been recently constructed in~\cite{BGH2019spacetime}. As we will see in the sequel, the LBC approach will also provide all the tools we need to this aim. Let us recall that the $\kappa$-AdS$_\ka$ Poisson homogeneous spacetime (PHS) was obtained in~\cite{BGH2019spacetime} through the Sklyanin bracket coming from the $r$-matrix (\ref{ba}) by restricting it to the Poisson subalgebra generated by the local spacetime coordinates $x^0$ and $x^a$, which are the group parameters of the time translation $P_0$ and space translations $P_a$, respectively. The resulting semiclassical $\kappa$-AdS$_\ka$ spacetime is given by the following Poisson brackets \begin{align} \begin{split} \label{da} &\{x^1,x^0\} =\frac{1}{\kappa}\, \frac{\tanh (\ro x^1)}{\ro \cosh^2(\ro x^2) \cosh^2(\ro x^3)} ,\\ &\{x^2,x^0\} =\frac{1}{\kappa}\,\frac{\tanh (\ro x^2)}{\ro \cosh^2(\ro x^3)} ,\\ &\{x^3,x^0\} =\frac{1}{\kappa}\,\frac{\tanh (\ro x^3)}{\ro}, \end{split} \end{align} \begin{align} \begin{split} \label{db} &\{x^1,x^2\} =-\frac{1}{\kappa}\,\frac{\cosh (\ro x^1) \tanh ^2(\ro x^3)}{\ro} ,\\ &\{x^1,x^3\} =\frac{1}{\kappa}\,\frac{\cosh (\ro x^1) \tanh (\ro x^2) \tanh (\ro x^3)}{\ro} ,\\ &\{x^2,x^3\} =-\frac{1}{\kappa}\,\frac{\sinh (\ro x^1) \tanh (\ro x^3)}{\ro} \, , \end{split} \end{align} which is a nonlinear deformation of the $\kappa$-Minkowski spacetime, which is recovered as the $\ro\to 0$ limit of the previous expressions, namely \be \{x^a,x^0\} =\frac{1}{\kappa}\, x^a ,\qquad \{x^a,x^b\} = 0. \label{dc} \ee We stress that since the Poisson brackets among the space sector (\ref{db}) do not vanish when $\ro\ne 0$, the quantization of the $\kappa$-AdS$_\ka$ PHS has to be performed by fixing a precise ordering. As it was shown in~\cite{BGH2019spacetime}, a suitable ordering allows the quantum $\kappa$-AdS$_\ka$ noncommutative spacetime to be expressed as a homogeneous quadratic algebra, provided that ambient coordinates are considered. Let us now focus on the obtention of the $\kappa$-Newtonian noncommutative spacetimes, whose associated Poisson--Hopf algebras were obtained in Section~\ref{sec3}. Since these are non-coboundary deformations with no underlying classical $r$-matrix, the use of the Sklyanin bracket in order to get the noncommutative spacetimes is precluded. Nevertheless the corresponding PHS can be deduced as the non-relativistic limit of the $\kappa$-AdS$_\ka$ one, since it is well-known that the clue for performing the contraction at the group level is to impose that the pairing between local coordinates and algebra generators has to be preserved (in other words, products $x^0P_0$ and $x^aP_a$ must be invariant under the contraction mapping, see~\cite{BHOS1995jmp} for details). In this way the contraction map (\ref{add}) induces the following transformation law for the Newtonian coordinates \be x^0\to x^0,\qquad x^a\to c \, x^a, \label{dd} \ee and the non-relativistic limit of the Poisson brackets (\ref{da}) and (\ref{db}) can be obtained by applying (\ref{dd}) onto them, and recalling that in this fundamental LBC the quantum deformation parameter $\kappa$ is not affected by the contraction. In this way, the $c\to\infty$ limit of the transformed Poisson brackets leads to the $\kappa$-Newtonian Poisson homogeneous spacetimes: \be \{x^a,x^0\} =\frac{1}{\kappa}\, x^a ,\quad\ \{x^1,x^2\} =-\frac{\ro}{\kappa}\, (x^3)^2 ,\quad\ \{x^1,x^3\} =\frac{\ro}{\kappa}\, x^2 x^3 ,\quad\ \{x^2,x^3\} =-\frac{\ro}{\kappa}\, x^1 x^3 . \label{ddd} \ee Note that~\eqref{ddd} are identical to the first-order in $\ro$ of the full $\kappa$-AdS$_\ka$ expressions (\ref{da})--(\ref{db}), which means that the non-relativistic contraction eliminates all higher order contributions in the cosmological constant parameter. Furthermore, the non-relativistic limit does not produce a complete separation between the space and time coordinates, as happens in the classical case. Here, quantum effects produce a residual mixing in the form of non-trivial brackets between the time and space sectors of spacetime. This is something that was already noticed in the Snyder noncommutative spacetime model \cite{SnyderG}. Furthermore, the joint effect of the quantum deformation and curvature parameters is still visible in the spatial coordinates brackets. Further comments on this are below. Now, by resorting to~\cite{BGH2019spacetime} we realize that this quadratic algebra can straightforwardly be quantized by considering the ordered monomials $(\hat x^1)^l\,(\hat x^3)^m\,(\hat x^2)^n$, and the quantum $\kappa$-Newtonian noncommutative spacetimes are thus given by the commutation rules \be [\hat x^a,\hat x^0] =\frac{1}{\kappa}\, \hat x^a,\qquad [\hat x^1,\hat x^2] =-\,\frac{\ro}{\kappa}\,(\hat x^3)^2, \qquad [\hat x^1,\hat x^3] =\,\frac{\ro}{\kappa}\,\hat x^3 \hat x^2,\qquad [\hat x^2,\hat x^3] =-\,\frac{\ro}{\kappa}\,\hat x^1\hat x^3 , \label{de} \ee which define a noncommutative and associative (Jacobi identities hold) homogeneous quadratic algebra. We stress that the time-space commutators $ [\hat x^a,\hat x^0] $ in~\eqref{de} are the same as in the $\kappa$-Minkowski spacetime $\>{M}^{3+1}_\kappa$ (\ref{dc}), but the noncommutative space coordinates define a quadratic subalgebra which, in the same manner as in the $\kappa$-AdS$_\ka$ case, can be identified~\cite{Grabowski1990brno,Grabowski1995} with a subalgebra of Woronowicz's quantum ${\rm SU}(2)$ group~\cite{Woronowicz1987tsu2,Woronowicz1987cmp,VaksmanSoibelman1988,ChaichianDemichev1996book}. In fact, there does exist a Casimir operator for the Newtonian space subalgebra, which is given by \be \hat S_{\eta/\kappa}=(\hat x^1)^2 + (\hat x^2)^2 + (\hat x^3)^2 + \frac{\ro}{\kappa}\, \hat x^1 \hat x^2,\qquad [\hat S_{\ro/\kappa} , \hat x^a]=0, \label{df} \ee that can be interpreted as the definition of a ``quantum sphere" in the 3-space, onto which the time operator $\hat x^0$ acts as a dilation: \be [ \hat x^0,\hat S_{\ro/\kappa}]=-\frac 2\kappa\, \hat S_{\eta/\kappa} . \label{dg} \ee We remark that the ultimate responsible of the non-vanishing spatial commutators and of the existence of the operator $\hat S_{\eta/\kappa}$ is the quantum $\mathfrak{su}(2)\simeq \mathfrak{so}(3)$ subalgebra generated by the rotation generators $\>J$ (see (\ref{be}) and (\ref{bi})) appearing in the $\kappa$-Newtonian deformation, which is a characteristic feature of the $\kappa$-AdS$_\ka$ deformation and turns out to be preserved under the non-relativistic limit. Note also that the $\kappa$-Galilei noncommutative spacetime $\>{G}^{3+1}_\kappa$ with $\ro=0$ is much more degenerate and has the very same abelian spatial sector as the $\kappa$-Minkowski spacetime (\ref{dc}). As far as the $\kappa$-Carrollian Poisson--Hopf algebras deduced in Section~\ref{sec4} are concerned, there are two different but equivalent ways to obtain their associated PHS. On the one hand, since they are coboundary deformations one can make use of the Sklyanin bracket with classical $r$-matrix (\ref{cc}), which requires left- and right-invariant vector fields to be computed. On the other hand, one can directly perform the ultra-relativistic limit of the $\kappa$-AdS$_\ka$ PHS by following the same approach as in the Newtonian cases, based on the fundamental LBC (which in this case coincides with the coboundary one). Therefore, from the LBC maps (\ref{aj}) and (\ref{cb}) the following transformation on the group parameters are induced \be x^0\to c^{-1}\, x^0,\qquad x^a\to x^a,\qquad \kappa\to c\,\kappa . \label{dh} \ee If we apply~\eqref{dh} to the $\kappa$-AdS$_\ka$ Poisson brackets (\ref{da}) and (\ref{db}) and compute the limit $c\to 0$, the $\kappa$-Carrollian PHS are obtained, namely \begin{align} \begin{split} \label{di} &\{x^1,x^0\} =\frac{1}{\kappa}\, \frac{\tanh (\ro x^1)}{\ro \cosh^2(\ro x^2) \cosh^2(\ro x^3)} ,\\ &\{x^2,x^0\} =\frac{1}{\kappa}\,\frac{\tanh (\ro x^2)}{\ro \cosh^2(\ro x^3)} ,\\ &\{x^3,x^0\} =\frac{1}{\kappa}\,\frac{\tanh (\ro x^3)}{\ro},\\ &\{x^a,x^b\} =0. \end{split} \end{align} We stress that within these spaces the spatial coordinates $x^a$ do Poisson commute, which implies that no ordering ambiguities appear and the $\kappa$-Carrollian noncommutative spacetime is just given by (\ref{di}) by replacing the Poisson brackets by commutators. Notice that in this case there is no quantum $\mathfrak{su}(2)\simeq \mathfrak{so}(3)$ subalgebra and the rotation generators remain non-deformed (see (\ref{ce}) and (\ref{cf})). This is due to the fact that the term $J_1\wedge J_2$ in the $r$-matrix (\ref{ba}) disappears under the ultra-relativistic contraction to the $r$-matrix (\ref{cc}). Moreover, once more, the proper $\kappa$-Carroll noncommutative spacetime $\>{C}^{3+1}_\kappa$ with $\ro =0$ coincides with $\>{M}^{3+1}_\kappa$ (\ref{dc}). \section{Concluding remarks} \label{sec6} While it has been understood for a while that Planck scale effects might have a non-trivial interplay with curvature effects, their relation to relativistic effects was not investigated before. Here, we provide a first characterization of such interplay by studying $\kappa$-deformed spacetime models with non-vanishing cosmological constant (and correspondingly deformed symmetries) in the non-relativistic $c\to\infty$ and ultra-relativistic $c\to0$ limits. In this way we can categorize the effects that are of purely quantum origin (i.e.~dependent on the quantum deformation parameter $\kappa$), those that are due to the interplay between the quantum deformation and curvature parameter $\eta$, and those that are also affected by the speed of light parameter $c$. The first kinds of effects are those that are well-known from the study of the $\kappa$-Minkowski spacetime and $\kappa$-Poincar\'e symmetries. Most prominently, the noncommutativity between the time and space coordinates and a deformation of the algebra of boosts and spatial translations. Effects that are due to the interplay between $\kappa$ and $\eta$ are the deformation of the algebra of rotations and the noncommutativity between spatial coordinates. In this paper, we found that the speed of light does not affect those effects that are purely quantum (since they do not appear in qualitatively different ways in either the non-relativistic limit or the ultra-relativistic limit). In particular, the noncommutativity between time and space coordinates is still present in the two limits, despite the fact that classically one would expect a complete separation between the time and space sectors of spacetime. In this sense, quantum effects are ``stronger'' than relativistic effects. However, $c$ does enter into the picture in a significant way when considering the joint $\kappa$ and $\eta$ effects. For example, the induced modification of the algebra of rotation and the noncommutativity between spatial coordinates are both preserved in the non-relativistic limit, but are lost in the ultra-relativistic limit. It is worth stressing that the three flat cases with $\ka=\ro =0$ (Minkowski $\>{M}^{3+1}_\kappa$, Galilei $\>{G}^{3+1}_\kappa$ and Carroll $\>{C}^{3+1}_\kappa$) share the same noncommutative spacetime algebra. Therefore it seems rather natural to wonder about the additional symmetry structure that could allow us to distinguish the different features of these three models from a phenomenological perspective. Indeed, the three Hopf algebra structures providing the quantum group invariance of the three spaces are different, and this will be reflected in the properties of their associated curved momentum spaces, as it has been recently analysed in~\cite{Flavio}. Another framework that might allow to distinguish the three cases is that of noncommutative spaces of worldlines associated to a given quantum deformation. This space has been recently constructed for the $\kappa$-Minkowski spacetime in~\cite{BGH2019worldlinesplb}. In fact, the construction of the $\kappa$-Galilei and $\kappa$-Carroll noncommutative spaces of geodesics might give rise to very different geometric structures, which are worth to be studied. Work on this line is currently in progress and will be presented elsewhere. \section*{Acknowledgements} \small This work has been partially supported by Ministerio de Ciencia, Innovaci\'on y Universidades (Spain) under grant MTM2016-79639-P (AEI/FEDER, UE) and by Junta de Castilla y Le\'on (Spain) under grants BU229P18 and BU091G19. The authors acknowledge the contribution of the COST Action CA18108. \small

proofpile-arXiv_069-13951

\section{Introduction} This paper studies the cohomology theory, abelian extensions and formal deformations for differential algebras of any weights. \subsection{Differential algebras, old and new} Classically, a differential algebra is a commutative algebra equipped with a linear operator satisfying the Leibniz rule, modeled after the differential operator in analysis. In fact, the origin of differential algebras is the algebraic study of differential equations pioneered by Ritt in the 1930s~\cite{Rit,Rit1}. Through the work of many mathematicians~\cite{Kol,Mag,PS} in the following decades, the subject has been fully developed into a vast area in mathematics, comprising of differential Galois theory, differential algebraic geometry and differential algebraic groups, with broad connections to other areas in mathematics such as arithmetic geometry and logic, as well as computer science (mechanical proof of geometric theorems) and mathematical physics (renormalization in quantum field theory)~\cite{CM,FLS,MS,Wu,Wu2}. Another broadly used notion of differential operators is the derivations on (co)chain complexes. There the operator $d$ is assumed to satisfy the nilpotent condition $d^2=0$, in which case a differential algebra induces an associative diassociative algebra structure~\mcite{Lo01}, in analog to the case of Rota-Baxter algebras that induce dendriform algebras and tridendriform algebras. In recent years, differential algebras without the commutative or nilpotent conditions have been considered, to include naturally arisen algebras such as path algebras and to have a more meaningful differential Lie algebra theory generalizing the classical relationship between associative algebras and Lie algebras~\cite{GL,Poi,Poi1}. In~\mcite{Loday} differential algebra is studied from an operadic point of view. Differential algebras have also been applied to control theory and gauge theory through the BV-formalism~\mcite{AKT,Va}. In~\cite{ATW}, a notion of differential algebras were generalized to non(anti)commutative superspace by deformation in the study of instantons in string theory. In another direction, the Leibniz rule is generalized to include the difference quotient $\frac{f(x+\lambda)-f(x)}{\lambda}$ before taking the limit $\lambda\mapsto 0$, leading to the notion of a differential algebra of weight $\lambda$~\cite{GK}. This generalized notion of differential algebra provides a framework for a uniform approach of the differential algebra (corresponding to the case when $\lambda=0$ and another extensively studied algebraic structure, the difference algebra (corresponding to the case when $\lambda=1$)~\cite{Coh,Lev,PS1}, as an algebraic study of difference equations. This notion also furnishes an algebraic context for the study of quantum calculus~\mcite{KC}. Differential operators with weights on Virasoro algebras were also investigated~\mcite{LGG}. \subsection{Homology and deformations of differential algebras} As noted above, nilpotent differential operators are fundamental notions for complexes to define cohomology which in turn plays a key role in deformation theory, either for specific algebraic structures, starting with the seminal works of Gerstenhaber for associative algebras and of Nijenhuis and Richardson for Lie algebras~\mcite{Ge0,Ge,NR}, or for the general context of operads, culminated in the monographs~\mcite{MSS,LV}. As a further step in this direction, studies of deformations and the related cohomology have recently emerged for algebras with linear operators, including Rota-Baxter operators and differential operators on Lie algebras~\mcite{TBGS,TFS}. The importance of differential (associative) algebras makes it compelling to develop their cohomology theory, in both the zero weight case and nonzero weight case. The natural role played by nilpotent differential operators in cohomology makes it even more fascinating in this study. The purpose of this paper is to develop such a theory for differential (associative) algebras of any weight, and give its applications in the study of abelian extensions and formal deformations of differential algebras. In comparison with the recent work~\mcite{TFS} on cohomology and deformation of differential Lie algebras, we note that the derivations there are of weight zero in which case its approach can be adapted for differential (associative) algebras. Our main emphasis on differential algebras in this paper is for the nonzero weight case, which is needed in order to study difference operators and difference algebras, but for which a different approach has to be taken. See comments in the outline below and Remark~\mref{rk:nonzero}. \subsection{Layout of the paper} The paper is organized as follows. In Section \ref{sec:bimod}, we introduce the notion of a bimodule over a differential algebra of nonzero weight, and provide its characterization in terms of a monoid object in slice categories. A differential algebra is the combination of the underlying algebra and the differential operator. In this light we build the cohomology theory of a differential algebra by combining its components from the algebra and from the differential operator. Thus in Section \ref{sec:cohomologydo}, we establish the cohomology theory for differential operators of any weights, which is quite different from the one for the underlying algebra unless the weight is zero. In Section \ref{sec:cohomologydasub}, we combine the Hochschild cohomology for associative algebras and the just established cohomology for differential operators of any weights to define the cohomology of differential algebras of any weights, with the cochain maps again posing extra challenges when the weight is not zero. Finally in Section \ref{sec:cohomologyrelation}, we establish a close relationship among these cohomologies. More precisely, we show that there is a short exact sequence of cochain complexes for the algebra, the differential operator and the differential algebra. The resulting long exact sequence gives linear maps from the cohomology groups of the differential algebra to those of the algebra, with the error terms (kernels and cokernels) controlled by the cohomology groups of the differential operator. As applications and further justification of our cohomology theory for differential algebras, in Section \ref{sec:ext}, we apply the theory to study abelian extensions of differential algebras of any weights, and show that abelian extensions are classified by the second cohomology group of the differential algebras. Further, in Section \ref{sec:def}, we apply the above cohomology theory to study formal deformations of differential algebras of any weights. In particular, we show that if the second cohomology group of a differential algebra with coefficients in the regular representation is trivial, then this differential algebra is rigid. \smallskip \noindent {\bf Notation.} Throughout this paper, $\bk$ denotes a field of characteristic zero. All the vector spaces, algebras, linear maps and tensor products are taken over $\bk$ unless otherwise specified. \section{Differential algebras and their bimodules} \label{sec:bimod} This section gives background on differential algebras and first results on their bimodules, with an interpretation in the general context of monoid objects in slice categories~\mcite{DMZ}. \subsection{The category of bimodules over differential algebras} \begin{defn} (\cite{GK}) Let $\lambda\in \bk$ be a fixed element. A {\bf differential algebra of weight $\lambda$} (also called a {\bf $\lambda$-differential algebra}) is an associative algebra $A$ together with a linear operator $d_A : A\rar A$ such that \begin{equation}\label{diff} d_A(xy)=d_A(x)y+x d_A(y)+\lambda d_A(x)d_A(y), \quad\forall x,y\in A. \end{equation} If $A$ is unital, it further requires that \begin{equation}\label{difu} d_A(1_A)=0. \end{equation} Such an operator is called a {\bf differential operator of weight $\lambda$} or a {\bf derivation of weight $\lambda$}. It is also called a {\bf $\lambda$-differential operator} or a {\bf $\lambda$-derivation}. Given two differential algebras $(A, d_A),\,(B,d_B)$ of the same weight $\lambda$, a {\bf homomorphism of differential algebras} from $(A,d_A)$ to $(B,d_B)$ is an algebra homomorphism $\varphi:A\lon B$ such that $\varphi\circ d_A=d_B\circ\varphi$. We denote by $\Diffl$ the category of $\lambda$-differential algebras. \end{defn} To simply notations, for all the above notions, we will often suppress the mentioning of the weight $\lambda$ unless it needs to be specified. Recall that a bimodule of an associative algebra $A$ is a triple $(V,\rho_l,\rho_r)$, where $V$ is a vector space, $\rho_l: A\to \mathrm{End}_\bk(V),~ x\mapsto (v\mapsto xv)$ and $\rho_r: A\to \mathrm{End}_\bk(V),~ x\mapsto (v\mapsto vx)$ are homomorphism and anti-homomorphism of associative algebras respectively such that $(xv)y=x(vy)$ for all $x,y\in A$ and $v\in V$. \begin{defn} Let $(A, d_A)$ be a differential algebra. \begin{itemize} \item[{\rm (i)}] A {\bf bimodule} over the differential algebra $(A, d_A)$ is a quadruple $(V,\rho_l,\rho_r,d_V)$, where $d_V\in \mathrm{End}_\bk(V)$, and $(V,\rho_l,\rho_r)$ is a bimodule over the associative algebra $A$, such that for all $x,y\in A, v\in V,$ the following equalities hold: \begin{eqnarray*} d_V(xv)&=&d_A(x)v+xd_V(v)+\lambda d_A(x)d_V(v),\\ d_V(vx)&=&v d_A(x)+d_V(v)x+\lambda d_V(v)d_A(x). \end{eqnarray*} \item[{\rm (ii)}] Given two bimodules $(U,\rho^U_l,\rho^U_r, d_U),\,(V,\rho^V_l,\rho^V_r, d_V)$ over $(A, d_A)$, a linear map $f:U\lon V$ is called a {\bf homomorphism} of bimodules, if $f\circ d_U=d_V\circ f$ and $$f\circ\rho^U_l(x)=\rho^V_l(x)\circ f,\quad f\circ\rho^U_r(x)=\rho^V_r(x)\circ f,\quad \forall x\in A.$$ \end{itemize} \end{defn} We denote by $(A,d_A)\bim$ the category of bimodules over the differential algebra $(A, d_A)$. \begin{exam} Any differential algebra $(A, d_A)$ is a bimodule over itself with $$\rho_l:A\to \mathrm{End}_\bk(A),\,x\mapsto(y\mapsto xy),\quad \rho_r: A\to \mathrm{End}_\bk(A),\,x\mapsto(y\mapsto yx).$$ It is called the {\bf regular bimodule} over the differential algebra $(A, d_A)$. \end{exam} It is straightforward to obtain the following result. \begin{prop} Let $(V,\rho_l,\rho_r, d_V)$ be a bimodule of the differential algebra $(A, d_A)$. Then $(A\oplus V, d_A\oplus d_V)$ is a differential algebra, where the associative algebra structure on $A\oplus V$ is given by $$ (x+u)(y+v)=xy+xv+uy,\quad \forall x,y\in A,~u,v\in V. $$ \end{prop} \subsection{Differential bimodules in terms of monoid objects in slice categories} We now show that the above definition of bimodules for differential algebras coincides with the notion obtained by applying monoid objects in certain slice categories. \subsubsection{Monoid objects in slice categories} We first recall some general concepts~\mcite{DMZ}. \begin{defn} For a category $\calc$ and an object $A$ in $\calc$. The {\bf slice category} $\calc/A$ is the category whose \begin{itemize} \item objects $(B,\pi)$ are $\calc$-morphisms $\pi:B\,\rightarrow\, A,B\in\calc$, and \item morphisms $(B',\pi')\stackrel{f}{\,\rightarrow\,}(B'',\pi'')$ are commutative diagrams of $\calc$-morphisms: \[\begin{CD} B'@>f>>B''\\ @V\pi' VV @VV\pi''V \\ A@=A. \end{CD}\] \end{itemize} \end{defn} \begin{defn} Let $\calc$ be a category with finite products and a terminal object $T$. A {\bf monoid} object in $\calc$ is an object $X\in \ob(\calc)$ together with two morphisms $\mu:X\times X\lon X$ and $\eta:T\lon X$ such that following diagrams commute: \begin{itemize} \item the {\bf associativity} of $\mu:$ \[\begin{CD}\label{neut} X\times X\times X@>\mu\times {\Id}_X>>X\times X\\ @V{\Id}_X\times \mu VV @VV\mu V \\ X\times X@>>\mu >X, \end{CD}\] \item the {\bf neutrality} of $\eta:$ \[\begin{CD} X\times X@>\mu>> X @<\mu<< X\times X \\ @A{\Id}_X\times \eta AA @| @AA\eta\times {\Id}_X A \\ X\times T@<<({{\Id}_X},t_X) < X @>>(t_X,{{\Id}_X})> T\times X, \end{CD}\] \end{itemize} where $t_X:X\lon T$ is the unique morphism. Let $\calc_\m$ be the category whose objects are monoid objects $(X,\mu,\eta)$ in $\calc$ as above and the hom-set $\Hom_{\calc_\m}((X,\mu,\eta),(X',\mu',\eta'))$ is the set of all $f\in\Hom_{\calc}(X,X')$ for which $\mu'\circ (f\times f)=f\circ \mu$ and $\eta'=f\circ\eta$. \end{defn} \subsubsection{The differential algebra case} Fix a differential algebra $(A,d)$, and consider the slice category $\Diffl/A$. The terminal object in $\Diffl/A$ is $T=A\stackrel{\Id}{\lon}A$. Given $(A_1,d_1),\,(A_2,d_2)\in\Diffl$ and $X_1=(A_1,\varphi_1),\,X_2=(A_2,\varphi_2)\in\Diffl/A$, the product $X_1\times X_2$ is given by $(A_1\times_A A_2,\bar{\varphi})$, where \[A_1\times_A A_2:=\{(a,a')\in A_1\times A_2\,|\,\varphi_1(a)=\varphi_2(a')\},\] and \[\bar{\varphi}:A_1\times_A A_2\lon A,\quad (a,a')\mapsto \varphi_1(a).\] For any $(A',d')\in\Diffl$, $X=A'\stackrel{\varphi}{\lon}A\in\Diffl/A$ is a monoid object if and only if there exist differential algebra homomorphisms $M:A'\times_A A'\lon A'$ and $\iota:A\lon A'$ such that \begin{align*} \bar{\varphi}=\varphi M,\quad \varphi\iota=\Id_A,\quad M(M(a,b),c)=M(a,M(b,c)),\quad M(\iota(\varphi(a)),b)=b,\quad M(a,\iota(\varphi(b)))=a. \end{align*} for any $(a,b,c)\in A'\times_A A'\times_A A'$. Consequently, \begin{align*} M(a,a')&=M(a-\iota(\varphi(a)),0)+M(0,a'-\iota(\varphi(a'))) +M(\iota(\varphi(a)),\iota(\varphi(a')))\\ &=a-\iota(\varphi(a))+a'-\iota(\varphi(a')) +\iota(\varphi(a'))\\ &=a+a'-\iota(\varphi(a)), \end{align*} for any $(a,a')\in A'\times_A A'$. Let $V:=\ker\varphi$ as a differential ideal of $A'$. Then $A'=\iota(A)\oplus V$. Since $M$ is also a differential algebra homomorphism, we have \[uv=M(u,0)M(0,v)=M((u,0)(0,v))=M(0,0)=0,\] for any $u,v\in V$. On the other hand, as $\iota d=d'\iota$ and $\varphi d'=d\varphi$, it is clear that $V$ is an $A$-bimodule with differential $d_V:=d'|_V$ by letting \[\rho_l(x)v=\iota(x)v,\,\rho_r(x)v=v\iota(x),\] for any $x\in A,\,v\in V$. Conversely, given any bimodule $(V,d_V)$ of differential algebra $(A,d_A)$, one can define a differential algebra structure on $A\oplus V$ naturally with differential operator $(d_A,d_V)$ by letting \[(x,u)(y,v)=(xy,xv+uy),\quad \forall x,y\in A,\,u,v\in V.\] We say this differential algebra the {\bf semi-direct product} of $A$ and $V$, and denote it as $(A\ltimes V,d_\ltimes)$. Let $p:A\ltimes V\lon A$ be the canonical projection, and associate it with \[(A\ltimes V)\times_A(A\ltimes V)=\{((x,u),(x,v))\in (A\ltimes V)\times(A\ltimes V)\,|\,x\in A,\,u,v\in V\}.\] Define differential algebra homomorphisms $$M_\ltimes:(A\ltimes V)\times_A(A\ltimes V)\lon A\ltimes V,\,((x,u),(x,v))\mapsto(x,u+v)$$ and $$i_\ltimes:A\lon A\ltimes V,\,x\mapsto(x,0).$$ Then the following lemma is easy to check. \begin{lemma} For any $A$-bimodule $V$, $X=A\ltimes V\stackrel{p}{\lon}A$ with morphisms $\mu_\ltimes,\eta_\ltimes$ determined by $M_\ltimes,i_\ltimes$ respectively is a monoid object in $\Diffl/A$. \end{lemma} \begin{theorem} The functors $\xymatrix@=2em{(\Diffl/A)_\m \ar@/^/[r]^-{\ker} &(A,d_A)\bim\ar@/^/[l]^-{\ltimes} }$ induce an equivalence of categories. \mlabel{thm:monoid} \end{theorem} \begin{proof} First note that $\ker\circ\,\ltimes=\Id_{(A,d_A)\bim}$. Define $\Psi:\Id_{(\Diffl/A)_\m}\lon\ltimes\circ\ker$ by letting \[\Psi_X:(A'\stackrel{\varphi}{\lon}A,M,\iota)\lon (A\ltimes\ker \varphi\stackrel{p}{\lon}A,M_\ltimes,i_\ltimes),\quad a\mapsto(\varphi(a),a-\iota(\varphi(a))).\] for any $X=A'\stackrel{\varphi}{\lon}A\in(\Diffl/A)_\m$. By definition $\varphi=p\circ\Psi_X$, thus $\Psi_X$ is a morphism in $(\Diffl/A)_\m$. The inverse of $\Psi_X$ is \[\Psi_X^{-1}:(A\ltimes\ker \varphi\stackrel{p}{\lon}A,M_\ltimes,i_\ltimes)\lon(A'\stackrel{\varphi}{\lon}A,M,\iota),\quad (x,u)\mapsto u+\iota(x).\] Also, for any $(a,a')\in A'\times_A A'$, $x\in A$, we have \begin{eqnarray*} M_\ltimes(\Psi_X(a),\Psi_X(a')) &=&M_\ltimes((\varphi(a),a-\iota(\varphi(a))),(\varphi(a'),a'-\iota(\varphi(a'))))\\ &=&(\varphi(a),a+a'-2\iota(\varphi(a)))\\ &=&\Psi_X(a+a'-\iota(\varphi(a)))=\Psi_X(M(a,a')),\\ \Psi_X(\iota(x))&=&(\iota(x),\iota(x)-\iota(\varphi(\iota(x))))=(\iota(x),0)=i_\ltimes(x),\\ \Psi_X(d'(a))&=&(\varphi(d'(a)),d'(a)-\iota(\varphi(d'(a))))=(d(\varphi(a)),d'(a-\iota(\varphi(a))))\\ &=&(d,d')\Psi_X(a). \end{eqnarray*} For any $X_1=A_1\stackrel{\varphi_1}{\lon}A,\,X_2=A_2\stackrel{\varphi_2}{\lon}A\in(\Diffl/A)_\m$ and morphism $f:X_1\lon X_2$, we have \begin{eqnarray*} \Psi_{X_2} (f(a))&=&(\varphi_2(f(a)),f(a)-\iota_2(\varphi_2(f(a))))\\ &=&(\varphi_1(a),f(a)-\iota_2(\varphi_1(a)))\\ &=&(\varphi_1(a),f(a-\iota_1(\varphi_1(a))))\\ &=&(\Id_A,f)(\Psi_{X_1}(a)). \end{eqnarray*} Hence, $\Psi$ is a natural isomorphism. The categories $(\Diffl/A)_\m$ and $(A,d_A)\bim$ are equivalent to each other. \end{proof} \section{Cohomology of differential algebras}\label{sec:cohomologyda} Let $V$ be a bimodule of an associative algebra $A$. Denote by $C^n_\alg(A,V)=\Hom(\otimes^n A,V)$. In particular, $C^0_\alg(A,V)=V.$ Recall that the Hochschild cochain complex is the cochain complex $(C^*_\alg(A,V)=\oplus_{n=0}^\infty C^n_\alg(A,V),\partial)$, where the coboundary operator $$\partial: C^n_\alg(A, V)\longrightarrow C^{n+1}_\alg(A, V)$$ is given by \[\begin{split} \partial f(x_1,\dots,x_{n+1})&=x_1 f(x_2,\dots,x_{n+1})+\sum_{i=1}^n(-1)^if(x_1,\dots,x_ix_{i+1},\dots,x_{n+1})\\ &+(-1)^{n+1}f(x_1,\dots,x_n) x_{n+1} \end{split}\] for all $f\in C^n_\alg(A, V),~x_1,\dots,x_{n+1}\in A$. The corresponding Hochschild cohomology is denoted by $HH^*_\alg(A, V)$ \subsection{Cohomology of differential operators}\label{sec:cohomologydo} Let $(A,d_A)$ be a differential algebra of weight $\lambda$ and let $(V, d_V)$ be a bimodule over $(A, d_A)$. In this subsection, we define the cohomology of differential operators. First we make the following observation, noting that the bimodule structure coincides with the regular bimodule when the weight $\lambda$ is zero. \begin{lemma}\label{le:bmd} A differential algebra $(A,d_A)$ admits a new bimodule structure on $(V,d_V)$ given by: \[x\vdash_\lambda v=(x+\lambda d_A(x))v,\quad v\dashv_\lambda x=v(x+\lambda d_A(x)),\quad \forall x \in A, v\in V.\] \end{lemma} \begin{proof} Given $x,y\in A$ and $v\in V$, we have \[\begin{split} x\vdash_\lambda (y\vdash_\lambda v)&=(x+\lambda d_A(x))((y+\lambda d_A(y))v)\\ &=(xy+\lambda(xd_A(y)+d_A(x)y+\lambda d_A(x)d_A(y)))v\\ &=(xy+\lambda d_A(xy))v\\ &=(xy)\vdash_\lambda v. \end{split}\] Similarly, $(v\dashv_\lambda x)\dashv_\lambda y=v\dashv_\lambda (xy)$. Thus $(V,\vdash_\lambda,\dashv_\lambda)$ is a bimodule over the associative algebra $A$. For $x\in A$ and $v\in V$, \[\begin{split} d_V(x\vdash_\lambda v)&=d_V((x+\lambda d_A(x))v)\\ &=d_V(xv)+\lambda d_V(d_A(x)v)\\ &=d_A(x)v+xd_V(v)+\lambda d_A(x)d_V(v)+\lambda (d_A(d_A(x))v+d_A(x)d_V(v)+\lambda d_A(d_A(x))d_V(v))\\ &=d_A(x)v+\lambda d_A(d_A(x))v+xd_V(v)+\lambda d_A(x)d_V(v)+\lambda ( d_A(x)d_V(v)+\lambda d_A(d_A(x))d_V(v))\\ &= d_A(x)\vdash_\lambda v+x\vdash_\lambda d_V(v)+\lambda d_A(x)\vdash_\lambda d_V(v). \end{split}\] Similarly, one shows the equality $$d_V( v\dashv_\lambda x)=v \dashv_\lambda d_A(x)+ d_V(v)\dashv_\lambda x+\lambda d_V(v) \dashv_\lambda d_A(x).$$ Also, it is obvious that $(x\vdash_\lambda v)\dashv_\lambda y=x\vdash_\lambda (v\dashv_\lambda y)$. Thus, $(V,\vdash_\lambda,\dashv_\lambda,d_V)$ is a bimodule over the differential algebra $(A,d_A)$. \end{proof} For distinction, we let $V_\lambda$ denote the new bimodule structure over $(A, d_A)$ given in Lemma~\ref{le:bmd}. Denote by $C^n_\diffo(d_A,d_V)=\Hom(\otimes^n A,V)$, which is called the space of $n$-chains of the differential operator $d_A$ with coefficients in the bimodule $(V,d_V)$. \begin{defn}The cohomology of the cochain complex $(C^*_\diffo(d_A,d_V)=\oplus_{n=0}^\infty C^n_\diffo(d_A,d_V),\partial_\lambda)$, denoted by $H^*_\diffo(d_A,d_V)$, is called the {\bf cohomology of the differential operator} $d_A$ with coefficients in the bimodule $(V,d_V)$, where $$\partial_\lambda:C^n_\diffo(d_A,d_V)\longrightarrow C^{n+1}_\diffo(d_A,d_V)$$ is the Hochschild coboundary operator of the associative algebra $A$ with the coefficients in the bimodule $V_\lambda$ given in Lemma \ref{le:bmd}. More precisely, we have \[\begin{split} \partial_\lambda f(x_1,\dots,x_{n+1})&=x_1\vdash_\lambda f(x_2,\dots,x_{n+1})+\sum_{i=1}^n(-1)^if(x_1,\dots,x_ix_{i+1},\dots,x_{n+1})\\ &+(-1)^{n+1}f(x_1,\dots,x_n)\dashv_\lambda x_{n+1} \end{split}\] for all $f\in C^n_\diffo(d_A, d_V),~x_1,\dots,x_{n+1}\in A$. \end{defn} \subsection{Cohomology of differential algebras} \label{sec:cohomologydasub} We now combine the classical Hochschild cohomology of associative algebras and the newly defined cohomology of differential operators to define the cohomology of the differential algebra $(A,d_A)$ with coefficients in the bimodule $(V,d_V)$. Define the set of $n$-cochains by \begin{equation}\label{eq:dac} C_{\Diffl}^n(A,V):= \begin{cases} C^n_\alg(A,V)\oplus C^{n-1}_\diffo(d_A,d_V),&n\geq1,\\ C^0_\alg(A,V)=V,&n=0. \end{cases} \end{equation} Define a linear map $\partial_{\Diffl}:C_{\Diffl}^n(A,V)\rar C_{\Diffl}^{n+1}(A,V)$ by \begin{eqnarray}\label{eq:pda} \partial_{\Diffl}(f,g)&:=&(\partial f,\partial_\lambda g+(-1)^n\delta f),\quad \forall f\in C^n_\alg(A,V),\,g\in C^{n-1}_\diffo(d_A,d_V),\quad n\geq 1,\\ \label{eq:pda2}\partial_{\Diffl} {v}&:=&(\partial {v}, \delta v),\quad\forall {v}\in C^0_\alg(A,V)=V, \end{eqnarray} where the linear map $\delta:C^n_\alg(A,V)\rar C^n_\diffo(d_A,d_V)$ is defined by \begin{align*} \delta f(x_1,\dots,x_n):=\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_k}),\dots,x_n)-d_V f(x_1,\dots,x_n), \end{align*} for any $f\in C^n_\alg(A,V)$, $ n\geq 1$ and $$\delta v= {-} d_V {(v)},\quad \forall v\in C^0_\alg(A,V)=V.$$ \begin{prop}\mlabel{prop:delta} The linear map $\delta$ is a cochain map from the cochain complex $(C^*_\alg(A,V),\partial)$ to $(C^*_\diffo(d_A,d_V),\partial_\lambda)$. \end{prop} The rather long and technical proof of this result is is postponed to the appendix in order not to interrupt the flow of the presentation. \begin{remark} Note that $C^n_\diffo(d_A,d_V)$ equals to $C^n_\alg(A,V)$ as linear spaces but they are equal as cochain complexes only when $\lambda=0$. When $\lambda$ is not zero, a new bimodule structure is needed to define $\partial_\lambda$ which eventually leads to the rather long and technical argument in order to establish the cochain map in Proposition~\mref{prop:delta}. \mlabel{rk:nonzero} \end{remark} \begin{theorem}\label{thm: cochain complex for differential algebras} The pair $(C_{\Diffl}^*(A, V),\partial_{\Diffl})$ is a cochain complex. So $\partial_{\Diffl}^2=0.$ \end{theorem} \begin{proof} For any $v\in C^0_\alg(A, V)$, by Proposition~\mref{prop:delta} we have \[\partial_{\Diffl}^2 v=\partial_{\Diffl}(\partial v,\delta v)=(\partial^2 v,\partial_\lambda\delta v-\delta\partial v)=0.\] Given any $f\in C^n_\alg(A,V),\,g\in C^{n-1}_\diffo(d_A,d_V)$ with $n\geq1$, also by Proposition~\mref{prop:delta} we have \[\partial_{\Diffl}^2(f,g)=\partial_{\Diffl}(\partial f,\partial_\lambda g+(-1)^n\delta f)=(\partial^2 f,\partial_\lambda(\partial_\lambda g+(-1)^n\delta f)+(-1)^{n+1}\delta\partial f)=0.\] Therefore, $(C_{\Diffl}^*(A, V),\partial_{\Diffl})$ is a cochain complex. \end{proof} \begin{defn} The cohomology of the cochain complex $(C_{\Diffl}^*(A, V),\partial_{\Diffl})$, denoted by $H_{\Diffl}^*(A, V)$, is called the {\bf cohomology of the differential algebra} $(A,d_A)$ with coefficients in the bimodule $(V,d_V)$. \end{defn} To end this subsection, we compute 0-cocycles, 1-cocycles and 2-cocycles of the cochain complex $(C_{\Diffl}^*(A,V),\partial_{\Diffl})$. It is obvious that for all $v\in V$, $\partial_{\Diffl}v=0$ if and only if $$\partial v=0,\quad d_V(v)=0.$$ For all $(f,v)\in\Hom (A,V)\oplus V$, $\partial_{\Diffl}(f,v)=0$ if and only if $\partial f=0,$ and $$ x\vdash_\lambda v-v\dashv_\lambda x=f(d_A(x))-d_V(f(x)),\quad \forall x\in A. $$ For all $(f,g)\in\Hom (\otimes^2A,V)\oplus\Hom (A,V) $, $\partial_{\Diffl}(f,g)=0$ if and only if $\partial f=0,$ and \begin{align*} x\vdash_\lambda g(y)-g(xy)+g(x)\dashv_\lambda y= -\lambda f(d_A(x),d(y))-f(d_A(x),y)-f(x,d_A(y))+d_V(f(x,y)),\ \end{align*} for all $x,y\in A.$ In the next two sections, we shall need a subcomplex of the cochain complex $C_{\Diffl}^*(A, V)$. Let \begin{equation}\label{eq:dacsub} \tilde{C}_{\Diffl}^n(A,V):= \begin{cases} C^n_\alg(A,V)\oplus C^{n-1}_\diffo(d_A,d_V),&n\geq2,\\ C^1_\alg(A,V),&n=1,\\ 0,&n=0. \end{cases} \end{equation} Then it is obvious that $(\tilde{C}_{\Diffl}^*(A,V)=\oplus_{n=0}^\infty\tilde{C}_{\Diffl}^n(A,V),\partial_{\Diffl} )$ is a subcomplex of the cochain complex $(C_{\Diffl}^*(A, V),\partial_{\Diffl})$. We denote its cohomology by $\tilde{H}_{\Diffl}^*(A,V)$. Obviously, $\tilde{H}_{\Diffl}^n(A,V)= {H}_{\Diffl}^n(A,V)$ for $n>2$. \subsection{Relationship among the cohomologies}\label{sec:cohomologyrelation} The coboundary operator $\partial_{\Diffl}$ can be illustrated by the following diagram: \[ \small{ \xymatrix{ \cdots \longrightarrow C^n_\alg(A,V)\ar[dr]^{(-1)^n\delta} \ar[r]^{\qquad\partial} & C^{n+1}_\alg(A,V) \ar[dr]^{(-1)^{n+1}\delta} \ar[r]^{\partial\qquad} &C^{n+2}_\alg(A,V)\longrightarrow\cdots \\ \cdots\longrightarrow C^{n-1}_\diffo(d_A,d_V) \ar[r]^{\qquad\partial_\lambda} &C^{n}_\diffo(d_A,d_V)\ar[r]^{\partial_\lambda\qquad}&C^{n+1}_\diffo(d_A,d_V)\longrightarrow \cdots.} } \] Thus we have \begin{prop}\label{pro:exactsequence} There exists an exact sequence of cochain complexes, \[0\rar C^{*-1}_\diffo(d_A, d_V)\stackrel{\iota}{\rar} C_{\Diffl}^*(A, V)\stackrel{\pi}{\rar} C^*_\alg(A, V)\rar 0,\] where $\iota$ and $\pi$ are the inclusion and the projection respectively. \end{prop} The relations among the various cohomology groups are given by the following theorem. \begin{theorem} We have the following long exact sequence of cohomology groups, \[\cdots \rar H^{n-1}_\diffo(d_A, d_V)\stackrel{\bar{\iota}}{\rar} H_{\Diffl}^n(A, V)\stackrel{\bar{\pi}}{\rar} HH^n_\alg(A, V)\stackrel{(-1)^n\bar{\delta}}{\rar} H^n_\diffo(d_A, d_V)\rar \cdots,\] where $\bar{\delta} :HH^n_\alg(A, V)\rar H^n_\diffo(d_A, d_V)$ is given by $\bar{\delta}[f]=[\delta f]$. Here $[f]$ and $ [\delta f]$ denote the cohomological classes of $f\in C^n_\alg(A,V)$ and $\delta f\in C^n_\diffo(d_A,d_V)$. \end{theorem} Thus the linear maps $\bar{\pi}$ establish a relationship between the cohomology groups of the differential algebra and those of the underlying algebra, with the error terms controlled by the cohomology groups of the differential operator. This is resemblance of the Mayer-Vietoris sequence. \begin{proof} By Proposition \ref{pro:exactsequence} and the Snake Lemma, we have the long exact sequence \[\cdots \rar H^{n-1}_\diffo(d_A, d_V)\stackrel{\bar{\iota}}{\rar} H_{\Diffl}^n(A, V)\stackrel{\bar{\pi}}{\rar} HH^n_\alg(A, V)\stackrel{\Delta_n}{\rar} H^n_\diffo(d_A, d_V)\rar \cdots.\] It remains to prove that the connecting homomorphism $\Delta_n:HH^n_\alg(A, V)\rar H^n_\diffo(d_A, d_V)$ are exactly $(-1)^n\bar{\delta} $. Indeed, by the construction of $\Delta_n$ and Eq.~\eqref{eq:pda}, for any $[f]\in HH^n_\alg(A, V)$, we have \[\bar{\iota}\Delta_n([f])=[\partial_{\Diffl}(f,0)] = [(0,(-1)^n\delta f)],\] which implies that $\Delta_n=(-1)^n\bar{\delta}$. \end{proof} \section{Abelian extensions of differential algebras} \label{sec:ext} In this section, we study abelian extensions of differential algebras and show that they are classified by the second cohomology, as one would expect of a good cohomology theory. \begin{defn} An {\bf abelian extension} of differential algebras is a short exact sequence of homomorphisms of differential algebras \[\begin{CD 0@>>> {V} @>i >> \hat{A} @>p >> A @>>>0\\ @. @V {d_V} VV @V d_{\bar{A}} VV @V d_A VV @.\\ 0@>>> {V} @>i >> \hat{A} @>p >> A @>>>0 \end{CD}\] such that $uv=0$ for all $u,v\in V.$ \end{defn} We will call $(\hat{A},d_{\hat{A}})$ an abelian extension of $(A,d_A)$ by $(V,d_V)$. \begin{defn} Let $(\hat{A}_1,d_{\hat{A}_1})$ and $(\hat{A}_2,d_{\hat{A}_2})$ be two abelian extensions of $(A,d_A)$ by $(V,d_V)$. They are said to be {\bf isomorphic} if there exists an isomorphism of differential algebras $\zeta:(\hat{A}_1,d_{\hat{A}_1})\rar (\hat{A}_2,d_{\hat{A}_2})$ such that the following commutative diagram holds: \[\begin{CD 0@>>> {(V,d_V)} @>i >> (\hat{A}_1,d_{\hat{A}_1}) @>p >> (A,d_A) @>>>0\\ @. @| @V \zeta VV @| @.\\ 0@>>> {(V,d_V)} @>i >> (\hat{A}_2,d_{\hat{A}_2}) @>p >> (A,d_A) @>>>0. \end{CD}\] \end{defn} A {\bf section} of an abelian extension $(\hat{A},d_{\hat{A}})$ of $(A,d_A)$ by $(V,d_V)$ is a linear map $s:A\rar \hat{A}$ such that $p\circ s=\Id_A$. Now for an abelian extension $(\hat{A},d_{\hat{A}})$ of $(A,d_A)$ by $(V,d_V)$ with a section $s:A\rar \hat{A}$, we define linear maps $\rho_l: A\to \mathrm{End}_\bk(V),~ x\mapsto (v\mapsto xv)$ and $\rho_r: A\to \mathrm{End}_\bk(V),~ x\mapsto (v\mapsto vx)$ respectively by $$ xv:=s(x)v,\quad vx:=vs(x), \quad \forall x\in A, v\in V. $$ \begin{prop} With the above notations, $(V,\rho_l,\rho_r,d_V)$ is a bimodule over the differential algebra $(A,d_A)$. \end{prop} \begin{proof} For any $x,y\in A,\,v\in V$, since $s(xy)-s(x)s(y)\in V$ implies $s(xy)v=s(x)s(y)v$, we have \[\rho_l(xy)(v)=s(xy)v=s(x)s(y)v=\rho_l(x)\circ\rho_l(y)(v).\] Hence, $\rho_l$ is an algebra homomorphism. Similarly, $\rho_r$ is an algebra anti-homomorphism. Moreover, $d_{\hat{A}}(s(x))-s(d_A(x))\in V$ means that $d_{\hat{A}}(s(x))v=s(d_A(x))v$. Thus we have \begin{align*} d_V(xv)&=d_V(s(x)v)=d_{\hat{A}}(s(x)v)= d_{\hat{A}}(s(x))v+s(x)d_{\hat{A}}(v)+\lambda d_{\hat{A}}(s(x))d_{\hat{A}}(v)\\ &=s(d_A(x))v+s(x)d_V(v)+\lambda s(d_A(x))d_V(v)\\ &=d_A(x)v+xd_V(v)+\lambda d_A(x)d_V(v). \end{align*} Hence, $(V,\rho_l,\rho_r,d_V)$ is a bimodule over $(A,d_A)$. \end{proof} We further define linear maps $\psi:A\otimes A\rar V$ and $\chi:A\rar V$ respectively by \begin{align*} \psi(x,y)&=s(x)s(y)-s(xy),\quad\forall x,y\in A,\\ \chi(x)&=d_{\hat{A}}(s(x))-s(d_A(x)),\quad\forall x\in A. \end{align*} We transfer the differential algebra structure on $\hat{A}$ to $A\oplus V$ by endowing $A\oplus V$ with a multiplication $\cdot_\psi$ and a differential operator $d_\chi$ defined by \begin{align} \label{eq:mul}(x,u)\cdot_\psi(y,v)&=(xy,xv+uy+\psi(x,y)),\,\forall x,y\in A,\,u,v\in V,\\ \label{eq:dif}d_\chi(x,v)&=(d_A(x),\chi(x)+d_V(v)),\,\forall x\in A,\,v\in V. \end{align} \begin{prop}\label{prop:2-cocycle} The triple $(A\oplus V,\cdot_\psi,d_\chi)$ is a differential algebra if and only if $(\psi,\chi)$ is a 2-cocycle of the differential algebra $(A,d_A)$ with the coefficient in $(V,d_V)$. \end{prop} \begin{proof} If $(A\oplus V,\cdot_\psi,d_\chi)$ is a differential algebra, then the associativity of $\cdot_\psi$ implies \begin{equation} \label{eq:mc}x\psi(y,z)-\psi(xy,z)+\psi(x,yz)-\psi(x,y)z=0. \end{equation}Since $d_\chi$ satisfies \eqref{diff}, we deduce that \begin{equation} \label{eq:dc}\chi(xy)-x\vdash_\lambda\chi(y)-\chi(x)\dashv_\lambda y +d_V(\psi(x,y))-\psi(d_A(x),y)-\psi(x,d_A(y))-\lambda \psi(d_A(x),d_A(y))=0. \end{equation} Hence, $(\psi,\chi)$ is a 2-cocycle. Conversely, if $(\psi,\chi)$ satisfies equalities~\eqref{eq:mc} and \eqref{eq:dc}, one can easily check that $(A\oplus V,\cdot_\psi,d_\chi)$ is a differential algebra. \end{proof} Now we are ready to classify abelian extensions of a differential algebra. \begin{theorem} Let $V$ be a vector space and $d_V\in\End_\bk(V)$. Then abelian extensions of a differential algebra $(A,d_A)$ by $(V,d_V)$ are classified by the second cohomology group ${\tilde{H}}_{\Diffl}^2(A,V)$ of $(A,d_A)$ with coefficients in the bimodule $(V,d_V)$. \end{theorem} \begin{proof} Let $(\hat{A},d_{\hat{A}})$ be an abelian extension of $(A,d_A)$ by $(V,d_V)$. We choose a section $s:A\rar \hat{A}$ to obtain a 2-cocycle $(\psi,\chi)$ by Proposition~\ref{prop:2-cocycle}. We first show that the cohomological class of $(\psi,\chi)$ does not depend on the choice of sections. Indeed, let $s_1$ and $s_2$ be two distinct sections providing 2-cocycles $(\psi_1,\chi_1)$ and $(\psi_2,\chi_2)$ respectively. We define $\phi:A\rar V$ by $\phi(x)=s_1(x)-s_2(x)$. Then \begin{align*} \psi_1(x,y)&=s_1(x)s_1(y)-s_1(xy)\\ &=(s_2(x)+\phi(x))(s_2(y)+\phi(y))-(s_2(xy)+\phi(xy))\\ &=(s_2(x)s_2(y)-s_2(xy))+s_2(x)\phi(y)+\phi(x)s_2(y)-\phi(xy)\\ &=(s_2(x)s_2(y)-s_2(xy))+x\phi(y)+\phi(x)y-\phi(xy)\\ &=\psi_2(x,y)+\partial\phi(x,y) \end{align*} and \begin{align*} \chi_1(x)&=d_{\hat{A}}(s_1(x))-s_1(d_A(x))\\ &=d_{\hat{A}}(s_2(x)+\phi(x))-(s_2(d_A(x))+\phi(d_A(x)))\\ &=(d_{\hat{A}}(s_2(x))-s_2(d_A(x)))+d_{\hat{A}}(\phi(x))-\phi(d_A(x))\\ &=\chi_2(x)+d_V(\phi(x))-\phi(d_A(x))\\ &=\chi_2(x)-\delta\phi(x). \end{align*} That is, $(\psi_1,\chi_1)=(\psi_2,\chi_2)+\partial_{\Diffl}(\phi)$. Thus $(\psi_1,\chi_1)$ and $(\psi_2,\chi_2)$ are in the same cohomological class {in $\tilde{H}_{\Diffl}^2(A,V)$}. Next we prove that isomorphic abelian extensions give rise to the same element in {$\tilde{H}_{\Diffl}^2(A,V)$.} Assume that $(\hat{A}_1,d_{\hat{A}_1})$ and $(\hat{A}_2,d_{\hat{A}_2})$ are two isomorphic abelian extensions of $(A,d_A)$ by $(V,d_V)$ with the associated homomorphism $\zeta:(\hat{A}_1,d_{\hat{A}_1})\rar (\hat{A}_2,d_{\hat{A}_2})$. Let $s_1$ be a section of $(\hat{A}_1,d_{\hat{A}_1})$. As $p_2\circ\zeta=p_1$, we have \[p_2\circ(\zeta\circ s_1)=p_1\circ s_1=\Id_{A}.\] Therefore, $\zeta\circ s_1$ is a section of $(\hat{A}_2,d_{\hat{A}_2})$. Denote $s_2:=\zeta\circ s_1$. Since $\zeta$ is a homomorphism of differential algebras such that $\zeta|_V=\Id_V$, we have \begin{align*} \psi_2(x,y)&=s_2(x)s_2(y)-s_2(xy)=\zeta(s_1(x))\zeta(s_1(y))-\zeta(s_1(xy))\\ &=\zeta(s_1(x)s_1(y)-s_1(xy))=\zeta(\psi_1(x,y))\\ &=\psi_1(x,y) \end{align*} and \begin{align*} \chi_2(x)&=d_{\hat{A}_2}(s_2(x))-s_2(d_A(x))=d_{\hat{A}_2}(\zeta(s_1(x)))-\zeta(s_1(d_A(x)))\\ &=\zeta(d_{\hat{A}_1}(s_1(x))-s_1(d_A(x)))=\zeta(\chi_1(x))\\ &=\chi_1(x). \end{align*} Consequently, all isomorphic abelian extensions give rise to the same element in {$\tilde{H}_{\Diffl}^2(A,V)$}. Conversely, given two 2-cocycles $(\psi_1,\chi_1)$ and $(\psi_2,\chi_2)$, we can construct two abelian extensions $(A\oplus V,\cdot_{\psi_1},d_{\chi_1})$ and $(A\oplus V,\cdot_{\psi_2},d_{\chi_2})$ via equalities~\eqref{eq:mul} and \eqref{eq:dif}. If they represent the same cohomological class {in $\tilde{H}_{\Diffl}^2(A,V)$}, then there exists a linear map $\phi:A\to V$ such that $$(\psi_1,\chi_1)=(\psi_2,\chi_2)+\partial_{\Diffl}(\phi).$$ Define $\zeta:A\oplus V\rar A\oplus V$ by \[\zeta(x,v):=(x,\phi(x)+v).\] Then $\zeta$ is an isomorphism of these two abelian extensions. \end{proof} \begin{remark} In particular, any vector space $V$ with linear endomorphism $d_V$ can serve as a trivial bimodule of $(A,d_A)$. In this situation, central extensions of $(A,d_A)$ by $(V,d_V)$ are classified by the second cohomology group $H_{\Diffl}^2(A,V)$ of $(A,d_A)$ with the coefficient in the trivial bimodule $(V,d_V)$. Note that for a trivial bimodule $(V,d_V)$, since $\partial_\lambda v=0$ for all $v\in V$, we have $$H_{\Diffl}^2(A,V)=\tilde{H}_{\Diffl}^2(A,V).$$ \end{remark} \section{Deformations of differential algebras}\label{sec:def} In this section, we study formal deformations of a differential algebra. In particular, we show that if the second cohomology group $\tilde{H}^2_{\Diffl}(A,A)=0$, then the differential algebra $(A,d_A)$ is rigid. Let $(A,d_A)$ be a differential algebra. Denote by $\mu_A$ the multiplication of $A$. Consider the 1-parameterized family $$\mu_t=\sum_{i=0}^{\infty} \mu_i t^i, \, \, \mu_i\in C^2_{\Diffl}(A, A),\quad d_t=\sum_{i=0}^{\infty} d_i t^i, \, \, d_i\in C^1_\diffo(d_A, d_A).$$ \begin{defn} A {\bf 1-parameter formal deformation} of a differential algebra $(A, d_A)$ is a pair $(\mu_t, d_t)$ which endows the $\bk[[t]]$-module $(A[[t]], \mu_t, d_t)$ with the differential algebra structure {over $\bk[[t]]$} such that $(\mu_0, d_0)=(\mu_A, d_A)$. \end{defn} Given any differential algebra $(A,d_A)$, interpret $\mu_A$ and $d_A$ as the formal power series $\mu_t$ and $d_t$ with $\mu_i=\delta_{i,0}\mu_A$ and $d_i=\delta_{i,0}d_A$ respectively for all $i\geq0$. Then $(A[[t]],\mu_A,d_A)$ is a 1-parameter formal deformation of $(A, d_A)$. The pair $(\mu_t, d_t)$ generates a 1-parameter formal deformation of the differential algebra $(A, d_A)$ if and only if for all $x, y, z\in A$, the following equalities hold: \begin{eqnarray}\label{equation: ass nonexpanded} \mu_t(\mu_t(x, y), z)&=&\mu_t(x, \mu_t(y, z)),\\ \label{equation: derivation nonexpanded} d_t(\mu_t(x, y))&=&\mu_t(d_t(x), y)+\mu_t(x, d_t(y))+\lambda \mu_t(d_t(x), d_t(y)). \end{eqnarray} Expanding these equations and collecting coefficients of $t^n$, we see that Eqs.~\eqref{equation: ass nonexpanded} and \eqref{equation: derivation nonexpanded} are equivalent to the systems of equations: \begin{eqnarray}\label{equation: ass} \sum_i^n \mu_i(\mu_{n-i}(x,y),z)&=&\sum_i^n\mu_i(x,\mu_{n-i}(y,z)),\\ \label{df} \sum_{k,l\geq0\atop k+l=n}d_l\mu_k(x,y)&=&\sum_{k,l\geq0\atop k+l=n}\left(\mu_k(d_l(x),y)+\mu_k(x,d_l(y))\right)+\lambda\sum_{k,l,m\geq0\atop k+l+m=n}\mu_k(d_l(x),d_m(y)). \end{eqnarray} \begin{remark}For $n=0$, Eq.~\eqref{equation: ass} is equivalent to the associativity of $\mu_A$, and Eq.~\eqref{df} is equivalent to the fact that $d_A$ is a $\lambda$-derivation. \end{remark} \begin{prop}\label{prop:fddco} Let $(A[[t]], \mu_t, d_t)$ be a $1$-parameter formal deformation of a differential algebra $(A,d_A)$. Then $(\mu_1, d_1)$ is a 2-cocycle of the differential algebra $(A,d_A)$ with the coefficient in the regular bimodule $(A,d_A)$. \end{prop} \begin{proof} For $n =1$, Eq.~\eqref{equation: ass} is equivalent to $\partial \mu_1=0$, and Eq.~\eqref{df} is equivalent to $$\partial_\lambda d_1+\delta \mu_1=0.$$ Thus for $n=1$, Eqs.~\eqref{equation: ass} and \eqref{df} imply that $(\mu_1,d_1)$ is a 2-cocycle. \end{proof} If $\mu_t=\mu_A$ in the above $1$-parameter formal deformation of the differential algebra $(A,d_A)$, we obtain a $1$-parameter formal deformation of the differential operator $d_A$. Consequently, we have \begin{coro}\label{coro:fdo} Let $ d_t $ be a $1$-parameter formal deformation of the differential operator $ d_A $. Then $d_1$ is a 1-cocycle of the differential operator $d_A$ with coefficients in the regular bimodule $(A,d_A)$. \end{coro} \begin{proof} In the special case when $n =1$, Eq.~\eqref{df} is equivalent to $\partial_\lambda d_1=0$, which implies that $d_1$ is a 1-cocycle of the differential operator $d_A$ with coefficients in the regular bimodule $(A,d_A)$. \end{proof} \begin{defn} The $2$-cocycle $(\mu_1,d_1)$ is called the {\bf infinitesimal} of the $1$-parameter formal deformation $(A[[t]],\mu_t,d_t)$ of $(A,d_A)$. \end{defn} \begin{defn} Let $(A[[t]],\mu_t,d_t)$ and $(A[[t]],\bar{\mu}_t,\bar{d}_t)$ be $1$-parameter formal deformations of $(A,d_A)$. A {\bf formal isomorphism} from $(A[[t]],\bar{\mu}_t,\bar{d}_t)$ to $(A[[t]],\mu_t,d_t)$ is a power series $\Phi_t=\sum_{i\geq0}\phi_it^i:A[[t]]\lon A[[t]]$, where $\phi_i:A\to A$ are linear maps with $\phi_0=\Id_A$, such that \begin{eqnarray} \Phi_t\circ\bar{\mu}_t&=& \mu_t\circ(\Phi_t\times\Phi_t),\\ \Phi_t\circ\bar{d}_t&=&d_t\circ\Phi_t. \end{eqnarray} Two $1$-parameter formal deformations $(A[[t]],\mu_t,d_t)$ and $(A[[t]],\bar{\mu}_t,\bar{d}_t)$ are said to be {\bf equivalent} if there exists a formal isomorphism $\Phi_t:(A[[t]],\bar{\mu}_t,\bar{d}_t) \lon (A[[t]],\mu_t,d_t)$. \end{defn} \begin{theorem} The infinitesimals of two equivalent $1$-parameter formal deformations of $(A,d_A)$ are in the same cohomology class {in $\tilde{H}_{\Diffl}^2(A,A)$}. \end{theorem} \begin{proof} Let $\Phi_t:(A[[t]],\bar{\mu}_t,\bar{d}_t)\lon(A[[t]],\mu_t,d_t)$ be a formal isomorphism. For all $x,y\in A$, we have \begin{eqnarray*} \Phi_t\circ\bar{\mu}_t(x,y)&=& \mu_t\circ(\Phi_t\times\Phi_t)(x,y),\\ \Phi_t\circ\bar{d}_t(x)&=& d_t\circ\Phi_t (x). \end{eqnarray*} Expanding the above identities and comparing the coefficients of $t$, we obtain \begin{eqnarray*} \bar{\mu}_1(x,y)&=&\mu_1(x,y)+\phi_1(x)y+x\phi_1(y)-\phi_1(xy),\\ \bar{d}_1(x)&=&d_1(x)+d_A(\phi_1(x))-\phi_1(d_A(x)). \end{eqnarray*} Thus, we have $$(\bar{\mu}_1,\bar{d}_1)=(\mu_1,d_1)+\partial_{\Diffl}(\phi_1),$$ which implies that $[(\bar{\mu}_1,\bar{d}_1)]=[(\mu_1,d_1)]$ in $\tilde{H}^2_{\Diffl}(A,A)$. \end{proof} \begin{defn} A $1$-parameter formal deformation $(A[[t]],\mu_t,d_t)$ of $(A,d_A)$ is said to be {\bf trivial} if it is equivalent to the deformation $(A[[t]],\mu_A,d_A)$, that is, there exists $\Phi_t=\sum_{i\geq0}\phi_it^i:A[[t]]\lon A[[t]]$, where $\phi_i:A\to A$ are linear maps with $\phi_0=\Id_A$, such that \begin{eqnarray} \Phi_t\circ\mu_t&=&\mu_A\circ(\Phi_t\times\Phi_t),\\ \Phi_t \circ d_t&=&d_A\circ\Phi_t. \end{eqnarray} \end{defn} \begin{defn} A differential algebra $(A,d_A)$ is said to be {\bf rigid} if every $1$-parameter formal deformation is trivial. \end{defn} \begin{theorem} Regarding $(A,d_A)$ as the regular bimodule over itself, if $\tilde{H}^2_{\Diffl}(A,A)=0$, the differential algebra $(A,d_A)$ is rigid. \end{theorem} \begin{proof} Let $(A[[t]],\mu_t,d_t)$ be a $1$-parameter formal deformation of $(A,d_A)$. By Proposition ~\ref{prop:fddco}, $(\mu_1,d_1)$ is a 2-cocycle. By $\tilde{H}^2_{\Diffl}(A,A)=0$, there exists a 1-cochain $\phi_1\in C^1_\alg(A,A)$ such that \begin{eqnarray} \label{rigid}(\mu_1,d_1)=-\partial_{\Diffl}(\phi_1). \end{eqnarray} Then setting $\Phi_t=\Id_A+\phi_1 t$, we have a deformation $(A[[t]],\bar{\mu}_t,\bar{d}_t)$, where \begin{eqnarray*} \bar{\mu}_t(x,y)&=&\big(\Phi_t^{-1}\circ \mu_t\circ(\Phi_t\times\Phi_t)\big)(x,y),\\ \bar{d}_t(x)&=&\big(\Phi_t^{-1}\circ d_t\circ\Phi_t\big)(x). \end{eqnarray*} Thus, $(A[[t]],\bar{\mu}_t,\bar{d}_t)$ is equivalent to $(A[[t]],\mu_t,d_t)$. Moreover, we have \begin{eqnarray*} \bar{\mu}_t(x,y)&=&(\Id_A-\phi_1t+\phi_1^2t^{2}+\cdots+(-1)^i\phi_1^it^{i}+\cdots)(\mu_t(x+\phi_1(x)t,y+\phi_1(y)t)),\\ \bar{d}_t(x)&=&(\Id_A-\phi_1t+\phi_1^2t^{2}+\cdots+(-1)^i\phi_1^it^{i}+\cdots)(d_t(x+\phi_1(x)t)). \end{eqnarray*} Therefore, \begin{eqnarray*} \bar{\mu}_t(x,y)&=&xy+(\mu_1(x,y)+x\phi_1(y)+\phi_1(x)y-\phi_1(xy))t+\bar{\mu}_{2}(x,y)t^{2}+\cdots,\\ \bar{d}_t(x)&=&d_A(x)+(d_A(\phi_1(x))+d_1(x)-\phi_1(d_A(x)))t+\bar{d}_{2}(x)t^{2}+\cdots. \end{eqnarray*} By Eq.~\eqref{rigid}, we have \begin{eqnarray*} \bar{\mu}_t(x,y)&=&xy+\bar{\mu}_{2}(x,y)t^{2}+\cdots,\\ \bar{d}_t(x)&=&d_A(x)+\bar{d}_{2}(x)t^{2}+\cdots. \end{eqnarray*} Then by repeating the argument, we can show that $(A[[t]],\mu_t,d_t)$ is equivalent to $(A[[t]],\mu_A,d_A)$. Thus, $(A,d_A)$ is rigid. \end{proof} \section*{Appendix: Proof of Proposition~\mref{prop:delta}} To simplify that notations, we use the abbreviation $x_{i,j}:=x_i,\dots, x_j,\,i\leq j,$ with the convention $x_{i,j}=1$ if $i>j$. For any $1\leq i_1<\cdots<i_k\leq n$ and $f\in C_\alg^n(A,V)$, define a function $f^{(i_1,\dots,i_k)}$ by \begin{eqnarray*} \lefteqn{f^{(i_1,\dots,i_k)}(x_1,\dots,x_n)}\\ &:=&f(x_1,\dots, x_{i_1-1}, d_A(x_{i_1}), x_{i_1+1}\dots, x_{i_2-1}, d_A(x_{i_2}), x_{i_2+1}\dots x_{i_k-1}, d_A(x_{i_k}), x_{i_k+1},\dots,x_n). \end{eqnarray*} In preparation for the proof of Proposition~\mref{prop:delta}, we first give two technical lemmas. \begin{lemma}\label{lem:111} For any $f\in C_\alg^n(A,V),\,x_1,\dots,x_{n+1}\in A$ with $n\geq1$, we have \begin{align} \nonumber&\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}\sum_{j=1}^n(-1)^jf^{(i_1,\dots,i_k)}(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1}) \\ \nonumber =&\sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq j\leq n\atop j\neq i_r-1, i_r,\,\forall r}(-1)^jf(x_1, \dots, d_A(x_{i_1}), \dots, x_jx_{j+1},\dots, d_A(x_{i_k}), \dots, x_{n+1})\\ \nonumber &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k\atop i_r-1\neq i_{r-1}}(-1)^{i_r-1}f(x_1,\dots,d_A(x_{i_1}),\dots,x_{i_r-1}d_A(x_{i_r}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ \nonumber &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1\neq i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})x_{i_r+1},\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ \nonumber &+\sum_{k=2}^{n+1}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1= i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})d_A(x_{i_{r+1}}),\dots,d_A(x_{i_k}),\dots,x_{n+1}) \end{align} \end{lemma} \begin{proof} In the second line of this proof, by convention $i_0=0, i_{k+1}=n+2$. By Eq.~\eqref{diff}, we have \begin{align*} &\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}\sum_{j=1}^n(-1)^jf^{(i_1,\dots,i_k)}(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1})\\ =& \sum_{k=1}^n\hspace{-.1cm} \lambda^{k-1}\hspace{-.65cm} \sum_{1\leq i_1<\cdots<i_k\leq n}\hspace{-.2cm} \sum_{1\leq j\leq n\atop {i_s<j<i_{s+1} \atop 0\leq s\leq k}}\hspace{-.25cm} (-1)^j f( \dots, d_A(x_{i_1}), \dots, d_A(x_{i_s}), \dots, x_j x_{j+1}, \dots, d_A(x_{i_{s+1}+1}), \dots, d_A(x_{i_k+1}), \dots )\\ &+ \sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n} \sum_{r=1}^k (-1)^{i_r}f( \dots, d_A(x_{i_1}),\dots, d_A(x_{i_r}x_{i_r+1}),\dots, d_A(x_{i_k+1}), \dots )\\ =& \sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq j\leq n\atop j\neq i_r, i_r-1,\,\forall r} (-1)^j f( \cdots, d_A(x_{i_1}), \dots, x_j x_{j+1}, \dots, d_A(x_{i_k}), \dots )\\ &+ \sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n} \sum_{r=1}^k (-1)^{i_r}f(x_1,\dots, d_A(x_{i_1}),\dots, d_A(x_{i_r})x_{i_r+1},\dots, d_A(x_{i_k+1}), \dots, x_{n+1})\\ &+ \sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n} \sum_{r=1}^k (-1)^{i_r}f(x_1,\dots, d_A(x_{i_1}),\dots, x_{i_r}d_A(x_{i_r+1}),\dots, d_A(x_{i_k+1}), \dots, x_{n+1})\\ &+ \sum_{k=1}^n\lambda^{k}\sum_{1\leq i_1<\cdots<i_k\leq n} \sum_{r=1}^k (-1)^{i_r}f(x_1,\dots, d_A(x_{i_1}),\dots, d_A(x_{i_r})d_A(x_{i_r+1}),\dots, d_A(x_{i_k+1}), \dots, x_{n+1})\\ =& \sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq j\leq n\atop j\neq i_r, i_r-1,\,\forall r} (-1)^j f( \dots, d_A(x_{i_1}), \dots, x_j x_{j+1}, \dots, d_A(x_{i_k}), \dots )\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k\atop i_r-1\neq i_{r-1}}(-1)^{i_r-1}f(x_1,\dots,d_A(x_{i_1}),\dots,x_{i_r-1}d_A(x_{i_r}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1\neq i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})x_{i_r+1},\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=1}^{n}\hspace{-.1cm} \lambda^{k-1}\hspace{-.6cm} \sum_{1\leq i_1<\cdots<i_{k+1}\leq n+1}\sum_{1\leq r\leq k\atop i_r+1= i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})d_A(x_{i_{r+1}}),\dots,d_A(x_{i_{k+1}}),\dots,x_{n+1})\\ =& \sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq j\leq n\atop j\neq i_r, i_r-1,\,\forall r} (-1)^j f( \dots, d_A(x_{i_1}), \dots, x_j x_{j+1}, \dots, d_A(x_{i_k}), \dots )\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k\atop i_r-1\neq i_{r-1}}(-1)^{i_r-1}f(x_1,\dots,d_A(x_{i_1}),\dots,x_{i_r-1}d_A(x_{i_r}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1\neq i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})x_{i_r+1},\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=2}^{n+1}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1= i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})d_A(x_{i_{r+1}}),\dots,d_A(x_{i_k}),\dots,x_{n+1}). \qedhere\end{align*} \end{proof} \begin{lemma} For any $f\in C_\alg^n(A,V),\,x_1,\dots,x_{n+1}\in A$ with $n\geq1$, \begin{align} &\sum_{k=1}^{n+1}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}(\partial f)^{(i_1, \dots, i_k)}(x_{1,{n+1}})-d_V (\partial f(x_{1,n+1}))\label{eq2}\\ \nonumber =&\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}x_1\vdash_\lambda f^{(i_1,\dots,i_k)}(x_{2,n+1})\\ \nonumber &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}\sum_{j=1}^n(-1)^jf^{(i_1,\dots,i_k)}(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1})\\ \nonumber &+(-1)^{n+1}\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n} f^{(i_1,\dots,i_k)}(x_{1,n})\dashv_\lambda x_{n+1}-x_1\vdash_\lambda d_V (f(x_{2,n+1}))\\ \nonumber &+\sum_{j=1}^n(-1)^{j-1}d_V(f(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1})) +(-1)^n d_V(f(x_{1,n}))\dashv_\lambda x_{n+1}. \end{align} \end{lemma} \begin{proof} As $d_V(xv)=d_A(x)v+x\vdash_\lambda d_V(v),\,d_V(vx)=vd_A(x)+d_V(v)\dashv_\lambda x$ for any $v\in V,x\in A$, we have \begin{align*} d_V (\partial f(x_{1,n+1}))&=d_A(x_1) f(x_{2,{n+1}})+x_1\vdash_\lambda d_V (f(x_{2,n+1}))\\ &+\sum_{j=1}^n(-1)^jd_V(f(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1}))\\ &+(-1)^{n+1}f(x_{1,{n}})d_A(x_{n+1})+(-1)^{n+1} d_V(f(x_{1,n}))\dashv_\lambda x_{n+1}. \end{align*} Hence, we only need to check Eq.~\eqref{eq2} as follows. By Lemma \ref{lem:111}, we have \begin{align*} &\sum_{k=1}^{n+1}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}(\partial f)^{(i_1, \dots, i_k)}(x_{1,{n+1}}) \\ =& \lambda^n \partial f(d_A(x_1),\dots, d_A(x_{n+1})) +\sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}(\partial f)^{(i_1, \dots, i_k)}(x_{1,{n+1}})\\ =&\lambda^n d_A(x_1) f(d_A(x_2),\dots, d_A(x_{n+1})) +\lambda^n \sum_{i=1}^n (-1)^i f(d_A(x_1), \dots, d_A(x_i)d_A(x_{i+1}),\dots, d_A(x_{n+1})) \\ &+(-1)^{n+1} \lambda^n f(d_A(x_1),\dots, d_A(x_{n}))d_A(x_{n+1}) \\ &+d_A(x_1) f(x_{2,{n+1}})+\sum_{k=2}^{n}\lambda^{k-1}\sum_{2\leq i_2<\cdots<i_k\leq n+1}d_A(x_1) f^{(i_2-1, \dots, i_k-1)}(x_{2,{n+1}})\\ &+\sum_{k=1}^{n}\lambda^{k-1}\sum_{2\leq i_1<\cdots<i_k\leq n+1}x_1 f^{(i_1-1, \dots, i_k-1)}(x_{2,{n+1}})\\ &+\sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq j\leq n\atop j\neq i_r-1, i_r,\,\forall r}(-1)^jf(x_1, \dots, d_A(x_{i_1}), \dots, x_jx_{j+1},\dots, d_A(x_{i_k}), \dots, x_{n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k\atop i_r-1\neq i_{r-1}}(-1)^{i_r-1}f(x_1,\dots,d_A(x_{i_1}),\dots,x_{i_r-1}d_A(x_{i_r}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1\neq i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})x_{i_r+1},\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1= i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})d_A(x_{i_{r+1}}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+(-1)^{n+1}f(x_{1,{n}})d_A(x_{n+1}) +\sum_{k=2}^{n}\lambda^{k-1}(-1)^{n+1}\sum_{1\leq i_1<\cdots<i_{k-1}\leq n} f^{(i_1, \dots, i_{k-1})}(x_{1,{n}})d_A(x_{n+1})\\ &+\sum_{k=1}^{n}\lambda^{k-1}(-1)^{n+1}\sum_{1\leq i_1<\cdots<i_k\leq n} f^{(i_1, \dots, i_k)}(x_{1,{n}})x_{n+1}\\ =& \lambda^n d_A(x_1) f(d_A(x_2),\dots, d_A(x_{n+1})) +(-1)^{n+1} \lambda^n f(d_A(x_1),\dots, d_A(x_{n}))d_A(x_{n+1}) \\ &+d_A(x_1) f(x_{2,{n+1}})+\sum_{k=1}^{n-1}\lambda^{k}\sum_{1\leq i_1<\cdots<i_k\leq n}d_A(x_1) f^{(i_1, \dots, i_k)}(x_{2,{n+1}})\\ &+\sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}x_1 f^{(i_1, \dots, i_k)}(x_{2,{n+1}})\\ &+ \sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq j\leq n\atop j\neq i_r-1, i_r,\,\forall r}(-1)^jf(x_1, \dots, d_A(x_{i_1}), \dots, x_jx_{j+1},\dots, d_A(x_{i_k}), \dots, x_{n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k\atop i_r-1\neq i_{r-1}}(-1)^{i_r-1}f(x_1,\dots,d_A(x_{i_1}),\dots,x_{i_r-1}d_A(x_{i_r}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1\neq i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})x_{i_r+1},\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=1}^{n+1}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1= i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})d_A(x_{i_{r+1}}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+(-1)^{n+1}f(x_{1,{n}})d_A(x_{n+1}) +\sum_{k=1}^{n-1}\lambda^{k}(-1)^{n+1}\sum_{1\leq i_1<\cdots<i_{k}\leq n} f^{(i_1, \dots, i_{k})}(x_{1,{n}})d_A(x_{n+1})\\ &+\sum_{k=1}^{n}\lambda^{k-1}(-1)^{n+1}\sum_{1\leq i_1<\cdots<i_k\leq n} f^{(i_1, \dots, i_k)}(x_{1,{n}})x_{n+1}\\ =&d_A(x_1) f(x_{2,{n+1}})+ (-1)^{n+1}f(x_{1,{n}})d_A(x_{n+1}) \\ &+\sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}x_1 f^{(i_1, \dots, i_k)}(x_{2,{n+1}})\\ & + \sum_{k=1}^{n}\lambda^{k}\sum_{1\leq i_1<\cdots<i_k\leq n}d_A(x_1) f^{(i_1, \dots, i_k)}(x_{2,{n+1}})\\ &+\sum_{k=1}^{n}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq j\leq n\atop j\neq i_r-1, i_r,\,\forall r}(-1)^jf(x_1, \dots, d_A(x_{i_1}), \dots, x_jx_{j+1},\dots, d_A(x_{i_k}), \dots, x_{n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k\atop i_r-1\neq i_{r-1}}(-1)^{i_r-1}f(x_1,\dots,d_A(x_{i_1}),\dots,x_{i_r-1}d_A(x_{i_r}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1\neq i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})x_{i_r+1},\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ &+\sum_{k=2}^{n+1}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}\sum_{1\leq r\leq k-1\atop i_r+1= i_{r+1}}(-1)^{i_r}f(x_1,\dots,d_A(x_{i_1}),\dots,d_A(x_{i_r})d_A(x_{i_{r+1}}),\dots,d_A(x_{i_k}),\dots,x_{n+1})\\ & +\sum_{k=1}^{n}\lambda^{k}(-1)^{n+1}\sum_{1\leq i_1<\cdots<i_{k}\leq n} f^{(i_1, \dots, i_{k})}(x_{1,{n}})d_A(x_{n+1})\\ &+\sum_{k=1}^{n}\lambda^{k-1}(-1)^{n+1}\sum_{1\leq i_1<\cdots<i_k\leq n} f^{(i_1, \dots, i_k)}(x_{1,{n}})x_{n+1}\\ =&d_A(x_1) f(x_{2,{n+1}})+ (-1)^{n+1}f(x_{1,{n}})d_A(x_{n+1}) \\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}x_1 \vdash_\lambda f^{(i_1,\dots,i_k)}(x_{2,n+1})\\ &+\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}\sum_{j=1}^n(-1)^jf^{(i_1,\dots,i_k)}(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1})\\ &+(-1)^{n+1}\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n} f^{(i_1,\dots,i_k)}(x_{1,n})\dashv_\lambda x_{n+1}. \qedhere\end{align*} \end{proof} {\bf Proof of Proposition~\mref{prop:delta}.} For any $v\in C_\alg^0(A,V)=V$ and $x\in A$, we have \begin{align*} \delta(\partial v)(x)&=\partial v(d_A(x))-d_V(\partial v(x))\\ &=d_A(x)v-vd_A(x)-d_V(x v-v x)\\ &=d_A(x)v-vd_A(x)\\ &\ \ \ \ -d_A(x) v-xd_V(v)-\lambda d_A(x)d_V(v)+d_V(v)x+v d_A(x)+\lambda d_V(v)d_A(x)\\ &=-xd_V(v)-\lambda d_A(x)d_V(v)+d_V(v)x+ \lambda d_V(v)d_A(x)\\ &=-x\vdash_\lambda d_V(v)+d_V(v)\dashv_\lambda x\\ &=x\vdash_\lambda\delta v-\delta v\dashv_\lambda x\\ &=\partial_\lambda (\delta v)(x). \end{align*} For any $f\in C_\alg^n(A,V),\,x_1,\dots,x_{n+1}\in A$ with $n\geq1$, \begin{align*} \partial_\lambda(\delta f)(x_{1,n+1})=&x_1\vdash_\lambda\delta f(x_{2,n+1})+\sum_{j=1}^n(-1)^j\delta f(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1}) +(-1)^{n+1}\delta f(x_{1,n})\dashv_\lambda x_{n+1}\\ =&\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}x_1\vdash_\lambda f^{(i_1,\dots,i_k)}(x_{2,n+1})\\ &+\sum_{j=1}^n\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n}(-1)^jf^{(i_1,\dots,i_k)}(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1})\\ &+(-1)^{n+1}\sum_{k=1}^n\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n} f^{(i_1,\dots,i_k)}(x_{1,n})\dashv_\lambda x_{n+1} -x_1\vdash_\lambda d_V(f(x_{2,n+1}))\\ &+\sum_{j=1}^n(-1)^{j-1}d_V(f(x_{1,j-1},x_jx_{j+1},x_{j+2,n+1}))+(-1)^n d_V (f(x_{1,n}))\dashv_\lambda x_{n+1}. \end{align*} On the other hand, we have \begin{align*} \delta(\partial f)(x_{1,n+1})=&\sum_{k=1}^{n+1}\lambda^{k-1}\sum_{1\leq i_1<\cdots<i_k\leq n+1}(\partial f)^{(i_1, \dots, i_k)}(x_{1,{n+1}})-d_V (\partial f(x_{1,n+1})). \end{align*} Hence, by Eq.~\eqref{eq2}, $\partial_\lambda \delta=\delta\partial$. \smallskip \noindent {{\bf Acknowledgments.} This work is supported in part by Natural Science Foundation of China (Grant Nos. 11501214, 11771142, 11771190, 11671139) and STCSM (Grant Nos. 13dz2260400). We give our warmest thanks to Rong Tang for useful comments.

proofpile-arXiv_069-13976

\section{Introduction} This is the first in a sequence of papers where we define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the {\it nonnegative configuration space}. In this paper, we focus on the combinatorics and topology of this space. In a sequel \cite{ALS2}, we will further study the geometry and its relations to cluster algebras, canonical bases, and scattering amplitudes. Some of the applications of our work to ${\mathcal N}=4$ super Yang-Mills amplitudes were announced in the note \cite{ALS}. \subsection{} Lusztig \cite{Lus} and Postnikov \cite{Pos} defined the totally nonnegative Grassmannian ${\rm Gr}(k,n)_{\geq 0}$ (informally, the positive Grassmannian), the closed subspace of the real Grassmannian ${\rm Gr}(k,n)({\mathbb R})$ of $k$-planes in ${\mathbb R}^n$ cut out by the condition that all Pl\"ucker coordinates are nonnegative. Postnikov \cite{Pos} (see also \cite{Rie}) studied the stratification of ${\rm Gr}(k,n)_{\geq 0}$ by positroid cells $\Pi_{{\mathcal{M}},>0}$ and Galashin, Karp, and Lam \cite{GKL,GKL2} showed that positroid cells endow ${\rm Gr}(k,n)_{\geq 0}$ with the structure of a regular CW-complex homeomorphic to a closed ball. Positroid cells are indexed by a class of matroids called positroids. Positroids have been completely classified \cite{Oh} and are in bijection with Grassmann necklaces \cite{Pos} and bounded affine permutations \cite{KLS}, among other combinatorial objects. This is in stark contrast to the situation of matroids. There is no explicit classification of (realizable) matroids, and the geometry of matroid strata \cite{GGMS} is notoriously complicated \cite{Mn}. Recently, the positive Grassmannian has made a prominent appearance in the study of scattering amplitudes \cite{book, AT}, where the boundary structure of ${\rm Gr}(k,n)_{\geq 0}$ was connected to the singularities of tree-level scattering amplitudes in maximally supersymmetric Yang-Mills theory. Part of this connection is formalized in the statement that ${\rm Gr}(k,n)_{\geq 0}$ is a positive geometry \cite{ABL}, and one driving force in this developing subject, and of the present work, is to find positive geometries that have relations to physical problems. \subsection{} The Grassmannian ${\rm Gr}(k,n)$ has a natural action of a torus $T$ that acts by scaling the basis vectors of the underlying vector space ${\mathbb C}^n$. The quotient ${\rm Gr}(k,n)/T$ is closely related to the configuration space ${\operatorname{Conf}}(k,n)$ of $n$ points in ${\mathbb P}^{k-1}$. Kapranov \cite{Kap} studied the Chow quotient ${\operatorname{Ch}}(k,n)$ of the Grassmannian, which is a compactification of the subspace ${\mathring{\Conf}}(k,n) \subset {\operatorname{Conf}}(k,n)$ of generic configurations. In the case $k = 2$, the Chow quotient ${\operatorname{Ch}}(2,n)$ is isomorphic to the Deligne-Knudsen-Mumford space $\overline{\M}_{0,n}$ of $n$-pointed stable rational curves. The image of the positive Grassmannian ${\rm Gr}(k,n)_{>0}$ in ${\mathring{\Conf}}(k,n) \subset {\operatorname{Ch}}(k,n)$ is the positive component ${\operatorname{Conf}}(k,n)_{>0}$ of generic configuration space. We define the {\it totally nonnegative part of the Chow quotient of the Grassmannian}, or simply {\it nonnegative configuration space} ${\operatorname{Ch}}(k,n)_{\geq 0}$, as the closure of this positive component inside ${\operatorname{Ch}}(k,n)$. As we shall explain in more detail in a sequel \cite{ALS2}, see also \cite{ALS,GGSVV}, scattering amplitudes can be thought of as ``functions" on ${\rm Gr}(4,n)/T$. Just as the combinatorics of ${\rm Gr}(k,n)_{\geq 0}$ controls the singularities of the amplitude at tree-level, we expect the combinatorics of ${\operatorname{Ch}}(4,n)_{\geq 0}$ to be closely related to the singularities of the full scattering amplitude (integrated, and at all loops). Some connections of (the closely related) tropical Grassmannians to amplitudes have also been discussed in \cite{CEGM,CR,DFOK1,SG,DFOK2,HP,DFOK3}. \subsection{} Our first aim in this work is to study the combinatorics of a natural stratification $\{\Theta_{{\tilde \Delta},>0}\}$ of ${\operatorname{Ch}}(k,n)_{\geq 0}$, extending the combinatorics associated to positroid cells of ${\rm Gr}(k,n)_{\geq 0}$. We call these strata {$\{\Theta_{{\tilde \Delta},>0}\}$} {\it positive Chow cells}. \begin{theorem}\label{thm:introbij} There are canonical bijections between the following sets. \begin{enumerate} \item The set $\{\Theta_{{\tilde \Delta},>0}\}$ of positive Chow cells of ${\operatorname{Ch}}(k,n)_{\geq 0}$. \item The set ${\mathcal{D}}(k,n)$ of regular subdivisions of the hypersimplex $\Delta(k,n)$ into positroid polytopes. \item The set of cones in the positive tropical Grassmannian ${\operatorname{Trop}}_{>0}{\rm Gr}(k,n)$, the space of valuations of positive Puiseux series points ${\rm Gr}(k,n)({\mathcal{R}}_{>0})$. \item The set of cones in the space ${\rm Dr}(k,n)_{>0} \subset {\mathbb R}^{\binom{[n]}{k}}$ (called the positive Dressian) of vectors satisfying the positive tropical (three-term) Pl\"ucker relations. \end{enumerate} \end{theorem} Our second main result is a description of the topology of ${\operatorname{Ch}}(k,n)_{\geq 0}$, a variant of the results of \cite{Pos,GKL,GKL2}. Somewhat surprisingly, while the geometry of the Chow quotient is considerably more complicated than that of the Grassmannian, the following result is easier than its Grassmannian counterpart. \begin{theorem}\label{thm:introtop} There is a stratification-preserving homeomorphism from nonnegative configuration space to a polytope. In particular, each positive Chow cell $\Theta_{{\tilde \Delta},>0} \subset {\operatorname{Ch}}(k,n)_{\geq 0}$ is homeomorphic to an open ball. \end{theorem} We remark that ${\rm Gr}(k,n)_{\geq 0}$ is {\it not} homeomorphic to a polytope as a stratified space. We now explain each of the objects in Theorem~\ref{thm:introbij} in turn. Let $[n]:= \{1,2,\ldots,n\}$ and let $\binom{[n]}{k}$ denote the set of all $k$-element subsets of $[n]$. \subsection{} Each point $X \in {\operatorname{Ch}}(k,n)$ is represented by an algebraic cycle inside the Grassmannian. If $X \in {\mathring{\Conf}}(k,n) \subset {\operatorname{Ch}}(k,n)$ then $X$ is represented by the torus orbit closure $\overline{T \cdot V}$ of a generic point $V$ in the Grassmannian. The toric variety $T \cdot V$ is isomorphic to the projective toric variety $X_{\Delta(k,n)}$ associated to the hypersimplex $\Delta(k,n)$, the convex hull of all vectors $e_I \in {\mathbb R}^n$, $I \in \binom{[n]}{k}$ with $k$ 0-s and $(n-k)$ 1-s. A general point $X \in {\operatorname{Ch}}(k,n)$ is represented by a union of toric varieties $X_{P_1}, \ldots,X_{P_m}$, where $P_1,\ldots,P_m$ are polytopes that form a regular subdivision of the hypersimplex into matroid polytopes (see \S\ref{sec:matroid}). We thus obtain a stratification of ${\operatorname{Ch}}(k,n)$ by matroid subdivisions of the hypersimplex, see \cite{Kap,KT,Laf}. However, it is a difficult question to describe which matroid subdivisions of the hypersimplex occur in this way. This is a variant of the (also difficult) question of which matroids are realizable. When $X \in {\operatorname{Ch}}(k,n)_{\geq 0}$ is nonnegative, the matroid polytopes $P_1,\ldots,P_m$ are positroid polytopes (Proposition~\ref{prop:conf}). Since positroids have been completely classified, subdivisions of the hypersimplex by positroid polytopes are far more tractable. Indeed, Theorem~\ref{thm:introbij} states that any regular subdivision of the hypersimplex $\Delta(k,n)$ into positroid polytopes appears in nonnegative configuration space. \subsection{} By definition, a regular subdivision of the hypersimplex arises from a weight vector $p_\bullet \in {\mathbb R}^{\binom{[n]}{k}}$: we lift each vertex $e_I$ of $\Delta(k,n)$ to height $p_I$, and project the lower faces of the resulting convex hull down to obtain a subdivision ${\tilde \Delta}(p_\bullet)$. Speyer \cite{Spe} showed that the subdivision ${\tilde \Delta}(p_\bullet)$ is into matroid polytopes if and only if $p_\bullet$ satisfies the three-term tropical Pl\"ucker relations. We say that $p_\bullet$ satisfies the three-term positive tropical Pl\"ucker relations if for every $S$ and $a<b<c<d$ not contained in $S$, we have that \begin{equation}\label{eq:trop} p_{Sac} + p_{Sbd} = \min(p_{Sab} + p_{Scd} , p_{Sad} + p_{Sbc}). \end{equation} We show in Proposition~\ref{prop:posdiv} that the subdivision ${\tilde \Delta}(p_\bullet)$ is into positroid polytopes if and only if $p_\bullet$ satisfies \eqref{eq:trop}. Following \cite{HJJS,HJS,OPS2}, we call the space of vectors satisfying \eqref{eq:trop} the positive Dressian, and denote it by ${\rm Dr}(k,n)_{>0}$. More generally, for each positroid ${\mathcal{M}}$, we have a positive local Dressian ${\rm Dr}({\mathcal{M}})_{>0}$. \subsection{} Speyer and Sturmfels \cite{SS} studied the tropical Grassmannian ${\operatorname{Trop}}\, {\rm Gr}(k,n)$ which parametrizes tropical linear spaces. Every point $p_\bullet \in {\operatorname{Trop}}\, {\rm Gr}(k,n)$ satisfies the three-term tropical Pl\"ucker relations, but in general the converse is not true \cite{HJJS}. Speyer and Williams \cite{SW} defined the positive tropical Grassmannian ${\operatorname{Trop}}_{>0}{\rm Gr}(k,n) \subset {\operatorname{Trop}} \,{\rm Gr}(k,n)$. Let ${\mathcal{R}}= \bigcup_{n=1}^\infty {\mathbb R}((t^{1/n})) $ denote the field of Puiseux series and ${\mathcal{R}}_{>0} \subset {\mathcal{R}}$ those Puiseux series whose leading (lowest) coefficient is positive. Then ${\operatorname{Trop}}_{>0}{\rm Gr}(k,n)$ is defined to be the closure of the set of valuations $p_\bullet = (p_I = {\rm val}( \Delta_I(V)) \mid I \in \binom{[n]}{k})$ for $V \in {\rm Gr}(k,n)({\mathcal{R}}_{>0})$. We generalize this by also considering the positive tropical positroid cell ${\operatorname{Trop}}_{>0} \Pi_{\mathcal{M}}$. It is immediate that every point $p_\bullet \in {\operatorname{Trop}}_{>0} {\rm Gr}(k,n)$ or $p_\bullet \in {\operatorname{Trop}}_{>0} \Pi_{\mathcal{M}}$ satisfies the three-term positive tropical Pl\"ucker relations \eqref{eq:trop}. A key technical result is that the converse holds (Theorem~\ref{thm:main}). \begin{theorem}\label{thm:mainintro} Every rational positive tropical Pl\"ucker vector is realizable. Thus $$ {\operatorname{Trop}}_{> 0}{\rm Gr}(k,n) = {\rm Dr}(k,n)_{>0}, \;\;\;{\operatorname{Trop}}_{>0} \Pi_{{\mathcal{M}}} = {\rm Dr}({\mathcal{M}})_{>0}. $$ \end{theorem} We give two proofs of Theorem~\ref{thm:mainintro}. The first one uses \eqref{eq:trop} directly. The second one uses a tropical bridge reduction (Section \S\ref{sec:bridge}) for positive tropical Pl\"ucker vectors, a variant of the bridge reduction algorithm for points $V \in {\rm Gr}(k,n)_{\geq 0}$ in \cite{LamCDM}. Whereas usual bridge reduction gives parametrizations of $\Pi_{{\mathcal{M}},>0}$, tropical bridge reduction gives parametrizations of ${\rm Dr}({\mathcal{M}})_{>0} = {\operatorname{Trop}}_{>0}\Pi_{\mathcal{M}}$. The space ${\rm Dr}({\mathcal{M}})_{>0}$ has a number of different fan structures. Two of them, the secondary fan structure and the Pl\"ucker fan structure, were shown to agree in \cite{OPS2}. We study a class of positive fan structures (generalizing \cite{SW}) coming from positive parametrizations of $\Pi_{{\mathcal{M}},>0}$ and show in Theorem~\ref{thm:coincide} that all the fan structures coincide. \subsection{} Let us give an example illustrating the bijections of Theorem~\ref{thm:introbij}. Full definitions are given in the main text. Consider the curve $V: [0,\infty) \to {\rm Gr}(2,5)$ given by $$ V(t) = \begin{bmatrix} 0 & 1& 1& 1&1\\ -1 & 0& 1& 1+ t & 1+2t \end{bmatrix} $$ which has Pl\"ucker coordinates $\Delta_{12}=\Delta_{13}=\Delta_{14}=\Delta_{15}=\Delta_{23}=1$ and $$ \Delta_{24}=1+t \qquad \Delta_{25} = 1+2t \qquad \Delta_{34} = t \qquad \Delta_{35}=2t \qquad \Delta_{45}=t. $$ Thus the curve $V(t)$ lies in the positive Grassmannian ${\rm Gr}(2,5)_{>0}$ for $t > 0$, and defines a curve $X(t) \in {\operatorname{Conf}}(2,5)_{>0} = {\rm Gr}(2,5)_{>0}/T_{>0}$ in the positive component of generic configuration space. Let $X:= \lim_{t \to 0} X(t) \in {\operatorname{Ch}}(2,5)_{\geq 0}$. We have \begin{align*} V_1&:=\lim_{t \to 0} V(t) = \begin{bmatrix} 0 & 1& 1& 1&1\\ -1 & 0& 1& 1 & 1 \end{bmatrix} \end{align*} and setting $V'(t) = V(t) \cdot {\rm diag}(t^{1/2},t^{1/2},t^{-1/2},t^{-1/2},t^{-1/2}) $, we have \begin{align*} V_2:= \lim_{t \to 0}V'(t) &= \lim_{t\to 0}\begin{bmatrix} 0 & \sqrt{t}& 1/\sqrt{t}& 1/\sqrt{t}&1/\sqrt{t}\\ -\sqrt{t} & 0& 1/\sqrt{t}& (1+t)/\sqrt{t} & (1+2t)/\sqrt{t} \end{bmatrix} \\ &=\lim_{t\to 0}\begin{bmatrix} 1 & 1& 0& -1&-2\\ -t & 0& 1& 1+t& 1+2t \end{bmatrix}\\ &=\begin{bmatrix} 1 & 1& 0& -1&-2\\ 0& 0& 1& 1& 1 \end{bmatrix}. \end{align*} The points $V_1$ and $V_2$ have matroids $$ {\mathcal{M}}_1:=\mbox{$\binom{[5]}{2}$} \setminus \{34,35,45\} \qquad \text{and} \qquad {\mathcal{M}}_2 :=\mbox{$\binom{[5]}{2}$} \setminus \{12\} $$ respectively. The matroid polytopes $P_{{\mathcal{M}}_1}$ and $P_{{\mathcal{M}}_2}$ give a decomposition of the hypersimplex $\Delta(2,5)$ into positroid polytopes: this decomposition comes from slicing $\Delta(2,5)$ using the hyperplane $x_1+x_2 =1$ (or, equivalently $x_3+x_4+x_5 = 1$). It follows that $X \in {\operatorname{Ch}}(2,5)_{\geq 0}$ is represented by the union of the two toric varieties $\overline{T \cdot V_1}$ and $\overline{T \cdot V_2}$ (inside ${\rm Gr}(2,5)$), which have moment polytopes $P_{{\mathcal{M}}_1}$ and $P_{{\mathcal{M}}_2}$ respectively. The valuation ${\rm val}(f(t))$ of a polynomial (or formal power series) $f(t)$ is the degree of the lowest term in $f(t)$. Defining $p_I:={\rm val}(\Delta_I(V(t)))$ we obtain a vector $p_\bullet \in {\mathbb R}^{\binom{[5]}{2}}$, given by $p_{34}=p_{35}=p_{45}=1$ and $p_J = 0$ for $J \notin \{34,35,45\}$. By definition, $p_\bullet$ lies the positive tropical Grassmannian. It is also easy to check that $p_\bullet$ satisfies \eqref{eq:trop}. For example, $0 = p_{13}+p_{25} = \min(p_{12}+p_{35},p_{15}+p_{23}) = \min(1,0) = 0$. This induces the bijections of Theorem~\ref{thm:introbij}. \subsection{} Finally, let us explain some ingredients of the proof of Theorem~\ref{thm:introtop}. We use the notion of nearly convergent functions on ${\operatorname{Ch}}(k,n)$ (the nomenclature comes from the stringy integrals of \cite{AHL1}). These are certain $T$-invariant, subtraction-free, rational functions on the Grassmannian whose tropicalizations take nonnegative values. The ring ${\mathbb C}[\Gamma]$ generated by nearly convergent functions is isomorphic to the coordinate ring of an affine open subset $X'_{P(k,n)}$ of a projective toric variety $X_{P(k,n)}$ (Proposition~\ref{prop:XPkn}) associated to some polytope $P(k,n)$. We obtain a morphism $$ \varphi: {\widetilde{\Ch}}(k,n) \longrightarrow X'_{P(k,n)} $$ from an open subset ${\widetilde{\Ch}}(k,n) \subset {\operatorname{Ch}}(k,n)$ of the Chow quotient to $X'_{P(k,n)}$. We show that the restriction of $\varphi$ to the nonnegative part ${\operatorname{Ch}}(k,n)_{\geq 0}$ is a homeomorphism onto the nonnegative part $X_{P(k,n),\geq 0}$ of the toric variety, which is known to be homeomorphic to the polytope $P(k,n)$. \smallskip \noindent {\bf Organization.} In \S\ref{sec:matroid}, we discuss matroids, positroids, and their matroid polytopes. In \S\ref{sec:Gr}, we discuss the Grassmannian, configuration space, and the positroid stratification. In \S\ref{sec:cluster}, we review cluster parametrizations of positroid cells. In \S\ref{sec:def} and \S\ref{sec:strata}, we introduce our main object of interest: the nonnegative configuration space and its stratification by positive Chow cells. In \S\ref{sec:Plucker} and \S\ref{sec:hypersimplex}, we study positive tropical vectors and positroid subdivisions of the hypersimplex. In \S\ref{sec:trop}, we show that the positive Dressian and the positive tropical Grassmannian agree. In \S\ref{sec:fan} we show that a number of fan structures on the positive Dressian coincide. In \S\ref{sec:Gamma}, we introduce nearly convergent functions. In \S\ref{sec:ball}, we prove that ${\operatorname{Ch}}(k,n)_{\geq 0}$ is homeomorphic to a ball. The case $k = 2$ is studied as an example in \S\ref{sec:M0n}. In \S\ref{sec:bridge}, we introduce and study tropical bridge operations. \S\ref{sec:connected} contains some technical statements concerning connected positroids. Appendix~\ref{sec:examples} contains data for the cases $(k,n) = (3,6), (3,7), (3,8)$. \smallskip \noindent {\bf Acknowledgements.} We thank Song He for conversations related to this work. T.L. was supported by NSF DMS-1464693, NSF DMS-1953852, and by a von Neumann Fellowship from the Institute for Advanced Study. N.A-H. and M.S. were supported by DOE grants DE-SC0009988 and DE-SC0010010 respectively. \begin{remark} Many of the results in this work were announced at Amplitudes 2019 \cite{Namp}. Some of the results in \S\ref{sec:Plucker}--\S\ref{sec:trop} regarding the positive tropical Grassmannian, the positive Dressian, and positroidal subdivisions of the hypersimplex are not surprising to experts and overlap with independent recent work in \cite{Ola,Ear,Ear2,LPW,SW2}. For instance, Proposition \ref{prop:posSpeyer} is closely related to \cite[Theorem 3.8]{LPW} and Proposition \ref{prop:posdiv}(2) is \cite[Theorem 9.12]{LPW}. \end{remark} \section{Matroids and positroids}\label{sec:matroid} \subsection{} A matroid ${\mathcal{M}} \subset \binom{[n]}{k}$ of rank $k$ on $[n]$ is a nonempty collection of $k$-element subsets of $[n]$, called {\it bases}, satisfying the {\it exchange axiom}: \begin{equation}\label{eq:exchange} \mbox{if $I, J \in {\mathcal{M}}$ and $i \in I$ then there exists $j \in J$ such that $I \setminus \{i\} \cup \{j\} \in {\mathcal{M}}$.} \end{equation} The {\it uniform matroid} is the collection ${\mathcal{M}} = \binom{[n]}{k}$ of all $k$-element subsets of $[n]$. \subsection{} The {\it matroid polytope} $P_{\mathcal{M}}$ of a matroid in ${\mathcal{M}}$ is the convex hull of the vectors $e_I$, for $I \in {\mathcal{M}}$. Here, $e_I = e_{i_1} + e_{i_2} + \cdots + e_{i_k}$ is the sum of $k$ basis vectors, where $I = \{i_1,\ldots,i_k\}$. Thus the matroid polytope of the uniform matroid is the hypersimplex $\Delta(k,n)$, whose vertices are exactly the $0$-$1$ vectors with $k$ $1$-s and $(n-k)$ $0$-s. We have the following characterization of matroid polytopes. \begin{proposition}[\cite{GGMS}] \label{prop:GGMS} A polytope $P \subset {\mathbb R}^n$ is the matroid polytope of a matroid of rank $k$ on $[n]$ if and only if its vertex set is a subset of $\{e_I \mid I \in \binom{[n]}{k}\}$, and all edges of $P$ are in the direction of $e_i - e_j$, for $i \neq j$. \end{proposition} \subsection{} If ${\mathcal{M}}_1$ is a matroid on a set $S_1$ and ${\mathcal{M}}_2$ a matroid on $S_2$, then the {\it direct sum }${\mathcal{M}}_1 \oplus {\mathcal{M}}_2$ is a matroid on the disjoint union $S_1 \sqcup S_2$, given by $$ {\mathcal{M}}_1 \oplus {\mathcal{M}}_2 = \{I_1 \sqcup I_2 \mid I_i \in {\mathcal{M}}_i\}. $$ We say that a matroid ${\mathcal{M}}$ of rank $k$ on $[n]$ is \emph{connected} if the matroid polytope $P_{\mathcal{M}}$ is of full dimension, that is, has dimension $n-1$. This is equivalent to the condition that ${\mathcal{M}}$ is not a non-trivial direct sum of smaller matroids. \subsection{} The {\it Bruhat partial order} on $\binom{[n]}{k}$ is defined as follows. For two subsets $I, J \in \binom{[n]}{k}$, we write $I \leq J$ if $I = \{i_1 < i_2 <\cdots < i_k\}$, $J = \{j_1 < j_2 < \cdots < j_k\}$ and we have $i_r \leq j_r$ for $r = 1,2,\ldots,k$. For $I \in \binom{[n]}{k}$, the {\it Schubert matroid} ${\mathcal{S}}_I$ is defined as $$ {\mathcal{S}}_I:= \{J \in {\mbox{$\binom{[n]}{k}$}} \mid I \leq J\} $$ and has minimal element $I$. For $a \in [n]$, let $\leq_a$ denote the cyclically rotated order on $[n]$ with minimum $a$, which induces a partial order $\leq_a$ on $\binom{[n]}{k}$. Let ${\mathcal{S}}_{I,a} := \{J \in \binom{[n]}{k} \mid I \leq_a J\}$ denote the cyclically rotated Schubert matroid. \subsection{} A {\it $(k,n)$-Grassmann necklace} ${\mathcal{I}} = (I_1,I_2,\ldots,I_n)$ \cite{Pos} is a $n$-tuple of $k$-element subsets of $[n]$ satisfying the following condition: for each $a \in [n]$, we have \begin{enumerate} \item $I_{a+1} = I_a$ if $a \notin I_a$, \item $I_{a+1} = I_a - \{a\} \cup \{a'\}$ if $a \in I_a$, \end{enumerate} with indices taken modulo $n$. A {\it positroid} ${\mathcal{M}}$ is the matroid of a totally nonnegative point in the Grassmannian, and are in bijection with Grassmann necklaces. \begin{proposition}[\cite{Oh, Pos}] \label{prop:Oh} Let ${\mathcal{I}}= (I_1,I_2,\ldots,I_n)$ be a $(k,n)$-Grassmann necklace. Then the intersection of cyclically rotated Schubert matroids \begin{equation}\label{eq:MI} {\mathcal{M}}_{\mathcal{I}} = {\mathcal{S}}_{I_1,1} \cap {\mathcal{S}}_{I_2,2} \cap \cdots \cap {\mathcal{S}}_{I_n,n} \end{equation} is a positroid, and the map ${\mathcal{I}} \mapsto {\mathcal{M}}_{\mathcal{I}}$ gives a bijection between $(k,n)$-Grassmann necklaces and positroids of rank $k$ on $[n]$. \end{proposition} \subsection{} Let ${\mathcal{M}}$ be an arbitrary matroid of rank $k$ on $[n]$ and $a \in [n]$. Then ${\mathcal{M}}$ has a minimum with respect to $\leq_a$, which is denoted $I_a({\mathcal{M}})$. \begin{proposition} The $n$-tuple ${\mathcal{I}}({\mathcal{M}})=(I_1({\mathcal{M}}),I_2({\mathcal{M}}),\ldots,I_n({\mathcal{M}}))$ is a $(k,n)$-Grassmann necklace. \end{proposition} We call ${\mathcal{M}}_{{\mathcal{I}}({\mathcal{M}})}$ the {\it positroid envelope} of ${\mathcal{M}}$ \cite{Pos,KLS}. A matroid ${\mathcal{M}}$ equals its positroid envelope if and only if ${\mathcal{M}}$ is a positroid. \subsection{} A {\it $(k,n)$-bounded affine permutation} is a bijection $f:{\mathbb Z} \to {\mathbb Z}$ satisfying the conditions \begin{enumerate} \item $f(i+n) = f(i) + n$ for all $i \in {\mathbb Z}$, \item $i \leq f(i) \leq i+n$ for all $i \in {\mathbb Z}$, \item $\sum_{i=1}^n (f(i)-i) = kn$. \end{enumerate} Given a $(k,n)$-Grassmann necklace ${\mathcal{I}} = (I_1,I_2,\ldots,I_n)$, we define $f_{\mathcal{I}}:{\mathbb Z} \to {\mathbb Z}$ by $$ f_{\mathcal{I}}(a) = \begin{cases} a & \mbox{if $a \notin I_a$} \\ a+n & \mbox{if $a \in I_a \cap I_{a+1}$} \\ a' & \mbox{if $I_{a+1} = I_a - \{a\} \cup \{a'\}$ and $a' > a$}\\ a'+n& \mbox{if $I_{a+1} = I_a - \{a\} \cup \{a'\} $ and $a' < a$} \end{cases} $$ for $a =1,2,\ldots,n$, and extending the domain to ${\mathbb Z}$ by setting $f(i+n) = f(i) + n$ for all $i \in {\mathbb Z}$. \begin{proposition}\cite{KLS} \label{prop:KLS} For any $(k,n)$-Grassmann necklace ${\mathcal{I}}$, the function $f_{\mathcal{I}}$ is a $(k,n)$-bounded affine permutation. The map ${\mathcal{I}} \mapsto f_{\mathcal{I}}$ gives a bijection between $(k,n)$-Grassmann necklaces and $(k,n)$-bounded affine permutations. \end{proposition} Thus we have bijections between positroids of rank $k$ on $[n]$, and $(k,n)$-Grassmann necklaces, and $(k,n)$-bounded affine permutations. We write $f_{\mathcal{M}}:= f_{{\mathcal{I}}({\mathcal{M}})}$. \subsection{} A polytope $P$ in $\{(x_1,\ldots,x_n) \mid x_1+x_2+\cdots+x_n = k\} \subset {\mathbb R}^n$ is called {\it alcoved} \cite{LP1} if it is given by the intersection of half spaces of the form $$ H = \{(x_1,\ldots,x_n) \mid \sum_{i \in [a,b]} x_i \geq c\} $$ where $[a,b] \subset [n]$ is a cyclic interval. The following two results are a special case of the theory of polypositroids \cite{LP}, see also \cite[Proposition 5.5 and Corollary 5.4]{ARW}. \begin{proposition}[\cite{LP}] \label{prop:posfacet} Let ${\mathcal{M}}$ be a matroid with matroid polytope $P_{\mathcal{M}}$. Then $P_{\mathcal{M}}$ is alcoved if and only if ${\mathcal{M}}$ is a positroid. \end{proposition} \begin{proof} The matroid polytope $P_{{\mathcal{S}}_{I,a}}$ of the rotated Schubert matroid ${\mathcal{S}}_I$ is the intersection of the hypersimplex $\Delta(k,n)$ with the inequalities $$ x_a + x_{a+1} + \cdots + x_b \geq \#(I \cap [a,b]) $$ for $i = 1,2,\ldots,n$. By definition, this is an alcoved polytope, and so is the intersection $P_{{\mathcal{S}}_{I_1,1}} \cap P_{{\mathcal{S}}_{I_2,2}} \cap \cdots \cap P_{{\mathcal{S}}_{I_n,n}}$. Since every positroid is of the form \eqref{eq:MI}, every positroid polytope is alcoved. Now let $P_{\mathcal{M}}$ be the matroid polytope of an arbitrary matroid. Then the smallest alcoved polytope $P$ containing $P_{\mathcal{M}}$ is the intersection of the rotated Schubert matroid polytopes $P_{{\mathcal{S}}_{I_a({\mathcal{M}}),a}}$ for $a =1,2,\ldots,n$. Thus $P$ is the matroid polytope of the positroid envelope of ${\mathcal{M}}$. In particular, if ${\mathcal{M}}$ is itself a positroid then $P_{\mathcal{M}}$ is alcoved. \end{proof} \begin{corollary} Every face of a positroid polytope is itself a positroid polytope. \end{corollary} \begin{proof} Every face of a matroid polytope (resp. alcoved polytope) is a matroid polytope (resp. alcoved polytope). \end{proof} A {\it noncrossing partition} $(S_1,\ldots,S_r)$ of $[n]$ is a partition of $[n]$ into disjoint sets such that there do not exist $a<b<c<d$ such that $a,c \in S_i$ and $b,d \in S_j$ for $i \neq j$. \begin{proposition}[{\cite[Theorem 7.6]{ARW}}] \label{prop:disconnect} Let $f = f_{\mathcal{M}}$ be the bounded affine permutation associated to a positroid ${\mathcal{M}}$. Suppose that ${\mathcal{M}} = {\mathcal{M}}_1 \oplus {\mathcal{M}}_2 \oplus \cdots \oplus {\mathcal{M}}_r$, where ${\mathcal{M}}_i$ is a positroid on the ground set $S_i \subset [n]$. Then $(S_1,\ldots,S_r)$ form a noncrossing partition of $[n]$, and $(f_{\mathcal{M}}(S_i) \mod n) = S_i$ for $i = 1,2,\ldots,r$. In particular, the connected components of the positroid ${\mathcal{M}}$ are themselves positroids. Conversely, let $(S_1,\ldots,S_r)$ be a noncrossing partition of $[n]$ and ${\mathcal{M}}_i$ be a positroid on the ground set $S_i$. Then the direct sum ${\mathcal{M}}_1 \oplus {\mathcal{M}}_2 \oplus \cdots \oplus {\mathcal{M}}_r$ is a positroid. \end{proposition} \section{Grassmannians and configuration spaces}\label{sec:Gr} \subsection{} Let ${\rm Gr}(k,n)$ denote the Grassmannian of $k$-planes in ${\mathbb C}^n$. For $V \in {\rm Gr}(k,n)$ we let $\Delta_I(V)$, for $I \in \binom{[n]}{k}$ denote its {\it Pl\"ucker coordinates}. These Pl\"ucker coordinates satisfy the Pl\"ucker relations and are defined up to a common scalar. We refer the reader to \cite{LamCDM} for further details. The most important relation for us will be the three-term Pl\"ucker relation \begin{equation}\label{eq:threeterm} \Delta_{Sac}\Delta_{Sbd} = \Delta_{Sab}\Delta_{Scd} + \Delta_{Sad} \Delta_{Sbc} \end{equation} where $S \subset [n]$ is of size $k-2$ and $a<b<c<d$ are not contained in $S$. It is convenient to also work with the affine cone $\widehat{\Gr}(k,n)$ over ${\rm Gr}(k,n)$. A point $V$ in $\widehat{\Gr}(k,n)$ is a collection of Pl\"ucker coordinates $\Delta_I(V)$ that satisfy the Pl\"ucker relations. But now scaling Pl\"ucker coordinates give different points, and $\widehat{\Gr}(k,n)$ also contains a distinguished cone point $0$, where all Pl\"ucker coordinates vanish. \subsection{} The {\it matroid} ${\mathcal{M}}(V)$ of $V \in {\rm Gr}(k,n)$ is defined as $$ {\mathcal{M}}(V) := \left\{I \in {\mbox{$\binom{[n]}{k}$}} \mid \Delta_I(V) \neq 0\right\}. $$ Similarly, one can define ${\mathcal{M}}(V)$ for $V \in \widehat{\Gr}(k,n) \setminus \{0\}$. If $V = 0$ is the cone point, then ${\mathcal{M}}(V)$ is not defined. For a matroid ${\mathcal{M}}$, define the matroid strata $$ {\mathcal{G}}_{\mathcal{M}}:= \{V \in {\rm Gr}(k,n) \mid {\mathcal{M}}(V) = {\mathcal{M}}\}$$ For the uniform matroid, we have $$ {\mathring{\Gr}}(k,n):={\mathcal{G}}_{\binom{[n]}{k}} = \left\{V \in {\rm Gr}(k,n) \mid \Delta_I(V) \neq 0 \text{ for all } I \in {\mbox{$\binom{[n]}{k}$}}\right\}. $$ \subsection{} The torus $({\mathbb C}^\times)^n = \{(t_1,t_2,\ldots,t_n) \mid t_i \in {\mathbb C}^\times\}$ acts on ${\rm Gr}(k,n)$ by scaling the $i$-th column of a representing matrix by $t_i$. For $t \in {\mathbb C}^\times$, the element $(t,t,\ldots,t) \in ({\mathbb C}^\times)^n$ scales all Pl\"ucker coordinates by $t^k$, and thus the action of $({\mathbb C}^\times)^n$ factors through the torus $$ T := ({\mathbb C}^\times)^n/{\mathbb C}^\times \simeq ({\mathbb C}^\times)^{n-1}. $$ An element $(t_1,\ldots,t_n) \in ({\mathbb C}^\times)^n$ acts on $V \in \widehat{\Gr}(k,n)$ by $$ \Delta_I((t_1,\ldots,t_n) \cdot V) = \prod_{i\in I} t_i \, \Delta_I(V). $$ This action factors through the quotient torus $$ {\widehat{ T}} = ({\mathbb C}^\times)^{n}/({\mathbb Z}/k{\mathbb Z}) $$ where ${\mathbb Z}/k{\mathbb Z} = \{(\zeta,\ldots,\zeta) \mid \zeta^k = 1\}$ is a cyclic group of order $k$. The character lattices $X(T)$ and $X({\widehat{ T}})$ of $T$ and ${\widehat{ T}}$ are naturally identified with sublattices of ${\mathbb Z}^n = X(({\mathbb C}^\times)^n)$: \begin{align} \begin{split}\label{eq:character} X(T) &= \{(x_1,\ldots,x_n) \in {\mathbb Z}^n \mid \sum x_i = 0\} \\ X({\widehat{ T}}) &= \{(x_1,\ldots,x_n) \in {\mathbb Z}^n \mid k \text{ divides } \sum x_i \} \end{split}. \end{align} \begin{lemma}\label{lem:connected} The matroid ${\mathcal{M}}$ is connected if and only if the action of $T$ on ${\mathcal{G}}_{\mathcal{M}}$ is free. \end{lemma} \begin{proof} Suppose ${\mathcal{M}}$ is connected. We may find a vertex $e_I$ of $P_{\mathcal{M}}$ and vertices $e_{J_1},\ldots,e_{J_{n-1}}$ connected to $e_I$ via an edge of $P_{\mathcal{M}}$, so that the span of $\{e_I,e_{J_1},\ldots,e_{J_{n-1}}\}$ is linearly independent. By Proposition~\ref{prop:GGMS}, the edges of $P_{\mathcal{M}}$ are roots, i.e., vectors of the form $e_i - e_j$. A collection of linearly independent roots is easily seen to be unimodular, i.e., their integral span is the lattice $\{(x_1,\ldots,x_n) \in {\mathbb Z}^n \mid \sum_i x_i = 0\}$, which is the character lattice $X(T)$ of $T$. Points $V \in {\mathcal{G}}_{\mathcal{M}}$ can be gauge-fixed to satisfy $\Delta_I(V) = 1$. Under this gauge-fix, the torus $T$ acts on the Pl\"ucker coordinates $\Delta_{J_1},\ldots,\Delta_{J_{n-1}}$ with weights $e_{J_1}-e_I,\ldots,e_{J_{n-1}}-e_I$. We have just argued that these weights span the character lattice $X(T)$, and thus $T$ must act freely on ${\mathcal{G}}_{\mathcal{M}}$. The ``if" direction is similar. \end{proof} \subsection{} The torus $T$ acts freely on the complex manifold ${\mathring{\Gr}}(k,n)$, and thus the quotient ${\mathring{\Gr}}(k,n)/T$ is a manifold of dimension $k(n-k) - (n-1)$. The space ${\mathring{\Gr}}(k,n)/T$ has the following alternative description. Let ${\operatorname{Conf}}(k,n)$ be the space $({\mathbb P}^{k-1})^n/{\rm GL}(n)$ of ${\rm GL}(n)$-orbits of $n$ points in the projective space ${\mathbb P}^{k-1}$. A configuration $p = (p_1,\ldots,p_n)$ is called {\it generic} if any $r$ of the points, for $r \leq k$, affinely span a subspace of ${\mathbb P}^{k-1}$ of dimension $r-1$. We denote the space of generic configurations by ${\mathring{\Conf}}(k,n) \subset {\operatorname{Conf}}(k,n)$. We have an isomorphism ${\mathring{\Gr}}(k,n)/T \simeq {\mathring{\Conf}}(k,n)$. \subsection{} The {\it totally nonnegative Grassmannian} ${\rm Gr}(k,n)_{\geq 0}$ is the subspace of ${\rm Gr}(k,n)$ consisting of points $V \in {\rm Gr}(k,n)({\mathbb R})$ all of whose Pl\"ucker coordinates are nonnegative. The {\it totally positive Grassmannian} ${\rm Gr}(k,n)_{>0}$ consists of points all of whose Pl\"ucker coordinates are positive. These spaces were defined by Lusztig \cite{Lus} and Postnikov \cite{Pos}. The totally nonnegative Grassmannian ${\rm Gr}(k,n)_{\geq 0}$ is homeomorphic to a closed ball \cite{GKL}. The matroid of a point $V \in {\rm Gr}(k,n)_{\geq 0}$ is called a {\it positroid}. Positroids can be indexed by $(k,n)$-Grassmann necklaces (Proposition~\ref{prop:Oh}) and $(k,n)$-bounded affine permutations (Proposition~\ref{prop:KLS}). The matroid of $V \in {\rm Gr}(k,n)_{>0}$ is the uniform matroid, and thus we have ${\rm Gr}(k,n)_{>0} \subset {\mathring{\Gr}}(k,n)$. Indeed, ${\rm Gr}(k,n)_{>0}$ is a connected component of the manifold ${\mathring{\Gr}}(k,n)$, and is diffeomorphic to an open ball of dimension $k(n-k) -(n-1)$. The image of ${\rm Gr}(k,n)_{>0}$ in ${\mathring{\Gr}}(k,n)/T \simeq {\mathring{\Conf}}(k,n)$ is called the {\it positive component} of (generic) configuration space, and denoted ${\operatorname{Conf}}(k,n)_{>0}$. \subsection{} For a positroid ${\mathcal{M}}$, define the {\it positroid cell} $$ \Pi_{{\mathcal{M}},>0} := \{V \in {\rm Gr}(k,n)_{\geq 0} \mid {\mathcal{M}}(V) = {\mathcal{M}}\}. $$ By \cite{Pos}, $\Pi_{{\mathcal{M}},>0}$ is homeomorphic to an open ball. We define $\dim({\mathcal{M}})$ to be the dimension of this ball. We have the disjoint union \cite{Pos} \begin{equation}\label{eq:positroiddecomp} {\rm Gr}(k,n)_{\geq 0} = \bigsqcup_{\mathcal{M}} \Pi_{{\mathcal{M}},>0} \end{equation} as ${\mathcal{M}}$ varies over positroids of rank $k$ on $[n]$. The closure $\Pi_{{\mathcal{M}},\geq 0}:= \overline{\Pi_{{\mathcal{M}},>0}}$ is a union $ \Pi_{{\mathcal{M}},\geq 0} = \bigsqcup_{{\mathcal{M}}' \subseteq {\mathcal{M}}} \Pi_{{\mathcal{M}}',>0} $ of positroid cells. The decomposition \eqref{eq:positroiddecomp} gives ${\rm Gr}(k,n)_{\geq 0}$ the structure of a regular CW complex \cite{GKL2}. \subsection{} The group $T_{>0}:= {\mathbb R}_{>0}^n/{\mathbb R}_{>0} \subset ({\mathbb C}^\times)^{n}/({\mathbb C}^\times) = T$ acts on ${\rm Gr}(k,n)_{\geq 0}$, preserving the positroid cells $\Pi_{{\mathcal{M}},>0}$. From Lemma \ref{lem:connected}, we have the following. \begin{lemma} Suppose that ${\mathcal{M}}$ is a connected positroid. Then $T_{>0}$ acts freely on $\Pi_{{\mathcal{M}},>0}$. \end{lemma} The quotient $\Pi_{{\mathcal{M}},>0}/T_{>0}$ is a real manifold of dimension $\dim({\mathcal{M}})-(n-1)$. In particular, if ${\mathcal{M}}$ is a connected positroid then $\dim({\mathcal{M}}) \geq n-1$. If ${\mathcal{M}}$ is a connected positroid and $\dim({\mathcal{M}}) = n-1$, then we call ${\mathcal{M}}$ a {\it minimal connected positroid} (see Lemma~\ref{lem:mintree}). In this case, $\Pi_{{\mathcal{M}},>0}/T_{>0}$ is a single point. \subsection{} For a positroid ${\mathcal{M}}$, define the {\it positroid variety} $\Pi_{\mathcal{M}}$ to be the Zariski-closure of $\Pi_{{\mathcal{M}},>0}$ inside ${\rm Gr}(k,n)$. The variety $\Pi_{\mathcal{M}}$ is an irreducible, normal subvariety of ${\rm Gr}(k,n)$ of dimension $\dim({\mathcal{M}})$ \cite{KLS}. Positroid varieties give a stratification of ${\rm Gr}(k,n)$. We define the {\it open positroid variety} ${\mathring{\Pi}}_{\mathcal{M}} \subset \Pi_{\mathcal{M}}$ so that $ \Pi_{{\mathcal{M}}} = \bigsqcup_{{\mathcal{M}}' \subseteq {\mathcal{M}}} {\mathring{\Pi}}_{{\mathcal{M}}'}. $ We have ${\rm Gr}(k,n)_{\geq 0} \cap {\mathring{\Pi}}_{{\mathcal{M}}'} = \Pi_{{\mathcal{M}}',>0}$. We caution that ${\mathring{\Pi}}_{\mathcal{M}}$ is not equal to ${\mathcal{G}}_{\mathcal{M}}$. Instead, ${\mathring{\Pi}}_{\mathcal{M}}$ is the union of ${\mathcal{G}}_{{\mathcal{M}}'}$ over all matroids ${\mathcal{M}}'$ such that the positroid envelope of ${\mathcal{M}}'$ is equal to ${\mathcal{M}}$. \subsection{}\label{ssec:bridgereduce} We recall the bridge parametrizations of $\Pi_{\mathcal{M}}$ from \cite{LamCDM}, see also \cite{Kar}. The group ${\rm GL}(n)$ acts on ${\rm Gr}(k,n)$ by right multiplication. For $i = 1,2,\ldots, n-1$, let $x_i(t) \in {\rm GL}(n)$ be the matrix that differs from the identity in a single entry equal to $t$ in the $(i,i+1)$-th matrix entry. For $i = n$, we let $x_n(t) \in {\rm GL}(n)$ be the matrix that differs from the identity in a single entry equal to $(-1)^{k-1} t$ in the $(n,1)$-th matrix entry. Thus $x_i(t)$ acts on a $k \times n$ matrix $V$ by adding the $i$-th column to the $(i+1)$-th column (with a sign if $i = n$). The action of $x_i(t)$ can be written in Pl\"ucker coordinates as (cf. \cite[Lemma 7.6]{LamCDM}) \begin{equation}\label{eq:Deltax} \Delta_I(V \cdot x_i(t)) = \begin{cases} \Delta_I(V) + t \Delta_{I \setminus \{i+1\} \cup \{i\}}(V) &\mbox{if $i+1 \in I$ but $i \notin I$} \\ \Delta_I(V) & \mbox{otherwise.} \end{cases} \end{equation} More generally, for $\gamma=(i,j)$ let $x_\gamma(t)$ be the matrix that differs from the identity in a single entry equal to $\pm t$ in the $(i,j)$-th matrix entry, taking the positive sign if $i < j$ and the sign $(-1)^{k-1}$ if $i > j$. The following result allows us to reduce totally nonnegative points recursively. \begin{proposition}[\cite{LamCDM}]\label{prop:bridgedecomp} Let $V \in \Pi_{{\mathcal{M}},>0} \subset {\rm Gr}(k,n)_{\geq 0}$ where $k > 0$. Then at least one of the following holds: \begin{enumerate} \item For some $i \in [n]$, we have $f_{\mathcal{M}}(i) = i$. Then $V$ is in the image of the map $\kappa_i: {\rm Gr}(k,n-1)_{\geq 0} \hookrightarrow {\rm Gr}(k,n)_{\geq 0}$ obtained by adding an $i$-th column equal to 0. \item For some $i \in [n]$, we have $f_{\mathcal{M}}(i) = i+n$. Then $V$ is in the image of the map $\eta_i: {\rm Gr}(k-1,n-1)_{\geq 0} \hookrightarrow {\rm Gr}(k,n)_{\geq 0}$ obtained by adding an extra first row and an extra $i$-th column, placing 0-s in all the new entries except for a $1$ in the $(1,i)$-th entry, and finally multiplying the columns $1,2,\ldots,i-1$ by $(-1)^{k-1}$. \item For some $i \in [n]$, we have $i+1 \leq f_{\mathcal{M}}(i) < f_{\mathcal{M}}(i+1) \leq i+n$. Then $V = V' \cdot x_i(a)$ where $V' \in \Pi_{{\mathcal{M}}',>0}$ with ${\mathcal{M}}'$ the positroid satisfying $f_{{\mathcal{M}}'} = f_{\mathcal{M}} s_i$ (where $s_i$ is the simple transposition swapping $i$ and $i+1$), and $$ a = \frac{\Delta_{I_{i+1}}(V)}{\Delta_{I_{i+1} \setminus\{i+1\} \cup \{i\}}(V)} >0. $$ \end{enumerate} \end{proposition} Proposition~\ref{prop:bridgedecomp} allows us to reduce any $V \in {\rm Gr}(k,n)_{\geq 0}$ to a positroid stratum that has dimension 0. The recursion can be chosen to only depend on ${\mathcal{M}} = {\mathcal{M}}(V)$, with the real numbers in (3) taken to be parameters. Parametrizations of ${\mathbb R}_{>0}^{\dim({\mathcal{M}})} \simeq \Pi_{{\mathcal{M}},>0}$ obtained in this way are called {\it bridge parametrizations}. Note that $\kappa_{i+1} \circ x_i(t) = x_{i,i+2}(t) \circ \kappa_{i+1}$, so in general we need to use the matrices $x_\gamma(t)$. Bridge parametrizations are of the form \begin{equation}\label{eq:bridgeparam} {\mathbb R}_{>0}^d \ni (t_1,t_2,\ldots,t_d) \longmapsto x_{\gamma_1}(t_1) \cdots x_{\gamma_d}(t_d) \cdot x_I \in \Pi_{{\mathcal{M}},>0} \end{equation} where $d = \dim({\mathcal{M}})$ and $x_{i_1,\ldots,i_k} = {\rm span}(e_{i_1},\ldots,e_{i_k}) \in {\rm Gr}(k,n)^T$ is a torus fixed point. We caution that in general $x_\gamma(t)$ for $t > 0$ does not preserve total nonnegativity. They do preserve total nonnegativity when used in a bridge parametrization. \section{Clusters for positroids}\label{sec:cluster} \subsection{}\label{ssec:weaksep} Let ${\mathcal{M}}$ be a positroid. Recall that ${\mathring{\Pi}}_{\mathcal{M}} \subset {\rm Gr}(k,n)$ denotes the open positroid variety. Let ${\widetilde{\Pi}}_{\mathcal{M}} \subset \widehat{\Gr}(k,n)$ denote the cone over ${\mathring{\Pi}}_{\mathcal{M}}$. By \cite{GL}, the coordinate rings ${\mathbb C}[{\mathring{\Pi}}_{\mathcal{M}}]$ and ${\mathbb C}[{\widetilde{\Pi}}_{\mathcal{M}}]$ are isomorphic to cluster algebras.\footnote{In \cite{GL}, the ring ${\mathbb C}[{\mathring{\Pi}}_{\mathcal{M}}]$ is considered. Working with ${\mathbb C}[{\widetilde{\Pi}}_{\mathcal{M}}]$ allows us to avoid gauge-fixing a Pl\"ucker coordinate to equal to 1.} A {\it cluster ${\mathcal{C}}$ for ${\mathcal{M}}$} is a subset of ${\mathcal{M}}$ that indexes a seed in the cluster structure of ${\mathbb C}[{\widetilde{\Pi}}_{\mathcal{M}}]$. If ${\mathcal{M}} = \binom{[n]}{k}$ is the uniform matroid, then we simply call ${\mathcal{C}}$ a {\it cluster}. Any cluster ${\mathcal{C}}$ for ${\mathcal{M}}$ has cardinality $|{\mathcal{C}}| = \dim({\mathcal{M}}) + 1$. (We consider only clusters where the cluster variables are Pl\"ucker coordinates coming from face labels of a plabic graph.) Clusters for ${\mathcal{M}}$ can be described via weak separation \cite{OPS}. We say that two subsets $I, J \subset [n]$ are {\it weakly-separated} if we cannot find $1\leq a < b < c < d \leq n$ such that $a,c \in I \setminus J$ and $b,d \in J \setminus I$ (or with $I$ and $J$ swapped). The following result can be taken to be the definition of a cluster. \begin{proposition}[\cite{OPS}]\label{prop:OPS} A subset ${\mathcal{C}} \subset {\mathcal{M}}$ is a cluster if it is pairwise weakly-separated, has size $\dim({\mathcal{M}}) + 1$, and contains the Grassmann necklace ${\mathcal{I}}$ of ${\mathcal{M}}$. Any pairwise weakly-separated subset of $\binom{[n]}{k}$ can be extended to a cluster. \end{proposition} Every $I \in \binom{[n]}{k}$ belongs to some cluster ${\mathcal{C}} \subset \binom{[n]}{k}$, but this is not true with an arbitrary positroid ${\mathcal{M}}$ replacing $\binom{[n]}{k}$. \subsection{} \label{ssec:mutation} Let $S \subset [n]$ be of size $k-2$ and $a<b<c<d$ numbers not contained in $S$. Let ${\mathcal{C}} \in {\mathcal{M}}$ be a cluster. If $Sac, Sab,Scd,Sad,Sbc \in {\mathcal{C}}$ (resp. $Sbd,Sab,Scd,Sad,Sbc \in {\mathcal{C}}$) then we can {\it mutate} ${\mathcal{C}}$ at $Sac$ (resp. $Sbd$) to produce another cluster ${\mathcal{C}}' \subset {\mathcal{M}}$ where $Sac$ has been replaced by $Sbd$ (resp. $Sbd$ has been replaced by $Sac$). The Pl\"ucker variables of ${\mathcal{C}}$ and ${\mathcal{C}}'$ are then related by \eqref{eq:threeterm}. \begin{proposition}[\cite{OPS}] \label{prop:mutation} Any two clusters ${\mathcal{C}},{\mathcal{C}}' \subset {\mathcal{M}}$ (as in Proposition~\ref{prop:OPS}) are related by a sequence of mutations. \end{proposition} By a positive Laurent polynomial we mean a Laurent polynomial such that the coefficient of every monomial is nonnegative. The following result is a special case of general positivity results of cluster algebras \cite{LS}. \begin{proposition}\label{prop:Laurent} For $J \in \binom{[n]}{k}$, and a cluster ${\mathcal{C}} \subset \binom{[n]}{k}$, the Pl\"ucker variable $\Delta_J$ is a positive Laurent polynomial in $\{\Delta_I \mid I \in {\mathcal{C}}\}$. \end{proposition} For an arbitrary positroid ${\mathcal{M}}$, we have the following weaker statement. \begin{proposition}\label{prop:Plucksub} For $J \in {\mathcal{M}}$, and a cluster ${\mathcal{C}} \subset {\mathcal{M}}$, the Pl\"ucker variable $\Delta_J$ is a subtraction-free rational expression in $\{\Delta_I \mid I \in {\mathcal{C}}\}$. \end{proposition} \begin{proof} This result follows, for example, from the formulae in \cite{MS}. It also follows from the proof of Theorem~\ref{thm:param} we give below. Namely, in that proof we show that a formula for $\Delta_J$ in terms of $\{\Delta_I \mid I \in {\mathcal{C}}\}$ can be obtained by iteratively applying the three-term Pl\"ucker relation \eqref{eq:threeterm}. In other words, we iteratively substitute $\Delta_{Sac} = (\Delta_{Sab}\Delta_{Scd} + \Delta_{Sad} \Delta_{Sbc})/\Delta_{Sbd}$, without ever dividing by 0. \end{proof} The following conjecture is likely known to many experts. It does not immediately follow from the identification \cite{GL} of ${\mathbb C}[{\mathring{\Pi}}_{\mathcal{M}}]$ with a cluster algebra because there are Pl\"ucker variables $\Delta_I$, $I \in {\mathcal{M}}$ that are not cluster variables in any cluster. \begin{conjecture}\label{conj:Laurent} For $J \in {\mathcal{M}}$, and a cluster ${\mathcal{C}} \subset {\mathcal{M}}$, the Pl\"ucker variable $\Delta_J$ is a positive Laurent polynomial in $\{\Delta_I \mid I \in {\mathcal{C}}\}$. \end{conjecture} \subsection{}\label{ssec:posparam} Let ${\mathcal{M}}$ be a positroid and set $d =\dim({\mathcal{M}})$. Let $({\mathbb C}^\times)^d$ be a torus with coordinate functions $x_1,\ldots,x_d$, so that the coordinate ring is ${\mathbb C}[x_1^{\pm 1},\ldots,x_d^{\pm 1}]$. A rational map $({\mathbb C}^\times)^d \to \Pi_{\mathcal{M}}$ is called a {\it positive parametrization} of $\Pi_{\mathcal{M}}$ if it is birational, the restriction to ${\mathbb R}_{>0}^d$ is a homeomorphism onto ${\rm Gr}(k,n)_{>0}$, and every Pl\"ucker coordinate is a subtraction-free rational expression $\Delta_I({\mathbf{x}})$ in $x_1,\ldots,x_m$. Any choice of cluster ${\mathcal{C}}$ gives a positive parametrization after setting one of the Pl\"ucker coordinates to 1: the map $T({\mathcal{C}}) = ({\mathbb C}^\times)^d \to \Pi_{\mathcal{M}}$ is simply the inclusion of the cluster torus ${\mathcal T}({\mathcal{C}})$ indexed by ${\mathcal{C}}$ into the positroid variety. This map comes from the inclusion ${\mathbb C}[{\widetilde{\Pi}}_{\mathcal{M}}] \subset {\mathbb C}[\Delta_J^{\pm 1} \mid J \in {\mathcal{C}}]$, called the {\it Laurent phenomenon}. \subsection{}\label{ssec:gaugefix} We now consider a simple-minded notion of ``cluster" for $\Pi_{\mathcal{M}}/T$. Let ${\mathcal{C}}$ be a cluster for a connected positroid ${\mathcal{M}}$. A {\it gauge-fix} ${\mathcal{G}} = \{J_1,\ldots,J_n\} \subset {\mathcal{C}}$ is a subset such that the integral span of $e_{J_1},\ldots,e_{J_n}$ inside ${\mathbb Z}^n$ is the $n$-dimensional lattice $X({\widehat{ T}}) = \{(x_1,\ldots,x_n) \in {\mathbb Z}^n \mid k \text{ divides } \sum x_i \}$. We shall prove the following result in \S\ref{ssec:proofspan}. \begin{lemma}\label{lem:span} Let ${\mathcal{M}}$ be a connected positroid. Then there exists a cluster ${\mathcal{C}} \subset {\mathcal{M}}$ such that a gauge-fix ${\mathcal{G}} \subset {\mathcal{C}}$ exists. \end{lemma} If ${\mathcal{G}}$ is a gauge-fix then the action of ${\widehat{ T}}$ can uniquely fix $\Delta_J = 1$ for all $J \in {\mathcal{G}}$ (assuming that $\Delta_J \neq 0$ for all $J \in {\mathcal{G}}$). We thus have a canonical identification \begin{equation}\label{eq:posgf} \Pi_{{\mathcal{M}},>0}/T_{>0} = \{V \in \Pi_{{\mathcal{M}},>0} \mid \Delta_J(V) = 1 \mbox{ for all } J \in {\mathcal{G}}\}. \end{equation} \section{Nonnegative configuration space}\label{sec:def} The goal of this section is to construct a compactification of ${\operatorname{Conf}}(k,n)_{>0}$. \subsection{} Let $X \subset {\mathbb P}^{k-1}$ be an irreducible subvariety of dimension $r-1$. The degree $\deg(X)$ of $X$ is equal to the number $\#(L \cap X)$ of intersection points of $X$ with a generic hyperplane $L \subset {\mathbb P}^{k-1}$ of dimension $(k-r-1)$. Let ${\mathcal{Z}}(X) \subset {\rm Gr}(k-r,k)$ denote the subvariety of $(k-r-1)$-dimensional projective subspaces $L \subset {\mathbb P}^{k-1}$ that intersect $X$. It is an irreducible hypersurface of degree $d$ in ${\rm Gr}(k-r,k)$ (\cite[Proposition 2.2]{GKZ}). Let $R_d(k-r,k)$ denote the degree $d$ component of the coordinate ring of the Grassmannian. We define the Chow form of $X$, denoted $R_X \in R_d(k-r,k)$, to be the unique up to scalar non-zero element of $R_d(k-r,k)$ that vanishes on ${\mathcal{Z}}(X)$. The variety $X$ can be recovered from $R_X$. \subsection{} An $(r-1)$-dimensional algebraic cycle in ${\mathbb P}^{k-1}$ is a formal finite linear combination $X = \sum m_i X_i$, where $m_i$ are nonnegative integers and $X_i \subset {\mathbb P}^{k-1}$ are irreducible closed subvarieties of dimension $(r-1)$. We define $\deg(X) = \sum m_i \deg(X_i)$. Let $C(r,d,k)$ denote the set of all $(r-1)$-dimensional algebraic cycles in ${\mathbb P}^{k-1}$ of degree $d$. Then $C(r,1,k)$ can naturally be identified with the Grassmannian ${\rm Gr}(r,k)$ of $r$-planes in ${\mathbb R}^k$. The set $C(r,d,k)$ acquires the structure of an algebraic variety, called the {\it Chow variety}, via the following result of Chow and van der Waerden. \begin{theorem}\label{thm:Chow} The map $C(r,d,k) \to {\mathbb P}(R_d(k-r,k))$ given by $ X \mapsto \prod_i R_{X_i}^{m_i} $ defines an embedding of $C(r,d,k)$ as closed subvariety of ${\mathbb P}(R_d(k-r,k))$. \end{theorem} \subsection{} A special case of Theorem \ref{thm:Chow} is the statement that ${\rm Gr}(k,n)$ is a closed subvariety of ${\mathbb P}^{\binom{[n]}{k}-1}$. Let $X = \overline{T \cdot V} \subset {\rm Gr}(k,n)$ be a torus orbit closure in ${\rm Gr}(k,n)$, where $V \in {\mathring{\Gr}}(k,n)$. Since $T \cdot V \simeq T$, the variety $X$ has dimension $(n-1)$. It is a toric variety with moment polytope equal to the hypersimplex. It follows that the degree of $X$ inside ${\rm Gr}(k,n)$ and inside ${\mathbb P}^{\binom{[n]}{k}-1}$ is equal to the volume ${\rm Vol}(\Delta(k,n))$, the Eulerian number $A_{n,k}$. We thus have a natural injection ${\mathring{\Conf}}(k,n) \hookrightarrow C(n-1,A_{n,k},\binom{n}{k})$ sending a point $V \in {\mathring{\Conf}}(k,n)$ to the algebraic cycle $\overline{T \cdot V}$. The closure of ${\mathring{\Conf}}(k,n)$ in $C(n-1,A_{n,k},\binom{n}{k})$ is called the {\it Chow quotient of the Grassmannian}, and denoted $\Ch(k,n)$. It is a projective algebraic variety. We define the {\it totally nonnegative part of the Chow quotient of the Grassmannian} or {\it nonnegative configuration space}, denoted ${\operatorname{Ch}}(k,n)_{\geq 0}$, to be the closure of the image of ${\operatorname{Conf}}(k,n)_{>0}$ in $\Ch(k,n)$. It is a compact Hausdorff topological space. \begin{remark} For a positroid ${\mathcal{M}}$, we can also define $\Ch({\mathcal{M}})_{\geq 0}$ as the closure of $\Pi_{{\mathcal{M}},>0}/T_{>0}$ inside an appropriate Chow variety. \end{remark} \section{Positive Chow cells}\label{sec:strata} A point $X \in \Ch(k,n)$ is an algebraic cycle in ${\rm Gr}(k,n)$ of dimension $(n-1)$ and degree $A_{n,k}$. We have $X = \sum_{i=1}^s m_i X_i$ where $X_i$ are toric varieties that are torus-orbit closures for the same torus $T$, and we assume $m_i$ are positive. Let $P_i$ denote the moment polytope of $X_i$. By \cite{Kap}, we have the following constraints on $X$: \begin{enumerate} \item we have $m_i = 1$ for all $i$, \item each $P_i = P_{{\mathcal{M}}_i}$ is a matroid polytope, and \item the polytopes $P_1,P_2,\ldots,P_s$ form a regular polyhedral subdivision of the hypersimplex $\Delta(k,n)$. \end{enumerate} For $X \in {\operatorname{Ch}}(k,n)_{\geq 0}$, we strengthen this result as follows. \begin{proposition}\label{prop:conf} Let $X = \sum_{i=1}^s X_i \in {\operatorname{Ch}}(k,n)_{\geq 0}$. Then $X_i$ is a toric variety with moment polytope equal to a positroid polytope $P_i = P_{{\mathcal{M}}_i}$. The positroid polytopes $P_1,\ldots,P_s$ form a regular polyhedral subdivision of the hypersimplex $\Delta(k,n)$. \end{proposition} \begin{proof} Let $X = \lim_{t \to 0} X(t)$ where $X(t) = \overline{T \cdot V(t)} \in {\operatorname{Conf}}(k,n)_{>0}$. For each $i = 1,2,\ldots,s$, we have $X_i = \overline{T \cdot V_i}$ for some $V_i \in {\rm Gr}(k,n)$. A generic point $p \in T \cdot V_i$ is thus the limit of points $p(t_1),p(t_2),\ldots$ where $p(t_j) \in T \cdot V(t_j)$, where $V(t_j) \in {\rm Gr}(k,n)_{>0}$. Let $({\mathbb Z}/2{\mathbb Z})^{n-1} := \{+1,-1\}^{n}/\{+1,-1\}$ denote the group of components of $T$. The $2^{n-1}$ points $g \cdot p$ are distinct for $g \in ({\mathbb Z}/2{\mathbb Z})^{n-1}$. For at least one of these points $q = g \cdot p \in X_i$, the infinite sequence $g \cdot p(t_1),g \cdot p(t_2), \ldots$ contains a subsequence $q(t'_1),q(t'_2),\ldots$ which all belong to ${\rm Gr}(k,n)_{>0}$. It follows that $q \in {\rm Gr}(k,n)_{\geq 0}$, and thus the matroid of $X_i$ is a positroid. \end{proof} Let ${\mathcal{D}}(k,n)$ denote the set of regular polyhedral subdivisions of the hypersimplex into positroid polytopes. For $X \in {\operatorname{Ch}}(k,n)_{\geq 0}$, let ${\tilde \Delta}(X)$ denote the positroid decomposition from Proposition~\ref{prop:conf}. We have a stratification \begin{align*} {\operatorname{Ch}}(k,n)_{\geq 0} &= \bigsqcup_{{\tilde \Delta} \in {\mathcal{D}}(k,n)} \Theta_{{\tilde \Delta}, >0} \\ \Theta_{{\tilde \Delta},>0} &:= \{X \in {\operatorname{Ch}}(k,n)_{\geq 0} \mid {\tilde \Delta}(X) = {\tilde \Delta}\}. \end{align*} Define $\Theta_{{\tilde \Delta},\geq 0}$ to be closure $\overline{\Theta_{{\tilde \Delta},>0} }$ in the analytic topology. The spaces $\Theta_{{\tilde \Delta},>0}$ and $\Theta_{{\tilde \Delta},\geq 0}$ are analogues of the open and closed positroid cells $\Pi_{{\mathcal{M}},>0}$ and $\Pi_{{\mathcal{M}},\geq 0}$ respectively. Define a partial order on ${\mathcal{D}}(k,n)$ by ${\tilde \Delta}' \leq {\tilde \Delta}$ if ${\tilde \Delta}'$ is a refinement of ${\tilde \Delta}$. \begin{theorem}\label{thm:ball} There is a stratification-preserving homeomorphism between ${\operatorname{Ch}}(k,n)_{\geq 0}$ and a polytope $P(k,n)$ of dimension $r = k(n-k)-(n-1)$. Each stratum $\Theta_{{\tilde \Delta},>0}$ (resp. $\Theta_{{\tilde \Delta},\geq 0}$) is non-empty and homeomorphic to an open ball (resp. closed ball) of dimension $\dim({\tilde \Delta})$, given by \eqref{eq:dimDelta}. The closed face $\Theta_{{\tilde \Delta},\geq 0}$ is the union of relatively open faces $\Theta_{{\tilde \Delta}',>0}$ as ${\tilde \Delta}'$ varies over subdivisions that refine ${\tilde \Delta}$. \end{theorem} The proof of Theorem~\ref{thm:ball} is delayed to \S\ref{ssec:homeo}. Theorem~\ref{thm:ball} generalizes various results concerning the topology of positroid cells \cite{Pos,GKL,GKL2}. Finally, we define $\Theta_{{\tilde \Delta}}$ to be the Zariski closure of $\Theta_{{\tilde \Delta},>0} $ inside $\Ch(k,n)$, and define ${\mathring{\Theta}}_{{\tilde \Delta}}$ to be the complement of $\{\Theta_{{\tilde \Delta}'} \mid {\tilde \Delta}' < {\tilde \Delta}\}$ in $\Theta_{{\tilde \Delta}}$. The varieties ${\mathring{\Theta}}_{{\tilde \Delta}}$ and $\Theta_{{\tilde \Delta}}$ are analogues of the open and closed positroid varieties ${\mathring{\Pi}}_{\mathcal{M}}$ and $\Pi_{\mathcal{M}}$. \section{Positive tropical Pl\"ucker vectors}\label{sec:Plucker} \subsection{} Let $p_\bullet = \{p_I \mid I \in \binom{[n]}{k}\}$ be a collection of ``numbers" where $p_I \in {\mathbb R} \cup \{\infty\}$, and not all $p_I$ are equal to infinity. We say that $p_\bullet$ satisfies the {\it tropical Pl\"ucker relations} \cite{Spe} if for every $S$ of size $k-2$ and $a<b<c<d$ not contained in $S$, the minimum of the three quantities $$\{p_{Sac} + p_{Sbd}, p_{Sab} + p_{Scd} , p_{Sad} + p_{Sbc}\}$$ is attained twice, and we call $p_\bullet$ a {\it tropical Pl\"ucker vector}. We say that $p_\bullet$ satisfies the {\it positive tropical Pl\"ucker relations} if for every $S$ and $a<b<c<d$ not contained in $S$, the equation \eqref{eq:trop} holds. We then call $p_\bullet$ a {\it positive tropical Pl\"ucker vector}. It is clear that every positive tropical Pl\"ucker vector is a tropical Pl\"ucker vector. A tropical Pl\"ucker vector is called {\it integral} if $p_I \in {\mathbb Z}\cup \{\infty\}$ and {\it rational} if $p_I \in {\mathbb Q} \cup \{\infty\}$. \subsection{} {\it Tropicalization} takes a subtraction-free rational expression to a piecewise-linear expression under the substitution \begin{equation}\label{eq:tropsubs} (+,\times,\div) \longmapsto (\min, +, -). \end{equation} For example, the rational function $\dfrac{x^3+y+1}{2xy+y^2}$ tropicalizes to the piecewise-linear function $\min(3X,Y)-\min(X+Y,2Y)$. The equation \eqref{eq:trop} is obtained from \eqref{eq:threeterm} by applying \eqref{eq:tropsubs}, and sending the variables $\Delta_I$ to the variables $p_I$. \subsection{} The support of $p_\bullet$ is the collection ${\rm Supp}(p) = \{I \mid p_I <\infty\} \subset \binom{[n]}{k}$. For a positroid ${\mathcal{M}}$, let ${\rm Dr}({\mathcal{M}})_{>0}$ denote the set of positive tropical Pl\"ucker vectors with support ${\mathcal{M}}$. Let ${\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})$ (resp. ${\rm Dr}({\mathcal{M}})_{>0}({\mathbb Q})$) denote those vectors that are integral (resp. rational). We write ${\rm Dr}(k,n)_{>0}$ when ${\mathcal{M}}$ is the uniform matroid and call $p_\bullet \in {\rm Dr}(k,n)_{>0}$ a {\it finite} positive tropical Pl\"ucker vector. In \cite{HJJS}, the set of tropical Pl\"ucker vectors is called the {\it Dressian}, and following their terminology, we call ${\rm Dr}(k,n)_{>0}$ the {\it positive Dressian}. \subsection{} The following result can be found in \cite{Spe}. \begin{proposition}\label{prop:Speyer} If $p_\bullet$ satisfies the tropical Pl\"ucker relations then ${\mathcal{M}}={\rm Supp}(p_\bullet)$ is a matroid. \end{proposition} If $p_\bullet$ satisfies the positive tropical Pl\"ucker relations, then in addition to \eqref{eq:exchange}, ${\mathcal{M}} = {\rm Supp}(p_\bullet)$ satisfies the following positive 3-term exchange relation: \begin{equation}\label{eq:3ex} (Sab,Scd \in {\mathcal{M}}) \text{ or } (Sad, Sbc \in {\mathcal{M}}) \implies (Sac,Sbd \in {\mathcal{M}}) \end{equation} for $a<b<c<d$ not contained in $S \subset [n]$. \begin{proposition}\label{prop:posSpeyer} If ${\mathcal{M}}$ satisfies the positive 3-term exchange relation then it is a positroid. Thus if $p_\bullet$ satisfies the positive tropical Pl\"ucker relations then ${\mathcal{M}}={\rm Supp}(p_\bullet)$ is a positroid. \end{proposition} \begin{proof} We establish this result by induction on $n$. Suppose that ${\mathcal{M}}$ is disconnected, so ${\mathcal{M}} = {\mathcal{M}}_1 \oplus {\mathcal{M}}_2$ on disjoint ground sets $S_1$ and $S_2$, such that $S_1 \cup S_2 =[n]$. If $S_1,S_2$ are cyclic intervals, then ${\mathcal{M}}_i$ satisfies positive 3-term exchange relation within $S_i$. Thus by induction ${\mathcal{M}}$ is the direct sum of two positroids on disjoint cyclic intervals, and is thus a positroid by Proposition~\ref{prop:disconnect}. Otherwise, we can find direct summands ${\mathcal{M}}',{\mathcal{M}}''$ of ${\mathcal{M}}$ which are two connected matroids on subsets $A \sqcup C,B \sqcup D$ that are crossing, i.e. $A,B,C,D$ are cyclic intervals occurring in cyclic order. Since ${\mathcal{M}}'$ is connected, there are bases $I_1,I_2$ of ${\mathcal{M}}'$ such that $|I_1 \cap A| \neq |I_2 \cap A|$. By repeated application of the basis exchange axiom, we see that ${\mathcal{M}}'$ contains two bases $I',J'$ such that $J' = I' \cup\{c\} -\{a\}$ with $a \in A$ and $c \in C$. Similarly, we have bases $I'',J''$ of ${\mathcal{M}}''$ such that $J'' = I'' \cup\{d\} -\{b\}$ with $b \in A$ and $d \in C$. We can thus find bases $Tab,Tcd$ of ${\mathcal{M}}$, while $Tac, Tbd$ are not bases, a contradiction. We now assume that ${\mathcal{M}}$ is a connected matroid. If it is not a positroid, then by Lemma \ref{prop:posfacet}, its matroid polytope $P_{\mathcal{M}}$ has a facet cut out by an equation $H = \{\sum_{s\in S} x_s = r\}$, where $S= S_1 \sqcup S_2$ is cyclically disconnected. Let ${\mathcal{M}}'$ be the matroid whose matroid polytope is $P_{{\mathcal{M}}'} = H \cap P_{\mathcal{M}}$. Now ${\mathcal{M}}' = {\mathcal{M}}_1 \oplus {\mathcal{M}}_2$ where ${\mathcal{M}}_1$ has ground set $S$ and ${\mathcal{M}}_2$ has ground set $[n] \setminus S$. The assumption that $\dim(P_{{\mathcal{M}}'}) = \dim(P_{\mathcal{M}}) - 1$ together with $|S|, |[n]\setminus S| \geq 2$ implies that both ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ are non-trivial. Since the ground sets $S$ and $[n] \setminus S$ are crossing, our earlier argument implies that ${\mathcal{M}}'$ cannot satisfy the positive 3-term exchange relation. After a cyclic relabeling we may assume that $Tab,Tcd \in {\mathcal{M}}'$, while at least one of $Tac$ and $Tbd$ is not in ${\mathcal{M}}'$. However, both $Tac,Tbd$ are in ${\mathcal{M}}$, so this is only possible if both $e_{Tac}$ and $e_{Tbd}$ do not lie on the hyperplane $H$. Indeed, the two vertices $e_{Tac}$ and $e_{Tbd}$ lie on opposite sides of $H$, and this contradicts the assumption that $H$ is a facet. \end{proof} \subsection{} Whereas the space of tropical Pl\"ucker vectors has a very complicated polyhedral structure \cite{SS,HJJS}, the situation is much simpler for positive tropical Pl\"ucker vectors. \begin{theorem}\label{thm:param} Let ${\mathcal{M}}$ be a positroid and ${\mathcal{C}}$ be a cluster for ${\mathcal{M}}$. The maps \begin{align}\label{eq:cluster} {\rm Dr}({\mathcal{M}})_{>0} &\longrightarrow {\mathbb R}^{{\mathcal{C}}}, \qquad &p_\bullet &\longmapsto (p_I \mid I \in {\mathcal{C}}) \\ {\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z}) &\longrightarrow {\mathbb Z}^{{\mathcal{C}}}, \qquad &p_\bullet &\longmapsto (p_I \mid I \in {\mathcal{C}}) \end{align} are bijections. \end{theorem} \begin{proof} Fix a cluster ${\mathcal{C}} \subset {\mathcal{M}}$. We show that $p_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}$ is determined by the values of $p_I$, $I \in {\mathcal{C}}$. Recall from \S\ref{ssec:weaksep} that $I,J$ are {\it weakly separated} if there does not exist cyclically ordered $a<b<c<d$ such that $a,c \in I \setminus J$ and $b,d \in J \setminus I$. Let $(I_1,\ldots,I_n)$ denote the Grassmann necklace of ${\mathcal{M}}$. For each $J \in {\mathcal{M}}$, define two integers $w(J)$ and $d(J)$ by \begin{align*} w(J)&:= \#\{a \mid (I_a, J) \text{ are not weakly separated } \} \\ d(J) &:= \begin{cases} \min_{a : \; (I_a, J) \text{ are not weakly separated}}(|I_a\setminus J|) & \mbox{if $w(J) > 0$} \\ 0 & \mbox{otherwise.} \end{cases} \end{align*} We show that $p_J$ is determined by $p_I$, $I \in {\mathcal{C}}$ by induction first on $w(J)$, then on $d(J)$. If $w(J) = 0$ then $J$ is weakly separated with $(I_1,\ldots,I_n)$ and by Proposition~\ref{prop:OPS} $J$ belongs to some cluster ${\mathcal{C}}$ for ${\mathcal{M}}$. By Proposition~\ref{prop:mutation}, all clusters are related by mutation, so we can express $p_J$ in terms of $p_I$, $I \in {\mathcal{C}}$ by repeated application of the positive tropical Pl\"ucker relations. Thus all $J \in {\mathcal{M}}$ satisfying $w(J)=0$ is determined by $p_I$, $I \in {\mathcal{C}}$. Now suppose $w = w(J) > 0$. Then we have $d = d(J) \geq 2$. We suppose by induction that the result has been proven for all $J'$ with $w(J') < w$, or $w(J') =w$ and $d(J') < d$. Choose $a \in [n]$ so that $(I_a,J)$ is not weakly separated and $I_a \setminus J = \{i_1,\ldots,i_d\}$ and $J \setminus I_a = \{j_1,\ldots,j_d\}$. Since $J \geq_a I_a$, there is a unique noncrossing matching on these $2d$ points, which after reindexing we assume to be $\{(i_1,j_1),(i_2,j_2),\ldots,(i_d,j_d)\}$ where $i_r <_a j_r$ for all $r$. Here, $<_a$ denotes the cyclic rotation of the total order where $a$ is minimal. Since $(I_a,J)$ is not weakly separated, we can find $(i,j)$ and $(i',j')$ in this matching so that $i <_a j <_a i' <_a j'$ and there are no elements of $(I_a \setminus J) \cup (J \setminus I_a)$ in the open cyclic intervals $(i,j)$ and $(i',j')$. Let $S = J \setminus \{j,j'\}$. Define $$ K_0 = Sii' \qquad K_1 = S ij \qquad K_2= Si'j' \qquad K_3 = Sij' \qquad K_4 = Sji' $$ so that we have a positive tropical Pl\"ucker relation $$ p_{J} + p_{K_0}= \min( p_{K_1} + p_{K_2}, p_{K_3} + p_{K_4}). $$ We make the following claims: \begin{enumerate} \item if $(I_b,J)$ is weakly separated, then $(I_b,K_t)$ is weakly separated, for $t=0,1,2,3,4$, \item $K_0,K_3,K_4 \in {\mathcal{M}}$, and \item for each $t =0,1,2,3,4$, one of the following holds: $K_t \notin {\mathcal{M}}$, or $w(J) > w(K_t)$, or ($w(J) = w(K_t)$ and $d(K_t) < d(J)$). \end{enumerate} \noindent {\bf Proof of (1).} First note that if $I_b \leq_b L$ and $(I_b,L)$ are weakly separated, then for some $c$ we have $I_b \setminus L \subset [b,c-1]$ and $L \setminus I_b \subset [c,b-1]$. In particular, for $L = I_a$, we have \begin{equation}\label{eq:Iab} I_b \setminus I_a \subset [b,a-1] \qquad \text{and} \qquad I_a \setminus I_b \subset [a,b-1] \end{equation} and for $L = J$, we have for some $c$, \begin{equation}\label{eq:IbJ} I_b \setminus J \subset [b,c-1] \qquad \text{and} \qquad J \setminus I_b \subset [c,b-1] \end{equation} We say that $(x,y) \in [n]^2$ {\it crosses} $(u,v) \in [n]^2$ if all of $x,y,u,v$ are distinct and the two line segments $\overline{xy}$ and $\overline{uv}$ cross when $1,2,\ldots,n$ are arranged in order around a circle. Suppose that $(I_b,K_t)$ is not weakly separated. Then we have $x,y \in I_b \setminus K_t$ and $u,v \in K_t \setminus I_b$ such that $(x,y)$ crosses $(u,v)$. Case (a): we have $i,j,i',j' \in [b,a-1]$. Then $i,i' \in I_b$ so neither $u$ or $v$ is equal to $i,i'$, and so the claim follows from $(I_b,J)$ being weakly separated. Case (b): we have $i \in [a,b-1]$ and $j,i',j' \in [b,a-1]$. We may assume that $u = i$ and since $i' \in I_b$, we have $v \in J \setminus I_b$. Neither $x$ nor $y$ lies in $[i+1,b-1] \subset (i,j)$ since they would have to belong to $I_a$ and thus also to $J$. Thus $v \in [b,i-1]$ and we have $(\{x,y\} \cap [v+1,i-1]) \neq \emptyset$ which contradicts \eqref{eq:IbJ}. Case (c): we have $i,j \in [a,b-1]$ and $i',j' \in [b,a-1]$. We may assume that $u =i$ and since $i' \in I_b$, we have $v \in J \setminus I_b$. Neither $x$ nor $y$ lies in $(i,j)$ since they would have to belong to $I_a$ and thus also to $J$. But $j \in J \setminus I_a$ and by \eqref{eq:Iab}, $j \notin I_b$. If $(x,y)$ crosses $(i,v)$ then it must also cross $(j,v)$, contradicting $(I_b,J)$ being weakly separated. Case (d): we have $i,j,i' \in [a,b-1]$ and $j' \in [b,a-1]$. If one of $u,v$ is equal to $i$ and the other is in $J \setminus I_b$ then the argument is the same as for Case (c). Also, if $(x,y)$ crosses $(i,i')$ then $(x,y)$ crosses $(j,i')$ as well since \eqref{eq:Iab} implies that $(\{x,y\} \cap (i,j) )= \emptyset$. We may thus assume that $u = i'$ and $v \in J \setminus I_b$. Similarly, $(\{x,y\} \cap [i+1,b-1]) = \emptyset$ so $v \in [b,i-1]$ and we have $(\{x,y\} \cap [v+1,i-1]) \neq \emptyset$ which contradicts \eqref{eq:IbJ}. Case (e): we have $i,j,i',j' \in [a,b-1]$. By \eqref{eq:Iab}, we have $(\{x,y\} \cap (i,j)) = \emptyset = (\{x,y\} \cap (i',j'))$. At least one of $u,v$ is equal to $i,i'$. Replacing $i$ by $j$ or $i'$ by $j'$ does not change whether $(x,y)$ crosses $(u,v)$. Thus $(x,y)$ crosses something of the form $(j,r)$ or $(j',r)$ where $r \in J \setminus I_b$. This contradicts $(I_b,J)$ being weakly separated. \noindent {\bf Proof of (2).} We use the description of ${\mathcal{M}}$ as an intersection of Schubert matroids (Proposition~\ref{prop:Oh}). We prove that $K_3,K_4 \in {\mathcal{M}}$ (and this immediately implies $K_0 \in {\mathcal{M}}$). Indeed, we show that swapping any $i<_a j$ where the open interval $(i,j)$ contains nothing in $(I_a \setminus J) \cup (J \setminus I_a)$ works. So let $J'$ be the result of such a swap. We need to show that $J' \geq_b I_b$ for all $b$, and it suffices to show this for $b \in J'$. Let $L = (i,j) \cap I_a = (i,j) \cap J = (i,j) \cap I_a \cap J$. Case (a): Suppose $b \leq_a i$. The claim follows from $J \geq_b I_b$ together with $\{i\} \cup L \subset I_b$, which holds since $\{i\} \cup L \subset I_a$, using \eqref{eq:Iab}. Case (b): Suppose $b >_a j >_a i$. We have that $I_b$ is the disjoint union of sets $A,B,C$ such that $A \leq_b J \cap [b,i)$ and $B\leq_b L \cup \{j\}$ and $C \leq_b J \cap (j,b-1]$. We claim that $B \leq_b L \cup \{i\}$. This follows from $j \notin I_a \implies j \notin I_b \implies j \notin B$. Case (c): Suppose $b >_a i$ but $b \leq_a j$. The claim follows from $J' \geq_b J \geq_b I_b$. \noindent {\bf Proof of (3).} Suppose that $K_t \in {\mathcal{M}}$. Then from (1) we have $w(K_t) \leq w(J)$, and equality can only happen if $(I_a,K_t)$ is not weakly separated. But by construction we have $|I_a \setminus K_t| > |I_a \setminus J|$, so if $w(K_t) = w(J)$ we have $d(K_t) < d(J)$. We have proved all the claims (1),(2),(3). By induction, all of the $p_{K_t}$ are determined by $p_I$, $I \in {\mathcal{C}}$. The formula $p_{J}= \min( p_{K_1} + p_{K_2}, p_{K_3} + p_{K_4}) - p_{K_0}$ shows that $p_J$ is also determined by $p_I$, $I \in {\mathcal{C}}$. By induction, $p_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}$ is determined by $p_I$, $I \in {\mathcal{C}}$. Furthermore, it is clear that if $p_I \in {\mathbb Z}$ for $I \in {\mathcal{C}}$ then $p_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})$. The theorem is proven. \end{proof} Another proof of Theorem \ref{thm:param} is given after Theorem \ref{thm:bridge}. Let $X^\vee({\widehat{ T}}) \subset {\mathbb R}^n$ denote the lattice generated by ${\mathbb Z}^n$ and the vector $(1/k,1/k,\ldots,1/k) \in {\mathbb R}^n$. The lattice $X^\vee({\widehat{ T}})$ is dual to the lattice $X({\widehat{ T}}) \subset {\mathbb Z}^n$ of \eqref{eq:character}. Restricting the action \eqref{eq:equiv} of ${\mathbb R}^n$ on ${\rm Dr}({\mathcal{M}})_{>0}$, we obtain an action of $X^\vee({\widehat{ T}})$ on ${\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})$. The orbits of this action are denoted ${\rm Dr}({\mathcal{M}})_{>0}/\!\!\sim$ and ${\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})/\!\!\sim$ respectively. Let us now parametrize ${\rm Dr}({\mathcal{M}})_{>0}/\!\!\sim$ and ${\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})/\!\!\sim$. Recall from \S\ref{ssec:gaugefix} the notion of a gauge-fix ${\mathcal{G}}\subset {\mathcal{C}}$. \begin{corollary}\label{cor:param} Let ${\mathcal{M}}$ be a connected positroid, ${\mathcal{C}}$ be a cluster for ${\mathcal{M}}$ and ${\mathcal{G}} \subset {\mathcal{C}}$ be a gauge-fix. The maps \begin{align*} {\rm Dr}({\mathcal{M}})_{>0}/\!\!\sim&\longrightarrow {\mathbb R}^{{\mathcal{C}} \setminus {\mathcal{G}}}, \qquad &p_\bullet &\longmapsto (p'_I \mid I \in {\mathcal{C}} \setminus {\mathcal{G}}) \\ {\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})/\!\!\sim &\longrightarrow {\mathbb Z}^{{\mathcal{C}}\setminus {\mathcal{G}}}, \qquad &p_\bullet &\longmapsto (p'_I \mid I \in {\mathcal{C}} \setminus {\mathcal{G}}) \end{align*} are bijections, where $p'_\bullet \sim p_\bullet$ is the unique vector in the equivalence class of $p_\bullet$ satisfying $p'_J = 0$ for $J \in {\mathcal{G}}$. \end{corollary} \begin{proof} We prove the ${\mathbb Z}$ case. The statement follows from Theorem \ref{thm:param} and the following claim: given $p_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})$ and integers $(c_J \mid J \in {\mathcal{G}}) \in {\mathbb Z}^{{\mathcal{G}}}$, there is a unique $p'_\bullet \sim p_\bullet$ such that $p'_J = c_J$ for all $J \in {\mathcal{G}}$. This claim follows from the following statement: for any $(c_J \mid J \in {\mathcal{G}}) \in {\mathbb Z}^{{\mathcal{G}}}$, there exists ${\mathbf{a}} \in X^\vee({\widehat{ T}})$ such that ${\mathbf{a}} \cdot e_J = c_J$ for all $J \in {\mathcal{G}}$, which in turn follows from the definition of gauge-fix in \S\ref{ssec:gaugefix}. \end{proof} \section{Subdivisions of the hypersimplex}\label{sec:hypersimplex} \subsection{} \def{\tilde P}{{\tilde P}} Let $P$ be a polytope. A {\it subdivision} of $P$ is a collection ${\tilde P} = \{Q\}$ of polytopes $Q$ (called {\it faces} of ${\tilde P}$) such that \begin{enumerate} \item each face of $Q \in {\tilde P}$ is in ${\tilde P}$, \item the intersection $Q \cap Q'$ for $Q,Q' \in {\tilde P}$ is a face of both $Q$ and $Q'$, \item the union of all $Q \in {\tilde P}$ is equal to $P$. \end{enumerate} We typically give a subdivision by only listing the polytopes $Q$ of maximal dimension. \subsection{} Given any $p_\bullet \in {\mathbb R}^{\binom{n}{k}}$ we obtain a subdivision of the hypersimplex as follows. We lift each vertex $e_I \in \Delta(k,n)$ to the point $e'_I = (e_I, p_I)$ in one higher dimension. Then we project the lower faces of the convex hull ${\operatorname{Conv}}(e'_I)$ back into $\Delta(k,n)$. These faces will give us a polyhedral subdivision of $\Delta(k,n)$ denoted ${\tilde \Delta}(p_\bullet)$, and these subdivisions are called {\it regular}. More generally, for $p_\bullet \in ({\mathbb R} \cup \{\infty\})^{\binom{n}{k}}$ (not all equal to $\infty$), we obtain a regular subdivision ${\tilde \Delta}(p_\bullet)$ of the polytope $P_{{\rm Supp}(p_\bullet)} := {\operatorname{Conv}}(e_I \mid I \in {\rm Supp}(p_\bullet))$. By lifting some vertices $e_I$ to $\infty$, the resulting convex hull ${\operatorname{Conf}}(e'_I)$ becomes a polyhedron with many ``vertical" faces. The projection of the lower faces of ${\operatorname{Conv}}(e'_I)$ will only cover $P_{{\rm Supp}(p_\bullet)}$. Let us describe the faces of ${\tilde \Delta}(p_\bullet)$ more explicitly. Faces $F$ of ${\tilde \Delta}(p_\bullet)$ are convex polytopes whose vertices are a subset of the vectors $\{e_I \mid I \in \binom{[n]}{k}\}$. Abusing notation, we will also consider $F$ as a subset of $\binom{[n]}{k}$. We call $p_\bullet$ and $p'_\bullet$ {\it equivalent}, and write $p_\bullet \sim p'_\bullet$ if there exists a vector ${\mathbf{a}} = (a_1,\ldots,a_n) \in {\mathbb R}^n$ such that for all $I \in \binom{[n]}{k}$, \begin{equation}\label{eq:equiv} p'_I = p_I + \sum_{i \in I} a_i. \end{equation} We write $p'_\bullet = {\mathbf{a}} \cdot p_\bullet$. If $p_\bullet \sim p'_\bullet$, then we have ${\tilde \Delta}(p_\bullet) = {\tilde \Delta}(p'_\bullet)$. The faces of ${\tilde \Delta}(p_\bullet)$ are the bottom faces \begin{equation}\label{eq:tDelta} F = F(p'_\bullet) := \{I \mid p'_I = \min(p'_\bullet)\} \subseteq {\mbox{$\binom{[n]}{k}$}} \end{equation} for $p'_\bullet \sim p_\bullet$. Here, $\min(p'_\bullet) \in {\mathbb R}$ is the minimum value in the vector $p'_\bullet$. By \eqref{eq:equiv}, we can always assume that $p'_\bullet \geq 0$ and $\min(p'_\bullet) = 0$. Note that if $p'_\bullet$ is chosen generically, we expect $F(p'_\bullet)$ to be a single vertex of the hypersimplex. \begin{lemma}\label{lem:facezero} Suppose that $F$ is a full-dimensional face of ${\tilde \Delta}(p_\bullet)$. Then there is a unique $p'_\bullet \sim p_\bullet$ such that $p'_I =0$ for $I \in F$. This $p'_\bullet$ satisfies $p'_J >0$ for $J \notin F$. \end{lemma} \begin{proof} If $F$ is full-dimensional, then the vectors $\{e_I \mid I \in F\}$ span ${\mathbb R}^n$. Thus at most one $p'_\bullet$ equivalent to $p_\bullet$ satisfies the condition $p'_I =0$ for $I \in F$. The existence and the last conclusion follows from the assumption that $F$ is a face of ${\tilde \Delta}(p_\bullet)$. \end{proof} The following result is immediate. \begin{lemma} The condition that $p_\bullet$ is a tropical Pl\"ucker vector (resp. positive tropical Pl\"ucker vector) is a property of the equivalence class of $p_\bullet$. \end{lemma} \subsection{} Suppose that $p_\bullet$ is a tropical Pl\"ucker vector. Then by Proposition \ref{prop:Speyer}, ${\tilde \Delta}(p_\bullet)$ is a regular subdivision of the matroid polytope $P_{{\rm Supp}(p_\bullet)}$. Suppose that $p_\bullet$ is a positive tropical Pl\"ucker vector. Then by Proposition \ref{prop:posSpeyer}, ${\tilde \Delta}(p_\bullet)$ is a regular subdivision of the positroid polytope $P_{{\rm Supp}(p_\bullet)}$. \begin{proposition}\label{prop:posdiv}\ \begin{enumerate} \item $p_\bullet$ satisfies the tropical Pl\"ucker relations if and only if each face of the regular subdivision ${\tilde \Delta}(p_\bullet)$ is a matroid polytope. \item $p_\bullet$ satisfies the positive tropical Pl\"ucker relations if and only if each face of the regular subdivision ${\tilde \Delta}(p_\bullet)$ is a positroid polytope. \end{enumerate} \end{proposition} \begin{proof} We prove (2) (the proof of (1) \cite{Spe} is similar). Suppose that ${\tilde \Delta}(p_\bullet)$ is a positroid subdivision. Let us fix $S \subset [n]$ of size $k-2$ and $a<b<c<d$ not contained in $S$. The intersection of $\Delta(k,n)$ with the hyperplanes $\{x_i = 0 \mid i \notin Sabcd\}$ and $\{x_i = 1 \mid i \in S\}$ is a face $F = F(S;abcd)$ of $\Delta(k,n)$. This face $F$ is an octahedron with six vertices $e_{Sab}, e_{Sac}, e_{Sad}, e_{Sbc}, e_{Sbd}, e_{Scd}$. The intersection of the positroid subdivision ${\tilde \Delta}(p_\bullet)$ with $F$ gives a subdivision of ${\tilde F}$ (or a subpolytope of ${\tilde F}$), which must be a positroid subdivision. Thus it suffices to observe that the positive tropical Pl\"ucker \eqref{eq:trop} holds for the six ``numbers" $p_{Sab}, p_{Sac}, p_{Sad}, p_{Sbc}, p_{Sbd}, p_{Scd}$ if they induce a positroid subdivision of a subpositroid of ${\tilde F}$. This is a straightforward case-by-case analysis. Suppose that $p_\bullet$ satisfies the positive tropical Pl\"ucker relation. Let $F$ be a face of the subdivision ${\tilde \Delta}(p_\bullet)$. Then we can find $p'_\bullet \sim p_\bullet$ satisfying $p'_\bullet \geq 0$ and $\min(p'_\bullet) = 0$ so that $F = \{I \mid p'_I = 0\}$. Let $q_\bullet$ be defined by $$ q_I = \begin{cases} 0 & \mbox{if $p'_I = 0$} \\ \infty & \mbox{if $p'_I > 0$.} \end{cases} $$ By Proposition \ref{prop:posSpeyer}, to show that $F$ is a positroid, it suffices to show that $q_\bullet$ is a positive tropical Pl\"ucker vector. Consider any $S$ and $a<b<c<d$ as in \eqref{eq:trop}. If both sides of \eqref{eq:trop} are equal to 0 (resp. positive) for $p'_\bullet$, then both sides of \eqref{eq:trop} are equal to 0 (resp. equal to $\infty$) for $q_\bullet$. Thus \eqref{eq:trop} holds for $q_\bullet$ if it holds for $p'_\bullet$. \end{proof} \section{Positive tropical Grassmannian}\label{sec:trop} \subsection{}\label{sec:real} Let ${\mathcal{R}} = \bigcup_{n=1}^\infty {\mathbb R}((t^{1/n})) $ denote the field of Puiseux series over ${\mathbb R}$. We define ${\rm val}: {\mathcal{R}} \to {\mathbb R} \cup \{\infty\}$ by ${\rm val}(0) = \infty$ and ${\rm val}(x(t)) = r$ if the lowest term of $x(t)$ is equal to $\alpha t^r$ where $\alpha \in {\mathbb R}^\times$. We define ${\mathcal{R}}_{>0} \subset {\mathcal{R}}$ to be the semifield consisting of Puiseux series $x(t)$ that are non-zero and such that coefficient of the lowest term is a positive real number. We let ${\mathcal{R}}_{\geq 0} :={\mathcal{R}}_{>0} \cup \{0\}$. A point $V(t)$ in the Grassmannian ${\rm Gr}(k,n)({\mathcal{R}})$ is, as usual, determined by its Pl\"ucker coordinates $\Delta_I(V) \in {\mathcal{R}}$, which satisfy the Pl\"ucker relations and are defined up to a common scalar. We let $ {\rm Gr}(k,n)({\mathcal{R}}_{\geq 0}) \subset {\rm Gr}(k,n)({\mathcal{R}})$ (resp. $ {\rm Gr}(k,n)({\mathcal{R}}_{> 0}) \subset {\rm Gr}(k,n)({\mathcal{R}})$) denote the subset of points $V(t)$ whose Pl\"ucker coordinates lie in ${\mathcal{R}}_{\geq 0}$ (resp. ${\mathcal{R}}_{>0}$). Similarly, we define $\widehat{\Gr}(k,n)({\mathcal{R}})$, $\widehat{\Gr}(k,n)({\mathcal{R}}_{\geq 0})$, and $\widehat{\Gr}(k,n)({\mathcal{R}}_{>0})$. The following result follows easily from the three-term Pl\"ucker relations \eqref{eq:threeterm}. \begin{lemma}\label{lem:valtrop} For $V(t) \in \widehat{\Gr}(k,n)({\mathcal{R}}_{\geq 0})$, the vector $p_I = {\rm val}(\Delta_I(V(t)))$ satisfies the positive tropical Pl\"ucker relations. \end{lemma} For a positroid ${\mathcal{M}}$, we also define $$ \Pi_{\mathcal{M}}({\mathcal{R}}_{>0}):= \{V(t) \in {\rm Gr}(k,n)({\mathcal{R}}_{\geq 0}) \mid \Delta_I(V(t)) \in {\mathcal{R}}_{>0} \text{ if and only if } I \in {\mathcal{M}}\}. $$ By Proposition~\ref{prop:posSpeyer} and Lemma~\ref{lem:valtrop}, we have the disjoint union \begin{equation}\label{eq:RRdecomp} {\rm Gr}(k,n)({\mathcal{R}}_{\geq 0}) = \bigsqcup_{\mathcal{M}} \Pi_{\mathcal{M}}({\mathcal{R}}_{>0}) \end{equation} as ${\mathcal{M}}$ varies over positroids of rank $k$ on $[n]$, an analogue of \eqref{eq:positroiddecomp}. \subsection{} We now connect ${\rm Gr}(k,n)({\mathcal{R}}_{>0})$ to the nonnegative part ${\operatorname{Ch}}(k,n)_{\geq 0}$ of the Chow quotient. Let $\gamma(t)$ be a curve in ${\rm Gr}(k,n)$ that is analytic near $t = 0$. Thus each Pl\"ucker coordinate $\Delta_I(\gamma(t))$ has a Taylor expansion $g_I(t) \in {\mathbb R}[[t]]$ that converges near $t = 0$. Suppose that for some $s > 0$, we have that $\gamma((0,s)) \in {\rm Gr}(k,n)_{>0}$. Then for every $I$, we have that $g_I(t) \neq 0$ and the coefficient of the lowest term of $g_I(t)$ must be positive. In particular, $\gamma(t)$ ``agrees" with a point $V(t) \in {\rm Gr}(k,n)({\mathcal{R}}_{>0})$. Furthermore, $\gamma((0,s))$ determines a curve in ${\operatorname{Conf}}(k,n)_{>0} ={\rm Gr}(k,n)_{>0}/T_{>0}$, and thus $\lim_{t \to 0} \gamma(t)$ is a point in $X = \sum_i X_i \in {\operatorname{Ch}}(k,n)_{\geq 0}$. The hypersimplex decomposition ${\tilde \Delta}(X)$ is given by ${\tilde \Delta}(X) = {\tilde \Delta}({\rm val}(V(t)))$. Let $d({\mathbf{a}}) = {\rm diag}(t^{a_1},\ldots,t^{a_n}) \in {\rm GL}(n)({\mathcal{R}})$ be an ${\mathcal{R}}$-valued point in the torus acting on ${\rm Gr}(k,n)$. Then $p_\bullet = {\rm val}(V)$ and $p'_\bullet = {\rm val}(V \cdot d({\mathbf{a}}))$ are related by \eqref{eq:equiv}. Let $V'(t) = V(t) \cdot d({\mathbf{a}})$, and suppose that $p'_\bullet = {\rm val}(V'(t)) \geq 0$ and at least one $p_I$ is equal to 0. Then $V_0 = \lim_{t \to 0} V'(t)$ lies in $\Pi_{{\mathcal{M}},>0}$ for some positroid ${\mathcal{M}}$. The point $V_0$ belongs to (at least) one of the toric varieties $X_i = \overline{T \cdot V_i}$, and the positroid polytope $P_{\mathcal{M}}$ is one of the faces of ${\tilde \Delta}(X)$. To obtain the maximal faces, one has to pick the vector ${\mathbf{a}}$ carefully, just as \eqref{eq:tDelta} typically gives lower-dimensional faces. We refer the reader to \cite{KT} for further details from this perspective. \subsection{} Following \cite{SW}, we define the {\it positive tropical Grassmannian} $$ {\operatorname{Trop}}_{> 0} {\rm Gr}(k,n) :=\overline{{\rm val}(\widehat{\Gr}(k,n)({\mathcal{R}}_{> 0}))} \subset {\rm Dr}(k,n)_{>0} $$ as the closure of ${\rm val}$ on $\widehat{\Gr}(k,n)({\mathcal{R}}_{> 0})$, and the {\it nonnegative tropical Grassmannian} $$ {\operatorname{Trop}}_{\geq 0} {\rm Gr}(k,n) := \overline{{\rm val} ( \widehat{\Gr}(k,n)({\mathcal{R}}_{\geq 0}) \setminus \{0\})} \subset {\rm Dr}(k,n)_{\geq 0} $$ where $\{0\}$ denotes the cone point of $\widehat{\Gr}(k,n)({\mathcal{R}})$ (by convention, this cone point does not lie in $\widehat{\Gr}(k,n)({\mathcal{R}}_{>0})$). For a positroid ${\mathcal{M}}$, we also define ${\operatorname{Trop}}_{>0} \Pi_{{\mathcal{M}}} := \overline{{\rm val} ({\widehat{\Pi}}_{\mathcal{M}}({\mathcal{R}}_{>0}))}$. By \eqref{eq:RRdecomp}, we have \begin{equation}\label{eq:Tropecomp} {\operatorname{Trop}}_{\geq 0} {\rm Gr}(k,n) = \bigsqcup_{\mathcal{M}} {\operatorname{Trop}}_{>0} \Pi_{{\mathcal{M}}} . \end{equation} \subsection{} We call a positive tropical Pl\"ucker vector $p_\bullet$ {\it realizable} if it arises as ${\rm val}(\Delta_I(V))$ for some $V \in {\rm Gr}(k,n)({\mathcal{R}}_{\geq 0})$. It is not immediately clear that every $p_\bullet \in {\rm Dr}(k,n)_{\geq 0}$ is realizable because a priori ${\rm val}(\Delta_I(V))$ satisfies relations beyond \eqref{eq:trop}: for example such a relation exists for any (not necessarily three-term) Pl\"ucker relation for ${\rm Gr}(k,n)$. \begin{theorem}\label{thm:main} Every rational positive tropical Pl\"ucker vector is realizable. Thus $$ {\operatorname{Trop}}_{\geq 0}{\rm Gr}(k,n) = {\rm Dr}(k,n)_{\geq 0}, \;\;\; {\operatorname{Trop}}_{> 0}{\rm Gr}(k,n) = {\rm Dr}(k,n)_{>0}, \;\;\;{\operatorname{Trop}}_{>0} \Pi_{{\mathcal{M}}} = {\rm Dr}({\mathcal{M}})_{>0}. $$ \end{theorem} \begin{proof} Let ${\mathcal{M}}$ be a positroid, ${\mathcal{C}} \subset {\mathcal{M}}$ be a cluster, and $d = \dim({\mathcal{M}})$. By \S\ref{ssec:posparam}, we have a positive parametrization $T({\mathcal{C}}) = ({\mathbb C}^\times)^d \hookrightarrow \Pi_{\mathcal{M}}$. This map induces a map $\iota_{\mathcal{C}}: {\mathcal{R}}_{>0}^d \to \Pi_{\mathcal{M}}({\mathcal{R}})$ where if $f=(f_I(t) \mid I \in {\mathcal{C}}) \in {\mathcal{R}}_{>0}^d$, we have $\Delta_I(\iota_{\mathcal{C}}((f)) = f_I(t)$. By Proposition~\ref{prop:Plucksub}, the image of $\iota_{\mathcal{C}}$ lies in $\Pi_{\mathcal{M}}({\mathcal{R}}_{>0})$. Let $\nu_{\mathcal{C}}: {\rm Dr}({\mathcal{M}})_{>0} \to {\mathbb R}^{\mathcal{C}}$ denote the projection $p_\bullet \mapsto (p_I \mid I \in {\mathcal{C}})$ of Theorem~\ref{thm:param}. Then the composition $\nu_{\mathcal{C}} \circ {\rm val} \circ \iota_{\mathcal{C}}$ sends ${\mathcal{R}}_{>0}^d$ surjectively onto ${\mathbb Q}^d$. By Theorem~\ref{thm:param} we have $\nu_{\mathcal{C}}: {\rm Dr}(M)_{>0}({\mathbb Q}) \to {\mathbb Q}^d$ is an isomorphism, we see that every rational positive tropical Pl\"ucker vector is realizable. Taking the closure in ${\mathbb R}^d$, we obtain ${\operatorname{Trop}}_{>0} \Pi_{{\mathcal{M}}} = {\rm Dr}({\mathcal{M}})_{>0}$. \end{proof} \subsection{} We give another proof of Theorem \ref{thm:main} using bridge decompositions, which we believe is of independent interest. Recall from \eqref{eq:bridgeparam} the notion of bridge parametrizations of $\Pi_{{\mathcal{M}},>0}$. In \S\ref{sec:bridge}, we define tropical bridges $T_i(a)$, $a\in {\mathbb Z}$ (and more generally $T_\gamma(t)$ for $\gamma =(i,j)$) acting on the space of positive tropical Pl\"ucker vectors, with the following property: for $a(t) \in {\mathcal{R}}_{>0}$ and $V(t) \in {\rm Gr}(k,n)({\mathcal{R}}_{\geq 0})$ $$ {\rm val}( V(t) \cdot x_i(a(t))) = T_i({\rm val}(a(t))) \cdot {\rm val}(V(t)). $$ In particular, if $p_\bullet$ is representable, it follows immediately that $T_i(a) \cdot p_\bullet$ is representable for any $a \in {\mathbb Q}$. \begin{theorem}\label{thm:bridge} For any bridge parametrization \eqref{eq:bridgeparam} of $ \Pi_{\mathcal{M}}({\mathbb R}_{>0})$, we have a tropical bridge parametrization of ${\rm Dr}({\mathcal{M}})_{>0}$, given by $$ {\mathbb R}^{d+1} \simeq {\rm Dr}({\mathcal{M}})_{>0}, \qquad (z_0,z_1,z_2,\ldots,z_d) \longmapsto T_\gamma({\mathbf{z}}) \cdot p(I,z_0)_\bullet := T_{\gamma_1}(z_1) \cdots T_{\gamma_d}(z_d) \cdot p(I,z_0)_\bullet $$ where $d = \dim({\mathcal{M}})$ and $p(I)_\bullet \in {\rm Dr}(\{I\})_{>0}$ is defined by $p(I,z_0)_I = z_0$ and $p(I,z_0)_J =\infty$ for $J \neq I$. The tropical bridge parametrization maps ${\mathbb Z}^{d+1}$ (resp. ${\mathbb Q}^{d+1}$) isomorphically onto ${\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})$ (resp. ${\rm Dr}({\mathcal{M}})_{>0}({\mathbb Q})$). \end{theorem} The proof of Theorem \ref{thm:bridge} is given in \S\ref{sec:bridge}. Theorem \ref{thm:main} follows immediately from Theorem \ref{thm:bridge}. We give another proof of Theorem~\ref{thm:param}. \begin{proof}[Second proof of Theorem \ref{thm:param}] Any two bridge parametrizations of $\Pi_{{\mathcal{M}},>0}$ are related by an invertible rational transformation that is subtraction-free in both directions, see \cite{Pos}. For particular choices of bridge parametrizations and cluster parametrizations, \cite{MS} gives explicit invertible subtraction-free rational expressions between them; for one direction we have \cite[Theorem 3.3]{MS} and for the other combine \cite[Theorem 7.1]{MS} with \cite[Proposition 7.10]{MS}. Tropicalizing these subtraction-free rational transformations (using \eqref{eq:tropsubs}), we conclude that ${\mathbf{z}} \in {\mathbb R}^d$ and $(p_I \mid I \in {\mathcal{C}}) \in {\mathbb R}^{\mathcal{C}}$ are related by an invertible piece-wise linear transformation, and this holds even integrally. Thus the bijections of \eqref{eq:cluster} follow from Theorem \ref{thm:bridge}. \end{proof} \section{Fan structure}\label{sec:fan} \subsection{} A (polyhedral) {\it fan} ${\mathcal F} = \{C\}$ in a vector space ${\mathbb R}^d$ is a finite collection of closed polyhedral cones $C \subset {\mathbb R}^d$ satisfying the conditions: \begin{enumerate} \item If $C, C' \in {\mathcal F}$ then $C \cap C' \in {\mathcal F}$. \item For $C \in {\mathcal F}$, every face $C' \subset C$ is in ${\mathcal F}$. \end{enumerate} We say that ${\mathcal F}$ is {\it complete} if the union of the cones in ${\mathcal F}$ is equal to ${\mathbb R}^d$. In the case that ${\mathcal F}$ is complete, or if ${\mathcal F}$ is pure of some dimension $d'$ (i.e., all maximal cones have dimension $d'$), we call the maximal cones {\it chambers}. \subsection{} The spaces ${\rm Dr}({\mathcal{M}})_{>0}$ and ${\rm Dr}(k,n)_{>0}$ have a number of different {\it fan structures} that we now define. All of these fan structures are compatible with equivalence: if $p_\bullet\sim p'_\bullet$ then they belong to the same cone. Thus we can also think of these as complete fan structures on the vector spaces ${\rm Dr}({\mathcal{M}})_{>0}/\!\!\sim$ and ${\rm Dr}(k,n)_{>0}/\!\!\sim$. If we consider ${\rm Dr}({\mathcal{M}})_{>0}$ as a subset of ${\mathbb R}^{\binom{[n]}{k}}$, then these fan structures are collections of cones of dimension less than or equal to $\dim({\mathcal{M}})+1 = |{\mathcal{C}}|$. If we consider ${\rm Dr}({\mathcal{M}})_{>0} \simeq {\mathbb R}^{{\mathcal{C}}}$ as a vector space (using Theorem \ref{thm:param}), then we have complete fan structures on ${\mathbb R}^{{\mathcal{C}}}$. \subsection{} We begin by defining the {\it secondary fan structure}. Two vectors $p_\bullet, q_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}$ belong to the same (relatively open) cone of the secondary fan structure if and only if they induce the same positroid polytope subdivision ${\tilde \Delta}$ of $P_{\mathcal{M}}$ i.e. if ${\tilde \Delta}(p_\bullet) = {\tilde \Delta}(q_\bullet)$. For a regular subdivision ${\tilde \Delta}$ of $P_{\mathcal{M}}$ into positroid polytopes, we denote by $C({\tilde \Delta}) \subset {\rm Dr}({\mathcal{M}})_{>0}$ the corresponding (closed) cone. The faces of a $C({\tilde \Delta}) \subset {\rm Dr}({\mathcal{M}})_{>0}$ are the cones $C({\tilde \Delta}')$ as ${\tilde \Delta}'$ varies over all subdivisions that refine ${\tilde \Delta}$. We define an integer $\dim({\tilde \Delta})$ for a regular polyhedral subdivision ${\tilde \Delta}$ of $P_{\mathcal{M}}$ into positroid polytopes: \begin{equation} \label{eq:dimDelta} \dim({\tilde \Delta}):= \dim({\mathcal{M}})+1 - \dim(C({\tilde \Delta})). \end{equation} Thus if ${\tilde \Delta}_0$ is the trivial subdivision of $\Delta(k,n)$, then $\dim({\tilde \Delta}_0) = k(n-k)-(n-1)$. \subsection{} For every $S \subset [n]$ and $a<b<c<d$ not contained in $S$, we have the three term Pl\"ucker fan whose two chambers are given by \begin{align*} p_{Sac} + p_{Sbd} &= p_{Sab} + p_{Scd} \leq p_{Sad} + p_{Sbc}\\ p_{Sac} + p_{Sbd} &= p_{Sad} + p_{Sbc} \leq p_{Sab} + p_{Scd} \end{align*} This fan has exactly three cones: the two chambers above and a third cone where $p_{Sac} + p_{Sbd} = p_{Sad} + p_{Sbc} = p_{Sab} + p_{Scd}$, the intersection of the two chambers. The {\it Pl\"ucker fan structure} on ${\rm Dr}({\mathcal{M}})_{>0}$ is the common refinement of all the three term Pl\"ucker fans. In other words, two vectors $p_\bullet$ and $q_\bullet$ belong to the same relatively open cone of the Pl\"ucker fan structure if exactly which of the three quantities $p_{Sac} + p_{Sbd}, p_{Sad} + p_{Sbc}, p_{Sab} + p_{Scd}$ is minimal is the same for $p_\bullet$ and for $q_\bullet$. \subsection{} The {\it positive fan structure} is the fan whose cones are the images of the domains of linearity for a positive parametrization by a cluster. To be precise, pick a cluster ${\mathcal{C}}$ for ${\mathcal{M}}$. By Theorem \ref{thm:param} we identify $p_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}$ with a point in ${\mathbb R}^{{\mathcal{C}}}$. For each $J \in {\mathcal{M}}$, the function $p_J$ is a piecewise-linear function on ${\mathbb R}^{{\mathcal{C}}}$. The common domains of linearity (see \S\ref{ssec:domain} for further discussion) for all the functions $p_J$, $J \in {\mathcal{M}}$ is the positive fan structure of ${\rm Dr}({\mathcal{M}})_{>0}$ with respect to the cluster ${\mathcal{C}}$. \begin{proposition} The positive fan structures on ${\rm Dr}({\mathcal{M}})_{>0}$ induced by two different clusters ${\mathcal{C}}$ and ${\mathcal{C}}'$ are naturally identified via the isomorphisms of Theorem \ref{thm:param}. \end{proposition} \begin{proof} In each chamber $A$ of the positive fan structure on ${\mathbb R}^{{\mathcal{C}}}$, all the functions $p_J$, $J\in {\mathcal{M}}$ are linear. The composition and inverse of linear functions is linear, so under the piecewise-linear isomorphism ${\mathbb R}^{{\mathcal{C}}} \to {\mathbb R}^{{\mathcal{C}}'}$ the cone $A$ will be sent inside some chamber $A'$ of the positive fan structure on ${\mathbb R}^{{\mathcal{C}}'}$, and reversing the roles of ${\mathcal{C}}$ and ${\mathcal{C}}'$ we see that $A$ and $A'$ are isomorphic. \end{proof} We may thus speak of {\it the} positive fan structure on ${\rm Dr}({\mathcal{M}})_{>0}$. \subsection{} We shall show that the secondary fan structure, the Pl\"ucker fan structure, and the positive fan structure coincide. \begin{proposition}\label{prop:pure} With any of the three fan structures, ${\rm Dr}({\mathcal{M}})_{>0}$ is a polyhedral fan pure of dimension $\dim({\mathcal{M}})+1$. \end{proposition} \begin{proof} Any complete finite fan structure on ${\mathbb R}^{{\mathcal{C}}}$ is pure of dimension $|{\mathcal{C}}|$, i.e., ${\mathbb R}^{{\mathcal{C}}}$ is the union of the closed cones of dimension exactly $|{\mathcal{C}}|$. By Theorem \ref{thm:param}, the set ${\rm Dr}({\mathcal{M}})_{>0}$ is the image of ${\mathbb R}^{{\mathcal{C}}}$ under a continuous piecewise-linear map. Thus ${\rm Dr}({\mathcal{M}})_{>0}$ is a polyhedral fan pure of dimension $\dim({\mathcal{M}})+1$. \end{proof} The secondary fan structure and the Pl\"ucker fan structure are shown to coincide in \cite{OPS2}. \begin{theorem}\label{thm:coincide} The three fan structures on ${\rm Dr}({\mathcal{M}})_{>0}$ coincide. \end{theorem} \begin{proof} We show that the Pl\"ucker fan and the positive fan agree. By Proposition \ref{prop:pure}, it suffices to compare cones of maximal dimension $\dim(\Pi_{\mathcal{M}})$. Let $A \subset {\mathbb R}^{{\mathcal{C}}}$ be such a cone in the positive fan structure for a cluster ${\mathcal{C}}$ for ${\mathcal{M}}$. Then all the functions $p_J$ are linear functions on $A$: we have $p_J = \sum_I \alpha^A_{J,I} p_I$, where $\alpha^A_{J,I} \in {\mathbb Z}$. Substituting these expressions into $(p_{Sac} + p_{Sbd}) - (p_{Sab} + p_{Scd})$ or $(p_{Sac} + p_{Sbd}) - (p_{Sad} + p_{Sbc})$ we obtain two linear functions. For each point of $A$, at least one of these linear functions vanishes. Since $A$ has maximal dimension, one of the two linear functions is identically 0 on $A$; it follows that $A$ is completely contained in some chamber of the Pl\"ucker fan. We have shown that the positive fan structure is a refinement of the Pl\"ucker fan structure. Next let $A'$ be a chamber of the Pl\"ucker fan structure. The proof of Theorem~\ref{thm:param} shows that each $p_J$ can be written as a piecewise-linear function of $p_I$, $I \in {\mathcal{C}}$, by iteratively applying the equality \eqref{eq:trop}. Within the chamber $A'$, the RHS of \eqref{eq:trop} is a linear function instead of a piecewise-linear function. It follows that the restriction of $p_J$ to $A'$ is a linear function. We have shown that the Pl\"ucker fan structure is a refinement of the positive fan structure. \end{proof} \begin{remark} In \cite{SW} a particular fan structure on ${\operatorname{Trop}}_{>0} {\rm Gr}(k,n)$ is studied. The positive parametrization in \cite{SW} is related by an invertible monomial transformation to a gauge-fix of the cluster parametrization for the cluster $$ {\mathcal{C}} = \{ \{1,2\ldots,i-1\} \cup \{i+j-k,\ldots,j\} \mid (i,j) \in [1,k] \times [k+1,n]\}. $$ The tropicalization of a monomial transformation is a linear map, so the fan structure of \cite{SW} agrees with the one in Theorem~\ref{thm:coincide}. \end{remark} \begin{remark} The twist map $\eta_{\mathcal{M}}$ \cite{MS} is a subtraction-free birational map that induces a birational isomorphism $\eta_{\mathcal{M}}: \Pi_{\mathcal{M}} \simeq \Pi_{\mathcal{M}}$. It also induces a piecewise-linear isomorphism ${\operatorname{Trop}}(\eta_{\mathcal{M}}) : {\rm Dr}({\mathcal{M}})_{>0} \simeq{\rm Dr}({\mathcal{M}})_{>0}$. It is likely that this map preserves the fan structure i.e. ${\operatorname{Trop}}(\eta_{\mathcal{M}})$ restricts to a linear map on each cone of the fan ${\rm Dr}({\mathcal{M}})_{>0}$. \end{remark} \subsection{} Theorem~\ref{thm:coincide} allows us to parametrize ${\mathcal{D}}(k,n)$ with collections of planar trees, see also \cite[Section 4]{HJJS} and \cite{BC,CGUZ}. For a subset $S \subset [n]$, let $H_{S=1}$ denote the subspace given by intersecting $\{x_s = 1\}$ for $s \in S$. Note that the face $H_{S=1} \cap \Delta(k,n)$ is isomorphic to $\Delta(k-|S|,n-|S|)$. Let ${\tilde \Delta} \in {\mathcal{D}}(k,n)$. Intersecting ${\tilde \Delta}$ with the face $H_{S=1}$ for $|S|=k-2$ gives a subdivision of $\Delta(2,[n] \setminus S)$, i.e. an element ${\tilde \Delta}_S \in {\mathcal{D}}(2,[n] \setminus S)$. It is well known, and explained in \S\ref{sec:M0n}, that ${\mathcal{D}}(2,[n] \setminus S)$ is in bijection with the set of planar trees $\{T_{[n] \setminus S}\}$ with leaves labeled cyclically by $[n] \setminus S$. We obtain a map \begin{equation}\label{eq:planartree} {\tilde \Delta} \longmapsto {\mathcal T}=\{T_{[n] \setminus S} \mid S \in \mbox{$\binom{[n]}{k-2}$}\} \end{equation} sending a hypersimplex subdivision to a collection of planar trees. \begin{corollary} The map \eqref{eq:planartree} is injective. \end{corollary} \begin{proof} Suppose ${\tilde \Delta} = {\tilde \Delta}(p_\bullet)$ where $p_\bullet \in {\rm Dr}(k,n)_{>0}$. To determine ${\tilde \Delta}$, by Theorem~\ref{thm:coincide} we need to know which cone of the Pl\"ucker fan structure $p_\bullet$ lies in. Thus for every $S \subset [n]$ and $a<b<c<d$ not contained in $S$, we need to know which of the following situations we are in:\begin{align*} p_{Sac} + p_{Sbd} &= p_{Sab} + p_{Scd} < p_{Sad} + p_{Sbc}\\ p_{Sac} + p_{Sbd} &= p_{Sab} + p_{Scd} = p_{Sad} + p_{Sbc}\\ p_{Sac} + p_{Sbd} &= p_{Sad} + p_{Sbc} < p_{Sab} + p_{Scd}. \end{align*} Again, by Theorem~\ref{thm:coincide}, this is determined by ${\tilde \Delta}_S \in {\mathcal{D}}(2,[n] \setminus S)$ which in turn is determined by $T_S$. \end{proof} Note however that not every collection $\{T_{[n] \setminus S} \mid S \in \binom{[n]}{k-2}\} $ of planar trees are in the image of \eqref{eq:planartree}. \section{Nearly convergent functions}\label{sec:Gamma} For more details on the material of this section we refer the reader to \cite{AHL1}. \subsection{}\label{ssec:domain} Let $f({\mathbf{x}}) = p({\mathbf{x}})/q({\mathbf{x}}) \in {\mathbb R}({\mathbf{x}}):={\mathbb R}(x_1,\ldots,x_r)$ be a subtraction-free rational function i.e., both $p({\mathbf{x}})$ and $q({\mathbf{x}})$ are polynomials with positive coefficients. The piecewise-linear function ${\operatorname{Trop}}(f)$ on ${\mathbb R}^r$ is obtained by the substitution \eqref{eq:tropsubs}. Suppose now that $f({\mathbf{x}}) \in {\mathbb R}_{\geq 0}[x_1^{\pm 1}, \ldots, x_r^{\pm 1}]$ is a positive Laurent polynomial. Let ${\bf N}[f({\mathbf{x}})]$ denote the Newton polytope of $f({\mathbf{x}})$ inside ${\mathbb R}^r$. This is the lattice polytope given by the convex hull of ${\mathbf{v}} \in {\mathbb Z}^r$, as ${\mathbf{v}}$ varies over lattice points such that $\alpha {\mathbf{x}}^{\mathbf{v}}$ is a monomial appearing in $f({\mathbf{x}})$. The following result is well-known. \begin{lemma}\label{lem:Newton} For a positive Laurent polynomial $f({\mathbf{x}})$, the domains of linearity of ${\operatorname{Trop}}(f)$ consist of the chambers of a complete fan ${\mathcal F}$, and ${\mathcal F}$ is equal to the inner normal fan of the polytope ${\bf N}[f({\mathbf{x}})]$. \end{lemma} The normal fan of the Minkowski sum of two polytopes is the common refinement of the two normal fans. Thus if $f_1({\mathbf{x}}),f_2({\mathbf{x}}),\ldots,f_a({\mathbf{x}})$ are positive Laurent polynomials, then the common domains of linearity give a complete fan that is equal to the inner normal fan of the Minkowski sum ${\bf N}[f_1({\mathbf{x}})]+\cdots +{\bf N}[f_a({\mathbf{x}})]$. \subsection{} A subtraction-free rational function $f({\mathbf{x}}) \in {\mathbb R}({\mathbf{x}})$ is called {\it nearly convergent} if the function ${\operatorname{Trop}}(f({\mathbf{x}}))$ takes nonnegative values on the whole of ${\mathbb R}^r$. \begin{lemma}\label{lem:nearlyconverge} A subtraction-free rational function $f({\mathbf{x}}) = p({\mathbf{x}})/q({\mathbf{x}})$ is nearly convergent if and only if ${\bf N}[p({\mathbf{x}})] \subset {\bf N}[q({\mathbf{x}})]$. \end{lemma} \begin{proof} Write $p({\mathbf{x}}) = \sum_{\mathbf{v}} \alpha_{\mathbf{v}} {\mathbf{x}}^{\mathbf{v}}$. Then $${\operatorname{Trop}}(f({\mathbf{x}})) = {\operatorname{Trop}}(\sum_{\mathbf{v}} \alpha_{\mathbf{v}} {\mathbf{x}}^{\mathbf{v}}/q({\mathbf{x}}) ) = \min_{\mathbf{v}}({\operatorname{Trop}}({\mathbf{x}}^{\mathbf{v}}/q({\mathbf{x}})).$$ Thus ${\operatorname{Trop}}(f({\mathbf{x}}))$ is nonnegative if and only if ${\operatorname{Trop}}({\mathbf{x}}^{\mathbf{v}}/q({\mathbf{x}}))$ is nonnegative for all ${\mathbf{v}}$ such that a monomial $\alpha_{\mathbf{v}} {\mathbf{x}}^{\mathbf{v}}$ occurs in $p({\mathbf{x}})$. Thus it suffices to establish the claim for the case $f({\mathbf{x}}) = {\mathbf{x}}^{\mathbf{v}}/q({\mathbf{x}})$, i.e., ${\operatorname{Trop}}(f({\mathbf{x}}))$ is nonnegative if and only if ${\mathbf{v}} \in {\bf N}[q({\mathbf{x}})]$. This follows from the following observation: a point ${\mathbf{v}} \in {\mathbb Z}^r$ is outside a polytope $P$ if and only if there exists a vector ${\mathbf{a}} \in {\mathbb R}^r$ such that ${\mathbf{a}} \cdot {\mathbf{v}} < \min({\mathbf{a}} \cdot {\mathbf{u}} \mid {\mathbf{u}} \mbox{ a vertex of $P$})$. \end{proof} \subsection{}\label{ssec:topgauge} Set $d = k(n-k)$ and $r = d-(n-1)$. Fix ${\mathcal{C}} \subset \binom{[n]}{k}$ a cluster and ${\mathcal{G}} \subset {\mathcal{C}}$ a gauge-fix as in \S\ref{ssec:gaugefix}. Let ${\mathcal{C}} \setminus {\mathcal{G}} = \{J_1,J_2,\ldots,J_r\}$ and denote $x_i:= \Delta_{J_i}$. Let ${\mathcal T}({\mathcal{C}},{\mathcal{G}}) \simeq ({\mathbb C}^\times)^{r}$ be the subtorus of ${\mathcal T}({\mathcal{C}}) := \operatorname{Spec}({\mathbb C}[\Delta_I \mid I \in {\mathcal{C}}])$ satisfying $\Delta_J = 1$ for all $J \in {\mathcal{G}}$. Let ${\rm Gr}(k,n)_{\mathcal{G}} \subset {\rm Gr}(k,n)$ be the subspace where $\Delta_J = 1$ for all $J \in {\mathcal{G}}$. We have a rational map $\pi_{{\mathcal{C}},{\mathcal{G}}}: {\mathcal T}({\mathcal{C}},{\mathcal{G}}) \to {\rm Gr}(k,n)_{\mathcal{G}}$ induced by the positive parametrization ${\mathcal T}({\mathcal{C}}) \to \widehat{\Gr}(k,n)$ in \S\ref{ssec:posparam}. We identify ${\rm Gr}(k,n)_{\mathcal{G}}$ birationally with ${\operatorname{Conf}}(k,n)$. The following result follows from \eqref{eq:posgf} and Proposition~\ref{prop:Laurent}. \begin{proposition}\label{prop:picg} The map $\pi_{{\mathcal{C}},{\mathcal{G}}}$ is birational, the restriction to ${\mathbb R}_{>0}^{d-(n-1)}$ is a homeomorphism onto ${\operatorname{Conf}}(k,n)_{>0}$, and every Pl\"ucker coordinate $\Delta_I$ pulls back under $\pi_{{\mathcal{C}},{\mathcal{G}}}$ to a positive Laurent polynomial $\Delta_I({\mathbf{x}})$ in $x_1,\ldots,x_r$. \end{proposition} \subsection{} A Laurent monomial \begin{equation}\label{eq:M} M= \prod_{I\in \binom{[n]}{k}} \Delta_I^{a_I}, \qquad a_I \in {\mathbb Z} \end{equation} in the Pl\"ucker coordinates is $T$-invariant if it has weight 0, i.e., $\sum_I a_I e_I = 0$. For fixed $({\mathcal{C}},{\mathcal{G}})$, each Laurent monomial $M$ \eqref{eq:M} pulls back to a subtraction-free rational function $M({\mathbf{x}})$ by Proposition~\ref{prop:picg}. We say that $M$ is nearly convergent with respect to $({\mathcal{C}},{\mathcal{G}})$ if $M({\mathbf{x}})$ is nearly converegent. \begin{lemma} Let $M$ be a $T$-invariant monomial and let $({\mathcal{C}},{\mathcal{G}})$ and $({\mathcal{C}}',{\mathcal{G}}')$ be two choices of cluster and gauge-fix. Then $M$ is nearly convergent with respect to $({\mathcal{C}},{\mathcal{G}})$ if and only if it is nearly convergent with respect to $({\mathcal{C}}',{\mathcal{G}}')$. \end{lemma} \begin{proof} Pulling $M$ back to ${\mathcal T}({\mathcal{C}})$ we have the notion of $M$ being nearly convergent with respect to ${\mathcal{C}}$. The $T$-invariance of $M$ implies that $M$ is nearly convergent with respect to ${\mathcal{C}}$ if and only if it is nearly convergent with respect to some $({\mathcal{C}},{\mathcal{G}})$. The positive parametrizations of two clusters ${\mathcal{C}}$ and ${\mathcal{C}}'$ are related by invertible subtraction-free rational transformations, and this implies that the notion of nearly convergent does not depend on cluster. \end{proof} \subsection{}\label{ssec:Px} Let $P({\mathbf{x}}) = P(x_1,\ldots,x_r) = \prod_I \Delta_I(x_1,\ldots,x_r)$ denote the product of all the Pl\"ucker variables, considered as a Laurent polynomial in $x_1,\ldots,x_r$. Let $P(k,n) = {\bf N}[P({\mathbf{x}})] \subset {\mathbb R}^r$ be the Newton polytope of $P({\mathbf{x}})$. Thus $P(k,n)$ is the Minkowski sum of the Newton polytopes ${\bf N}[\Delta_I({\mathbf{x}})]$. By Theorem~\ref{thm:coincide} (specifically, the equivalence of the secondary fan structure and the positive fan structure) and Lemma~\ref{lem:Newton}, we have the following. \begin{proposition}\label{prop:Ft} There is a bijection $F \mapsto {\tilde \Delta}(F)$ between the set of faces $\{F \subset P(k,n)\}$ of $P(k,n)$ and the set ${\mathcal{D}}(k,n)$ of regular subdivisions of the hypersimplex into positroid polytopes. \end{proposition} \subsection{}\label{ssec:Gamma} Let $$ \Gamma=\{M \mid \mbox{$M$ is $T$-invariant and nearly convergent}\} $$ denote the (finitely-generated) monoid of nearly convergent, $T$-invariant, Laurent monomials in $\Delta_I$. Let ${\mathbb C}[\Gamma] \subset {\mathbb C}({\rm Gr}(k,n))^T$ be the subring of $T$-invariant rational functions on the Grassmannian generated by $M \in \Gamma$. We will also identify ${\mathbb C}[\Gamma]$ with the subring ${\mathbb C}[M({\mathbf{x}}) \mid M \in \Gamma] \subset {\mathbb C}({\mathbf{x}})$ of rational functions on the torus ${\mathcal T}({\mathcal{C}},{\mathcal{G}})$. The following result follows from Lemma~\ref{lem:nearlyconverge} and the fact that each variable $x_i$ is the image in ${\mathbb C}({\mathbf{x}})$ of some $T$-invariant monomial $M$. See also \cite[Section 10]{AHL1}. \begin{lemma}\label{lem:vP} The ring ${\mathbb C}[\Gamma] \subset {\mathbb C}({\mathbf{x}})$ is spanned by the rational functions ${\mathbf{x}}^{\mathbf{v}}/P({\mathbf{x}})^\ell$ for an integer $\ell \geq 1$ and ${\mathbf{v}} \in \ell \,P(k,n) \cap {\mathbb Z}^r$. \end{lemma} \begin{lemma}\label{lem:bound} Let $f \in {\mathbb C}[\Gamma]$. Then there exists a constant $D\in {\mathbb R}$ such that $|f({\mathbf{x}})| \leq D$ for all ${\mathbf{x}} \in {\mathbb R}_{>0}^r$. \end{lemma} \begin{proof} By Lemma~\ref{lem:vP}, it is enough to show that ${\mathbf{x}}^{\mathbf{v}}/P({\mathbf{x}})^\ell$ is bounded above on ${\mathbb R}_{>0}^r$, where ${\mathbf{v}}$ is a lattice point in the polytope $\ell P(k,n)$. We may write $c {\mathbf{v}} = \sum_i c_i {\mathbf{v}}_i$ where ${\mathbf{v}}_i$ are the vertices of $\ell\,P(k,n)$ and $c, c_i$ are nonnegative integers satisfying $c>0$ and $c = \sum_i c_i$. Then ${\mathbf{x}}^{c{\mathbf{v}}} = \prod_i ({\mathbf{x}}^{{\mathbf{v}}_i})^{c_i} \leq (\sum_i {\mathbf{x}}^{{\mathbf{v}}_i})^c$ and it follows that ${\mathbf{x}}^{\mathbf{v}}/P({\mathbf{x}})^\ell \leq 1$ on ${\mathbb R}_{>0}^r$. \end{proof} \subsection{} For a lattice polytope $P \subset {\mathbb R}^r$, one has a proper normal toric variety $X_{P}$ which depends only on the normal fan of $P$ \cite{Ful}. The toric variety $X_P$ contains a dense torus ${\mathcal T} = ({\mathbb C}^\times)^r \subset X_P$. The subspace $X_{P,>0}:={\mathbb R}_{>0}^r \subset X_P({\mathbb R})$ is called the {\it positive part of $X_P$} and its closure $X_{P,\geq 0}$ is called the {\it nonnegative part of $X_P$}. The torus orbits of ${\mathcal T}$ on $X_P$ stratify $X_P$ and $X_{P,\geq 0}$, and the strata are in bijection with faces of $P$. We have a stratification-preserving homeomorphism between $X_{P, \geq 0}$ and the polytope $P$ with its face stratification. For a face $F \subset P$, let $X_F$ denote the corresponding closed stratum, which is itself isomorphic to the toric variety for the polytope $F$. Thus the closed (resp. relatively open) faces of $X_{P,\geq 0}$ are exactly the $X_{F,\geq 0}$ (resp. $X_{F,>0}$) as $F$ varies over faces of $P$. \begin{proposition}[see \cite{AHL1}]\label{prop:XPkn}\ \begin{enumerate} \item The variety $\operatorname{Spec}({\mathbb C}[\Gamma])$ is isomorphic to an affine open subset $X'_{P(k,n)}$ of the projective (and normal) toric variety $X_{P(k,n)}$ associated to the normal fan of $P(k,n)$. \item The affine open subset $X'_{P(k,n)}$ contains the nonnegative part $X_{P(k,n),\geq 0}$. \item Let ${\mathbf{v}} \in \ell \,P(k,n) \cap {\mathbb Z}^r$ and $F \subset P(k,n)$ be a face. The function ${\mathbf{x}}^{\mathbf{v}}/P({\mathbf{x}})^\ell \in {\mathbb C}[\Gamma]$ vanishes identically on $X_{F,\geq 0}$ if and only if ${\mathbf{v}} \notin \ell F$. The subspace $X_{F,\geq 0}$ is cut out of $X'_{P(k,n)}$ by the vanishing of such functions. \end{enumerate} \end{proposition} \begin{proof} (1) is proven in \cite[Section 10]{AHL1}; we sketch the main idea. Recall that a full-dimensional lattice polytope $Q$ is called {\it very ample} if for sufficiently large integers $s > 0$, every lattice point in $rQ$ is a sum of $s$ (not necessarily distinct) lattice points in $Q$. Given $Q$, it is known that some dilate $\ell Q$ of $Q$ is very ample. We then have a projective embedding of $X_Q$ given by the closure of the image of the map \begin{equation}\label{eq:closure} {\mathbf{x}} \mapsto [{\mathbf{x}}^{{\mathbf{v}}_0}: {\mathbf{x}}^{{\mathbf{v}}_1}: \cdots : {\mathbf{x}}^{{\mathbf{v}}_m}] \in {\mathbb P}^m \end{equation} where $\ell Q \cap {\mathbb Z}^r = \{{\mathbf{v}}_0,{\mathbf{v}}_1,\ldots,{\mathbf{v}}_m\}$ and ${\mathbf{x}} \in ({\mathbb C}^\times)^r$. We apply this construction to $X_{P(k,n)}$, supposing that $\ell P(k,n)$ is very ample. Let $P({\mathbf{x}})^\ell = \sum_{i=0}^m c_i {\mathbf{x}}^{{\mathbf{v}}_m}$, where $\ell P(k,n) \cap {\mathbb Z}^r = \{{\mathbf{v}}_0,\ldots,{\mathbf{v}}_m\}$. Consider the linear section $H = \{c_0y_0 + \cdots + c_m y_m = 0\} \subset {\mathbb P}^m$, where $y_i$ are the homogeneous coordinates on ${\mathbb P}^m$. The complement $X'_{P(k,n)} := X_{P(k,n)} \setminus H$ is an open subset of $X_{P(k,n)}$ that is an affine variety. The coordinate ring ${\mathbb C}[X'_{P(k,n)}]$ is generated by the images of $y_i/(c_0y_0 + \cdots + c_m y_m)$, that is, ${\mathbf{x}}^{{\mathbf{v}}_i}/P({\mathbf{x}})^\ell$. By Lemma~\ref{lem:vP}, this gives the isomorphism $\operatorname{Spec}({\mathbb C}[\Gamma])\simeq X'_{P(k,n)}$, establishing (1). Since all the coefficients of $P({\mathbf{x}})$ are positive, the linear section does not intersect $X_{P(k,n),\geq 0}$, giving (2). The torus orbit closure $X_F$ is cut out from $X_P$ by the vanishing of the homogeneous coordinates $\{{\mathbf{x}}^{{\mathbf{v}}} \mid {\mathbf{v}} \notin F \cap {\mathbb Z}^r\}$ in the embedding \eqref{eq:closure}. This gives (3). \end{proof} \section{Topology of nonnegative configuration space}\label{sec:ball} \subsection{} We continue the notation of \S\ref{ssec:topgauge}. In particular, a cluster ${\mathcal{C}} \subset \binom{[n]}{k}$ and a gauge-fix ${\mathcal{G}} \subset {\mathcal{C}}$ are fixed. Since ${\operatorname{Conf}}(k,n) \simeq {\mathring{\Gr}}(k,n)/T$, each $T$-invariant monomial extends to a rational function on $\Ch(k,n)$. Let ${\widetilde{\Ch}}(k,n) \subset \Ch(k,n)$ denote the locus where all nearly convergent monomials are regular i.e. where the polar locus of each $M \in \Gamma$ has been removed. Then we have a natural morphism \begin{equation}\label{eq:varphi} \varphi: {\widetilde{\Ch}}(k,n) \longrightarrow \operatorname{Spec}({\mathbb C}[\Gamma]) = X'_{P(k,n)}. \end{equation} It follows from Lemma~\ref{lem:bound} that ${\operatorname{Ch}}(k,n)_{\geq 0} \subset {\widetilde{\Ch}}(k,n)$. Recall the bijection $F \mapsto {\tilde \Delta}(F)$ in Proposition~\ref{prop:Ft}. \begin{proposition}\label{prop:strata} Let $F \subset P(k,n)$ be a face. If $X \in \Theta_{{\tilde \Delta}(F)}(k,n)$, then $\varphi(X) \in X_{F,> 0} \subset X'_{P(k,n)}$. \end{proposition} \begin{proof} Identify ${\rm Dr}(k,n)_{>0}/\sim$ with a complete fan on ${\mathbb R}^r$. Via Proposition~\ref{prop:Ft}, the cones $C(F) = C({\tilde \Delta}(F))$ are indexed by either faces $F$ of $P(k,n)$ or by ${\tilde \Delta}(F) \in {\mathcal{D}}(k,n)$. Let $\gamma(t)$ be an (analytic) curve in ${\rm Gr}(k,n)_{>0}/T_{>0}$ such that $\lim_{t \to 0} \gamma(t) = X \in \Theta_{{\tilde \Delta}(F)}(k,n)$. Then $\gamma(t)$ gives rise to a formal curve, and thus a Puiseux curve $V(t) \in {\rm Gr}(k,n)({\mathcal{R}}_{>0})$. The positive tropical Pl\"ucker vector ${\rm val}(V(t))$ lies in the cone $C({\tilde \Delta}(F))$. Now let ${\mathbf{v}} \in (\ell P(k,n) \cap {\mathbb Z}^r) \setminus (\ell F \cap {\mathbb Z}^r)$. Let $f({\mathbf{x}}) = {\mathbf{x}}^{\mathbf{v}}/P({\mathbf{x}})^\ell$. The function ${\operatorname{Trop}}( {\mathbf{x}}^{\mathbf{v}}/P({\mathbf{x}})^\ell)$ is nonnegative on ${\mathbb R}^r$ and strictly positive on the cone $C(F)$ in the fan ${\rm Dr}(k,n)_{>0}$. Considering $f({\mathbf{x}})$ as a $T$-invariant rational function on ${\rm Gr}(k,n)$, we conclude that ${\rm val}(f(V(t)))>0$. Thus $f({\mathbf{x}})$ vanishes at $X$ and by Proposition~\ref{prop:XPkn}(3), $\varphi(X) \in X_{F,\geq 0}$. The same argument shows that we must actually have $\varphi(X) \in X_{F,> 0}$. \end{proof} \subsection{}\label{ssec:homeo} We can now prove Theorem~\ref{thm:ball}. \begin{theorem}\label{thm:homeo} There is a stratification-preserving homeomorphism ${\operatorname{Ch}}(k,n)_{\geq 0} \simeq P(k,n)$. In particular, each stratum $\Theta_{{\tilde \Delta}, \geq 0}$ (resp. $\Theta_{{\tilde \Delta}, > 0}$) is homeomorphic to a closed ball (resp. open ball) of dimension $\dim({\tilde \Delta})$. \end{theorem} We shall show that the map $\varphi$ restricts to a stratified bijection $\varphi: {\operatorname{Ch}}(k,n)_{\geq 0} \to X_{P(k,n),\geq 0}$. Since ${\operatorname{Ch}}(k,n)_{\geq 0}$ is compact and $X_{P(k,n),\geq 0}$ is Hausdorff, this claim would establish Theorem \ref{thm:homeo}. Proposition~\ref{prop:strata} shows that $\varphi$ is stratification-preserving. It is easy to see that $\varphi:{\operatorname{Conf}}_{\geq 0} \to X_{P(k,n),\geq 0}$ is surjective. If $p \in X_{P(k,n),\geq 0}$ then $p = \lim_{t \to 0} p(t)$ where $p(t) \in X_{P(k,n),> 0}$ for $t >0$. The curve $p(t)$ can be lifted to a curve $X(t) \in {\operatorname{Conf}}(k,n)_{>0}$, such that $\varphi(X(t)) = p(t)$ for $t>0$. Then $\varphi(\lim_{t \to 0} X(t) ) = p$. It remains to argue that the map $\varphi$ is injective. A point $X = \sum_{i=1}^r X_i \in {\operatorname{Ch}}(k,n)_{\geq 0}$ is determined by the torus orbit closures $X_i = \overline{T \cdot V_i}$ where $V_i \in \Pi_{{\mathcal{M}}_i, >0}/T$ and $P_{{\mathcal{M}}_1},\ldots,P_{{\mathcal{M}}_r}$ are positroid polytopes that form a regular polyhedral subdivision of the hypersimplex. The point $X$ is uniquely determined by the collection of points $\{V_1,\ldots,V_r\}$. Thus, the injectivity of $\varphi$ follows from the following proposition which completes the proof of Theorem~\ref{thm:homeo}. \begin{proposition}\label{prop:TV} Let $X \in \sum_{i=1}^r X_i \Theta_{{\tilde \Delta},\geq 0}$ with $X_i = \overline{T \cdot V_i}$ where $V_i \in \Pi_{{\mathcal{M}}_i, >0}/T_{>0}$. Then $V_1,\ldots,V_r$ is determined by the values $f(V_i)$ for varying $f \in {\mathbb C}[\Gamma]$. \end{proposition} \begin{proof} Note that for $f \in {\mathbb C}[\Gamma]$, we have $f(V_i) = f(V_j)$ whenever $f$ makes sense on both $V_i$ and $V_j$. Let us first consider the special case where ${\mathcal{M}}=\binom{[n]}{k}$, and suppose $V \in {\rm Gr}(k,n)_{>0}/T_{>0}$. By Lemma~\ref{lem:vP}, the function ${\mathbf{x}}^{\mathbf{v}}/P({\mathbf{x}})^\ell$ is nearly convergent when ${\mathbf{v}} \in \ell P(k,n)$. For sufficiently large $\ell$, we can find lattice points ${\mathbf{v}}$ and ${\mathbf{v}}+e_i$ inside $\ell P(k,n)$, and the ratio of $\frac{{\mathbf{x}}^{\mathbf{v}}}{P({\mathbf{x}})^\ell}$ and $ \frac{{\mathbf{x}}^{{\mathbf{v}}+e_i}}{P({\mathbf{x}})^\ell}$ is equal to $x_i$, so the value of $x_i(V)$ is determined by $f(V)$, for $f \in \Gamma$. (Note that the ratios $\frac{{\mathbf{x}}^{\mathbf{v}}}{P({\mathbf{x}})^\ell}$ are always positive because all Pl\"ucker variables are positive on $V$.) By Proposition~\ref{prop:picg}, $V \in {\rm Gr}(k,n)_{>0}/T_{>0}$ is determined by the positive parameters $x_1,\ldots,x_r \in {\mathbb R}_{>0}$, and thus we have recovered the point $V$. Now suppose that $V \in \Pi_{{\mathcal{M}}, >0}/T_{>0}$ where ${\mathcal{M}}$ is an arbitrary connected positroid. Applying Lemma~\ref{lem:span} we find a cluster ${\mathcal{C}}' \subset {\mathcal{M}}$ and a gauge-fix ${\mathcal{G}} \subset {\mathcal{C}}'$. By Proposition~\ref{prop:OPS}, we can find a cluster ${\mathcal{C}}$ of $\binom{[n]}{k}$ that contains ${\mathcal{C}}'$. We use $({\mathcal{C}},{\mathcal{G}})$ as our positive parametrization of ${\operatorname{Conf}}(k,n)_{>0}$, and we suppose that $x_1,\ldots,x_b$ belong to ${\mathcal{C}}' \setminus {\mathcal{G}}$ while $x_{b+1},\ldots,x_r$ belong to ${\mathcal{C}} \setminus {\mathcal{C}}'$. Here, $b = \dim({\mathcal{M}})-(n-1)$. To determine $V$, we need to determine $x_1(V),\ldots,x_b(V)$. The same argument as in the previous paragraph applies, except we need to ensure that the ratios $\frac{{\mathbf{x}}^{\mathbf{v}}}{P({\mathbf{x}})^\ell}$ and $ \frac{{\mathbf{x}}^{{\mathbf{v}}+e_i}}{P({\mathbf{x}})^\ell}$ used do not vanish on $\Pi_{{\mathcal{M}},>0}/T_{>0}$. As in the proof of Proposition~\ref{prop:strata}, the function $x^{\mathbf{v}}/P({\mathbf{x}})^\ell$ vanishes on $\Theta_{{\tilde \Delta}}$ exactly when ${\mathbf{v}}$ does not belong to the face $\ell F$ of $\ell P(k,n)$. We claim that for each $i = 1,2,\ldots,b$, the face $\ell F$ contains lattice points ${\mathbf{v}}$ and ${\mathbf{v}}+e_i$ for some ${\mathbf{v}}$. In other words, if ${\mathbf{y}} = (y_1,\ldots,y_r) \in {\mathbb R}^r$ is a vector in the normal cone $C(F)$ to $F$ then $y_i = 0$ for $i = 1,2,\ldots,b$. Let $p_\bullet$ be a positive tropical Pl\"ucker vector such that ${\tilde \Delta}(p_\bullet) = {\tilde \Delta}$. By Lemma~\ref{lem:span}, there is a unique $p'_\bullet$ equivalent to $p_\bullet$ such that $p'_I = 0$ for all $I \in {\mathcal{G}}$, and by Lemma~\ref{lem:facezero}, it must be the case that $p'_J = 0$ for all $J \in {\mathcal{M}}$, since $P_{\mathcal{M}}$ is one of the full-dimensional pieces in ${\tilde \Delta}$. By Corollary~\ref{cor:param}, setting $y_i = p'_{I_i}$ for $I_i \in {\mathcal{C}}$ gives a vector in $C(F)$, and any vector in the relative interior of $C(F)$ is obtained in this way. All these vectors satisfy $y_1 = y_2 = \cdots = y_b = 0$ since $p'_J =0$ for all $J \in {\mathcal{M}}$. \end{proof} Let us illustrate the proof of Proposition~\ref{prop:TV} with an example for $(k,n) = (3,6)$. Take the positive parametrization \begin{equation}\label{eq:36param1} (z_1,z_2,z_3,z_4) \in ({\mathbb C}^\times)^4 \mapsto \begin{bmatrix} 0 & 0 & -1 & -1 & -1 & -1 \\ 0 & 1 & 0 & -1 & -1-z_1 & -1-z_1z_2-z_2 \\ 1 & 0 & 0 & 1 & 1+z_1+z_3 & 1+z_1+z_2+z_4+\frac{z_3 z_3}{z_1}+\frac{z_4}{z_1} \\ \end{bmatrix} \end{equation} This is the positive parametrization associated to the gauge-fix and cluster $${\mathcal{G}}:=\{123,124,125,126,134,234\} \text{ and } {\mathcal{C}} := {\mathcal{G}} \cup \{z_1=145,z_2=156,z_3=345,z_4=456\}. $$ Consider the two-piece hypersimplex decomposition ${\tilde \Delta}$ (see \S\ref{ssec:36}) that slices $\Delta(3,6)$ with the hyperplane $x_4+x_5 +x_6= 2$. On one side we have the positroid polytope for ${\mathcal{M}} = \binom{[6]}{3}\setminus \{4,5,6\}$, which has a cluster $$ {\mathcal{C}} \supset {\mathcal{C}}' := \{123,124,125,126,134,234,145,156,345\} \supset {\mathcal{G}}. $$ Now, the normal cone $C(F) = C({\tilde \Delta})$ is a ray and it is generated by the vector ${\mathbf{y}} = r'_6=(0,0,0,1)$. (This $r'_6$ is the image of $r_6$ in \S\ref{ssec:36} under the linear transformation that tropicalizes the monomial transformation $(x_1,x_2,x_3,x_4) \to (z_1,z_3/z_1, z_2/z_1, z_4/(z_2z_3))$.) Agreeing with the proof of Proposition~\ref{prop:TV}, we have $y_1=y_2=y_3=0$. \subsection{} We end this section with the following question. \begin{question} Is the morphism $\varphi$ of \eqref{eq:varphi} an isomorphism of algebraic varieties? \end{question} We note that $\operatorname{Spec}({\mathbb C}[\Gamma])$ is a normal variety, while the Chow quotient $\Ch(k,n)$ has rather complicated geometry (c.f \cite{KT,Laf}). \section{\texorpdfstring{${\mathcal{M}}_{0,n}$ and the case $k=2$}{M0n and the case k=2}}\label{sec:M0n} \subsection{} It is well-known \cite{Kap} that we have ${\operatorname{Conf}}(2,n) = {\mathring{\Conf}}(2,n) = {\mathcal{M}}_{0,n}$ and $\Ch(2,n) = \overline{\M}_{0,n}$, the moduli space of $n$ points on ${\mathbb P}^1$ and its Deligne-Knudsen-Mumford compactification respectively. The space ${\mathcal{M}}_{0,n}({\mathbb R})$ consists of $n$ points $z_1,\ldots,z_n$ on a circle $S^1$. It is a smooth open manifold with $(n{-}1)!/2$ connected components, each of which is diffeomorphic to an open ball of dimension $n{-}3$. Each connected component is given as the subspace where the $n$ points are in a fixed dihedral ordering. Fixing such an ordering $z_1 < z_2 < \cdots < z_n$ (up to dihedral symmetries) we obtain the positive component $({\mathcal{M}}_{0,n})_{>0} \subset {\mathcal{M}}_{0,n}({\mathbb R})$. It is well-known that the closure ${\operatorname{Conf}}(2,n)_{\geq 0} = ({\mathcal{M}}_{0,n})_{\geq 0}$ of $({\mathcal{M}}_{0,n})_{>0}$ in ${\mathcal{M}}_{0,n}({\mathbb R})$ is homeomorphic as a stratified space to the associahedron ${\mathcal{A}}_n$, agreeing with Theorem \ref{thm:homeo}. The affine variety ${\widetilde{\M}}_{0,n} := {\widetilde{\Ch}}(2,n)$ sits in between ${\mathcal{M}}_{0,n}$ and $\overline{\M}_{0,n}$. It can be obtained from $\overline{\M}_{0,n}$ by removing all boundary divisors of $\overline{\M}_{0,n}$ that do not intersect $ ({\mathcal{M}}_{0,n})_{\geq 0}$ in codimension one, see \cite{Brown}. \subsection{} Let us now spell out our combinatorial constructions in this case. A positroid ${\mathcal{M}}$ of rank $2$ on $[n]$ is given by a collection of conditions of the following form: \begin{enumerate} \item For some $i \in [n]$, we have $i \in I$ for all $I \in {\mathcal{M}}$. \item For some $j \in [n]$, we have $j \notin I$ for all $I \in {\mathcal{M}}$. \item For some cyclic interval $[a,b]$, we have $|[a,b] \cap I| \leq 1$ for all $I \in {\mathcal{M}}$. \end{enumerate} For ${\mathcal{M}}$ to be connected we must have none of the conditions of the form (1) or (2). Such a connected positroid ${\mathcal{M}}$ is then determined by a decomposition $[n] = \bigsqcup_{i=1}^r [a_i,b_i]$ of $[n]$ into at least three cyclic intervals such that ${\mathcal{M}}$ is given by $$ {\mathcal{M}}([a_i,b_i]) = \left\{I \in \mbox{$\binom{[n]}{2}$} \mid |I \cap [a_i,b_i]| \leq 1 \text{ for } i = 1,2,\ldots,r\right\}. $$ We have the formula $\dim({\mathcal{M}}) = n+r-4$. For example, if ${\mathcal{M}}$ is the uniform matroid then $r = n$ and $\dim({\mathcal{M}}) = 2n-4 = 2(n-2)$. If $r = 3$ then $\dim(M) = n-1$ and ${\mathcal{M}}$ is a minimal connected positroid. For a connected positroid ${\mathcal{M}} = {\mathcal{M}}([a_i,b_i])$, it is not difficult to see that there is a canonical isomorphism ${\mathring{\Pi}}_{\mathcal{M}}/T \simeq {\mathcal{M}}_{0,r}$ that sends $\Pi_{{\mathcal{M}},>0}/T_{>0}$ to $({\mathcal{M}}_{0,r})_{>0}$, where $r$ is the number of cyclic intervals. If $r = 3$, then ${\mathcal{M}}_{0,3}$ is a point, as expected. \subsection{} The faces of the associahedron ${\mathcal{A}}_n$ are labeled by planar trees $T$ with $n$ cyclically ordered leaves $1,2,\ldots,n$, with internal vertices having degree at least $3$. Some such trees for $n =5$ are drawn below: $$ \begin{tikzpicture} \draw (0:1.2) node {$3$}; \draw (72:1.2) node {$2$}; \draw (144:1.2) node {$1$}; \draw (216:1.2) node {$5$}; \draw (288:1.2) node {$4$}; \draw (72:1)--(106:0.5)--(144:1); \draw (216:1)--(252:0.5)--(288:1); \draw (106:0.5)--(0:0.5)--(0:1); \draw (252:0.5)--(0:0.5); \begin{scope}[shift={(4,0)}] \draw (0:1.2) node {$3$}; \draw (72:1.2) node {$2$}; \draw (144:1.2) node {$1$}; \draw (216:1.2) node {$5$}; \draw (288:1.2) node {$4$}; \draw (72:1)--(72:0.1)--(144:1); \draw (72:0.1)--(0:1); \draw (216:1)--(252:0.5)--(288:1); \draw (252:0.5)--(72:0.1); \end{scope} \begin{scope}[shift={(8,0)}] \draw (0:1.2) node {$3$}; \draw (72:1.2) node {$2$}; \draw (144:1.2) node {$1$}; \draw (216:1.2) node {$5$}; \draw (288:1.2) node {$4$}; \draw (72:1)--(0:0)--(144:1); \draw (0:1)--(0:0)--(216:1); \draw (288:1)--(0:0); \end{scope} \end{tikzpicture} $$ Let ${\rm Vert}(T)$ denote the set of internal vertices of $T$. Let $T$ be such a planar tree and $v \in {\rm Vert}(T)$ be an internal vertex of degree $r \geq 3$. Removing $v$ from $T$ we obtain a forest with $r$ components, and this decomposes $[n]$ into $r$ cyclic intervals $[a_1,b_1],\ldots,[a_r,b_r]$, giving a positroid ${\mathcal{M}}(v)$. Thus $I = \{i,j\}$ is a basis of the positroid ${\mathcal{M}}(v)$ if and only if the unique path from $i$ to $j$ in $T$ passes through $v$. \begin{proposition}[\cite{Kap}] The map $$ T \longmapsto {\tilde \Delta}_T:=\{P_{{\mathcal{M}}(v)} \mid v \in {\rm Vert}(T)\} $$ gives a bijection between planar trees with $n$ leaves and regular subdivisions of the hypersimplex $\Delta(2,n)$ into positroid polytopes. \end{proposition} These hypersimplex decompositions can be obtained by intersecting $\Delta(k,n)$ with the hyperplanes $H_e:=\{x_a+x_{a+1}+\cdots+x_b = 1\}$, one for each each internal $e$ of $T$, where for an internal edge $e$, we set $[a,b]$ to be the cyclic interval of leaves on one side of $e$. The positroid polytope $P_{{\mathcal{M}}(v)}$ has as interior facets the $H_e$ where $v$ is incident to $e$. If $T$ is the star with a single internal vertex $v$ and no edges then ${\tilde \Delta}_T = \{P_{{\mathcal{M}}(v)} =\Delta(2,n)\}$ is the trivial decomposition. \subsection{} Now let $p_\bullet \in {\rm Dr}(2,n)_{>0}$ be a (finite) positive tropical Pl\"ucker vector. We assign a planar tree $T(p_\bullet)$ to $p_\bullet$ as follows. For $1 \leq a<b<c<d \leq n$, we consider which of the following three possibilities holds: \begin{enumerate} \item $p_{ac} + p_{bd} = p_{ab} + p_{cd} < p_{ad} + p_{bc}$, \item $p_{ac} + p_{bd} = p_{ad} + p_{bc}< p_{ab} + p_{cd}$, \item $p_{ac} + p_{bd} = p_{ab} + p_{cd} = p_{ad} + p_{bc}$. \end{enumerate} The tree $T(p_\bullet)$ has the property that \begin{enumerate} \item the shortest path from leaf $a$ to leaf $d$ does not intersect the shortest path from leaf $b$ to leaf $c$, \item the shortest path from leaf $a$ to leaf $b$ does not intersect the shortest path from leaf $c$ to leaf $d$, \item there is an internal vertex $v$ such that any shortest path between two of the vertices $a,b,c,d$ passes through $v$, \end{enumerate} respectively. \begin{proposition} The map $p_\bullet \mapsto T(p_\bullet)$ defines a fan structure on ${\rm Dr}(2,n)$ that agrees with the ones in \S\ref{sec:fan}. \end{proposition} \subsection{} Let $T$ be a planar tree and ${\tilde \Delta}_T$ be the corresponding hypersimplex subdivision. Suppose that ${\tilde \Delta}_T = \{P_{{\mathcal{M}}(v)}\}$. We have a homeomorphism \begin{equation}\label{eq:Ch} \Theta_{{\tilde \Delta}_T,>0} \simeq \prod_{v \in {\rm Vert}(T)} \Pi_{{\mathcal{M}}(v),>0}/T_{>0} \end{equation} and $\dim({\tilde \Delta}_T) = \dim(\Theta_{{\tilde \Delta}_T,>0}) = n{-}2{-}|{\rm Vert}(T)|$. This agrees with $\Pi_{{\mathcal{M}}(v),>0}/T_{>0} \simeq {\mathbb R}^{\deg(v) - 3}$, since $\sum_{v \in {\rm Vert}(T)} (\deg(v){-} 3) = n{-}2{-}|{\rm Vert}(T)|$. For $k > 2$, we do not know an easy estimate for $\dim({\tilde \Delta})$, nor does the ``factorization" in \eqref{eq:Ch} hold. We will study of the geometry of $\Theta_{{\tilde \Delta},>0}$ in future work \cite{ALS2}. \subsection{} We shall use the following positive parametrization of ${\operatorname{Conf}}(2,n)$: \begin{equation}\label{eq:Markparam} \begin{bmatrix} 0 & 1& 1& 1&1& \cdots & 1 \\ -1 & 0& 1& 1+x_1 & 1+x_1+x_1x_2 & \cdots & 1+x_1+x_1x_2+ \cdots +x_1x_2\cdots x_{n-3} \end{bmatrix} \end{equation} In this positive parametrization, the Pl\"ucker coordinates for $12,13,\ldots,1n$ and $23$ have been gauge-fixed to 1, while the positive parameters $(x_1,\ldots,x_{n-3})$ are given by $x_i = \Delta_{i+2,i+3}/\Delta_{i+1,i+2}$. Thus, our positive parameterization is related to the one for $$ ({\mathcal{C}},{\mathcal{G}}) = (\{12,13,\ldots,1n,23,34,\ldots,(n-1)n\},\{12,13,\ldots,1n,23\}) $$ by a monomial transformation. Our reasons for choosing this parametrization will be explained in \cite{ALS2}. An explicit description of the fan structure in these positive coordinates is given in \cite{SW}. Let us take $n = 5$. Then the non-monomial Pl\"ucker variables are $\Delta_{24} = 1+x_1$, $\Delta_{25} = 1+x_1+x_1x_2$, and $\Delta_{35} = x_1+x_1x_2$. The common domains of linearity of ${\operatorname{Trop}}(\Delta_{24}) = \min(0,X_1)$, ${\operatorname{Trop}}(\Delta_{25}) = \min(0,X_1,X_1+X_2)$ and ${\operatorname{Trop}}(\Delta_{35}) = X_1 + \min(0,X_2)$ give the following fan: $$ \begin{tikzpicture} \draw [help lines, step=1cm] (-2,-2) grid (2,2); \node[fill=black,circle, inner sep=0pt,minimum size=5pt] at (0,0) {}; \draw[thick] (0,0) --(2,0); \draw[thick] (0,0) --(-2,0); \draw[thick] (0,0) --(0,-2); \draw[thick] (0,0) --(0,2); \draw[thick] (0,0) --(2,-2); \end{tikzpicture} $$ As an example, let us consider the integer vector $(1,-1)$ lying on the southeast pointing ray, and substitute $(x_1,x_2) = (t,1/t)$ into \eqref{eq:Markparam} to obtain $$ V(t) = \begin{bmatrix} 0 & 1& 1& 1&1\\ -1 & 0& 1& 1+ t & 2+t \end{bmatrix} $$ The tropical Pl\"ucker vector $p_\bullet ={\rm val}( \Delta_\bullet(V(t)))$ is given by $p_{34} = 1$ and $p_J = 0$ for $J \neq 34$. Taking ${\mathbf{a}} = (1/2,1/2,-1/2,-1/2,1/2)$ in \eqref{eq:equiv}, we see that $p_\bullet \sim p'_\bullet$, where $p'_{12}=p'_{15}=p'_{25} = 1$ while $p_J =0$ for $J \notin \{12,15,25\}$. Thus the corresponding hypersimplex subdivision ${\tilde \Delta}(p_\bullet)$ consists of two positroid polytopes $P_{{\mathcal{M}}_1}$ and $P_{{\mathcal{M}}_2}$ where $$ {\mathcal{M}}_1 = \mbox{$\binom{[5]}{2}$} \setminus \{34\} \qquad \text{and} \qquad {\mathcal{M}}_2=\{13,14,23,24,34,35,45\} $$ separated by the hyperplane $x_3+x_4 = 1$. The planar tree is $$ \begin{tikzpicture} \node at (180:2.5) {$T(p_\bullet)=$}; \draw (0:1.2) node {$3$}; \draw (72:1.2) node {$2$}; \draw (144:1.2) node {$1$}; \draw (216:1.2) node {$5$}; \draw (288:1.2) node {$4$}; \draw (72:1)--(144:0.1)--(144:1); \draw (144:0.1)--(216:1); \draw (288:1)--(-36:0.5)--(0:1); \draw (-36:0.5)--(144:0.1); \end{tikzpicture} $$ \subsection{} Let us fix a cluster ${\mathcal{C}}\subset \binom{[n]}{2}$. In the case $k=2$, the polytope $P(2,n)$ has the special feature that it is simple, and thus every cone of the normal fan ${\mathcal F}$ is a simplicial cone. Let us denote the (first integer point on the) rays of ${\mathcal F}$ by $r_{ij}$, corresponding to the tree with a single interior edge separating leaves $i+1,\ldots,j$ from $j+1,\ldots,i-1,i$, where $(i,j)$ varies over the diagonals of a polygon with vertices $1,2,\ldots,n$. Also write $p^{ij}_\bullet$ for the tropical Pl\"ucker vector that maps to $r_{ij}$ under Theorem~\ref{thm:param}. As explained in \cite{AHL1}, a consequence of the simplicial-ness of ${\mathcal F}$ is that the ring ${\mathbb C}[\Gamma]$ has some particularly nice generators. For $(i,j)$ a diagonal of the polygon with vertices $1,2,\ldots,n$, define the rational function $$ u_{ij} = \frac{\Delta_{i,j+1} \Delta_{i+1,j} }{\Delta_{ij} \Delta_{i+1,j+1}} = \frac{(z_i-z_{j+1})(z_{i+1}-z_j)}{(z_i-z_j)(z_{i+1}-z_{j+1})} $$ on ${\operatorname{Conf}}(2,n)$, which can be interpreted as a cross ratio of the four points $z_i,z_{i+1},z_j,z_{j+1}$ on ${\mathbb P}^1$. The functions $u_{ij}$ have the following special property. \begin{proposition}[\cite{AHL1,AHL2}] The ring ${\mathbb C}[\Gamma]$ of \S\ref{sec:Gamma} is the subring of ${\mathbb C}({\rm Gr}(k,n))$ generated by the $u_{ij}$. For two diagonals $(i,j)$ and $(i',j')$, we have $${\operatorname{Trop}}(u_{ij})(r_{i'j'}) = {\operatorname{Trop}}(u_{ij})(p^{i'j'}_\bullet) = \delta_{ij,i'j'}.$$ \end{proposition} Note that ${\operatorname{Trop}}(u_{ij}) = p_{i,j+1} + p_{i+1,j} - p_{ij} - p_{i+1,j+1}$, so it is easy to see that it takes nonnegative values on any tropical Pl\"ucker vector $p_\bullet$, and thus $u_{ij}$ is nearly convergent. It is not hard to see that all the inequalities from \eqref{eq:trop} are positive linear combinations of ${\operatorname{Trop}}(u_{ij})$ for various $i,j$. Let ${\mathcal{G}} \subset {\mathcal{C}}$ be a gauge-fix. Setting $\Delta_I = 1$ for $I \in {\mathcal{G}}$, it is not difficult to see that there is an invertible monomial transformation between the $n(n{-}3)/2$ functions $u_{ij}$ and the functions $\Delta_J$ with $J \notin {\mathcal{C}} \setminus {\mathcal{G}}$. The fan structure on ${\rm Dr}(2,n)_{>0}$ is given by intersecting the cones ${\operatorname{Trop}}(u_{ij}) = 0$ and ${\operatorname{Trop}}(u_{ij}) > 0$ as $i,j$ vary. For $n= 5$ and the parametrization \eqref{eq:Markparam}, the $u_{ij}$ functions are \begin{align*} u_{13} &= \frac{1}{1+x_1},& u_{14}&= \frac{1 + x_1}{1 + x_1 + x_1 x_2},& u_{24}&=\frac{x_1 (1 + x_1 + x_1 x_2)}{(1 + x_1) (x_1 + x_1 x_2)} \\ u_{25}&= \frac{x_1 + x_1 x_2}{1 + x_1 + x_1 x_2}, &u_{35}&=\frac{x_1 x_2}{x_1 + x_1 x_2} \end{align*} which are easily seen to belong to $\Gamma$ using Lemma~\ref{lem:nearlyconverge}. \section{Tropical bridge reduction} \label{sec:bridge} We will frequently use the following easy result without mention. \begin{lemma} We have $I \leq J$ if and only if $(I \setminus J) \leq (J \setminus I)$. \end{lemma} \subsection{} Let $q_\bullet$ be a positive tropical Pl\"ucker vector. Pick $a \in {\mathbb Z}$ and $i \neq j \in [n]$. We define the tropical bridge $T_\gamma(a) = T_{i,j}(a)$ acting on ${\mathbb R}^{\binom{[n]}{k}}$ by $q'_\bullet = T_\gamma(a) \cdot q_\bullet$ where $$ q'_I = \begin{cases} \min(q_I, q_{I \setminus \{j\} \cup \{i\}} + a)& \mbox{if $j \in I$ but $i \notin I$} \\ q_I & \mbox{otherwise.} \end{cases} $$ This formula is the tropicalization of \eqref{eq:Deltax}. When $j=i+1$, we write $T_i(a):= T_{i,i+1}(a)$. \begin{proposition}\label{prop:bridge} If $q_\bullet \in {\rm Dr}(k,n)_{\geq 0}$ then $T_i(a) \cdot q_\bullet \in {\rm Dr}(k,n)_{\geq 0}$. \end{proposition} \begin{proof} By Theorem~\ref{thm:main}, every $q_\bullet \in {\rm Dr}(k,n)_{\geq 0}$ is representable by $V \in {\rm Gr}(k,n)({\mathcal{R}}_{\geq 0})$. The claim then follows immediately from \eqref{eq:Deltax}. \end{proof} \begin{remark} Our proof below gives proofs of Theorem~\ref{thm:main} and of Proposition~\ref{prop:bridge} that are independent of our earlier proof of Theorem~\ref{thm:main}. We will only apply Proposition~\ref{prop:bridge} to $q_\bullet \in {\rm Dr}(k,n)_{\geq 0}$ that we separately know to be representable. And once Theorem \ref{thm:main} is established, Proposition~\ref{prop:bridge} follows for arbitrary $q_\bullet \in {\rm Dr}(k,n)_{\geq 0}$. \end{remark} \subsection{} We shall show that bridge reduction (Proposition~\ref{prop:bridgedecomp}) holds for positive tropical Pl\"ucker vectors. \begin{proposition}\label{prop:tropreduce} Let $p_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}$ where ${\mathcal{M}}$ is a rank $k >0$ positroid on $[n]$. Then we show that at least one of the following holds: \begin{enumerate} \item For some $i \in [n]$, we have $f_{\mathcal{M}}(i) = i$. Define $\epsilon_i: [n-1] \to [n]$ by $\epsilon_i(a) = a$ if $a <i$ and $\epsilon_i(a) = a+1$ if $a \geq i$. Then $p_\bullet$ is in the image of the map $\epsilon_i:{\rm Dr}(k,n-1)_{\geq 0} \hookrightarrow {\rm Dr}(k,n)_{\geq 0}$ given by $$ \epsilon_i(q_\bullet)_I = \begin{cases} \infty & \mbox{if $i \in I$} \\ q_J & \mbox{if $I = \epsilon_i(J)$.} \end{cases} $$ \item For some $i \in [n]$, we have $f_{\mathcal{M}}(i) = i+n$. Define $\epsilon_i: [n-1] \to [n]$ by $\epsilon_i(a) = a$ if $a <i$ and $\epsilon_i(a) = a+1$ if $a \geq i$. Then $p_\bullet$ is in the image of the map $\varepsilon_i:{\rm Dr}(k-1,n-1)_{\geq 0} \hookrightarrow {\rm Dr}(k,n)_{\geq 0}$ given by $$ \varepsilon_i(q_\bullet)_I = \begin{cases} \infty & \mbox{if $i \notin I$} \\ q_J & \mbox{if $I = \epsilon_i(J) \cup\{i\}$.} \end{cases} $$ \item For some $i \in [n]$, we have $i+1 \leq f_{\mathcal{M}}(i) < f_{\mathcal{M}}(i+1) \leq i+n$. Then $p_\bullet = T_i(a) \cdot q_\bullet$ where $q_\bullet \in {\rm Dr}({\mathcal{M}}')_{>0}$ with ${\mathcal{M}}'$ the positroid satisfying $f_{{\mathcal{M}}'} = f_{\mathcal{M}} s_i$, and $$ a = p_{I_{i+1}} - p_{I_{i+1} \setminus\{i+1\} \cup \{i\}}. $$ \end{enumerate} \end{proposition} It is easy to see that $f_{\mathcal{M}}$ satisfies at least one of the three stated conditions in Proposition~\ref{prop:tropreduce}. In Case (1), if $f_{\mathcal{M}}(i) = i$ then $I \notin {\mathcal{M}}$ for all $I$ containing $i$. Thus $p_I = \infty$ whenever $i \in I$. It is clear that $p_\bullet$ is in the image of $\epsilon_i$, and that the image of $\epsilon_i$ lies inside ${\rm Dr}(k,n)_{\geq 0}$. Case (2) is similar. \subsection{} We now consider Case (3) of Proposition~\ref{prop:tropreduce}. Let $f = f_{\mathcal{M}}$. We suppose that $i \in [n]$ satisifies $i+1 \leq f_{\mathcal{M}}(i) < f_{\mathcal{M}}(i+1) \leq i+n$. To simplify the notation, we assume that $i = 1$ so we have $2 \leq f(1) < f(2) \leq 1+n$. First consider the case $f(1) = 2$. Define $q_\bullet$ by $$ q_I = \begin{cases} \infty & \mbox{if $2 \in I$} \\ p_I & \mbox{if $2 \notin I$} \end{cases} $$ which clearly satisfies the positive tropical Pl\"ucker relations. \begin{lemma}\label{lem:f12} We have $p_\bullet = T_i(a) \cdot q_\bullet$ where $a = p_{I_2} - p_{I_{2} \setminus\{2\} \cup \{1\}}$. \end{lemma} \begin{proof} By induction on ${\mathcal{M}}$, we may assume that $q_\bullet$ is representable and thus $p'_\bullet:= T_i(a) \cdot q_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}$ by Proposition \ref{prop:bridge}. The assumption $f(1) = 2$ implies that $I \notin {\mathcal{M}}$ for any $I$ containing $\{1,2\}$. It suffices to show that $p'_I = p_I$ for $2 \in I$ and $I \in {\mathcal{M}}$. Let $I_2 = 2J := J \sqcup \{2\}$ where $J \subset [3,n]$. We claim that \begin{equation}\label{eq:12K} p_{2K} = p_{1K} + a. \end{equation} for $K \subset [3,n]$ such that $2K \in {\mathcal{M}}$. To show this we proceed by (downward) induction on $|K \cap J|$, the case $|K\cap J| = k-1$ being tautological. If $|K \cap J| < k-1$ then by the exchange relation there exists $K'$ with $2K' \in {\mathcal{M}}$ such that $|K' \cap J| = |K \cap J| + 1$ and $K' = K \setminus \{a\} \cup \{b\}$. Setting $L = K \setminus \{a\}$, we have by \eqref{eq:trop} the equality $$ p_{1La}+p_{2Lb} = \min(p_{12L}+p_{Lab}, p_{1Lb}+p_{2La}) = p_{1Lb}+p_{2La} $$ if $a < b$, using $p_{12L} =\infty$. The same equality $p_{1La}+p_{2Lb} = p_{1Lb}+p_{2La}$ holds if $a>b$. Thus by induction we have $p_{2K}-p_{1K} = p_{2K'}-p_{1K'} = a$, establishing \eqref{eq:12K}. By definition $p'_{2K} = \min(q_{2K}, q_{1K}+a) = p_{1K}+a$. We conclude that $p_\bullet = p'_\bullet$. \end{proof} \subsection{} For the remainder of the proof we assume that $f(1) = j > 2$, and we set $f(2) = \ell> j$. (It is possible for $\ell$ to equal $1+n$.) Let the Grassmann necklace of ${\mathcal{M}}$ be $(I_1,I_2,\ldots,I_n)$. Then $I_1 = 12I$, $I_2 = 2I j $, $I_3 = I j \ell$. Define ${\mathcal{M}}'$ to be the positroid with bounded affine permutation given by $f'(1) = \ell$ and $f'(2) = j$ and $f'(a) = f(a)$ for $a \in [3,n]$. We define $q_\bullet$ by $$ q_J = \begin{cases} \mbox{recursion given below} & \mbox{if $1 \notin J$ and $2 \in J$} \\ p_J & \mbox{otherwise.} \end{cases} $$ It follows immediately that instances of \eqref{eq:trop} for $q_\bullet$ immediately hold whenever all subsets involved either contain $1$ or do not contain $2$. We now give the formula for $q_{K2}$, $K \subset [3,n]$ recursively. At every step, we will check that instances of \eqref{eq:trop} of the form \begin{equation}\label{eq:ind} q_{L2a} + q_{L1b} = \min(q_{L12} + q_{Lba}, q_{L1a}+ q_{L2b}) \end{equation} hold, where $3 \leq b < a \leq n$ and $K = La \subset [3,n]$. We say that \eqref{eq:ind} is associated to the pair $\{L2a,L1b\}$ of subsets, which uniquely determines \eqref{eq:ind}. \noindent (a) If $K2 \notin {\mathcal{M}}$, for example if $K < Ij$ in dominance order, then we already have $p_{K2} = \infty$. We then set $q_{K2} = \infty$ as well. \eqref{eq:ind} will hold since both sides were already equal to $\infty$ for $p_\bullet$ (and thus also for $q_\bullet$). \noindent (b) If $K = I j$, then we set $q_{I2j} = \infty$. The equation $\eqref{eq:ind}$ holds because the LHS is already $\infty$ for $p_\bullet$. To see this, observe that for the pair $\{I2j, I1b\}$ with $b < j$, we have that $I1 b \not \geq_3 Ij\ell = I_3$; for the pair $\{I2j = L'2aj, L'1bj\}$ where $b < a$, either $b < j$ and $L'1bj \not \geq_3 L'ajk = I_3$, or $b > j$ and $L'1bj \not \geq_1 L'12a = I_1$. In both cases $I1b$ (resp. $L'1bj$) does not belong to ${\mathcal{M}}$. \noindent (c) If $K2 \in {\mathcal{M}}$ and $K1 \notin {\mathcal{M}}$, then we set $q_{K2} = p_{K2}$. We call the subset $K2$ of {\it parallel type}. Setting $K = La$, we see that \eqref{eq:ind} holds because $$ p_{L2a} + p_{L1b} = \min(p_{L12}+p_{Lab}, p_{L1a}+ p_{L2b}) = p_{L12}+p_{Lab} $$ and all the terms on both sides are unchanged in $q_\bullet$. \noindent (d) Suppose finally that $K2, K1 \in {\mathcal{M}}$, and $K2 \neq I2j$. We call such minors {\it general type}. We inductively assume that the value of $q_{K'2}$ has been defined for all $K'2 < K2$. Suppose that we have found $2 < b < a \leq n$ such that \begin{equation}\label{eq:pair} \mbox{$K = La$ and $L1b \in {\mathcal{M}}$.} \end{equation} (The existence of such a pair $(a,b)$ follows from an exchange relation argument similar to the proof of Lemma~\ref{lem:f12}, see \cite[Lemma 7.11]{LamCDM}.) We then define \begin{equation}\label{eq:ab} q_{K2} =q_{L2a} := \min(q_{L12} + q_{Lba}, q_{L1a}+q_{L2b})- q_{L1b}. \end{equation} Since $L2b< L2a$, we may assume that $q_{L2b}$ has already been defined. \begin{lemma} Equation \eqref{eq:ab} well-defines the value of $q_{K2}$, regardless of the choice of $a$ and $b$. \end{lemma} \begin{proof} Suppose we have two pairs $(b < a)$ and $(b' < a')$ such that \eqref{eq:pair} holds. Case 1: If $a = a'$, we assume that $b' < b$, and compute as follows. \begin{align*} &q_{L2a} + q_{L1b} + q_{L1b'} \\ &= \min(q_{L12} + q_{Lba} +q_{L1b'}, q_{L1a}+q_{L2b} + q_{L1b'}) & \mbox{using \eqref{eq:ab} for $(b < a)$} \\ &= \min(q_{L12} + q_{Lba} +q_{L1b'}, q_{L1a}+q_{L12} + q_{Lb'b}, q_{L1a}+q_{L1b} + q_{L2b'}) \\ &=\min(q_{L12} + q_{Lb'a} +q_{L1b}, q_{L1a}+q_{L2b'} + q_{L1b}) \end{align*} Noting that $q_{L1b} + q_{L1b'} < \infty$, we conclude that \eqref{eq:ab} gives the same result using $(b<a)$ and $(b'<a)$. Case 2: We have $b = b'$. Similar to Case 1. Case 3: The four numbers $a,b,a',b'$ are distinct. Set $K = Maa'$, so by assumption $M1ab', M1ba' \in {\mathcal{M}}$. Let us assume that $b < a < a'$. If $b< b' < a$, then $M1ba,M1b'a' \in {\mathcal{M}}$ as well by \eqref{eq:trop}, and we can reduce to Case 1. In the other cases, we may conclude from \eqref{eq:trop} that $M1bb', M1aa' \in {\mathcal{M}}$. We now assume that $M1bb',M1aa' \in {\mathcal{M}}$ and first consider the case $b < a < b' < a'$. To simplify notation, we omit $M$, and suppose that $(b,a,b',a') = (3,4,5,6)$. So $145,136,135,146 \in {\mathcal{M}}$. In the following, we underline the terms where \eqref{eq:trop} has been applied. Using \eqref{eq:ab} for the pair $(246,145)$ we have \begin{align*} &\underline{q_{246}} + \underline{q_{145}} + q_{136} \\ &= \min(\underline{q_{124}} + q_{456} + \underline{q_{136}}, q_{146}+q_{245} + q_{136}) \\ &= \min(q_{123} + q_{456} + q_{146}, q_{456} + q_{126} + q_{134}, q_{146}+q_{245} + q_{136}) \end{align*} Using \eqref{eq:ab} for the pair $(246,136)$ we have \begin{align*} &\underline{q_{246}} + q_{145} + \underline{q_{136}} \\ &= \min(q_{126} + \underline{q_{346}} + \underline{q_{145}}, q_{146}+q_{236} + q_{145}) \\ &= \min(q_{126} + q_{134} + q_{456},q_{126} + q_{146} + q_{345} ,q_{146}+q_{236} + q_{145}). \end{align*} The quantity $q_{126}+q_{134}+q_{456}$ appears in both minima, so it suffices to show that $$ \min(q_{123}+q_{456},q_{245}+q_{136}) = \min(q_{126}+q_{345},q_{236}+q_{145}). $$ Adding $q_{135}$ (which is $<\infty$), we compute \begin{align*} &\min(q_{123}+q_{456}+q_{135},\underline{q_{245}}+q_{136}+\underline{q_{135}}) \\ &=\min(q_{123}+q_{456}+q_{135},\underline{q_{125}}+q_{345}+\underline{q_{136}},q_{145}+q_{235}+q_{136}) \\ &=\min(q_{123}+q_{456}+q_{135},q_{345}+q_{123}+q_{156},q_{345}+q_{126}+q_{135},q_{145}+q_{235}+q_{136}) \end{align*} and \begin{align*} &\min(q_{126}+q_{345}+q_{135},\underline{q_{236}}+q_{145}+\underline{q_{135}}) \\ &=\min(q_{126}+q_{345}+q_{135},\underline{q_{145}}+q_{123}+\underline{q_{356}},q_{145}+q_{136}+q_{235}) \\ &=\min(q_{126}+q_{345}+q_{135},q_{123}+q_{135}+q_{456},q_{123}+q_{156}+q_{345},q_{145}+q_{136}+q_{235}). \end{align*} This shows that \eqref{eq:ab} for the pair $(246,145)$ gives the same result as for the pair $(246,136)$. Note that we have used \eqref{eq:ab} for $K2 = 245$ and $K2 = 236$, which may assume to hold since they are both less than $246$ in dominance order. The other case $b' < b < a < a'$ is similar. \end{proof} We have now completely defined the vector $q_\bullet$. \begin{lemma}\label{lem:qbullet} We have $q_\bullet \in {\rm Dr}({\mathcal{M}}')_{>0}$. \end{lemma} We first show that $q_\bullet$ has the correct support. \begin{lemma}\label{lem:supp} We have ${\rm Supp}(q_\bullet) = {\mathcal{M}}'$. \end{lemma} \begin{proof} We have ${\rm Supp}(p_\bullet) = {\mathcal{M}}$. Let $J \in {\mathcal{M}} \setminus {\mathcal{M}}'$ be such that $p_J < \infty$. We must show that $q_J = \infty$. Note that $J = K2$ must be of general type. Apply \eqref{eq:ab}. If $q_J < \infty$, then by induction at least one of the two pairs $\{L12,Lba\}$ and $\{L1a,L2b\}$ must be contained in ${\mathcal{M}}'$, while $J = K2$ is not. This contradicts the fact that ${\mathcal{M}}'$ satisfies the 3-term positive exchange relation. A similar argument shows that for $J \in {\mathcal{M}}'$ we have $q_J < \infty$. \end{proof} By Lemma \ref{lem:supp}, to prove Lemma \ref{lem:qbullet} only need to check relations where the LHS of \eqref{eq:trop} is finite i.e. both terms on the LHS are indexed by elements in ${\mathcal{M}}'$. \begin{lemma}\label{lem:notgeneral} Suppose that $J,K \in {\mathcal{M}}'$ appear on the LHS of \eqref{eq:trop}, and both $J$ and $K$ are not of general type. Then all four subsets on the RHS of \eqref{eq:trop} are also not of general type. \end{lemma} \begin{proof} Let $X$ be any point in $\Pi_{{\mathcal{M}},>0}$ and $a>0$ be the unique value such that $X' = x_i(-a) \cdot X \in \Pi_{{\mathcal{M}}',>0}$ as in \cite{LamCDM}. We have $\Delta_J(X) = \Delta_{J}(X')$ and $\Delta_K(X) = \Delta_K(X')$. But we also have $\Delta_I(X') \leq \Delta_I(X)$ for all $I$. It follows that on the RHS of the three-term Pl\"ucker relation for the matrix $X$, all subsets $S$ that appear must satisfy $\Delta_S(X) = \Delta_S(X')$. In particular, all subsets $S$ that appear in the relation are not of general type. \end{proof} Thus \eqref{eq:trop} holds whenever the LHS involves $q_{T2}$ where $T2$ is not of general type. It thus suffices to consider instances of \eqref{eq:trop} involving $q_{T2}$ where $T2$ is of general type. \noindent (a) We have already verified all relations of the form \eqref{eq:ind}. \noindent (b) Let us verify relations of the form \begin{equation}\label{eq:caseb} q_{S2b} + q_{Sac} = \min(q_{S2a} + q_{Sbc} , q_{S2c} + q_{Sab}) \end{equation} where $2 < a < b < c \leq n$ and $S2b$ is of general type. Thus $S1b \in {\mathcal{M}}$ and we add $q_{S1b} < \infty$ to both sides. The LHS becomes \begin{align*} &\min(q_{S2b}+q_{S1a} + q_{Sbc},q_{S2b}+q_{S1c}+q_{Sab}) \\ &= \min(q_{S12}+q_{Sab}+q_{Sbc},q_{S1b}+q_{S2a}+q_{Sbc},q_{S2b}+q_{S1c}+q_{Sab}) \end{align*} which is equal to (RHS of \eqref{eq:caseb} $+q_{S1b}$), using only instances of \eqref{eq:trop} for $q_\bullet$ that we know must hold. \noindent (c) Let us verify relations of the form $$ q_{S2ac} + q_{S12b} = \min(q_{S12a} + q_{S2bc} , q_{S12c} + q_{S2ab}) $$ where $2 < a < b < c \leq n$ and $S2ac$ is of general type. Thus $S1ac \in{\mathcal{M}}$ and we add $q_{S1ac} < \infty$ to both sides. The calculation is similar to (b). \noindent (d) Let us verify relations of the form \begin{equation}\label{eq:verify} q_{S2ac} + q_{S2bd} = \min(q_{S2ab} + q_{S2cd} , q_{S2ad} + q_{S2bc}) \end{equation} where $2<a<b<c<d \leq n$ are disjoint from $S2$ and both $S2ac, S2bd$ are of general type. By assumption, $S1ac, S1bd \in {\mathcal{M}}$, so it follows that either both $S1ad, S1bc \in {\mathcal{M}}$ or $S1ab, S1cd \in {\mathcal{M}}$. In the former case, add $q_{S1ad}+q_{S1ac} < \infty$ to both sides. Apply previously established instances of \eqref{eq:trop} successively to the pairs $(2bd,1ad)$, $(abd,1ac)$, $(1ab,2ac)$, $(1ac,1bd)$, $(1ab,2ad)$, $(2ac,abd)$ on the LHS, and $(1ad,2cd)$, $(1ac,2bc)$, $(12c,2ad)$, $(1ac,2ad)$ on the RHS. The resulting formulae are the same, proving \eqref{eq:verify}. In the latter case, add $q_{S1ab} + q_{S1bd} < \infty$ to both sides. Apply previously established instances of \eqref{eq:trop} successively to the pairs $(2ac,1ab)$, $(1ac,1bd)$, $(1bc,2bd)$, $(12b,1ad)$ on the LHS, and $(1ab,2ad)$, $(2cd,1bd)$, $(2bc,abd)$ on the RHS. The resulting formulae are the same, proving \eqref{eq:verify}. \noindent (e) Let us verify the relation \eqref{eq:verify} where one of $S2ac, S2bd$ is of general type, and the other one is of parallel type. By an argument similar to the proof of Lemma \ref{lem:notgeneral}, at least one of $S2ab,S2cd,S2bc,S2ad$ is of general type. We have $S2bd,S2ac \in {\mathcal{M}}$. If one of $S1ad$, $S1cd$, or $S1bc$ is in ${\mathcal{M}}$ then \eqref{eq:3ex} implies that $S1ac \in {\mathcal{M}}$. So if $S2ac$ is of parallel type, we may assume that $S1bd \in {\mathcal{M}}$. Thus $q_{S1ab}+q_{S1bd} <\infty$ and the argument reduces to that in Case (d). Otherwise, we have $S2ac$ of general type. If $S1cd \in {\mathcal{M}}$ then \eqref{eq:3ex} implies that $S1bd \in {\mathcal{M}}$, a contradiction. If $S1ad \in {\mathcal{M}}$ then again the argument reduces to that in Case (d). The remaining cases are very similar, and can be simplified by using the assumption that $q_{S1bd}, q_{S1ad}, q_{S1cd}$ are equal to $\infty$. We have now completed the proof of Lemma~\ref{lem:qbullet}. Finally, let us check that indeed $p_\bullet \to q_\bullet$ is a tropical bridge reduction. \begin{lemma} Let $a = p_{I2j} - p_{I1j}$. Then $p_\bullet = T_i(a) \cdot q_\bullet$. \end{lemma} \begin{proof} Let $p'_\bullet =T_i(a) \cdot q_\bullet$. By induction on ${\mathcal{M}}$, we may suppose that $q_\bullet$ is representable, so Proposition \ref{prop:bridge} applies and $p'_\bullet$ is a positive tropical Pl\"ucker vector. We have that ${\rm Supp}(p'_\bullet) = {\mathcal{M}}$ and $p'_J = p_J$ except when $J = K2 > I_2$ is of general type. But the recursion we used to define $q_{K2}$ can also be applied to $p'_\bullet$ (resp. $p_\bullet$), so we conclude that the rest of the values of $p_\bullet$ and $p'_\bullet$ agree. \end{proof} This completes the proof of Proposition~\ref{prop:tropreduce}. \subsection{Proof of Theorem \ref{thm:bridge}} Since ${\rm Dr}({\mathcal{M}})_{>0}$ has the structure of a rational polyhedral complex, the statement over ${\mathbb Z}$ implies the other statements. If ${\mathcal{M}} = \{I\}$ is a singleton, then ${\rm Dr}(\{I\})_{>0}({\mathbb Z})$ consists of the vectors $p(I,z)_\bullet$, $z \in {\mathbb Z}$ given by $$ p(I,z)_J = \begin{cases} z & \mbox{if $I = J$,} \\ \infty & \mbox{otherwise.} \end{cases} $$ Thus the result holds for ${\mathcal{M}} = \{I\}$. Otherwise, by using the bridge reduction moves, we have $f_{\mathcal{M}} = f_{{\mathcal{M}}'} (i,j)$ for $\dim({\mathcal{M}}') = \dim({\mathcal{M}}) -1$ and some transposition $\gamma = (i,j)$. The effect of using the embeddings $\epsilon_a:{\rm Dr}(k,n-1)_{\geq 0} \hookrightarrow {\rm Dr}(k,n)_{\geq 0}$ and $\varepsilon_a:{\rm Dr}(k-1,n-1)_{\geq 0} \hookrightarrow {\rm Dr}(k,n)_{\geq 0}$ is that $(i,i+1)$ is changed to $(i,j)$ (extra numbers are inserted in between $i$ and $i+1$). We have shown that each $p_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})$ is equal to $T_\gamma(a) \cdot q_\bullet$ for a uniquely specified $q_\bullet \in {\rm Dr}({\mathcal{M}}')_{>0}({\mathbb Z})$ and $a \in {\mathbb Z}$. Conversely, by induction every point $q_\bullet \in {\rm Dr}({\mathcal{M}}')_{>0}({\mathbb Z})$ is representable so $T_\gamma(a) \cdot q_\bullet \in {\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})$ by Proposition \ref{prop:bridge}. We conclude that we have a bijection ${\mathbb Z} \times {\rm Dr}({\mathcal{M}}')_{>0}({\mathbb Z}) \simeq {\rm Dr}({\mathcal{M}})_{>0}({\mathbb Z})$. The stated tropical bridge parametrization follows. \section{Connected positroids}\label{sec:connected} \subsection{} We collect some facts concerning connected positroids here. \begin{lemma}\label{lem:connectbridge} Let ${\mathcal{M}}$ be a connected positroid with bounded affine permutation $f = f_{\mathcal{M}}$. Then at least one of the following holds: \begin{enumerate} \item $f(i) \in \{i,i+1,i+n-1,i+n\}$ for some $i \in [n]$; \item there exists $i \in [n]$ so that $i < f(i) < f(i+1) < i+n$ such that $f' = f s_i$ is a bounded affine permutation of a connected positroid. \end{enumerate} \end{lemma} \begin{proof} Suppose that $f = f_{\mathcal{M}}$ is a counterexample. Since no $i \in [n]$ satisfies (1), we must have some $i \in [n]$ such that $i < f(i) < f(i+1) <i+n$. We have $f'(j) = f(j)$ if $j \neq i,i+1$ and $f'(i) = f(i+1)$ and $f'(i+1) = f(i)$. Let ${\mathcal{M}}'$ satisfy $f_{{\mathcal{M}}'} = f'$. Write $\pi,\pi':[n] \to [n]$ for the permutations that are reductions of $f,f'$ modulo $n$. Since (2) fails, ${\mathcal{M}}'$ is not connected. Then by Lemma~\ref{prop:disconnect}, we must have a decomposition $[n] = A \cup B$ into disjoint cyclic intervals $A=A[i],B=B[i]$ so that $$ \pi'(A) = A \qquad \text{and} \qquad \pi'(B) = B. $$ But ${\mathcal{M}}$ is connected so after renaming $A$ and $B$ we must have $i,\pi(i+1) \in A=[j+1,i]$ and $i+1,\pi(i) \in B = [i+1,j]$ where we assume that $j$ has been chosen so that $|B|$ is minimal. We now assume that out of all $i \in [n]$ satisfying $i < f(i) < f(i+1) <i+n$ we have chosen $i$ so that $|B[i]|$ is minimal. Suppose that $f(r) <f(r+1)$ for some $r,r+1 \in [i+2,j]$. Since $\pi(r),\pi(r+1) \in B$, the minimality assumption on $|B|$ implies that $\pi(r+1), r , r+1, \pi(r)$ are in order within (the totally ordered cyclic interval) $B$. Set $f''=fs_r$ and let $\pi''$ be the reduction of $f''$ modulo $n$. Now let $B[r] = [r+1,j']$, so that $\pi''(B[r]) = B[r]$. Since $|B[r]| \geq |B|$ by assumption, we must have $B[r] \cap A \neq \emptyset$. If $\pi(i+1) \in (B[r]\cap A)$, we would contradict $\pi''(B[r]) = B[r]$, since $i+1 \notin B[r]$. Thus $i \notin B[r]$, and we must have $\pi''(A \cap B[r]) \subset A$ as well, so that $\pi''(A \cap B[r]) = (A \cap B[r])$. This contradicts our assumption that $B[r] = [r+1,j']$ is chosen so that $|B[r]|$ is minimal. Thus we must have $f(i+2) > f(i+3) > \cdots > f(j)$, and all these values lie inside $B$ modulo $n$. It is easy to see that this is impossible (since (1) is never satisfied) and we have arrived at a contradiction.\end{proof} \subsection{} Let ${\mathcal{C}} \subset {\mathcal{M}}$ be a cluster for a positroid ${\mathcal{M}}$. The plabic graph construction \cite{Pos,OPS} implies that there exists a bipartite graph $G({\mathcal{C}})$ embedded into the disk, whose faces are in bijection with ${\mathcal{C}}$ (write $F(J)$ for the face indexed by $J \in {\mathcal{C}}$). The graph $G({\mathcal{C}})$ is essentially unique if we assume that interior vertices have degree $> 2$, except for vertices connected to the boundary vertices. (To fix conventions, let us use ``target-labels" for the faces, in agreement with \cite{OPS}.) The graph $G({\mathcal{C}})$ has the following properties: \begin{enumerate} \item every connected component of $G({\mathcal{C}})$ is connected to the boundary, and the connected components of $G({\mathcal{C}})$ are naturally in bijection with the connected components of ${\mathcal{M}}$; \item for each interior face $F(J)$, we have $e_{J_1}+e_{J_3}+\cdots+e_{J_{2r-1}} = e_{J_2}+e_{J_4}+\cdots+e_{J_{2r}}$, where $F(J_1),F(J_2),\ldots,F(J_{2r})$ are the faces edge adjacent to $F(J)$ arranged in cyclic order. \end{enumerate} \begin{lemma}\label{lem:mintree} A connected positroid ${\mathcal{M}}$ is minimal if and only if $G({\mathcal{C}})$ is a tree, for some cluster ${\mathcal{C}}$ of ${\mathcal{M}}$. In this case, ${\mathcal{M}}$ has a unique cluster ${\mathcal{C}} \subset {\mathcal{M}}$ and we denote by $T_{\mathcal{M}}:=G({\mathcal{C}})$ this tree. \end{lemma} The positroid polytope $P_{\mathcal{M}}$ of a minimal connected positroid ${\mathcal{M}}$ can be described as follows. We assume that $T_{\mathcal{M}}$ is chosen so that it is a bipartite tree all of whose vertices have degree $> 2$. Then the facets of $P_{\mathcal{M}}$ are in bijection with the edges of $T_{\mathcal{M}}$. There are $n$ edges of $T_{\mathcal{M}}$ that connect to boundary points. These correspond to external facets: these are facets of $P_{\mathcal{M}}$ that are supported on the same hyperplane $\{x_i = 0\}$ or $\{x_i = 1\}$ as a facet of the hypersimplex (the color of the internal vertex determines which of $\{x_i = 0\}$ or $\{x_i = 1\}$ occurs). Interior edges $e$ of $T_{\mathcal{M}}$ correspond to internal facets of $P_{\mathcal{M}}$. The edge $e$ divides $T_{\mathcal{M}}$ into two components, and thus $[n] = [a,b-1] \cup [b,a-1]$ into two cyclic intervals. The internal facet of $P_{\mathcal{M}}$ is supported on a hyperplane of the form $\sum_{i\in[a,b-1]} x_i = c$ for some $c$. \subsection{Proof of Lemma~\ref{lem:span}}\label{ssec:proofspan} We first argue that $\operatorname{span}_{\mathbb Z}\{e_J \mid J \in {\mathcal{C}}\} = X({\widehat{ T}})$ for any cluster ${\mathcal{C}}$. Since ${\mathcal{M}}$ is connected, Lemma~\ref{lem:connected} says that the action of $T$ on ${\mathring{\Pi}}_{\mathcal{M}}$ is faithful and thus the action of ${\widehat{ T}}$ on ${\widetilde{\Pi}}_{\mathcal{M}}$ is faithful. On the other hand, if the integral span of $\{e_J \mid J \in {\mathcal{C}}\}$ is strictly smaller than $X({\widehat{ T}})$, then there is a nontrivial subtorus $\{1\} \subsetneq S \subset {\widehat{ T}}$ that acts on the rational function field ${\mathbb C}(\Delta_{J} \mid J \in {\mathcal{C}})$ as the identity. But we have an inclusion ${\mathbb C}[{\widetilde{\Pi}}_{\mathcal{M}}] \subset {\mathbb C}(\Delta_{J} \mid J \in {\mathcal{C}})$ coming from the cluster structure on ${\widetilde{\Pi}}_{\mathcal{M}}$, and thus $S$ must act trivially on ${\widetilde{\Pi}}_{\mathcal{M}}$, a contradiction. Suppose that ${\mathcal{M}}$ is a connected positroid. Then $i<f_{\mathcal{M}}(i)<i+n$ for all $i$. By Lemma~\ref{lem:connectbridge}, we must be in one of the following situations: \begin{enumerate} \item for some $i$, we have $f_{\mathcal{M}}(i) = i+1$ or $f_{\mathcal{M}}(i) = i+n-1$; \item for some $i$, we have $i < f_{\mathcal{M}}(i) < f_{\mathcal{M}}(i+1) < i+n$ and $ f s_i = f_{{\mathcal{M}}'}$ where ${\mathcal{M}}'$ is connected. \end{enumerate} Suppose (1) holds. We assume $f_{\mathcal{M}}(i) = i+n-1$; the other case is similar. For any cluster ${\mathcal{C}} \subset {\mathcal{M}}$ the graph $G({\mathcal{C}})$ the vertices $i-1,i$ are connected to the same white interior vertex. Thus there is $J \in {\mathcal{C}}$ such that $i-1 \in J$ but $i-1\notin J'$ for all $J '\in {\mathcal{C}}':={\mathcal{C}} \setminus \{J\}$. The set ${\mathcal{C}}'$ is a cluster for some connected positroid ${\mathcal{M}}'$ on $[n] \setminus \{i-1\}$ (not depending on ${\mathcal{C}}$) of the same rank as ${\mathcal{M}}$ and all clusters ${\mathcal{C}}'$ for ${\mathcal{M}}'$ occur in this way. By induction, there exists a gauge-fix and cluster ${\mathcal{G}}' \subset {\mathcal{C}}'$ for ${\mathcal{M}}'$ i.e. $|{\mathcal{G}}'| = n-1$ and $$\operatorname{span}_{\mathbb Z}\{e_J \mid J \in {\mathcal{C}}'\} = \{(x_1,\ldots,x_n) \in {\mathbb Z}^n \mid x_{i-1} = 0 \text{ and } k \text{ divides } \sum x_i \}.$$ It follows that ${\mathcal{G}} := ({\mathcal{G}}' \cup \{J\} )\subset {\mathcal{C}}:=({\mathcal{C}}' \cup \{J\})$ is a gauge-fix. Suppose (2) holds. Let ${\mathcal{G}}' \subset {\mathcal{C}}' \subset {\mathcal{M}}'$ be a gauge-fix and a cluster for ${\mathcal{M}}'$. It is possible to add an edge to the planar bipartite graph $G({\mathcal{C}}')$ to obtain the planar bipartite graph $G({\mathcal{C}})$ for some cluster ${\mathcal{C}}$ of ${\mathcal{M}}$, see \cite[Section 7.4]{LamCDM}. We have ${\mathcal{C}} = {\mathcal{C}}' \cup \{J\}$ where $J$ is the face label of the new face in $G({\mathcal{C}})$. Thus ${\mathcal{G}}' \subset {\mathcal{C}}$ is a gauge-fix.

proofpile-arXiv_069-14023

\section{Introduction} Modelling the mass distribution of M31 is a classical problem that has seen many past iterations (e.g., Einasto 1972; Kent, Huchra \& Stauffer 1989; Klypin, Zhoa \& Somerville 2002; Widrow, Perrett \& Suyu 2003; Geehan et al.\ 2006). Recently, a Spitzer 3.6 $\mu$m mosaic image of M31 has become available (Barmby et al.\ 2006). This allows for a more accurate determination of the baryonic mass distribution in M31 than has been determined previously. Accurate mass modelling of M31 is an important tool for both exploring our ideas of cosmology, and for understanding the dynamics of newly discovered Local Group dwarf galaxies (e.g., Majewski et al.\ 2007; Chapman et al.\ 2007). Because M31 can be studied in exquisite detail, it provides a crucial testing ground for our ideas of galaxy formation (Kent 1989; Evans \& Wilkinson 2000). A problem of particular relevance for Cosmological constant plus Cold Dark Matter ($\Lambda$CDM) cosmology is the Tully-Fisher zero-point problem, which refers to the fact that standard models cannot reproduce the relation between galaxy luminosity and circular velocity (Tully \& Fisher 1977) without over-producing the number density of galaxies at fixed luminosity (e.g.\ Gonzalez et al.\ 2000; Cole et al.\ 2000; Benson et al.\ 2003; Yang, Mo \& van den Bosch 2003). Another problem of related concern is the cusp/concentration problem -- namely that dark-matter dominated galaxy rotation curves seem to rise more slowly than predicted in $\Lambda$CDM (Moore 1994; Flores \& Primack 1994; Moore et al.\ 1999; van den Bosch \& Swaters 2001; Blais-Ouellette, Amram \& Carignan 2001; Alam, Bullock \& Weinberg 2002; Swaters et al.\ 2003; Simon et al.\ 2005; Dutton et al.\ 2005; Kuzio de Naray et al.\ 2006). It remains to be seen whether these problems point to some unaccounted for process in galaxy formation (e.g.\ Dutton et al.\ 2007) or to some new physics of cosmological relevance (Kaplinghat, Knox \& Turner 2000; Zentner \& Bullock 2002; Kaplinghat 2005; Cembranos et al.\ 2005; Strigari, Kaplinghat \& Bullock 2007; Gnedin et al.\ 2007). Recently, Dutton et al.\ (2007) and Gnedin et al.\ (2007) revisited the Tully-Fisher problem using two large well-defined samples of disc-dominated (late-type) galaxies. They both took as a starting point the ``standard'' model of disc formation, which assumes that initial Navarro, Frenk \& White (1997; hereafter NFW) dark haloes respond to disc formation via adiabatic contraction (AC) (Blumenthal et al.\ 1986; Gnedin et al.\ 2004; Sellwood \& McGaugh 2005; Choi et al.\ 2006). Both groups conclude that the Tully-Fisher zero point cannot be explained if initial NFW haloes have concentrations as high as those expected for $\Lambda$CDM with $\sigma_8 \simeq 0.9$ (with $c \sim 12$ for M31-size haloes, e.g.\ Bullock et al.\ 2001a, Macci\`o et al.\ 2006). Both sets of authors agree that the problem could be alleviated if AC did not operate (e.g.\ Somerville \& Primack 1999) and Dutton et al.\ (2007) go on to advocate a model where disc formation induces an {\em expansion} in the underlying halo density structure. In contrast, Gnedin et al.\ (2007) argue that AC is a fundamental prediction in galaxy formation and instead suggest that the cosmology be changed to favor lower halo concentrations (e.g. Zentner \& Bullock 2002) or that the IMF is lighter (with a stellar mass-to-light ratio $M_*/L$ lower by 0.15 dex) than the standard Kroupa (2001) assumption. Given the fundamental issues at hand, the question of whether AC occurs in nature is of significant interest. Theory certainly favours the idea that haloes contract. Indeed, halo contraction must occur when the infall of baryons is smooth and adiabatic (e.g.\ Blumenthal et al.\ 1986; Ryden \& Gunn 1987; Sellwood \& McGaugh 2005) and simulations suggest that dark haloes will contract even when galaxies or galaxy clusters form quickly from an irregular collapse (Barnes 1987; Flores et al.\ 1993; Jesseit, Naab \& Burkert 2002; Gnedin et al.\ 2004; Sellwood \& McGaugh 2005; Choi et al.\ 2006; Weinberg et al.\ 2008). On the other hand, there is very little observational evidence that halo contraction actually occurs in nature. For example, Zappacosta et al.\ (2006) performed a detailed XMM study of the radio-quiet galaxy cluster Abell 2589 and conclude that an NFW halo + AC model cannot explain the data but that a pure NFW halo provides a remarkable fit down to $\sim 1 \%$ of the halo's virial radius. Similarly, an investigation of seven elliptical galaxies with {\em Chandra} by the same group (Humphrey et al.\ 2006) finds that AC degrades the mass profile fits significantly unless strong deviations from a Kroupa IMF are allowed. Also, Kassin, de Jong \& Pogge (2006a) and Kassin, de Jong \& Weiner (2006b) found that the rotation curves of 32 out of 34 of the bright spiral galaxies they study are better fit without adiabatic contraction. Klypin et al.\ (2002) have shown that the observed rotation curve of M31 is best explained with a halo model that includes AC. However, other recent M31 mass modelling papers (e.g., Widrow et al.\ 2003; Geehan et al.\ 2006) have not directly addressed the question of AC. Furthermore, M31 is essentially the only well-studied galaxy that has been shown to require AC (Klypin et al.\ 2002). Most other dynamical studies of galaxies tend to disfavour AC (e.g., Kassin et al.\ 2006a, b). Given the availability of a Spitzer 3.6 $\mu$m image, it is now appropriate to revisit this question and test the conclusion that M31 requires AC. In what follows we show that the rotation curve of M31 can only be explained without AC if an unusually high NFW concentration, $c_{\rm vir} \sim 50$, is adopted. Such a concentration is well outside the range expected in $\Lambda$CDM haloes, and therefore this model is disfavoured. Moreover, the best-fit AC model has a slightly high NFW concentration, $c_{\rm vir}=20$, for typical, $\sigma_8= 0.9$, $\Lambda$CDM haloes (Bullock et al.\ 2001a, Macci\`o et al.\ 2006), but it is within 1.5$\sigma$ of the expected value for an M31 size halo. We speculate on the implications of these results in the conclusion section. Our approach is unique compared to other recently published mass profiles derived from modelling the rotation curve of M31. While our method bears most similarity to that adopted by Klypin et al.\ (2002), we use an updated baryonic mass component, which is derived using a 3.6 $\mu$m Spitzer image. Geehan et al.\ (2006) used a pure NFW halo profile, and did not consider AC. Tempel et al.\ (2007) also use the 3.6 $\mu$m Spitzer image, yet once again, AC is not considered. Lastly, Widrow et al.\ (2003) used a different functional form for the halo density profile (see Evans 1993). The Evans profile is based upon an isothermal distribution function. Unlike the NFW profile, the Evans profile was not derived by fitting models to dark matter haloes produced in $\Lambda$CDM simulations. Widrow et al.\ (2003) then go on to fit an NFW model to their best-fitting Evans (1993) model. However, we have chosen to use the cosmologically motivated NFW halo density profile model directly. Throughout this paper we assume a flat $\Lambda$CDM cosmology with $\Omega_m=0.27$ and a Hubble constant of $H_0=75$ km s$^{-1}$ Mpc$^{-1}$. We relate virial masses ($M_{\rm vir}$) and radii ($R_{\rm vir}$) assuming a virial over-density relative to {\em average} of $\Delta_{\rm vir}=347$\footnote{Specifically we use $R_{\rm vir}= 206h^{-1} (M_{\rm vir}/10^{12} h^{-1} {\rm M}_{\odot})^{1/3}$ kpc.}. At times we quote baryon fractions relative to the universal value with $f_b=\Omega_b/\Omega_m=0.16$. We adopt a distance of 784 kpc to M31 (Holland 1998). As a result, an angular distance of $1^{\prime}$ is equivalent to a distance of 228 pc. \section{Data} \label{RSS} We make use of the Spitzer/IRAC 3.6 $\mu$m image of M31 (Barmby et al.\ 2006). The IRAC observations of M31 were taken as part of {\em Spitzer} General Observer program 3126 in 2005 August. Fifteen Astronomical Observation Requests (AORs) were used to map a region approximately $3\fdg7\times1\fdg6$ on the sky. The central $1\fdg6\times0\fdg4$ was covered by three AORs, each having two 12 s frames per position. The outer regions were covered by two AORs, each with two dithered 30 s frames per position. The pixel scale of IRAC at 3.6 $\mu$m is 1.221 arcsec, with a point spread function (PSF) that has a FWHM $\sim$1.2 arcsec. The 3.6 $\mu$m filter is 1.2 $\mu$m broad, spanning the wavelengths 2.965--4.165 $\mu$m. For a description of the IRAC instrument see Fazio et al.\ (2004). The Spitzer data are preferred over 2MASS $K_s$-band data for a few reasons. The regular 2MASS $K_s$-band image is simply not deep enough to obtain an accurate bulge/disc decomposition. Furthermore, this same 2MASS image suffers from over subtraction of the sky by about 0.5 magnitudes (T.\ Jarrett, private communication). A 6$\times$ 2MASS $K_s$-band image is also available (where the data were observed with an integration time 6 times longer than the regular 2MASS survey), but it is only about 1 magnitude deeper and also hampered by uncertain sky background. The Spitzer image has a larger field of view, better depth, and a more accurate determination of the sky level. We also use the $B-R$ colour profile of Walterbos \& Kennicutt (1987), in order to determine the stellar mass-to-light ratio as a function of radius. We have corrected this $B-R$ profile has for Galactic foreground extinction, which is $E(B-V)=0.062$ mags in the direction of M31 (Schlegel, Finkbeiner \& Davis 1998). This is converted to a correction for the $B-R$ colour by using the transformation, $E(B-R)=1.64E(B-V)$ from Schlegel et al.\ (1998), i.e.\ $E(B-R)=0.102$ for M31. As a comparison the amount of extinction at 3.6 $\mu$m is 1.5 per cent of that at $B$, and about 34 per cent of that in the $K_s$ band. We adopt the rotation curve data from several sources. In one case (M1) we adopt the H$\alpha$ rotation data out to 25 kpc from Rubin \& Ford (1970)\footnote{Figure 9 of Rubin \& Ford (1970) also includes rotation velocities calculated from a narrow [N{\tt II}] $\lambda$6583 emission feature, for the region within 3 kpc. However, these velocities are not tabulated in their paper, and so we do not include them here.} and extend the rotation curve to 35 kpc using H{\tt I} data from Carignan et al.\ (2006). In another case (M2) we use the CO rotational velocities from Loinard, Allen \& Lequeux (1995) and the H{\tt I} from Brinks \& Burton (1984) to construct an observed rotation curve out to a 30 kpc radius. This was the rotation curve adopted by Klypin et al.\ (2002) in their M31 model. We adopt M1 as our fiducial case here because Rubin \& Ford published errors on their observed velocities. However, as we show below, both cases yield consistent results. In both cases we take into account the presence of a $10^{8} M_{\odot}$ supermassive black hole (Bender et al.\ 2005) and the H{\tt I} gas distribution (Carignan et al.\ 2006). \section{Mass modelling of M31} \label{MM} \subsection{The baryonic contribution} \label{bruzual} Our goal is to determine the best possible mass model for M31. We perform a bulge-disc decomposition in order to estimate the baryonic contribution to the rotation curve. We then determine several possible mass models. The best fit model is determined by minimizing the reduced-$\chi^2$ in a fit to the observed rotation curve. \begin{figure} \special{psfile=f1.eps hscale=45 vscale=45 hoffset=-10 voffset=-305 angle=0} \vspace*{8.5cm} \caption{The Spitzer 3.6 $\mu$m surface brightness profile of M31 (without correction for inclination) with decomposition into bulge and disc components. The bulge has been fitted with a S\'ersic model (short dashed line) and the disc has been fitted with an exponential model (long dashed line).} \label{SB} \end{figure} \begin{figure*} \special{psfile=f2.eps hscale=85 vscale=85 hoffset=0 voffset=-380 angle=0} \vspace*{8.5cm} \caption{$K-$[3.6] colour as a function of $B-R$ colour for Bruzual \& Charlot (2003) population synthesis models ({\em Left Panel}) and Maraston (2005) population synthesis models ({\em Right Panel}). In both cases, dashed lines represent a constant age stellar population and solid lines represent constant metallicity. The dotted lines represent the typical $B-R$ colour of the inner disc ($B-R=1.79$) and the outer disc ($B-R=1.40$) from Walterbos \& Kennicutt (1987).} \label{synphot} \end{figure*} We first extract the surface brightness profile of M31 using the Spitzer 3.6 $\mu$m image and the IRAF {\tt ellipse} routine, which fits ellipses to an image using an iterative method described by Jedrzejewski (1987). In order to mask out foreground stars and satellite galaxies SExtractor (Bertin \& Arnouts 1996) was used. An inclination correction was then applied to the surface brightness profile (see de Jong 1996; Seigar \& James 1998a) as follows \begin{equation} \mu_i = \mu - 2.5C\log{\left(\frac{a}{b}\right)} \end{equation} where $\mu_i$ is the surface brightness when viewed at some inclination, $i$, $\mu$ is the corrected surface brightness, $a$ is the major axis, $b$ is the minor axis and $C$ is a factor dependent on whether the galaxy is optically thick or thin; if $C=1$ then the galaxy is optically thin; if $C=0$ then the galaxy is optically thick. Since we are using infrared data, the correction for inclination is small, and in many cases completely ignored (see e.g.\ Seigar \& James 1998a; de Jong 1996). However, Graham (2001) showed that $C=0.91$ is a good typical value to use for the near-infrared $K_s$-band. If one adopts a simple reddening law, where extinction falls as the square of wavelength, then the value of $C=0.97$ is appropriate at 3.6 $\mu$m, and we adopt this value here. The resulting surface brightness profile (Figure 1) reaches a surface brightness of $\mu_{3.6} \sim 21.2$ mag arcsec$^{-2}$ at a radius of $1\fdg5$. This is very similar to the surface brightness found by Barmby et al.\ (2006). Furthermore, the PNe counts of Merrett et al.\ (2006) predict an R band surface brightness of $\sim 23.8$ mag arcsec$^{-2}$. At these radii the metallicity of M31 is $0.4 Z_{\odot}$ (Brewer, Richer \& Crabtree 1995; Worthey et al.\ 2005). Using the Maraston (2005) population synthesis codes, one can see that the expected $R-$[3.6] color at this radius is $R-$[3.6]$\sim$2.5, very similar to our measured surface brightness. From this surface brightness profile, we perform a one-dimensional bulge-disc decomposition, which employs the S\'ersic model for the bulge component and an exponential law for the disc component (e.g.\ Andredakis et al.\ 1995; Seigar \& James 1998a; Khosroshahi, Wadadekar \& Kembhavi 2000; D'Onofrio 2001; Graham 2001b; M\"ollenhoff \& Heidt 2001; see also Graham \& Driver 2005 for a review). The S\'ersic (1963, 1968) $R^{1/n}$ model is most commonly expressed as a surface brightness profile, such that \begin{equation} \mu(R)=\mu_{e}\exp{\left\{-b_{n}\left[\left(\frac{R}{R_{e}}\right)^{1/n}-1\right]\right\}}, \end{equation} where $\mu_{e}$ is the surface brightness at the effective radius $R_{e}$ that encloses half of the total light from the model (Ciotti 1991; Caon, Capaccioli \& D'Onofrio 1993). The constant $b_{n}$ is defined in terms of the parameter $n$, which describes the overall shape of the light profile. When $n=4$ the S\'ersic model is equivalent to a de Vaucouleurs (1948, 1959) $R^{1/4}$ model and when $n=1$ it is equivalent to an exponential model. The parameter $b_{n}$ has been approximated by $b_{n}=1.9992n-0.3271$, for $0.5<n<10$ (Capaccioli 1989; Prugniel \& Simien 1997). The exponential model for the disc surface brightness profile can be written as follows, \begin{equation} \mu(R)=\mu_{0}\exp{(-R/h)}, \end{equation} where $\mu_{0}$ is the disc central surface brightness and $h$ is the disc scale length. Our bulge-disc decomposition ignores the inner $4^{\prime\prime}$ of M31, which is dominated by an independent nuclear feature (Light, Danielson \& Schwarschild 1974). The results of the bulge-disc decomposition can be seen in Figure \ref{SB}. Note that although there appears to be a slight break in the disc surface brightness profile at $\sim$4000$^{\prime\prime}$, the residuals are still very small. This break may appear due to the presence of spiral structure, evident at similar radii in the 24 $\mu$m of M31. Observational properties of M31 are derived from this bulge-disc decomposition and these are listed in Table 1. \begin{table*} \label{BDtab} \caption{M31 Observational Data. Hubble type taken from de Vaucouleurs et al.\ (1991). Distance in kpc taken from Holland et al.\ (1998).} \begin{tabular}{ll} \hline Parameter & Measurement \\ \hline Hubble type & SA(s)b \\ Distance (kpc) & 784 \\ Position Angle of major axis (degrees) & 45 \\ Bulge effective radius, $R_e$ (arcmin) & 8.45$\pm$0.43 \\ Bulge effective radius, $R_e$ (kpc) & 1.93$\pm$0.10 \\ Bulge surface brightness at the effective radius, $\mu_e$ (3.6 $\mu$m-mag arcsec$^{-2}$) & 16.42$\pm$0.83 \\ Bulge S\'ersic index, $n$ & 1.71$\pm$0.11 \\ Disc central surface brightness, $\mu_0$ (3.6 $\mu$m-mag arcsec$^{-2}$) & 17.35$\pm$0.93 \\ Inclination corrected disc central surface brightness, $\mu_{0_{i}}$ (3.6 $\mu$m-mag arcsec$^{-2}$) & 18.94$\pm$0.93 \\ Disc scalelength, $h$ (arcmin) & 25.92$\pm$1.28 \\ Disc scalelength, $h$ (kpc) & 5.91$\pm$0.27 \\ Disc luminosity, $L_{disc}$ ($L_{\odot}$) & (6.08$\pm$0.60)$\times$10$^{10}$ \\ Bulge-to-disc ratio, $B/D$ & 0.57$\pm$0.02 \\ \hline \end{tabular} \end{table*} \begin{figure} \special{psfile=f3.eps hscale=40 vscale=40 hoffset=0 voffset=-280 angle=0} \vspace*{8cm} \caption{$B-R$ colour profile for M31 from Walterbos \& Kennicutt (1987) corrected for foreground Galactic extinction.} \label{colour} \end{figure} With a disc scalelength of $25.92^{\prime}$ and a surface brightness profile extending to a radius of $1\fdg5$, we are tracing the disc profile out to $\sim$3.5 scalelengths. Assuming that the exponential disc profile extrapolates to infinity, our profile includes 86.1 per cent of the total disc light. Below, where we assign a mass to the disc, we extrapolate the disc profile to 6 scalelengths, which would contain 98.2 per cent of the disc light. Since Ibata et al.\ (2005) have shown that the disc of M31 extends to very large radii ($\sim$70 kpc) with the same scalelength, our above calculation assumes no disc truncation. Both the 10 kpc ring and the 2.6 kpc bar (Beaton et al.\ 2007) show up as small perturbations in the surface brightness profile (Fig.\ 1) at radii of 46.67$^{\prime}$ and 26.67$^{\prime}$ respectively. We now assign masses to the disc and bulge of M31. The stellar mass-to-light ratio in the $K_s$-band is a well calibrated quantity (see Bell et al.\ 2003) which depends on $B-R$ colour. However, a similar calibration does not exist for the Spitzer 3.6 $\mu$m waveband. An alternative method would be to estimate the $K_s$-band surface brightness profile, given the 3.6 $\mu$m profile, and since we know the $B-R$ colour profile, we have chosen to determine the $K-$[3.6] colour dependence on $B-R$ colour using both the Bruzual \& Charlot (2003; hereafter BC03) stellar population synthesis models and the Maraston (2005; hereafter M05) stellar population synthesis models. The BC03 models use the stellar library of Pickles (1998), which includes stars in a wide range of spectral types (05-M10) and luminosity classes (I-V) and three metallicity groups (metal poor, metal rich and solar). The Pickles (1998) library has wavelength coverage from 0.1 $\mu$m to 2.5 $\mu$m. BaSeL 3.1 spectra (see BC03 Tables 2 and 3) are then used to extend the Pickles (1998) spectra into the infrared from 2.5 $\mu$m to 160 $\mu$m. The spectra of M giant stars are the only ones not observed in the Pickles (1998) library. As a result the BC03 models use synthetic M giant spectra calculated by Fluks et al.\ (1994). These spectra are extended into the infrared using the model atmospheres of Schultheis et al.\ (1999), which cover a spectral range from 0.5 $\mu$m to 10 $\mu$m, and 10 equally spaced stellar temperatures in the range $2600<T_{\rm eff}<4400$ K. The main difference between the BC03 and the M05 models is the way in which thermally-pulsing AGB (TP-AGB) stars are treated. The M05 models provide a better treatment of TP-AGB stars than the BC03 models. Furthermore, Maraston et al.\ (2006) find that in general the M05 models provide better fits to observations of infrared spectra of galaxies than the BC03 models. They indicate systematically lower ages and, on average, masses that are 60 per cent lower for the stellar population samples in these galaxies. Figure \ref{synphot} shows the result of both the BC03 ({\em left panel}) and M05 ({\em right panel}) models. In both cases, the dotted lines in Figure \ref{synphot} represent the $B-R$ colour of the inner and outer disc of M31 (Walterbos \& Kennicutt 1987 and Figure \ref{colour}) after correction for Galactic foreground extinction. Since the M05 models provide an updated treatment of TP-AGB stars and generally provide better fits to observations than the BC03 models, from here on we concentrate on the M05 results. However, it is interesting to note that the $K-$[3.6] colour index predicted by the BC03 and M05 models differs by $<$0.03 mag (see Figure 2). Lines of constant metallicity (solid lines) and age (dashed lines) are shown. The typical metallicity in the inner few kpc of M31 is $\sim$2.4 $Z_{\odot}$, whereas in the outer disc this falls to $\sim$0.4 $Z_{\odot}$ (Brewer, Richer \& Crabtree 1995; Worthey et al.\ 2005). From this metallicity gradient and the M05 models (Figure \ref{synphot} {\em right panel}) it can be seen that the $B-R$ colour gradient in M31 is almost entirely due to changes in metallicity, and that the $K-$[3.6] colour difference is $<0.1$ mag. Furthermore, we can now use the $B-R$ gradient (Figure \ref{colour}) and the metallicity gradient of M31 to estimate the $K-$[3.6] colour as a function of radius. We find that the inner region of M31 has a colour $K-$[3.6]$\simeq 0.25$, and the outer disc (at 26 kpc) has a colour $K-$[3.6]$\sim 0.18$. Given the $B-R$ colour profile and the K-band mass-to-light ratio calibration from Bell et al. (2003) we calculate a central mass-to-light ratio, $M/L_K=1.0\pm0.1$ and a mass-to-light ratio at 26 kpc, $M/L_K=0.75\pm0.1$. From the $K-$[3.6] colours determined in our stellar population synthesis analysis, the above mass-to-light ratios are equivalent to 3.6 $\mu$m mass-to-light ratios of $M/L_{3.6}\simeq 1.25\pm0.10$ in the centre and $M/L_{3.6}\simeq 0.89\pm0.10$ at 26 kpc (equivalent to a $M/L$ gradient 0.13 kpc$^{-1}$). In our analysis, we prefer to keep the central $(M/L_{3.6})$ value a free parameter, and as such we allow it to vary in the range $0.05<(M/L_{3.6})<1.5$. This is a standard approach to adopt in mass modelling of galaxies (see Humphrey et al.\ 2006; Zappacosta et al.\ 2006) and as discussed at the end of section 3.2 and shown in Table 2, when the $M/L_{3.6}$ is left as a free parameter, the best-fit value found is $M/L_{3.6}=1.15$, i.e.\ consistent with the value of $M/L_{3.6}$ derived here. We use our derived value for the 3.6 $\mu$m disc luminosity of $L_{3.6}=(6.08\pm0.60)\times10^{10}$ $L_{\odot}$. We then use the derived disc and bulge light profiles $L_{3.6}=L_{\rm disc}+L_{\rm bulge}$ to determine the stellar mass contribution to the rotation curve, $M_*=(M/L_{3.6})L_{3.6}$. A concern in using the 3.6 $\mu$m Spitzer waveband to determine the underlying stellar mass, is the effect of emission from hot dust in this waveband, although this is probably only important in or near H{\tt II} regions. In order to place some constraint on this, we have chosen to explore the emission from dust in the the near-infared $K$-band at 2.2 $\mu$m. Using near-infrared spectroscopy at 2.2 $\mu$m it has been shown that hot dust can account for up to 30 per cent of the continuum light observed at this wavelength in areas of active star formation, i.e., spiral arms (James \& Seigar 1999). When averaged over the entire disc of a galaxy, this reduces to a $<$2 per cent effect, if one assumes that spiral arms can be up to 12$^{\circ}$ in width. At 3.6 $\mu$m this would therefore result in $<$3 per cent of emitted light from dust. Another concern for the 3.6 $\mu$m waveband, would be the contribution from the Polycyclic Aromatic Hydrocarbon (PAH) emission feature at 3.3 $\mu$m. However, ISO spectra covering the wavelength range 5.15 to 16.5 $\mu$m have shown that PAH emission in M31 seems to be very weak (e.g., Cesarsky et al.\ 1998). Furthermore, in a survey of actively star forming galaxies, Helou et al.\ (2000) found that the 3.3 $\mu$m feature was also very weak when they analysed the average $2.5-11.6$ $\mu$m spectrum of 45 galaxies. The contribution of the PAH feature to the 3.6 $\mu$m Spitzer waveband, is therefore not a major concern. In the following section, where we model the dark matter halo, we have included a bulge surface brightness model using a deprojected S\'ersic function. The function we use is from Trujillo et al.\ (2002) who show that the deprojected mass density profile is given by, \begin{equation} \rho(R)=\frac{f^{1/2}I(0)b_{n}2^{(n-1)/2n}}{R_{e}n\pi}\Upsilon\frac{h^{p(1/n-1)}K_{\nu}(b_{n}h^{1/n})}{1-C(h)} \label{dep} \end{equation} where $R_e$ is the effective radius, $I(0)$ is the central intensity, $\Upsilon$ is the stellar mass-to-light ratio, $h=R/R_e$, $C(h)=h_1(\log{h})^{2}+h_2\log{h}+h_3$ and $K_{\nu}$ is the $\nu^{\rm th}$-order modified Bessel function of the third kind. Trujillo et al.\ (2002) give values for $\nu$, $p$, $h_1$, $h_2$ and $h_3$ for different values of the S\'ersic parameter, $n$. In our case, where $n=1.71$, we adopt values of $\nu=0.45394$, $p=0.67930$, $h_1=-0.0162935$, $h_2=-0.18161$ and $h_3=0.04435$. Finally, the constant $f^{1/2}$ depends on the three-dimensional spatial orientation of the bulge. In our case, we assume the bulge is intrinsically spherical, and therefore $f^{1/2}=1$. It should be noted that this version of the deprojected S\'ersic function (equation \ref{dep}) is an approximation, but it is good enough for our purposes. \subsection{Modelling the dark matter halo} \label{halo} A range of allowed dark matter halo masses and density profiles are now explored, using two models for disc galaxy formation. In the first we assume that the dark matter halo surrounding M31 has not responded significantly to the formation of the disc, i.e., adiabatic contraction does not occur (the ``no-AC'' model). In this case the dark matter contribution to the rotation curve is described by a density profile similar to those found in dissipationless dark matter simulations (the NFW profile): \begin{equation} \rho(R)=\frac{\rho_{s}}{(R/R_{s})(1+R/R_{s})^{2}} \end{equation} where $R_s$ is a characteristic ``inner'' radius, and $\rho_s$ is a corresponding inner density. This is a two parameter function and is completely specified by choosing two independent parameters, e.g., the virial mass $M_{\rm vir}$ (or virial radius $R_{\rm vir}$) and concentration $c_{\rm vir}=R_{\rm vir}/R_s$ (see Bullock et al.\ 2001a for a discussion). Similarly, given a virial mass $M_{\rm vir}$ and the dark matter circular velocity at any radius, the halo concentration $c_{\rm vir}$ is completely determined. In the second class of models we adopt the scenario of adiabatic contraction (AC) and specifically use the original scheme discussed by Blumenthal et al.\ (1986, hereafter B86; see also Bullock et al.\ 2001b and Pizagno et al.\ 2005). We also investigate the slightly revised scheme presented in Gnedin et al.\ (2004, hereafter G04). As we discuss below, the G04 scheme reproduces the outer slope of the observed rotation curve of M31, but needs a very high concentration to do so. \begin{figure*} \special{psfile=f4.eps hscale=110 vscale=110 hoffset=20 voffset=-710 angle=0} \vspace*{18.5cm} \caption{{\em Top left}: Rotation curve data (solid square points) from Rubin \& Ford (1970) with the best fit model, with adiabatic contraction (M1 B86 model), overlaid (solid line). For comparison the H{\tt I} rotation velocities (grey open triangles) from Carignan et al.\ (2006) are shown (the black point corresponds to the average velocity and radius of the Carignan data that we make use of). The best fit rotation curve is decomposed into four components, the contribution from dark matter (short-dashed line), the contribution from baryonic matter (long-dashed line), the H{\tt I} contribution (dotted line) from Carignan et al.\ (2006) and the contribution from the central $10^{8}M_{\odot}$ black hole (dot-dashed line) from Bender et al.\ (2005). {\em Middle left}: Rotation curve CO and H{\tt I} data (solid square points) from Loinard et al.\ (1995) and Brinks \& Burton (1984) respectively, as used by Klypin et al.\ (2002). The best fit model with adiabatic contraction (M2 B86 model) is overlaid (solid line). It is decomposed into four components as in the top left panel. {\em Bottom left}: Rotation curve data from Rubin \& Ford (1970; solid points) and Carignan et al.\ (2006; triangles) and the data as used by Klypin et al.\ (2002; hollow circles), with adiabatic contraction model (M1+M2 B86), overlaid. {\em Top right}: The same as the top left panel, but with the best fitting model using a modified version of adiabatic contraction (M1 G04 model) as in Gnedin et al.\ (2004). The long-dashed curve includes the total baryonic mass, including the H{\tt I} mass and the central black hole. {\em Middle right}: The same as the top left panel, but with the best fitting model without an adiabatically contracted halo (M1 no-AC model). In all five cases the NFW concentration, $c_{\rm vir}$, and virial mass, $M_{\rm vir}$, are shown.} \label{rcurve} \end{figure*} \begin{table*} \label{modTab} \caption{M31 best fitting models. ``M1 B86'' is the best-fit model to the H$\alpha$ rotation curve from Rubin \& Ford (1970) (M1) using the AC prescription of B86. ``M2 B86'' is the best-fit B86 model to a combination of the CO rotation velocities from Loinard et al.\ (1995) and the H{\tt I} velocities from Brinks \& Burton (1984) (M2). ``M1+M2 B86'' is the best-fit B86 model to the H$\alpha$ rotation curve from Rubin \& Ford (1970) (M1) combined with the CO rotation velocities from Loinard et al.\ (1995) and the H{\tt I} velocities from Brinks \& Burton (1984) (M2). ``M1 G04'' corresponds to the M1 data fit using the modified adiabatic contraction model of Gnedin et al.\ (2004) and ``M1 no-AC'', uses the same data without adiabatic contraction.} \begin{tabular}{lccccc} \hline Parameter & M1 & M2 & M1+M2 & M1 & M1 \\ & B86 & B86 & B86 & G04 & no-AC \\ \hline Shear & 0.51 & 0.50 & 0.51 & 0.54 & 0.50 \\ \\ $V_{2.2}$ (km s$^{-1}$) & 275$\pm$5 & 275$\pm$5 & 275$\pm$5 & 275$\pm$5 & 275$\pm$5 \\ \\ Disc luminosity, $L_{disc}$ ($L_{\odot}$) & $(4.9\pm0.6)\times 10^{10}$ & $(6.1\pm0.6)\times 10^{10}$ & $(5.5\pm0.6)\times 10^{10}$ & $(4.9\pm0.6)\times 10^{10}$ & $(6.1\pm0.6)\times 10^{10}$\\ \\ Disc scale length, $h$ (kpc) & 5.09$\pm$0.06 & 5.91$\pm$0.06 & 5.50$\pm$0.06 & 5.50$\pm$0.06 & 5.91$\pm$0.06 \\ \\ Central disc mass-to-light ratio, & 1.15$\pm$0.10 & 1.00$\pm$0.10 & 1.05$\pm$0.10 & 1.05$\pm$0.10 & 0.95$\pm$0.10 \\ $M/L$ (3.6 $\mu$m solar units)\\ \\ Mass-to-light ratio gradient, & 0.013 & 0.013 & 0.013 & 0.013 & 0.013 \\ $d(M/L)/dR$ (kpc$^{-1}$)\\ \\ Initial NFW concentration, $c_{\rm vir}$ & 20.0$\pm$1.1 & 16.0$\pm$1.1 & 18.0$\pm$1.1 & 40.0$\pm$1.1 & 51.0$\pm$1.1 \\ \\ Bulge-to-disc ratio, $B/D$ & 0.57$\pm$0.02 & 0.57$\pm$0.02 & 0.57$\pm$0.02 & 0.57$\pm$0.02 & 0.57$\pm$0.02 \\ \\ Virial mass, $M_{\rm vir}$ ($M_{\odot}$) & $(8.2\pm0.2)\times10^{11}$ & $(8.5\pm0.2)\times10^{11}$ & $(8.4\pm0.2)\times10^{11}$ & $(8.9\pm0.2)\times10^{11}$ & $(7.3\pm0.2)\times10^{11}$\\ \\ Dark matter concentration, & 14.9$\pm$1.4\% & 14.4$\pm$1.5\% & 14.7$\pm$1.5\% & 9.9$\pm$1.3\% & 14.1$\pm$2.0\% \\ $c_{\rm DM}$, $R<10$ kpc \\ \\ Total mass concentration, & 26.2$\pm$2.5\% & 25.4$\pm$2.4\% & 25.9$\pm$2.4\% & 23.8$\pm$2.7\% & 26.5$\pm$2.6\% \\ $c_{\rm tot}$, $R<10$ kpc\\ \\ Stellar baryon fraction, $f_{*}$ & 56.7$\pm$3.9\% & 59.2$\pm$4.5\% & 58.1$\pm$4.2\% & 56.3$\pm$6.0\% & 63.5$\pm$5.6\% \\ \\ $\chi^2$ & 38.75 & 40.63 & 80.31 & 46.50 & 41.58 \\ \\ Degrees of freedom, $\nu$ & 31 & 27 & 64 & 31 & 31 \\ \hline \end{tabular} \end{table*} For each AC algorithm (B86 and G04) and for the no-AC model we generate a grid of final rotation curves. We vary the baryonic contribution to the rotation curve by allowing the bulge-disc decomposition parameters for $h$ and $L_{\rm disc}$ to range over their $\pm 2\sigma$ values (see Table 1 and the previous section). The central mass-to-light ratio of the galaxy is allowed to vary over the range $0.05<(M/L)<1.5$. From this we determine the bulge and disc contribution to the rotation curve. For the bulge a simplified spherical model is used. For the disc the potential is derived assuming zero thickness and modified Bessel functions as described by Binney \& Tremaine (1987). For each set of baryonic parameters we generate a range of dark halo models with total circular velocities at $2.2$ disc scalelength ($V_{2.2}$) that span the best-fit value for M31 ($V_{2.2}=270$ km s$^{-1}$) quoted by Rubin \& Ford (1970) by a wide margin: $200<V_{2.2}<340$ km s$^{-1}$. In practice we achieve each $V_{2.2}$ value in our models by allowing the initial halo concentration to vary over $>4\sigma$ of its expected range $c_{\rm vir}=3-60$ (Bullock et al.\ 2001a; Wechsler et al.\ 2002; Macci\`o et al.\ 2006) and then setting the halo virial mass $M_{\rm vir}$ necessary to reproduce the desired value of $V_{2.2}$. Finally, we determine the implied fraction of the mass in the system in the form of stars compared to that expected from the Universal baryon fraction, $f_*=M_*/(f_{b}M_{\rm vir})$ and demand that $f_*$ obeys $0.01f_b<f_*<f_b$. From this grid of choices we derive best-fitting dark halo parameters by minimizing the reduced-$\chi^2$ for our two choices of rotation curve data. These results are summarized in Table 2. For the most part, when minimizing $\chi^2$, we assume that the errors on the observational data (i.e., the rotation velocities) are not correlated. Instead we assume that they are Gaussian in nature. However, this is not the case for the Carignan et al.\ (2006) H{\tt I} data (see Figure \ref{rcurve} top row and bottom right beyond 25 kpc), where the errors obviously are correlated. In order to account for the strange behaviour of the errors on these H{\tt I} rotation velocities we have treated all of this data as an average rotation velocity of 226.3 km s$^{-1}$ at 31.16 kpc. This is an idealized simplification but allows a quantitative estimate of the relative goodness of fit between models. Furthermore, although error bars are not available for the ``M2'' curve, we assume 10 per cent errors in the model fitting, as this is similar to the size of the errors quoted for the Rubin \& Ford (1970) H$\alpha$ rotation curve. We therefore quote a $\chi^2$ for the ``M2'' curve in Table 2 that is consistent with 10 per cent errors. We then determine how consistent all the derived rotation curves are with each other, by adopting the formal best fitting rotation curve as our fiducial mass model. For our ``M1'' curve, the number of degrees of freedom in the fit is $\nu=31$. The number of data points we fit to is $ n_{\rm data}=37$, and 6 parameters are determined from the fit, the NFW concentration $c_{\rm vir}$, the halo virial mass $ M_{\rm vir}$, the stellar mass-to-light ratio $M/L$, the disc luminosity $L_{\rm disc}$, the disc scalelength $h$ and the bulge-to-disc ratio $B/D$, thus leaving the degrees of freedom in the fit as $\nu=31$. The M1 rotation curve from Rubin \& Ford (1970) is shown by the solid squares with error bars in Figure \ref{rcurve} ({\em top left, top right, middle right, bottom left}) along with our best-fitting rotation curve models (solid lines). The contribution from stellar mass (short-dashed) and dark matter (long-dash) in each case is shown along with the rather minor contributions from neutral gas (dotted) and the central black hole (dot-dashed). Overlaid in the {\em top left} panel of Figure \ref{rcurve} is a model that includes adiabatic contraction (overall reduced-$\chi^2=1.25$). The dark halo in this case {\em has undergone} adiabatic contraction according to the B86 prescription. Our best-fitting parameters are summarized in Table 2 under ``M1 B86''. We determine the halo of M31 to have a total (dark+baryonic) virial mass of $M_{\rm vir}=(8.2\pm0.2)\times10^{11}M_{\odot}$ and an initial concentration $c_{\rm vir}=20.0\pm1.1$. Our derived virial mass is almost a factor of 2 smaller than that quoted by Klypin et al.\ (2002). We also find a concentration that is higher than both Klypin et al.\ (2002), who find $c_{\rm vir}=12$ and Widrow et al.\ (2003), who find $c_{\rm vir}=10$. In the case of Widrow et al.\ (2003) it should be noted that they initially fit an isothermal model given by Evans (1993). They then fit an NFW profile to this model. Given that the isothermal model favors a constant density core, the derived NFW concentration will be articifially low and fitting an NFW model in a more direct manner may result in a higher concentration. The model rotation curve fits the data adequately within the uncertainties. The innermost data point from the observational rotation curve is significantly lower than the model. This can be explained by the recent strong evidence for a bar, which dominates the near-infrared light out to a radius of $\sim$2.6 kpc, in M31 (Athanassoula \& Beaton 2006; Beaton et al.\ 2007). This bar has been used to explain the minimum seen in the observed rotation curve at a radius of $\sim2.7$ kpc (Athanassoula \& Beaton 2006). This bar may also affect the outer rotation velocities as well, although probably by less than 10 per cent. This would therefore be within the errors associated with the observed data. Indeed, shifting the outer rotation velocities systematically upward by $\sim$10 per cent changes the best-fit models by an amount that remains within the quoted error range of the original fit (see Table 2). It should also be noted that by a radius of $\sim26-35$ kpc the rotational velocity has dropped to $\sim 225-230$ km s$^{-1}$, which is similar to the rotational velocities seen at radii of $25-35$ kpc in the H{\tt I} rotation curve (Carignan et al.\ 2006; open triangles). \begin{figure} \special{psfile=f5.eps hscale=40 vscale=40 hoffset=0 voffset=-280 angle=0} \vspace*{8cm} \caption{Virial mass as a function of NFW concentration from our modelling with our 1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence contours shown as solid lines for our ``M1 B86'' model and as dashed lines for our ``M1 no-AC''.} \label{chi2} \end{figure} The best-fitting model without adiabatic contraction is shown by the solid line in the {\em middle right} panel of Figure \ref{rcurve}. This, the no-AC model, was determined in exactly the same way as our best-fit B86 AC model. As listed in Table 2, the no-AC assumption requires a much higher NFW concentration, $c_{\rm vir}=51.0\pm1.1$ and a lower virial mass, $M_{\rm vir}=(7.3\pm0.2)\times10^{11}M_{\odot}$. This model also produces a reasonable fit, with a reduced-$\chi^2=1.35$, but it should be noted that the required concentration is, at best, only marginally acceptable within a $\Lambda$CDM context. This would correspond to a $\sim 4.4 \sigma$ outlier from the expected concentration distribution of M31-sized haloes (e.g., Bullock et al.\ 2001a). In Figure \ref{chi2} we plot the NFW concentration, $c_{\rm vir}$, as a function of the virial mass, $M_{\rm vir}$, for both the ``M1 B86'' case ({\em solid line}) and the ``M1 no-AC'' case ({\em dashed line}). The contours represent the 1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence levels for the concentration and virial mass for both models. From figure \ref{chi2}, one can see that the ``M1 no-AC'' model is inconsistent with any reasonable concentration of $c_{\rm vir}<30$ at the 5$\sigma$ level at least. However, since it is possible to produce a reasonable fit with a pure NFW model, albeit with a very large concentration, we do not rule out our ``M1 no-AC'' case, although it is clearly not consistent with standard $\Lambda$CDM cosmology. The confidence contours shown in Figure \ref{chi2} have also allowed us to estimate errors on the derived best-fit masses and concentrations. The 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours are drawn as the boundaries where $\chi^2$ increases by 1.0, 2.71 and 6.63 relative to its minimum value. For our best-fit `M1 B86' model, $\chi^2$=38.75. There is also a `M1 B86' model for which the $\chi^2$=39.75, i.e.\ 1$\sigma$ higher than the best fit. For this model, the concentration is $c_{\rm vir}$=21.1 and the virial mass is $M_{\rm vir}=8.0\times10^{11} M_{\odot}$. There is a change in the concentration of $\Delta c_{\rm vir}$=1.1 and a change in the virial mass of $\Delta M_{\rm vir}=0.2\times10^{11} M_{\odot}$ at the 1$\sigma$ level, i.e. when the value of $\chi^2$ is increased by 1. The values of $\Delta c_{\rm vir}$ and $\Delta M_{\rm vir}$ are therefore adopted as our 1$\sigma$ error on $c_{\rm vir}$ and $M_{\rm vir}$ respectively, and these are the errors that we report in Table 2. An intermediate result is shown in the {\em top right} panel of Figure \ref{rcurve}. Here we use the AC model of G04, which produces less contraction than the B86 prescription. If the initial NFW concentration is held fixed at the same value as the best-fitting B86 case, i.e., $c_{\rm vir}=20$, then the G04 model overestimates the observed rotation curve at $r>15$ kpc. The G04 model can only reproduce this outer slope when the initial halo has $c_{\rm vir}=40$. With this high concentration the reduced-$\chi^2$ of the G04 model is 1.50. This model cannot be ruled out, but (as in the pure NFW case) its halo concentration is higher than expected in $\Lambda$CDM (Bullock et al.\ 2001a, Macci\`o et al.\ 2006). Therefore, from here on, we chose to focus on the B86 AC case, but provide the best-fitting model results for the G04 case in Table 2. For completeness, we also perform a fit to the same observed rotation curve that was used by Klypin et al.\ (2002) in their M31 mass decomposition. Our best fitting mass model is shown in the {\em middle left} panel of Figure \ref{rcurve} (``M2 B86'' in Table 2). Here we have adopted the B86 AC model. The initial NFW concentration in this case ($c_{\rm vir}=18$) is higher than that the value of $c_{\rm vir}=12$ favored by Klypin et al.\ (2002) and lower than our ``M1 B86'' case. However, it is not a large difference, and at face value our ``M1 B86'' and ``M2 B86'' models are consistent with each other. As we discuss below, the difference between our result and that of Klypin et al. (2002) is driven mainly by our updated baryonic model, which seems to provide a less concentrated bulge. For the M2 case, we calculate a $\chi^2$, assuming 10 per cent errors. As a final consistency check, we also perform a fit of the B86 AC model to a combination of the M1 and M2 data ({\em bottom left} of Figure \ref{rcurve}; ``M1+M2 B86'' in Table 2). This produces a result consistent with ``M1 B86'' and ``M2 B86''. \begin{figure} \special{psfile=f6.eps hscale=40 vscale=40 hoffset=0 voffset=-280 angle=0} \vspace*{8cm} \caption{Enclosed mass as a function of radius for model M1 B86. The solid line indicates the total mass. Also indicated are the HI mass out to a 35 kpc radius (dotted line), the bulge+BH mass (short-dashed line), the disc mass (long-dashed line) and the halo mass (dot-dashed line). The data points indicate masses derived using other observational methods at fixed radii of 10 kpc, 31 kpc, 35 kpc and 125 kpc from Rubin \& Ford (1970), Evans \& Wilkinson (2000), Carignan et al.\ (2006) and Fardal et al.\ (2006) respectively.} \label{mass} \end{figure} Figure \ref{mass} shows the enclosed mass as a function of radius for our fiducial model (M1 B86). The fraction of mass contained within the central 10 kpc (the ``central mass concentration'' hereafter, as originally defined by Seigar et al.\ 2006) is $c_{\rm tot}=26.2\pm2.5$ per cent. In absolute terms, the mass contained within 10 kpc is $(21.5\pm2.6)\times10^{10} M_{\odot}$. This is about a factor of 2 larger than that calculated by Rubin \& Ford (1970), but it should be noted that they did not take into account non-circular orbits due to the presence of a bar, which we now know exists in M31. This would have the effect of increasing the mass contained within the bar region. The mass calculated within a 35 kpc radius from the H{\tt I} rotation curve by Carignan et al.\ (2006) is $3.4 \times 10^{11} M_{\odot}$, similar to the mass derived from our best fit model within a 35 kpc radius of $(3.4\pm0.5)\times10^{11} M_{\odot}$. Within a 31 kpc radius we find a mass of $(3.1\pm0.4) \times 10^{11} M_{\odot}$, which is similar to the mass of $2.8 \times 10^{11} M_{\odot}$ within the same radius found using kinematic data of planetary nebulae (Evans \& Wilkinson 2000). The total implied stellar baryon fraction of this model is $f_{*}=56.7\pm3.9$ per cent. Other mass models of M31 (Klypin et al.\ 2002; Geehan et al.\ 2006; Widrow et al.\ 2003) have used bulge velocity dispersions as inputs into their models. For example, Klypin et al.\ (2002) used the line-of-sight velocity dispersion of M31 from Kormendy \& Bender (1999) to constrain mass models of M31 within the bulge region. In the bulge region, the velocity dispersion, $\sigma$ is dominant (c.\ 150 km s$^{-1}$) at least out to twice the effective radius, i.e.\ $2R_e=$3.96 kpc. Geehan et al.\ (2006) assume a Hernquist (1990) model for the bulge, and compare the expected velocity dispersion profile from Jeans equation modelling with the observations of Kormendy (1988). Our approach differs from this. We have chosen to assume a S\'ersic law for the Bulge and we then perform a mass-to-light conversion. In our approach, we essentially ignore any kinematics within the inner few kpc, because in that region the kinematics are likely to be perturbed by the bar or affected by streaming motions along the bar. Instead, we assume that stellar light traces stellar mass. Figure 4 shows that the bar kinematics are strongest within the inner $\sim$6 kpc. Beyond 10 kpc, the bar kinematics are not likely to affect the disc kinematics. Furthermore, we assume that the bulge is spherically symmetric (an assumption also made in the papers by Klypin, Geehan and Widrow). The bulge in our model has a mass of $M_{bulge}\simeq3.5\times10^{10}$ $M_{\odot}$. It is reassuring to note that the Jeans modelling of Geehan et al.\ (2006) derives a bulge mass of $M_{bulge}=3.2\times10^{10}$ $M_{\odot}$, very similar to our bulge mass. We note that the bulge component in our models is less concentrated than that derived by some other groups. Klypin et al.\ (2002) have a more concentrated bulge, but that is probably due to a combination of the fact that they use a Hernquist model (equivalent to a S\'ersic index of $n=4$) for the M31 bulge and optical data to determine the bulge characteristics. Geehan et al.\ (2006) also use a Hernquist model. This type of model will naturally produce a more concentrated bulge than a S\'ersic model with index $n\simeq1.7$. The use of the $M/L$ ratios from Bell \& de Jong (2001) also gives Klypin et al.\ (2002) an overall more massive bulge. The more recent models by Tamm et al.\ (2007) and Tempel et al.\ (2007) make use of a model that is similar to a S\'ersic law, but they find an index of $n\simeq3$ compared to our index of $n\simeq1.7$. As a result, their bulge model is also naturally more concentrated than ours. It is also interesting to note that replacing our S\'ersic bulge with a bulge profile modelled using a Hernquist or exponential law does not affect the overall dark matter halo profile by more than 10 per cent. This may be because the overall bulge mass does not change; It is just the way in which the bulge mass is distrbuted that changes. The issue of turbulent gas motions is very important for low-mass galaxies with small rotation velocities (e.g., Valenzuela et al.\ 2007), as this requires a correction for pressure support or asymmetric drift in the mass model. The correction to the total mass is $\sim(\sigma_{\rm gas}/V_{\rm rot})^2$. Since the H$\alpha$ and H{\tt I} rotation curves are measured from gas, it is important to understand what effect the turbulent gas motions may have on our mass model. Since, M31 is a galaxy with a high rotation velocity ($\sim 275$ km s$^{-1}$) and a turbulent velocity dispersion in galaxy discs is typically $<30$ km s$^{-1}$ (i.e.\ around 10 per cent of the rotation velocity) this effect is small. Indeed, if we assume that the scatter of the H$\alpha$ Rubin \& Ford (1970) rotation curve around our best fitting model is due to turbulent gas motions, then we can derive an estimate for their typical value in M31. We can do this because the Rubin \& Ford (1970) rotation curve is derived from H$\alpha$ spectroscopic observations of individual H{\tt II} regions in M31. In this case, we have to assume that the errors on the Rubin \& Ford (1970) rotation curve are uncorrelated, and we derive a turbulent velocity dispersion of $\sigma_{\rm gas}=18.3$ km s$^{-1}$. Adding this in quadrature to the maximum rotation velocity from the Rubin \& Ford (1970), results in a mass difference of $\sim$0.4 per cent for the {\em total} (baryonic + halo) galaxy mass. As mentioned above, the preferred total (dark+baryonic) virial mass for M31 is $(8.2\pm0.2) \times 10^{11} M_{\odot}$. This is similar to the estimate of $\sim (7.9\pm0.5) \times 10^{11} M_{\odot}$ derived from satellite kinematics (C\^ot\'e et al.\ 2000) and the total mass of $\sim 8 \times 10^{11} M_{\odot}$ from kinematics of the Andromeda Stream (Fardal et al.\ 2006; Geehan et al.\ 2006). It is also close to the lower limit of $9\times10^{11}M_{\odot}$ derived from kinematics of halo stars (Chapman et al.\ 2006). Our favored virial mass is about a factor of 2 smaller than the value favored by the Klypin et al.\ (2002) analysis of the M31 rotation curve. This can be attributed to three differences in our analysis. First, Klypin et al.\ (2002) use the H{\tt I} rotation curve of Brinks \& Burton (1984) to find their best fitting model. In this study we use the H$\alpha$ rotation velocities of Rubin \& Ford (1970). However, we find that this is not the major driver. We find that the model rotation velocities are consistent with those from the more recent H{\tt I} study of Carignan et al.\ (2006), and when we fit to the same observed rotation curve as Klypin et al.\ (2002), we find that our best-fit model is consistent with our best-fit model to the Rubin \& Ford (1970) curve. Furthermore, if we try and model a combination of data from Rubin \& Ford (1970) and Klypin et al.\ (2002) we also find a best-fitting rotation curve consistent with our fiducial case. More importantly, Klypin et al.\ (2002) used the $M/L$ ratios of Bell \& de Jong (2002) to determine the stellar mass from their bulge-disc decomposition. In this study we allow $M/L$ to vary freely in the range $0.05<(M/L)<1.50$, and find a best-fitting value close to the expected 3.6 $\mu$m mass-to-light ratios (see section \ref{bruzual}). Possibly the most significant difference is that Klypin et al.\ (2002) perform a bulge-disc decomposition using an $R$ band image, whereas we use a Spitzer 3.6 $\mu$m image. Using a 3.6 $\mu$m image provides a better trace of the stellar mass (e.g., Barmby et al.\ 2006 and references therein). Overall, we should therefore have a more accurate estimate of the baryonic mass profile in M31. Indeed, if we adopt the baryonic mass profile of Klypin et al.\ (2002) and try to fit either to the H{\tt I} or to the H$\alpha$ rotation curve, we find a best fitting model that is consistent with their result. Typically, the best-fitting $M/L$ values we find are in the range 0.95--1.15, depending on the model used. The lowest value of $M/L=0.95$ is difficult to reconcile with the expected value, which is $M/L_{3.6}=1.2\pm0.1$, for a galaxy with the central $B-R$ colour of M31. This expected value was determined by combining the Bell et al.\ (2003) $K_s$-band $M/L$ with the $K-$[3.6] colours derived from our stellar population synthesis described in section 3.1 (Figure \ref{synphot}). The lowest value we find ($M/L=0.95$) is for our no-AC model. We also find that this model is difficult to explain in the context of $\Lambda$CDM cosmology, due to its large concentration. Our M1 B86 model reproduced the highest value of $M/L=1.15$ that we find. This is consistent with the expected range of $M/L$ values. \section{Summary} In this paper we have derived new mass models for M31, based on an updated baryonic contribution to the rotation curve. The baryonic contribution is derived using a Spitzer 3.6 $\mu$m image. We find that both a pure NFW and an adiabatically contracted NFW profile can produce reasonable fits to the observed rotation curve of M31. However, the best-fit concentration of M31 for a pure NFW model is $c_{\rm vir}=51\pm1.1$, which is disfavoured when compared to the expected value of $\log_{10}(c_{\rm vir})=1.08\pm0.14$ (Wechsler et al.\ 2002; Maccio et al.\ 2006) in a $\Lambda$CDM cosmology (our best fitting concentration is a $4.4\sigma$ outlier). The best fit produced by the AC recipe of Gnedin et al.\ (2004) brings down the concentration to $c_{\rm vir}=40$, or $\log_{10}(c_{\rm vir})=1.60$, and is thus marginally consistent with $\Lambda$CDM as a $\sim$3.7$\sigma$ outlier. If one accepts $\Lambda$CDM cosmology, then the most consistent fit to the observed rotation curve seems to be that produced by the AC recipe of Blumenthal et al.\ (1986). In this case the concentration, $c_{\rm vir}=20\pm1.1$, and this is consistent with the expectations of $\Lambda$CDM, to within 1.5$\sigma$. For all of the types of model we use, the derived best-fits are in good agreement with the mass distributions derived from various other observational methods. Our estimate of the halo virial mass lies in the range $M_{\rm vir}=(7.3\pm0.2)\times10^{11} M_{\odot}$ (for the M1 no-AC case) and $M_{\rm vir}=(8.9\pm0.2)\times10^{11} M_{\odot}$ (for the M1 G04 case). This is in close agreement with the virial mass of $\sim 8\times 10^{11} M_{\odot}$ found via kinematics of satellite galaxies (C\^ot\'e et al.\ 2000) and the Andromeda Stream (Fardal et al.\ 2006; Geehan et al.\ 2006), and the lower limit of $9\times10^{11} M_{\odot}$ found from kinematics of halo stars (Chapman et al.\ 2006). Our M1 B86 model, which is in closest agreement with the expectations of $\Lambda$CDM, lies in the middle of the range with a virial mass of $M_{\rm vir}=8.2\pm0.2\times10^{11}M_{\odot}$. Another recent M31 mass model by Tamm, Tempel \& Tenjes (2007) and Tempel, Tamm \& Tenjes (2007) also uses Spitzer 3.6 $\mu$m imaging to determine the total mass of M31, and they find a mass of $1.1\times10^{12} M_{\odot}$ for a pure NFW model, i.e.\ a slightly higher mass than for our best fit. However, they did not look at the question of AC. Also, their bulge seems to have a more concentrated density profile (see Figure 6 in Tempel et al.\ 2007), which is probably driven by the use of rotation velocities inside 5 kpc. This may be driving the differences between our model and the model presented in Tempel et al.\ (2007). We have been cautious not to include rotation data within the inner few kpc in our model, since it is unlcear how non-circular motions induced in this barred region will affect the overall mass model. It is interesting that, while it has been shown that adiabatically contracted halo models do not generally fit the observed rotation curves of galaxies (e.g.\ Dutton et al.\ 2005, 2007; Kassin et al.\ 2006a, b; Pizagno et al.\ 2005), the observed rotation curve of a well-studied galaxy such as M31 can be well described by a halo that has undergone adiabatic contraction. Indeed, one either has to adopt {\em (i)} a pure NFW profile with an unphysically large concentration, within the context of $\Lambda$CDM, or {\em (ii)} an adiabatically contracted NFW profile and a more reasonable concentration. Such a scenario is in agreement with the results of Klypin et al.\ (2002), who claim that an adiabatically contracted halo is favoured for M31. Much of the power in this analysis came from including the normalization {\em and} outer slope of M31's rotation curve in our fits (where many similar studies of AC attempt to fit to just the rotation velocity at 2.2 disc scalelengths, $V_{2.2}$). This highlights the usefulness of extended rotation curve data in constraining general models of galaxy formation. Having rotation curve data out to large radii is essential for determining dark matter density profiles. Studies that test the applicability of adiabatic contraction often make use of large samples of galaxies that are dominated by late-type disc-dominated galaxies, with small bulges or no bulge at all (e.g.\ Pizagno et al.\ 2005; Dutton et al.\ 2005, 2007; Gnedin et al.\ 2006; Kassin et al.\ 2006a, b). These ``bulgeless'' galaxies also tend to have very flat (or even rising) rotation curves, and it is therefore easy to explain these galaxy rotation curves without any need for adiabatic contraction. However, we suggest that early-type disc galaxies (e.g.\ Hubble types Sa--Sb), which have bulge-dominated centres, and rotation curves that fall-off due to the presence of a large bulge, may require adiabatic contraction. Although we have studied very few galaxies so far (M31 in this paper and two other galaxies in Seigar et al.\ 2006) our results indicate that in cases where rotation curves have a fall-off (two out of three cases), adiabatic contraction is favoured. This may be true for all galaxies with large bulges. If so, it may favour a secular (rather than merger-driven) origin for these bulge-dominated systems, as the gradual accumulation of central mass increases the likelihood that AC will operate. It should also be noted that in our best-fit model, while it is consistent with the current H{\tt I} data from Carignan et al.\ (2006), it is unclear if the rotation curve is declining or remaining flat past 30 kpc, due to the large error bars. The most important requirement for future kinematical studies of galaxies would be improved H{\tt I} measurements at large radii. This will enable us to put better constraints on the halo properties of M31 and other spiral galaxies. \section*{Acknowledgments} Support for this work was provided by NASA through grant number HST-AR-10685.01-A from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. MSS acknowledges partial support from a Gary McCue Fellowship through the Center for Cosmology at UC Irvine. MSS also acknowledges the support of the College of Science and Mathematics at the University of Arkansas at Little Rock. Research by AJB is supported in part by NSF grant AST-0548198. Research by JSB is supported by NSF grants AST-0507916 and AST-0607377. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. 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. The authors wish to thank Tom Jarrett, Luis Ho, Stephane Courteau and Steve Majewski for their discussions. We also wish to express gratitude to Pauline Barmby, for supplying us with the most recent mosaic of the Spitzer 3.6 $\mu$m image of M31. Finally, the authors wish to thank the anonymous referees for comments which improved the content of this paper.

proofpile-arXiv_069-14186

\section{Introduction} Sine-Gordon (SG) models and their Coulomb gas representations appear as a low energy effective theory for many types of physical systems. The so-called ``bulk'' SG model describes the Berezinskii-Kosterlitz-Thouless (BKT) transition in two-dimensional superfluids \cite{Berezinskii,KT,Tsvelik} and the superfluid to insulator transition of Cooper pairs in a chain of Josephson junctions \cite{junctions}. The so-called ``boundary'' SG model \cite{boundarySG} can be used to describe the Chakravarty-Schmid transition in a single Josephson junction with dissipation \cite{Chakravarty-Schmid} and a quantum impurity problem \cite{kanefisher} in one dimension (1D). Powerful theoretical techniques have been developed for studying such SG models including Bethe ansatz solutions \cite{SGBA}, renormalization group analysis (see e.g., Ref. \cite{Tsvelik}), and functional renormalization group \cite{wegner}. Recent theoretical work suggested that there is another class of SG models which is important for several kinds of physical systems. Such models can be described by the action \begin{eqnarray} S(g)=\pi K \int_\Omega \, (\vec \nabla \phi )^2 dx\,d\tau + 2\int_\omega g \, \cos{\left(2\pi\phi\right)} dx\,d\tau \label{s_Gordon_finite}, \end{eqnarray} where the interaction term $\cos{\left(2\pi\phi\right)}$ is present in the spatial region $\omega$ (or space-time region for quantum problems), which is only a part of the domain $\Omega$ over which the noninteracting part $(\vec \nabla \phi )^2$ is defined. This should be contrasted to the bulk SG models, in which the interaction term is present in the entire region $\Omega$, and to the boundary SG models, in which the interaction term is present on a line. We note that models with inhomogeneous $g(x,\tau)$ can be considered using methods developed in the paper as well, but for concreteness we will consider only constant $g.$ Model (\ref{s_Gordon_finite}) interpolates between the bulk and boundary SG models, and we will refer to it as the interior sine-Gordon (ISG) model. Here are a few examples of physical systems that can be described by such models. The first example is the problem of ''interwire coherence`` \cite{AdyStern} in which wires are brought together over a finite length $l$ and separated on both ends. The $\cos$-term describes the correlated umklapp electron scattering in the two wires ($+2k_F$ scattering in one wire accompanied by $-2k_F$ scattering in the other wire). The quantum space-time action of this system has the form of Eq. (\ref{s_Gordon_finite}) with $\Omega = [-\infty,\infty]_x \times [0,\beta]_\tau$ and $\omega = [-l/2,l/2]_x \times [0,\beta]_\tau$. Another example comes from a system of quantum particles in one dimension (e.g., electrons or Cooper pairs in a wire, or ultracold atoms in a weak optical trap) with a periodic potential present in a finite region of the system. The $\cos$-term comes from the umklapp scattering on the external potential and is limited to the finite region in the interior of the system. The third example is the problem of distribution functions of interference fringe amplitudes (DFIFA) for a pair of independent low-dimensional condensates \cite{Schumm,Hofferberth,Hadzibabic}. Individual moments of the distribution function can be represented as a microcanonical partition function of Coulomb gases \cite{pnas,Gritsev,fcslong}, with positions of Coulomb charges restricted to the part of the system from which interference patterns have been extracted. The latter is typically smaller than the total system size. For example, in the case of large 2D condensates we have $\Omega = [-\infty,\infty]_x \times [-\infty,\infty]_\tau$, and when the interference pattern is extracted from the area $l \times l$, we have $\omega= [-l/2,l/2]_x \times [-l/2,l/2]_\tau$. The relation between the DFIFA and the partition function of the ISG model will be outlined below and has also been discussed in Refs. \cite{pnas,Gritsev,fcslong}. In this paper we develop a nonperturbative approach to calculating partition functions of a wide class of SG models and Coulomb gases, which relies on the mapping of their partition functions to certain problems of statistics of random surfaces. We point out that our method does not rely on the existence of the exact solutions of SG models, but uses the structure of the multi-point correlation functions in the absence of interactions. We also note that a suitable extension of our method can be used to compute correlation functions of SG models in equilibrium and non-equilibrium situations. The particular strength of our approach is that it can be applied to study ISG models, which {\it cannot be analyzed by other theoretical methods}. As a concrete application of our method we calculate DFIFA for both 1D and 2D condensates. We point out that earlier theoretical work on interference experiments focused on 1D systems with periodic boundary conditions (PBC) \cite{Gritsev}. While these boundary conditions are extremely artificial from the point of view of realistic experiments, they allow one to relate the DFIFA to the quantum impurity problem \cite{kanefisher} and use certain exact results about the latter \cite{BLZ1-3}. Methods used in Ref. \cite{Gritsev} can not be generalized either to the realistic case of open boundary conditions (OBC) (i.e., interference patterns extracted from the interior of a large system) or 2D systems, but these cases can be analyzed using the method discussed in this paper. We emphasize, however, that the main goal of this paper is to introduce an approach to the analysis of SG models and the problem of DFIFA is just one example that illustrates the power of the new method. \section{Mapping} The partition function corresponding to the action (\ref{s_Gordon_finite}) is given by $Z(g)=\int {\cal D}\phi e^{-S(g)}/\int {\cal D}\phi e^{-S(0)}$. By expanding $Z(g)$ in powers of $g$ we arrive at the grand canonical partition function of the Coulomb gas \cite{ChuiLee} \begin{widetext} \begin{equation} Z(g)=\sum_{n=0}^{n=\infty}\frac{g^{2n}}{(n!)^2}Z_{2n} , \;\mbox{where}\; Z_{2n}= \underset{\omega}{\int}...\underset{\omega}{\int} d^2 \vec u_1...d^2 \vec v_n e^{\frac1K\left(\underset{i<j}{\sum}G(\vec u_i,\vec u_j) + \underset{i<j}{\sum}G(\vec v_i ,\vec v_j)- \underset{ij}{\sum}G(\vec u_i,\vec v_j) \right)}. \label{Z2n} \end{equation} \end{widetext} Here, $Z_{2n}$ is a microcanonical partition function of a classical two-component neutral Coloumb gas of $2n$ particles, and $G(\vec x,\vec y)$ is an interaction potential, which is proportional to Green's function of the Laplace operator on $\Omega.$ The most familiar case is when $\Omega = [-\infty,\infty]_x \times [-\infty,\infty]_\tau$ and $G(\vec x,\vec y)=\ln{|\vec x-\vec y|}$. To evaluate $Z(g)$ nonperturbatively in $g,$ we introduce an auxiliary function $W(\alpha)$, which should be understood as a certain distribution function, and is defined in such a way that its $n\mbox{th}$ moment equals $Z_{2n}.$ \begin{eqnarray} Z_{2n}=\int W(\alpha) \alpha^n d\alpha.\label{walpha} \end{eqnarray} One can use the Hankel transformation \cite{hankel} to compute $Z(g)$ from $W(\alpha)$ as \begin{equation} Z(g)=\int_0^\infty W(\alpha)I_0(2g\sqrt{\alpha}) d\alpha. \label{hankeleq} \end{equation} This equation equation can be verified using the Taylor expansion of the modified Bessel function $I_0(x)$ and Eq. (\ref{walpha}). Formulation of the auxiliary ''problem of moments`` allows one to avoid calculating $Z(g)$ order by order, and can be viewed as a tool to sum the perturbation series in Eq.~(\ref{Z2n}) to all orders. Function $G(\vec x,\vec y)$ is real and symmetric, so it can be diagonalized on $\omega$ by solving the eigenvalue equations \begin{eqnarray} \int_{\omega} G(\vec x,\vec y) \Psi_{f}(\vec y) d^2\vec y=G(f)\Psi_{f}(\vec x). \label{inteqs} \end{eqnarray} Here $f$ is an integer index, which goes from $1$ to $\infty.$ $\Psi_{f}(\vec x)$ can be chosen to be real and normalized according to $\int_{\omega}\Psi_{f}(\vec x)\Psi_{k}(\vec x) d^2\vec x=\delta(f,k).$ Then, $G(\vec x,\vec y)$ is given by $ G(\vec x,\vec y)=\sum_{f=1}^{f=\infty}G(f) \Psi_{f}(\vec x) \Psi_{f}(\vec y). $ Such decomposition is similar to the diagonalization of a symmetric matrix using its eigenvectors and eigenvalues. We have \begin{widetext} \begin{eqnarray} Z_{2n}= \int_{\omega}...\int_{\omega} d^2\vec u_1 ... d^2\vec v_n e^{\sum_f\frac{ G(f)}{2K}\left[\left(\sum_{i=1}^{i=n}\Psi_{f}(\vec u_i)-\Psi_{f}(\vec v_i)\right)^2 -\sum_{i=1}^{i=n}\left(\Psi_{f}(\vec u_i)^2+\Psi_{f}(\vec v_i)^2 \right)\right]}= \nonumber\\ \int_{\omega}...\int_{\omega} d^2\vec u_1 ... d^2\vec v_n\prod_{f=1}^{f=\infty} \frac{ \int_{-\infty}^{\infty}d t_{f}e^{-\frac{t_{f}^2}2} e^{\sum_{i} t_f\sqrt{\frac{G(f)}{K}}\left(\Psi_{f}(\vec u_i)-\Psi_{f}(\vec v_i) \right)-\frac{G(f)}{2K}\left(\Psi_{f}(\vec u_i)^2+\Psi_{f}(\vec v_i)^2 \right)}} {\sqrt{2\pi}}. \label{decoupling} \end{eqnarray} \end{widetext} To go from the first to the second line in Eq. (\ref{decoupling}) we introduced the Hubbard-Stratonovich variables $t_f.$ Integration over $d^2\vec u_1 ... d^2\vec v_n,$ is now straightforward since all $\vec u-$ and $\vec v-$ integrals are identical. \begin{eqnarray} Z_{2n}=\left(\prod_{f=1}^{f=\infty}\frac{\int_{-\infty}^{\infty} e^{-\frac{t_{f}^2}2}d t_{f}}{\sqrt{2\pi}}\right)g(\{t_f\})^ng(\{-t_f\})^n, \label{10} \end{eqnarray} where \begin{eqnarray} g(\{t_f\})=\int_{\omega}d\vec x\; e^{\sum_f t_f\sqrt{\frac{G(f)}{K}}\Psi_{f}(\vec x)-\frac{G(f)}{2K}\Psi_{f}(\vec x)^2}.\label{gt} \end{eqnarray} From a comparison of Eqs. (\ref{walpha}) and (\ref{10}) we obtain \begin{eqnarray} W(\alpha)=\prod_{f=1}^{f=\infty}\frac{\int_{-\infty}^{\infty} e^{-\frac{t_{f}^2}2}d t_{f}}{\sqrt{2\pi}}\delta\left[\alpha-g(\{t_f\})g(\{-t_f\})\right]. \label{Walpha} \end{eqnarray} Equations (\ref{gt}) and (\ref{Walpha}) have a simple physical interpretation. Consider $\Psi_{f}(\vec x)$ to be the eigenmodes of the surface vibrations, $t_f$ the fluctuating mode amplitudes, and $|G(f)|$ the noise power. Infinite dimensional integral over $\{t_f\}$ variables can be understood as an averaging over fluctuations of the surface. For a particular realization of noise variables $\{t_f\},$ a complex valued surface coordinate at point $\vec x$ is given by $ h(\vec x;\{t_f\})= \sum_f t_f \Psi_{f}(\vec x)\sqrt{G(f)/K}-G(f)\Psi_{f}(\vec x)^2/2K. $ For each realization of a random surface $\{t_f\},$ $g(\{t_f\})$ is obtained as an integral (\ref{gt}), which can also be written as $g(\{t_f\})=\int_\omega d^2\vec x e^{h(\vec x;\{t_f\})}.$ Hence Eq.~(\ref{Walpha}) can be interpreted as the mapping between the partition function of the SG model (\ref{s_Gordon_finite}) and the statistics of random surfaces subject to classical noise. This mapping is the central result of this paper. In general, $g(\{t_f\})$ is a complex number, thus in general, $\alpha$ is defined on a complex plane. Simplifications occur: if all eigenvalues $G(f)$ are negative, then $ g(\{-t_f\})= g(\{t_f\})^*, g(\{t_f\})g(\{-t_f\})=|g(\{t_f\})|^2, $ and $\alpha$ is always real and positive. Function $W(\alpha)$ can be computed efficiently using Eq.~(\ref{Walpha}) and Monte Carlo simulations. In the first step one solves integral equations~(\ref{inteqs}) numerically to obtain eigenfunctions $\Psi_{f}(\vec x)$ and eigenvectors $G(f).$ Then one samples random numbers $\{t_f\}$ from the Gaussian ensemble, and plots the histogram of the results for $g(\{t_f\})g(\{-t_f\}).$ Each point on a histogram requires a computation of only two integrals, and $W(\alpha)$ can be evaluated to arbitrary precision. Note, that this simple numerical evaluation of $W(\alpha)$ allows one to extract $Z(g)$ for {\it all} values of $g$ using Eq. (\ref{hankeleq}). While conventional large-scale Monte Carlo simulations \cite{SGMC1,SGMC2} can be used to extract properties of SG models, they require separate simulation for each value of $g.$ Application of such methods to the calculation of DFIFA described below would also require analytic continuation of numerical results to imaginary values of $g$ \cite{Gritsev}, which is numerically unstable. In addition, the mapping in Eqs.~(\ref{gt}) and (\ref{Walpha}) does not only simplify numerical simulations, but can also be used to obtain analytical results (see below). \section{Applications to interference of low-dimensional gases} We now apply a general formalism developed to a particular problem of the interference of low-dimensional Bose gases \cite{Schumm,Hofferberth,Hadzibabic,pnas,Gritsev,fcslong}. Typical experimental setup for interference of low-dimensional gases with open boundary conditions (OBC) is shown in Fig. \ref{setup}. Two parallel condensates are extended in the $x$ direction. After atoms are released from the trap, clouds expand predominantly in the transverse direction. After sufficient time of flight clouds overlap, and the laser beam propagating along the $z$ axis takes an absorption image. Fluctuations of the relative phase result in fluctuations of the minima positions for different $x.$ For each $y,$ the image can be integrated along the $x$ direction to obtain the integrated fringe amplitude $A.$ One experimental image can be used to extract information for different values of $L.$ Many images are still required to obtain distribution functions for each $L.$ \begin{figure} \psfig{file=setup.eps} \caption{\label{setup} Simplified setup of interference experiments with 1D Bose liquids (see, e.g., Refs. \cite{Schumm,Hofferberth}). Two parallel condensates are extended in the $x$ direction. After atoms are released from the trap, clouds are imaged by the laser beam propagating along the $z$ axis. Meandering structure of the interference pattern arises from phase fluctuations along the condensates. The net interference amplitude $A$ is defined from the density integrated along the section of length $L.$ For the analogous 2D setup, see, e.g., Fig.~2 of Ref. \cite{pnas}.} \end{figure} For two identical 1D clouds higher moments of the fringe amplitude $A$ can be written as \cite{pnas} \begin{eqnarray} \langle |A|^{2n}\rangle =A_0^{2n} Z_{2n}, \; \mbox{where} \; A_0=\sqrt{ C \rho^2 \xi_h^{1/K} L^{2-1/K}}.\label{A0} \end{eqnarray} Here $\rho$ is the density, $L$ is the imaging length, $\xi_h$ is the healing length, and $C$ is a constant of the order of unity. For OBC $Z_{2n}$ is given \cite{pnas} by Eq. (\ref{Z2n}) with $\Omega=[-\infty,\infty]_x \times[0,\beta]_\tau$ (and periodic boundary conditions in $\tau$) and $\omega=[0,1]_x$ (and fixed $\tau$). Equation (\ref{A0}) has been derived neglecting the shot noise, which arises due to a finite number of particles in the interfering clouds \cite{shotnoise,fcslong}. In what follows, we will be interested in the distribution functions $W(\alpha)$ of a positive variable $\alpha=|A|^{2}/A_0^2$ defined by Eq. (\ref{walpha}), or of its normalized version $\tilde \alpha=|A|^{2}/\langle A^2\rangle,$ defined by $Z_{2n}/Z_2^n=\int_0^\infty \tilde W(\tilde \alpha)\tilde \alpha^n d\tilde \alpha.$ For zero temperature $Z_{2n}$ depends only on the Luttinger parameter $K,$ which describes \cite{Haldane,Cazalilla} the long-distance behavior of boson correlation functions, given by $\langle a^{\dagger}(x)a(0)\rangle \sim \rho \left(\xi_h/x\right)^{1/2K}$. For bosons $K$ ranges from $K=1$ (strong interactions) to $K=\infty$ (weak interactions). For OBC and zero temperature $G(\vec x,\vec y)$ equals $G(\vec x,\vec y)=\log{|\vec x-\vec y|},$ while for PBC considered in Ref.~\cite{Gritsev}, $G^{per}(\vec x,\vec y)=\ln{\frac1\pi\sin{\pi|\vec x-\vec y|}}.$ While $A_0$ depends on $L,$ for zero temperature distribution $\tilde W(\tilde \alpha)$ does not depend on $L,$ but depends only on $K.$ For nonzero temperature, $Z_{2n}$ depends on $K$ and the thermal length $\xi_T=\hbar v_s/(k_B T),$ where $v_s$ is the sound velocity: $ G(\vec x,\vec y,\xi_T/L)=\ln{\left(\frac{\xi_T}{\pi L}\sinh{\frac{\pi|\vec x-\vec y|L}{\xi_T}}\right)}. $ For 2D, one can use a similar approach to describe the contrast distribution at finite temperature below the BKT transition. In this case, correlation functions are given by $ \langle a^{\dagger}(r)a(0)\rangle \sim \rho \left(\xi_h/r\right)^{\eta(T)}, $ where $\eta(T)=m T/(2\pi\hbar^2 \rho_s(T))$ depends on the temperature and the superfluid density $\rho_s(T).$ The BKT transition happens at the universal value $\eta_c(T_c)=1/4.$ To keep a connection to the 1D case, we will use $K=1/(2\eta(T)),$ and restrict our attention to $K>K_c=2.$ For 2D with the aspect ratio of the imaging area equal to unity, $\vec u_i$ and $\vec v_i$ in Eq. (\ref{Z2n}) are defined on a square $\omega=[0,1]_{x}\times[0,1]_{\tau}$ with $G(\vec x,\vec y)=\ln{|\vec x-\vec y|}.$ \begin{figure} \includegraphics[scale=0.8,angle=0,bb=50 10 200 150]{1dzeroT.eps} \caption{\label{1dzeroT}Distribution functions of the normalized interference amplitude $\tilde W(\tilde \alpha)$ at $T=0$ for 1D gases with open boundary conditions, shown for Luttinger parameters $K=2\;\mbox{(dashed)}, K=3\;\mbox{(dotted)}, $ and $ K=5 \;\mbox{(solid)}.$ The inset shows a comparison between open (solid) and periodic (dashed) boundary conditions for $K=5$.} \end{figure} In the limit $K\rightarrow \infty,$ one can expand the exponent of Eq. (\ref{gt}) in the Taylor series. Then $\alpha=g(\{t_f\})g(\{-t_f\})$ is linearly related to roughness, or mean square fluctuation of the surface, as defined in Ref.~\cite{1fnoise}. For PBC the noise has a $1/f$ power spectrum, since $G^{per}(x,y)=\ln{\frac1\pi\sin{\pi|x-y|}}=-\ln{2\pi}-\sum_{f=1}^{f=\infty}(\cos{2\pi f x}\cos{2\pi f y}+\sin{2\pi f x}\sin{2\pi f y})/f.$ This results in the Gumbel statistics of the roughness \cite{Gritsev,1fnoise, BertinClusel,fcslong}: $\tilde W_{G}(\tilde\alpha)=K\;\exp(x-e^{x}),$ where $x=K(\tilde{\alpha}-1)-\gamma$ and $\gamma=0.577$ is the Euler constant. This provides the {\it analytical} proof of the conjecture made in Ref. \cite{Gritsev}, that the distribution function in this case is given by one of the extreme value statistical distributions, the Gumbel function \cite{gumbelbook}. In what follows we perform simulations of $W(\alpha)$ with up to $N=10^6-10^7$ realizations of $\{t_f\}$ and smoothen the data. We use a finite value of $f_{max}$ and check for convergence with $f_{max},$ typically $\sim 30.$ $\langle \alpha\rangle $ is always kept within $1\%$ from its expected value. For most of the presented results, all eigenvalues $G(f)$ are negative, and Eq. (\ref{Walpha}) can be directly applied. Special care should be taken of the 1D case with nonzero temperatures, since one of the eigenvalues can be positive. This situation can be handled by subtracting a sufficiently large positive constant $C$ from $G(\vec x,\vec y,\xi_T/L),$ which makes all eigenvalues negative. According to Eqs. (\ref{Z2n}) and (\ref{walpha}), this leads to rescaling of $\alpha$ by a factor $e^{-C/K},$ which can be easily taken into account. In Fig.~\ref{1dzeroT} we show distribution functions of the normalized interference amplitude $\tilde W(\tilde \alpha)$ at $T=0$ for 1D gases with OBC for various $K.$ The inset shows a comparison between OBC and PBC for $K=5.$ In Fig. \ref{1dnztfig}, we show distribution functions of the normalized interference amplitude for a 1D gas with OBC at nonzero temperature and $K=5.$ For $\xi_T K/L\ll1$ distribution is Poissonian \cite{pnas, Gritsev, fcslong} and wide, while for $K\gg1$ and $\xi_T K/L\gg1$ it is very narrow. Evolution of the full distribution function of the visibilities as $L$ is varied can be used to precisely measure the thermal length $\xi_T,$ and to extract the temperature. As seen in Fig. \ref{1dnztfig}, at $T\neq 0$ the distribution function has characteristic features, i.e., it is generally nonsymmetric and can have a {\it minimum}. These features can be used to distinguish the intrinsic noise due to fluctuations of the phase from technical noise. Finally, in Fig.~\ref{2dfig} we show distribution functions of the normalized interference amplitude for a 2D gas with an aspect ratio of imaging area equal to unity and OBC below the BKT temperature. Above the BKT temperature, distribution functions become Poissonian for $L\gg \xi,$ where $\xi$ is the correlation length. In 2D one cannot describe the crossover at $L\sim \xi$ similar to 1D, since the action which describes the fluctuations of the phase is not quadratic in this region, and Eq. (\ref{Z2n}) does not hold. \begin{figure} \includegraphics[scale=0.8,angle=0,bb=50 10 200 150]{1dnztfig.eps} \caption{\label{1dnztfig} Distribution functions of the normalized interference amplitude $\tilde W(\tilde \alpha)$ for a 1D Bose gas with open boundary conditions at nonzero temperature and $K=5.$ Different curves correspond to ratios $K \xi_T/L =\infty\;\mbox{(solid)}, K \xi_T/L=1\;\mbox{(dotted)},$ and $K \xi_T/L=0.25 \;\mbox{(dashed)}.$ $\xi_T$ is the thermal correlation length, $K$ is the Luttinger parameter, and $L$ is the imaging length.} \end{figure} \begin{figure} \includegraphics[scale=0.8,angle=0,bb=50 10 200 150]{2dfignew.eps} \caption{\label{2dfig}Distribution functions of the normalized interference amplitude $\tilde W(\tilde \alpha)$ for a two-dimensional Bose gas with the aspect ratio of the imaging area equal to unity and open boundary conditions. Temperature is below the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature. Different curves correspond to $\eta(T)=\eta_c(T_c)=1/4$(the BKT transition point, solid), $\eta(T)=1/6$ (dashed line), and $\eta(T)=1/10$ (dotted line). Above the BKT transition temperature the distribution function is Poissonian (dot-dashed line). } \end{figure} \section{Conclusions} To summarize, we introduced a class of sine-Gordon models, for which an interaction term is present in a spatial region different from the domain over which the noninteracting part is defined. We developed a general mapping of such sine-Gordon models and related Coulomb gases to statistical properties of random surfaces, which can be used to calculate their partition functions {\it nonperturbatively}. As a specific application of our approach, we considered interference experiments with two independent low-dimensional Bose gases. We calculated full distribution functions of the interference amplitude for 1D and 2D gases with open boundary conditions and nonzero temperatures. Full distribution functions of interference fringe visibilities can be used for thermometry. We thank M.~Mueller and A.~Polkovnikov for useful discussions. This work was partially supported by NSF Grant No. DMR-0705472, MIT-Harvard CUA, and AFOSR. V.G. is also partially supported by Swiss National Science Foundation, Grant No. PBFR2-110423.

proofpile-arXiv_069-14244

\section{Introduction} Ultraviolet and soft X-ray observations by {\it{Copernicus}}, {\it{Voyager}}, {\it{HUT}}, {\it{ORFEUS}}, {\it{FUSE}}, {\it{SPEAR}}, the Wisconsin All Sky Survey, {\it{HEAO}}, {\it{ROSAT}}, {\it{Chandra}}, and {\it{XMM}} detected \oxysix\ ions and soft X-ray emission from the interstellar medium above the Milky Way's disk \citep{jenkins_1978b,murthy_etal,davidsen,hurwitz_bowyer,wakker_etal, sheltonetal01, dixonetal, korpela_etal, mccammon_sanders, snowden_etal_91,yao_wang,breitschwerdt_cox,henley_etal}. If the gas is in collisional ionizational equilibrium (CIE), then the diffuse \oxysix\ ions and their 1032, 1038 \AA\ resonance line emission trace $3 \times 10^5$~K gas, while the diffuse 1/4 keV and 3/4 keV soft X-rays trace $10^6$ to $10^7$~K gas \citep{mazzotta_etal,raymond_smith}. It is natural to anticipate that the \oxysix-rich and 1/4 keV emitting plasmas are causally, and perhaps physically, associated because million degree gas eventually cools into the 300,000 K regime. As it cools, its \oxyseven\ ions recombine with electrons to make \oxysix\ ions. Furthermore, as time goes by, interfaces between $10^6$~K gas and warm ionized gas ($\sim10^4$~K) should develop at temperatures that are between these two values and exhibit intermediate levels of ionization. Thus, it is reasonable to expect the 1/4~keV emitting gas to be clad in an \oxysix-rich sheath, irrespective of whether the hot extraplanar gas is due to fountains, superbubbles, or extraplanar supernova (SN) explosions. Such bubble-sheath geometries are present in the disk simulations of \citet{avillez_breitschwerdt}, as are $T \sim 3 \times 10^5$~K bubbles without $T \sim 10^6$~K cores. In this paper, we focus on hot gas above the Galactic disk. Even though the nearest part of this region may be only a few hundred parsecs above the midplane, we will follow the X-ray community's tradition by referring to this region as the halo. There are a number of fundamental issues that we plan to address about the character of the material that creates \oxysix\ and X-ray emissions: 1.) Are the source structures in the halo young, middle aged, or ancient? 2.) What is the temperature distribution function of this gas? and 3.) How long is this plasma's cooling timescale and how does the energy loss rate compare with the rate of energy injection from supernova explosions in both the disk and halo? Our analyses are based on sets of {\it{Far Ultraviolet Spectroscopic Explorer}} (\fuse) \oxysix\ and {\it{Roentgen Satellite}} (\rosat) 1/4 keV emission data for $l=278.7^{\rm{o}},b=-47.1^{\rm{o}}$ and $l=278.6^{\rm{o}},b=-45.3^{\rm{o}}$. We focus much of our attention on the O VI emission data, which can be compared to the soft X-ray emission to synthesize a long baseline spectrum. In turn, this spectrum defines the shape of the plasma's temperature distribution function and can be compared with models of supernova remnant (SNR) evolution to constrain the age of a remnant, on the assumption that the hot gas in our line of sight is dominated by the products of one such remnant. We have no direct measurement of the \oxysix\ column density along our line of sight. Nevertheless, observations of \oxysix\ absorption features toward bright extragalactic sources at only moderate angular separations from our direction give an approximate guide on the value of the \oxysix\ column density, $N_{\rm OVI}$. This information is useful, since both \oxysix\ and soft X-ray emission intensities are equal to $(1/4\pi)\int n_en_i<\sigma v>(T)dl$, where $n_e$ is the local electron density, $n_i$ is the local density of ions, and $<\sigma v>(T)$ is a temperature-dependent emissivity function. In contrast, the \oxysix\ column density is simply equal to the line of sight integral of the ion density. The column density integral is not additionally weighted by electron density, and its temperature dependence is simply that due to the changes in the ion fraction. As a result, the ratio of the \oxysix\ intensity to column density can be used to determine the electron density and from that, the thermal pressure. Every observation of the halo's intensity is contaminated by photons produced elsewhere along the line of sight. The most obvious non-halo sources are the heliosphere \citep{lallement} and the Local Bubble (also called the Local Hot Bubble), where the latter is an irregularly shaped pocket of X-ray emissive, interstellar gas surrounding the Solar neighborhood out to a distance of $\sim$100~pc \citep{cox_reynolds}. The combined Local Bubble and heliospheric emission must be subtracted from that found for observations of the high latitude sky in order to determine the halo's intensity. To accomplish this, we selected two adjacent lines of sight, one that is nearly free of foreground obscuration and another that has a cloud of neutral gas, which blocks the more distant halo emission but not the foreground emission. By evaluating the difference between the two intensities, we can measure the emission that arises only from the halo. This procedure is called the ``shadowing technique.'' For our shadowed line of sight, we used the strength of \oxysix\ emission found along the path to a nearby, opaque filament. The filament is located $230\pm30$ pc from the Sun \citep{penprase_etal}, so is positioned beyond, but not far beyond the Local Bubble. \fuse\ \ has already observed a dense part of the filament at $l=278.6^{\rm{o}},b=-45.3^{\rm{o}}$. The observation yielded a 2$\sigma$ upper limit on the \oxysix\ doublet's intensity, including the systematic uncertainties, of 800 photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ \citep{shelton_2003}. Ignoring the systematic uncertainties and using the $1\sigma$ statistical uncertainties as error bars leads to a best fit value of $30 \pm {340}$ photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ (as determined by one of us (RLS) while preparing the original article). To measure the combined halo and Local Bubble \oxysix\ intensity for this paper, we used \fuse\ to observe an unobscured line of sight only 2$^{\rm{o}}$ from the filament ($l=278.7^{\rm{o}},b=-47.1^{\rm{o}}$; see Figure~\ref{fig:diras}). In order to estimate the halo's soft X-ray count rate, we perform a similar shadowing analysis using \rosat\ data for the same pointing directions. We also subtract the extragalactic soft X-ray contribution and account for the variation in optical depth with photon energy. By carefully isolating the halo emission, we expect to circumvent the amalgamation problem seen in plots of total \oxysix\ intensity versus total soft X-ray countrate. In Section~\ref{sect:observations} of this paper we present our measurements of the \oxysix\ emission produced along the unobscured, off-filament line of sight. In Subsection~\ref{subsect:oviintensity}, we subtract the local \oxysix\ intensity from our measurement of the unobscured line of sight, yielding the intensity originating in the halo. We determine the halo's \oxysix\ column density from published data for nearby directions in Subsection~\ref{NOVI_estimate}. In Subsection~\ref{subsect:xraybrightness}, we estimate of the halo's 1/4~keV surface brightness by performing a shadowing analysis with \rosat\ data from the filament and off-filament directions. We then use the halo's \oxysix\ intensity and column density determinations and treat the \oxysix-rich gas as if it is isothermal in order to estimate the density and pressure of the emitting material in Subsection~\ref{subsect:physicalproperties}. We compare the \oxysix\ and 1/4~keV results with simulation predictions in order to estimate the plasma's maturity in Subsections~\ref{subsect:halosnr}. We compare the \oxysix\ and 1/4~keV results with analytic functions in order to estimate the plasma's temperature distribution function, pressure, and cooling rate in Subsection~\ref{temp_dist}. The results are summarized in Section~\ref{sect:summary}. \section{Observations, Data Reduction, Spectral Analysis, and Determination of the Halo's O VI Intensity} \label{sect:observations} \subsection{Observations} \fuse\ observed the blank sky toward $l=278.69^{\rm{o}},b=-47.08^{\rm{o}}$ (R.A. = $03^{\rm{h}} 20^{\rm{m}} 10^{\rm{s}}$, decl. = $-62^{\rm{o}} 26' 29''$) once in July and twice in November of 2002, for Guest Observer program C153. These observations exposed the LiF 1A detector segment, the segment used for the subsequent \oxysix\ signal extraction, for 53, 104, and 61~ksec, respectively. Of these durations, 6, 61, and 26~ksec, respectively, occurred while the satellite was in the night portion of its orbit. \fuse\ also made a non-proprietary ``Early Release Observation'' of this direction in December 1999 and archived the data under program identification number X0270301. However, detector 1 was de-powered during most of the observation time. \subsection{Data Reduction} During most of the Guest Observer exposures, data were taken on all four detector segments (1A, 1B, 2A, and 2B). The data from each detector segment were taken in time-tag mode where the photoevents from each detector are listed in the sequence that they were recorded, with their times and locations on the detectors where they were found. We concatenated the photon lists obtained from the individual exposures, creating lists for each detector segment and observation. We then ran the concatenated photon lists through version 3.1.1 of the CalFUSE pipeline \citep{sahnow_etal,datahandbook}, with corrections enabling us to use the extended source extraction window. We did not use automatic background subtraction. For the LiF 1A extractions, we set the lower and upper pulse height limits to 4 and 25, respectively. For the other detector segments, we used the default pulse height limits. This process yielded several spectra, because in addition to recording emission with four detector segments, \fuse\ records light reflected off of each of two types of coatings (lithium fluoride (henceforth called LiF) and silicon carbide (henceforth called SiC)) and four gratings. The multiplicity enables the telescope to see eight overlapping wavelength regimes with various efficiencies. Furthermore, each mirror's focal plane assembly has four apertures. Thus, each detector segment is able to receive light from each of the four apertures simultaneously and without overlap. The spectra observed through the largest aperture (LWRS: 30'' $\times$ 30'') in the LiF1A channel provides the greatest signal to noise and the greatest effective area in the 1030 to 1040 \AA\ regime. We will use these spectra for the following \oxysix\ data reduction, and will use the spectra from the other channels when searching for other cosmic emission lines outside of the 1030 to 1040 \AA\ bandpass. Our avoidance of the SiC channels overcomes the solar \oxysix\ line contamination problem discussed by \citet{lecavelier_etal}. The wavelength scales for each spectrum require re-calibration, which we determined by comparing the reported heliocentric wavelengths of the observed geocoronal emission lines around 1027 and 1040 \AA\ with their rest wavelengths \citep{morton} converted from the geocentric to the heliocentric reference frame. We fit a linear function of wavelength to the difference between the expected and observed heliocentric wavelengths, then subtracted this function from the observed wavelengths. We then combined like spectra and shifted to the Local Standard of Rest (LSR) reference frame. \subsection{Spectral Analysis} Figure~\ref{fig:spectrum} displays the satellite-night and day$+$night LiF 1A spectra for wavelengths between 1027 and 1045 \AA. The spectra reveal \oxysix\ emission in the 1032 and 1038~\AA\ lines (rest wavelengths: 1031.93 and 1037.62 \AA) and \cartwostar\ emission at 1037~\AA\ (rest wavelength: 1037.02 \AA). An additional emission feature appears in the day+night spectrum at 1031 \AA, but does not appear in the night-only spectrum. It also appears in other long-duration \fuse\ daytime spectra but not in their nighttime counterparts. We suspect that the 1031~\AA\ feature is the second order diffraction peak associated with atmospheric \heone, whose rest wavelength is 515.62~\AA. Other \heone\ second order lines in our bandpass include $2 \times 522.21$~\AA\ and $2 \times 537.03$~\AA, both of which appear in our daytime spectrum but not our nighttime spectrum. Along with the emission lines is also a continuum, which arises from radioactive decay within the detector, high energy particles, and scattered terrestrial airglow photons. The last of these is strongest during the daytime portion of the orbit. We used two methods to measure the strengths of the interstellar emission features. With the first method, which parallels that of \citet{sheltonetal01}, we fit a second order polynomial to the continuum of the counts spectrum. (Note that ``counts'' are called ``weights'' in CALFUSE version 3.1 output files.) We then subtracted the continuum from the spectrum and searched for wavelength regions with large numbers of counts. Due to continuum subtraction and statistical variation, a few of the pixels within weak emission features have negative numbers of counts and some pixels outside of the emission features have positive numbers of counts. Although this noise complicated our effort to determine the boundaries of emission features, we set the boundaries of the emission features so as to exclude neighboring regions exhibiting near random fluctuations. We then summed the residual counts in each emission feature, divided by the instrumental effective area, and incorporated geometric factors in order to convert the measurements to units of intensity. Each measurement's statistical uncertainty was calculated from the square root of the number of counts in the original spectra between the upper and lower wavelength range of the feature. In the day$+$night spectrum, the \oxysix\ 1032~\AA\ emission line and the daytime 1031 \AA\ feature are too close together to be separated with this algorithm. Therefore, when we analyzed the 1032~\AA\ emission line in the day$+$night data, we used the wavelength range found from the nighttime data (1031.64 \AA\ to 1032.45 \AA). As can be seen in Figure~\ref{fig:spectrum}, the lower end of this range overlaps very little with the \heone\ feature. Furthermore, the measured intensity in the day$+$night 1032 \AA\ \oxysix\ feature is less than that of the night time feature, leading us to believe that little or no \heone\ emission was attributed to the day$+$night 1032 \AA\ feature. Table~\ref{table:intensityresults} lists the measured intensities and 1 sigma statistical uncertainties of the \oxysix\ 1032 and 1038 \AA\ lines, the \cartwostar\ line, and the anomalous 1031 \AA\ feature. For presentation purposes, we round the intensities as well as the Local Bubble intensity in \S\ref{subsect:oviintensity} to 10s of photons sec $^{-1}$ cm$^{-2}$ sr$^{-1}$. We use the unrounded values when calculating resulting quantities. Given the uncertainties in aperture size and flux calibrations, additional systematic uncertainties of $14\%$ are expected \citep{sheltonetal01}. Placement of the continuum is another source of uncertainty. Since the continuum was independently determined during each analysis, the variation within our set of measurements provides an estimate of the size of this uncertainty. The standard deviation among the 4 measurements of the 1032 \AA\ emission line (Table~\ref{table:intensityresults}) is 220 photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$ while the standard deviation for the 1038 \AA\ line is 470 photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$. In order to determine the central wavelengths of our irregular, weak emission features, we fit each residual spectrum with a Gaussian and a quadratic. Generally, the fitting routine modeled the emission line with the Gaussian and the residual background with the quadratic. Occasionally, as occurred in the \oxysix\ 1032~\AA\ fits, the Gaussian extended beyond the bulk of the emission line in order to include nearby weak residual background emission. Thus, the reported wavelengths for the day$+$night \oxysix\ 1032~\AA\ Gaussian fit and the nighttime \oxysix\ 1038~\AA\ Gaussian fit are higher and lower, respectively, than one would estimate by eye. See Table~\ref{table:velocityresults} for the tabulated results and note that the corrected \fuse\ scale is accurate to only $\sim 10$ km sec$^{-1}$. In the second method, which parallels that of \citet{dixonetal}, we fit the intensity spectrum's continuum with a straight line and its emission features with Gaussian shapes that had been convolved with a 106 km sec$^{-1}$ top-hat function (simulating the finite width of the LWRS aperture). Because the \oxysix\ 1032~\AA\ emission line and the daytime 1031 \AA\ emission feature are so close together in the day$+$night spectrum, they were fit simultaneously. The \oxysix\ 1038~\AA\ emission line and the \cartwostar\ emission line required simultaneous fitting for the same reason. The fits occasionally included outlying emission, which shifted the reported centroid wavelengths and increased the areas under the curves. The $1 \sigma$ error bars for each model parameter were determined as described in \citet{dixonetal}. See Tables~\ref{table:intensityresults} and \ref{table:velocityresults} for the measured intensities and velocities. The \oxysix\ and \cartwostar\ appear to be moving very slowly relative to the Local Standard of Rest. However, the 1032 \AA\ signal is offset by about 30 km s$^{-1}$ from the 1038 \AA\ signal, even in the fits that were not corrupted by outlying emission, the night-only 1032 \AA\ and the day+night 1038 \AA\ fits. The difference may be due to inaccuracies in the wavelength scale, being as the observed difference is somewhat more than twice the uncertainty in the velocity of each feature. It could also be due to variations in self-absorption optical depth as a function of wavelength. Larger differences have been seen in other spectra, for example the Virgo and Coma spectra of \citet{dixonetal}. In the night-only 1032 \AA\ and the day+night 1038 \AA\ fits, the $\sigma$ of the cosmic emission lines' Gaussian profiles were 47 and 39~km~s$^{-1}$, respectively. The resulting full widths at half maximum are 111 and 92 km~s$^{-1}$, respectively, while the Doppler parameters are 65 and 55~km~s$^{-1}$, respectively. The true widths of the cosmic emission lines should be slightly narrower than these values because the fitting method accounts for only the broadening due to a finite slit width and not any other form of instrumental broadening. As a result of applying two analysis methods to two dataset variants (the day $+$ night data and the night only data), we have 4 measurements for each of the \oxysix\ 1032 \AA, \oxysix\ 1038 \AA, and \cartwostar\ 1037 \AA\ emission lines. For the most part, the $ 1 \sigma$ measurements overlap. However, for each cosmic line, 3 of the 4 intensity measurements are similar to each other while the 4th measurement is an outlier. From the 3 clustered and unrounded measurements, we calculate average intensities of the \oxysix\ 1032~\AA\ line, 1038~\AA\ line, and the \cartwostar\ line. They are 3190$\pm$450$, 1520\pm$350, and 1600$\pm$330 photons sec$^{-1}$ cm$^{-2}$ sr$^{-1}$, respectively. Our \oxysix\ signals are neither especially bright nor especially dim when compared with the mid latitude ($35\arcdeg \leq b \leq 60\arcdeg$) \fuse\ detections in the \citet{otte_dixon} and \citet{dixon_etal_06} survey. The larger of these compilations, that of \citet{dixon_etal_06}, lists 17 sight lines having 2 $\sigma$ and 3 $\sigma$ detections of moderate velocity (-100 km s$^{-1}$ $\leq v \leq$ 100 km s$^{-1}$) \oxysix\ 1032 \AA\ emission. Of these 17 signals, 10 were brighter and 7 were dimmer than our 1032 \AA\ signal. We also examined the data for signs of other cosmic emission lines. No significant signals were found. Our search for \carthree\ emission at 977.02 \AA\ in the satellite-night SiC 1B spectrum yielded an intensity $\pm 1\sigma$ of 1300 $\pm$ 1740 photons sec$^{-1}$ cm$^{-2}$ sr$^{-1}$. Similarly, our search for \nittwo\ emission at 916.71 \AA\ in the same spectrum yielded 510 $\pm$ 1250 photons sec$^{-1}$ cm$^{-2}$ sr$^{-1}$. Although neither species is observed in our dataset, both are observed at the $\geq2\sigma$ level in the on-filament observation (\carthree: 4700 $\pm$ 1300 and 2600 $\pm$ 1000 photons sec$^{-1}$ cm$^{-2}$ sr$^{-1}$ in the nighttime SiC 1B and SiC 2A spectra, respectively; \nittwo: 2300 $\pm$ 1100 photons sec$^{-1}$ cm$^{-2}$ sr$^{-1}$ in the night time SiC 1B spectrum, \citet{shelton_2003}). Perhaps the observed photons came from the filament itself. Though, if this is the case, then the \carthree\ intensity is remarkably bright, considering that theoretical predictions for evaporating clouds \citep{slavin}, predict an order of magnitude dimmer emission. \subsection{Determination of the \oxysix\ Intensity from the Halo}\label{subsect:oviintensity} Along our line of sight, the combined intensity arising from both members of the \oxysix\ doublet is 4710 $\pm$ 570 photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$. The observed intensity can be apportioned between the Local Bubble (plus heliospheric) and `halo' components if we assume that the local emission is approximately constant over small angular separations. The observation of an opaque filament on a nearby direction, $\ell=278.6^{\rm{o}},b=-45.3^{\rm{o}}$, set upper limits on the local \oxysix\ intensity \citep{shelton_2003}. For the local contribution, we take the tightest 1 $\sigma$ upper limit (due to random variation only) on the doublet from the day$+$night data for the 1032 \AA\ emission line and the assumption that the 1038 \AA\ line is half as strong as the 1032 \AA\ line. This doublet intensity is $30\pm{340}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$. However, since the Local Bubble cannot produce negative photons, we adjust the estimated intensity arising within 230 $\pm$ 30~pc of the Earth (thus within 160 $\pm$ 20~pc of the Galactic plane) to $30^{+340}_{-30}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$. Thus, for our sight line, the remainder and vast majority of the observed intensity ($4680^{+570}_{-660}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$) is produced in the halo. Note that the filament data were later included in the \citet{dixon_etal_06} catalog. To understand the \citet{dixon_etal_06} reanalysis, the reader needs to first know that the filament was observed in 5 sets of grouped LWRS exposures, taken over a 2 year period, towards 3 very similar directions on the sky (($\ell=278.58^{\rm{o}},b=-45.31^{\rm{o}}$), ($\ell=278.59^{\rm{o}},b=-45.30^{\rm{o}}$), and ($\ell=278.63^{\rm{o}},b=-45.31^{\rm{o}}$)). Figure 3 of \citet{shelton_2003} shows that all three directions point toward a dense knot in an infrared-bright filament. \citet{shelton_2003} separately plotted the spectra for each of the three directions, found no important systematic differences between the spectra, then added the spectra and searched for emission in the \oxysix\ 1032 and 1038 \AA\ lines. No emission was found near the Milky Way's Local Standard of Rest (LSR) velocity in either line, in either the data taken during the satellite night portion of the orbit or data taken during the the satellite day + night portions of the orbit. In our shadowing analysis (previous paragraph), we have adopted the tightest $1\sigma$ upper limit for the doublet, excluding systematic uncertainties, from the \citet{shelton_2003} analysis. It was derived from the day+night data for the \oxysix\ 1032 \AA\ region. The later analysis by \citet{dixon_etal_06} differed from that of \citet{shelton_2003} because it included only data taken during orbital night, searched for only the 1032 \AA\ emission line, and used a different version of the pipeline, different choices of pulse height cutoffs, and a different spectral fitting algorithm. They noticed a high velocity feature in their combined dataset, but determined that it was primarily associated with their data taken towards $\ell=278.59^{\rm{o}},b=-45.30^{\rm{o}}$. In that direction, their \oxysix\ 1032 \AA\ feature's velocity and intensity are $206\pm13$ km~s$^{-1}$, and $3.0 \pm 0.6 \times 10^3$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$, respectively. In contrast, neither of their spectra in the other directions revealed statistically significant \oxysix\ 1032 \AA\ emission. From this, \citet{dixon_etal_06} concluded that \oxysix\ in the Magellanic Stream moving at $\sim200$~km~s$^{-1}$ relative to the LSR had been viewed through a previously unknown hole in the filament and that the other observations viewed opaque portions of the filament. When \citet{dixon_etal_06} excluded the red-shifted feature from their analysis of the combined dataset, they derived a $2\sigma$ upper limit on the \oxysix\ 1032 \AA\ emission of $600$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$ (see the last paragraph of their Appendix A.2 and note that their naming convention differs from that of \citet{shelton_2003}), which is roughly similar to the $2\sigma$ upper limit reported by \citet{shelton_2003}. Given that they did not find LSR rest frame emission, they did not modify the conclusions of \citet{shelton_2003} regarding the physical conditions in the Local Bubble. If the high velocity \oxysix\ reported by \citet{dixon_etal_06} were to lie along the off-filament line of sight analyzed in this paper, then its 1032 \AA\ emission would be $90\%$ as bright as and 0.6~\AA\ longwards of the \oxysix\ 1032 \AA\ feature we observed. As can be seen in Fig.~\ref{fig:spectrum}, our spectra are dim around $\lambda = 1032.64$~\AA, the wavelength of \oxysix\ traveling at 206~km~s$^{-1}$. Our wavelength calibration in this part of the spectrum is good to within $\sim$0.034~\AA\ ($\sim$10~km~s$^{-1}$); therefore, the observed velocity of the \oxysix\ 1032 \AA\ feature in our spectrum is inconsistent with the reported velocity of the \citet{dixon_etal_06} feature. Thus, we conclude that the high velocity, presumably extra-galactic feature seen by \citet{dixon_etal_06} in the I2050501 + I2050510 data has not appeared in the off-filament data and does not affect our analysis of the halo's emission. Next we consider the effects of extinction. Our off-filament line of sight is only mildly extincted, but we obtain somewhat different results from different datasets. According to the {\it{DIRBE}} and {\it{IRAS}} data \citep{schlegel_etal}, its color excess is E(B-V) = 0.0217, implying that $N({\rm H~I})= 1.06\times 10^{20}{\rm cm}^{-2}$ if we use the empirical relation $N({\rm H~I})/E(B-V)=4.93\times 10^{21}{\rm cm}^{-2}$ derived by \citet{diplas_savage}. Thus, we expect that 25$\%$ of the photons originating beyond the obscuring material have been extincted \citep{fitzpatrick}. However, according to the Leiden-Argentine-Bonn Survey \citep{kalberla_etal}, the column density of neutral hydrogen, $N_{\rm HI}$, is $2.04 \times 10^{20}$~cm$^{-2}$ on the two nearest lines of sight ($(l,b) = 278.69, -47.08$ and $278.80, -47.00$), equating to a color excess of 0.0414 \citep{diplas_savage} and an extinction loss of $42\%$ \citep{fitzpatrick}. We recognize that this may slightly overestimate the amount of hydrogen along our line of sight because some small portion of the $0\fdg 6$ (HPBW) diameter beam could be responding to the emission from the filament. We will take $N_{\rm HI} = 2.0 \times 10^{20}$~cm$^{-2}$ as the upper end of the possible range. For a lower limit to the absorption, we consider the extreme case that a large fraction of the neutral hydrogen might be beyond the \oxysix-emitting gas (almost all of which must be more distant than the shadowing filament). A limit to that fraction is defined by an estimate for the amount of H~I that exists within the Local Bubble. \citet{lallement_etal_2003} found that well in front of the distance to the filament (230 $\pm$ 30 pc) the equivalent widths of intervening Na {\small{I}} D absorption in the spectra of stars in the general vicinity of our sight line are at least 20 m\AA\ and perhaps even as large as 50 m\AA\ (corresponding to $N_{\rm HI}= 0.5$ to $2.0 \times 10^{20}$~cm$^{-2}$, respectively.) Thus, there must be some extinction between our location and the \oxysix\ ions. Allowing for small scale variations in ISM column densities and for possible revisions in the distance estimates, we take $N_{\rm HI}= 0.5 \times 10^{20}$~cm$^{-2}$ as our lower limit. Also, we take $N_{\rm HI} = 1.0 \times 10^{20}$~cm$^{-2}$ as our nominal value. For $N_{\rm HI}= 1.0 \times 10^{20}$~cm$^{-2}$, the intrinsic intensity, $I_{\rm OVI}$, is $6110^{+740}_{-860}$ ($11.8^{+1.4}_{-1.7} \times 10^{-8}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$). For the extreme range of possible extincting column densities, $N_{\rm HI}= 0.5$ to $2.0 \times 10^{20}$~cm$^{-2}$, the intrinsic intensity, $I_{\rm OVI}$, is $5350^{+650}_{-750}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$ ($10.3^{+1.2}_{-1.5} \times 10^{-8}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$) to $7960^{+970}_{-1120}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$ ($15.3^{+1.9}_{-2.2} \times 10^{-8}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$). \subsection{An Estimate for the Halo's \oxysix\ Column Density}\label{NOVI_estimate} In order to determine a characteristic electron density for the hot gas, we must know the halo's \oxysix\ column density ($N_{\rm OVI}$). To estimate $N_{\rm OVI}$ along our line of sight, we have at our disposal only determinations made in other directions about $15\arcdeg$ away \citep{wakker_etal}. We make use of 4 such measurements and evaluate their average column density. This average equals $2.34 \times 10^{14}$~cm$^{-2}$ towards (1) NGC~1705, (2) Fairall~9, (3) several targets within the LMC, which we treat as a single data point, and (4) several targets within the SMC, which we also treat as a single data point \citep{savage-etal-2003}. While this mean value represents our best estimate for our direction, we recognize that \oxysix\ column densities vary markedly over angular separations of $15\arcdeg$, and thus our adopted value could deviate substantially from the true value. We now estimate the size of the error in our determination. By examining Figure~11 of \citet{savage-etal-2003}, we can better estimate the expected deviation from the parent population than simply evaluating the internal dispersion of the nearest 4 lines of sight considered here. The average difference in column density of individual sight lines in pairs with $\sim 15\arcdeg$ separations in the data set of \citet{savage-etal-2003} is $35\%$ of the mean of each pair. Thus, on average, each determination differs from the mean of the two by $17.5\%$. The root-mean-square ($rms$) deviations should be $\sqrt{\pi/2}$ times this value if the distribution is Gaussian, and individual samples drawn from the parent population have dispersions that are a factor of $\sqrt{2}$ larger, leading to a fundamental $rms$ deviation of $31\%$ from what we could consider to be a ``true'' overall mean value over some arbitrarily large sector of the sky. The mean of 4 randomly drawn samples from the parent population should deviate from the overall mean by only half as much ($15.5\%$). Thus, our estimate for $N_{\rm OVI}$ along our line of sight based on the other 4 measurements could be in error by an amount equal to the error in the mean ($15.5\%$) combined in quadrature with the intrinsic fluctuations in the individual lines of sight ($31\%$), giving a total uncertainty of $34.7\%$. Thus, we estimate the \oxysix\ column density and 1 $\sigma$ error for our line of sight should equal $2.34 \pm 0.81 \times 10^{14}$~cm$^{-2}$. From this value, we must subtract the Local Bubble's column density, which we estimate from its \oxysix\ volume density and radius. Three of the white dwarfs in the \citet{savage_lehner} survey of nearby stars are within 30$^{\rm{o}}$ of our pointing direction. The volume densities on these short and nearby lines of sight range from $2.01$ to $10.6 \times 10^{-8}$~\oxysix~ions~cm$^{-3}$. From Figure~6 in \citet{lallement_etal_2003}, we take the radius of the Local Bubble in our direction to be 80 to 140~pc. From the minimum radius and \oxysix\ volume density, we estimate the minimum Local Bubble \oxysix\ column density to be $5.0 \times 10^{12}$~cm$^{-2}$, and from the maximum radius and \oxysix\ volume density, we estimate it to be $4.58 \times 10^{13}$~cm$^{-2}$. Conservatively adopting these values as the Local Bubble's average column density $\pm$ 1 $\sigma$ and subtracting them from the column density toward extragalactic targets yields the halo column density, $N_{\rm OVI} = 2.09 \pm 0.84 \times 10^{14}$~cm$^{-2 }$ \subsection{An Estimate for the Halo's X-Ray Brightness}\label{subsect:xraybrightness} We now use results from the ROSAT All Sky Survey to find X-ray intensities that apply to our two lines of sight. We extract the 1/4~keV X-ray count rate from the survey data for the R1 and R2 bands ($\sim110$ to 284~eV and $\sim140$ to 284~eV, respectively). A 0.4$^{\rm{o}}$ radius disk centered on our direction $l=278.7^{\rm{o}},b=-47.1^{\rm{o}}$, has a combined \rosat\ R1 $+$ R2 countrate of 1322 $\pm$ 55 $\times 10^{-6}$ (\citet{snowden_etal_97}, retrieved with the X-Ray Background Tool: http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/xraybf/xraybg.pl). This countrate includes contributions from the halo, Local Bubble, heliospheric charge exchange, and extragalactic objects. Assuming that the Local Bubble contribution is roughly uniform over small angular scales and that the heliospheric contribution is roughly time invariant, we take the Local Bubble plus heliospheric contributions to equal the countrate seen toward the nearby filament (534 $\pm$ 131 $\times 10^{-6}$ counts s$^{-1}$ arcmin$^{-2}$ for a 0.1$^{\rm{o}}$ disk centered on $l=278.6^{\rm{o}},b=-45.3^{\rm{o}}$). In order to estimate the extragalactic contribution, we draw upon the parameterizations of Miyaji et al. (1998). They found that $I(E) = 10.0 \pm 0.5 \times E^{-1.42 \pm 0.03}$ and $I(E) = 11.5 \pm 0.7 \times E^{-1.42 \pm 0.03}$ photons cm$^{-2}$ k$^{-1}$ keV$^{-1}$ sr$^{-1}$ for photon energies, $E$, between 0.1 and 10 keV for the two fields they studied. We take the average: $I(E) = 10.75 \times E^{-1.42}$ photons cm$^{-2}$ k$^{-1}$ keV$^{-1}$ sr$^{-1}$. When absorbed by galactic material (here we take the {\it{total}} column to be as high as $N_{\rm{HI}} = 2.0 \times 10^{20}$~cm$^{-2}$) and convolved with the \rosat\ response function, such a spectrum yields 55.6 $\times 10^{-6}$ counts s$^{-1}$ arcmin$^{-2}$ in the R1$+$R2 bandpass, hereafter termed the R12 bandpass. Subtracting the local and extragalactic contributions from the observed countrate along our line of sight yields a halo countrate of 732 $\pm$ 142 $\times 10^{-6}$ counts s$^{-1}$ arcmin$^{-2}$. This rate applies if emission originates below all of the observed neutral hydrogen. However, we believe that the halo emission originates above an absorbing column of 0.5 to $2.0 \times 10^{20}$~cm$^{-2}$. In this case, the intrinsic countrate would be $779 \pm 151$ to $4770 \pm 930$ counts s$^{-1}$ arcmin$^{-2}$. For the analysis that will be presented in \S~\ref{temp_dist}, we will also require the halo's R6 + R7 band (hereafter termed the R67 band) countrate. The filament is too optically thin at 1.5 keV for a useful shadowing analysis. However, \citet{henley_etal} have performed spectral fits with the \rosat\ and {\it{XMM}} data for the on-filament and off-filament directions. Their analysis takes into account the extragalactic and Local Bubble contributions, as well as instrumental and other sources of noise. According to David Henley (personal communication), the halo's intrinsic intensity in the R67 band is $\sim30$ counts s$^{-1}$ arcmin$^{-2}$. \section{Discussion} \subsection{The Heavy Element Abundances}\label{subsect:abundances} Many of the calculations performed in this paper are dependent, to varying degrees, on the assumed abundance ratios of heavy elements to hydrogen, since we reported on observations of atomic transitions of either oxygen (in the case of \oxysix\ emission or absorption) or an ensemble that includes many other heavy elements (the main source of soft x-ray emission). Beyond an application to our observations, heavy element abundances also influence some theoretical aspects of our subject matter, such as all processes that depend the rate of radiative cooling of the gas. While we recognize that the abundances can vary with galactocentric distance \citep{peimbert,martin-hernandez,daflon_cunha,esteban_etal}, and even from one place to the next at a given radius from the center \citep{edvardsson_etal,rolleston_etal}, we adopt the simplest interpretation that the abundances agree with the solar values. Even here, however, there are choices to be made. While the outcomes for determinations of the solar abundances of elements heavier than oxygen have been reasonably stable through the years \citep{grevesse_sauval}, very recently there have been some substantial downward revisions for C, N and O, based on interpretations of line strengths in the context of detailed models of line formation in a convective atmosphere \citep{holweger,allendeprieto_etal,asplund_etal_04}. Of particular relevance to our work is the change in the solar abundance of oxygen relative to hydrogen, which has recently declined by 0.27~dex. While there has been some independent support for the new oxygen abundance, based on abundances found for O- and B-type stars \citep{daflon_etal}, it is discordant with models of the sound speed inside the Sun and the depth of its convective zone, based on the interpretations of helioseismological data \citep{bahcall_serenelli,bahcall_etal,antia_basu_06} coupled with models that incorporated more accurate calculations of atomic opacities \citep{badnell_etal}. The problem might be solved with a higher abundance of Ne to compensate for the decreased abundances of C, N and O \citep{antia_basu_05,drake_testa,cunha_etal}, but this proposal has been met with some skepticism \citep{asplund_etal_05}. In view of the fact the solar abundance of oxygen might still be questioned, in various sections of this paper we will discuss the consequences of adopting either the old or new values. In cases where we make our own calculations, we favor the new abundances, but our use of some older, relatively complex models discussed in \S\ref{subsect:halosnr} and \S\ref{temp_dist} were based on the old abundances. \subsection{The Physical Properties of \oxysix-bearing Gas (Isothermal Case)}\label{subsect:physicalproperties} To introduce the basic concepts of our analysis of some relevant physical parameters, we start with the simplest case where the hot gas is isothermal and at the temperature that corresponds to that where \oxysix\ has its maximum ion fraction when the gas is in CIE ($3.2 \times 10^5$~K). The density of electrons ($n_e$) in the plasma that bears the \oxysix\ ions can be calculated from the halo's \oxysix\ column density derived in \S\ref{NOVI_estimate}, intrinsic doublet intensity derived in \S\ref{subsect:oviintensity}, and temperature ($T$, assumed to be the CIE temperature, $3.2 \times 10^5$~K, but the calculation is relatively insensitive to temperature if $10^5$~K $> T > 10^6$~K). We use Equation 5 in \citet{shull-slavin}: $n_e = (4 \pi I_{\rm OVI}) / (<\sigma \nu>_e N_{\rm OVI})$, where $<\sigma \nu>_e$ is the electron-impact excitation rate coefficient. If the observed emission originates beyond an extincting column of $N_{\rm HI} = 1.0 \times 10^{20}$~cm$^{-2}$, then $n_e = 12.5^{+5.2}_{-5.3}\times 10^{-3} $ cm$^{-3}$, but if it lies beyond our extreme estimates, $N_{\rm HI} = 0.5$ or $2.5 \times 10^{20}$~cm$^{-2}$, respectively, then $n_e = 11.0^{+4.6}_{-4.7}$ or $n_e = 16.3^{+6.8}_{-6.9} \times 10^{-3}$ cm$^{-3}$ (see Table~\ref{table:parameters}.) Table~\ref{table:parameters} also lists the thermal pressure and the depth of the emission region. The thermal pressure, $p_{th}$, is calculated from the ideal gas law. Taking the cosmic abundance of the elements into account results in $p_{th}/k = 1.92 n_e T$. The depth of the emitting region, $\Delta l$, is equal to the column density of \oxysix\ divided by the product of the electron density and three calculated ratios: (H/$e$), which equals 0.833 in a fully-ionized plasma with a cosmic composition, $({\rm O/H})_\odot$, for which we adopt a value $4.57\times 10^{-4}$ given by \citet{asplund_etal_04}, and the peak value of the fractional ion concentration arising from the balance of collisional ionizations and various recombination processes, $f_{\rm O~VI}(T_{\rm max})$, calculated by \citet{nahar_pradhan}. Our result for $\Delta l = N_{\rm{OVI}} / \{ n_e \times ({\rm{H}}/$e$) \times ({\rm O/H})_\odot \times f_{\rm O~VI}(T_{\rm max}) \}$ is nearly a factor of 2 larger than that presented in previous papers (i.e. \citet{sheltonetal01}), because the oxygen abundance estimates of \citet{asplund_etal_04} are nearly a factor of 2 smaller than those of \citet{grevesse_anders}. Lastly, we calculate the time required for the gas to cool, if it were to cool solely by the emission of \oxysix\ resonance line photons. (See Table~\ref{table:parameters}.) The timescale estimates are inversely proportional to the oxygen abundance. The quoted estimates are approximately twice as large as they would have been if we had used the \citet{grevesse_anders} oxygen to hydrogen ratio. Furthermore, the quoted timescales are upper limits on the true cooling timescales for the present \oxysix-rich gas and do not include cooling of nearby, hotter gas that may, in the future, evolve through an \oxysix-rich stage. In the following subsections, we will compare these simple predictions with those of more comprehensive models. \subsection{The Extended Baseline Spectrum}\label{subsect:xrayspectrum} In this subsection, we create a long baseline spectrum from the halo's \oxysix\ and soft X-ray emission. For our comparison between the 1/4~keV and \oxysix\ intensities in \S~\ref{subsect:halosnr}, we must convert the \rosat\ R12 countrate to units of intensity. In order to estimate the conversion rate factor, we take the X-ray spectrum as that of a $T = 10^6$~K, CIE plasma, as determined by \citet{bloch_etal} and \citet{pietz_etal}; \citet{kuntz_snowden} found this temperature to be the dominant temperature component in their analysis of the halo's soft X-ray emission (the other component is $T = 3\times 10^6$~K). As a result, the halo's intensity in $\sim110$ to 284~eV photons is $2.2 \pm 0.4 \times 10^{-8}$ ergs s$^{-1}$ cm$^{-2}$ sr$^{-1}$ if the emission originates above $N_{\rm{HI}}= 0.5 \times 10^{20}$~cm$^{-2}$ and $13.4 \pm 2.6 \times 10^{-8}$ ergs s$^{-1}$ cm$^{-2}$ sr$^{-1}$ if it originates above $N_{\rm{HI}}= 2.0 \times 10^{20}$~cm$^{-2}$. The intensity of the entire 1/4~keV band amounts to only 21$\%$ of the intensity in the \oxysix\ doublet if the halo emission originates above extincting material having $N_{\rm{HI}} = 0.5 \times 10^{20}$~cm$^{-2}$ and 87$\%$ of the \oxysix\ doublet intensity if the halo emission originates above $N_{\rm{HI}} = 2.0 \times 10^{20}$~cm$^{-2}$. As an aside, it is of interest to compare these numbers with the local (Local Bubble $+$ heliospheric) X-ray to \oxysix\ ratio, which is at least 1.4 for the on-filament line of sight. Thus, the local region preferentially sheds energy via the soft X-ray emission lines while the halo preferentially sheds energy via the \oxysix\ emission lines. We note that interpreting the relationship between the \oxysix\ and X-ray emission rests on the assumption that the same regions of space are being sampled. The OVI and X-ray data are from the same directions, but have differing fields of view and differing optical depths due to the dependence of optical depth on photon energy. As a result, the assumption is not strictly correct. The extent to which this is an issue depends in part on the small scale structure in the ISM, but is anticipated to affect our results less than the other simplifications involved in our modeling. Although a collisional ionizational equilibrium spectrum at $T = 10^6$~K has been found to fit the soft X-ray data, it underpredicts the \oxysix\ intensity (even when we assume that the halo emission has been absorbed by $N_{\rm{HI}} = 2.0 \times 10^{20}$~cm$^{-2}$.) As the model's temperature is lowered, its soft X-ray emission decreases more rapidly than its \oxysix\ emission. Therefore, it is possible to find a single temperature spectrum that matches both the halo's \oxysix\ and the 1/4 keV intensities. However, the assumed temperature is significantly less than $10^6$~K, and so does not produce the soft X-ray band ratios observed across much of the high latitude sky. Therefore, we will move on to more complex models which are inspired by physically conceivable events. \subsection{The Expected Outcome for a Halo Supernova Remnant}\label{subsect:halosnr} We now examine the expected consequences for a line of sight in the halo that is dominated by the effects of a single SN event. Our rationale is that in most explanations for the hot halo gas, the gas had been heated suddenly by an energetic event in the past. Irrespective of whether the energetic events occurred above the Galactic disk (i.e. extraplanar supernova explosions and collisions between infalling clouds and the Milky Way) or in the disk and the hot gas later rose into the halo, we expect the plasma to evolve from a recently shock heated and underionized state to a tepid and overionized state; and we expect the observationally determined \oxysix\ to soft X-ray ratio to be a useful diagnostic. In the following halo SNR simulations, we found that the ratio of \oxysix\ to 1/4 keV emission rises almost monotonically throughout the remnant's life, making it a diagnostic of the remnant's maturity. We calculated the ratio of \oxysix\ to \rosat\ 1/4~keV intensities from a simulated, extraplanar supernova remnant. For the simulation, we set the ambient density, explosion energy, and ambient nonthermal pressure to 0.01 atoms cm$^{-3}$, $0.5 \times 10^{51}$~ergs, and 1800 K cm$^{-3}$, respectively. As in \citet{shelton_1999}, the simulation included thermal conduction and non-CIE radiative cooling. The ionization and recombination rates were calculated using the \citet{gaetz_edgar_chevalier} tables and the spectra were calculated from the plasma's non equilibrium ionization level populations and the \citet{raymond_smith} algorithm. We used the \citet{grevesse_anders} abundance tables, in which the oxygen to hydrogen ratio is approximately a factor of 2 greater then that found by \citet{asplund_etal_04}. The predicted \oxysix\ intensity roughly scales with the adopted oxygen abundance, while the 1/4 keV soft X-ray spectrum is mostly unaffected. Here, we describe the plasma's evolution and the consequent \oxysix\ to X-ray ratio's evolution. When the halo supernova remnant is very young (age $<$ 10,000 yrs), its most emissive portion is the hot, dense gas immediately behind the shock front. Due to the rapid ionizations, the recently shock-heated gas in this zone contains \oxysix\ and higher ions. Although this plasma emits both \oxysix\ resonance line and soft X-ray photons, its temperature ($T > 10^7$~K) is far too high for optimum \oxysix\ resonance line emission. As a result, the ratio of \oxysix\ to soft X-ray intensities is less than the observed ratio (see Figure~\ref{fig:SNRratio}). As time passes, the shock front slows, thus heating the gas in and just behind the shock to lesser temperatures and leading to a slight increase in the \oxysix\ to X-ray ratio. Before the remnant is $\sim50,000$ years old, both the hot dense gas near the shock front and the hot rarefied gas in its interior are drastically out of CIE. When the remnant is between 50,000 and 100,000~years old, most of its plasma comes into CIE, but not in the usual manner. Rather than maturing sufficiently for its ionization levels to come into equilibrium with the gas temperature, the gas temperature drops sufficiently as to match the ionization levels (see Figure~4 of \citet{shelton_1999}). At this time, the most important gas, that immediately behind the shock, is close to, but not yet in CIE. During or just before this era, the simulation's \oxysix\ to 1/4~keV ratio crosses that of the observed ratio (1.1, assuming that the intrinsic emission had been absorbed by a $2.0 \times 10^{20}$~cm$^{-2}$ column, to 4.7, assuming that the intrinsic emission had been absorbed by a $0.5 \times 10^{20}$~cm$^{-2}$ column). These observationally determined ratios are marked on Figure~\ref{fig:SNRratio} by horizontal dashed and dot-dashed lines which cross the SNR ratio curve when the remnant is $\sim$40,000 and $\sim$70,000~years old, respectively. As the remnant continues to cool by adiabatic expansion and radiative cooling, its atoms begin to recombine. When the remnant is between 100,000 and 250,000~years, its shock front becomes too cool to produce much \oxysix\ or soft X-ray emission. Thus the shock front no longer outshines the bubble's interior. The remnant's \oxysix\ to \rosat\ 1/4~keV ratio, now primarily due to the remnant's ``overionized'' interior, rises. Henceforth, recombinations from \oxyeight\ to \oxyseven\ to \oxysix\ provide the remnant's center with \oxysix\ ions while reducing its supply of higher, more X-ray emissive, ions. In time, the temperature drops from $T \sim 10^6$~K down to $T \sim 10^5$~K, causing the soft X-ray emission function to plummet (although the \oxysix\ resonance line emission function remains nearly constant.) As a result, the \oxysix\ versus 1/4~keV ratio continues to rise. In its final million years, the remnant is tepid ($T =$ several $\times 10^4$~K), contains some \oxysix\ ions, but few higher ions. Thus it produces an enormous \oxysix\ to 1/4~keV ratio, as shown in the plot. We observe similar trends in simulated supernova remnants having greater ambient densities, ambient nonthermal pressures, and/or explosion energies, though with some variation in the age when the model matches the observational ratio. Assuming that the observed gas can be compared to that in an undisturbed, extraplanar supernova remnant bubble, the time since heating and the lifetime of this gas are on the order of $10^4 - 10^5$ and $10^7$~years, respectively. The cooling timescale exceeds that calculated directly from the \oxysix\ data (Table~\ref{table:parameters}) because the SNR contains a reservoir of hotter, more highly ionized gas that will eventually transition through the \oxysix\ level. Assuming that the other possible sources of hot gas behave similarly to simulated halo SNRs, we suggest that the resulting structure is middle aged. \subsection{The Volume Distribution Function of Temperatures}\label{temp_dist} \subsubsection{The Basic Assumptions}\label{subsubsect:assumptions} We move on to a more generalized picture and propose that the hot gas in the halo of our Galaxy is a heterogeneous mixture of regions having plasmas at different temperatures, created possibly by many SN events whose influences on the halo medium have merged together. A convenient characterization of the temperature distribution function over volume can take on the form of a power-law, one that extends from $10^5\,$K (above which appreciable amounts of \oxysix\ are expected in CIE) to a sharp cutoff at some high temperature $T_{\rm{cut}}$. Within the paradigm that the hot gas is created by supernova shock waves that are produced in the halo or that escape from the disk, the origin of $T_{\rm{cut}}$ can be interpreted to arise from either a generalized limit on the supernova shock velocities or, alternatively, from our recognition that gases with temperatures that are too high may escape very rapidly in the form of a very low density, high velocity wind, which is difficult to detect. Since all of the observable effects of the hot gas represent line integrals of various physical quantities through the Galactic halo, there is some benefit in our starting with a formulation that describes how the many differential length segments $dl$ within the population of discrete, homogeneous gas regions are distributed over temperature. We do this by specifying a transformation between $dl$ and $d\ln T$, \begin{eqnarray}\label{dl} dl&=&BT^\beta d\ln T~{\rm for}~T<T_{\rm cut}\cr &=&0~{\rm for}~T>T_{\rm cut}~, \end{eqnarray} where $B$ is a distance scale factor. In a broad, statistical sense, this distribution function describes how temperatures are weighted according to volume fractions, but it says nothing about the internal electron densities $n_e$ within the length segments. In the development of our interpretation, we impose a simplifying constraint that the Galactic halo is approximately isobaric. We use this universal pressure constraint to tie the electron density to temperature and thermal pressure, i.e., $n_e=p_{th}/(1.92kT)$. While we must accept the reality that some pressure variations can exist from one location to another, we can assume that the magnitude of such variations are small compared to the vastly different temperatures that we adopt in our model. Our representative pressure $p_{th}$ is a free parameter that we will determine from the ratio of $I_{\rm OVI}/N_{\rm OVI}$ after we have solved for the power-law coefficient $\beta$ and temperature limit $T_{\rm cut}$. \subsubsection{How the Observations Relate to the Model}\label{subsubsect:relate} The column density of \oxysix\ is given by the expression \begin{equation}\label{NOVI_dl} N_{\rm OVI}=\left( {{\rm O}\over {\rm H}}\right)_\odot \left( {{\rm H}\over e}\right)~\int f_{\rm OVI}(T) n_e dl \end{equation} where $(\rm O/H)_\odot$, $({\rm H}/e)$, and $f_{\rm OVI}(T)$ were first used in \S\ref{subsect:physicalproperties}. We can rewrite Eq.~\ref{NOVI_dl} in terms of an integral over $\ln T$ of the temperature distribution we adopted, \begin{equation}\label{NOVI_dlnT} N_{\rm OVI}=\left( {{\rm O}\over {\rm H}}\right)_\odot \left( {{\rm H}\over e}\right)~{p_{th}B\over 1.92k} \int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}}f_{\rm OVI}(T)T^{\beta -1}d\ln T \end{equation} by making a substitution of the terms in Eq.~\ref{dl} for $dl$, using the relation $n_e=p_{th}/(1.92kT)$, and noting that \oxysix\ is rare at temperatures below $10^5$~K. We can develop similar equations for the expected strengths of the line radiation from \oxysix\ or the intensities of soft X-rays. For the emission from both members of the \oxysix\ doublet, we anticipate that \begin{equation}\label{IOVI_dl} I_{\rm OVI}=\int r_{\rm OVI}(T)n_e^2 dl~, \end{equation} where the emission rate coefficient $r_{\rm OVI}(T)$ per unit emission measure is given by \begin{equation} \label{rOVI} r_{\rm OVI}(T)=3.09\times 10^{18}\left( {{\rm O}\over {\rm H}} \right)_\odot \left( {{\rm H}\over e}\right)f_{\rm OVI}{<\sigma v>(T) \over 4\pi}~{\rm photons~cm}^{-2}{\rm s}^{-1}{\rm sr}^{-1}({\rm cm}^{-6} \,{\rm pc})^{-1}~. \end{equation} As we did in \S\ref{subsect:physicalproperties}, we adopt an analytical formulation of $<\sigma v>(T)/(4\pi)$ that was specified by Shull \& Slavin (1994). Again after substituting the expression in Eq.~\ref{dl} for $dl$ and $p_{th}/(1.92kT)$ for $n_e$ we find that \begin{equation}\label{IOVI_dlnT} I_{\rm OVI}={p_{th}^2B\over (1.92k)^2} \int_{\ln 10^5 \rm{K}}^{\ln T_{cut}} r_{\rm OVI}(T)T^{\beta -2}d\ln T~ \end{equation} A similar equation can be expressed for the soft X-ray emission, \begin{equation}\label{R12_dlnT} R_{12}={p_{th}^2B\over (1.92k)^2}\int_{\ln 10^5 \rm{K}}^{\ln T_{cut}} r_{12}[N_{\rm HI},T] T^{\beta -2}d\ln T~, \end{equation} where $r_{12}[N_{\rm HI},T]$ is the emission rate coefficient as a function of temperature for 0.25~keV X-rays, matched to the responses of the \rosat\ bands 1 and 2 and expressed in the units ${\rm counts~cm}^{-2}\,{\rm s}^{-1}\,{\rm arcmin}^{-2}({\rm cm}^{-6}\, {\rm pc})^{-1}$ (Snowden et al. 1997).\footnote{The cosmic abundances adopted by Snowden et al. (1997) did not incorporate the recent downward revisions of the solar photospheric abundances of C (Allende Prieto et al. 2002), N (Holweger 2001) or O (Asplund et al. 2004). The correction to allow for these abundance changes should be rather small, since the line emission over the energy range of interest is dominated by lines from other, much heavier elements, such as Si, S, Mg and Fe; see, e. g., Kato (1976).} This coefficient includes the effect of an energy-dependent foreground absorption represented by $N_{\rm HI}$. \subsubsection{Evaluation of Parameters}\label{subsubsect:parameters} To evaluate the free parameters $\beta$, $T_{\rm cut}$ and $p_{th}/k$, we compare three ratios of observable quantities with their expectations within our formalism. First, we may constrain the value of $\beta$ by matching the observed ratio of $I_{\rm OVI}$ to $R_{12}$ to the expression \begin{equation}\label{ratio_IOVI_R12} {I_{\rm OVI}\over R_{12}}={\int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}} r_{\rm OVI}(T)T^{\beta -2}d \ln T\over \int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}} r_{12}[ N_{\rm HI},T] T^{\beta -2}d \ln T}~. \end{equation} While the result depends on an adopted value of $T_{\rm cut}$, for reasonably high values of this quantity the effect is small. We find that for $I_{\rm OVI}=4680\,{\rm photons~cm}^{-2}{\rm s}^{-1}{\rm sr}^{-1}$ and $R_{12}=0.000733\,{\rm counts~cm}^{-2}{\rm s}^{-1}{\rm arcmin}^{-2}$ we obtain $I_{\rm OVI}/R_{12}=6.4\times 10^6$ (in the same units) with an uncertainty of 24\% using the errors stated in \S\ref{sect:observations}. The first row of Table~\ref{pwr_law_parameters} lists the derived values of $\beta$ for the best and limiting values of foreground H~I absorption. The top row of panels in Fig.~\ref{multipanel} shows how $I_{\rm O~VI}/R_{12}$ changes with $\beta$ and demonstrates that the dependence on $T_{\rm cut}$ is not important for $\beta < 1.5$ and is a small effect for $\beta < 2$. A strong response to $T_{\rm cut}$ arises from the ratio of high energy X-ray emission to the intensity at lower energies, as indicated in the second row of panels in Fig.~\ref{multipanel}, where we have made use of an equation that is identical in form to Eq.~\ref{ratio_IOVI_R12} except that it compares the ratio of X-ray emission at 1.5~keV (the sum of intensities recorded in \rosat\ Bands 6 and 7) to that at 0.25~keV (the sum in \rosat\ Bands 1 and 2), \begin{equation}\label{ratio_R67_R12} {R_{67}\over R_{12}}={\int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}}r_{67} [N_{\rm H~I},T]T^{\beta -2}d \ln T\over \int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}}r_{12} [N_{\rm H~I},T]T^{\beta -2}d \ln T}~ \end{equation} From the observed value of $R_{67}/R_{12}=0.04$ (\S\ref{subsect:xraybrightness}), we arrive at values of $\log T_{\rm cut}$ given in the second row of Table~\ref{pwr_law_parameters}. Finally, we solve for $p_{th}/k$ by matching the observed ratio of $I_{\rm OVI}$ to $N_{\rm OVI}$ to the formula \begin{equation}\label{ratio_IOVI_NOVI} {I_{\rm OVI}\over N_{\rm OVI}}= {p_{th}\over 1.92k}~{\int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}}f_{\rm OVI} [<\sigma v>(T)/(4\pi)]T^{\beta -2}d \ln T\over \int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}}f_{\rm OVI}T^{\beta -1}d \ln T}~ \end{equation} Values of $p_{th}$, found for $I_{\rm OVI}/N_{\rm OVI }=2.24\times 10^{-11}$ photons s$^{-1}$ sr$^{-1}$ with an uncertainty of about 50\%, are listed in the third row of Table~\ref{pwr_law_parameters}. From the flatness of the curves shown in the third row of panels in Fig.~\ref{multipanel}, the value of $p_{th}$ is not strongly influenced by the choice of $\beta$ (and has virtually no dependence on $T_{\rm cut}$). The dominant uncertainty in $p_{th}$ arises from the expected error in $I_{\rm OVI} / N_{\rm OVI}$. It is important to check that the computed width for thermal Doppler broadening of \oxysix\ in the model does not exceed the observed widths of the actual absorption profiles observed in the Galactic halo. Our expectation is that \begin{equation}\label{vel_var} <v^2>={\int n({\rm OVI})<v({\rm OVI})^2> dl\over \int n({\rm OVI})dl} = {k\over 16m_{\rm p}}~{\int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}} f_{\rm OVI}(T)T^\beta d\ln T\over \int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}} f_{\rm OVI}(T)T^{\beta -1}d \ln T}~, \end{equation} where $m_{\rm p}$ is the mass of a nucleon. For our derived values of $\beta$, we find that the Doppler parameter $b=(2<v^2>)^{\onehalf}$ ranges from 19 to $23\,{\rm km~s}^{-1}$. These values are smaller than the representative observed values, which are of order $60\,{\rm km~s}^{-1}$. Evidently, even with our allowance for some \oxysix\ arising from temperatures above $10^6\,$K, the expected profile widths are still much less than the kinematic broadening that takes place in the halo. As with our findings on $p_{th}/k$, the bottom row of panels in Fig.~\ref{multipanel} shows that the outcome from Eq.~\ref{vel_var} is nearly independent of $T_{\rm cut}$. So far, all of our evaluations of various ratios of certain quantities have not had to incorporate the constant $B$ that appears in Eq.~\ref{dl}, since this term canceled out in each case. This constant is important, however, if we wish to relate the results from the observations to either a total length scale along our line of sight, $L=\int dl$, or the values for the electron density, $n_e(T)$. By solving either Eqs.~\ref{NOVI_dlnT}, \ref{IOVI_dlnT} or \ref{R12_dlnT} one can derive this constant, which varies markedly with the adopted value of $\beta$ (as it should). Values of $B$ for the various combinations of $\beta$ and $N({\rm H~I})$ are listed in Table~\ref{pwr_law_parameters}. From an appropriate value of $B$ we can evaluate the total length of hot gas along our path \begin{equation}\label{L} L=B\int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}}T^\beta d\ln T=B(T_{\rm cut}^\beta-10^{5\beta})/\beta~. \end{equation} Outcomes for $\log L$ (in pc) are listed in Table~\ref{pwr_law_parameters}. If the gas above and below the plane is stratified in layers parallel to the plane, then a representative vertical sight line (i.e. one perpendicular to the galactic disk) intersects less material than our sight line at an intermediate galactic latitude of $b = -47.1\arcdeg$. Thus, calculations for the vertical sight line would require that the total length scale, $L$ in Eq.~\ref{L} be foreshortened by a factor of $\sin(|b|)$. Similarly, $N_{\rm{OVI}}$ in Eq.~\ref{NOVI_dlnT}, $I_{\rm{OVI}}$ in Eq.~\ref{IOVI_dlnT}, $R_{12}$ in Eq.~\ref{R12_dlnT}, together with $N_e$ and $dEM(T)$ in the upcoming Eqs.~\ref{Ne} \ref{EM}, would require reduction by the same factor. On the one hand, it is important to note that $L$ represents a minimum length scale, since parcels of gas could be scattered over a longer distance with some unseen gas phase filling in the intervening gaps. On the other hand, the fraction of $L$ that has an appreciable concentration of \oxysix\ is small. Figure~\ref{plt2} illustrates what the concentration of \oxysix\ would look like if all of the gas parcels at different temperatures were sequenced along a line in order of increasing temperature. Most of the \oxysix-bearing gas is confined within a region that is less than about 100~pc thick, with the remaining, much greater length filled with a plasma that is too hot to contain much O VI. This thickness is considerably less than the 2.3~kpc scale height found for \oxysix\ in the Galactic halo by Savage et al. (2003). Evidently, our large observed ratio of $I_{\rm OVI}/N_{\rm OVI}$ shows that the gas with temperatures of around 300,000~K is highly clumped, even when one acknowledges that there can be a broad distribution of temperatures. The internal densities $n({\rm OVI})$ within each clump, which in our model can reach as high as $\sim10^{-6}~{\rm cm}^{-3}$, is about two orders of magnitude higher than an overall average $<n({\rm OVI})>=1.7\times 10^{-8}{\rm cm}^{-3}$ that was found in a \fuse\ survey of the Galactic disk \citep{bowen_etal}. The column density of electrons along our line of sight is given by \begin{equation}\label{Ne} N_e=B\int_{\ln 10^5 \rm{K}}^{\ln T_{\rm cut}}n_e(T)T^\beta d\ln T ={p\over 1.92k}~{B(T_{\rm cut}^{\beta -1}-T^{5(\beta-1)})\over \beta -1}~, \end{equation} and we list outcomes for the appropriate values of $\beta$ and $N_{\rm HI}$ in Table~\ref{pwr_law_parameters}. In \S\ref{subsect:abundances} we discussed the possibility that the abundance of oxygen might be higher than the one that we adopted from \citet{asplund_etal_04}. If we were to adopt the old, higher abundance value given by \citet{grevesse_sauval} (larger by 0.27~dex), the values of $\beta$ listed in Table~\ref{pwr_law_parameters} would increase by about 0.5 because the relative proportion of the plasma at temperatures that emit \oxysix\ line radiation is reduced, when compared to the higher temperature material that emits soft X-rays. (Recall that the predicted X-ray emission is not strongly influenced by changes in the abundance of oxygen, as we stated in an earlier footnote.) For the three values of foreground $N$(H~I), (0.5, 1.0, $2.0\times 10^{20}\,{\rm cm}^{-2}$), the old oxygen abundance changes $\log p_{th}/k$ by (+0.03, +0.03, +0.05)~dex, respectively. Values of $\log L$ change by (+0.23, +0.09, +0.17)~dex, while $\log N_e$ changes by (+0.05, +0.01, +0.03)~dex. \subsubsection{The Total Radiative Loss Rate for the Halo}\label{subsubsect:lossrate} Here, we assume that the observed line of sight provides a fair representation of the halo as a whole. This assumption allows us to calculate the total radiative energy loss rate associated with hot gas in the halo, a rate that can be usefully compared with the energy injection rate. To compute the radiative energy loss rate from the hot gas in the halo, we must know how the differential emission measure $dEM=n_e^2(T)dl$ varies with temperature. With our reformulation of $dl$ in terms of $d\ln T$ we find that \begin{equation}\label{EM} dEM(T)=\left({p_{th}\over 1.92k}\right)^2 B T^{\beta -2}d\ln T \end{equation} along our line of sight. The second to the last group of numbers in Table~\ref{pwr_law_parameters} shows the logarithms of the coefficient in front of the $T^{\beta - 2}$ term in the equation. With our expression for the emission measure as a function of $T$, we are now prepared to estimate the power radiated by the gas at temperatures above $10^5\,$K. For a cooling function $\Lambda_N$ that is normalized to the local product of electron and ion densities $n_en_i$, we adopt the values listed by \citet{sutherland_dopita} that apply to a plasma that has a solar composition and nonequilibrium ion fractions in a regime of radiative, isochoric cooling, starting with an initial temperature of $10^{7.5}\,$K. If we were to suppose that our line of sight shows a fair representation of the cooling per unit area $dU/dt$ by hot gas on both sides of the Galactic plane in our region of the Galaxy, we find that \begin{equation}\label{dEdt} dU/dt=2\sin(|b|)(0.917) \int_{\ln 10^5\rm{K}}^{\ln T_{\rm cut}}EM(T)\Lambda_N d\ln T \end{equation} where $b$ is the Galactic latitude of our line of sight ($-47\fdg 1$), the factor 2 accounts for both sides of the plane, and the factor 0.917 allows for the fact that our expression for $EM(T)$ is cast in terms of $n_e^2$ whereas the normalization of the function expressed by \citet{sutherland_dopita} is normalized to $n_en_i$. Values of $\log dU/dT$ are listed in the last group of numbers in Table~\ref{pwr_law_parameters}. Several important qualifications must be expressed about these numbers. First, we have truncated the calculation at a lower limit $T=10^5\,$K because we have no ability to sense gas at lower temperatures. If an extrapolation of our power-law representation seems plausible, we should expect to find some additional energy radiated below $10^5\,$K. This is demonstrated in Fig.~\ref{plt3}, where we have plotted the shape of the integrand in Eq.~\ref{dEdt} as a function of $\log T$. Second, \citet{sutherland_dopita} used the solar abundances given by \citet{anders_grevesse} instead of the more modern values of \citet{allendeprieto_etal} that we adopted. Thus, if the function $\Lambda_N$ were to be recalculated using the newer abundances, we would find a somewhat lower cooling rate, but by a factor that is less severe than the changes in the C, N and O abundances, since other elements are also important coolants. If indeed the older abundances are correct, the ripple effect from the changes in the parameters derived in \S\ref{temp_dist} above could reduce the emission power by up to -0.4~dex. Aside from these possible offsets, our determination should be free of bias, and indeed the values of \oxysix\ and X-ray emission found here are fairly typical of those found along other high latitude sight lines. Nevertheless, it is important to note that the distribution of hot gas in the Galactic halo is extremely uneven; hence our single line of sight does not give a very accurate representation of the average $EM$ on either side of the Galactic plane. The rate of cooling, $dU/dt$, is similar to the rate of energy input from supernova explosions and pre-supernova winds. The average massive SN progenitor star releases $1.4$ to $2.0 \times 10^{50}$~ergs in wind energy before it explodes, according to calculations in \citet{ferriere_1998} and \citet{leitherer_etal}. At the Sun's galactocentric radius, 18.6 massive stars and 2.6 white dwarfs explode per Myr per kpc$^{2}$ cross-sectional disk area, though many of these stars explode above the disk \citep{ferriere_1998}. If each explosion releases $10^{51}$~ergs of energy, then SN and pre-SN winds inject $7.66$ to $8.06 \times 10^{38}$ erg kpc$^{-2}$ s$^{-1}$ into the ISM. A comparison with the calculated energy loss rate for our nominal case ($5.37^{+0.80}_{-0.69} \times 10^{38}$ erg kpc$^{-2}$ s$^{-1}$) shows that the majority of the injected SN and pre-SN energy is later radiated away by hot halo gas. Since the halo cooling rate accounts for $\sim70\%$ of the SN and pre-SN wind energy injection rate, little remains to power other activities, such as large scale galactic winds. Half of the photons emitted by the halo will travel toward the galactic midplane and likely be absorbed, while the other half will travel upwards. Their absorption rate depends on the column density of gas residing above the emitting gas, which is expected to be less than $\sim1 \times 10^{20}$~cm$^{-20}$. Thus, most of the 1032 \AA\ and shorter photons will leave the system. \section{Summary}\label{sect:summary} We analyzed \fuse\ LWRS data for $\ell = 278.7^{\rm{o}}, b = -47.1^{\rm{o}}$, finding an \oxysix\ doublet intensity of 4710 $\pm$ 570 photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$. Our pointing direction is only mildly extincted, so the observed intensity included contributions from the Local Bubble and the Galactic halo. Only $2^{\rm{o}}$ from our pointing direction, the sky is heavily extincted by a filament residing 230 pc from the Earth. A previous observation of that direction indicated the Local Bubble's contribution. The difference between the intensities observed on these two sight lines, $4680^{+570}_{-660}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$, can be attributed to the halo. Given the extinction along our line of sight due to an $N_{\rm{HI}}$ of 0.5 to $2.0 \times 10^{20}$~cm$^{-2}$, the halo's intrinsic intensity is $5350^{+650}_{-750}$ to $7960^{+970}_{-1120}$ photons s$^{-1}$ cm$^{-2}$ sr$^{-1}$. We estimated the halo's \oxysix\ column density from absorption lines seen in the UV spectra of extragalactic objects. We averaged values toward the 4 nearest sight lines and then subtracted an estimate for the expected contribution from the Local Bubble. When we used it and our intrinsic \oxysix\ intensity range and treated the \oxysix-bearing gas as if it were isothermal, we found that the electron density and thermal pressure in the halo's \oxysix-rich gas are 0.01 to 0.02~cm$^{-3}$ and 7000 to 10,000~K~cm$^{-3}$, respectively. By performing a similar shadowing analysis with the \rosat\ 1/4~keV data, we determined the 1/4 keV count rate attributable to the Galactic halo along $\ell = 278.7^{\rm{o}}, b = -47.1^{\rm{o}}$. It is 732 $\pm$ 142 $\times 10^{-6}$ counts s$^{-1}$ arcmin$^{-2}$, before accounting for line of sight extinction. After accounting for extinction, the intrinsic R12 countrate is $779 \pm 151$ to $4770 \pm 930$ counts s$^{-1}$ arcmin$^{-2}$. The \oxysix\ vs 1/4~keV intensity ratio is 4.7 to 1.1, depending on whether the emitting gas is beyond $N_{\rm{HI}} = 0.5$ or $2.0 \times 10^{20}$~cm$^{-2}$, respectively. Thus, more energy leaves the system through the \oxysix\ resonance lines than through the \rosat\ 1/4~keV bandpass. In contrast, the opposite is true of the local region (Local Bubble $+$ heliospheric), where roughly twice as much energy (at least) is radiated by the 1/4~keV emission lines than by the \oxysix\ resonance line doublet. In order to estimate the maturity of the emitting plasma, we compared the \oxysix\ to soft X-ray ratio with predictions for a simulated supernova remnant at various times in its life. Around the time that the thermal temperature throughout most of the remnant approached the plasma's collisional ionizational equilibrium temperature, the spectrum produced a 4.7 to 1 ratio, coincident with the observational ratio assuming minimal extinction. Earlier in its life, the simulated remnant had produced a 1.1 to 1 ratio, coincident with the observational ratio assuming mild extinction. Specifically, these ratios were achieved when the SNR was 70,000 and 40,000 years old, respectively. We suggest that other possible hot gas structures would be similarly mature when they, too, produce a similar \oxysix\ to soft X-ray intensity ratio. Using the \oxysix\ to soft X-ray ratio, we were also able to parameterize the hot halo plasma with a power law temperature distribution ($dl = BT^\beta d\ln T~{\rm for}~10^5\,{\rm K}<T<T_{\rm cut}$). Given the nominal estimates of foreground absorption ($N_{\rm HI} = 1.0 \times 10^{20}$ cm$^{-2}$) and \oxysix\ column density ($N_{\rm OVI} = 2.09 \times 10^{14}$ cm$^{-2}$) along our line of sight, we found $\beta = 1.48 \pm 0.18$, $T_{\rm cut} = 10^{6.6}$ K, and $B = 10^{-6.16 \pm 0.37}$ K$^{-\beta}$ pc. Hot gas with a temperature between $10^5$ K and $T_{\rm cut}$ occupies $\int dl = 10^{3.44\pm0.37}$ pc along our line of sight, but the \oxysix-rich gas occupies a small fraction of this length. Assuming that our line of sight is typical of high latitude sight lines, we found the cooling rate for the halo (both sides of the plane beyond the Local Bubble) per unit cross sectional area to be $dU/dt = 10^{38.73\pm0.06}$ erg s$^{-1}$ kpc$^{-2}$. At the Sun's galactocentric radius, the hot halo's radiative cooling accounts for $\sim70\%$ of the energy injected into the ISM from SNe and pre-SN winds in the galactic disk and halo. The remaining $\sim30\%$ of the injected energy must be split between all other energy loss processes. \acknowledgements We appreciate K.D. Kuntz's assistance with the DIRBE corrected IRAS data, D. Henley's comments on X-ray observations of the Local Bubble and Galactic halo, and S. Snowden's provisions of the digitized versions of the \rosat\ response curves shown in Fig. 7 of Snowden et al. (1997). This work was funded through NASA grants NNG04GD77G and NNG04GD78G to the University of Georgia and grant NAG5-12519 to Princeton University. This paper utilized observations obtained by the NASA-CNES-CSA {\it Far Ultraviolet Spectroscopic Explorer (FUSE)} mission operated by Johns Hopkins University, supported by NASA contract NAS5-32985. \\

proofpile-arXiv_069-14245

\section{\label{sec:introduction}Introduction} Many authors have studied energies of large\cite{Vi61,HiTo85,Ad89,VePeDe91,BaGeIaNi91,EmHi89,DzBaAb96,BaKhSh97,% dCPe98,PoFoDeBaKl00,My01,Mu02,SeEr02,Ka04,KaLaSy03} and small\cite{ChSc76,AlRa81,CoEcSo84,RaTh92,smallbipolarons,AlMo94,Al96,AlBr99, AlKo02,Marsiglio95, FeRoWeMi95,WeRoFe96,FeLoWe00,LaMaPu97,FiKu97,PrAu99,FrWa99,Sil99,ZhJeWh99, BoKaTr00,BoTr01, ElShBoKuTr03}\newline\cite{IaPeCadF01,deFiCaIaMaPeVe01,Ma04,HoAivdL04,HovdL05,Zh05} bipolarons in various numbers of dimensions, and some have made calculations of bipolaron effective masses at the band minimum.\cite{EmHi89,dCPe98,PoFoDeBaKl00,CoEcSo84,LaMaPu97,Sil99,BoKaTr00, BoTr01, ElShBoKuTr03} However, we are not aware of published calculations of bipolaron energies as a function of wave vector which extend to large wave vectors except in the case of small bipolarons when the electron-phonon coupling is strong.\cite{FeRoWeMi95,WeRoFe96,FeLoWe00} In this paper we take up a study of this problem in a one-dimensional model with local interactions by use of a variational method used by Gurari for the single-polaron problem.\cite{Gu53} The method is appropriate for intermediate electron-phonon coupling strengths. It was discussed in some detail in a review article by Fr\"{o}hlich,\cite{Fr54} and, for large polarons, gives the same results for binding energies and effective masses as those obtained by Lee \etal\cite{LePi52,LeFlPi53} using different approaches. The Hamiltonian we shall use for the bipolaron problem is similar but not identical to that of the Hubbard-Holstein model,\cite{Hu63,Ho59a,Al96,AlKo02} and will be formulated in terms of center-of-mass and relative coordinates of two electrons rather than in terms of electron creation and annihilation operators. This permits us to follow the variational method for the single-polaron problem with only minor modifications. The biggest difference of our model from the Hubbard-Holstein model is our assumption of a constant bare mass for the electron. We have not yet found a way to apply the Gurari method to the Hubbard-Holstein model itself. The original motivation for this work was to help to find out whether large enhancements of electron-electron attractions mediated via phonons or other intermediate bosons predicted in the simplest perturbation approach to interactions\cite{Pa59,Ea66,Ea94,Ea94_2,Ho65} may still occur when complications due to intermediate coupling are included. For metals with Fermi energies large compared with the energy of any boson mediating electron-electron attractions, Eliashberg theory has been used to show that the net effect of enhancements of attractions is unlikely to lead to high-temperature superconductivity at high currents either in three-dimensional\cite{Ho65} or quasi one-dimensional systems.\cite{Ea94_2} However, for small Fermi energies, infinite enhancements are predicted by the simplest perturbation approach at some drift velocities in one dimension.\cite{Ea94,Ea94_2} Study of the bipolaron problem in one dimension should give some insight into how this conclusion is modified by effects beyond perturbation theory in the limit of low concentrations of electrons. Although we shall use the term ``phonon'' for the boson mediating the attraction, we have in mind that interactions mediated by plasmons may be of practical importance for understanding possible high-temperature superconductivity associated with bipolarons. There have been many published studies of bipolarons in one dimension (including studies of two-site models),\cite{RaTh92,PoFoDeBaKl00,CoEcSo84,Al96, AlKo02, Marsiglio95,FeRoWeMi95,WeRoFe96,FeLoWe00,FiKu97,FrWa99,Sil99,ZhJeWh99, BoKaTr00, BoTr01,ElShBoKuTr03,deFiCaIaMaPeVe01,HoAivdL04,HovdL05} but these mostly concentrate on finding the energies for the ground state as a function of coupling strength and Coulomb repulsion. Some calculate the bipolaron effective mass at the bottom of the band, but most do not discuss how energies vary with center-of-mass wave vector well away from the band minimum, except for strong electron-phonon coupling. Hohenadler \etal\cite{HoAivdL04} give graphical results for the spectral function of bipolarons as a function of wave vector for various cases where the coupling is not very strong, and approximate $E(k)$ curves can be deduced from these. An easier comparison to make is with some unpublished calculations performed by S. El Shawish for the Holstein model for some parameters corresponding roughly to some of those we have used. These values correspond to high ratios of phonon energy to electronic transfer integral $t$. In Sec.~\ref{sec:model} we introduce the Hamiltonian and our variational method. Some numerical results are presented in Sec.~\ref{sec:results}, and some discussion is given in Sec.~\ref{sec:discussion}. \section{\label{sec:model}Hamiltonian and variational method} With a notation somewhat similar to that of Ref.~\onlinecite{BaGeIaNi91} but modified to apply to one dimension and for short-range interactions, we write the Hamiltonian $H$ for the bipolaron problem with constant bare effective masses in the form \begin{equation}\label{eq:model} H = H_X + H_u +H_p +H_\text{e-p}. \end{equation} Here $H_X$ is the center-of-mass kinetic energy given by \begin{equation} H_X= -\mbox{\small$\frac{1}{2}$}\nabla_X^2, \end{equation} where the center-of-mass coordinate \begin{equation} X = \mbox{\small$\frac{1}{2}$}(x_1+x_2), \end{equation} and $x_1$, $x_2$ are the coordinates of the two electrons; $H_u$ is the Hamiltonian for relative motion, with \begin{equation}\label{eq:Hu} H_u = -2\nabla_u^2 + W(u), \end{equation} where \begin{equation} u = x_1-x_2 \end{equation} is the relative coordinate and \begin{equation} W(u) = \begin{cases}\label{eq:P} P &-\mbox{\small$\frac{1}{2}$}a < u <\mbox{\small$\frac{1}{2}$}a\,,\\ 0 &\text{otherwise}\,, \end{cases} \end{equation} with $a$ the lattice constant; the phonon Hamiltonian $H_\text{p}$ and the electron-phonon interaction $H_\text{e-p}$ are given by \begin{equation} H_\text{p} = \sum_k a_k^{\dagger} a^{\phantom{\dagger}}_k \end{equation} and \begin{equation}\label{eq:H_e-p} H_\text{e-p} = -iV(a/L)^{\frac{1}{2}} \sum_k [2 \cos(\mbox{\small$\frac{1}{2}$}ku) e^{\mathrm{i} kX}a_k] + \text{H.c.}, \end{equation} where $a_k^{\dagger}$ and $a^{\phantom{\dagger}}_k$ are creation and destruction operators for phonons of wave number $k$ and $L$ is the length of the crystal. We use the usual reduced units, with units of energy, length and mass equal to $\hbar \omega$, $(\hbar/2 m \omega)^{\frac{1}{2}}$ and $2m$ respectively, where $\omega$ is the phonon angular frequency and $m$ is the bare electron mass. The form of the potential term in $H_u$ is similar to but not the same as that in the Hubbard model because, with the form used, two electrons within a unit cell do not always interact, but this is compensated by interactions between electrons which are in neighbouring cells but separated by less than $a$. We do not include any spin-dependent terms in the Hamiltonian, and in the following we shall not include any terms involving electron-spin wave functions. For bipolarons we implicitly assume that the two electrons in the pair have opposite spin by choosing a wave function for relative motion which is even in the relative coordinates. For a given center-of-mass vector $Q$, we adopt a trial wave function $\Psi$ of the form \begin{equation} \Psi = L^{-\frac{1}{2}} e^{\mathrm{i} QX} \phi_Q(u) \prod_k \psi(Q,k,u) \chi, \end{equation} where $\chi$ is the phonon or other boson vacuum, \begin{eqnarray} \psi(Q,k,u) &=& N(Q,k,u)\\ &&\times\left[ 1 + L^{-\frac{1}{2}} c(Q,k) \cos(\mbox{\small$\frac{1}{2}$} ku) e^{-\mathrm{i} kX} a_k^\dagger \right],\nonumber \end{eqnarray} where $c(Q,k)$ are variational parameters, $\phi_Q(u)$ is a normalised even function, \begin{equation} \int |\phi_Q(u)|^2 du = 1, \end{equation} depending on one or more parameters, and $N(Q,k,u)$ is a normalisation constant given by \begin{equation} N(Q,k,u) = \left[ 1 + L^{-1} |c(Q,k)|^2 \cos^2 (\mbox{\small$\frac{1}{2}$} ku) \right]^{- \frac{1}{2}} \simeq 1. \end{equation} With such a trial wave function, calculations similar to those in Sec.~4 of Fr\"{o}hlich's review article\cite{Fr54} for single polarons give the expectation values of the different terms in $H$. After replacing sums by integrals with use of the replacement \begin{equation} \sum_k \mapsto \frac{L}{2\pi} \int_{-\pi/a}^{\pi/a} dk, \end{equation} we find \begin{equation}\label{eq:<H_p>} \langle H_\text{p} \rangle = \frac{1}{2\pi}\int_{-\pi/a}^{\pi/a} |c(Q,k)|^2 h_k dk, \end{equation} where \begin{equation}\label{eq:h_k} h_k = \int |\phi_Q(u)|^2 \cos^2 (\mbox{\small$\frac{1}{2}$}ku) \: du, \end{equation} \begin{equation}\label{eq:<H_p>2} \langle H_\text{e-p} \rangle = \frac{i}{\pi} V a^\frac{1}{2} \int_{-\pi/a}^{\pi/a} [c^*(Q,k)-c(Q,k)] h_k dk, \end{equation} \begin{eqnarray}\label{eq:<H_X>} \langle H_X \rangle &=& \frac{1}{2} Q^2 - \frac{1}{2\pi} \int_{-\pi/a}^{\pi/a} Qk |c(Q,k)|^2 h_k \\\nonumber &+& \frac{1}{4\pi} \int_{-\pi/a}^{\pi/a} |c(Q,k)|^2 k^2 h_k dk\\\nonumber &+& \frac{1}{8\pi^2} {\int\int}_{-\pi/a}^{\pi/a} kk^\prime |c(Q,k)|^2 |c(Q,k^\prime)|^2 h_{kk^\prime} dk dk^\prime \,. \end{eqnarray} Here \begin{equation}\label{eq:hkk} h_{kk^\prime} = \int |\phi_Q(u)|^2 \cos^2 (\mbox{\small$\frac{1}{2}$}ku) \cos^2 (\mbox{\small$\frac{1}{2}$} k^\prime u) \: du \end{equation} and \begin{equation}\label{eq:<H_u>} \las H_u \ras = E_u + \frac{1}{4\pi} \int_{-\pi/a}^{\pi/a} |c(Q,k)|^2 k^2 (1-h_k) dk, \end{equation} where the first term is independent of the variational coefficients $c(Q,k)$. Going back to sums over k in Eqs.~(\ref{eq:<H_p>}),~(\ref{eq:<H_p>2}),~(\ref{eq:<H_X>}) and~(\ref{eq:<H_u>}), by minimisation of $\las H\ras$ with respect to $c(Q,k)$ and $c^*(Q,k)$, we find \begin{equation} c(Q,k) = \frac{-2iV a^\frac{1}{2}}{1 - (Q - h_k^{-1} G_k)k + \mbox{\small$\frac{1}{2}$}h_k^{-1} k^2}, \end{equation} where \begin{eqnarray} G_k & = & \frac{1}{2\pi} \int_{-\pi/a}^{\pi/a} k^\prime h_{kk^\prime} |c(Q,k^\prime)|^2 dk^\prime \\ & = &\nonumber \frac{2V^2 a}{\pi} \int_{-\pi/a}^{\pi/a} \frac{k^\prime h_{kk^\prime}} {\left[1-(Q - h_{k^\prime}^{-1} G_{k^\prime})k^\prime + \mbox{\small$\frac{1}{2}$} h_{k^\prime}^{-1} k^{\prime \:2}\right]^2} dk^\prime. \end{eqnarray} From Eqs.~(\ref{eq:h_k}) and~(\ref{eq:hkk}), \begin{equation}\label{eq:h_kk2} h_{kk'}=\mbox{\small$\frac{1}{2}$}h_k+\mbox{\small$\frac{1}{2}$}h_{k'}+\mbox{\small$\frac{1}{4}$} h_{k+k'}+\mbox{\small$\frac{1}{4}$}h_{k-k'}-\mbox{\small$\frac{1}{2}$}. \end{equation} For the purpose of determining the coefficients we will make the following approximation \begin{equation}\label{eq:h_kk3} h_{kk'} \simeq h_k h_{k'}. \end{equation} This gives \begin{equation}\label{eq:c(Q,k)} c(Q,k) \simeq \frac{-2iV a^\frac{1}{2}}{1 - (Q - g)k + \mbox{\small$\frac{1}{2}$} h_k^{-1} k^2}, \end{equation} where \begin{equation}\label{eq:g} g= \frac{2V^2 a}{\pi} \int_{-\pi/a}^{\pi/a} \frac{k^\prime h_{k^\prime}}{\left[1- S k^\prime + \mbox{\small$\frac{1}{2}$} h_{k^\prime}^{-1} k^{\prime \:2}\right]^2} dk^\prime, \end{equation} with \begin{equation} S = Q - g. \end{equation} From Eqs.~(\ref{eq:model}),~(\ref{eq:<H_p>}),~(\ref{eq:<H_p>2}),~(\ref{eq:<H_X>}),~(\ref{eq:<H_u>}) and~(\ref{eq:c(Q,k)}), we find that the expectation value of $H$, which we write as $E(Q)$, is given by \begin{widetext} \begin{eqnarray}\label{eq:E(Q)}\nonumber E(Q) &=& \mbox{\small$\frac{1}{2}$} (Q^2-g^2) -\frac{2V^2 a}{\pi} \int_{-\pi/a}^{\pi/a} \frac{h_k}{1 - Sk + \frac{1}{2} h_{k}^{-1} k^2} dk \\ &&+ \frac{2V^4 a^2 }{\pi^2} \int_{-\pi/a}^{\pi/a} \int_{-\pi/a}^{\pi/a} \frac{kk^\prime (h_{kk^\prime}-h_k h_{k\prime})} {(1- Sk + \frac{1}{2} h_{k}^{-1} k^2)^2 (1-Sk^\prime + \frac{1}{2}h_{k^\prime}^{-1} k^{'\:2})^2} dk dk^\prime + E_u\,. \end{eqnarray} \end{widetext} Although the variational coefficients determined in this approximation do not represent the optimal choice, nonetheless the expectation value of $H$, $E(Q)$, yields an upper bound to the exact energy. The part $E_u$ of the expectation value of $H_u$ in Eq.~(\ref{eq:<H_u>}) which is independent of $c(Q,k)$ can be written as \begin{equation}\label{eq:E_u} E_u=T_u+V_u. \end{equation} Here the relative kinetic energy $T_u$ is given by \begin{equation}\label{eq:T_u} T_u = \frac{2}{2\pi}\int |f_k|^2 k^2 dk, \end{equation} where $f_k$ is given by \begin{equation}\label{eq:f_k} f_k= \int\phi(u) e^{\mathrm{i} ku}du. \end{equation} In Eq.~(\ref{eq:T_u}) the integral over $k$ is, in principle, unrestricted, \ie is an integral over $k$ from $-\infty$ to $\infty$. However, we shall restrict our trial wave functions for relative electron motion to those that do not have Fourier components for $k>k_m=\pi/a$, and for such trial wave functions we use an integral from $-k_m$ to $k_m$. Writing \begin{equation}\label{eq:d_k} \int_{-\infty}^{\infty}|\phi(u)|^2 e^{\mathrm{i} ku}du=d_k \end{equation} we have, for a trial function for which $f_k=0$ for $|k|>k_m$, \begin{equation}\label{eq:d_k2} d_k = \frac{1}{2\pi} \int_{-(k_m-\frac{1}{2}|k|)}^{k_m-\frac{1}{2}|k|} f^{*}_{k'-\frac{1}{2}k}f_{k'+\frac{1}{2}k}dk'. \end{equation} Using \begin{equation} \int_{-\frac{1}{2}a}^{\frac{1}{2}a} e^{-\mathrm{i} ku}du = \frac{2}{|k|}{\rm sin}(\mbox{\small$\frac{1}{2}$}|k|a), \end{equation} from Eq.~(\ref{eq:P}) we find that the potential term $V_u$ in Eq.~(\ref{eq:E_u}) is given by \begin{equation}\label{eq:V_u} V_u = \frac{2P}{2\pi} \int_{-2k_m}^{2k_m} d_k[{\rm sin} (\mbox{\small$\frac{1}{2}$}ka)/k]dk \end{equation} We restrict the integral to the limits shown in Eq.~(\ref{eq:V_u}) because $d_k$ vanishes for $|k|>2k_m$. Remembering that \begin{equation} \cos^2(\mbox{\small$\frac{1}{2}$}ku) = \mbox{\small$\frac{1}{2}$} [\cos(ku)+1], \end{equation} Eqs.~(\ref{eq:h_k}) and~(\ref{eq:d_k}) enable us to write \begin{equation}\label{eq:h_k2} h_k=\mbox{\small$\frac{1}{2}$}+\mbox{\small$\frac{1}{2}$}d_k; \end{equation} $h_{kk'}$ is then determined from Eq.~(\ref{eq:h_kk2}). We now consider a two-parameter trial function for the relative motion, with an assumed function in real space modified by replacement of all Fourier components with wave vectors of magnitude greater than $k_m$ replaced by zero. \subsection*{Pseudo real-space function for relative motion} We consider a function which has Fourier transforms up to $|k|=k_m$ of the same form, up to a proportionality factor, as the transforms of $\phi$, where \begin{equation}\label{eq:phi(u)} \phi(u) = N_0(1+b|u|)e^{-\lambda|u|}, \end{equation} with $b$ and $\lambda$ adjustable parameters, and $N_0$ a normalisation factor. If $b$ in the trial function is small, then the maximum of $\phi$ is at $u=0$, whereas if $b>\lambda$ there are two maxima at finite $|u|$. For $|k|>k_m$, we assume that the Fourier transform of $\phi$ is zero. Thus, using Eq.~(\ref{eq:phi(u)}), $f_k$ of Eq.~(\ref{eq:f_k}) is given by \begin{equation} f_k = \begin{cases} N\left[ \frac{2\lambda}{(\lambda^2+k^2)} +\frac{2b(\lambda^2-k^2)}{(\lambda^2+k^2)^2} \right] &\text{if $|k|<k_m$\,,}\\ 0 &\text{if $|k|>k_m$}\,, \end{cases} \end{equation} where $N$ is a normalisation factor. From $(1/2\pi)\int_{-k_m}^{k_m} |f_k|^2=1$, we find $N$ is given by \begin{equation}\label{eq:N^2} N^2 =B^{-1} N_0^2, \end{equation} where \begin{equation}\label{eq:N_0} N_0=(1/\lambda + b/\lambda^2 +b^2/2\lambda^3)^{-\frac{1}{2}} \end{equation} is the normalisation factor of a wave function given by Eq.~(\ref{eq:phi(u)}) before imposing a restriction on the Fourier components for $|k|>k_m$, and \begin{equation}\label{eq:B} B = \frac{1}{2\pi}\int_{-k_m}^{k_m}|f_{0k}|^2, \end{equation} where $f_{0k}$ is the same as $f_k$ but with the normalisation factor replaced by $N_0$. The departure from unity of the integral of Eq.~(\ref{eq:B}) gives the fractional change of the square of the normalisation factor due to putting the Fourier transforms of $\phi$ for $|k|>k_m$ as zero. From Eqs.~(\ref{eq:h_k}) and~(\ref{eq:phi(u)}), if one were to suppose that the assumption of zero Fourier transforms of $\phi$ for $|k|>k_m$ only affected $h_k$ for $|k|<k_m$ via the change of normalisation factor, we would find \begin{equation}\label{eq:h_k/N^2} \frac{h_k}{N^2} = \frac{2\lambda}{4\lambda^2+k^2}+\frac{2b(4\lambda^2-k^2)}{(4\lambda^2+k^2)^2} +\frac{2b^2(8\lambda^3-6\lambda k^2)}{(4\lambda^2+k^2)^3}+\mbox{\small$\frac{1}{2}$} \end{equation} for $|k|\leq k_m$. We have verified that, if the integral in Eq.~(\ref{eq:d_k2}) is extended to be from $-\infty$ to $\infty$ then Eq.~(\ref{eq:h_k2}) gives the result of Eq.~(\ref{eq:h_k/N^2}). The kinetic term $T_u$ in Eq.~(\ref{eq:E_u}) is given by Eq.~(\ref{eq:T_u}). If $\phi$ were taken as in Eq.~(\ref{eq:phi(u)}) without removal of the Fourier components for $|k|>k_m$, then using Eqs.~(\ref{eq:Hu}),~(\ref{eq:P}) and~(\ref{eq:phi(u)}), the potential term $V_u$ in Eq.~(\ref{eq:E_u}) would be \begin{eqnarray}\label{eq:V_u2}\nonumber V_u &=& P\int_{-\frac{1}{2}a}^{\frac{1}{2}a}{|\phi|^2 du}\\\nonumber &=& P N_0^2[(1/\lambda+b/\lambda^2+b^2/2\lambda^3)(1-e^{-\lambda a})\\ &&-(ab/\lambda+b^2 a/2\lambda^2-b^2 a^2/4\lambda)e^{-\lambda a}]\,. \end{eqnarray} By comparison of numerical results obtained from Eq.~(\ref{eq:V_u}) and~(\ref{eq:h_k2}) with those from Eqs.~(\ref{eq:h_k/N^2}) and~(\ref{eq:V_u2}), we can find the effect of putting Fourier components of $\phi$ for $|k|>k_m$ on $h_k$ and $V_u$ for given values of the parameters. \section{\label{sec:results}Numerical results} Marsiglio\cite{Marsiglio95} states that the physical region of the Hubbard-Holstein model requires a condition $\alpha^2/K < U$, where $\alpha$ is a factor multiplying local vibrational coordinates $x_i$ in the electron-phonon interaction, and $K$ appears in the expression $\frac{1}{2}Kx_i^2$ for the vibrational potential energy. If we identify $Pa$ in our model with $U$, although this correspondence is not exact, the same type of condition would require that \begin{equation}\label{eq:V-Pa} 2 V^2 \leq Pa. \end{equation} However, in attractive Hubbard models ($U<0$), the negative $U$ is usually thought to be brought about by effects of electron-phonon interactions overcoming a positive Hubbard $U$, and Marsiglio's condition would appear to imply that negative $U$ Hubbard models are unphysical. Therefore we shall not restrict our numerical work to the region defined by Eq.~(\ref{eq:V-Pa}), although in cases in which we are using our model with short-range forces as a simple approximation for effects of long-range forces we shall need to take the condition more seriously. For our numerical calculations we consider two types of cases. First we choose parameters which may be applicable to oxidised atactic polypropylene (OAPP),\cite{fn0,fn} and make use of some of the parameters used previously in the model for superconductivity in channels composed of arrays of quasi one-dimensional filaments.\cite{Ea94_2} As in Ref.~\onlinecite{Ea94_2}, we take the cross section of individual filaments to be 0.25 nm$^2$. However, because the periodic potential due to aligned dipoles near the strings of charges forming the filaments according to the model of Grigorov \etal\cite{Gr90,GrAnSm90,Gr91} for individual filaments may not have very deep minima, we assume here that the bare electron mass is $m_\text{e}$ and not $2m_\text{e}$ as in Ref.~\onlinecite{Ea94_2}. We consider two variants of parameters which could be applicable to OAPP. One, as in Ref.~\onlinecite{Ea94_2}, where the excitations mediating the electron-electron attraction are plasmons, and the other where optical phonons of average energy 0.36 eV mediate the attraction. For the plasmon-induced interaction, we should strictly have long-ranged forces, but we hope that the model used here with short-range forces will give a first approximation to the real situation. We choose 0.36 eV for a phonon energy because there are several branches of the phonon spectrum associated with \chem{C-H_2} and \chem{C-H_3} stretching vibrations whose energies at long wavelengths lie between 0.35 and 0.37 eV.\cite{McWa61} For given values of $V^2$, $a$, $Q$ and $P$, we solve for $g$ of Eq.~(\ref{eq:g}) and then minimise the total energy with respect to $\lambda$ and $b$, using our full expressions of Eqs.~(\ref{eq:d_k2}) and~(\ref{eq:h_k2}) for $h_k$, Eq.~(\ref{eq:V_u}) for $V_u$, and the normalisation factor of Eq.~(\ref{eq:N^2}) for $f_k$. For computational purposes, we made use of programs or modifications of programs from a book.\cite{numrec} In the model for strings or nanofilaments of Refs.~\onlinecite{Gr90,GrAnSm90,Gr91}, if there is a periodic potential acting on the electrons in the string, it is due to groups of about three aligned dipoles surrounding each electron, and so the period of the potential, or lattice constant, $a$, is equal to the inverse of the linear electron concentration $c = nd^2$, where $n$ is the three-dimensional concentration in the filament and $d^2$ is its cross section. Thus we may not be free to choose the carrier concentration and lattice constant independently. However, we note that a recent theoretical study of channels through films of oxidised atactic polypropylene making use of Bose condensation of bosons in an array of nanofilaments with an $E(K)$ curve for bosons consisting of a combination of linear and quadratic terms,\cite{Ea05} as indicated to occur in studies of Cooper pair dispersion,\cite{AdCaPuRiFoSoDLVaRo00} did not assume a periodic potential acting on the electrons in the nanofilaments. Another constraint was imposed in Ref.~\onlinecite{Ea05} because it was thought that it was probably necessary for the Fermi energy to be smaller than a quarter of the energy of the excitation mediating the electron-electron attraction in order to have the possibility of large enhancements of interactions at high drift velocities.\cite{Ea94_2} However, in view of the results of the present paper, these enhancements may not occur when calculations of electron-electron interactions go beyond second-order perturbation theory. Bearing in mind the possible constraints, and assuming $d^2=0.25$ nm$^2$ and $m_\text{b} = m_\text{e}$ as discussed above, we find two values of $a$ and the related carrier concentrations corresponding to our two possible choices of phonons or plasmons to mediate the attraction. In both cases we use parameters such that the ratio of $a$ to the polaron radius $r_\text{p} = (\hbar/2m\omega)^{\frac {1}{2}}$ satisfies $a/r_\text{p}=4$. Using a value for the high-frequency dielectric constant of 2.2,\cite{dielectric} we find, for plasmons mediating the attraction, that $a/r_\text{p} = 4$ implies $a= 0.53$ nm, the linear concentration $c=1/a=1.9\times 10^7$ cm$^{-1}$, the three-dimensional carrier concentration $n$ within a filament is $n= 7.6 \times 10^{21}$ cm$^{-3}$, the plasmon energy $\hbar \omega_\text{p}$ calculated using a three-dimensional formula appropriate for not too long wavelengths is $\hbar \omega_\text{p} =$ 2.2 eV, the polaron radius $r_\text{p} = 0.132$ nm, and, from a one-dimensional formula, the bare Fermi energy $\epsilon_F = 0.34$ eV. The values of most of these quantities are only slightly different from those used in Ref~\onlinecite{Ea94_2}. For phonons of energy $0.36$ eV, $a/r_\text{p} = 4$ implies $a = 1.3$ nm, $r_\text{p} = 0.325$ nm, $n = 3.1 \times 10^{21}$ cm$^{-3}$ and $\epsilon_F=0.056$ eV. We also consider a different type of case appropriate for a quantum wire of a crystalline material. If, \eg, we assume that $m=2m_\text{e}$ and $\hbar \omega$ = 0.05 eV, then the polaron radius is 0.62 nm, and a ratio $a/r_\text{p} = 0.5$ would then imply a plausible value of the lattice constant of 0.31 nm. We note that, for such a small value of $a/r_\text{p}$, there is no point in doing calculations for very large values of $Q$, since the type of variational method we are using will not be appropriate when the single-polaron energy lies more than $\hbar \omega$ above the bottom of the band.\cite{Fr54} Note that, when the polaron or bipolaron energy above the bottom of the band becomes close to $\hbar \omega$, the discrepancy in energy from the types of states more generally discussed (see \eg Refs.~\onlinecite{FeRoWeMi95,WeRoFe96,FeLoWe00,ElShBoKuTr03}) becomes large. This is because our method requires one or two electrons (for polarons or bipolarons) whose average wave vector or centre-of-mass wave vector is equal to the centre-of-mass wave vector of the polaron or bipolaron to be present whatever the energy of the state, whereas more commonly used methods find the lowest energy of the electron-phonon system for a given centre-of-mass wave vector. Such wave functions have only a small electron component at the wave vector concerned when the threshold energy for emission of phonons is approached. We think that our method is concerned with states of more physical interest than the states usually discussed in this energy region. These states, with the wave vector provided by phonons, and electrons at the bottom of the band, do not help in describing what happens when a polaron or bipolaron is accelerated rapidly past the threshold for emission of phonons. Since we are using $a/r_\text{p} =4$ for both possible parameter choices for OAPP, the same computer calculations can be used for both plasmon or phonon-induced interactions, with only the values of quantities obtained in real units being different. Figures~\ref{fig:fig1} and~~\ref{fig:fig3} show values of the bipolaron energy $E(Q)$, the energy $2E_s$ of two widely separated polarons each with wave vector $0.5Q$, and the parameters $\lambda$ and $b$ in the bipolaron trial function, for two values of $V^2$ and two related values of $P$ for each $V$. as a function of $Q/Q_m$, where \begin{equation}\label{eq:Q_m} Q_m=2k_m=2\pi/a. \end{equation} For small $b$, $1/\lambda$ is the bipolaron radius in units of the polaron radius. Figures~\ref{fig:fig5}(a) and~(b) show results of similar calculations as a function of $Q$ for a single value of $V^2$ for a more limited range of $Q$ for the much smaller value of $a/r_\text{p}=0.5$ which may be appropriate for a quantum wire of crystalline material. The figures also show the values of $g$ of Eq.~(\ref{eq:g}) and of the parameters $\lambda$ and $b$ which are found in the numerical work. We notice that in all cases shown in Figs.~\ref{fig:fig1}\,--\,\ref{fig:fig5}, there is a monotonic rise of energies with $Q$. We have also considered two other types of trial functions for relative motion, a Gaussian type of function with a second parameter in the prefactor, and a wave function constant in $k$-space for wave vectors with magnitudes between minimum and maximum values $k_1$ and $k_2$. Both these types of trial function gave poorer results for the bipolaron energies than the function used here. Also, in many cases $k_1$ turned out to be zero in the second type of function, and so our second parameter often did not improve matters. \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{fig1a.eps} \includegraphics[width=0.45\textwidth]{fig1b.eps} \caption{\label{fig:fig1} Bipolaron energies $E(Q)$, the energy $2E_\text{s}$ of two widely separated single polarons, the parameters $\lambda$ and $b$ of the trial wave function for relative motion of Eq.~(\ref{eq:phi(u)}), and the quantity $g$ of Eq.~(\ref{eq:g}) as a function of bipolaron wave vector $Q$, for $V^2=0.125$, $a=4$, and (a) $P=0$, (b) $P=0.125$. Energies are measured with respect to the bottom of the bare band, in units of the phonon energy. $Q_m$ is defined by Eq.~(\ref{eq:Q_m}).} \end{center} \end{figure} \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{fig2a.eps} \includegraphics[width=0.45\textwidth]{fig2b.eps} \caption{\label{fig:fig3} As for Fig.~\ref{fig:fig1}, but with $V^2=0.25$, $a=4$ and (a) $P=0$, (b) $P=0.25$. } \end{center} \end{figure} \begin{figure} \begin{center} \includegraphics[width=0.45\textwidth]{fig3a.eps} \includegraphics[width=0.45\textwidth]{fig3b.eps} \caption{\label{fig:fig5} As for Fig.~\ref{fig:fig1}, but with $V^2=1$, $a=0.5$, and (a) $P=0$, (b) $P=1$. } \end{center} \end{figure} \section{\label{sec:discussion}Discussion} Bipolaron energies measured from the bottom of the bare bands for the Holstein model obtained from unpublished results of S. El Shawish for the case of $\hbar\omega/t=16$, $g^2=0.125$ and $\hbar\omega/t=4$, $g^2=0.5$, corresponding to $a=4$, $V^2=0.125$ and $a=2$, $V^2=0.5$ in our model, are equal to -$0.335\hbar\omega$ and -$1.22\hbar\omega$ compared with -$0.29\hbar\omega$ and -$0.92\hbar\omega$ obtained by us, and so our results are 13\% and 25\% smaller than those of El Shawish for the magnitudes of this quantity. The energies of two single polarons in our model for the same parameters are -$0.21\hbar\omega$ and -$0.64\hbar\omega$, and so our bipolaron binding energies are $0.08\hbar\omega$ and $0.28\hbar\omega$. Average effective masses (strictly the reciprocals of the average reciprocal masses) of bipolarons up to $Q=0.1Q_m$ in our model for the same pairs of parameters are $2.13m_\text{b}$ and $2.69m_\text{b}$, whereas the average masses over the same region of wave vectors inferred from the results of El Shawish are $2.88m_\text{b}$ and $3.05m_\text{b}$. Thus the departure of the mass from twice the single-particle mass is much larger in the results of El Shawish. For $a=4$, with $V^2=0.25$ and 0.375, bipolaron binding energies in our model are $0.22 \hbar\omega$ and $0.38\hbar\omega$ respectively, while bipolaron masses for these two cases are $2.31 m_b$ and $2.50 m_b$. Thus we are able to get bipolaron binding energies of over a third of the phonon energy without very large effective masses. In the limit that $a\rightarrow 0$ (with $ta^2$ constant), our method gives single-polaron binding energies $E_\text{s}=0.5 V^2a\hbar \omega$ and effective masses $m_\text{s}$ such that $m_\text{s}/m_\text{b}=1+0.25V^2a$. These results may be compared with those of Chen \etal\cite{ChReJi98} for a continuum model with short-range interactions with a single dispersionless phonon for values of $V^2a$ up to three (corresponding to $\alpha$ up to 1.5 in their notation). For $V^2a=1$ ($\alpha=0.5$) and $V^2a=3$ ($\alpha=1.5$), from Figs.~5 and 8 of their paper we find $E_\text{s}\approx 0.52\hbar \omega$ and $E_\text{s}\approx 1.69\hbar \omega$, while their masses $m_\text{s}$ satisfy $m_\text{s}/m_\text{b}\approx 1.2$ and 3.7 for the two values of coupling. Thus $E_\text{s}$ for the two models differ by only 13\% for $V^2a=3$, whereas masses for the two models are within 4\% of each other for $V^2a=1$ (closer to 20\% for the departure of $m/m_\text{b}$ from unity), but for $V^2a=3$ the discrepancy in masses is large ($m_\text{s}/m_\text{b}=3.7$ in the model of Ref.~\onlinecite{ChReJi98}, but 1.75 for our model). The larger discrepancy for masses between the two models than for binding energies is related to the fact that intermediate-coupling methods are accurate for binding energies up to higher values of coupling constants $\alpha$ in the Fr\"{o}hlich model than for masses.\cite{LeFlPi53,Sc59} Another case for which we may make comparison with other authors is that of $a=1$, $V^2=1$, corresponding to $\omega/t=1$ and $g^2=1$ in the Hubbard-Holstein model. Here we find that the bipolaron energy measured from the bottom of the bare band is $-1.10\hbar \omega$, and the energy of two single polarons is $-0.80\hbar\omega$. Thus the bipolaron binding energy with respect to two single polarons is $0.30\hbar \omega$ in our model, compared with about $0.5\hbar \omega$ which can be inferred from the inset to Fig.~3 of a paper by Bon\v{c}a \etal\cite{BoKaTr00} For this case we have also found the value of $Pa$ in our model for which the bipolaron binding energies for the above values of parameters vanish. We find the bipolaron energy vanishes for $Pa\approx 1.9$. The result of a vanishing of the binding energy at $Pa\approx 1.9$ may be compared with $U\approx 1.6$ for the binding energy to vanish which may be inferred from the inset to Fig.~3 of Ref.~\onlinecite{BoKaTr00}. The smaller binding energies and masses, and larger partial bandwidths cf. what are probably accurate results for the Hubbard-Holstein model are thought to be due to a combination of (i) limitations of our variational method to quite weak couplings, (ii) the use of only a two-parameter model for the relative motion, and (iii) to differences between a Holstein model and a model with constant bare mass. We do not know which of the three causes of discrepancy is most important at present. A lower limit for the percentage change due to departure from weak coupling of the expectation value of the part of the Hamiltonian which does not depend on relative motion of the two electrons can be obtained by looking at the percentage change in single-polaron energies at the same value of coupling when this is known. An approximate upper limit to the percentage change in the same quantity can be obtained from the percentage change in the single-polaron binding energy when the coupling $g^2$ is twice as large, since, in the limit when the relative wave function is very small, the coupling to phonons is twice as large for a bipolaron as for a polaron. For example, for the case $g=1$, $t/\hbar \omega =1$, in the Holstein model, corresponding to $V^2=1$ and $a=1$ in our model, one can estimate from Fig.~1 of Ref.~\onlinecite{RoBrLi99I} that both the lower and upper limits in the errors in the bipolaron energy below the bottom of the bare band due to use of a weak-coupling method are very small. A simple way to estimate errors due to the change from the constant bare-mass model to the Holstein model is to compare the weak-coupling result for the single-polaron model in the Holstein model given in Eq.~(8) of Ref.~\onlinecite{RoBrLi99I}, which gives a polaron binding energy $E_\text{bs}$ of $g^2\hbar\omega/ (1+4J/\hbar\omega)^{\frac{1}{2}}$ in their notation.\cite{n3} In our notation, $J/\hbar\omega$ corresponds to $1/a^2$, and $g$ to $V$. Thus their result in our notation is \begin{equation}\label{eq:E_bs} E_\text{bs}=V^2 a \hbar\omega/(4+a^2)^{\frac{1}{2}}. \end{equation} Our method gives binding energies in units of $\hbar\omega$ for single polarons in the constant bare-mass model of $0.21 V^2a$, $0.32 V^2a$ and $0.40 V^2a$ for $a$=4, 2 and 1, which may be compared with $0.22 V^2a$, $0.35 V^2a$ and $0.45 V^2a$ for the Holstein model for weak coupling from Eq.~(\ref{eq:E_bs}). Thus the fractional changes in binding energies at weak coupling due to use of the Holstein model appear to be approximately 0.05, 0.09 and 0.12 for the three values of $a$ considered. The results presented above are for single-polaron theory, and what changes would be expected for bipolarons is not obvious. However, if we take the results based on single-polaron theory seriously for bipolarons, then we expect, \eg for $a=4$, $V^2=0.125$, that the magnitude of the difference in the energy of the bottom of the bare and bipolaron bands to be increased from about 0.29$\hbar\omega$ for the constant bare-mass model to between 0.30 and 0.31$\hbar \omega$ for the Holstein model, compared with $0.335\hbar\omega$ obtained by El Shawish. Thus in this case it appears probable that errors due to the restriction of our method to weak or intermediate coupling and due to use of a two-parameter trial function for relative motion may be of the order of 10\% for bipolaron energies at $Q=0$. Besides differences from the Hubbard model, we also discuss briefly how far our method is likely to be fairly accurate within the framework of the constant-bare-mass model that we have used. We note the following points: 1. From the results mentioned after Eq.~(\ref{eq:E_bs}) and values of parameters used for the figures and other values mentioned in the first paragraph of this section, the single-polaron binding energy lies between about 0.105$\hbar\omega$ (for $V^2=0.125$ and $a=4$) and 0.315$\hbar\omega$ (for $V^2=0.375$ and $a=4$). The type of variational method we use is accurate to about 4\% for single-polaron binding energies in the Fr\"{o}hlich model and to within 20\% for masses\cite{Sc59} up to polaron couping constants $\alpha=3$, corresponding to polaron binding energies of about 3$\hbar\omega$. Our binding energies are so far below this value that our method can be expected to be accurate for single polarons, and also for bipolarons if one assumes that the coupling constant for bipolarons to have the same percentage errors in masses as for single polarons is at least half as large as that for single polarons. (See point 3 below for further discussion of this). 2. There is also the question of whether we are anywhere near a transition between large and small polarons. Toyozawa\cite{To61} was the first to study such transitions, for the case of interactions between electrons and acoustical phonons, and found a sudden transition. However, since we are dealing with optical phonons here, his work does not have much relevance for our problem. For interactions with optical phonons such transitions (between large and nearly-small polarons) were first discussed by one of the present authors.\cite{Ea69,n2} Emin\cite{Em73} considers transitions between large and small polarons for a 3-D molecular-crystal model. For the case where the bare half bandwidth is ten times as large as the phonon energy, he finds a transition between types for the lowest-energy state as a function of the binding energy of a small polaron which would exist in the case of zero bare bandwidth. He calls this energy $E_b$. From his Fig.2 one finds that the transition at $T=0$ for the above value of bare half-bandwidth occurs when the polaron binding energy (not $E_b$) is about $2\hbar\omega$, although both types of solutions exist for weak-coupling polaron binding energies between about $\hbar\omega$ and 3$\hbar\omega$. Even the lower bound on these values is considerably larger than the values of polaron binding energies of up to 0.315$\hbar\omega$ which we have discussed in the present paper. Thus it is probable that our results are not influenced by any proximity to such a transition. However, Emin does not discuss in detail results in the opposite limit which we have considered when bare band half bandwidths are small compared with phonon energies. For a model with Fr\"ohlich electron-phonon interactions, it appears that a large to small polaron transition occurs for coupling constants $\alpha$ near 3 to 5,\cite{Le92,Ia98} with details depending on the degree of adiabaticity. The transition is fairly sharp for bare half bandwidths greater than phonon energies, but more gradual for the opposite case. For the smallest bare half bandwidth of $\hbar\omega$ considered in Ref.\onlinecite{Le92}, their Fig.3 indicates only a fairly small departure of masses from a linear dependence on coupling constant $\alpha$ up to one. The cases we concentrate on are $a=4$ and $a=2$, corresponding to bare half bandwidths of 0.31$\hbar\omega$ and $1.23\hbar\omega$ for a constant bare mass up to the edge of the Brillouin zone at $\pi/a$, or to 0.125$\hbar\omega$ and 0.5$\hbar\omega$ if we convert masses to obtain transfer integrals $t$ for a tight-binding model with the same bare mass at the bottom of the band. Thus, although our coupling constants correspond to polaron binding energies considerably smaller than $\hbar\omega$ in the Fr\"ohlich model, we could not completely rule out a larger mass rise than we find by our method because of the beginning of a transition between polaron types if our band structure were similar to that of a tight-binding model as considered in Refs.\onlinecite{Le92,Ia98}. However, in our case we are assuming that any effect of the lattice periodicity on the bare electrons is small, and so in such a case any transitions between polaron types will correspond to transitions in the continuum model, i.e. to a change from the lattice following the instantaneous position of the electron for weak coupling to responding to some average position for stronger coupling, and the effects of such transitions are already included in calculations such as those reported in Ref.\onlinecite{Sc59}. For our case where there is no significant effect of periodicity of the potential on the bare-electron wave function, there is no such thing as a small polaron in the sense of a state which is a linear combination of states with the electron on one lattice site and surrounded by the appropriate lattice polarisation, since lattice sites are almost indistinguishable from positions in between them. In this case the fact that our single-polaron radius for $a=4$ is only a quarter of the lattice constant does not imply small polarons in the usual sense. The only way that the lattice constant comes into our model is by a cut-off in the phonon wave vector. Thus in this sense it is similar to the continuum-polarisation model with a cut-off considered by Schultz.\cite{Sc59} 3. Iadonisi et al.\cite{IaPeCadF01} also consider transitions between types for bipolarons. For a case shown in Fig.1 of their paper, corresponding to a bare bandwidth equal to twenty times the phonon energy, the transition for bipolarons occurs at about an 8\% smaller values of the coupling constant than for polarons. Thus our guess in point 1 above that a given percentage error of mass may occur at a value of coupling constant of about half that of polarons may be pessimistic. We get no confirmation in this work of our conjectures based on perturbation theory of great enhancements of pair binding energies at certain large centre-of-mass wave vectors.\cite{Ea94,Ea94_2} Also, unpublished calculations of El Shawish up to bipolaron wave vectors of $\pi/a$ do not give us much reason to expect that suggestions of a cusp-like minimum at $2\pi/a$ (in an extended zone scheme) indicated by early attempts to extend results of our variational method to the Hubbard-Holstein model as reported in Ref.~\onlinecite{Ea99}, are likely to occur in accurate calculations. However, we still cannot rule out the possibility that a dip in bipolaron energies would be obtained near certain wave vectors if we were to use a different type of variational wave function for relative motion which could take better advantage of the small denominators in the integrands in Eq.~(\ref{eq:E(Q)}) for suitable values of $Q$ and fairly weak coupling. Our calculations indicate that there are parameter values where bipolarons do not have excessively high masses while having binding energies with respect to two single-polaron energies greater than a few tenths of the relevant boson energy. If the bosons are plasmons of energy of the order of 2 eV, then this permits bipolaron binding energies of the order of 0.5 eV without too great increases in masses, whereas for phonons of energies of about 0.36 eV, bipolaron binding energies of 0.1 eV can be obtained without too large mass increases. A binding energy of at least 0.1 eV is a minimum requirement for room-temperature superconductivity, assuming pair binding energies must be at least about $4k_\text{B}T$ for superconductivity at temperature $T$. The masses must not be too high in order to be able to have a high Bose-Einstein condensation temperature for bipolarons without excessively high bipolaron concentrations. Previous calculations of condensation temperatures for bosons with a quadratic $E(Q)$ curve\cite{Ea98} in arrays of nanofilaments have recently been extended\cite{Ea05} to cases with a dispersion approximated by a sum of linear and quadratic terms, as indicated to occur for Cooper pairs.\cite{AdCaPuRiFoSoDLVaRo00} We hope to modify the calculations of Ref.~\onlinecite{Ea05} soon by use of a Bogoliubov-type of dispersion for pairs (see \eg Ref.~\onlinecite{PiSt05}), which we now think is more appropriate than that based on a Cooper-pair model for the strongly coupled pairs at fairly low carrier concentrations in which we were interested in Ref.~\onlinecite{Ea05}. \section{\label{sec:conclusions}Conclusions} No support comes from our variational method for previous results based on perturbation theory that great enhancements of binding energies of pairs can be obtained at appropriate large centre-of-mass velocities. However, parameters have been found such that bipolaron masses are smaller than about $3m_\text{e}$ while keeping binding energies with respect to energies of two single polarons greater than 0.1 eV. Thus bipolarons in one dimension may provide a basis for an explanation of probable room-temperature superconductivity in narrow channels through films of oxidised atactic polypropylene and other polymers, but for different reasons than conjectured in earlier papers. \begin{acknowledgments} One of us (DME) would like to thank Professor J. Robles-Dominguez for weekly discussions over a period of a few months on bipolarons and related subjects, Dr. S. El Shawish for sending results on bipolaron dispersion of the Holstein model for some parameters of interest, and Dr. M. Hohenadler for help in preparation of the manuscript. \end{acknowledgments}

proofpile-arXiv_069-14424

\section{Introduction} Modern theoretical physics analysis of strongly correlated systems deals with a wide range of numerical methods, because no serious progress has yet been made in formulating a regular analytic description. In particular, progress has been achieved in Dynamical Mean Field Theory (DMFT)\cite{RevModPhys.68.13,kotliar:865}, as well as in its extensions and generalizations\cite{maier:1027}. In the DMFT approach strongly-correlated systems are reduced to an effective impurity problem. Virtually, a single atom or small cluster is removed from the lattice but placed into a Gaussian bath defined in a self-consistent way. DMFT requires a so-called solver algorithm, which is intended to give an approximate evaluation of Green's functions for an effective impurity model. For the most cases, the Quantum Monte Carlo (QMC) class of solvers is used \cite{PhysRevLett.56.2521}, \cite{PhysRevB.40.4780}. These algorithms allow the calculation of electronic Green's functions for many models lying far beyond the regime where perturbation theory is valid. It is important to note that the output of QMC calculation belongs to the imaginary-time domain. This information is enough to evaluate DMFT loops, since DMFT equations are written in terms of fermionic Matsubara frequencies $\omega_n = (2n+1)\pi/\beta$, where $\beta$ is the inverse temperature. On the other hand, experimental data is, of course, obtained in the real-time or real-frequency domain. Consequently, it is evidently quite a difficult problem to extract physical information from a numerical data set, because analytic continuation is needed. In the canonical problem one has to obtain the spectral density function $A(\omega)$ from a noisy set of values for thermal Green's function $\mathcal{G}(\tau)$ or its Fourier transform $\mathcal{G}(i\omega_n) = \int_0^\beta d\tau\ e^{-i\omega_n\tau}\mathcal{G}(\tau)$, produced in a QMC simulation. The spectral density is proportional to imaginary part of retarded Green's function, from the definition: \begin{equation} A(\omega) = -\frac{1}{\pi}\Im G^R(\omega) \end{equation} Because of the analyticity of the retarded Green's function in the upper complex plane of frequency the following is true: \begin{equation}\label{Analyticity} G^R(\omega) = -\frac{1}{\pi}\int\limits_{-\infty}^{+\infty}\frac{\Im G^R(\omega')}{\omega - \omega' + i0}\ d\omega' = \int\limits_{-\infty}^{+\infty}\frac{A(\omega')\ d\omega'}{\omega - \omega' +i0} \end{equation} Substituting $\omega = i\omega_n$ one can derive the integral equation \begin{equation}\label{MaxEnt_problem} \mathcal{G}(\tau) = \int\limits_{-\infty}^{+\infty}d\omega \frac{e^{-\tau\omega}}{1 + e^{-\beta\omega}} A(\omega) \end{equation} which is a particular formulation of the analytic continuation problem. The kernel of this integral equation is exponentially small at large positive and negative frequencies, so tiny variations in $\mathcal{G}(\tau)$ may correspond to a very strong change of the spectrum at those frequencies. Uncertainty in $\mathcal{G}(\tau)$ is unavoidable due to the stochastic nature of QMC algorithms, so the problem of numerical analytic continuation is extremely ill-posed, similarly to a numerical inversion of a Laplace transform. The earliest attempts to solve the problem were based on Pade approximation method \cite{thirumalai:5029}. For a lot of cases, modifications of the standard least squares procedure has been proposed \cite{PhysRevLett.55.1204}, \cite{PhysRevLett.63.2504}. Currently the most common practice is to use the Maximum Entropy algorithm for the analytic continuation problem \cite{MaxEnt_Rev}. It's based on Bayesian inference concept and essentially uses \textit{a priori} knowledge about the properties of the spectral function, its positivity and sum rules. In other words, the MaxEnt algorithm is essentially a \textit{nonlinear} formulation. Existing realizations of MaxEnt do not allow the analytic continuation of functions, whose norm is not a known constant (self-energy, for instance), or do not have a definite sign (for example, off-diagonal components of Green's function). The aim of this paper is to introduce a new analytic continuation algorithm, which is linear with respect to input data and strong enough to be considered as an alternative for the MaxEnt method. Section II of the paper is devoted to the general idea behind the method without giving details of a particular realization. It explains optimal regularization principle, which is built on top of Tikhonov regularization ansatz. Some more specific realization details are described in sections III and IV. They, in principle, may be revised and adjusted to a certain extent for specific problem. In Section V we present practical results of analytic continuation for the Hubbard model on the Bethe lattice, which is then compared with corresponding results from the MaxEnt algorithm. In Section VI we conclude the paper. \section{The optimal regularization matrix} Mathematically speaking, analytic continuation requires a solution of a linear integral equation for unknown function $F(\omega)$ \begin{equation}\label{REG_problem} \hat M F(\omega) = F(i\omega) \end{equation} using certain {\it a priori} information. Here $\hat M$ is the ill-posed linear operator which analytically continues from the real to the imaginary axis, and function $F(i\omega)$ is approximately known from a QMC simulation. If we have introduced certain basis of orthogonal functions to represent $F(\omega)$ and $F(i\omega)$, the above integral equation becomes a set of linear algebraic equations \begin{equation}\label{first_eq} \hat{M}x=y, \end{equation} where $M$ is badly conditioned matrix, and the right hand part $y$ is subject to some level of inaccuracy. Formally, the rank of the set is infinite. One would expect that for a properly chosen basis the effective reduction to a finite set is possible. Such a basis, suitable for the analytic-continuation problem is introduced in Section III. The staring point of the proposed method is a Tikhonov regularization of the problem \cite{Tikhonov}. Let us search for a vector $x$, which minimizes the Tikhonov functional: \begin{equation}\label{tikhonov} \mathcal{F}[x;\hat R] = ||\hat M x - y||^2 +(x,\hat R x) \end{equation} Here $\hat R$ is a regularizing Hermitian matrix. Vector $y$ is known approximately: $y = \bar y + \delta y$, where $\bar y$ is the mean value of the vector $y$, and the deviation $\delta y$ is a random quantity distributed with zero mean value and characterized by the covariation matrix $\hat K_y$, such that: \begin{equation}\label{K_def} \overline{\delta y} = 0, \quad \hat K_y = \overline{\delta y \delta y^\dag} \end{equation} (hereafter the line over an expression denotes the QMC expectation). Varying the functional $\mathcal{F}[x;\hat R]$ with respect to $x$, one obtains a condition for the vector $x$, which gives the minimum of the functional for given $y$ and $\hat R$: \begin{equation}\label{X_def} x = \hat X\hat M^\dag y, \quad \textrm{where} \ \hat X \equiv (\hat M^\dag \hat M + \hat R)^{-1} \end{equation} Assume that the vector $\bar{x}$ is an exact solution of the set of equations (\ref{first_eq}) with an exactly known right-hand part: $\hat M\bar x = \bar y$. We average the mean square deviation of $x$ from $\bar x$ over all possible values of the random vector $\delta y$ to obtain the following: \begin{equation}\label{deviation} \overline{||x - \bar x||^2} = \mathrm{Tr}\{\hat X \hat A \hat X - 2\hat X\hat B\} + \mathrm{Tr}\{\bar x \bar x^\dag\} \end{equation} \begin{equation}\label{A_def} \hat A \equiv \hat M^\dag \hat M\bar x\bar x^\dag\hat M^\dag \hat M + \hat M^\dag \hat K_y \hat M \end{equation} \begin{equation}\label{B_def} \hat B \equiv \hat M^\dag \hat M \bar x \bar x^\dag \end{equation} It should be obvious, that a proper choice of regularizing matrix $R$ is required to provide a satisfactory small value of $\overline{||x - \bar x||^2}$. On the other hand, a desired type of the behavior of a solution $\bar x$ cannot be determined or even estimated from the results of QMC simulation alone. Construction of a regularizing algorithm presumes utilization of certain \textit{a priori} information about the properties of the solution. Very generally, the use of {\it a priori} information means making an assumption that the result $\bar{x}$ is not arbitrary, but falls into certain classes of possible solutions. For example, one can suppose that the resulting function is smooth, not too large, etc. We denote by $\langle \ldots \rangle$ the average over possible class of the solutions. It is reasonable to require that the regularizing functional $R$ delivers the minimum of the deviation $\overline{||x - \bar x||^2}$ in the average over the set of possible solutions, \begin{equation} \label{mean_deviation} \langle\overline{||x - \bar x||^2}\rangle = \mathrm{Tr}(\hat X \langle\hat A\rangle \hat X - 2\hat X\langle\hat B\rangle) + \mathrm{Tr}\langle\bar x \bar x^\dag \rangle = \min_{\hat R} \end{equation} The main idea of the proposed method is to solve this minimum problem with respect to $R$ and then to obtain $x$ from (\ref{X_def}). A specific choice of the possible class of solutions is discussed in Section IV. However already at this point it is worth noting that we actually do not need a complete information about the possible solutions, but only the first and the second momenta of $x$, since only these quantities appear in (\ref{mean_deviation}) (recall expressions (\ref{A_def}, \ref{B_def}) for $A$ and $B$). Solving (\ref{mean_deviation}) with respect to $\hat R$ and taking into account the additional condition $\delta\hat R = \delta\hat R^\dag$, one obtains an equation for the matrix $\hat X$ (the trivial solution $\hat X = 0$ does not relate to the problem): \begin{equation}\label{opt_eq} \langle\hat A\rangle \hat X + \hat X \langle\hat A\rangle = \langle\hat B\rangle + \langle\hat B^\dag\rangle \end{equation} In this way we formally found a system of $N(N+1)/2$ linear equations for all elements of the matrix $\hat X$ ($where N$ is the dimension of the square matrix $\langle\hat B\rangle$). However, there is a more efficient way to solve the obtained equation (\ref{opt_eq}). Let us denote eigenvalues of the matrix $\langle\hat A\rangle$ by $\lambda_i$ and perform an unitary transformation to a primed basis, in which $\langle\hat A\rangle$ is diagonal. In this basis a solution of equation (\ref{opt_eq}) can be expressed explicitly: \begin{equation}\label{opt_eq_sol} {x_{ij}' = \frac{b_{ij}' + (b_{ji}^*)'}{\lambda_i + \lambda_j} \quad 1 \leq i,j \leq N} \end{equation} Changing to the eigenbasis of the matrix $\langle\hat A\rangle$ allows us to organize the numerical solving of the system (\ref{opt_eq}) in an efficient way. The search procedure for eigenvalues and eigenvectors of $\langle\hat A\rangle$ is quite stable because $\langle\hat A\rangle$ is Hermitian. The advantage of such an approach in comparison with the direct solving of (\ref{opt_eq}) is that instead of solving a system of $N(N+1)/2$ equations ($\propto N^6$ operations) it is enough to diagonalize the $N \times N$ matrix $\langle\hat A\rangle$ ($\propto N^3$ operations). We should note that, mathematically, formula (\ref{opt_eq_sol}) unambiguously gives finite expressions for all the elements of $\hat X$, since its denominator is positive. Indeed the matrix $\hat M^\dag\hat M$ is nonnegative definite. Matrix $\langle\bar x \bar x^\dag\rangle$ is a correlation matrix, which is also essentially positive definite. Finally, the matrix $\hat K_y$ is also positive definite, if the vector $y$ is defined with nonzero inaccuracy (all components having nonzero dispersion). In this way we ensure that the matrix $\langle\hat A\rangle$ is positively defined and therefore all its eigenvalues are positive. On the other hand, the practical realization of the method faces certain difficulties, because the numerical solution suffers from round-off errors in our calculations. The singular value decomposition procedure does not help much. To get rid of these round-off errors, we have switched to performing the calculations with to arbitrary number of digits, using CLN library version 1.1.13\cite{CLN}. \section{Choice of representation. Correlation matrix} Now we apply results of the previous Section directly to the analytic continuation of a function $F(\omega)$ from the imaginary axis to the real one, assuming $F(\omega)$ to be analytic in the upper plane. QMC simulation gives values of the function $F$ at Matsubara frequencies $i \omega_k$ on the imaginary axis: \begin{equation} F_k = F(i\omega_k), \quad k=\overline{1,K} \end{equation} Let $\omega_0$ be a typical frequency scale of the problem. Let us introduce a conformal mapping of the upper frequency plane to a circle of a unitary radius with the center at zero point: \begin{equation}\label{conformal} \omega \rightarrow z: \quad z = \frac{\omega - i\omega_0}{\omega + i\omega_0} , \quad \omega = i\omega_0\frac{1+z}{1-z} \end{equation} \begin{figure} \includegraphics[width=0.8\columnwidth]{conformal.eps} \caption{Schematic view of the conformal mapping of upper frequency plane to a circle.} \label{fig:conformal} \end{figure} The mapping is illustrated by figure \ref{fig:conformal}. In this case all imaginary frequencies are mapped to a segment $z\in[-1;1]$; all real frequencies correspond to the circle with radius 1. If $F$ is analytic in the upper plane as a function of complex frequency, it has the same property in the unit circle in the $z$-plane and can be expanded in the Taylor series at point $z=0$: \begin{equation}\label{F_expansion} F(\omega(z)) = \sum_{n=0}^\infty f_nz^n, \quad f_n = \frac{1}{2\pi i}\oint\limits_{|z|=1}\frac{F(\omega(z))}{z^{n+1}}dz \end{equation} Assuming that the function $F(z)$ is smooth, we can then take into account a finite number of terms in this expansion. The proper number $N$ of terms to be kept can be estimated from test runs of the program for a known function $F$. After the expansion coefficients $f_n$ are determined, we can then sum up the series at any point of the circle $|z|=1$ and thus restore values of $F(\omega)$ on the real axis. The chosen representation $\hat M$ entering equation (\ref{first_eq}) is then a $K\times N$ matrix with elements given by formula \begin{equation} M_{kn} = \left(\frac{\omega_k-\omega_0}{\omega_k+\omega_0}\right)^n \end{equation} Using some \textit{a priori} information about the expected solutions $F(\omega\in\mathbb{R})$, we can then choose a correlation matrix $\langle F(\omega)F^*(\omega')\rangle\mid_{\omega,\omega'\in\Re}$. This correlation matrix also can be expanded into the double Taylor series at zero point, similarly to (\ref{F_expansion}): \begin{equation}\label{FF_expansion} \langle F(\omega)F^*(\omega')\rangle\mid_{\omega,\omega'\in\mathbb{R}} \approx \sum_{n,n'=0}^{N-1} \langle f_n f_{n'}^*\rangle z^n (z')^{-n'} \end{equation} \begin{multline}\label{ff_expansion}\tiny \langle f_n f_{n'}^*\rangle =\\= \frac{1}{(2\pi i)^2}\oint\limits_{|z|,|z'|=1} \langle F(\omega(z))F^*(\omega(z'))\rangle z^{-(n+1)}z'^{(n'-1)}dz dz' \normalsize \end{multline} Practically, it was found that $N\approx 30-50$ terms is enough for the calculation. A value of $\omega_0$ was not found to be really important; it is sufficient just to take $\omega_0$ of the right order of magnitude. \section{Correlation matrix of Lorentzian peaks} The explicit form of the correlator $\langle F(\omega) F^*(\omega')\rangle$, leading to satisfactory results, can be obtained from the following procedure. Let us assume that function $F$ is a superposition of several Lorentzian peaks having the same width $\gamma$ \begin{equation} F(\omega) = \sum_{j=1}^J \frac{Z_j}{\omega-\Omega_j+i\gamma} \end{equation} The positions of the peaks $\Omega_j$ and their magnitudes $Z_i$ are random quantities with known statistical properties \begin{equation}\label{FF} \langle F(\omega_1)F^*(\omega_2)\rangle = \sum_{j,j'=1}^J \left\langle \frac{Z_j}{\omega_1-\Omega_j+i\gamma}\frac{Z_{j'}}{\omega_2-\Omega_{j'}-i\gamma}\right\rangle_{Z,\Omega} \end{equation} We assume that all $Z_j$ are equally distributed and uncorrelated with each other, so that two model parameters $\langle Z^2 \rangle$ and $\langle Z \rangle$ completely determine the distribution of $Z_j$: \begin{widetext} \begin{equation}\label{FF_final} \langle F(\omega_1)F^*(\omega_2)\rangle = \langle Z^2 \rangle\sum_{j=1}^J \left\langle \frac{1}{\omega_1-\Omega_j+i\gamma}\frac{1}{\omega_2-\Omega_j-i\gamma}\right\rangle_\Omega \\ + \langle Z \rangle^2 \sum_{j\neq j'}^J \left\langle \frac{1}{\omega_1-\Omega_j+i\gamma}\frac{1}{\omega_2-\Omega_{j'}-i\gamma}\right\rangle_\Omega \end{equation} Finally we assume, that the values $\Omega_j$ are distributed independently with certain model distribution, for example the Lorentzian: \begin{equation} P(\Omega_j) = \frac{1}{\pi\Omega_M}\frac{1}{1+(\Omega_j/\Omega_M)^2} \end{equation} where $\Omega_M$ is the scale of the spectrum. For example, half of the Hubbard $U$ is a good guess for this quantity in Hubbard-like models. Combination of formulas (\ref{ff_expansion}) and (\ref{FF_final}) yields a result suitable for a practical calculations: \begin{multline} \langle f_n f_{n'}^*\rangle = J \langle Z^2 \rangle \int_{-\infty}^{+\infty} P(\Omega) I(n,\gamma,\Omega) I^*(n',\gamma,\Omega)\ d\Omega + \\ + J(J-1)\langle Z \rangle^2 \int_{-\infty}^{+\infty} P(\Omega_1) I(n,\gamma,\Omega_1)\ d\Omega_1 \int_{-\infty}^{+\infty} P(\Omega_2) I(n',\gamma,\Omega_2)\ d\Omega_2 \end{multline} \end{widetext} \[I(n,\gamma,\Omega) \equiv \left\{ \begin{array}{ll} \frac{1}{i\gamma -\Omega + i\omega_0}, & n = 0 \\ -2i\omega_0 \frac{(i\gamma - \Omega - i\omega_0)^{n-1}}{(i\gamma - \Omega + i\omega_0)^{n+1}}, & n \neq 0 \end{array} \right. \] Although these integrals could be calculated analytically for a particular Lorentzian form of $P(\Omega)$, the answer is very complicated and has significant computational complexity. It is therefore much more practical to perform an integration numerically for every pair of $n$ and $n'$. \section{Mott transition at Bethe lattice: Practical calculation of DOS} \begin{figure}[htp] \subfigure[]{ \includegraphics[scale=0.75]{input.eps} } \subfigure[]{ \includegraphics[scale=0.75]{output.eps} } \caption{(a) Input data for $\Sigma(i\omega_n)-\Delta(i\omega_n)$ at $U=2.3$. (b) Reconstructed real and imaginary parts of $\Sigma(\omega)-\Delta(\omega)$. 300 Matsubara frequencies were used, with an estimated errorbar in $\Sigma-\Delta$ is about $10^{-2}$, the noise at different Matsubara frequencies was assumed to be uncorrelated. In the regularization procedure, the eigenvalues of $40\times 40$ matrix have been obtained, using an accuracy to 80 decimal digits.} \label{fig:in_out} \end{figure} To illustrate the potential of the method, we reconstruct the density of states for the half-filled single-band Hubbard model defined on the Bethe lattice. Near-neighbor hopping constant for the model was equal to 0.5, and the Hubbard $U$ ranged from 1.0 to 3.0. Inverse temperature in the simulations was $\beta=50$. The Mott transition was observed between $U=2.4$ and $U=2.5$. Data for $\mathcal{G}(\tau)$ and $\mathcal{G}(i\omega_n)$ was produced by the DMFT self-consistent loop. The continuous-time QMC code \cite{PhysRevB.72.035122} was used as an impurity-problem solver. This version of the QMC solver is particularly suitable, because it can produce high-accuracy data for $\mathcal{G}(i\omega_n)$ directly in $i \omega$-domain, up to quite high Matsubara frequencies. \begin{figure}[htp] \subfigure[]{ \includegraphics[height=5.1cm]{dos10.eps} } \subfigure[]{ \includegraphics[height=5.1cm]{dos22.eps} } \subfigure[]{ \includegraphics[height=5.1cm]{dos23.eps} } \caption{Reconstructed densities of states for $U=1.0$, $U=2.2$ and $U=2.3$. Parameters of the regularization procedure are the same as in Fig.2. Results of the regularization are represented by solid lines. MaxEnt predictions are shown with dots.} \label{fig:dos} \end{figure} The regularization algorithm uses the model described in the previous section with the Gaussian distribution of $\Omega_j$. An important peculiarity of the approach is that we work not with the Green's function, but with the quantity $\Sigma-\Delta$, defined by the relation \begin{equation} \mathcal{G}(i\omega_n) = \frac{1}{i\omega_n +\Delta(i\omega_n)-\Sigma(i\omega_n)} \end{equation} Practically such a method has an advantage over the direct continuation of $\mathcal{G}(i\omega_n)$, because $\Sigma-\Delta$ does not have trivial $1/(i\omega)$ asymptotics. Therefore, a smaller number of $z$-harmonics is required for an adequate representation of this function. This also provides us with a physically important property $\int^\infty_{-\infty} A(\omega) d\omega=1$, assuming that $\Delta-\Sigma$ remains finite as $\omega\to\infty$. Figure \ref{fig:in_out} illustrates the method, and Figure \ref{fig:dos} shows the results for several values of $U$ corresponding to metallic behavior. Unfortunately, the procedure works well only for the metal phase, because the insulator gap in DOS requires an infinite $\Sigma-\Delta$, which is clearly an impossible task for the continuation procedure. For comparison, we present the result of MaxEnt calculation for the same QMC data. We have used a relatively simple MaxEnt program \cite{PhysRevB.57.10287}. This code does not diagonalize the covariance matrix and uses a full search algorithm. As can be seen from Figure 3, our procedure produces the data with much more details. Firstly, in all graphs the DOS decreases more rapidly at large energies. This is quite physical, one would expect that energy bands should be quite well confined. Secondly, in our case the shape of the dips at small energy in Figure \ref{fig:dos} b,c is more similar to a ``pre-formed'' insulator gap. Finally, we were able to resolve very pronounced DOS shoulders near the onsets of the Hubbard bands. This agrees well the with the predictions of dynamical density-matrix renormalization-group theory \cite{0953-8984-16-39-038}. Of course, it should be noted that the particular appearance of the resulting graphs depends on a choice of {\it a priori} parameters $\Omega_M$ and $\gamma$. However, all qualitative peculiarities discussed in the previous paragraph are found to be quite robust against the variation of $\Omega_M$ and $\gamma$ in a reasonable interval (approximately, from 0.1 to 2). Values of $\omega_0$, $\langle Z\rangle^2$ and $\langle Z^2\rangle$ do not affect the result considerably, and so we put these quantities to 1 in the calculations. \section{Discussion and conclusions} To conclude, we developed a new regularization approach for the analytic continuation of numerical data from imaginary to real axis. The method is the best possible regularization in sense of (\ref{mean_deviation}), so that we produce an optimal regularization matrix given a class of possible solutions and errorbar of the input data. An important property of the optimal regularization approach is its linearity with respect to the input data. This makes possible the continuation of the self-energy and off-diagonal Green's function components. In principle, we could follow MaxEnt paradigm and put additional constrains like the requirement $A(\omega)\ge 0$. However this would make our scheme nonlinear and seriously reduce its flexibility. There is also negative consequences of the linearity of our approach: unphysical regions of negative DOS can occur for certain input data. It should also be noted that, accordingly to our observations, our method requires higher accuracy and larger number of Matsubara frequencies. In this case, our scheme has an advantage, as it is illustrated by the practical calculation for the Mott transition. Authors are thankful to A.I. Lichtenstein and M.I. Katsnelson for useful discussions, and to P. Crompton for reading the text. The work was supported by ``Dynasty'' foundation and NWO grant 047.016.005.

proofpile-arXiv_069-14522

\section{INTRODUCTION} The measurement of transverse single spin asymmetries gives us an opportunity to probe the quark and gluon structure of transversely polarized nucleons. Large transverse single spin asymmetries of up to 20\% - 40\% were observed for pions produced at large $x_F$ at $\sqrt{s}$ = 20 GeV\cite{E704} and have been found to persist at $\sqrt{s}$ = 200 GeV by the STAR \cite{Star} and BRAHMS\cite{BRAHMS} experiments, although they were expected to vanish in pQCD calculations at the leading order. A number of pQCD based models have been developed to explain this phenomenon. Among them are the Sivers effect (transversely asymmetric $k_T$ quark and gluon distributions)\cite{Sivers}, the Collins effect (transversity distribution in combination with spin-dependent fragmentation function)\cite{Collins}, and the higher twist effect (interference between quark and gluon fields in the initial or final state)\cite{Twist1,Twist2}. At RHIC energy, heavy flavor production is dominated by gluon-gluon interaction, thus Collins effect has minimum impact on $A_N$ as the gluon's transversity is zero. Therefore, the production of heavy flavor particles in transversely polarized p+p collisions at the PHENIX experiment offers a good opportunity to probe the gluon's Sivers effect. \section{MEASUREMENT AND ASYMMETRY CALCULATION} In this report we present the latest results of transverse single spin asymmetry in J/$\Psi$ production in polarized p+p collisions at RHIC. The PHENIX muon spectrometers have measured J/$\Psi$ yields through the J/$\Psi\rightarrow \mu^+\mu^-$ channel at forward rapidities ($1.2<|\eta|<2.4$). During the recent 2006 polarized pp run, the PHENIX experiment has collected 2.7 pb$^{-1}$ data with transverse beam polarization of about 56\%. The Level-2 filterd data sample has been used in this analysis. The J/$\Psi$ yield is obtained by fitting the dimuon mass spectrum with a single exponential background plus two gaussian functions (for J/$\Psi$ and $\Psi^{\prime}$) in the mass range 2.3 GeV - 4.5 GeV (as shown in figure 1). The total Number of J/$\Psi$ from this fit is 5236 $\pm$ 81. The Monto Carlo simulation shows the systematic error in determining J/$\Psi$ yield is less than 2\%. \begin{figure}[tb] \includegraphics[height=.3\textheight]{mass} \caption{Dimuon invariant mass distribution.} \label{mass} \end{figure} The transverse single spin asymmetry $A_N$ can be determined by: \begin{equation} A_N= {1\over P_b}{{\sigma^\uparrow-\sigma^\downarrow} \over{\sigma^\uparrow+\sigma^\downarrow}} = {1\over P_b}{{N^\uparrow-R N^\downarrow} \over{N^\uparrow+R N^\downarrow}}, \label{lum} \end{equation} where $P_b$ is the beam polarization, $\sigma^\uparrow(\sigma^\downarrow)$ is the production cross section, $N^\uparrow(N^\downarrow)$ is the J/$\Psi$ yield from up(down) polarized bunches, and $R=L^\uparrow / L^\downarrow$ is the relative luminosity of bunch crossings of opposite polarization sign. The relative luminosity is measured by two global detectors, BBC and ZDC. Alternatively, we also use square root formula \begin{equation} A_N={1\over P_b}{{\sqrt{N_L^\uparrow\cdot N_R^\downarrow}-\sqrt{N_L^\downarrow\cdot N_R^\uparrow}} \over{\sqrt{N_L^\uparrow\cdot N_R^\downarrow}+\sqrt{N_L^\downarrow\cdot N_R^\uparrow}}}, \label{sqrt} \end{equation} to cross check our results. The asymmetry of J/$\Psi$ was determined for each fill using Eq. (\ref{lum}), then averaged over all fills. The raw J/$\Psi$ sample is contaminated by other background, mainly comes from Drell-Yan process, open charm and light hadrons decay. The $A_N^{J/\Psi}$ was corrected for the contribution of background by using \begin{equation} \label{bg-1} A_N^{J/\Psi}={{A_N^{incl}-r\cdot A_N^{BG}}\over{1-r}}, \end{equation} \begin{equation} \label{bg-2} \delta A_N^{J/\Psi}={{\sqrt{(\delta A_N^{incl})^2+r^2\cdot (\delta A_N^{BG})^2}}\over{1-r}}, \end{equation} where $r$ is the background fraction under the J/$\Psi$ peak, $A_N^{BG}$ is the averaged background asymmetry. $A_N^{BG}$ is obtained using the like (unlike) sign dimuon pairs within 2.0 GeV$ < M_{\mu\mu} < 2.5$ GeV, and like sign dimuon pairs within 2.8 GeV$ < M_{\mu\mu} < 3.5$ GeV. The systematic uncertainty has been checked with bunch shuffling technique with randomly assigned polarization direction for bunches in each fill. The bunch-to-bunch and fill-to-fill systematic errors are much smaller than the statistical errors for this data set. \section{RESULTS} The asymmetry was calculated for two $x_F$ bins. Figure \ref{fig1} shows the preliminary result of the single spin asymmetry as a function of $x_F$ in J/$\Psi$ production from the 2006 RHIC transverse run. A scale uncertainty of 20\% due to the absolute polarization values is not included in this analysis. Within the limits of errors, $A_N$ is consistent with zero over the measured $x_F$ range. \vskip 0.3cm \begin{figure}[h] \includegraphics[height=.3\textheight]{AN_xF} \caption{Transverse single spin asymmetry vs. $x_F$ for J/$\Psi$ at forward rapidities. A scale uncertainty of 20\% is not included.} \label{fig1} \end{figure} \vskip 0.3cm Currently there is no theoretical calculation of J/$\Psi$'s $A_N$. However, there is a prediction of maximum values of $|A_N|$ in D meson production at fixed transverse momentum $p_T=1.5$ GeV/c at RHIC energy, as shown in figure \ref{fig2}\cite{Anselmino}. The solid line shows $|A_N|_{max}$ when the gluon Sivers function is set to its maximum with the quark Sivers function set to zero, and the dashed line corresponds to a maximized quark Sivers function with the gluon Sivers function set to zero. If the transverse single spin asymmetry comes from the initial state (the J/$\Psi$ production mechanism does not play an important role), then we expect $A_N$ in J/$\Psi$ and D meson production to be similar. The fact that there is no sizable asymmetry observed in figure \ref{fig1} may indicate our results disfavor the maximum contribution of gluon's Sivers function. \begin{figure}[h] \includegraphics[height=.3\textheight]{AN_xF_D} \caption{Maximized values of $|A_N|$ for the process p+p$\rightarrow$DX as a function of $x_F$ at fixed transverse momentum $p_T=1.5$ GeV/c at RHIC energy. } \label{fig2} \end{figure} \section{SUMMARY} The transverse single spin asymmetry $A_N$ for J/$\Psi$ has been measured for the first time at forward rapidity and $\sqrt{s}$ = 200 GeV with the PHENIX muon spectrometers. Using Level-2 dimuon triggered data sample (corresponds to 60\% of the whole 2006 transverse data set), no sizable $A_N$ of J/$\Psi$ has been observed which indicates our results disfavor the maximum gluon Sivers contribution. \bibliographystyle{aipproc} \section{General Overview} The \aipcls{} is a \LaTeXe{} document class for conference proceedings of the American Institute of Physics and other documents with similar layout requirements. Your file will be used to reproduce your paper as is, the only modifications done by the publisher are adding appropriate page numbers and the copyright line. It is therefore essential that you embed all fonts when saving your file. This version of the guide explains all features of the class some of which are only applicable to certain proceeding layouts. The class provides essentially the same markup as implemented by \LaTeX's standard \texttt{article} class. 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These commands differ from those used by standard \LaTeX's \texttt{article} class. Thus, if an article already written is adapted to be used with the \aipcls{}, the front matter has to be modified somewhat. Some of the commands below are required only for certain proceedings. Declarations that are not required will be silently ignored. \BDefC{title}[om]{short title}{title text} In standard \LaTeX{} this command has no optional argument. In the \aipcls{} one can specify an abbreviated title text which is used, for example, in the running footer in draft mode. \BDefC{author}[mm]{author name}{author information} In standard \LaTeX{} this command had only one argument containing both author name and address information. In this class it has two arguments and the second argument contains data structured using key/value pairs separated by commas. For example, the authors of this paper have been specified as: \begin{verbatim} \author{F. Mittelbach}{ address={Zedernweg 62, Mainz}, ,email= {frank.mittelbach@latex-project.org}} \author{D. P. Carlisle}{ address={Willow House, Souldern}, ,email={david@dcarlisle.demon.co.uk}} \end{verbatim} Supported keywords will be \texttt{address}, \texttt{email}, \texttt{altaddress}, \texttt{homepage}, and \texttt{thanks}. (With release 1.3 of \aipcls{} only \texttt{address}, \texttt{altaddress} and \texttt{email} should be used; support for the other keywords will be added later.) Depending on the layout of the target proceedings some of the keys may get ignored! \BDefC{classification}[m]{data} Some proceedings require classification data, e.g., PACS numbers. If not, this declaration is ignored. \BDefC{keywords}[m]{data} Some layouts require keyword data. If not, this declaration is ignored. \BDefC{copyrightholder}[m]{name} Some layouts require copyright information. Normally a default is provided by the class. With this declaration the copyright holder can be overwritten. \BDefC{copyrightyear}[m]{year} Some layouts require copyright data. With this declaration the copyright year can be specified. (If such data is required the current year is provided as default). \BDefE{abstract} In contrast to standard \LaTeX{} the abstract environment has to appear before the |\maketitle| command. \BDefC{maketitle} This command inserts the actual front matter data. It has to follow the above declarations. \subsubsection{Multiple authors} Multiple authors are entered by specifying one |\author| command per author. Care needs to be taken when specifying shared addresses: they have to be absolutely identical. Depending on the chosen layout the class will merge such addresses but will recognize them only as identical, if the input including spaces is the same! The |\and| command as defined in the \texttt{article} class to separate multiple authors is not supported. \subsubsection{Dates} \BDefC{received}[m]{date} \BDefC{revised}[m]{date} \BDefC{accepted}[m]{date} Some layouts require specification of date of arrival, revision, and/or acceptance. The above declarations provide a way to specify such dates if necessary. \BDefC{date}[m]{date} The article class provides the |\date| command which is not used by \aipcls. If supplied it will be ignored unless the \texttt{draft} option is specified in which case it will show up in a footer line together with the title and the page number to ease document development. \subsubsection{Other front matter commands} The |\tableofcontents|, |\listoffigures|, and |\listoftables| commands are provided but produce (beside output) an error message unless the \texttt{draft} option was selected. This is done since the \aipcls{} does not support page numbering and thus the above commands essentially produce incorrect data. \subsection{Headings} The \aipcls{} officially supports three heading levels, i.e., |\section|, |\subsection|, and |\subsubsection|. It also supports the commands |\paragraph| and |\subparagraph| although the latter heading levels are not part of the \aipcls{} specification and are therefore discouraged. In some layouts |\section| headings are changed to UPPERCASE. Special care is taken not to uppercase math material, but this support is only available if the package |textcase| is part of the \LaTeX{} distribution. \subsection{Cross-references}\label{xref} Cross-references to page numbers are not possible with the \aipcls{} as the page numbers are determined after production. For this reason the |\pageref| command of \LaTeX{} is disabled by default. Since headings in most layouts do not carry numbers they can't be referenced either. References to tables, figures, and equations are possible using the \LaTeX{} commands |\label| and |\ref|. However if the class option \texttt{varioref} or \texttt{nonvarioref} is used, references to page numbers are possible again as they will generate textual references of the form ``on the following page'' or ``on an earlier page'' etc. The produced strings are customizable as described in detail in the \texttt{varioref} package documentation or in \cite[p.68ff]{A-W:MG04}. The class defaults are as follows and can be changed with |\renewcommand| in the document preamble. The \texttt{varioref} package normally distinguishes between reference to facing pages and references to pages that need turning over using different strings in these cases. However, since with \aipcls{} class page numbers are not determined at the time of production no assumption can be made that page $x$ and $x+1$ actually fall onto the same double spread. For this reason the defaults used here do not produce strings containing the word ``facing'' or ``opposite''. \begin{verbatim} \renewcommand\reftextfaceafter {on the next page} \renewcommand\reftextfacebefore {on the \reftextvario{previous} {preceding} page} \renewcommand\reftextafter {on the \reftextvario{next} {following} page} \renewcommand\reftextbefore {on the \reftextvario{previous page}{page before}} \renewcommand\reftextcurrent {on \reftextvario{this} {the current} page} \end{verbatim} Normally, text for references which are ``far away'' are produced using |\reftextfaraway| in \texttt{varioref}. However, to produce textual references without referring to actual page numbers even in this case, this command was hijacked in the \aipcls{} and redefined to determine whether or not this is a reference to some earlier or later page. So instead of changing this command the class provides the following two commands for customization: \begin{verbatim} \renewcommand\reftextearlier {\reftextvario{on an earlier page}{earlier on}} \renewcommand\reftextlater {\reftextvario{later on} {further down}} \end{verbatim} To illustrate the result of this package all references in this document are made using |\vref| or |\vpageref|, e.g., references to Figure~\vref{fig:b} and Figure~\vref{fig:a}. These commands work best if used only for important references. Be careful when using them several times close to each other as the automatically generated texts then may sound strange (as they do in the example in this paragraph). \BDefC{eqref}[m]{label} For reference to equation numbers |\eqref| can be used instead of the standard |\ref| command. The |\eqref| command will automatically add any frills required by the layout style, while |\ref| will only typeset the plain number. For example, in the \texttt{arlo} style it will print ``Eq.~(1)'' while |\ref| would result in ``1''. \subsection{Lists} The \aipcls{} supports all standard list environments like \texttt{itemize}, \texttt{enumerate}, etc. \subsection{Graphics support} Support for including and manipulating graphics is provided as the standard \LaTeX{} \texttt{graphicx} package is automatically loaded by the \aipcls. For detailed descriptions of the commands made available by this package see~\cite{A-W:GMR97} or the package documentation coming with the \LaTeX{} release. A sufficient introduction is also given by~\cite{A-W:LLa94} although there only the \texttt{graphics} package (a subset of the \texttt{graphicx} package) is described. A typical application is given in the following example where a picture is resized to span 70\% of one column: \begin{verbatim} \begin{figure}[!b] \resizebox{.7\columnwidth}{!} {\includegraphics{escher}} \source{Guy Shaw} \caption{An illustration taken from~\cite{A-W:MG04}} \label{fig:a} \end{figure} \end{verbatim} resulting in figure \vref{fig:a}. \begin{figure}[!b] \resizebox{.7\columnwidth}{!} {\includegraphics[draft=false]{escher}} \source{Guy Shaw} \caption{An illustration taken from~\cite{A-W:MG04}} \label{fig:a} \end{figure} \subsection{Floats} Floats are objects which do not have to stay in sync with the running text but are allowed to move from their original place to some other position where they fit better for page breaking reasons. Such objects they are typically numbered so that they can be referenced from within the running text. \LaTeX{} by default supports two float types: figures and tables. These float types are also supported by the \aipcls{} although their internal implementation is quite different resulting in a number of important differences in behavior:\footnote{There exist packages that extend the number of float types. (This information is given as a footnote to show that footnotes in this class come out below a bottom float.)} \begin{itemize} \item The position of the float caption is determined automatically, independently of the placement of the |\caption| command within the float body. \item Depending on its width the float automatically spans two columns. In case of a table the whole object (including its caption) might be rotated automatically if its exceeds |\textwidth|. \item The body of the float environments are processed in L-R mode and not in paragraph mode as in standard \LaTeX. This is necessary for measuring its width. Thus if paragraph mode is needed one has to put a \texttt{minipage} environment of the appropriate width (e.g., |\columnwidth|) into the body. \item Only one |\caption| command per float is allowed. \end{itemize} \subsubsection{Figures} \BDefE{figure}[o]{pos} Like with standard \LaTeX{} the optional \Larg{pos} argument can be used to specify into which float areas this float is allowed to migrate (default is |tbp|). The environment \texttt{figure*} is not supported as figures that need to span both columns are automatically recognized in two column mode. \BDefC{source}[m]{text} Command to specify the origin of the picture shown. The \Larg{text} will be printed in small italics below the illustration. A typical example of a figure float would be \begin{verbatim} \begin{figure} \resizebox{.8\textwidth}{!} {\includegraphics{outline}} \caption{PostScript example taken from~\cite{A-W:MG04}} \label{fig:b} \source{F. Mittelbach} \end{figure} \end{verbatim} The result is shown in Figure~\vref{fig:b}. \begin{figure} \resizebox{.8\textwidth}{!}{\includegraphics[draft=false]{outline}} \caption{PostScript example taken from~\cite{A-W:MG04}} \label{fig:b} \source{F. Mittelbach} \end{figure} \BDefC{spaceforfigure}[mm]{horizontal}{vertical} If the illustration is to be manually pasted into the final document one can leave the right amount of space by using this command as follows: \begin{verbatim} \begin{figure} \spaceforfigure{2in}{1cm} \caption{Caption for a figure to be pasted in later} \label{fig:3} \source{F. Mittelbach} \end{figure} \end{verbatim} All standard \TeX{} units can be used to specify the space needed. The above example make room for an illustration that is two inches wide and one centimeter high. The result is shown as Figure~\vref{fig:3}. \begin{figure} \spaceforfigure{2in}{1cm} \caption{Caption for a figure to be pasted in later} \label{fig:3} \source{F. Mittelbach} \end{figure} \subsubsection{Tables} \BDefE{table}[o]{pos} Like with standard \LaTeX{} the optional \Larg{pos} argument can be used to specify into which float areas this float is allowed to migrate (default is |tbp|). The environment \texttt{table*} is not supported as tables that need to span both columns are automatically recognized in two column mode. Typically the body of the environment would consist of a \texttt{tabular} environment responsible for producing the actual table including the table and stub headers. \BDefC{tablehead}[mmmm]{cols}{h-pos}{v-pos}{heading text} To ease the production of tables the command |\tablehead| is provided which is essentially and abbreviation for a |\multicolumn| command that additionally boldens its text argument. I.e., \Larg{cols} specifies the number of columns the \Larg{heading text} should span and \Larg{h-pos} defines the horizontal positioning of the text of the column(s), e.g., |l|, |r|, |c|, or |p{...}|. In contrast to a simple |\multicolumn| command the \Larg{heading text} can be split vertically by using |\\| to denote the line breaks. The \Larg{v-pos} argument should contain either |t|, |c|, or |b| denoting the vertical placement of the text in relation to other cells of that row. It is only relevant if the \Larg{heading text} consists of more than one line. See the example table \vpageref[below]{tab:source} that demonstrates the use of this command. \BDefC{source}[m]{text} Command to specify the origin of the data given in the table. The \Larg{text} will be printed in small italics below the table. \BDefC{tablenote}[m]{text} Command to produce a note to the table. It can only be used within a \texttt{table} environment and should be used only at the right end of a table cell. The command produces a raised footnote symbol at the place used which sticks into the right margin. As far as \LaTeX{} is concerned this symbol does not occupy any space. Thus is will not modify the alignment of table columns. The \Larg{text} will appear below the table. In the current release notes to |\caption| or |\source| are not possible. \BDefC{tablenote*}[m]{text} Like |\tablenote| but this time the raised footnote symbol will occupy space. This version is intended to be used in the middle of cells. An example showing the use of all commands described above is shown in Table~\vref{tab:a}. It was produced by the following input:\label{tab:source} \begin{verbatim} \begin{table} \begin{tabular}{lrrrr} \hline &\tablehead{1}{r}{b}{Single\\outlet} &\tablehead{1}{r}{b}{Small\tablenote {2-9 retail outlets}\\multiple} &\tablehead{1}{r}{b}{Large\\multiple} &\tablehead{1}{r}{b}{Total} \\ \hline 1982 & 98 & 129 & 620 & 847\\ 1987 & 138 & 176 & 1000 & 1314\\ 1991 & 173 & 248 & 1230 & 1651\\ 1998\tablenote{predicted} & 200 & 300 & 1500 & 2000\\ \hline \end{tabular} \source{Central Statistical Office, UK} \caption{Average turnover per shop: by type of retail organisation} \label{tab:a} \end{table} \end{verbatim} \begin{table} \begin{tabular}{lrrrr} \hline & \tablehead{1}{r}{b}{Single\\outlet} & \tablehead{1}{r}{b}{Small\tablenote{2-9 retail outlets}\\multiple} & \tablehead{1}{r}{b}{Large\\multiple} & \tablehead{1}{r}{b}{Total} \\ \hline 1982 & 98 & 129 & 620 & 847\\ 1987 & 138 & 176 & 1000 & 1314\\ 1991 & 173 & 248 & 1230 & 1651\\ 1998\tablenote{predicted} & 200 & 300 & 1500 & 2000\\ \hline \end{tabular} \source{Central Statistical Office, UK} \caption{Average turnover per shop: by type of retail organisation} \label{tab:a} \end{table} \BDefC{setlength}[mm]{\texttt{\upshape\string\hlinesep}}{value} Vertical spacing between horizontal lines produced from |\hline| inside a tabular environment is controlled by the length parameter |\hlinesep| in this class. The default value (1pt) gives one point extra space above such lines and three times as much (i.e. 3pt) extra space below. This is done to implement the layout requirements for tables in the AIP proceedings (which are not supposed to have vertical lines in the tables). If tables with vertical lines are necessary for some reason, then the value of this parameter should be set to \texttt{0pt} either globally for the whole document or locally within the \texttt{table} environment. Otherwise the vertical lines will have strange gaps whenever a |\hline| command is used to produce a horizontal line. \subsubsection{Counters} The |\alph| and |\fnsymbol| commands to represent counter values have extended ranges. For example |\alph| will now count up to 52 (zz) and the |\fnsymbol| command will produce the following symbols \makeatletter \@fnsymbol{1}, \@fnsymbol{2}, \@fnsymbol{3}, \@fnsymbol{4}, \@fnsymbol{5}, \@fnsymbol{6}, \@fnsymbol{7}, \@fnsymbol{8}, \@fnsymbol{9}, \@fnsymbol{10}, \@fnsymbol{11}, \@fnsymbol{12}, \@fnsymbol{13}, \@fnsymbol{14}, \@fnsymbol{15}, and \@fnsymbol{16}. \makeatother This will allow for up to 16 table notes per table. For documents that need a larger number of table notes select the option \texttt{tnotealph} to switch to lower case alphabetic letters to mark such notes. \subsubsection{Long tables} Tables which are longer than one page cannot be placed into a \texttt{table} environment as floats cannot have a size larger than a page. Such tables are supported by the standard \LaTeX{} package \texttt{longtable} written by David Carlisle. However this package only works in single column mode. With two-column layouts, such as the one for the AIP 8x11 double column proceedings, such tables can only be added at the end of the paper by preceding the |longtable| environments with a |\onecolumn| declaration. The package is supported by the class in the sense that captions within a \texttt{longtable} environment will be formatted using the appropriate style; however in contrast to the \texttt{table} environment it is the responsibility of the user to place the caption at the top of the table. The commands |\source| and |\tablenote| are not supported within this environment, but the |\tablehead| command can be used to produce column heads if desired. Refer to the \texttt{longtable} package documentation or to \cite[p.122ff]{A-W:LLa94} for a detailed description of the syntax of the \texttt{longtable} environment. A possible alternative is the package \texttt{supertabular} written by Johannes Braams; however in this case no attempt has been made to ensure that a table produced with \texttt{supertabular} conforms to the layout specification for the \aipcls{}. Be aware that this package defines its own |\tablehead| command (with a completely different function). Refer to the package documentation for the syntax description. A detailed comparison between \texttt{supertabular} and \texttt{longtable} can be found in Chapter~5 of \cite{A-W:LLa94}. \subsubsection{Building floats manually} The original \LaTeX{} environments \texttt{figure} and \texttt{table} as well as their star forms are still available under the names \texttt{ltxfigure} and \texttt{ltxtable}. They should not be used in normal circumstances but are provided in case the automatism of the \aipcls{} needs overwriting. Please note that if these environments are used the position of the |\caption| command determines the placement of the caption within the float body and that the special commands for figures and tables, e.g., |\tablenote|, etc.\ as provided by this class are not available within these environments. \begin{table}[!t] \makeatletter \if8\expandafter\@car\selectedlayoutstyle\@nil\relax \else \fontsize{7}{8}\selectfont \fi \makeatother \begin{tabular}{rrrp{.6\textwidth}} \hline \tablehead{1}{r}{b}{File} & \tablehead{1}{c}{b}{Date} & \tablehead{1}{c}{b}{Version} & \tablehead{1}{c}{b}{Description} \\ \hline aipproc.cls & 2000/08/31 & v1.2a & AIP Proceedings (FMi) \\ fixltx2e.sty & 1999/12/01 & v1.0b & fixes to LaTeX \\ calc.sty & 1998/07/07 & v4.1b & Infix arithmetic (KKT,FJ) \\ ifthen.sty & 1999/09/10 & v1.1b & Standard LaTeX ifthen package (DPC) \\ graphicx.sty & 1999/02/16 & v1.0f & Enhanced LaTeX Graphics (DPC,SPQR) \\ keyval.sty & 1999/03/16 & v1.13 & key=value parser (DPC) \\ graphics.sty & 1999/02/16 & v1.0l & Standard LaTeX Graphics (DPC,SPQR) \\ trig.sty & 1999/03/16 & v1.09 & sin cos tan (DPC) \\ graphics.cfg & \\ dvips.def & 1999/02/16 & v3.0i & Driver-dependant file (DPC,SPQR) \\ url.sty & 1999/03/28 & ver 1.5x & Verb mode for urls, etc. \\ article.cls & 2000/05/19 & v1.4b & Standard LaTeX document class \\ size10.clo & 2000/05/19 & v1.4b & Standard LaTeX file (size option) \\ aipxfm.sty & \\ mathptm.sty & 2000/01/12 &PSNFSS-v8.1 &Times + math package (SPQR) \\ times.sty & 2000/01/12 &PSNFSS-v8.1 &Times font as default roman(SPQR) \\ ot1ptm.fd & 2000/01/12 &PSNFSS-v8.1 & font definitions for OT1/ptm. \\ fontenc.sty & \\ t1enc.def & 2000/08/30 & v1.91 &Standard LaTeX file \\ t1ptm.fd & 2000/01/12 &PSNFSS-v8.1 & font definitions for T1/ptm. \\ textcomp.sty & 2000/08/30 &v1.91 &Standard LaTeX package \\ ts1enc.def & 1998/06/12 & v3.0d & (jk/car/fm) Standard LaTeX file \\ varioref.sty & 1999/12/02 &v1.2c &package for extended references (FMi) \\ aip-8s.clo & \\ ttct0001.sty & \\ shortvrb.sty & 2000/07/04 &v2.0m & Standard LaTeX documentation package (FMi) \\ hyperref.sty & 2000/05/08 &v6.70f & Hypertext links for LaTeX \\ pd1enc.def & 2000/05/08 &v6.70f & Hyperref: PDFDocEncoding definition (HO) \\ hyperref.cfg & \\ hdvips.def & 2000/05/08 &v6.70f & Hyperref driver for dvips \\ pdfmark.def & 2000/05/08 &v6.70f & Hyperref definitions for pdfmark specials \\ ts1cmr.fd & 1999/05/25 &v2.5h & Standard LaTeX font definitions \\ nameref.sty & 2000/05/08 &v2.18 & Cross-referencing by name of section \\ t1pcr.fd & 2000/01/12 &PSNFSS-v8.1 & font definitions for T1/pcr. \\ ot1ptmcm.fd & 2000/01/03 &Fontinst v1.801 & font definitions for OT1/ptmcm. \\ omlptmcm.fd & 2000/01/03 &Fontinst v1.801 & font definitions for OML/ptmcm. \\ omspzccm.fd & 2000/01/03 &Fontinst v1.801 & font definitions for OMS/pzccm. \\ omxpsycm.fd & 2000/01/03 & Fontinst v1.801 &font definitions for OMX/psycm. \\ ts1ptm.fd & 2000/01/12 & PSNFSS-v8.1 &font definitions for TS1/ptm. \\ escher.eps & && Graphic file (type eps) \\ outline.eps & & & Graphic file (type eps) \\ \hline \end{tabular} \caption{Files used by the \aipcls{}} \label{tab:b} \source{Output of \texttt{\string\listfiles} when processing \texttt{aipguide.tex}} \end{table} \subsection{Urls} \BDefC{url}[m]{data} For documenting URLs and related data the |\url| command is provided. It allows breaking the URL in certain places and typesets it in an adequate font and format. Instead of using curly brackets the argument can be delimited by two identical characters not used in the argument. \subsection{Bibliography} Referring to other articles, books, etc.\ can be done using the |\cite| command of standard \LaTeX{}. The list of references itself can either be produced using standard \LaTeX{} methods or using \textsc{Bib}\TeX. If installed, the \aipcls{} class includes the \texttt{natbib} system which offers an extended set of citation commands. These commands have been originally developed to support author/year citation styles but are also useful with numerical citation styles. The \texttt{natbib} system has two basic citation commands, |\citet| and |\citep| for \emph{textual} and \emph{parenthetical} citations, respectively. There also exist the starred versions |\citet*| and |\citep*| that print the full author list, and not just the abbreviated one. All of these may take one or two optional arguments to add some text before and after the citation. Table~\vref{tab:natbib} shows some examples. \begin{table} \begin{tabular}{@{}l@{\quad$\Rightarrow$\quad}l} \hline \multicolumn{2}{@{}l}{\bfseries Author/year style} \\ \hline |\citet{jon90}| & Jones et al. (1990)\\ |\citet[chap.~2]{jon90}| & Jones et al. (1990, chap.~2)\\[0.5ex] |\citep{jon90}| & (Jones et al., 1990)\\ |\citep[chap.~2]{jon90}| & (Jones et al., 1990, chap.~2)\\ |\citep[see][]{jon90}| & (see Jones et al., 1990)\\ |\citep[see][chap.~2]{jon90}| & (see Jones et al., 1990, chap.~2)\\[0.5ex] |\citet*{jon90}| & Jones, Baker, and Williams (1990)\\ |\citep*{jon90}| & (Jones, Baker, and Williams, 1990) \\ \hline \multicolumn{2}{@{}l}{\bfseries Numerical style} \\ \hline |\citet{jon90}| & Jones et al. [21]\\ |\citet[chap.~2]{jon90}| & Jones et al. [21, chap.~2]\\[0.5ex] |\citep{jon90}| & [21]\\ |\citep[chap.~2]{jon90}| & [21, chap.~2]\\ |\citep[see][]{jon90}| & [see 21]\\ |\citep[see][chap.~2]{jon90}| & [see 21, chap.~2]\\[0.5ex] |\citep{jon90a,jon90b}| & [21, 32]\\ \hline \end{tabular} \caption{Example of \texttt{natbib} commands and their results} \label{tab:natbib} \end{table} There are many more commands and variants, see \cite{man:Daly99a} or \cite{man:Daly99b} for further details. \subsubsection{Bibliography produced manually} \BDefE{thebibliography}[m]{widest-label} Environment to hold the list of references. \BDefC{bibitem}[m]{label} Command to start a bibliographical entry having the label \Larg{label} for use in |\cite| commands. Refer to the publishers manual, e.g., \cite{man:aipproceed}, for information on how to lay out individual entries. For example: \begin{verbatim} \bibitem{Brown2000} M.~P. Brown and K. Austin, \emph{The New Physique}, Publisher Name, Publisher City, 2000, pp. 212--213. \end{verbatim} If commands from \texttt{natbib} (e.g., from table~\ref{tab:natbib}) should be usable, then additional information has to be passed to the |\bibitem| via an optional argument. \BDefC{bibitem}[om]{display-info}{label} The optional argument \Larg{display-info} should then, and only then, contain the author(s) name(s) followed by the year in parentheses without any spaces, for example: \begin{verbatim} \bibitem[Brown and Austin(2000)] {Brown2000} ... \end{verbatim} The essential feature is that the label (the part in brackets) consists of the author names, as they should appear in the citation, with the year in parentheses following. There must be no space before the opening parenthesis! This will be automatically produced if \BibTeX{} is used. \subsubsection{Bibliography produced using \textsc{Bib}\TeX} The \aipcls{} is accompanied by \BibTeX{} style files which can be used to produce compliant reference lists from \BibTeX{} database files. To use \BibTeX{} one first has to run the source file through \LaTeX{} then run \BibTeX{} and then rerun \LaTeX{} twice to get all references resolved. \BibTeX{} is described in more detail in appendix B of \cite{A-W:LLa94} and in chapter~13 of \cite{A-W:MG04}. \BDefC{bibliographystyle}[m]{style-name} This declaration specifies to \BibTeX{} that the style \Larg{style-name} should be used. It can be placed anywhere within the document but is usually positioned directly in front of the command described below. For a discussion which of the supplied \BibTeX{} styles should be used for which proceedings see the section ``Special requirements\ldots'' below. \BDefC{bibliography}[m]{bib-list} This command denotes the position where the reference list produced by \BibTeX{} will be included in the document. The \Larg{bib-list} is a comma separated list of \BibTeX{} database files. \section{General requirements and restrictions} This class was designed to work with \LaTeXe{} release 1999/06/01 or a later version. Earlier releases may work but have not been tested. With the exception of the packages \texttt{natbib} and \texttt{url} it only requires files which are part of a standard \LaTeX{} distribution, i.e., it should work if your installation contains the following components: \texttt{base}, \texttt{tools}, \texttt{graphics}, and \texttt{psnfss}, see \vref{tab:b} for files used to produce this document. The most recent \LaTeX{} distribution as well as \texttt{natbib} and \texttt{url} can be obtained from CTAN sites (Comprehensive \TeX{} Archive Network). Refer to \url{http://www.tug.org} for more information on CTAN and \TeX{} in general. A ready to run \TeX{} system for various platforms which has everything required is available on CD-ROM, look into \url{http://www.tug.org/texlive.html}. This \TeX{} implementation is also made available as an add-on to several books on \LaTeX, e.g., \cite{A-W:KD04,A-W:MG04}. For loading individual packages from a CTAN archive refer to \url{http://www.ctan.org} and search for the package name. Please omit extensions such as \texttt{.sty} when searching, e.g., search for \texttt{natbib} rather than \texttt{natbib.sty}, as such packages are often distributed in source form only, e.g., as a \texttt{.dtx} file. It is also possible to download a complete \TeX/\LaTeX{} installation from CTAN, e.g., Miktex + Winedit + Ghostview. Finally, it is also possible to download a CD-ROM image of the \TeX-live CD from CTAN (roughly 300MB): search for \texttt{texlive} (and make sure you select a suitable mirror near you). \section{Special requirements for individual layouts} \subsection{AIP proceeding layout 6x9} \begin{itemize} \raggedright \item The entire paper will be reduced 15\% in the printing process. 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proofpile-arXiv_069-14943

\section{Introduction} \label{sec:introduction} The Kondo effect in quantum dots has been an active research field over a decade.\cite{GR,NG,Goldharber,Cronenwett,Simmel,VanDerWiel} So far, the equilibrium and linear-response properties of a single quantum dot have been understood well, basically, based on the knowledge of the Kondo physics in dilute magnetic alloys.\cite{Hewson} The nonlinear transport under a finite bias voltage $V$,\cite{WM,HDW} however, is still not fully understood, although a number of previous works\cite{TheoriesII,SchillerHershfield,KNG,KSL,OguriA,OguriB} and recent ones\cite{FujiiUeda,GogolinKomnik,HB} have contributed to the continuous progress. One of the complications in the nonequilibrium system is that the density matrix is not a simple function of the Hamiltonian $H$. In thermal equilibrium, the density matrix has a universal form $\widehat{\rho}_{\rm eq}^{\phantom{0}} \propto e^{-\beta H}$. On the other hand, the nonequilibrium density matrix is not unique, even for the steady states. The Keldysh formalism has widely been used for constructing the nonequilibrium density matrix.\cite{Keldysh,Caroli} It has traditionally been described with a perturbative Green's function approach. However, in order to study comprehensively the nonequilibrium properties of strongly correlated electron systems, accurate nonperturbative approaches are also required. For instance, in the thermal equilibrium the approaches such as numerical renormalization group (NRG)\cite{KWW} and quantum Monte Carlo\cite{FH} methods played important roles to clarify the Kondo physics on the whole energy scale. On the other hand, we still have not had systematic ways to handle the nonequilibrium density matrix. It makes the construction of nonperturbative approaches difficult. Therefore, further considerations about the basic formulation itself seem to be still needed. In this paper, we describe a reformulation of a nonequilibrium steady state, using a reduced density matrix (RDM). It is obtained in the Fock space by integrating out one of the two channel degrees of freedom, which can be separated out from the dot by a reconstruction of the linear combination of the channels. The rest of the Hilbert space consists of the dot and the remaining channel, which are coupled with each other by a tunneling matrix element. The RDM takes a simple form in the noninteracting case, as shown in Eq.\ (\ref{eq:rho_s_0_T}). It enables us to treat the nonequilibrium averages using a single-channel Anderson model. The bias voltage appears in the subspace through an energy-dependent parameter $\mathcal{T}_{\varepsilon}$ given in Eq.\ (\ref{eq:T_eff}), which has a close relation to the usual nonequilibrium distribution function $f_\mathrm{eff}(\varepsilon)$. Specifically for the symmetric coupling $\Gamma_L=\Gamma_R$, the electrons distribute at zero temperature to the one-particle states between the two chemical potentials $\mu_L$ and $\mu_R$ completely at random. The Coulomb interaction is taken into account adiabatically, and then the nonequilibrium averages can be expressed in form of Eq.\ (\ref{eq:neq_average_u}), in terms of the interacting eigenstates and the RDM. This expression could be used for Hamiltonian-based nonperturbative calculations. It is also deduced that the nonequilibrium statistical weight spreads out in a rather wide energy region of the many-particle Hilbert space over the bias energy $eV$. The model is described in Sec.\ \ref{sec:formulation}. The derivation of the RDM is given in Sec.\ \ref{sec:RDM}. The mixed-state aspects of the RDM are examined for $\Gamma_L=\Gamma_R$ in Sec.\ \ref{sec:mixed-state}. The nonequilibrium distribution in the many-particle Hilbert space is discussed in Sec.\ \ref{sec:average}. Summary is given in Sec.\ \ref{sec:summary}. In the Appendixes, details of some calculations are provided. \section{Formulation} \label{sec:formulation} \subsection{Scattering states} We start with a single Anderson impurity connected to two leads on the left ($L$) and right ($R$). The Hamiltonian is given by $H = H_0 + H_{U}$ with $H_0 = H_{d} + H_{c} + H_T$, \begin{align} &H_{d}\, = \, \Bigl( \epsilon_d + \frac{U}{2}\, \Bigr)\, n_{d}^{\phantom 0} \,, \qquad H_{U} = \frac{U}{2} \Bigl(n_{d}^{\phantom 0} -1\Bigr)^2 , \label{eq:Hd} \\ & H_{c} \,= \sum_{\nu=L,R}\sum_{\sigma} \int \! d\varepsilon\, \varepsilon\, c^{\dagger}_{\varepsilon,\nu\sigma} \, c^{\phantom{\dagger}}_{\varepsilon,\nu\sigma} \;, \label{eq:Hc} \\ & H_T \, =\, \sum_{\nu=L,R}\sum_{\sigma} v_{\nu}^{\phantom{0}} \, \Bigl( \,d_{\sigma}^{\dagger} \,C_{\nu\sigma}^{\phantom{\dagger}} + C_{\nu\sigma}^{\dagger} \,d_{\sigma}^{\phantom{\dagger}} \Bigr) \;. \label{eq:Hmix} \end{align} Here, $d_{\sigma}^{\dagger}$ creates an electron with spin $\sigma$ in the dot, $n_{d}^{\phantom 0} = \sum_{\sigma} d^{\dagger}_{\sigma} d^{\phantom{\dagger}}_{\sigma}$, $\epsilon_d$ is the onsite potential, and $v_{\nu}^{\phantom{0}}$ is a tunneling matrix element between the dot and lead $\nu$. The conduction electrons are normalized as $\{ c^{\phantom{\dagger}}_{\varepsilon,\nu\sigma}, c^{\dagger}_{\varepsilon',\nu'\sigma'} \} = \delta_{\nu\nu'} \,\delta_{\sigma\sigma'}\, \delta(\varepsilon-\varepsilon')$, and $C_{\nu\sigma}^{\phantom{\dagger}} = \int d\varepsilon \sqrt{\rho(\varepsilon)} \, c^{\phantom{\dagger}}_{\varepsilon,\nu\sigma}$ with a condition $\int d\varepsilon\, \rho(\varepsilon) =1$; the same one-particle density of states $\rho(\varepsilon)$ is assumed for both of the leads. We are using explicitly the continuous form of the conduction bands\cite{KWW} in order to make the scattering states well defined. The Fermi level at equilibrium is taken to be $\mu = 0$. The two different chemical potentials, $\mu_L=-\mu_R=eV/2$, are introduced usually into the isolated leads for $H_T=0$.\cite{Caroli} Alternatively, as described by Hershfield,\cite{Hershfield} the same steady state can be constructed from $H_0$ including $H_T$ in the initial condition, and it has been applied to some special models.\cite{SchillerHershfield,Han} We use the later formulation in the following to set up the nonequilibrium density matrix, and assign $\mu_L$ and $\mu_R$ to the scattering states incident from left and right, respectively. We will use units $\hbar=1$, except for the current defined in Eq.\ (\ref{eq:current_op}). The two conduction bands can be classified into the $s$-wave and $p$-wave parts, defined by \begin{align} &s_{\varepsilon\sigma}^{\phantom{\dagger}} = \frac{ v_R^{\phantom{|}} c_{\varepsilon,R\sigma}^{\phantom{\dagger}} +v_L^{\phantom{|}} c_{\varepsilon,L\sigma}^{\phantom{\dagger}}}{\sqrt{v_R^2+v_L^2}} \,, \quad p_{\varepsilon\sigma}^{\phantom{\dagger}} = \frac{ v_L^{\phantom{|}} c_{\varepsilon,R\sigma}^{\phantom{\dagger}} -v_R^{\phantom{|}} c_{\varepsilon,L\sigma}^{\phantom{\dagger}}}{\sqrt{v_R^2+v_L^2}} \,. \label{eq:s_p_def} \end{align} The $s$-wave electrons couple to the dot via the tunneling matrix elements, as $H_T= \sum_{\sigma} \overline{v} \left( d_{\sigma}^{\dagger} S_{\nu\sigma}^{\phantom{\dagger}} + S_{\nu\sigma}^{\dagger} d_{\sigma}^{\phantom{\dagger}} \right) $, where $S_{\sigma}^{\phantom{\dagger}} = \int d\varepsilon \sqrt{\rho(\varepsilon)} \, s^{\phantom{\dagger}}_{\varepsilon\sigma}$ and $\overline{v} \equiv \sqrt{v_R^2+v_L^2}$. The $p$-wave electrons do not couple to the dot, but they contribute to the current through the impurity,\cite{KNG} \begin{align} \!\! J \equiv \frac{v_R^2 J_L + v_L^2 J_R }{v_R^2+v_L^2} = -i \frac{e}{\hbar} \frac{v_L^{\phantom{|}}v_R^{\phantom{|}}} {\overline{v}} \sum_{\sigma} \left( P_{\sigma}^{\dagger} d_{\sigma}^{\phantom{\dagger}} - d_{\sigma}^{\dagger} P_{\sigma}^{\phantom{\dagger}} \right) . \label{eq:current_op} \end{align} Here, $P_{\sigma}^{\phantom{\dagger}} = \int d\varepsilon \sqrt{\rho(\varepsilon)} \, p^{\phantom{\dagger}}_{\varepsilon\sigma}$, and $J_L$ ($J_R$) is the current flowing from the left lead (dot) to the dot (right lead). The scattering states in the $s$-wave part, $\alpha_{\varepsilon\sigma}^{\dagger}$, can be calculated explicitly for $H_U=0$ including all effects of $H_T$, \begin{align} &\alpha_{\varepsilon\sigma}^{\dagger} \,= \, s_{\varepsilon\sigma}^{\dagger} \,+\, \overline{v}\, \sqrt{\rho(\varepsilon)} \,G^{r}_{0}(\varepsilon) \,d_{\sigma}^{\dagger} \, \nonumber \\ & \qquad \qquad \ \ + \overline{v} \sqrt{\rho(\varepsilon)} \,G^{r}_{0}(\varepsilon)\! \int \! d\varepsilon'\, \frac{\sqrt{\rho(\varepsilon')}}{\varepsilon - \varepsilon' + i \delta} \ s_{\varepsilon'\sigma}^{\dagger} \,, \\ & G^{r}_{0}(\varepsilon) =\, \left[ \varepsilon- \left(\epsilon_d +\frac{U}{2}\right) -\overline{v}^2 \!\!\int\! d\varepsilon' \frac{\rho(\varepsilon')}{\varepsilon-\varepsilon' +i \delta} \right]^{-1} \! . \label{eq:G^r_0} \end{align} Note that the resonance width of the impurity level is given by $\Delta(\varepsilon) \equiv \pi \overline{v}^2 \rho(\varepsilon) =\Gamma_L+\Gamma_R$ with $\Gamma_{\nu} = \pi v_{\nu}^2 \rho(\varepsilon)$ in the noninteracting case. We will also be using the advanced Green's function $G^{a}_{0}(\varepsilon) \equiv \left\{ G^{r}_{0}(\varepsilon) \right\}^*$. The one-particle states $\alpha^{\phantom{\dagger}}_{\varepsilon\sigma}$ and $p^{\phantom{\dagger}}_{\varepsilon\sigma}$ constitute the eigenstates of $\,H_0 =H_0^S+H_0^P$, \begin{align} H_0^S = \sum_{\sigma} \! \int \!\! d\varepsilon\, \varepsilon\, \alpha^{\dagger}_{\varepsilon\sigma} \alpha^{\phantom{\dagger}}_{\varepsilon\sigma} , \quad \ H_0^P = \sum_{\sigma}\! \int \!\! d\varepsilon\, \varepsilon\, p^{\dagger}_{\varepsilon\sigma} p^{\phantom{\dagger}}_{\varepsilon\sigma} \, . \label{eq:H0_Jost1} \end{align} The scattering state is also normalized as $\{ \alpha^{\phantom{\dagger}}_{\varepsilon,\sigma}, \alpha^{\dagger}_{\varepsilon',\sigma'} \} = \delta_{\sigma\sigma'}\, \delta(\varepsilon-\varepsilon')$, and the impurity state $d_{\sigma}$ can be expressed in terms of $\alpha_{\varepsilon\sigma}$ as \begin{align} d_{\sigma} = \overline{v} \int d\varepsilon \sqrt{\rho(\varepsilon)}\, G_{0}^{r}(\varepsilon) \, \alpha^{\phantom{\dagger}}_{\varepsilon\sigma}\;. \label{eq:d} \end{align} Here, we have omitted, for simplicity, the possibility that bound states could emerge outside of the conduction band. It can be taken into account when it is necessary. \subsection{Nonequilibrium steady state} The noninteracting Hamiltonian can also be diagonalized with the scattering states moving towards left and right, \begin{align} &H_0 = \sum_{\nu=L,R} \sum_{\sigma} \int \! d\varepsilon\, \varepsilon \, \gamma^{\dagger}_{\varepsilon,\nu\sigma} \gamma^{\phantom{\dagger}}_{\varepsilon,\nu\sigma} \;, \label{eq:H0_Jost2} \\ &\gamma_{\varepsilon,R\sigma}^{\phantom{\dagger}} = \frac{ v_R^{\phantom{|}}\,\alpha_{\varepsilon\sigma}^{\phantom{\dagger}} + v_L^{\phantom{|}}\, p_{\varepsilon\sigma}^{\phantom{\dagger}}}{\sqrt{v_L^2+v_R^2}} \;, \quad \gamma_{\varepsilon,L\sigma}^{\phantom{\dagger}} = \frac{ v_L^{\phantom{|}}\, \alpha_{\varepsilon\sigma}^{\phantom{\dagger}} - v_R^{\phantom{|}}\, p_{\varepsilon\sigma}^{\phantom{\dagger}}}{\sqrt{v_L^2+v_R^2}} \;. \label{eq:scattering_state} \end{align} This propagating-wave form of $H_0$ are compatible with the standing-wave form Eq.\ (\ref{eq:H0_Jost1}), as the scattering states have two-fold degeneracy. The chemical potentials $\mu_L$ and $\mu_R$ can be assigned to these two propagating states $\gamma^{\dagger}_{\varepsilon,L\sigma}$ and $\gamma^{\dagger}_{\varepsilon,R\sigma}$, respectively, using\cite{Hershfield,KNG} \begin{align} Y_0 \equiv&\, \frac{eV}{2} \sum_{\sigma} \int\!d\varepsilon \left( \gamma^{\dagger}_{\varepsilon,L\sigma} \gamma^{\phantom{\dagger}}_{\varepsilon,L\sigma} - \gamma^{\dagger}_{\varepsilon,R\sigma} \gamma^{\phantom{\dagger}}_{\varepsilon,R\sigma} \right) \nonumber \\ =&\, -\, eV \, \frac{v_Lv_R}{v_L^2+v_R^2} \sum_{\sigma} \int\!d\varepsilon \, \Bigl( \alpha^{\dagger}_{\varepsilon\sigma} \,p^{\phantom{\dagger}}_{\varepsilon\sigma} + p^{\dagger}_{\varepsilon\sigma} \alpha^{\phantom{\dagger}}_{\varepsilon\sigma} \Bigr) \nonumber \\ &\, + \frac{eV}{2} \frac{v_L^2-v_R^2}{v_L^2+v_R^2} \sum_{\sigma} \int\!d\varepsilon \, \Bigl( \alpha^{\dagger}_{\varepsilon\sigma} \alpha^{\phantom{\dagger}}_{\varepsilon\sigma} - p^{\dagger}_{\varepsilon\sigma} p^{\phantom{\dagger}}_{\varepsilon\sigma} \Bigr) . \label{eq:Y_0} \end{align} With this operator, the noninteracting density matrix can be constructed as \begin{align} \widehat{\rho}_0^{\phantom{0}} = \frac{ e^{ -\beta (H_0- Y_0 )}}{\Xi_0} \;, \qquad \Xi_0 \equiv \mbox{Tr}\, e^{ -\beta (H_0- Y_0 )}\;. \label{eq:rho_0} \end{align} The statistical average is defined by $\langle \cdots\rangle_0 = \mbox{Tr}\,[\widehat{\rho}_0^{\phantom{0}} \cdots]$. For instance, the distribution function for $\gamma^{\phantom{\dagger}}_{\varepsilon,\nu\sigma}$ is given by \begin{align} \langle \gamma^{\dagger}_{\varepsilon,\nu\sigma} \gamma^{\phantom{\dagger}}_{\varepsilon',\nu'\sigma'} \rangle_0 = f_{\nu}(\varepsilon )\, \delta(\varepsilon-\varepsilon')\, \delta_{\nu\nu'}\, \delta_{\sigma\sigma'}\;, \label{eq:gamma_average} \end{align} where $f_{\nu}(\varepsilon) \equiv f(\varepsilon- \mu_{\nu})$, and $f(\varepsilon)=\left[ e^{\varepsilon/T}+1 \right]^{-1}$ is the Fermi function. Similarly, from Eqs.\ (\ref{eq:scattering_state}) and (\ref{eq:gamma_average}), we obtain \begin{align} \langle \alpha^{\dagger}_{\varepsilon\sigma} \alpha^{\phantom{\dagger}}_{\varepsilon'\sigma'} \rangle_0 \,=& \, f_\mathrm{eff}(\varepsilon)\, \delta(\varepsilon-\varepsilon')\, \delta_{\sigma\sigma'}\;, \label{eq:f_eff} \\ f_\mathrm{eff}(\varepsilon) \equiv& \, \frac{\Gamma_L\,f_L(\varepsilon) + \Gamma_R\,f_R(\varepsilon)} {\Gamma_L + \Gamma_R} \;, \label{eq:f_eff_def} \end{align} and \begin{align} & \langle \alpha^{\dagger}_{\varepsilon\sigma} p^{\phantom{\dagger}}_{\varepsilon'\sigma'} \rangle_0 \,=\, \langle p^{\dagger}_{\varepsilon'\sigma'} \alpha^{\phantom{\dagger}}_{\varepsilon\sigma} \rangle_0 \nonumber\\ & \quad = -\, \frac{v_L^{\phantom{|}}v_R^{\phantom{|}}} {v_R^2+v_L^2} \bigl[\, f_L(\varepsilon) - f_R(\varepsilon)\,\bigr]\, \delta(\varepsilon-\varepsilon')\, \delta_{\sigma\sigma'} . \label{eq:alpha_p} \end{align} Here, the distribution function $f_\mathrm{eff}(\varepsilon)$ appears not only for the impurity state but also for all the scattering states with the $s$-wave character.\cite{HDW} At zero temperature, $f_\mathrm{eff}(\varepsilon)$ takes a constant value $\Gamma_L/(\Gamma_R+\Gamma_L)$ for $\mu_R<\varepsilon<\mu_L$, as shown in Fig.\ \ref{fig:distribution}. Specifically, the constant becomes $1/2$ for the symmetric coupling $\Gamma_L=\Gamma_R$. This feature looks quite similar to a high-temperature behavior of the Fermi function $\left. f(\varepsilon)\right|_{T\to\infty} =1/2$, and it can be explained as a manifestation of the mixed-state nature of the RDM, which is given in Eq.\ (\ref{eq:rho_s_0_T}). \input{fig1_feff.tex} The Coulomb interaction is switched on adiabatically based on the Keldysh formalism,\cite{Keldysh} \begin{align} &\widehat{\rho} \equiv S(0,-\infty) \,\widehat{\rho}_0^{\phantom{0}}\, S(-\infty,0) \;, \label{eq:rho_org} \\ &S(t_2,t_1) \equiv \mbox{T} \exp \! \left[- i \! \int_{t_1}^{t_2} \! dt\, e^{iH_0^S t} H_U e^{-iH_0^S t} \right]. \label{eq:S} \end{align} Note that $\bigl[ S(0,-\infty), \, p_{\varepsilon\sigma}^{\phantom{\dagger}} \bigr]=0$, and thus the interaction affects only the $s$-wave part of the Hilbert space, which is described by the Hamiltonian \begin{equation} H^S= H_0^S+H_U \;. \label{eq:H_s} \end{equation} Hershfield has provided one possible way to construct an interacting operator $Y$, by which the full density matrix is written in the form $\widehat{\rho} \propto e^{-\beta(H-Y)}$.\cite{Hershfield} In the present study, however, we go back to the original definition of $\widehat{\rho}$ given in Eq.\ (\ref{eq:rho_org}) for a general description. The interacting average $\langle \cdots\rangle$ is taken with the full density matrix $\widehat{\rho}$. For instance, using Eq.\ (\ref{eq:d}), the average for the local charge can be written in the form \begin{align} \langle n_d^{\phantom{0}} \rangle =& \ \overline{v}^2 \sum_{\sigma} \!\int \!\!d\varepsilon\,d\varepsilon' \sqrt{\rho(\varepsilon')\rho(\varepsilon)} \, G^{0}_{a}(\varepsilon') G^{0}_{r}(\varepsilon) \bigl\langle \alpha_{\varepsilon' \sigma}^{\dagger} \alpha_{\varepsilon \sigma}^{\phantom{\dagger}} \bigr\rangle , \nonumber \\ =& \sum_{\sigma} \!\int \! \frac{d\omega}{2 \pi i} \, G^{<}(\omega) \;. \label{eq:nd_GF} \end{align} Here, $ G^{<}(\omega) \equiv i \int_{-\infty}^{\infty} \! dt \, e^{i \omega t} \bigl\langle d_{\sigma}^{\dagger}\, e^{iH^S t}d_{\sigma}^{\phantom{\dagger}}e^{-iH^S t} \bigr\rangle$ is the lesser Green's function, which for the noninteracting limit is given by $ G_{0}^{<}(\varepsilon) = -f_{\mathrm{eff}}(\varepsilon) \left[ G_{0}^r(\varepsilon) - G_{0}^a(\varepsilon) \right]$. The nonequilibrium current can also be expressed as an average defined in the $s$-wave subspace,\cite{MW} \begin{align} \langle J \rangle =& -i\, \frac{e\,v_L^{\phantom{|}} v_R^{\phantom{|}}}{\hbar} \sum_{\sigma} \int\!\! d\varepsilon \, d\varepsilon' \sqrt{\rho(\varepsilon') \rho(\varepsilon)} \nonumber \\ & \times \left[\, G_{0}^{r}(\varepsilon)\, \bigl\langle p_{\varepsilon'\sigma}^{\dagger} \alpha_{\varepsilon\sigma}^{\phantom{\dagger}} \bigr\rangle - G_{0}^{a}(\varepsilon)\, \bigl\langle \alpha_{\varepsilon\sigma}^{\dagger} p_{\varepsilon'\sigma}^{\phantom{\dagger}} \bigr\rangle \,\right] \;, \nonumber \\ =&\, \frac{2 e}{h} \int \!\! d\omega \, \bigl[ f_L(\omega) - f_R(\omega) \bigr] \frac{4 \Gamma_L \Gamma_R }{\Gamma_L +\Gamma_R } \left[\, - {\rm Im}\, G^r(\omega) \,\right] . \label{eq:caroli} \end{align} Here, $ G^r(\omega) \equiv -i \int_0^{\infty} \! dt \, e^{i(\omega+i\delta)t} \bigl\langle \bigl\{e^{iH^S t}d_{\sigma}^{\phantom{\dagger}}e^{-iH^S t}, d_{\sigma}^{\dagger} \bigr\} \bigr\rangle$ is the full retarded Green's function. Therefore, to calculate these averages, the $p$-wave degrees of freedom can be integrated out first before taking the trace over the correlated $s$-wave part. \section{Reduced Density Matrix} \label{sec:RDM} We describe in this section a derivation of the RDM. To this end, we introduce a discretized Hamiltonian corresponding to Eq.\ (\ref{eq:H0_Jost1}) as $\widetilde{H}_0 = \widetilde{H}_0^S + \widetilde{H}_0^P$, \begin{align} & \widetilde{H}_0^S = \! \sum_{\sigma} \sum_{m} \varepsilon_m \, \Bigl( \widetilde{\alpha}^{\dagger}_{\varepsilon_m,\sigma} \widetilde{\alpha}^{\phantom{\dagger}}_{\varepsilon_m,\sigma} - \langle \widetilde{\alpha}^{\dagger}_{\varepsilon_m,\sigma} \widetilde{\alpha}^{\phantom{\dagger}}_{\varepsilon_m,\sigma} \rangle^\mathrm{GS}_0 \Bigr), \label{eq:H_0_s_discrete} \\ & \widetilde{H}_0^P = \! \sum_{\sigma} \sum_{m} \varepsilon_m \, \Bigl( \widetilde{p}^{\dagger}_{\varepsilon_m,\sigma} \widetilde{p}^{\phantom{\dagger}}_{\varepsilon_m,\sigma} - \langle \widetilde{p}^{\dagger}_{\varepsilon_m,\sigma} \widetilde{p}^{\phantom{\dagger}}_{\varepsilon_m,\sigma} \rangle^{\mathrm{GS}}_0 \Bigr), \label{eq:H_0_p_discrete} \end{align} with the normalization $\{ \widetilde{\alpha}^{\phantom{\dagger}}_{\varepsilon_m,\sigma}, \widetilde{\alpha}^{\dagger}_{\varepsilon_{m'},\sigma'} \} = \delta_{mm'}\delta_{\sigma\sigma'}$, and $\{ \widetilde{p}^{\phantom{\dagger}}_{\varepsilon_m,\sigma}, \widetilde{p}^{\dagger}_{\varepsilon_{m'},\sigma'} \} = \delta_{mm'}\delta_{\sigma\sigma'}$. The constant term $\langle \cdots \rangle^\mathrm{GS}_0$, which denotes the noninteracting ground-state average in equilibrium, is subtracted so as to make the lowest energy of $\widetilde{H}_0^S$ and that of $\widetilde{H}_0^P$ zero, \begin{align} \!\! \langle \widetilde{\alpha}^{\dagger}_{\varepsilon_m,\sigma} \widetilde{\alpha}^{\phantom{\dagger}}_{\varepsilon_m,\sigma} \rangle^\mathrm{GS}_0 =\langle \widetilde{p}^{\dagger}_{\varepsilon_m,\sigma} \widetilde{p}^{\phantom{\dagger}}_{\varepsilon_m,\sigma} \rangle^\mathrm{GS}_0 = \left\{ \!\! \begin{array}{cl} 1, & \varepsilon_m < 0 , \\ 0, & \varepsilon_m > 0 . \end{array} \right. \end{align} Specifically, we assume a linear discretization with a finite level spacing $\delta \varepsilon$, such that $\varepsilon_m = (m+1/2) \,\delta \varepsilon$ for $m=0,\pm 1, \pm 2, \ldots$. The continuous spectrum can be recovered in the limit of $\delta \varepsilon \to 0$ with $\alpha_{\varepsilon,\sigma}^{\phantom{\dagger}} = \widetilde{\alpha}_{\varepsilon_m,\sigma}^{\phantom{\dagger}} / \sqrt{\delta \varepsilon}$ and $ p_{\varepsilon,\sigma}^{\phantom{\dagger}} = \widetilde{p}_{\varepsilon_m,\sigma}^{\phantom{\dagger}}/ \sqrt{\delta \varepsilon}$. The discretized operator is also introduced for the propagating states $\widetilde{\gamma}^{\phantom{\dagger}}_{\varepsilon_m,\nu\sigma}$ using Eq.\ (\ref{eq:scattering_state}), and then $\widetilde{H}_0$ can also be written in the form \begin{align} \widetilde{H}_0 = \! \sum_{\nu} \sum_{\sigma} \sum_{m} \varepsilon_m \, \Bigl( \widetilde{\gamma}^{\dagger}_{\varepsilon_m,\nu\sigma} \widetilde{\gamma}^{\phantom{\dagger}}_{\varepsilon_m,\nu\sigma} - \langle \widetilde{\gamma}^{\dagger}_{\varepsilon_m,\nu\sigma} \widetilde{\gamma}^{\phantom{\dagger}}_{\varepsilon_m,\nu\sigma} \rangle^\mathrm{GS}_0 \Bigr). \end{align} In this discretization, each discrete level $\varepsilon_m$ preserves the two-fold degeneracy due to the left-going and right-going scattering states. This degeneracy is essential to describe the current-carrying state with the density matrix. The trace over the Hilbert space can be carried out explicitly for the discretized model. The normalization factor $\widetilde{\Xi}_0 = \mbox{Tr}\, e^{ -\beta (\widetilde{H}_0- \widetilde{Y}_0 )}$, where $\widetilde{Y}_0$ is the discretized version of Eq.\ (\ref{eq:Y_0}), can be calculated as \begin{align} \widetilde{\Xi}_0=& \, \prod_{\nu=L,R} \prod_{\sigma} \prod_{\varepsilon_m} \left[1+e^{-\beta(\varepsilon_m-\mu_{\nu})\, \mathrm{sgn}\,\varepsilon_m} \right] \;. \label{eq:xi_0_expression} \end{align} Here, we have used a property $e^{\beta(\mu_L+\mu_R)}=1$ following from $\mu_L=-\mu_R=eV/2$. The corresponding {\em free energy} that is defined by $\widetilde{\Xi}_0=e^{-\beta F_0}$ takes the form \begin{align} \beta F_0 = - \frac{1}{\delta \varepsilon} \sum_{\nu=L,R} \!\sum_{\sigma} \!\int \!\! d\varepsilon \, \ln \left[1+e^{-\beta(\varepsilon-\mu_{\nu})\, \mathrm{sgn}\,\varepsilon } \right]. \end{align} In the continuum limit $\delta \varepsilon\to 0$, the factor $1/\delta \varepsilon$ should be replaced by the Dirac $\delta$-function $\delta(0)$. It appears because the energy of the whole system is proportional to the system size $N$. For instance, the level spacing is given by $\delta \varepsilon = 2D/N$ for a flat band with a half width $D$. Therefore the level spacing $\delta \varepsilon$ has the physical meanings for the averages of order $N$, although we have started from the continuous model in order to capture properly the two-fold degeneracy in the scattering states. The density matrix can also be calculated as, \begin{align} \widetilde{\rho}_0 \,\equiv& \ \frac{e^{-\beta(\widetilde{H}_0 -\widetilde{Y}_0)}} {\mbox{Tr}\,e^{-\beta(\widetilde{H}_0 -\widetilde{Y}_0)}} \\ =& \prod_{\nu=L,R}\! \prod_{\sigma}\prod_{\varepsilon} \frac{ 1+\left[\,e^{-\beta(\varepsilon-\mu_{\nu})}-1\,\right] \widetilde{\gamma}_{\varepsilon,\nu\sigma}^{\dagger} \widetilde{\gamma}_{\varepsilon,\nu\sigma}^{\phantom{\dagger}} }{1+e^{-\beta (\varepsilon-\mu_{\nu})}} \;. \label{eq:rho_0_expression} \end{align} Here, we have suppressed the suffix $m$ of the discrete frequency $\varepsilon_m$, for simplicity. Equation (\ref{eq:rho_0_expression}) can be rewritten in terms of $\widetilde{\alpha}_{\varepsilon,\sigma}^{\phantom{\dagger}}$ and $\widetilde{p}_{\varepsilon,\sigma}^{\phantom{\dagger}}$, by substituting Eq.\ (\ref{eq:scattering_state}), as \begin{align} & \!\!\!\!\!\! \widetilde{\rho}_0 = \prod_{\sigma}\prod_{\varepsilon} \bigl(1-f_L\bigr) \bigl(1-f_R\bigr) \biggl[ \ 1\ + \nonumber \\ & + \left( \frac{v_L^2}{v_L^2+v_R^2} \frac{f_L}{1-f_L} + \frac{v_R^2}{v_L^2+v_R^2} \frac{f_R}{1-f_R} -1 \right) \widetilde{\alpha}_{\varepsilon\sigma}^{\dagger} \widetilde{\alpha}_{\varepsilon\sigma}^{\phantom{\dagger}} \nonumber \\ & + \left( \frac{v_R^2}{v_L^2+v_R^2} \frac{f_L}{1-f_L} + \frac{v_L^2}{v_L^2+v_R^2} \frac{f_R}{1-f_R} -1 \right) \widetilde{p}_{\varepsilon\sigma}^{\dagger} \widetilde{p}_{\varepsilon\sigma}^{\phantom{\dagger}} \nonumber\\ & -\frac{v_Lv_R}{v_L^2+v_R^2}\frac{f_L-f_R}{(1-f_L)(1-f_R)} \left( \widetilde{\alpha}_{\varepsilon\sigma}^{\dagger} \widetilde{p}_{\varepsilon\sigma}^{\phantom{\dagger}} + \widetilde{p}_{\varepsilon\sigma}^{\dagger} \widetilde{\alpha}_{\varepsilon\sigma}^{\phantom{\dagger}} \right) \nonumber\\ & + \frac{(1-2f_L)(1-2f_R)}{(1-f_L)(1-f_R)}\, \widetilde{\alpha}_{\varepsilon\sigma}^{\dagger} \widetilde{\alpha}_{\varepsilon\sigma}^{\phantom{\dagger}} \, \widetilde{p}_{\varepsilon\sigma}^{\dagger} \widetilde{p}_{\varepsilon\sigma}^{\phantom{\dagger}} \,\biggr] . \label{eq:rho_0_sp} \end{align} Furthermore, the partial trace over the $p$-wave part can be carried out to yield the RDM for the $s$-wave subspace \begin{align} \mbox{Tr}_{P}\, \widetilde{\rho}_0 =& \prod_{\sigma}\prod_{\varepsilon} \left(1-f_\mathrm{eff}\right) \left[ 1 + \frac{2f_\mathrm{eff}-1}{1-f_\mathrm{eff}} \, \widetilde{\alpha}_{\varepsilon\sigma}^{\dagger} \widetilde{\alpha}_{\varepsilon\sigma}^{\phantom{\dagger}} \right]. \label{eq:rho_s_0_primitive} \end{align} It can also be expressed in an exponential form, \begin{align} & \widehat{\rho}_0^S \equiv \mbox{Tr}_{P}\, \widetilde{\rho}_0 \ = \frac{e^{-K_0^S}}{\mbox{Tr}_S\ e^{-K_0^S}}\;, \label{eq:rho_s_0_T} \\ & \!\!\! K_0^S \equiv \sum_{\sigma} \sum_{\varepsilon} \frac{\varepsilon}{\mathcal{T}_{\varepsilon}} \, \widetilde{\alpha}_{\varepsilon\sigma}^{\dagger} \widetilde{\alpha}_{\varepsilon\sigma}^{\phantom{\dagger}} \xrightarrow[\ \delta \varepsilon \to 0\ ]{} \sum_{\sigma} \int\! d\varepsilon\, \frac{\varepsilon}{\mathcal{T}_{\varepsilon}} \, \alpha_{\varepsilon\sigma}^{\dagger} \alpha_{\varepsilon\sigma}^{\phantom{\dagger}} , \label{eq:K_s} \\ &\frac{1}{\mathcal{T}_{\varepsilon}} \equiv \frac{1}{\varepsilon} \ln \! \left[ \frac{1-f_\mathrm{eff}(\varepsilon)} {f_\mathrm{eff}(\varepsilon)} \right] . \label{eq:T_eff} \end{align} \begin{figure}[t] \includegraphics[width= 0.78\linewidth]{beffz3.eps} \caption{ (color online). Parameter $\beta_\mathrm{eff} \equiv 1/\mathcal{T}_{\varepsilon}$, which determines the RDM as Eq.\ (\ref{eq:K_s}), is plotted as a function of $\varepsilon$ for several temperatures $T/(eV)=0.0,\, 0.05,\, 0.1$ and $0.2$, in the symmetric coupling case $\Gamma_L=\Gamma_R$. } \label{fig:beff} \end{figure} Then, the average of an operator $\widehat{\cal O}_S = \widehat{\cal O}_S(\alpha,\alpha^{\dagger})$, which is an arbitrary function of $\alpha^{\phantom{\dagger}}_{\varepsilon,\sigma}$ and $\alpha^{\dagger}_{\varepsilon,\sigma}$, can be calculated in the $s$-wave subspace using the RDM, \begin{align} &\langle \widehat{\cal O}_S \rangle_0 \equiv \mbox{Tr}_S^{\phantom{|}} \mbox{Tr}_P^{\phantom{|}} \! \left[ \widehat{\rho}_{0}^{\phantom{S}} \widehat{\cal O}_S \right] = \mbox{Tr}_S^{\phantom{|}} \! \left[ \widehat{\rho}_{0}^{S}\, \widehat{\cal O}_S \right] . \label{eq:with_rho_s_0} \end{align} Note that the impurity level has been included in the $s$-wave part. It is straight forward to confirm that $\langle \alpha^{\dagger}_{\varepsilon,\sigma} \alpha^{\phantom{\dagger}}_{\varepsilon',\sigma'} \rangle_0$ given in Eq.\ (\ref{eq:f_eff}) can be reproduced from Eq.\ (\ref{eq:with_rho_s_0}), and then the function $f_\mathrm{eff}(\varepsilon)$ defined in Eq.\ (\ref{eq:f_eff_def}) can be expressed in terms of $\mathcal{T}_{\varepsilon}$ as \begin{equation} f_\mathrm{eff}(\varepsilon) = \frac{1}{e^{\varepsilon/\mathcal{T}_{\varepsilon}} +1} \;. \end{equation} The bias dependence appears in the $s$-wave subspace through this energy-dependent parameter $\mathcal{T}_{\varepsilon}$. The inverse of this parameter, $\beta_\mathrm{eff} \equiv 1/\mathcal{T}_{\varepsilon}$, is plotted in Fig.\ \ref{fig:beff} as a function of $\varepsilon$ for several values of real temperature $T$ in the symmetric coupling $\Gamma_L=\Gamma_R$ case. The energy dependence of $\mathcal{T}_{\varepsilon}$ becomes weak with increasing $T$. At zero temperature, it takes the form, \begin{align} \mathcal{T}_{\varepsilon} \to \left\{ \! \begin{array}{cl} \infty, & \ \ |\varepsilon| < eV/2 , \\ 0, & \ \ |\varepsilon| > eV/2 . \end{array} \right. \end{align} Namely, $\mathcal{T}_{\varepsilon}$ reaches infinity inside the energy window $|\varepsilon| < eV/2$, while it remain zero on the outside. For this reason, in the limit of $eV \to \infty$ the nonequilibrium Green's functions at the impurity site coincide with the equilibrium ones for the high-temperature limit $T\to \infty$.\cite{OguriB} The partial trace over the $p$-wave part can be carried out also for $H_U \neq 0$, as the Coulomb interaction affects only the $s$-wave part of the Hilbert space. Therefore, the interacting RDM $\widehat{\rho}_{\phantom{0}}^{S}$ evolves adiabatically from the noninteracting RDM $\widehat{\rho}_0^{S}$ as \begin{align} \widehat{\rho}_{\phantom{0}}^{S} =& \, S(0,-\infty) \,\widehat{\rho}_0^{S} S(-\infty,0) \;, \label{eq:rho_s} \\ \langle \widehat{\cal O}_S \rangle \equiv& \, \mbox{Tr}_S^{\phantom{|}} \mbox{Tr}_P^{\phantom{|}}\! \left[\, \widehat{\rho} \ \widehat{\cal O}_S \right] = \mbox{Tr}_S^{\phantom{|}}\! \left[\, \widehat{\rho}_{\phantom{0}}^{S} \, \widehat{\cal O}_S \right] \;. \label{eq:neq_average_u2} \end{align} It enables us to treat the bias voltage within the $s$-wave part of the Hilbert space. Namely, one can start with the single-channel Anderson model $H^S$ in the continuous form, Eq.\ (\ref{eq:H_s}). Then, the nonequilibrium distribution can be taken into account through the initial RDM $\widehat{\rho}_{0}^{S}$, which evolves to the interacting one $\widehat{\rho}_{\phantom{0}}^{S}$. For the single-channel model, it is not necessary to use the same discretization scheme that was introduced to obtain Eq.\ (\ref{eq:rho_s_0_primitive}), because $\widehat{\rho}_{0}^{S}$ has a well-defined continuum form as Eq.\ (\ref{eq:K_s}). In addition to these points, the correlation functions containing a $p$-wave operator can also be calculated with the single-channel model as shown in Appendix \ref{sec:RDM_for_p}. \section{Mixed-state aspect for $\Gamma_L=\Gamma_R$} \label{sec:mixed-state} As we see in Fig.\ \ref{fig:beff}, effects of the bias voltage are pronounced significantly at zero temperature, especially when the system has an inversion symmetry $\Gamma_L=\Gamma_R$. We consider in the following the properties of the RDM in this particular case. At zero temperature $T=0$ the noninteracting density matrix $\widehat{\rho}_0^{\phantom{0}}$ for the whole system can be constructed as a pure state $\left| \Phi_{\rm neq} \right \rangle$, in which the scattering states incident from left and right are filled up to $\mu_L=eV/2$ and $\mu_R=-eV/2$, respectively, \begin{align} &\left| \Phi_{\rm neq} \right \rangle = \prod_{\varepsilon=0}^{\frac{eV}{2}} \prod_{\varepsilon'=-\frac{eV}{2}}^{0} \prod_{\sigma} \widetilde{\gamma}_{\varepsilon,L\sigma}^{\dagger} \widetilde{\gamma}_{\varepsilon',R\sigma}^{\phantom{\dagger}} \left| \Phi_{0} \right \rangle \;, \label{eq:Phi_0_neq} \\ &\left| \Phi_{0} \right \rangle \equiv \prod_{\varepsilon<0}\prod_{\sigma} \widetilde{\gamma}_{\varepsilon,L\sigma}^{\dagger} \widetilde{\gamma}_{\varepsilon,R\sigma}^{\dagger} \left|0\right\rangle = \prod_{\varepsilon<0}\prod_{\sigma} \widetilde{\alpha}_{\varepsilon\sigma}^{\dagger} \widetilde{p}_{\varepsilon\sigma}^{\dagger} \left|0\right\rangle . \label{eq:Phi_0} \end{align} Here, $\left|0\right\rangle$ is the vacuum state. The noninteracting ground state $\left| \Phi_{0}^{} \right \rangle$ can be decomposed into the Fermi sea of the $s$-wave part $\left| \Phi_{0}^{S}\right \rangle$ and that of the $p$-wave part $\left| \Phi_{0}^{P}\right \rangle$ as $\left| \Phi_{0}^{} \right \rangle = \left| \Phi_{0}^{S} \right \rangle \left| \Phi_{0}^{P} \right \rangle$. Therefore, Eq.\ (\ref{eq:Phi_0_neq}) can be rewritten, using Eq.\ (\ref{eq:scattering_state}) for $v_L=v_R$, as \begin{align} \left| \Phi_{\rm neq} \right \rangle &=\, \prod_{\varepsilon=0}^{\frac{eV}{2}} \prod_{\varepsilon'=-\frac{eV}{2}}^{0} \prod_{\sigma} \frac{(\widetilde{\alpha}_{\varepsilon\sigma}^{\dagger} - \widetilde{p}_{\varepsilon\sigma}^{\dagger})}{\sqrt{2}} \frac{(\widetilde{\alpha}_{\varepsilon'\sigma}^{\phantom{\dagger}} + \widetilde{p}_{\varepsilon'\sigma}^{\phantom{\dagger}})}{\sqrt{2}} \,\left| \Phi_{0} \right \rangle \nonumber \\ &=\, \frac{1}{\sqrt{Z_0}}\,{\sum_{i \in \mathcal{W} }}' \left| \Phi_{i}^{S} \right \rangle \left| \Phi_{i}^{P} \right \rangle \;. \label{rho_0_mixed} \end{align} Here, $\left| \Phi_{i}^{S} \right \rangle$ and $\left| \Phi_{i}^{P} \right \rangle$ are the excited states of the $s$-wave part and that of the $p$-wave part, respectively: the sign caused by exchanges of the fermion operators is absorbed into the global phase of each many-particle eigenstate in the definition. The primed sum for $i$ runs over a set $\mathcal{W}$, which consists of all the possible excited states that can be generated from $\left| \Phi_{0}^{} \right \rangle$ by changing the occupation of the one-particle states inside the biased energy region $|\varepsilon|<eV/2$. The factor $Z_0$ corresponds to the dimension of $\mathcal{W}$. The pure-state average of an arbitrary operator $\widehat{\cal O}_S = \widehat{\cal O}_S(\alpha,\alpha^{\dagger})$ that is defined in the $s$-wave space can be calculated, carrying out the summation over the $p$-wave part, as \begin{align} & \left\langle \Phi_{\rm neq} \right | \widehat{\cal O}_S \left| \Phi_{\rm neq} \right \rangle = \frac{1}{Z_0}\,{\sum_{ij\in\mathcal{W}}}' \! \left \langle \Phi_{i}^{P} \right| \!\! \left\langle \Phi_{i}^{S} \right | \widehat{\cal O}_S \left| \Phi_{j}^{S} \right \rangle \!\! \left| \Phi_{j}^{P} \right \rangle \nonumber \\ & \qquad \quad \ \ = \frac{1}{Z_0}\,{\sum_{i\in\mathcal{W}}}' \!\left\langle \Phi_{i}^{S} \right | \widehat{\cal O}_S \left| \Phi_{i}^{S} \right \rangle = \mbox{Tr}_S^{\phantom{|}}\! \left[\, \widehat{\rho}_0^{S} \, \widehat{\cal O}_S \right] . \label{eq:mixed_s_first} \end{align} Therefore in this case, the noninteracting RDM takes a mixed-state form $\widehat{\rho}_{0}^{S} = \sum_n \! \left|\Phi_{n}^{S}\right \rangle w_{0,n}^{S} \! \left\langle \Phi_{n}^{S}\right|$ with \begin{align} w_{0,n}^{S} = \left\{\! \begin{array}{cl} 1/Z_0, &\quad n \in \mathcal{W} , \\ 0, &\quad n \notin \mathcal{W} . \end{array} \right. \label{eq:w_n^S} \end{align} The summation with respect to $n$, or $\mbox{Tr}_S^{\phantom{|}}[\cdots]$, run over the whole region of the $s$-wave subspace. The statistical weight $w_{0,n}^{S}$ takes just two values $0$ or $1/Z_0$. However, it is not a simple function of the many-particle energy eigenvalue $E_{0,n}^S$ of $\widetilde{H}_0^S\left|\Phi_{n}^{S}\right \rangle = E_{0,n}^S\left|\Phi_{n}^{S}\right\rangle$. There is another important relation between $\left| \Phi_{i}^{S} \right \rangle$ and $\left| \Phi_{i}^{P} \right \rangle$, as these states are generated simultaneously in Eq.\ (\ref{rho_0_mixed}). If a one-particle level $\varepsilon$ in the biased energy region $|\varepsilon| < eV/2$ is being occupied by an $s$-wave electron $\widetilde{\alpha}_{\varepsilon\sigma}^{\dagger}$ in $\left| \Phi_{i}^{S} \right \rangle$, then the corresponding level has to be unoccupied in the $p$-wave part $\left| \Phi_{i}^{P} \right \rangle$, or vice versa. Therefore, there exists a pair of states, $i$ and $i'$, that show a relation $\widetilde{\alpha}_{\varepsilon\sigma}^{\phantom{\dagger}} \left| \Phi_{i}^{S} \right \rangle \!\left| \Phi_{i}^{P} \right \rangle = - \,\widetilde{p}_{\varepsilon\sigma}^{\phantom{\dagger}} \left| \Phi_{i'}^{S} \right \rangle \!\left| \Phi_{i'}^{P} \right \rangle $ for $|\varepsilon| < eV/2$. Here, the minus sign is caused by the order of the operators for the one-particle level $\varepsilon$. Using this property, a $p$-wave operator appearing in an average can be replaced by an $s$-wave operator as \begin{align} & \!\!\!\!\!\! \left\langle \Phi_{\rm neq} \right | \widehat{\cal O}_S \, p_{\varepsilon\sigma} \! \left| \Phi_{\rm neq} \right \rangle = \frac{1}{Z_0} {\sum_{ij\in\mathcal{W}}}' \! \left\langle \Phi_{i}^{P} \right | \!\! \left\langle \Phi_{i}^{S} \right | \widehat{\cal O}_S \, p_{\varepsilon\sigma} \! \left| \Phi_{j}^{S} \right \rangle \!\! \left| \Phi_{j}^{P} \right \rangle \nonumber \\ & \qquad \qquad \quad = -\, \left[ f_L(\varepsilon) - f_R(\varepsilon) \right]\, \mbox{Tr}_S^{\phantom{|}}\! \left[ \widehat{\rho}_0^{S} \, \widehat{\cal O}_S \, \alpha_{\varepsilon\sigma} \right] . \label{eq:with_p} \end{align} Similar relation holds also for the asymmetric coupling $\Gamma_L \neq \Gamma_R$ and at finite temperatures as shown in Appendix \ref{sec:RDM_for_p}. The interacting density matrix at $T=0$ is also constructed as a pure state with the wavefunction \begin{align} \left| \Psi_{\rm neq} \right \rangle \equiv S(0,-\infty) \left| \Phi_{\rm neq} \right \rangle = \frac{1}{\sqrt{Z_0}} {\sum_{i\in \mathcal{W}}}' \left|\Psi_{i}^{S}\right\rangle \left|\Phi_{i}^{P}\right\rangle \;. \label{eq:rho_mixed} \end{align} Here, $\left|\Psi_{n}^{S}\right\rangle \equiv S(0,-\infty) \left|\Phi_{n}^{S}\right\rangle$, and it is an interacting eigenstate,\cite{GML} $(\widetilde{H}_0^S+H_U)\left| \Psi_{n}^{S} \right \rangle = E_n^{S} \left| \Psi_{n}^{S} \right \rangle$. The wave function for the $p$-wave part keeps the noninteracting form, because $S(0,-\infty)$ commutes with the $p$-wave operators. Therefore, the summation over the $p$-wave part can be carried out in the same way as that in Eq.\ (\ref{eq:mixed_s_first}), and the interacting average for the operator $\widehat{\cal O}_S$ can be written in the form, \begin{align} \left\langle \Psi_{\rm neq} \right | \widehat{\cal O}_S \left| \Psi_{\rm neq} \right \rangle & = \frac{1}{Z_0} {\sum_{i\in \mathcal{W}}}' \left\langle \Psi_{i}^{S}\right| \widehat{\cal O}_S \left|\Psi_{i}^{S} \right\rangle \nonumber \\ &= \sum_n w_{0,n}^{S} \!\left\langle \Psi_{n}^{S}\right| \widehat{\cal O}_S \left|\Psi_{n}^{S} \right\rangle . \label{eq:neq_average_u} \end{align} The interacting RDM in this case also takes a simple form $ \widehat{\rho}_{\phantom{0}}^{S} = \sum_n \left|\Psi_{n}^{S}\right\rangle w_{0,n}^{S} \!\left\langle \Psi_{n}^{S}\right| $. Equation (\ref{eq:neq_average_u}) gives us much insight into the steady state. It will provide, obviously, an interacting equilibrium average if we replace $w_{0,n}^{S}$ by $\rho_{\mathrm{eq},n}^S \equiv e^{-\beta E_n^S}\!/\! \sum_n \! e^{-\beta E_n^S}$. Therefore, some of the nonequilibrium properties could be inferred from the difference between the statistical weights $w_{0,n}^{S}$ and $\rho_{\mathrm{eq},n}^S$. As Eq.\ (\ref{eq:neq_average_u}) is being expressed in terms of the interacting eigenstates, Hamiltonian-based nonperturbative approaches developed for equilibrium, such as exact diagonalization and NRG, can be applied to the matrix element $\left\langle \Psi_{n}^{S}\right| \!\widehat{\cal O}_S \!\left|\Psi_{n}^{S} \right\rangle$. Furthermore, the bias voltage appears just through $w_{0,n}^{S}$, which keeps the noninteracting form given in Eq.\ (\ref{eq:w_n^S}). Thus, what is necessary to carry out the summation with respect to $n$ in Eq.\ (\ref{eq:neq_average_u}) is the information about whether or not each interacting eigenstate $\left| \Psi_{n}^{S} \right \rangle$ belongs to the set $\mathcal{W}$ in the limit of $H_U \to 0$. Namely, the {\em correspondence\/} between $\left|\Psi_{i}^{S}\right\rangle$ and its noninteracting counterpart $\left|\Phi_{i}^{S}\right\rangle$ is an additional issue that one has to solve in the nonequilibrium case. At low energies, the {\em correspondence\/} can be treated correctly with the quasi-particles of a local Fermi liquid.\cite{Hewson,KNG,OguriA,FujiiUeda,GogolinKomnik,HB} The evolution of the wavefunctions is described formally with the time-dependent operator $S(0,-\infty)$. Alternatively, it can be expressed in a time-independent form as that in a stationary-state scattering theory,\cite{GML,GMGB} and several versions of the time-independent approaches have been examined with some different points of view. \cite{SchillerHershfield,KSL,KNG,Han,MehtaAndrei} In our description, the adiabatic evolution of the interacting eigenstates from the noninteracting ones could also be traced directly in the Hilbert space of a discretized version of the single-channel model, by changing the coupling constant $U$ continuously. \section{Energy distribution in the mixed states } \label{sec:average} We consider in the following the mixed-sate properties for the symmetric coupling $\Gamma_L=\Gamma_R$ further at $T=0$. The pure state $\left| \Phi_{\rm neq} \right \rangle$ given in Eq.\ (\ref{eq:Phi_0_neq}) is an eigenstate, $\widetilde{H}_0 \left| \Phi_{\rm neq} \right \rangle = E_0^\mathrm{neq} \left| \Phi_{\rm neq} \right \rangle$, of the noninteracting Hamiltonian with an excitation energy \begin{align} E_0^\mathrm{neq} \, = \, 2 \! \sum_{\varepsilon_m = -eV/2}^{eV/2} |\varepsilon_m| \xrightarrow[\ \delta \varepsilon \to 0 \ ]{} \frac{\left(eV\right)^2}{2\,\delta \varepsilon} \;. \label{eq:pure_state_energy} \end{align} Here, we retained only the dominant term that is proportional to the system size of order $N$. As mentioned in Sec.\ \ref{sec:RDM}, the level spacing is given by $\delta \varepsilon = 2D/N$ for a flat band with a half width $D$. The feature of the pure state reflects in the $s$-wave subspace through the RDM. The mean excitation energy in the subspace is equal to a half of the one in Eq.\ (\ref{eq:pure_state_energy}), $\overline{E}_0^S \equiv \mbox{Tr}_S^{\phantom{|}} \bigl[ \widehat{\rho}_0^S \widetilde{H}_0^S \bigr] = E_0^\mathrm{neq}/2$. Note that the contribution of the impurity level is of order $1$, namely a $1/N$ portion of the total energy. \begin{figure}[t] \includegraphics[width= 0.78\linewidth]{wapp5x.eps} \caption{ (color online). Averaged statistical weight $\overline{w}_0^{S}(E) \equiv W(E)\,/\,\Omega(E)$ vs many-particle excitation energy $E$, where $N_0 = eV/\delta\varepsilon$. } \label{fig:wapp} \end{figure} In order to clarify how the nonequilibrium weight $w_{0,n}^{S}$ distributes in the many-particle Hilbert space consisting of the eigenstates $\left| \Phi_{n}^{S} \right \rangle$, we examine an averaged weight $\overline{w}_0^{S}(E)$ over an equienergy surface; \begin{align} & \overline{w}_0^{S}(E) \, \equiv \, W(E)\,/\,\Omega(E) \;, \\ & \!\!\! W(E) = \! {\sum_{i\in\mathcal{W}}}' \! \delta (E-E_{0,i}^{S}) , \ \ \Omega(E) =\!\sum_n \delta (E-E_{0,n}^{S}). \label{eq:MB_DOS} \end{align} It is normalized as $Z_0 = \int \! dE\ \overline{w}_0^{S}(E) \Omega(E)$, and may be used as $ \left\langle \Psi_{\rm neq} \right | \! \widehat{\cal O}_S \! \left| \Psi_{\rm neq} \right \rangle \approx \! \sum_n \overline{w}_0^{S}(E_n^S) \left\langle \Psi_{n}^{S}\right| \! \widehat{\cal O}_S \! \left|\Psi_{n}^{S} \right\rangle \!/Z_0 $ for a rough approximation. We calculate $\overline{w}_0^{S}(E)$ for several finite $\delta \varepsilon$, and the results are shown in Fig.\ \ref{fig:wapp}: the parameter $N_0 \equiv eV/\delta \varepsilon$ corresponds to the number of one-particle states in the energy window $|\varepsilon|<eV/2$, and thus $Z_0 = 4^{N_0}$. The averaged weight $\overline{w}_0^{S}(E)$ becomes a constant for small $E$, as all the many-particle excited states for $0<E<eV/2$ belong to the set $\mathcal{W}$ defined in Eq.\ (\ref{rho_0_mixed}). This flat $E$-dependence is quite different from the exponential dependence of the Boltzmann factor $e^{-\beta E}$ in the thermal equilibrium. We also see that $\overline{w}_0^{S}(E)$ penetrates into the higher energy region over the bias energy $eV$. It spreads out, approximately, up to $E \sim c\, \overline{E}_0^S$ with a numerical constant $c$ of order $10^{-1}$. Note that the mean excitation energy is proportional to $N_0$ as $\overline{E}_0^S=N_0 eV/4$. We have also obtained the analytic expressions of $\Omega(E)$ and $W(E)$ for small $\delta \varepsilon$ as shown in Appendix \ref{sec:averaged_w_details}, and have checked that the same behavior could be seen for larger $N_0$ assuming a Gaussian form for $W(E)$ given in Eq.\ (\ref{eq:W_integral_app}), instead of the exact integration form Eq.\ (\ref{eq:W_integral}). The feature of $\overline{w}_0^{S}(E)$ seen in the $s$-wave subspace is caused by the fact that the bias voltage has been introduced not into the many-particle energy but into the one-particle states as the width of the energy window between the two chemical potentials $\mu_L$ and $\mu_R$. It is quite different from the way the temperature determines the equilibrium distribution $e^{- E/T}$, in which $T$ scales directly the many-particle energy $E$. The original weight $w_{0,n}^{S}$ must have the similar features in the $n$ dependence. However, whether or not the high-energy many-particle states give significant contributions to the nonequilibrium averages depends also on the properties of the operator $\widehat{\cal O}_S$. For instance, the noninteracting correlation functions, such as $\langle \alpha^{\dagger}_{\varepsilon,\sigma} \alpha^{\phantom{\dagger}}_{\varepsilon',\sigma'} \rangle_0$ and $G^{<}_{0}(\varepsilon)$, can be reproduced from Eq.\ (\ref{eq:mixed_s_first}). The feature of $w_{0,n}^{S}$ reflects in these correlation functions just through $f_\mathrm{eff}(\varepsilon)$. It should be regarded, however, those are the results that are determined by all the many-particle states belonging to the set $\mathcal{W}$. Therefore, it is necessary for calculating the interacting Green's functions with Eq.\ (\ref{eq:neq_average_u}) to take all these high-energy states into account. The Kondo resonance at finite bias voltages is being affected by these features of the nonequilibrium distribution. \section{Summary} \label{sec:summary} In conclusion, we have reexamined the nonequilibrium steady state for a quantum dot, checking out the statistical distribution in the many-particle Hilbert space. We have, first of all, discretized the conduction bands keeping the two-fold degeneracy due to the left-going and right-going scattering states. This degeneracy is essential to describe the current-carrying state driven by the bias voltage. Then, by integrating out the $p$-wave part of the channel degrees of freedom, the noninteracting RDM $\widehat{\rho}_0^S$ has been obtained explicitly as shown in Eq.\ (\ref{eq:rho_s_0_T}). It brings the information about the bias voltages to the remaining $s$-wave subspace, and determines the nonequilibrium distribution. The interacting RDM $\widehat{\rho}^S$, which is defined in Eq.\ (\ref{eq:rho_s}), evolves adiabatically from $\widehat{\rho}_0^S$. Using the RDM, the steady-state averages Eq.\ (\ref{eq:neq_average_u2}) can be calculated, in principle, with the single-channel Anderson model. We have studied further the properties of the RDM at zero temperature for the symmetric coupling $\Gamma_L=\Gamma_R$. In this case the nonequilibrium averages can be written in a Lehmann-representation form Eq.\ (\ref{eq:neq_average_u}) with a simple mixed-state distribution $w_{0,n}^{S}$ given in Eq.\ (\ref{eq:w_n^S}). It has also been deduced that the mixed states contain large portions of high-energy many-particle states. Our formulation could be used for constructing Hamiltonian-based nonperturbative approaches in future. \begin{acknowledgements} I wish to thank Y. Nisikawa, J. Bauer, and A. C. Hewson for valuable discussions. This work was supported by the Grant-in-Aid for Scientific Research from JSPS. \end{acknowledgements}

proofpile-arXiv_069-15027

\section{Introduction} At the top of the causal ladder of spacetimes \cite{minguzzi06c} \cite{hawking74,beem96,senovilla97} stands the property of {\em global hyperbolicity} which implies many good properties for the Lorentzian distance function. Indeed, in a globally hyperbolic spacetime $(M,g)$ the Lorentzian distance $d: M\times M \to[0,+\infty]$ is \begin{itemize} \item[(a)] finite \cite[Lemma 4.5]{beem96}, \item[(b)] maximized by a suitable connecting causal geodesic $\sigma$, $l(\sigma)=d(x,z)$, whenever $z \in J^{+}(x)$, $d(x,z)<+\infty$, (Avez-Seifert theorem \cite{avez63,seifert67}) \cite[Theor. 3.18, Prop. 10.39]{beem96}, \cite[Prop. 14.19]{oneill83}, \cite[Prop. 6.7.1]{hawking73}, \item[(c)] continuous \cite[Lemma 4.5]{beem96}. \end{itemize} All these properties are lost even in spacetimes sharing the causal property which stays immediately below global hyperbolicity in the causal ladder, i.e. {\em causal simplicity}. In particular even if $d(x,z)<+\infty$ there can be no connecting maximizing geodesic (see figure \ref{fig1} and figure 10 in \cite{minguzzi06c}). Apparently unrelated with the previous aspects of the Lorentzian distance function in causally simple spacetimes stands the problem of finding under which conditions the warped product spacetime $P=M \times_f H$, endowed with the warped metric $\hat{g}=g+f^2 h$, is causally simple. For other causal properties such as being chronological, causal, strongly causal, stably causal or globally hyperbolic, it has been proved \cite[Prop. 3.61,3.62,3.64,3.68]{beem96} that they are shared by $(P,\hat{g})$ if and only if they are shared by $(M,g)$. I shall prove that the same holds for non-total viciousness and the distinction property, but for causal continuity and causal simplicity this simple correspondence does not hold. For instance for causal simplicity in the simple case $f=1$, $(H,h)=(\mathbb{R},{\rm d} y^2)$, one has to require that $(M,g)$ satisfies also properties (b) and (c) above. I refer the reader to \cite{minguzzi06c} for most of the conventions used in this work. In particular, I denote with $(M,g)$ a $C^{r}$ spacetime (connected, time-oriented Lorentzian manifold), $r\in \{2, \dots, \infty\}$ of arbitrary dimension $n\geq 2$ and signature $(-,+,\dots,+)$. On $M\times M$ the usual product topology is defined. With $\bm{g}$ it is denoted the class of metrics conformal to $g$. With $(M,\bm{g})$ it is denoted the conformal structure i.e. the class of spacetimes $[(M,g)]$ on the same manifold $M$, with metrics related by a conformal rescaling, and the same time orientation. Sometimes I write ``spacetime $(M,\bm{g})$'' although by this I mean the conformal structure. With ``lightlike geodesic $\gamma$ of $(M,\bm{g})$'' I mean a curve which is a lightlike pregeodesic for a representative $g \in \bm{g}$ and hence for every element of the class $\bm{g}$. For convenience and generality I often use the causal relations on $M \times M$ in place of the more widespread point based relations $I^{+}(x)$, $J^{+}(x)$, $E^{+}(x)$ (and past versions). Recall \cite{minguzzi06c} the following definition of sets on $M \times M$ \[ I^{+}=\{(p,q) : p\ll q \} , \quad J^{+}=\{(p,q) : p\leq q \}, \quad E^{+}=\{(p,q) : p\rightarrow q \}. \] Clearly, $E^+={J^+}\backslash I^+$. Moreover, $I^{+}$ is open \cite[Chap. 14, Lemma 3]{oneill83} \cite[Prop. 2.16]{minguzzi06c}, $\bar{J}^+=\bar{I}^+$, $\textrm{Int}\, J^+=I^+$ and $\dot{J}^+=\dot{I}^+$ \cite[Prop. 2.17]{minguzzi06c}. Finally, $J^{+}$ is closed in causally simple spacetimes \cite[Prop. 3.68]{minguzzi06c}. The {\em vanishing distance set} is the set $(I^+)^C$ which is made of all the pairs of events at which the Lorentzian distance $d$ vanishes (this set is a conformal invariant concept although the Lorentzian distance is not). Recall that $d$ is lower semi-continuous \cite[Lemma 4.4]{beem96}, in particular it is continuous at those points where $d(x,z)=+\infty$ and in the open set $(\bar{I}^+)^C$, where it vanishes identically. Most of the causal vectors and curves that we shall encounter will be {\em future directed}, thus, for simplicity, I shall omit this adjective. \section{The spacetime $P=M\times H$ and warped products} Consider the spacetime $(P, \tilde{g})$, $P=M \times H$, $\tilde{g}=g+ h$, where $(H,h)$ is a complete Riemannian manifold and let $\rho: H\times H \to [0,+\infty)$ be the Riemannian distance (the assumption of completeness for $(H,h)$ is needed only for a subset of the results proved below). The length of a causal curve on $M$ will be denoted with $l$ while the length of a curve on $H$ will be denoted with $l_H$. The Riemannian distance $\rho$ is continuous \cite[p. 3]{beem96} while the Lorentzian distance $d: M\to [0,+\infty]$ is only lower semi-continuous in general. By the Hopf-Rinow (Heine-Borel) theorem any closed and bounded subset of the complete Riemannian manifold $H$ is compact, in particular if $\textrm{diam}(H,h)=\sup_{y_0,y_1 \in H} \rho(y_0,y_1)\, <+\infty$ then $H$ is compact and since $\rho$ is continuous it attains the supremum, that is, there are $y_0,y_1 \in H$ such that $\rho(y_0,y_1)=\textrm{diam}(H,h)$. One of the simplest choices for $(H,h)$ is $(\mathbb{R},{\rm d} y^{2})$, $y \in \mathbb{R}$, in which case $\rho(y_0,y_1)=\vert y_1-y_0 \vert$, is the absolute value of the difference of the extra coordinate. Denote with $\pi: P \to M$ the projection to $M$, with $\pi_{H}: P \to H$ the projection to $H$, and with $y\in H$, the generic point of $H$. The direct product structure defines at each point of $p \in P$, a natural splitting of $TP_p$ into horizontal and a vertical parts, namely a generalized connection. Given a vector $V \in TP_p$, $V$ is horizontal if $\pi_{H*}V=0$, while it is vertical if $\pi_{*}V=0$. The horizontal lift of $v\in TM_x$, to $p \in P$, $\pi(p)=x$, is defined as usual as the only horizontal vector $V$ which projects to $v$. The time orientation of $(P, \tilde{g})$ is obtained from the global timelike vector field obtained by taking the horizontal lift of the global, future directed, timelike vector field for $(M,g)$. It is easy to check that since $h$ is positive definite, the projection of a timelike (resp. causal) vector on $TP_p$, $p \in P$, is a timelike (resp. causal) vector on $TM_x$, $x =\pi(p)$. Note, however, that the projection of a lightlike vector $V$ is timelike unless $h(\pi_{H*}V,\pi_{H*}V)=0 \Rightarrow \pi_{H*}V=0$, i.e. unless it is the horizontal lift of its projection, in which case the projection is lightlike too. As a consequence the projection of a causal (resp. timelike) curve is a causal (resp. timelike) curve, thus $p_0 \le p_1$ (resp. $p_0 \ll p_1$) implies $\pi(p_0) \le \pi(p_1)$ (resp. $\pi(p_0) \ll \pi(p_1)$). Finally, since the horizontal lift of a causal (resp. timelike) curve on $M$ is a causal (resp. timelike) curve on $P$, it is $\pi(J^{+}(p))=J^{+}(\pi(p))$ (resp. $\pi(I^{+}(p))=I^{+}(\pi(p))$). For applications, the warped product $(P,\hat{g})$, where $\hat{g}= g+f^2(x) h $, and $f: M \to \mathbb{R}^{+}$, is far more interesting than the direct product, as many physical metrics are in fact obtained by the repeated application of warped products. Sometimes the notation $P= M\times_{f} H $ is used although the manifold $P$ does not depend on $f$. \begin{remark} The spacetime $P=M\times_{f} H$ with $M=(a,b)$, $-\infty\le a < b \le +\infty$, and $g=-{\rm d} t^2$, also known as generalized FRW metric, will not be considered in this work. The reason is that the causality of these spacetimes is trivial if $(H,h)$ is a complete Riemannian manifold. Indeed, it has been shown that $(P,g+f^2h)$ is globally hyperbolic iff $(H,h)$ is complete \cite[Th. 3.66]{beem96}. \end{remark} The next theorem proves that it is not restrictive to study the relation between the causality properties for $(M,g)$ and the direct product $(P,\tilde{g})$ as the results hold also for the warped product whatever the positive function $f$. \begin{theorem} \label{mjh} Let $(M,g)$ be a generic spacetime, $(H,h)$ a Riemannian manifold, $\tilde{g}$ the direct product metric on $P=M\times H$, and $\hat{g}$ a warped product metric for a positive function $f$. Let $\mathcal{P}$ and $\mathcal{P}'$ be two given conformal invariant properties. Consider the following logical statements \begin{itemize} \item[(a)] $(M,g)$ satisfies $\mathcal{P}$ $\Rightarrow $ $ (P,\tilde{g})$ satisfies $\mathcal{P}'$, \item[(b)] $(M,g)$ satisfies $\mathcal{P} \Rightarrow (P,\hat{g})$ satisfies $\mathcal{P}'$, \item[(c)] $(P,\tilde{g})$ satisfies $\mathcal{P}'$ $\Rightarrow $ $(M,g)$ satisfies $\mathcal{P}$, \item[(d)] $(P,\hat{g})$ satisfies $\mathcal{P}'$ $\Rightarrow $ $(M,g)$ satisfies $\mathcal{P}$, \end{itemize} Then (a) $\Leftrightarrow$ (b) and (c) $\Leftrightarrow$ (d). \end{theorem} \begin{proof} (a) $\Rightarrow$ (b). Let $(M,g)$ satisfy $\mathcal{P}$, then because $\mathcal{P}$ is a conformal invariant property, $(M,g/f^{2})$ satisfies $\mathcal{P}$, and because of the assumed implication, $(P,f^{-2} g+ h)$ satisfies $\mathcal{P}'$, and again because of conformal invariance, $(P,g+ f^2 h)$ satisfies $\mathcal{P}'$. (b) $\Rightarrow$ (a). Let $(M,g)$ satisfy $\mathcal{P}$, then because $\mathcal{P}$ is a conformal invariant property, $(M, f^2 g)$ satisfies $\mathcal{P}$, and because of the assumed implication, $(P,f^2 g+ f^2 h)$ satisfies $\mathcal{P}'$, and again because of conformal invariance, $(P,g+ h)$ satisfies $\mathcal{P}'$. (c) $\Rightarrow$ (d). Let $(P,\hat{g})$ satisfy $\mathcal{P}'$, then because $\mathcal{P}'$ is a conformal invariant property, $(P,f^{-2} g+ h )$ satisfies $\mathcal{P}'$, and because of the assumed implication, $(M,f^{-2}g)$ satisfies $\mathcal{P}$, and again because of conformal invariance, $(M,g)$ satisfies $\mathcal{P}$. (d) $\Rightarrow$ (c). Let $(P,\tilde{g})$ satisfy $\mathcal{P}'$, then because $\mathcal{P}'$ is a conformal invariant property, $(P,f^2 g+f^2 h)$ satisfies $\mathcal{P}'$, and because of the assumed implication, $(M,f^2 g)$ satisfies $\mathcal{P}$, and again because of conformal invariance, $(M,g)$ satisfies $\mathcal{P}$. \end{proof} Thus, assume to know, for instance, that if $(M,g)$ is distinguishing then $(P,\tilde{g})$ is distinguishing, then the previous theorem implies that $ M \times_{f} H$ is distinguishing too, whatever the positive function $f$. In what follows I shall clarify whether a result holds in the warped product case $ M \times_{f} H$ or only in the direct product case $ M \times_{} H$. Nevertheless, the proofs will be given only for the direct product case, the generalization to the warped product being trivial because of the previous theorem. The theorems which will not be generalizable to the warped product case are those which depend on non-conformal properties of the base spacetime $(M,g)$. Usually, these non-conformal invariant conditions are completeness conditions, geodesic connectedness conditions or continuity requirements on the Lorentzian distance function $d$. The properties of causal continuity and causal simplicity will present such a difficulty. From now on $(H,h)$ is a {\em complete} Riemannian manifold. The next lemma clarifies the relation between the additional dimensions and the Lorentzian distance on $M \times M$. In the convention of this work the inclusion $\subset$ is a reflexive relation $U\subset U$. \begin{lemma} \label{uty} Consider the direct product spacetime $(P,\tilde{g})$, $P=M\times H$. Let $p_0=(x_0,y_0) \in P$, for every $x_1 \in M$, it is \begin{equation} \label{iyr} I^{+}(p_0) \cap \pi^{-1}(x_1)=\{(x_1,y): \rho(y_0,y) < d(x_0,x_1)\} , \end{equation} moreover, $J^{+}(p_0) \cap \pi^{-1}(x_1) \ne \emptyset$ only if $x_1 \in J^{+}(x_0)$ and \begin{align} J^{+}(p_0) \cap \pi^{-1}(x_1)&\subset \{(x_1,y): \rho(y_0,y) \le d(x_0,x_1)\}. \label{iyr2} \end{align} If $d(x_0,x_1)=+\infty$, then $E^{+}(p_0) \cap \pi^{-1}(x_1)=\emptyset$, otherwise $d(x_0,x_1)<+\infty$ and \begin{align} E^{+}(p_0) \cap \pi^{-1}(x_1)&\subset \{(x_1,y): \rho(y_0,y) = d(x_0,x_1)\}. \label{iyr3} \end{align} Analogous past versions of these statements also hold. \end{lemma} \begin{proof} If the set $I^{+}(p_0) \cap \pi^{-1}(x_1)$ is not empty there is a timelike curve connecting $p_0$ with $x_1$'s fiber. Its projection is a timelike curve that connects $x_0$ to $x_1$ thus, $x_1 \in I^{+}(x_0)$ or the set $I^{+}(p_0) \cap \pi^{-1}(x_1)$ is empty. The right-hand side of Eq. (\ref{iyr}) gives an empty set if $x_1 \notin I^{+}(x_0)$ thus there remains to consider the case $x_1 \in I^{+}(x_0)$. Let $\sigma(\lambda)$, $\lambda \in [0,1]$ be any $(C^1)$ timelike curve from $x_0$ to $x_1$, let $y: [0,+\infty) \to H$, be a geodesic parametrized with respect to length starting from $y_0$, and consider for any given $\alpha \in [-1,1]$, the curve on $P$ \[ \gamma(\lambda)=(\sigma(\lambda), y(\alpha \int_{\sigma x_0}^{\sigma(\lambda)} {\rm d} s) ). \] It can be easily checked to be timelike for $\alpha =(-1,1)$ and lightlike for $\vert \alpha\vert=1$. Its second endpoint is $(x_1,y(\alpha l(\sigma)))$. Thus every event $(x_1,y_1)$, $ \rho(y_0,y_1) < d(x_0,x_1) $, can be reached by a timelike curve from $p_0$, simply choose $\sigma$ such that $l(\sigma) > \rho(y_0,y_1)$, the constant $\alpha$ so that $ \alpha l(\sigma)=\rho(y_0,y_1)$, and the geodesic $y$ to be a minimizing geodesic connecting $y_0$ to $y_1$. Finally, if $\gamma(\lambda)$ is a timelike (resp. causal) curve from $p_0$ to $x_1$'s fiber and $\sigma$ is its projection, the timelike (resp. causal) condition reads $ \sqrt{ h(\frac{{\rm d} y}{{\rm d} \lambda}, \frac{{\rm d} y}{{\rm d} \lambda})}< \frac{{\rm d} s}{{\rm d} \lambda}$ (resp. $\sqrt{ h(\frac{{\rm d} y}{{\rm d} \lambda}, \frac{{\rm d} y}{{\rm d} \lambda})} \le \frac{{\rm d} s}{{\rm d} \lambda}$), so that if $(x_1,y_1)$ is its second endpoint, $\rho(y_0,y_1) \le l_H(y) < l(\sigma)\le d(x_0,x_1)$ (resp. $\rho(y_0,y_1) \le l_H(y) \le l(\sigma)\le d(x_0,x_1)$). \end{proof} The next lemma summarizes some results basically due to Beem and Powell \cite{beem82}. I give a simple proof for the reader convenience \begin{lemma} \label{itg} Consider the direct product spacetime $(P,\tilde{g})$, $P=M\times H$. Let $\gamma(\lambda)$, $\lambda \in [0,1]$, be a causal (resp. timelike) geodesic on $(P,\tilde{g})$ connecting $p_0=(x_0,y_0)$ to $p_1=(x_1,y_1)$, then it decomposes as \[ \gamma(\lambda)=(\sigma(\lambda), y(\lambda)), \] where $\sigma(\lambda)$ is a causal (resp. timelike) geodesic on $(M,g)$, and $y(\lambda)$ is a geodesic on $H$ (possibly degenerated to the only point $y_0$) connecting $y_0$ to $y_1$ of length $l_H(y) \le l(\sigma)$ (resp. $l_H(y) < l(\sigma)$). Moreover, if $\gamma$ is a maximizing lightlike geodesic then $\sigma$ is a maximizing geodesic and $y$ is a minimizing geodesic. \end{lemma} \begin{proof} That the projection $\sigma$ is a causal (resp. timelike) geodesic follows from the direct product structure, and the same can be said for the geodesic nature of $y(\lambda)$. From the condition of causality (resp. chronology) for $\gamma$, that is $ -(\frac{{\rm d} s}{{\rm d} \lambda})^{2}+h(\frac{{\rm d} y}{{\rm d} \lambda},\frac{{\rm d} y}{{\rm d} \lambda}) \le 0$ $\Rightarrow$ $\sqrt{ h(\frac{{\rm d} y}{{\rm d} \lambda}, \frac{{\rm d} y}{{\rm d} \lambda})} \le \frac{{\rm d} s}{{\rm d} \lambda}$ (resp. $\sqrt{ h(\frac{{\rm d} y}{{\rm d} \lambda}, \frac{{\rm d} y}{{\rm d} \lambda})} < \frac{{\rm d} s}{{\rm d} \lambda}$), it follows $l_H(y) \le l(\sigma)$ (resp. $l_H(y) < l(\sigma)$). The last statement follows from $\rho(y_0,y_1) \le l_H(y) \le l(\sigma)\le d(x_0,x_1)$, and from the inclusion (\ref{iyr3}) because as $\gamma$ is lightlike and maximizing, $p_1 \in E^{+}(p_0)$, thus $d(x_0,x_1)=\rho(y_0,y_1)$, and hence $l(\sigma)=d(x_0,x_1)$, $l_H(y)=\rho(y_0,y_1)$. \end{proof} It is worth mentioning that Beem and Powell \cite{beem82} were able to extended some aspects of this direct product result to the warped product case. They proved that in this last case the projection $y$ on $H$ of a geodesic $\gamma$ on $P$ is a pregeodesic of $(H,h)$, and that if $\gamma$ is maximizing then $y$ is minimizing. However, the projection on $M$, $\sigma$, is not a geodesic. Although interesting, we shall need the simpler direct product case given above in what follows. The next theorem gives a key relation between all the distance functions involved in a direct product spacetime. \begin{theorem} \label{cfd} Consider the direct product spacetime $(P,\tilde{g})$, $P=M\times H$. Let $p_0=(x_0,y_0)$, $p_1=(x_1,y_1)$, be events in $(P,\tilde{g})$, then if $d^{(P)}$ is the Lorentzian distance on $P\times P$, \begin{equation} \label{ynb} d^{(P)}(p_0,p_1)=\sqrt{d(x_0,x_1)^2-\rho(y_0,y_1)^2} , \end{equation} whenever the argument of the square root is positive, otherwise $d^{(P)}(p_0,p_1)=0$. \end{theorem} \begin{proof} If $d(x_0,x_1)\le \rho(y_0,y_1)$ by lemma \ref{uty} $(p_0,p_1) \notin I^{+}$, thus $d^{(P)}(p_0,p_1)=0$, as claimed. Let $ \rho(y_0,y_1) < d(x_0,x_1)$ so that $d(x_0,x_1) >0$, and let $\sigma(\lambda)$, $\lambda \in [0,1]$, be a timelike curve connecting $x_0$ to $x_1$ such that $\rho(y_0,y_1)<l(\sigma)\le d(x_0,x_1)$. Let $y$ be a minimizing geodesic connecting $y_0$ to $y_1$, parametrized with respect to length, and starting from $y_0$. The curve \[ \gamma(\lambda)=(\sigma(\lambda),y(\frac{\rho(y_0,y_1)}{l(\sigma)}\int_{\sigma x_0}^{\sigma(\lambda)} {\rm d} s)) , \] is timelike, connects $p_0$ to $p_1$ and has length \[ \int_{\gamma} {\rm d} s^{(P)}=\int_{\sigma} \sqrt{1-(\frac{\rho(y_0,y_1)}{l(\sigma)})^2}\, {\rm d} s=\sqrt{l(\sigma)^2-\rho(y_0,y_1)^2}, \] thus, taking the lower upper bound over the space of timelike connecting curves on the base \[ d^{(P)}(p_0,p_1) \ge \sup_{\sigma} \sqrt{l(\sigma)^2-\rho(y_0,y_1)^2}=\sqrt{d(x_0,x_1)^2-\rho(y_0,y_1)^2}. \] But if it were $d^{(P)}(p_0,p_1)>\sqrt{d(x_0,x_1)^2-\rho(y_0,y_1)^2}$ there would be a timelike curve $\tilde\gamma(\lambda)=(\tilde\sigma(\lambda),y(\lambda))$ on $P$ of length greater than $\sqrt{d(x_0,x_1)^2-\rho(y_0,y_1)^2}$ \[ \sqrt{d(x_0,x_1)^2-\rho(y_0,y_1)^2}<\int_{\tilde\gamma} {\rm d} s^{(P)}=\int_{\tilde\sigma} \sqrt{1-h(\frac{{\rm d} y}{{\rm d} s}, \frac{{\rm d} y}{{\rm d} s})} \, {\rm d} s = \int_{\tilde\sigma} \sqrt{1-v^2(s)} \, {\rm d} s, \] where $v(s)= \sqrt{h(\frac{{\rm d} y}{{\rm d} s}, \frac{{\rm d} y}{{\rm d} s})}$ and $\int_{\tilde\sigma} v(s) {\rm d} s=l_H(y)$. It is well known from special relativity that the right-hand side is maximized if $v(s)$ is a constant necessarily equal to $l_H(y)/l(\tilde\sigma)$, thus \[ \sqrt{d(x_0,x_1)^2-\rho(y_0,y_1)^2}< \sqrt{l(\tilde\sigma)^2-l_H(y)^2} \le \sqrt{l(\tilde\sigma)^2-\rho(y_0,y_1)^2}, \] that is $d(x_0,x_1) <l(\tilde\sigma)$, a contradiction. \end{proof} \begin{lemma} \label{uty2} Consider the direct product spacetime $(P,\tilde{g})$, $P=M\times H$. Let $p_0=(x_0,y_0) \in P$, for every $x_1 \in M$, $\bar{J}^{+}(p_0) \cap \pi^{-1}(x_1) \ne \emptyset$ only if $x_1 \in \bar{J}^{+}(x_0)$, in which case \begin{equation} \bar{J}^{+}(p_0) \cap \pi^{-1}(x_1)=\{(x_1,y): \rho(y_0,y) \le S^{+}(x_0,x_1)\} ,\label{iyrt} \end{equation} where \begin{equation} S^{+}(x_0,x_1)=\inf_{U\ni x_1} \sup_{x \in U} d(x_0,x)\ge d(x_0,x_1),\label{iyrt2} \end{equation} ($U$ open set) and $S^{+}(x_0,x_1)-d(x_0,x_1)$ is the discontinuity of the restricted distance function $d(x_0,\cdot): M \to [0,+\infty]$, at $x_1$. \end{lemma} \begin{proof} Let $q=(x_1,y_1) \in \bar{J}^{+}(p_0) \cap \pi^{-1}(x_1) $ then there is a sequence $q_i=(x_i,y_i) \to (x_1,y_1)$, $q_i \in J^{+}(p_0)$, thus $x_i \in J^{+}(x_0)$ and $x_1 \in \bar{J}^{+}(x_0)$ as claimed. The inequality $S^{+}(x_0,x_1) \ge d(x_0,x_1)$ follows from the fact that for every open set $U\ni x_1$, $\sup_{x \in U} d(x_0,x) \ge d(x_0,x_1)$ because $x_1$ belongs to $U$. From the definition of $S^{+}(x_0,x_1)$ it also follows that for every sequence $x_i \to x_1$ then $\limsup_{i \to +\infty}d(x_0,x_i)\le S^{+}(x_0,x_1)$. Coming back to the sequence $q_i$, since, because of lemma \ref{uty}, $\rho(y_0,y_i) \le d(x_0,x_i)$, we have \[ \rho(y_0,y_1) = \limsup_{i \to +\infty} \rho(y_0,y_i) \le \limsup_{i \to +\infty}d(x_0,x_i) \le S^{+}(x_0,x_1), \] and thus $q$ stays in the set given by the right-hand side of Eq. (\ref{iyrt}). In order to prove the other inclusion, let $x_1 \in \bar{J}^{+}(x_0)$ and consider the two cases $S^{+}(x_0,x_1)=0$ and $S^{+}(x_0,x_1)>0$. In the former case, it is clear that the event $(x_1,y_0)$ belongs to $\bar{J}^{+}(p_0) \cap \pi^{-1}(x_1)$, indeed if $x_i \to x_1$, $x_0<x_i$, and $\sigma_i$ is a causal curve connecting $x_0$ to $x_i$, then its horizontal lift is a causal curve which connects $p_0$ to $q_i=(x_i, y_0) \to (x_1,y_0)$. In the latter case let $q=(x_1,y_1)$ with $\rho(y_0,y_1) \le S^{+}(x_0,x_1)$. From the definition of $S^{+}(x_0,x_1)$ it is not difficult to show that there is always a sequence $x_i \to x_1$ such that $\lim_{i \to +\infty} d(x_0,x_i)=S^{+}(x_0,x_1)$. Clearly we can assume $x_i \in I^{+}(x_0)$ since $S^{+}(x_0,x_1)>0$. Let $\sigma_i(s)$ be a timelike curve connecting $x_0=\sigma_i(0)$ to $x_i$, parametrized with proper time, and of length $l(\sigma_i)>d(x_0,x_i)-\frac{1}{i}$ if $d(x_0,x_i)<+\infty$, or $l(\sigma_i)>i$ if $d(x_0,x_i)=+\infty$. With this choice $\lim_{i \to +\infty} l(\sigma_i)=\lim_{i \to +\infty}d(x_0,x_i)=S^{+}(x_0,x_1)$. Let $y$ be a minimizing geodesic connecting $y_0$ to $y_1$, parametrized with respect to length and starting at $y_0$. If $\rho(y_0,y_1) < S^{+}(x_0,x_1)$ the curve \[ \gamma_i=(\sigma_i(s), y(\frac{\rho(y_0,y_1)}{l(\sigma_i)} \,s)), \] is causal for sufficiently large $i$ and connects $p_0$ to $(x_i,y_1)$ whose limit is $(x_1,y_1)$, thus $(x_1,y_1) \in \bar{J}^{+}(p_0)$. If $\rho(y_0,y_1) =S^{+}(x_0,x_1)$ the curve \[ \gamma_i=(\sigma_i(s), y( s)), \] is lightlike and hence causal and connects $p_0$ to $(x_i,y( l(\sigma_i)))$ whose limit is $(x_1,y(S^{+}(x_0,x_1)))=(x_1,y(\rho(y_0,y_1)))=(x_1,y_1)$, thus $(x_1,y_1) \in \bar{J}^{+}(p_0)$. \end{proof} \begin{remark} There is an analogous past version of lemma \ref{uty2}. Let $p_1=(x_1,y_1) \in P$, for every $x_0 \in M$, $\bar{J}^{-}(p_1) \cap \pi^{-1}(x_0) \ne \emptyset$ only if $x_0 \in \bar{J}^{-}(x_1)$, in which case \begin{equation} \bar{J}^{-}(p_1) \cap \pi^{-1}(x_0)=\{(x_0,y): \rho(y,y_1) \le S^{-}(x_0,x_1)\} , \end{equation} Here $S^{-}(x_0,x_1)=\inf_{U\ni x_0} \sup_{x \in U} d(x,x_1)\ge d(x_0,x_1)$, and $S^{-}(x_0,x_1)-d(x_0,x_1)$ is the discontinuity of the restricted distance function $d(\cdot,x_1): M \to [0,+\infty]$, at $x_0$. \end{remark} Lemma \ref{uty2} will be particularly important in connection with causal continuity. The next lemma will be useful in connection with causal simplicity. \begin{lemma} \label{uty3} Consider the direct product spacetime $(P,\tilde{g})$, $P=M\times H$. For every pair $x_0,x_1 \in M$, $\bar{J}^{+} \cap[ \pi^{-1}(x_0)\times \pi^{-1}(x_1)] \ne \emptyset$ only if $(x_0,x_1) \in \bar{J}^{+}$, in which case \begin{align} &\bar{J}^{+} \cap [ \pi^{-1}(x_0)\times \pi^{-1}(x_1)] \nonumber \\ &=\{(p_0,p_1): \ p_0=(x_0,y_0), p_1=(x_1,y_1) \textrm{ and }\rho(y_0,y_1) \le D(x_0,x_1)\} , \end{align} where \begin{equation} D(x_0,x_1)=\inf_{V\ni (x_0,x_1)} \sup_{(x,z) \in V} \! \! d(x,z)\, \ge d(x_0,x_1), \end{equation} ($V\subset M\times M$ open set) and $D(x_0,x_1)-d(x_0,x_1)$ is the discontinuity of the distance function $d: M\times M \to [0,+\infty]$, at $(x_0,x_1)$. \end{lemma} \begin{proof} Let $(p_0,p_1) \in \bar{J}^{+} \cap[ \pi^{-1}(x_0)\times \pi^{-1}(x_1)] $, $p_0=(x_0,y_0)$, $p_1=(x_1,y_1)$, then there is a converging sequence $(p_{0i},p_{1i}) \to (p_0,p_1)$ with $(p_{0i},p_{1i}) \in J^{+}$. Denoting $p_{0i}=(x_{0i},y_{0i})$ and $p_{1i}=(x_{1i},y_{1i})$, it follows $(x_{0i},x_{1i}) \in J^{+}$ and since $(x_{0i},x_{1i}) \to (x_0,x_1)$, $(x_0,x_1) \in \bar{J}^{+}$ as claimed. The inequality $D(x_0,x_1) \ge d(x_0,x_1)$ follows from the fact that for every open set $V\ni (x_0,x_1)$, $\sup_{(x,z) \in V} d(x,z) \ge d(x_0,x_1)$ because $(x_0,x_1)$ belongs to $V$. From the definition of $D(x_0,x_1)$ it also follows that for every sequence $(x_{0i},x_{1i}) \to (x_0,x_1)$ then $\limsup_{i \to +\infty}d(x_{0i},x_{1i})\le D(x_0,x_1)$. In particular for the sequence $(x_{0i},x_{1i})$ constructed above, since, because of lemma \ref{uty}, $\rho(y_{0i},y_{1i}) \le d(x_{0i},x_{1i})$, we have \[ \rho(y_0,y_1) = \limsup_{i \to +\infty} \rho(y_{0i},y_{1i}) \le \limsup_{i \to +\infty} d(x_{0i},x_{1i}) \le D(x_0,x_1), \] and thus $(p_0,p_1)$ stays in the set given by the right-hand side of Eq. (\ref{iyrt}). In order to prove the other inclusion, let $(x_0,x_1) \in \bar{J}^{+}$ and consider the two cases $D(x_0,x_1)=0$ and $D(x_0,x_1)>0$. In the former case, we have only to prove that, for any given $y_0$, $p_0=(x_0,y_0)$, the event $p_1=(x_1,y_0)$ is such that $(p_0,p_1) \in \bar{J}^{+}$, which is trivial because if $(x_{0i},x_{1i}) \to (x_0,x_1)$, $(x_{0i},x_{1i})\in J^{+}$, and $\sigma_i$ is a causal curve connecting $x_{0i}$ to $x_{1i}$, then its horizontal lift is a causal curve which connects $p_{0i}=(x_{0i},y_0)$ to $p_{1i}=(x_{1i},y_0)$, and $(p_{01},p_{1i}) \to (p_0,p_1)$. In the latter case let $p_0=(x_0,y_0)$ and $p_1=(x_1,y_1)$ with $\rho(y_0,y_1) \le D(x_0,x_1)$. From the definition of $D(x_0,x_1)$ it is not difficult to show that there is always a sequence $(x_{0i},x_{1i}) \to (x_0,x_1)$ such that $\lim_{i \to +\infty} d(x_{0i},x_{1i})=D(x_0,x_1)$. Clearly we can assume $(x_{0i},x_{1i}) \in I^{+}$ since $D(x_0,x_1)>0$. Let $\sigma_i(s)$ be a timelike curve connecting $x_{0i}=\sigma_i(0)$ to $x_{1i}$, parametrized with proper time, and of length $l(\sigma_i)>d(x_{0i},x_{1i})-\frac{1}{i}$ if $d(x_{0i},x_{1i})<+\infty$, or $l(\sigma_i)>i$ if $d(x_{0i},x_{1i})=+\infty$. With this choice $\lim_{i \to +\infty} l(\sigma_i)=\lim_{i \to +\infty}d(x_{0i},x_{1i})=D(x_0,x_1)$. Let $y$ be a minimizing geodesic connecting $y_0$ to $y_1$, parametrized with respect to length and starting at $y_0$. If $\rho(y_0,y_1) < D(x_0,x_1)$ the curve \[ \gamma_i=(\sigma_i(s), y(\frac{\rho(y_0,y_1)}{l(\sigma_i)}\, s)), \] is causal for sufficiently large $i$ and connects $p_{0i}=(x_{0i}, y_0)$ to $p_{1i}=(x_{1i},y_1)$, moreover $(p_{0i},p_{1i}) \to (p_0,p_1)$, thus $(p_0,p_1) \in \bar{J}^{+}$. If $\rho(y_0,y_1) =D(x_0,x_1)$ the curve \[ \gamma_i=(\sigma_i(s), y( s)), \] is lightlike and hence causal and connects $p_{0i}=(x_{0i},y_0)$ to $(x_{1i},y( l(\sigma_i)))$ whose limit is $(x_1,y(D(x_0,x_1)))=(x_1,y(\rho(y_0,y_1)))=(x_1,y_1)=p_1$, while $p_{0i} \to p_0$, thus $(p_0,p_1) \in \bar{J}^{+}$. \end{proof} \begin{theorem} \label{pcr} Whatever the value of $f$, $P=M\times_f H$, the warped product spacetime $(P,\bm{\hat{g}})$ is chronological (resp. causal, strongly causal, stably causal, globally hyperbolic) iff $(M,\bm{g})$ is chronological (resp. causal, strongly causal, stably causal, globally hyperbolic). \end{theorem} \begin{proof} Recall that the proof can be given in the more specialized direct product case because the involved properties are conformal invariant (see theorem \ref{mjh}). This result is proved for instance in \cite[Prop. 3.61,3.62,3.64,3.68]{beem96} (for the globally hyperbolic case see also \cite{walschap95,caponio03,minguzzi06}). I give explicitly the proof for the causal case as we will use it. If $(P,\bm{\tilde{g}})$ is not causal then there is a closed causal curve whose projection is a closed causal curve for $(M,\bm{g})$. Conversely, if $(M,\bm{g})$ admits a closed causal curve then its horizontal lift is a closed causal curve for $(P,\bm{\tilde{g}})$. \end{proof} Recall that a spacetime $(M,g)$ is totally vicious if for every pair of events $x,z \in M$, $d(x,z)=+\infty$. \begin{theorem} Whatever the value of $f$, $P=M\times_f H$, the warped product spacetime $(P,\bm{\hat{g}})$ is non-totally vicious iff $(M,\bm{g})$ is non-totally vicious. \end{theorem} \begin{proof} Recall that the proof can be given in the more specialized direct product case because the involved property is conformal invariant (see theorem \ref{mjh}). In this case the claim follows easily from Eq. (\ref{ynb}). \end{proof} Let us consider the distinguishing property. We need some preliminary results. \begin{lemma} \label{kijd} Let $(M,g)$ be a spacetime, if $I^{+}(x_2)\subset I^{+}(x_1)$, $x_1 \ne x_2$, then for every $z \in I^{+}(x_2)$, $d(x_2,z) \le d(x_1,z)$. In particular, if $I^{+}(x_1)=I^{+}(x_2)$, $x_1 \ne x_2$, then for every $z \in I^{+}(x_1)$, $d(x_1,z)=d(x_2,z)$. Analogous past versions of these statements also hold. \end{lemma} \begin{proof} Indeed, let $\sigma_2(s)$ be a timelike curve connecting $x_2$ to $z$ and let it be parametrized with respect to proper time. For every $0<\epsilon < l(\sigma_2)$, $\sigma_2(\epsilon) \in I^{+}(x_1)$ thus there is a timelike curve $\sigma_1$ which connects first $x_1$ to $\sigma_2(\epsilon)$ and then this event to $z$ following $\sigma_2$. Thus $l(\sigma_1)+\epsilon \ge l(\sigma_2)$, and taking the sup over the set of connecting timelike curves $\sigma_2$, $d(x_1,z)+\epsilon \ge d(x_2,z)$. As $\epsilon$ is arbitrary $d(x_1,z)\ge d(x_2,z)$, and analogously in the other direction by interchanging the roles of $x_1$ and $x_2$. \end{proof} The next result has an analog in the strongly causal case \cite[Cor. 4.28]{beem96}. \begin{theorem} \label{xdy} If $(M,g)$ is a future (resp. past) distinguishing spacetime then for any $x\in M$, there is an arbitrary small open neighborhood $U_x$ (which can be chosen globally hyperbolic) such that $d(x, \cdot): U_x \to [0,+\infty]$ is continuous and finite (resp. $d(\cdot,x): U_x \to [0,+\infty]$ is continuous and finite). \end{theorem} \begin{proof} Since $M$ is future distinguishing \cite[Lemma 3.10]{minguzzi06c} for every open set $U\ni x$ there is a neighborhood $V \subset U$, $V\ni x$ such that every timelike curve starting from $x$ and ending at $y \in V$, is necessarily contained in $V$. Moreover, the same proof \cite[Lemma 3.10]{minguzzi06c} shows that $V$ can be chosen globally hyperbolic when regarded as a spacetime with the induced metric. As a consequence $d(x, \cdot): V \to [0,+\infty]$ coincides with $d\vert_{V\times V}(x,\cdot)$, where $ d\vert_{V\times V}$ is the Lorentzian distance on the spacetime $(V,g\vert_{V})$. Since $V$ is globally hyperbolic $d\vert_{V\times V}(x,\cdot)$ is continuous and finite and so is $d(x, \cdot): V \to [0,+\infty]$. \end{proof} The next example shows that the assumption of distinction is needed in the previous lemma. \begin{example} Consider the spacetime $M=\mathbb{R}\times S^1\backslash \{o\}$, of coordinates $(t,\theta)$, $\theta \in (-\pi,+\pi]$, $o=(0,0)$, and metric $g=-\omega^{-} \otimes \omega^{+}-\omega^{+} \otimes \omega^{-}$, where $\omega^{-}=\cos\alpha(t){\rm d} t - \sin \alpha(t) {\rm d} \theta $ and $\omega^{+}=\sin\alpha(t){\rm d} t + \cos \alpha(t) {\rm d} \theta$, $\alpha(t)\ge 0 $, $\alpha(t)$ even function with $ \alpha'> 0$ in $(0,1)$, $\alpha(t)=\pi/4$ if $\vert t\vert \ge 1$, see figure \ref{dist}. \begin{figure} \centering \includegraphics[width=2.2cm]{distinction} \caption{A causal but non-distinguishing spacetime. A suitable metric compatible with the drawn conformal structure exists which makes the restricted Lorentzian distance $d(x, \cdot)$ neither finite nor continuous in neighborhoods of the events $x$ belonging to the middle circle.} \label{dist} \end{figure} Let $x=(0,\phi)$, $\phi\ne 0$, be a point in the middle circle, and let $z$ be a point such that $t(z)>0$. Consider the past inextendible timelike curve $\sigma$ of future endpoint $z$ and of equation ${\rm d} t/{\rm d} \theta=\tan \beta(t)$ where $\tan\beta=\tan\alpha+t^2$. As $\alpha<\beta<\pi/2$ the curve $\sigma$ is indeed timelike. Because of the differential equation it satisfies, the curve can not cross the middle circle, as a consequence it passes infinitely often on any neighborhood of $x$. Assume $\tan\alpha(t)=t^2$ for small $\vert t\vert$. The curve $\sigma$ can be parametrized with $t$ and it has infinite length because the integral \[ \int_{\epsilon}^{t(z)} \frac{{\rm d} \tau}{{\rm d} t} {\rm d} t\ge \int_{\epsilon}^{t(z)}\sqrt{2\cos^2\alpha(t) (1-\tan\alpha(t) \frac{{\rm d} \theta}{{\rm d} t}) \frac{{\rm d} \theta}{{\rm d} t}}\,{\rm d} t \sim \frac{1}{\sqrt{2}}\int_{\epsilon}^{t(z)} \frac{1}{t} {\rm d} t , \] diverges as $\epsilon \to 0^{+}$. As a consequence, for suffieciently large $n$, we can choose to cut the curve near $x$ so that it takes $n$ cycles around the cylinder to get to $z$, and then to join $x$ with a causal curve to the starting point of the found segment. The found sequence of causal curves $\gamma_n$ connecting $x$ to $z$ has increasing length which goes to infinity with $n$, thus $d(x,z)=+\infty$, where $z$ is a generic point with $t(z)>0$. Thus the Lorentzian distance is neither finite nor continuous in a neighborhood of $x$ as $d(x,x)=0$. \end{example} \begin{theorem} \label{pcr2} Whatever the value of $f$, $P=M\times_f H$, the warped product spacetime $(P,\bm{\hat{g}})$ is (future, past) distinguishing iff $(M,\bm{g})$ is (future, past) distinguishing. \end{theorem} \begin{proof} Since the distinction property is conformally invariant the proof can be restricted to the direct product spacetime case, see theorem \ref{mjh}. Assume $(M,g)$ is not future distinguishing, then there are $x_1$, $x_2\in M$, $x_1 \ne x_2$, such that $I^{+}(x_1)=I^{+}(x_2)$. Defined $p_1=(x_1,y_0)$ and $p_2=(x_2,y_0)$, by lemma \ref{kijd} and lemma \ref{uty}, $I^{+}(p_1)=I^{+}(p_2)$. Thus if $(P,{\tilde{g}})$ is future distinguishing then $(M,g)$ is future distinguishing. Conversely, if $(M,g)$ is future distinguishing, assume that there exist $p_1\ne p_2$, $I^{+}(p_1)=I^{+}(p_2)$. It follows $I^{+}(x_1)=I^{+}(x_2)$, and since $(M,g)$ is future distinguishing $x_1=x_2$, thus $p_1$ and $p_2$ stay in the same fiber. But since $(M,g)$ is chronological $d(x_1,x_1)=0$ and since $p_1 \ne p_2$ lemma \ref{uty2} implies that there is a discontinuity in the distance $d(x_1, \cdot)$ at $x_1$, in contradiction with theorem \ref{xdy}. \end{proof} \begin{lemma} The direct product spacetime $(P,\tilde{g})$ is geodesically connected if and only if $(M,g)$ is geodesically connected. \end{lemma} \begin{proof} Trivial taking into account that any geodesic $\gamma(\lambda)$ can be written \[ \gamma(\lambda)=(\sigma(\lambda),\, y(\lambda)) , \] where both $\sigma(\lambda)$ and $y(\lambda)$ are geodesics, and taking into account that $(H,h)$ is geodesically connected. \end{proof} \begin{definition} A spacetime $(M,g)$ will be said to be {\em maximizing geodesically connected} if for every pair of events, $x_1 \in J^{+}(x_0)\backslash\{x_0\}$, $d(x_0,x_1)<+\infty$, there is a maximizing causal geodesic $\sigma$ connecting the two events, $l(\sigma)=d(x_0,x_1)$. \end{definition} The next result, with a different (unpublished) proof and for the special case $(H,h)=(\mathbb{R},{\rm d} y^2)$, has also been obtained by M. S\'anchez. \begin{lemma} \label{cxz} The direct product spacetime $(P,\tilde{g})$ is maximizing geodesically connected if and only if $(M,g)$ is maximizing geodesically connected. \end{lemma} \begin{proof} Assume $(M,g)$ is maximizing geodesically connected. Let $p_0,p_1 \in P$, $p_0=(x_0,y_0)$, $p_1=(x_1,y_1)$, with $p_0 <p_1$. Then, since the projection of a causal curve is a causal curve, $x_0 \le x_1$. Let $\sigma(\lambda)$, $\lambda \in [0,1]$, be the maximizing geodesic connecting $x_0$ to $x_1$. If it is lightlike then $d(x_0,x_1)=0$, thus $y_1=y_0$ due to the inclusion (\ref{iyr2}), in particular there is no timelike curve connecting $p_0$ to $p_1$ thus $d^{(P)}(p_0,p_1)=0$. The horizontal lift of $\sigma$ gives the maximizing lightlike geodesic. Otherwise $d(x_0,x_1)>0$, let $y$ be a minimizing geodesic parametrized with respect to length starting at $y_0$ and ending at $y_1$. The curve on $P$ \[ \gamma(\lambda)=(\sigma(\lambda),y(\frac{\rho(y_0,y_1)}{d(x_0,x_1)} \int_{\sigma\, x_0}^{\sigma(\lambda)} {\rm d} s)) \] is causal and connects $p_0$ to $p_1$. Its length is $\int_{\gamma} {\rm d} s^{(P)}=\int_{\sigma} \sqrt{1-(\frac{\rho(y_0,y_1)}{d(x_0,x_1)})^2}\, {\rm d} s=\sqrt{d(x_0,x_1)^2-\rho(y_0,y_1)^2}$ and by theorem \ref{cfd}, $\gamma$ is a maximizing geodesic. Conversely, let $(P,\tilde{g})$ be maximizing geodesically connected. Let $x_0 <x_1$ and take $p_0=(x_0,y_0)$, $p_1=(x_1,y_0)$. The events $p_0$, $p_1$, are connected by the horizontal lift of any causal curve connecting $x_0$ to $x_1$, thus $p_0 <p_1$. By lemma \ref{itg} the maximizing geodesic connecting $p_0$ to $p_1$ reads $\gamma(\lambda)=(\sigma(\lambda), y_0)$ so that $l^{(P)}(\gamma)=l(\sigma)$ and by theorem \ref{cfd}, $d^{(P)}(p_0,p_1)=d(x_0,x_1)$, from which it follows $l(\sigma)=d(x_0,x_1)$, i.e. $\sigma$ is a maximizing connecting geodesic. \end{proof} \begin{lemma} \label{comp} The direct product spacetime $(P,\tilde{g})$ is (past, future) (timelike, causal) geodesically complete if and only if $(M,g)$ is (resp. past, future) (resp. timelike, causal) geodesically complete. \end{lemma} \begin{proof} Assume $M$ is geodesically complete. Given a geodesic $\gamma(\lambda)=(\sigma(\lambda),y(\lambda))$ on $P$, a complete extension of the projection $\sigma(\lambda)$ exists ($\sigma$ has the same causal character of $\gamma$), and the same can be said for $y(\lambda)$ as $(H,h)$ is complete. Thus the original geodesic can be extended to a complete geodesic using the same decomposition. Given $\sigma$ on $M$, the converse is proved taking the projection of the extension of its horizontal lift. \end{proof} \section{Causal continuity and causal simplicity} So far there has been a complete correspondence between causal properties of $(M,g)$ and causal properties of the warped product spacetime $(P,\hat{g})$. Only the levels of the causal ladder corresponding to causal continuity and causal simplicity were not included in the previous analysis. Indeed, as we shall see, such a simple correspondence does not hold for these properties. Interestingly, they are also the only levels of the causal ladder which are not preserved by causal mappings \cite{garciaparrado03,garciaparrado05,garciaparrado05b}. Recall that a causally continuous spacetime $(M,\bm{g})$ is a distinguishing spacetime which is moreover reflecting. \begin{theorem} \label{causc} The direct product spacetime $(P,{\tilde{g}})$ is causally continuous if and only if $(M,{g})$ is causally continuous and for every pair $x_1 \in \bar{J}^{+}(x_0)$ (or equivalently, by causal continuity, $x_0 \in \bar{J}^{-}(x_1)$), $S^{+}(x_0,x_1)=S^{-}(x_0,x_1)$ or \begin{equation} \label{inc} \textrm{diam}(H,h) \le \textrm{min}[S^{+}(x_0,x_1), S^{-}(x_0,x_1)] < +\infty. \end{equation} In particular, if $(M,{g})$ is causally continuous and $d:M\times M \to [0,+\infty]$ is continuous (at least) at the pairs of events belonging to the set $d^{-1}(C) \subset M \times M$ where $C=[0,\textrm{diam}\, (H,h))$, then the direct product spacetime $(P,{\tilde{g}})$ is causally continuous. \end{theorem} \begin{proof} Assume $(P,\tilde{g})$ is causally continuous. By theorem \ref{pcr2} $(M,g)$ is distinguishing, thus we have only to prove that it is past reflective, the future case being similar. If $(M,g)$ is not past reflecting there are events $x,z,w,$ such that $I^{+}(x) \supset I^{+}(z)$ but $w \in I^{-}(x)$ while $w \notin I^{-}(z)$. In particular, $d(w,z)=0$. Define $p=(x,y_0)$, $q=(z,y_0)$, $r=(w,y_0)$ for an arbitrary $y_0 \in H$. Let $q' \in I^+(q)$ then $z'=\pi(q') \in I^{+}(z)$ thus $z' \in I^{+}(x)$. By lemma \ref{kijd}, $\rho(\pi_H(q'),y_0)<d(z,z') \le d(x,z')$ and because of lemma \ref{uty}, $q' \in I^{+}(p)$, hence $I^{+}(p) \supset I^{+}(q)$. As the horizontal lift of a timelike curve is a timelike curve $r \in I^{-}(p)$, but $r \notin I^{-}(q)$ otherwise $w \in I^{-}(z)$ hence $(P,{\tilde{g}})$ is not past reflective, a contradiction. Let $x_1 \in \bar{J}^{+}(x_0)$. Note that if the equality $S^{+}(x_0,x_1)=S^{-}(x_0,x_1)$ does not hold then necessarily $\textrm{min}[S^{+}(x_0,x_1), S^{-}(x_0,x_1)] < +\infty$. Let us show that if $\textrm{diam}(H,h) > \textrm{min}[S^{+}(x_0,x_1), S^{-}(x_0,x_1)]$ then $S^{+}(x_0,x_1)=S^{-}(x_0,x_1)$. Indeed, if the equality were not satisfied, for instance $S^{+}(x_0,x_1)>S^{-}(x_0,x_1)$, taken $y_0 \ne y_1$ such that $S^{-}(x_0,x_1)<\rho(y_0,y_1)< \textrm{min}[\textrm{diam}\, H,S^{+}(x_0,x_1)]$ the event $p_1=(x_1, y_1)$ would belong to $\bar{J}^{+}(p_0)$ with $p_0=(x_0,y_0)$ by lemma \ref{uty2} but $p_0 \notin \bar{J}^{-}(p_1)$ by the same lemma, thus $(P,\tilde{g})$ would not be causally continuous (recall the definition of reflectivity \cite[Lemma 3.42]{minguzzi06c}). For the converse since $(M,g)$ is distinguishing, by theorem \ref{pcr2}, $(P,{\tilde{g}})$ is distinguishing, and we have only to show that it is reflective. Let $p_1=(x_1,y_1) \in \bar{J}^{+}(p_0)$, $p_0=(x_0,y_0)$, we have to show that $p_0\in \bar{J}^{-}(p_1)$ (this fact would prove past reflectivity, the proof for future reflectivity being analogous). But $p_1 \in \bar{J}^{+}(p_0)$ implies $x_1 \in \bar{J}^{+}(x_0)$, and by lemma \ref{uty2} $\rho(y_0,y_1) \le S^{+}(x_0,x_1)$ and by causal continuity on the base $x_0 \in \bar{J}^{-}(x_1)$. There are two cases. If the equality $S^{+}(x_0,x_1)=S^{-}(x_0,x_1)$ holds then $\rho(y_0,y_1)\le S^{-}(x_0,x_1)$, and by lemma \ref{uty2}, $p_0\in \bar{J}^{-}(p_1)$. Otherwise, $\rho(y_0,y_1)\le \textrm{diam}(H,h) \le S^{-}(x_0,x_1)$ and by lemma \ref{uty2}, $p_0\in \bar{J}^{-}(p_1)$. In conclusion $(P,\tilde{g})$ is past reflecting. The last statements follows because given $x_1 \in \bar{J}^{+}(x_0)$ if $d(x_0,x_1) <\textrm{diam}\, (H,h)$ then $d$ is continuous and hence $S^{+}(x_0,x_1)=d(x_0,x_1)=S^{-}(x_0,x_1)$. Thus $S^{+}(x_0,x_1)\ne S^{-}(x_0,x_1)$ only if $d(x_0,x_1) \ge \textrm{diam}\, (H,h)$, in which case (recall $S^{\pm}(x_0,x_1)\ge d(x_0,x_1)$) inequality (\ref{inc}) holds. In both cases the condition needed for the causal continuity of the direct product is satisfied. \end{proof} \begin{corollary} Whatever the value of $f$, $P=M\times_f H$, if the warped product spacetime $(P,\bm{\hat{g}})$ is causally continuous then $(M,\bm{g})$ is causally continuous. \end{corollary} Apart from the one already considered, a way to prove that in the case $\textrm{diam}\, (H,h)=+\infty$ the causal continuity of $(M,g)$ and the continuity of $d$ implies the causal continuity of $(P,\tilde{g})$ is as follows. In the first step the next result which improves a result by Beem and Ehrlich \cite[Th. 4.24]{beem96} \cite{beem77}, as it imposes continuity of $d$ only on the vanishing distance set, is obtained. \begin{theorem} \label{cdfq} Let $(M, \bm{g})$ be a conformal structure, and let a representative $g$ exist such that $d: M\times M \to [0,+\infty]$ is continuous on the vanishing distance set $I^{+C}$, then $(M,\bm{g})$ is a reflecting spacetime. \end{theorem} \begin{proof} If $(M,g)$ were not reflecting then it would not be either past or future reflecting. We can assume the first possibility as the other case can be treated similarly. Thus there is a pair $(x,z)$ and an event $y$ such that $I^{+}(x) \supset I^{+}(z)$ but $y \in I^{-}(x)$ while $y \notin I^{-}(z)$. In particular, $d(y,z)=0$. Since $I^{+}(z)\subset I^{+}(x)$, $z \in \bar{I}^{+}(x)$. Let $z_n \to z$, $z_{n} \in I^{+}(x)$, then \[ d(y,z_n)\ge d(y,x)+d(x,z_{n})>d(y,x)>0, \] thus there is a discontinuity at $(y,z)$, where $d(y,z)=0$, a contradiction. \end{proof} this result has the consequence \begin{corollary} \label{fyu} Let $(M,\bm{g})$ be a distinguishing spacetime and let a representative $(M,g)$ exist such that the Lorentzian distance $d$ is continuous on the vanishing distance set $I^{+C}$, then $(M,\bm{g})$ is causally continuous. \end{corollary} then, under the said assumptions, theorem \ref{pcr2} proves that $(P,{\tilde{g}})$ is distinguishing, Eq. (\ref{ynb}) proves that $d^{(P)}$ is continuous and from corollary \ref{fyu} the thesis follows. \begin{example} In order to construct an example of causally continuous spacetime $(M,g)$ such that $(P,\tilde{g})$ is not causally continuous one has only take $(H,h)=(\mathbb{R},{\rm d} y^2)$, so that $\textrm{diam}\, (H,h)=+\infty$, and to find a causally continuous spacetime $(M,g)$ in which $S^{+}(x_0,x_1)\ne S^{-}(x_0,x_1)$. To this end let $M$ be $\mathbb{R}^2$ without the origin, let $(t,x)$ be coordinates on $\mathbb{R}^2$, and consider the metric $g=\Omega^2(-{\rm d} t^2+{\rm d} x^2)$ where $\Omega^2=\frac{1}{t^{2}+\omega(x)}$, $\omega(x)=x^2$ for $x>0$, and $\omega(x)=4x^2$ for $x<0$. Chosen $x_0=(-1,1)$, $x_1=(1,-1)$, the reader may convince him or herself that $S^{+}(x_0,x_1)\ne S^{-}(x_0,x_1)$ (although a rigorous proof would require much more effort). \end{example} Recall that a spacetime $(M,g)$ is causally simple if it is causal \cite{bernal06b} and such that for every $x\in M$, $J^{+}(x)$ and $J^{-}(x)$, are closed (or, equivalenty, if it is causal and $J^{+} \subset M \times M$ is closed \cite[Prop. 3.68]{minguzzi06c}) \begin{theorem} \label{csim} Whatever the value of $f$, $P=M\times_f H$, if the warped product spacetime $(P,\bm{\hat{g}})$ is causally simple then $(M,\bm{g})$ is causally simple. \end{theorem} \begin{proof} The proof can be given in the direct product case, $f=1$. It has been already proved that $(M,g)$ is causal. Let $z \in \bar{J}^+(x)$ then there is a succession of points $z_{i} \to z$ such that $z_i \in J^{+}(x)$ and let $\sigma_i(\lambda)$ be a causal curve connecting $x$ with $z_i$ parametrized between 0 and 1. The horizontal lifts $\sigma_i^{*}$ staring from $p=(x,y_0)$, $y_0 \in H$, obtained by setting $\pi_{H}(\sigma_i^{*}(\lambda))=y_0$ are causal and their endpoints $q_{i}=(z_i,y_0)$ converge to $q=(z,y_0)$, thus since $P$ is causally simple there is a causal curve $\gamma(\lambda)$ connecting $p$ and $q$. Since the projection of a (timelike) causal curve is a (resp. timelike) causal curve the causal curve $\sigma(\lambda)=\pi \circ \gamma$ connects $x$ and $z$. Thus we have proved $\bar{J}^{+}(x)=J^{+}(x)$ for arbitrary $x$ and the past case is proved similarly. \end{proof} Recall \cite[Def. 11.1]{beem96} \begin{definition} The {\em timelike diameter} is, $\textrm{diam} (M,g)=\sup\{d(x,z):x,z \in M\}$ \end{definition} \begin{lemma} \label{sim} If the direct product spacetime $(P,\bm{\tilde{g}})$ is causally simple then any pair of causally related events on $M$, $(x_0,x_1) \in J^{+}$, such that $d(x_0,x_1) <+\infty$ and $d(x_0,x_1) \le \textrm{diam}(H,h)$ is connected by a maximizing causal geodesic. In particular, if $\textrm{diam}(M,g) \le \textrm{diam}(H,h)$ and $(P,\bm{\tilde{g}})$ is causally simple then $(M,g)$ is maximizing geodesically connected. \end{lemma} \begin{proof} Let $(x_0,x_1) \in J^{+}$ such that $d(x_0,x_1) <+\infty$ and $d(x_0,x_1) \le\textrm{diam}(H,h)$. We can find two points $y_0$ and $y_1$ on $H$ such that $\rho(y_0,y_1)=d(x_0,x_1)$ (because of the Hopf-Rinow theorem the distance on $H$ attains the supremum). By lemmas \ref{uty} and \ref{uty2}, $p_1 =(x_1,y_1)$ belongs to $E^{+}(p_0)$ where $p_0=(x_0,y_0)$, thus there is a lightlike geodesic $\gamma(\lambda)$, $\lambda \in [0,1]$, connecting $p_0$ to $p_1$. By lemma \ref{itg} is has the form \[ \gamma(\lambda)=(\sigma(\lambda), y(\lambda)), \] where $\rho(y_0,y_1)\le l_H(y) \le l(\sigma) \le d(x_0,x_1)$ thus in particular $l(\sigma)=d(x_0,x_1)$, that is, $\sigma$ is a maximizing geodesic. \end{proof} In particular the spacetime $(P,\tilde{g})$ with $(H,h)=(\mathbb{R},{\rm d} y^2)$, constructed above the manifold $M$ of figure \ref{fig1} is not causally simple. Indeed, given a point $p=(x_0,0)$ in the fiber of $x_0$ the point $q=(x_1, d(x_0,x_1))$ belongs to the boundary of the causal future of $p$ but is not causally related to it. In other words \begin{figure} \centering \psfrag{A}{$x$} \psfrag{B}{$t$} \psfrag{M}{$M$} \psfrag{C}{$x_0$} \psfrag{D}{$x_1$} \psfrag{R}{Remove} \includegraphics[width=5cm]{example1} \caption{A causally simple spacetime is not necessarily maximizing geodesically connected. If $(M,g)$ is causally simple $(M\times \mathbb{R},g+{\rm d} y^2)$ is not necessarily causally simple.} \label{fig1} \end{figure} \begin{remark} If $(M,g)$ is causally simple $(P,\tilde{g})$ is not necessarily causally simple. \end{remark} \begin{theorem} \label{pkh} Let $(M,\bm{g})$ be causally simple then for any representative $(M,g)$ the Lorentzian distance $d$ is continuous on the vanishing distance set $I^{+C}$. \end{theorem} \begin{proof} Let $d(x,z)=0$ and let $(x,z)$ be a discontinuity point for $d$, then there is a $\epsilon>0$ and a sequence $(x_n,z_n) \to (x,z)$, such that $d(x_n,z_n)>\epsilon>0$. In particular $(x_n,z_n) \in I^{+}$ and $(x,z) \in \bar{I}^{+}\backslash I^{+}=\dot{I}^{+}=E^{+}$, by causal simplicity \cite[Lemma 3.67]{minguzzi06c}. Let $\sigma_n$ be causal curves connecting $x_n$ to $z_n$ and such that $\limsup_{n \to +\infty} l(\sigma_n) \ge\epsilon$ (for instance let $l(\sigma_n)> d(x_n,z_n)-\frac{1}{n}$ if $d(x_n,z_n)<+\infty$ and $l(\sigma_n) >n$ if $d(x_n,z_n)=+\infty$). By \cite[Prop. 3.31]{beem96} there is a causal curve $\gamma$ passing through $x$ and a distinguishing subsequence $\sigma_j$ which converges to it. But by construction any event $y\ne x,z$ of $\gamma$ is the limit of events $y_j \in \sigma_j$, $ (x_j, y_j) \in J^{+}$, hence $(x,y) \in \bar{J}^{+}=J^{+}$ and analogously $(y,z) \in J^{+}$, thus $\gamma$ must be a lightlike geodesic connecting $x$ to $z$, otherwise $(x,z) \in I^{+}$. Finally, \[ d(x,z) \ge l(\gamma) \ge \limsup_{j \to +\infty} l(\sigma_j)\ge \epsilon>0. \] The contradiction concludes the proof. \end{proof} Thus, if $(M,g)$ is causally simple, any discontinuity point $(x,z)$ for the Lorentzian distance satisfies $0<d(x,z)<+\infty$. Due to lemma \ref{sim} it is natural to look for conditions on $(M,g)$ that guarantee the causal simplicity of $(P,\tilde{g})$. A useful observation is \begin{lemma} \label{bhy} Let $(M,g)$ be a spacetime, $P=M\times H$ the direct product spacetime, $p_0,p_1 \in P$ and set $x_0=\pi(p_0)$, $x_1=\pi(p_1)$. If $p_1 \in \bar{J}^{+}(p_0)\backslash {J}^{+}(p_0) (\subset \dot{I}^{+}(p_0) =\dot{J}^{+}(p_0))$ then $d(x_0,x_1) \le \rho(y_0,y_1)$ and one of the following three possibilities holds \begin{itemize} \item[(a)] $x_1 \in \bar{J}^{+}(x_0)\backslash{J}^{+}(x_0)$, \\ \item[(b)] $x_1\in J^{+}(x_0) \backslash {I}^{+}(x_0)$, and $d(x_0,\cdot): M \to [0, +\infty] $ is discontinuous at $x=x_1$, \item[(c)] $x_1 \in I^{+}(x_0)$. \end{itemize} Moreover, if $(M,\bm{g})$ is causally simple, only the possibility (c) holds. \end{lemma} \begin{proof} Let $p_0=(x_0,y_0)$, $p_1=(x_1,y_1)$. By lemma \ref{uty2} $x_1 \in \bar{J}^{+}(x_0)$. It can not be $d(x_0,x_1)> \rho(y_0,y_1)$ otherwise by lemma \ref{uty}, $p_1 \in I^{+}(p_0)$, a contradiction. Assume that the case (a) and (c) do not hold and, thus, consider the remaining case $x_1 \in J^{+}(x_0)\backslash {I}^{+}(x_0)$. If $x_1 \in J^{+}(x_0)\backslash {I}^{+}(x_0) \, (\subset \dot{J}^{+}(x_0))$, it is $d(x_0,x_1)=0$. If $d(x_0,\cdot)$ were continuous at $x_1$ then $S^{+}(x_0,x_1)=d(x_0,x_1)=0$, and by lemma \ref{uty2}, $\rho(y_0,y_1)=0$ thus $y_1=y_0$. However, since $x_1 \in E^{+}(x_0)$ there is a lightlike geodesic connecting $x_0$ to $x_1$ and its horizontal lift gives a causal curve connecting $p_0$ to $p_1$. The contradiction proves that $d(x_0,\cdot)$ is discontinuous. If $M$ is causally simple this case is ruled out due to lemma \ref{pkh} (case (a) is ruled out because of the definition of causal simplicity). \end{proof} \begin{theorem} \label{dcon} If the direct product spacetime $(P,\bm{\tilde{g}})$ is causally simple then the Lorentzian distance $d: M \times M \to [0,+\infty]$ on the spacetime $(M,g)$ is continuous at the pairs of events belonging to the set $d^{-1}(C\cup \{+\infty\})\subset M\times M$ where $C=[0,\textrm{diam}(H,h))$ (but it can be continuous in a larger set). In particular, if the direct product spacetime $(P,\bm{\tilde{g}})$ is causally simple and $\textrm{diam}(H,h)=+\infty$ then $d: M \times M \to [0,+\infty]$ is continuous. \end{theorem} \begin{proof}It is clear that $d$ is continuous at those points $(x_0,x_1)$ such that $d(x_0,x_1)=+\infty$ because there the Lorentzian distance is necessarily upper semi-continuous. It is also clear that if $(x_0,x_1) \notin \bar{J}^{+}$ then $d$ is continuous at $(x_0,x_1)$ because $(\bar{J}^{+})^C$ is an open set where $d$ vanishes. Assume $d(x_0,x_1)<+\infty$ and $(x_0,x_1) \in \bar{J}^{+}$. From lemma \ref{uty} and \ref{uty3} \begin{align} &{J}^{+} \cap [ \pi^{-1}(x_0)\times \pi^{-1}(x_1)] \nonumber \\ &\quad \subset \{(p_0,p_1): \ p_0=(x_0,y_0), p_1=(x_1,y_1) \textrm{ and }\rho(y_0,y_1) \le d(x_0,x_1)\} , \label{pis1} \\ &\bar{J}^{+} \cap [ \pi^{-1}(x_0)\times \pi^{-1}(x_1)] \nonumber \\ &\quad =\{(p_0,p_1): \ p_0=(x_0,y_0), p_1=(x_1,y_1) \textrm{ and }\rho(y_0,y_1) \le D(x_0,x_1)\} \label{pis2}, \end{align} thus if $d(x_0,x_1)<\textrm{diam}(H,h)$ then $d$ is continuous at $(x_0,x_1)$ otherwise, $D(x_0,x_1)>d(x_0,x_1)$ and a pair $(y_0,y_1) \in H \times H$ could be found such that $d(x_0,x_1)<\rho(y_0,y_1)<\textrm{min}(D(x_0,x_1),\textrm{diam}(H,h))$. Hence, because of (\ref{pis1}) and (\ref{pis2}), $p_0=(x_0,y_0)$ and $p_{1}=(x_1,y_1)$, would be such that $(p_0,p_1) \in \bar{J}^+$ but $(p_0,p_1) \notin J^{+}$ in contradiction with the simple causality of $P$. \end{proof} The next result proves that lemmas \ref{csim}, \ref{sim} and \ref{dcon} have a converse. Essentially, it proves the equivalence between the causal simplicity of $(P,\tilde{g})$ and the three properties of causal simplicity of $(M,g)$, the continuity of $d$ on $M$ and the maximizing geodesic connectedness of $M$. \begin{theorem} \label{koi2} The direct product spacetime $(P,\bm{\tilde{g}})$, $P=M\times H$, is causally simple if and only if the following three conditions hold \begin{itemize} \item[(i)] the spacetime $(M,g)$ is causally simple, \item[(ii)] the Lorentzian distance $d: M \times M \to [0,+\infty]$ on the spacetime $(M,g)$ is continuous at (least at) the pairs of events belonging to the set $d^{-1}(C)\subset M\times M$ where $C=[0,\textrm{diam}(H,h))$, \item[(iii)] any pair of causally related events, $(x_0,x_1) \in J^{+}$, such that $d(x_0,x_1) <+\infty$ and $d(x_0,x_1) \le \textrm{diam}(H,h)$ is connected by a maximizing causal geodesic $\sigma$, $l(\sigma)=d(x_0,x_1)$. \end{itemize} In particular if $(M,g)$ is causally simple, maximizing geodesically connected and $d$ is continuous then $(P,\bm{\tilde{g}})$ is casually simple whatever the choice of $(H,h)$. \end{theorem} \begin{proof} We have only to prove that (i),(ii) and (iii) imply that $(P,\tilde{g})$ is causally simple. If $(M,g)$ is causally simple then it is causal and hence $(P,\tilde{g})$ is causal too (theorem \ref{pcr}). We are going to show that if $(M,g)$ is also maximizing geodesically connected then points $p_1 \in \bar{J}^{+}(p_0)\backslash {J}^{+}(p_0)$ do not exist (the past case is analogous). Otherwise, by lemmas \ref{bhy} and \ref{uty2}, $x_1 \in I^{+}(x_0)$ and $d(x_0,x_1) \le \rho(y_0,y_1) \le S^+(x_0,x_1)$ where $p_0=(x_0,y_0)$, $p_1=(x_1,y_1)$. In particular $0<d(x_0,x_1)<+\infty$ and $d(x_0,x_1) \le\textrm{diam}(H,h)$. There are two cases, either (a) $d(x_0,x_1) =\textrm{diam}(H,h)$ and thus $d(x_0,x_1) =\rho(y_0,y_1)$ or (b) $d(x_0,x_1) <\textrm{diam}(H,h)$ and from (ii) $d$ is continuous at $(x_0,x_1)$, that is $ S^+(x_0,x_1)=d(x_0,x_1)$, thus $\rho(y_0,y_1)=d(x_0,x_1)$. In both cases $\rho(y_0,y_1)=d(x_0,x_1)$. Let $y$ be a minimizing geodesic starting at $y_0$, ending at $y_1$ and parametrized with respect to length. Since $d(x_0,x_1)=\rho(y_0,y_1)\le \textrm{diam}(H,h)$ there is, by assumption (iii) a maximizing geodesic $\sigma(s)$ connecting $x_0$ to $x_1$ parametrized with respect to proper time. The curve on $P$ \[ (\sigma(s), y(s)) \] is lightlike and connects $p_0$ to $p_1$, thus $p_1 \in {J}^{+}(p_0)$ a contradiction. \end{proof} \begin{remark} The optimality of the theorem would follow from the independence of conditions (i),(ii) and (iii). The next three spacetime examples prove this independence whatever the choice of the constant $\textrm{diam}(H,h)>0$. Thus they provide three distinct circumstances for which a direct product spacetime may fail to be causally simple. That (i) and (ii) does not imply (iii) is shown by the spacetime of figure \ref{fig1} where the events $x_0,x_1$, are not connected by a maximizing geodesic and can be chosen at arbitrary small Lorentzian distance. That (ii) and (iii) does not imply (i) can be proved in the spacetime $(M,g)$, $M=\Lambda\backslash\{o\}$ where $o=(0,0)$ is the origin of 1+1 Minkowski spacetime $\Lambda$, and the metric is $g=(t^2+x^2)^{-2}\eta$. Indeed $(M,g)$ is clearly non-causally simple, and if $(x_0,x_1)$ is such that $J^+(x_0,\Lambda)\cap J^{-}(x_1,\Lambda)$ contains $o$ then $d(x_0,x_1)=+\infty$ ($d$ distance in $(M,g)$) thus (ii) and (iii) are satisfied in this case. If $J^+(x_0,\Lambda)\cap J^{-}(x_1,\Lambda)$ does not contain $o$ then there are $\bar{x}_0\in I^{-}(x_0,\Lambda)$ $\bar{x}_1\in I^{+}(x_1,\Lambda)$, such that $(x_0,x_1) \in V= I^+(\bar{x}_0,\Lambda)\cap I^{-}(\bar{x}_1,\Lambda)$ and $o \notin V$. But $(V,\eta)$ is globally hyperbolic thus $(V,g)$ is globally hyperbolic too and has continuous and finite distance function (because $g$ is conformal to $\eta$) which coincides with the restriction of the distance function $d$ to $V$ thus $d$ is continous at $(x_0,x_1)$. Moreover, the global hyperbolicity of $(V,g)$ implies the existence of a maximizing geodesic connecting $x_0$ to $x_1$. Thus (ii) and (iii) hold. It remains to prove that (i) and (iii) does no imply (ii). Take $M=\Lambda\backslash\{(t,x): x\le 0\}$ and $g=\frac{1}{x} \, \eta $. The spacetime $(M,g)$ is clearly causally simple. The pairs of events of the form $(x_0,x_1)$ with $x_0=(b,k-b)$, $x_1=(b,k+b)$ with $b>0$, and $k$ arbitrary constants, have finite Lorentzian distance which is maximized by a connecting geodesic and which can be chosen arbitrarily small choosing $b$ sufficiently small. [All these statements follow from the fact that the maximizing geodesic can be explicitly calculated. It solves $2\ddot x+\dot x^2-1=0$, and has equation $x(t)=a+2\ln \cosh(\frac{t-k}{2})$ with $0<a=b-2\ln\cosh(b/2)$. The fact that the distance goes to zero as $b$ goes to zero follows from the fact that the length of the maximizing geodesic is bounded by $\frac{2b}{\sqrt{a}}$ which goes to zero.] Nevertheless, there is an infinite discontinuity for $d$ at each pair $(x_0,x_1)$ as above because given $\bar{x}_0\ll x_0$ and $\bar{x}_1 \gg x_1$, it is $d(\bar{x}_0,\bar{x}_1)=+\infty$ as there are sequences of causal curves connecting $\bar{x}_0$ to $\bar{x}_1$ which approach a finite vertical segment on the axis $x=0$. Thus (iii) holds but (ii) does not hold. \end{remark} \section{Conclusions} In this work I studied the correspondence between the causal properties of $(M,g)$ and those of the warped product $(P,\hat{g})$, $P=M\times H$, $\hat{g}=g+f^2 h$, with $(H,h)$ a complete Riemannian manifold. I showed that any statement involving only conformal properties can be reduced to the case $f=1$, relating $(M,g)$ with the direct product $(P,\tilde{g})$. An almost complete correspondence between the causal properties of the two spacetimes was found, in particular I found a correspondence for the properties of being distinguishing or non-totally vicious for which no previous result was available. In the process a formula for the Lorentzian distance on $(P,\tilde{g})$ in terms of the Lorentzian distance on $(M,g)$ was obtained. For causal continuity and causal simplicity the correspondence does not hold and indeed I gave an explicit counterexample in the latter case and suggested a possible counterexample for the former case. Distinct, non conformal invariant, and apparently unrelated properties must be required on $(M,g)$. The results which clarify this issue were theorems \ref{causc}, for causal continuity and theorem \ref{koi2}, for causal simplicity. Theorem \ref{koi2}, obtained here for a spacelike dimensional reduction geometry (the fibers $\pi^{-1}(x)$ are spacelike), has an interesting analog in the lightlike dimensional reduction case \cite{minguzzi06d}. In that work the role of the Lorentzian distance is replaced by a classical action functional on the base, and the upper semi-continuity of the Lorentzian distance is replaced by the lower semi-continuity of the action functional with respect to endpoints changes. Finally, some new results on the continuity of the Lorentzian distance on distinguishing, causally continuous and causally simple spacetimes were also obtained, see theorem \ref{xdy}, corollary \ref{fyu} and theorem \ref{pkh}. \section*{Acknowledgements} I warmly thank M. S\'anchez, this work, in its early stage of development, has benefited from his suggestions especially in connection with theorem \ref{pkh} and lemma \ref{cxz}.

proofpile-arXiv_069-15222

\section{Introduction} In this work~\cite{paper}, we are interested in the effect that axion-photon mixing can have on the polarisation of light coming from distant astronomical sources. In particular, the observations of redshift-dependent large-scale coherent orientations of AGN polarisation vectors can, at least qualitatively, even in very simple models, be reproduced as a result of such a mixing of incoming photons with extremely light axion-like particles in external magnetic fields. These observations, presented in the second edition of this conference, were based on good quality measurements of the linear polarisation for a sample of 355 measured quasars in visible light~\cite{hutsemekers}. This has been discussed in terms of axion-photon mixing by several authors, in the case of plane waves~\cite{planewaves} and a prediction from this mixing is an observable circular polarisation comparable to the linear one. Here, we present the case in which light is described by wave packets and show that the circular polarisation can be suppressed with respect to the plane wave case. \section{Axion-photon mixing using Gaussian wave packets} \subsection{The idea behind this} The mixing of axion-like particles with photons is usually discussed mathematically in terms of infinite plane waves. Using that description, the Stokes parameters can be computed and predictions of the polarisation of light from the interaction can be given; the main properties of such a mixing being \emph{dichroism} and \emph{birefringence} (see~\cite{reviewHLPW} for a review of this case). While dichroism would be an interesting way to produce linear polarisation and, in particular, to explain the observations concerning quasars, birefringence ---which is linked to the creation of circular polarisation--- would give a very clear signature of the mixing. Indeed, in this formalism of plane waves, except in extremely specific cases, the \emph{circular polarisation} predicted \emph{can be as large as the linear polarisation}\footnote{This is what one obtains if one does not assume very specific distributions of magnetic field orientations along the line of sight.}. The idea discussed here is to send wave packets into a region of uniform magnetic field and to compute the Stokes parameters. Before the magnetic field, the wave packets have the form: \begin{equation} E (x,t) = \int_{\omega_p}^{\infty} \frac{d\omega}{N} e^{-\frac{a^2}{4}(\omega-\omega0)^2} e^{i \sqrt{\omega^2 - \omega_p^2} (x-x_0)} e^{- i \omega (t-t_0)},\label{eq:packet} \end{equation} where $\omega_p$ is the plasma frequency of the medium and $a$ controls the initial width of the packet (in the limit $a\rightarrow +\infty$, this reduces to the plane wave case). The main motivation for considering this formalism comes from the measurements of circular polarisation of some of the quasars considered in~\cite{hutsemekers}. While axion-photon mixing would be an attractive explanation of the observations for linear polarisation, preliminary results show that circular polarisation of light from these AGN seems to be, in general, \emph{much smaller} than the linear polarisation~\cite{polcirc}. This means that if the creation of circular polarisation was really a smoking gun of ALP-photon mixing, no matter how refined the description, these observations would rule out the mixing mechanism and could only be used to constrain the existence of axion-like particles. For these reasons, it can be interesting to work with wave packets, as new effects will be taken into account, including dispersion, separation of packets and coherence; effects that might be of importance for the Stokes parameters. Note that calculating the propagation of packets of the form \eqref{eq:packet} is numerically\footnote{We use Multiple-Precision Floating-point library with correct Rounding: \url{www.mpfr.org}.} tricky, as the computation of the Stokes parameters requires a spatial resolution of the order of the width of the wave packets after a propagation over huge distances in the magnetic field (we will usually consider one magnetic field zone of 10~Mpc~\cite{vallee} and initial wave packets of width $\lesssim1\mu$m). \subsection{Results with wave packets} In the plane transverse to the direction of propagation, we choose a basis of two orthogonal linear polarisations, the same as the one used in the plane wave case, so that we will talk about polarisation parallel or perpendicular to the transverse external magnetic field $\vec{\mathcal{B}}$. This being done, we next choose the electric fields $E_{\parallel}(x,t)$ and $E_{\perp}(x,t)$ both initially described by a function of the form \eqref{eq:packet}. Then, we propagate these using the equations of motion for the electromagnetic field which take into account the interaction with pseudoscalar particles and find the expressions of the electric fields after a propagation, when axion-photon mixing is at work, inside a step-like magnetic field region. \begin{figure} \begin{center} \includegraphics[width=0.69\textwidth]{payez_alexandre.fig1.eps} \caption{Wave packets: illustration after a propagation time $T$ in an external magnetic field, in a strong mixing case ---here, the axion mass is $m=4.7~10^{-14}$~eV, $\omega_p = 3.7~10^{-14}$~eV and $g\mathcal{B}=5.5~10^{-29}$~eV. The initial width of the wave packet has been chosen $\simeq\lambda_0$.} \label{fig:packets} \end{center} \end{figure} We can then use the expressions of the Stokes parameters ---which are observables built on intensities--- that can, for example, be plotted as functions of $x$, the distance travelled inside the magnetic field, for a given propagation time, $T$. This is what is represented in Figure~\ref{fig:packets} which shows what the two packets look like (respectively $\mathcal{I}_{\parallel}(x,t=T)$ and $\mathcal{I}_{\perp}(x,t=T)$) but also the total intensity (which is just the sum of the two) and the unnormalised circular polarisation, $\mathcal{V}(x,t=T)$. This is for a beam with a central wavelength $\lambda_0 = 500$~nm, initially 100\% linearly polarised, with its polarisation plane making a 45$^\circ$ angle initially with the magnetic field direction (i.e. $u(0) = \frac{U(0)}{I(0)}= 1$; $q(0) = v(0) = 0$)\footnote{$u$ and $q$ are the two Stokes parameters required to describe fully the linear polarisation of a light beam, while $v$ accounts for the circular polarisation.}; this angle being, in fact, the most favourable one for the creation of circular polarisation, due to birefringence. Note also that the abscissa is $dx$, the shift in position with respect to a frame moving a the speed of light $c$ (namely, a maximum at $dx = 0$ corresponds to $|\vec{v}|=c$). From the observational point of view, there is a macroscopic exposure time over which one should integrate these functions to obtain, finally, the value of the observable Stokes parameters, e.g.: \begin{equation} I(x) = \int_\mathrm{exposure~time} dt~\mathcal{I}(x,t)\nonumber. \end{equation} From these integrals, we obtain that the wave packet formalism leads to a circular polarisation, $v=\frac{V}{I}$, \emph{lowered} with respect to plane wave case. Figure~\ref{fig:circpol} illustrates the plane wave ($a\rightarrow\infty$) result: it shows the amount of circular polarisation gained due to axion-photon mixing with different values of the coupling $g\mathcal{B}$ ($g$ being the axion-photon coupling constant). In that simpler case, it is known that $v=\frac{V}{I}$ oscillates between $-|u(0)|$ and $|u(0)|$, whereas in the wave packet case it is shown that there is a damping of these oscillations. It follows from this observation that \emph{$v$ is no longer expected to be as large as the linear polarisation in general}. \begin{figure} \begin{center} \includegraphics[width=0.69\textwidth]{payez_alexandre.fig2.eps} \caption{Circular polarisation for different values of the coupling in the case of initially partially linearly polarised light with $u(0)=0.01$. $\lambda_0$, $\omega_p$ ---and $a$, for the wave packet--- are the same as in Figure~\ref{fig:packets}, other parameters are: $T=10$~Mpc, $m=4.3~10^{-14}$~eV. } \label{fig:circpol} \end{center} \end{figure} \section{Conclusion} We have briefly presented axion-photon mixing with the use of wave packets. The main consequence of this treatment is the net decrease of circular polarisation with respect to what is predicted using plane waves. Hence, the lack of circular polarisation in the light from AGN does not rule out the ALP-photon mixing. \section*{Acknowledgements} A.~P. would like to thank the IISN for funding and to acknowledge constructive discussions on physical and numerical matters with Fredrik Sandin and Davide Mancusi. \begin{footnotesize}

proofpile-arXiv_069-15390

\section{Introduction} In this paper we study the existence and regularity for the solution of the inhomogeneous Cauchy-Riemann equations, or the $\overline{\partial}$-equation on product domains. When the product domain is a polydisc in ${\mathbb{C}}^n$, the solution to the $\overline{\partial}$-equation can be obtained by an inductive process from the solution in one variable given by the Cauchy integral formula for the disc. This is known as the Dolbeault-Grothendieck Lemma (see \cite[Theorem~2.1.6]{cs}. For other approaches, see \cite{nick,nw}.) which is the analog for the $\overline{\partial}$-operator of the Poincar\'{e} lemma for the exterior derivative $d$. We are interested here in the $\overline{\partial}$-problem in the $L^2$ setting. For a bounded pseudoconcovex domain in ${\mathbb{C}}^n$, or more generally in a Stein manifold, $L^2$ existence theorems have been established in H\"ormander \cite{Hor1}. We prove $L^2$ existence on a product, i.e., we show that $\overline{\partial}$ has closed range on a product provided that $\overline{\partial}$ has closed range on each factor domain . \begin{thm}\label{thm-main} For $j=1,...,N$, let $\Omega_j$ be a relatively compact domain with Lipschitz boundary in a complex hermitian manifold $M_j$ . Let $\Omega\subset M_1\times \dots\times M_N$ be the product domain $\Omega=\Omega_1\times\dots\times\Omega_N$. Suppose the $\overline{\partial}$ operator has closed range in $L^2(\Omega_j)$ for all degrees for each $j$, then the $\overline{\partial}$ operator has closed range for all degrees in $L^2(\Omega)$. Furthermore, the K\"{u}nneth formula holds for the $L^2$ cohomology: \[ H^*_{L^2}(\Omega)= H^*_{L^2}(\Omega_1){\widehat{\otimes}} \dots\ {\widehat{\otimes}} H^*_{L^2}(\Omega_N),\] where ${\widehat{\otimes}}$ denotes the Hilbert space tensor product. \end{thm} If the $L^2$ space on the domain $\Omega_j$ is defined with respect to a weight function $\phi_j$, i.e., if on $\Omega_j$ we use the norm $\int_{\Omega_j}\abs{f}^2e^{-\phi_j}dV$, the same statement holds if $\sum_{j=1}^N\phi_j$ is used as a weight on the product $\Omega$. A classical approach to the study of partial differential equations on product domains is by separation of variables and spectral representation. This method can be applied to the $\Box$-operator (the complex Laplacian $\overline{\partial}\dbar^*+\overline{\partial}^*\overline{\partial}$): see \cite[pp. 103ff]{gh} for the case of compact complex manifolds and \cite{fu} for the case of the polydisc. A proof of a general version of Theorem~\ref{thm-main} (without the assumption of relative compactness or boundary regularity of the $\Omega_j$) may be given using separation of variables and spectral theory (see \cite{cproc}.) However, it is difficult to use this method to draw conclusions about the regularity of the solution of the $\overline{\partial}$-equation. There is a different approach, using a direct construction of a solution operator on a product domain, used first in \cite{zuck} for the de Rham complex, and we use this approach to prove Theorem~\ref{thm-main}. Using this method we not only prove Theorem~\ref{thm-main}, but also obtain regularity results for the canonical solution of $\overline{\partial}$. The closed-range property given by Theorem~\ref{thm-main} has numerous applications. First it immediately gives that the Hodge decomposition holds for the product domain $\Omega$. Notice that it is not assumed that the domains $\Omega_j$ are non-compact: the theorem can be applied to the case when the product domain is a product $D\times M$ of a bounded pseudoconvex domain $D$ in ${\mathbb{C}}^n$ and a compact complex manifold $M$. Though the proof for Theorem~\ref{thm-main} is not difficult, it has not been stated explicitly in the literature. We obtain boundary regularity results for the canonical solution of the $\overline{\partial}$-equation on product domains in ${\mathbb{C}}^n$ or complex hermitian manifolds. The regularity for the canonical solution of the $\overline{\partial}$-equation and the $\overline{\partial}$-Neumann operator on a polydisc have been studied extensively (see \cite{eh1,eh2,eh3,jb} and the references in these works.) There is also a considerable amount of work for the $\overline{\partial}$-equation on domains with Lipschitz boundary or piecewise smooth domains (see \cite{ms4}). Notice that a product domain is only {\em piecewise} smooth even if each factor domain has smooth boundary. Thus the boundary is only Lipschitz. It is known that on a general Lipschitz domain, the $\overline{\partial}$-Neumann operator or even the Green's operator for the Dirichlet problem (see \cite{BV,mcs2}) is not regular near the singular part of the domain. Thus one cannot expect the $\overline{\partial}$-Neumann operator to be regular near the product of the boundaries of the factor domains. This is confirmed by the explicit computations in \cite{eh1,eh2,eh3}. One would expect that the canonical solution might also not be regular. An interesting feature is that while the $\overline{\partial}$-Neumann operator on product domains might not be well-behaved, the canonical solution still exhibits regularity on certain Sobolev spaces. Before our results here, only $\mathcal{C}^k$ estimates were known for the special case of the polydisc using an explicit integral formula \cite{lan}. In order to state precise regularity results on the canonical solution operator we introduce special Sobolev spaces, called {\em Partial Sobolev spaces}, denoted by $\widetilde{W}^k(\Omega)$ (for definition of $\widetilde{W}^k(\Omega)$, see $\S$\ref{sec-partialsobolev}.) If $W^k(\Omega)$ denotes the usual Sobolev space of functions having $L^2$-derivatives of order $k$ on $\Omega$, we have $W^{Nk}(\Omega)\subset \widetilde{W}^k(\Omega)\subset W^k(\Omega)$. We prove the following regularity result for the canonical solution operator in the partial Sobolev spaces on a product pseudoconvex domain. \begin{thm}\label{thm-reg} Let $\Omega$ be the same as in Theorem~\ref{thm-main}. Then the $\overline{\partial}$-Neumann operator $\mathsf{N}$ exists for all degrees on $(p,q)$-forms with $L^2$ coefficients. Assume further that for each $j$, the domain $\Omega_j$ is smoothly bounded, and the $\overline{\partial}$-Neumann operator on $\Omega_j$ preserves the space of forms with coefficients in $W^k(\Omega_j)$ for every integer $k\geq 0$. For any $p$ with $0\leq p\leq \dim_{\mathbb{C}}\Omega$, let $f$ be a $\overline{\partial}$-closed $(p,1)$-form on $\Omega$ orthogonal to the $(p,1)$-harmonic forms such that the coefficients of $f$ are in the partial Sobolev space $\widetilde{W}^l(\Omega)$, for some integer $l\geq 0$. Then, the canonical solution $u=\overline{\partial}^* \mathsf{N}f$ of the equation $\overline{\partial} u=f$ also has coefficients in $\widetilde{W}^l(\Omega)$. \end{thm} As with Theorem~\ref{thm-main}, if the $L^2$ space on $\Omega_j$ is defined with respect to a weight function $\phi_j$, the same conclusion holds if the $\overline{\partial}$-Neumann operator $\mathsf{N}_{\phi_j}$ with weight $\phi_j$ preserves $W^k(\Omega_j)$ forms, and the canonical solution on the product is taken with respect to the weight $\sum_{j=1}^N\phi_j$. Note that it follows from the inclusions $W^{Nk}(\Omega)\subset \widetilde{W}^k(\Omega)\subset W^k(\Omega)$ that if the $(p,1)$-form $f$ has coefficients in $W^{Nk}(\Omega)$, then the canonical solution $\overline{\partial}^* \mathsf{N}f$ has coefficients in $W^k(\Omega)$. Of course this loss of smoothness disappears on using the correct space $\widetilde{W}^k(\Omega)$. Also note that unless $D$ is a domain in ${\mathbb{C}}^n$, the $\overline{\partial}$-equation for $(p,q)$ forms on a domain $D$ in a complex manifold cannot be reduced to the $\overline{\partial}$-equation for $(0,q)$ forms. To use this result, we need to understand the regularity of the $\overline{\partial}$-Neumann operator on the factor domains. There is a vast literature on the regularity of the $\overline{\partial}$-Neumann operator on smooth and pseudoconvex domains. In particular, regularity is known when the boundary is strongly pseudoconvex (see \cite{Ko1}) or finite type (see \cite{Ca}), or if the boundary has a plurisubharmonic defining function (see \cite{BS}) or if the boundary has transverse symmetry ( see \cite{ba1}.) However, for each $s>0$, there exists a pseudoconvex domain with smooth boundary such that the $\overline{\partial}$-Neumann operator or the canonical solution is not regular in the Sobolev space $W^s$ (see \cite{ba2}). Even in this case, we can obtain regularity in a weighted Sobolev space (see \cite{Ko2}.) Using these results, one can draw many corollaries from Theorem~\ref{thm-reg} regarding the regularity of the solution of the $\overline{\partial}$-problem in Sobolev spaces or spaces of smooth forms (see Corollary~\ref{cor-main} below.) One example is the following: \begin{cor}\label{cor-strong} Suppose that the smoothly bounded pseudoconvex domains $\Omega_1,\dots,\Omega_N$ in hermitian manifolds of dimension $n_1,\dots,n_j$ respectively are such that for each $j$, and every $0\leq p\leq n_j$, the canonical solution operator on $\Omega_j$ maps the space $\mathcal{C}^\infty_{p,1}(\overline{\Omega_j})$ of $(p,1)$ forms smooth up to the boundary to $\mathcal{C}^\infty_{p,0}(\overline{\Omega_j})$. Let $\Omega=\Omega_1\times\dots\times\Omega_N$. For $0\leq \boldsymbol{p}\leq\sum_{j=1}^N n_j$, let $f$ be a $\overline{\partial}$-closed $(\boldsymbol{p},1)$ form with $\mathcal{C}^\infty(\overline{\Omega})$ coefficients. Then $\overline{\partial}^*\mathsf{N}f$ also has coefficients in $\mathcal{C}^\infty(\overline{\Omega})$, where $\mathsf{N}$ is the $\overline{\partial}$-Neumann operator on $\Omega$. Further, the Bergman projection on the space of functions $L^2(\Omega)$ preserves the space $\mathcal{C}^\infty(\overline{\Omega})$. \end{cor} Note that if $f$ is orthogonal to the harmonic forms, $\overline{\partial}^*\mathsf{N}f$ is the canonical solution to $\overline{\partial} u=f$. Also, Corollary~\ref{cor-strong} applies to the product of domains which are { strongly} pseudoconvex or more generally of finite type. In contrast, we note that when the domain is the intersection of two balls, the Bergman projection is not regular near the nongeneric points of the boundary (see \cite{BV}). Notice that on a smoothly bounded pseudoconvex domain in ${\mathbb{C}}^n$, if the canonical solution $\overline{\partial}^*\mathsf{N}$ is regular, it follows that the $\overline{\partial}$-Neumann operator $\mathsf{N}$, and the adjoint of the canonical solution operator $\overline{\partial}\mathsf{N}$ are all exact regular on Sobolev spaces (see \cite{cs}.) However, the same method cannot be applied to the adjoint of the canonical solution or to the $\overline{\partial}$-Neumann operator on a product domain. On a product domain, the canonical solution is regular, but neither the $\overline{\partial}$-Neumann operator $\mathsf{N}$ nor the operator $\overline{\partial}\mathsf{N}$ is regular near the boundary. The plan of this paper is as follows: in $\S$\ref{sec-l2setting} and in $\S$\ref{sec-diffforms} we establish terminology and notation regarding the $L^2$ $\overline{\partial}$-problem and tensor products of forms, and discuss some basic properties of the objects involved. We note here that although our results have been stated for general manifolds, in these sections, for simplicity of exposition and notation, we give the definitions for domains in Euclidean space ${\mathbb{C}}^n$. The generalization to manifolds is easy and left to the reader. Also, due to the nature of the proof, we need to consider spaces of forms of arbitrary degrees. For example, we denote by $L^2_*(D)$ the space of forms with square integrable coefficients on a domain $D$. The key observation in $\S$\ref{sec-l2setting} is that closure of the range of the $\overline{\partial}$-operator is a necessary as well as sufficient condition for representation of cohomology classes by harmonic forms (see Lemma~\ref{lem-j}.) The next $\S$\ref{sec-dbarl2} represents the central argument of the paper. Starting from the canonical solution operator and the harmonic projection on the factor domains, we write down a formula \eqref{eq-tnj} defining a solution operator $S$ on the product, which coincides with the canonical solution operator $\overline{\partial}^*\mathsf{N}$ on $(0,1)$-forms. Using $S$ we give a simple proof of Theorem~\ref{thm-main}. In $\S$\ref{sec-partialsobolev} we consider the tensor products of Sobolev spaces, which gives rise to the partial Sobolev spaces referred to above. This is used in the last $\S$\ref{sec-regularity} to prove regularity results. {\em Acknowledgements:} The authors would like to thank professors Carl De Boor, Dariush Ehsani, Sophia Vassiliadou for helpful discussions, and the anonymous referee for his comments. They especially would like to thank professors Xiuxiong Chen and Jianguo Cao for raising the questions on the closed range property for product domains, which arise naturally in many geometric problems. In particular, this paper answers affirmatively (see $\S$\ref{sec-chencao} below) on the closed-range property for the product domain of an annulus and a ball in ${\mathbb{C}}^n$ (which is not pseudoconvex, a question raised by X. Chen) and the product of $D\times {\mathbb{C}} P^1$ of the unit disc $D$ in ${\mathbb{C}}$ and the Riemann sphere (which is not Stein, a question raised by J. Cao.) \section{The $L^2$ Setting for the $\overline{\partial}$ problem}\label{sec-l2setting} \subsection{Spaces of forms on domains} We recall the definition and notation used in the $L^2$ theory of the $\overline{\partial}$-operator. Let $D$ be a bounded domain in ${\mathbb{C}}^n$, and let $\phi$ be a continuous function on $\overline{D}$. We denote by $L^2(D)$ the space of square integrable functions on $D$ with respect to weight $\phi$, which has the norm \[ \norm{f}= \int_D \abs{f}^2 e^{-\phi} dV,\] where $dV$ is the volume form on ${\mathbb{C}}^n$ induced by the standard hermitian metric. (Note that we have suppressed $\phi$ from the notation.) We denote by $L^2_*(D)$ the space of differential forms with coefficients in $L^2(D)$. More generally, for any space of functions $\mathcal{F}(D)$ on $D$, we will let $\mathcal{F}_*(D)$ denote the space of forms with coefficients in $\mathcal{F}$. Then $\mathcal{F}_*(D)$ can be thought of as a vector space direct sum \begin{equation}\label{eq-fstard} \mathcal{F}_*(D)= \bigoplus_{\substack{0\leq p\leq n\\0\leq q\leq n}}\mathcal{F}_{p,q}(D) \end{equation} of the spaces of forms of bidegree $(p,q)$. Often the space $\mathcal{F}(D)$ will be a Hilbert space. Then we can give $\mathcal{F}_*(D)$ a Hilbert space structure in the following way: first, we declare that forms of different bidegrees are orthogonal, so that the sum in \eqref{eq-fstard} is now an orthogonal direct sum of Hilbert Subspaces. Any form $f\in\mathcal{F}_{p,q}(D)$ can be uniquely represented as \[ f = \sum'_{I,J}{f_{I,J}}dz^I\wedge d\overline{z}^J,\] where $I=(i_1,\dots,i_p)\in \mathbb{N}^p$, and $J=(j_1,\dots,j_q)\in \mathbb{N}^q$ are multi-indices, $f_{I,J}\in\mathcal{F}(D)$, $dz^I=dz_{i_1}\wedge\dots\wedge dz_{i_p}$ and $d\bar{z}^J=d\bar{z}_{j_1}\wedge\dots\wedge d\bar{z}_{j_q}$, and the notation $\sum'$ means that the summation is over strictly increasing multi-indices only, i.e. $i_1<i_2<\dots<i_p$ and $j_1<j_2<\dots<j_q$. We define the norm of $f$ as \begin{equation}\label{eq-formhilbert} \norm{f}_{\mathcal{F}_*(D)}^2 = \sum'_{I,J}\norm{f_{I,J}}_{\mathcal{F}(D)}^2.\end{equation} In this paper, the Hilbert space $\mathcal{F}$ will be either a usual $L^2$ space (possibly with weight), a Sobolev space, or a partial Sobolev space on product domains (to be defined in $\S$\ref{sec-partialsobolev}.) These notions easily extend to spaces of forms on domains in hermitian manifolds (see \cite[Chapter~5]{cs}) using the natural pointwise inner-product on forms induced by the hermitian structure. \subsection{The $L^2$ Dolbeault Complex}\label{sec-l2defns} We now recall the definition of the the $\overline{\partial}$-operator on the Hilbert space $L^2_*(D)$ of forms with square integrable coefficients on $D$. The $\overline{\partial}$-operator is the closed, densely defined unbounded operator from $L^2_*(D)$ to itself which coincides with the usual $\overline{\partial}$ operator from $\mathcal{C}^\infty_*(\overline{D})$ to $\mathcal{C}^\infty_*(\overline{D})$, and which has been extended as a distributional operator to the dense domain of definition \[ \dm(\overline{\partial}) = \{ f\in L^2_*(D)\colon \overline{\partial} f \in L^2_*(D)\}.\] In the terminology of \cite{brun}, the operator $\overline{\partial}$ is the differential map of a {\em Hilbert Complex}, i.e., a cochain complex, in which the cochain space $\dm(\overline{\partial})$ is a dense subspace of a graded Hilbert Space, and the differential is a closed, densely defined unbounded linear map of the graded Hilbert space into itself. Note that the map $\overline{\partial}$ has bidegree $(0,1)$, i.e., it maps $(p,q)$ forms to $(p,q+1)$ forms. We denote by $\overline{\partial}^*$ the Hilbert space adjoint of $\overline{\partial}$. This is again a closed, densely defined operator on $L^2_*(\Omega)$. Its domain $\dm(\overline{\partial}^*)$ is in general very different from $\dm(\overline{\partial})$, because of the natural boundary conditions that the Hilbert space adjoint must satisfy. The map $\overline{\partial}^*$ is of bidegree $(0,-1)$ A form $f\in L^2_*(D)$ is said to be {\em harmonic}, if $\overline{\partial} f= \overline{\partial}^* f=0$. The harmonic forms $\mathcal{H}_*(D)$ form a closed subspace of $L^2_*(D)$. The orthogonal projection $P:L^2_*(D)\rightarrow \mathcal{H}_*(D)$ is called the {\em harmonic projection}, which is of course a map of bidegree $(0,0)$. Note that since $\overline{\partial}^*$ vanishes on $L^2_{0,0}(D)\equiv L^2(D)$, the space $\mathcal{H}_{0,0}(D)$ can be identified with the space $L^2(D)\cap \mathcal{O}(D)$ of square integrable holomorphic functions, the {\em Bergman space} associated to $D$. The operator $P_{0,0}$ is the {\em Bergman projection} onto square integrable holomorphic functions. \subsection{The closed range property and its consequences}\label{sec-canonicalsolution} Let $g\in L^2_*(D)$ be such that $\overline{\partial} g=0$. In order to solve the equation $\overline{\partial} u=g$ in the $L^2$ sense first we need to show that the $L^2$ $\overline{\partial}$-operator has closed range. In general, the closed-range property is not easy to establish, even with a smooth boundary. Subtle holomorphically invariant convexity properties of the boundary of $D$ control whether $\overline{\partial}$ has closed range on $D$ (see the example on p. 76 of \cite{fk}.) Note that, in contrast, for the $L^2$ $d$-complex on a Riemannian manifold the operator $d$ always has closed range when the boundary is $\mathcal{C}^2$ or even Lipschitz \cite{mcs1, mmt}. An important consequence of the closed range property on $D$ is the existence of the {\em Canonical-} or {\em Kohn's solution operator} $K$, which is a bounded map from $L^2_*(D)$ to itself of bidegree $(0,-1)$, and is a right-inverse of the operator $\overline{\partial}$. For every $f\in \im(\overline{\partial})$, we define $Kf$ to be the unique solution to $\overline{\partial} u=f$ which is orthogonal to $\ker(\overline{\partial})$. We then extend $K$ to all of $L^2_*(D)$ by setting $K\equiv 0$ on $(\im (\overline{\partial}))^\perp$ and extending linearly. The map $K$ is bounded by the closed graph theorem, and is represented in terms of the $\overline{\partial}$-Neumann operator $\mathsf{N}$ on $D$ as $K=\overline{\partial}^*\mathsf{N}$. We further have the following: \begin{lem}\label{lem-homotopy} If $\overline{\partial}$ has closed range, and $K$ is the canonical solution operator, then on $\dm(\overline{\partial})$ we have \begin{equation}\label{eq-homotopy} I-P=\overline{\partial} K+K\overline{\partial}.\end{equation} Further, the ranges of the three operators $\overline{\partial} K$, $K\overline{\partial}$ and $P$ are orthogonal. \end{lem} \begin{proof} Since $\im(\overline{\partial})$ is closed in $L^2_*(D)$, we have the Strong Hodge decomposition: \[ L^2_*(D)=(\im (\overline{\partial}^*))\oplus (\im (\overline{\partial}))\oplus\mathcal{H}_*(D),\] where $\mathcal{H}_*(D)$ is the Hilbert space of harmonic forms, and $\oplus$ means that the summands are orthogonal (see \cite{cs}). Let $Q$ and $R$ be the orthogonal projections from $L^2_*(D)$ onto the closed subspaces $\im(\overline{\partial})$ and $\im(\overline{\partial}^*)$ respectively. Then on $L^2_*(D)$, we have $I=P+Q+R$. From the definition of $K$, we have $\overline{\partial} K= Q$. Also, on $\dm(\overline{\partial})$ we have $K\overline{\partial}=R$ by noting that the left hand side is the identity on $\left(\ker(\overline{\partial})\right)^\perp=\im(\overline{\partial}^*)$ and zero on the orthogonal complement $\ker \overline{\partial}$. Equation~\eqref{eq-homotopy} now follows. The last statement follows from the method of proof. \end{proof} For any domain $D$, the {\em $L^2$ Dolbeault Cohomology} space is the graded vector space \[ {H}^*_{L^2}(D) = \frac{\ker(\overline{\partial})}{\im(\overline{\partial})}.\] Note that in the quotient topology, this is a Hilbert space if $\im(\overline{\partial})$ is closed (and not even Hausdorff if $\im(\overline{\partial})$ is not closed, see \cite[Chapter~1, $\S$2.3]{scha}.) If $\im(\overline{\partial})$ is closed, we have \begin{align*} H^*_{L^2}(D)&\cong(\ker(\overline{\partial}))\cap \left(\im(\overline{\partial})\right)^\perp\\ &= (\ker(\overline{\partial}))\cap(\ker(\overline{\partial}^*))\\ &=\mathcal{H}_*(D), \end{align*} so that the cohomology space is naturally isomorphic to the space of harmonic forms. We will now recall the less well-known converse to this statement, due to Kodaira (see \cite[p. 165]{dRh}. Let \[ [.]: \ker(\overline{\partial})\rightarrow H^*_{L^2}(D)\] denote the natural projection onto the quotient space. We have the following: \begin{lem}\label{lem-j} Let $\eta$ be the linear map from the vector space of harmonic forms $\mathcal{H}_*(D)$ to the cohomology vector space $H^*_{L^2}(D)$ given by $ \eta(f)= [f]$. Then \begin{itemize} \item[(i)] $\eta$ is injective. \item[(ii)] If $\eta$ is also surjective, then the range of $\overline{\partial}$ is closed. \end{itemize} \end{lem} \begin{proof}(i) For $(0,0)$-forms, i.e. functions, the space $\mathcal{H}_{0,0}(D)$ coincides by definition with the cohomology space $H_{L^2}^{0,0}(D)$. For forms of higher degree, a harmonic form in $\ker (\eta)$ is of the form $\overline{\partial} g$ with $\overline{\partial}^*(\overline{\partial} g)=0$ so that \begin{align*} 0&=(\overline{\partial}^*(\overline{\partial} g),g)\\ &=\norm{{\overline{\partial}} g}^2. \end{align*} (ii) Since $\eta$ is an isomorphism, we can identify $\mathcal{H}_*(D)$ with the cohomology space $H^*_{L^2}(D)$. Since $\mathcal{H}_*(D)$ is a closed subspace of the Hilbert Space $L^2_*(D)$, the space $H^*_{L^2}(D)$ becomes a Hilbert space in the natural way. Then the map $[\cdot]$ can be thought of as an operator from the Hilbert space $\ker(\overline{\partial})\subset L^2_*(D)$ to the Hilbert space $H^*_{L^2}(\Omega)$. Since $\eta$ is surjective, every element of $\ker(\overline{\partial})$ can be written as $f+{\overline{\partial}} g$, where $f\in \mathcal{H}^*(D)$. Then $[(f+\overline{\partial} g)]=f$, using the identification of $\mathcal{H}_*(D)$ and $H^*_{L^2}(D)$. Since $\norm{f+{\overline{\partial}} g}^2= \norm{f}^2+\norm{{\overline{\partial}} g}^2\geq \norm{f}^2$, so that $\norm{[f+\overline{\partial} g]}\leq \norm{f+\overline{\partial} g}$, it follows that $[\cdot]$ is actually a bounded map. Therefore, $\ker[\cdot]= \im(\overline{\partial})$ is closed. \end{proof} \section{Differential forms on product domains}\label{sec-diffforms} \subsection{Algebraic tensor product of spaces of forms} Let $\mathsf{H}_1$ and $\mathsf{H}_2$ be ${\mathbb{C}}$-vector spaces. We denote by $\mathsf{H}_1\otimes \mathsf{H}_2$ the {\em algebraic} tensor product (over ${\mathbb{C}}$) of $\mathsf{H}_1$ and $\mathsf{H}_2$ : then $\mathsf{H}_1\otimes \mathsf{H}_2$ can be thought of as the space of finite sums of elements of the type $x\otimes y$, where $x\in \mathsf{H}_1$ and $y\in \mathsf{H}_2$, where $\otimes: \mathsf{H}_1\times \mathsf{H}_2\rightarrow \mathsf{H}_1\otimes \mathsf{H}_2$ is the canonical bilinear map (see e.g. \cite[$\S$3.4]{weid} for the purely algebraic definition.) Similarly $\mathsf{H}=\mathsf{H}_1\otimes \mathsf{H}_2\otimes\dots\otimes \mathsf{H}_N$ denotes the algebraic tensor product of $N$ vector spaces $\mathsf{H}_1,\dots \mathsf{H}_N$. We call an element of $\mathsf{H}$ of the form $x_1\otimes\dots\otimes x_N$ a {\em simple tensor.} When $\mathsf{H}_1,\dots, \mathsf{H}_N$ are realized as spaces of forms on domains (or manifolds) $\Omega_1,\dots, \Omega_N$, there is a concrete realization of the algebraic tensor product $\mathsf{H}$ as a space of forms on the product domain $\Omega=\Omega_1\times\dots\times\Omega_N$. For $j=1,\dots, N$, let $f_j\in \mathsf{H}_j$, so that $f_j$ is a form on the domain $\Omega_j$ and let $\pi_j:\Omega\rightarrow \Omega_j$ denote the projection onto the $j$-th factor $\Omega_j$ from the product $\Omega$. We define a form on $\Omega$, the {\em tensor product} of the forms $f_1,\dots, f_N$, by setting \begin{equation}\label{eq-cross} f_1\otimes\dots\otimes f_N = \pi_1^*f_1\wedge\dots\wedge \pi_N^* f_N, \end{equation} which we will call a {\em simple decomposable} form. Then $\mathsf{H}_1\otimes\dots\otimes\mathsf{H}_N$ is the linear span of the simple decomposable forms. It is easy to verify that this construction gives rise to a vector space isomorphic to the usual algebraic definition of a tensor product by the universal property. \subsection{Hilbert tensor products}\label{sec-hilten} We now specialize to the case where the factors $\mathsf{H}_j$ are Hilbert spaces. For ease of exposition, we assume that $N=2$, and the general case should be obvious. We can define an inner product on the algebraic tensor product $\mathsf{H}_1\otimes \mathsf{H}_2$ defined above by setting \[ (x\otimes y, z\otimes w)= (x, z)_{\mathsf{H}_1}(y, w)_{\mathsf{H}_2},\] and extending bilinearly. This is well-defined thanks to the bilinearity of $\otimes$. This makes $\mathsf{H}_1\otimes \mathsf{H}_2$ into a pre-Hilbert space, and its completion is a Hilbert space denoted by $\mathsf{H}_1{\widehat{\otimes}} \mathsf{H}_2$, the {\em Hilbert tensor product} of the spaces $\mathsf{H}_1$ and $\mathsf{H}_2$. The algebraic tensor product $\mathsf{H}_1\otimes \mathsf{H}_2$ sits inside $\mathsf{H}_1{\widehat{\otimes}} \mathsf{H}_2$ as a dense subspace. We will refer to any element of $\mathsf{H}_1\otimes \mathsf{H}_2$ (thought of as a subspace of $\mathsf{H}_1{\widehat{\otimes}} \mathsf{H}_2$) as a {\em decomposable form.} For further details on Hilbert tensor products, see \cite[$\S$3.4]{weid}, or for a more intrinsic approach \cite[$\S$2.6, vol. 1]{kr}. Now let $\mathcal{F}(\Omega_1)$ and $\mathcal{G}(\Omega_2)$ be Hilbert Spaces of functions on $\Omega_1$ and $\Omega_2$ respectively, and let $\mathcal{F}_*(\Omega_1)$ and $\mathcal{G}_*(\Omega_2)$ be the Hilbert Spaces of forms with coefficients in $\mathcal{F}(\Omega_1)$ and $\mathcal{G}(\Omega_2)$ respectively, with the norm given by \eqref{eq-formhilbert}. It is easily verified from the definitions that there is an isometric equality of Hilbert spaces: \begin{equation}\label{eq-formhilbten} \mathcal{F}_*(\Omega_1){\widehat{\otimes}}\mathcal{G}_*(\Omega_2) =(\mathcal{F}{\widehat{\otimes}}\mathcal{G})_*(\Omega_1\times \Omega_2),\end{equation} with an obvious extension to the case of more than two factor domains. \subsection{Forms with square integrable coefficients} We now recall the most important case of the above constructions. Another example will be considered in $\S$\ref{sec-partialsobolev}. Recall the following classical fact, which we will use repeatedly: \begin{lem}[{\cite[p.~369]{hor}}]\label{lem-cartan} {\rm Let $\Omega_1, \Omega_2$ be domains in Euclidean spaces (or manifolds), and let $\Omega=\Omega_1\times \Omega_2$. Then every function in $\mathcal{C}^\infty_0(\Omega)$ can be approximated in the $\mathcal{C}^k$ norm (where $0\leq k\leq \infty$) by functions in the algebraic tensor product $\mathcal{C}^\infty_0(\Omega_1)\otimes \mathcal{C}^\infty_0(\Omega_2)$.} \end{lem} Now, $\mathcal{C}^\infty_0(\Omega)$ is dense in $L^2(\Omega)$, and the decomposable compactly supported smooth functions $\mathcal{C}^\infty_0(\Omega_1)\otimes \mathcal{C}^\infty_0(\Omega_2)$ are dense in the uniform norm (and therefore in the $L^2$ norm) in the space $\mathcal{C}^\infty_0(\Omega)$. It follows that: \[ L^2(\Omega_1){\widehat{\otimes}} L^2(\Omega_2)= L^2(\Omega_1\times \Omega_2).\] Combining with \eqref{eq-formhilbten} we have: \begin{equation}\label{eq-l2tensor} L^2_*(\Omega_1){\widehat{\otimes}} L^2_*(\Omega_2)= L^2_*(\Omega_1\times\Omega_2). \end{equation} \subsection{Tensor products of operators}\label{sec-optensor} Again, for clarity we confine ourselves to the case $N=2$. Let $\mathsf{H}_1,\mathsf{H}_2, \mathsf{H}_1',\mathsf{H}_2'$ be Hilbert Spaces. Given bounded linear operators $T_1: \mathsf{H}_1\rightarrow \mathsf{H}_1'$ and $T_2: \mathsf{H}_2\rightarrow \mathsf{H}_2'$, we can define an algebraic tensor product $T_1\otimes T_2$ which maps the algebraic tensor product $\mathsf{H}_1\otimes \mathsf{H}_2$ into $\mathsf{H}_1'\otimes \mathsf{H}_2'$ : on decomposable tensors it is given by $(T_1\otimes T_2)(x\otimes y)= T_1x \otimes T_2 y$ and extended linearly. Then $T_1\otimes T_2$ is bounded on the dense subspace $\mathsf{H}_1\otimes \mathsf{H}_2$ and therefore extends to a bounded linear operator $T_1{\widehat{\otimes}} T_2$ from $\mathsf{H}_1{\widehat{\otimes}} \mathsf{H}_2$ to $\mathsf{H}_1'{\widehat{\otimes}} \mathsf{H}_2'$. This construction can be extended to densely defined unbounded linear operators, provided they are {\em closed.} (see \cite[$\S$11.2, vol. 2]{kr}.) Given closed (or even closable) operators $T_1: \dm(T_1)\rightarrow \mathsf{H}_1'$ and $T_1: \dm(T_2)\rightarrow \mathsf{H}_2'$, where $\dm(T_1)$ and $\dm(T_2)$ are dense subspaces of the Hilbert spaces $\mathsf{H}_1$ and $\mathsf{H}_2$, the algebraic tensor product (which is densely defined on $\mathsf{H}_1{\widehat{\otimes}} \mathsf{H}_2$ with domain $\dm(T_1)\otimes \dm(T_2)$) is closable (see \cite[Proposition~11.2.7 (vol. 2)]{kr}.) Its closure, denoted by $T_1{\widehat{\otimes}} T_2$ is a closed densely defined operator from $\mathsf{H}_1{\widehat{\otimes}} \mathsf{H}_2$ to $\mathsf{H}_1'{\widehat{\otimes}} \mathsf{H}_2'$. Note that this definition agrees with the previous one, when both $T_1$ and $T_2$ are bounded. \section{$\overline{\partial}$ on a product domain in the $L^2$ sense} \label{sec-dbarl2} In this section we construct a solution operator to the $\overline{\partial}$-problem on a product domain in terms of the canonical solution operators on the factor domains, and show that the operator constructed in fact gives the canonical solution on $\ker\overline{\partial}_{p,1}$, the $\overline{\partial}$-closed $(p,1)$-forms. We use the following notation: for $j=1,\dots,N$, let $\Omega_j$ be a bounded domain in Euclidean space ${\mathbb{C}}^{n_j}$. All our arguments and results will have easy generalizations to relatively compact domains in hermitian manifolds, which we leave for the reader. We will assume that the boundary of each domain $\Omega_j$ is Lipschitz, i.e., it can be represented locally in holomorphic coordinates as the graph of a Lipschitz function. For each $j$, we also fix a weight function $\phi_j$ continuous on $\overline{\Omega_j}$. We use the $L^2$ space forms $L^2_*(\Omega_j)$ on the domain $\Omega_j$ with the weight $\phi_j$, i.e., the norm of a function $f$ is given by $\norm{f}^2=\int_{\Omega_j}\abs{f}^2 e^{-\phi_j} dV$, where $dV$ is the volume form induced by the hermitian metric. (If we want spaces without weights, we simply take $\phi_j\equiv 0$.) The product domain $\Omega=\Omega_1\times\cdots\times\Omega_N$ also has Lipschitz boundary. We will consider $L^2_*(\Omega)$ with the weight $\phi=\phi_1+\dots +\phi_N$ (and with the product hermitian metric.) The analog of formula \eqref{eq-l2tensor} holds with this choice of metric and weight: \[ L^2_*(\Omega)= L^2_*(\Omega_1){\widehat{\otimes}}\dots{\widehat{\otimes}} L^2_*(\Omega_N).\] Our fundamental assumption will be the following: {\em For each $j$, the $L^2$ $\overline{\partial}$-operator (with weight $\phi_j$) on $\Omega_j$ has closed range in each degree.} We remark that the closed range property is independent of the weight function $\phi_j$ as long as it is continuous to the boundary since the $L^2$ spaces are the same. We will show that the $\overline{\partial}$ operator on $\Omega$ (with weight $\phi$) also has closed range and deduce a formula for the canonical solution on $\ker(\overline{\partial})$. \subsection{ Construction of solution operator on smooth decomposable forms} \label{sec-kdecomposable} For simplicity of exposition, we from now on consider the case $N=2$, that is we have two domains $\Omega_1$ and $\Omega_2$ and we are trying to solve the $L^2$ $\overline{\partial}$-problem on the product $\Omega=\Omega_1\times\Omega_2$. In this section we write down some algebraic formulas which hold for smooth decomposable forms on $\Omega$. We first note that if $f\in\mathcal{C}^\infty_*(\overline{\Omega_1})$ and $g\in\mathcal{C}^\infty_*(\overline{\Omega_2})$, then we have \begin{equation}\label{eq-leib0} \overline{\partial}(f\otimes g) = \overline{\partial}_1 f\otimes g +\sigma_1 f\otimes \overline{\partial}_2 g, \end{equation} where $\overline{\partial}_1,\overline{\partial}_2,\overline{\partial}$ denote the $\overline{\partial}$ operator on the domains $\Omega_1,\Omega_2,\Omega$ respectively, and $\sigma_1$ is the map on $\mathcal{C}^\infty_*(\overline{\Omega_1})$ which is multiplication by $(-1)^{p+q}$ on $\mathcal{C}^\infty_{p,q}(\overline{\Omega_1})$. Note that if $T$ be any linear map of odd degree on the space $\mathcal{C}^\infty_*(\overline{\Omega_1})$ (i.e. the degrees of $Tf$ and $f$ differ by an odd integer) then we obviously have \begin{equation}\label{eq-sigma} \sigma_1 T= -T\sigma_1. \end{equation} Extending \eqref{eq-leib0} bilinearly to $\mathcal{C}^\infty_*(\overline{\Omega_1})\otimes\mathcal{C}^\infty_*(\overline{\Omega_2})$, we obtain the {\em Leibnitz formula} for smooth decomposable forms: \begin{equation}\label{eq-leibniz} \overline{\partial}=\overline{\partial}_1\otimes I_2+ \sigma_1\otimes \overline{\partial}_2. \end{equation} Let $K_1,K_2$ be the canonical solution operators on $\Omega_1,\Omega_2$ (see $\S$\ref{sec-canonicalsolution}.) We define an operator $S$ from $\mathcal{C}^\infty_*(\overline{\Omega_1})\otimes\mathcal{C}^\infty_*(\overline{\Omega_2})$ into $L^2_*(\Omega_1)\otimes L^2_*(\Omega_2)$ by the formula \begin{equation}\label{eq-kdef} S= K_1\otimes I_2 +\sigma_1 P_1\otimes K_2, \end{equation} where $P_j$ denotes the harmonic projection on the domain $\Omega_j$ (see $\S$\ref{sec-l2defns}.) It will be proved in the next section that $S$ extends to $L^2_*(\Omega)$, and coincides on $(0,1)$-forms with the canonical solution operator on the product $\Omega$. In this section, we take a first step in this direction by proving the following homotopy formula: \begin{lem}\label{lem-khomotopy} On the space of smooth decomposable forms $\mathcal{C}^\infty_*(\overline{\Omega_1})\otimes\mathcal{C}^\infty_*(\overline{\Omega_2})$, we have \begin{equation}\label{eq-khomotopy} \overline{\partial} S+ S\overline{\partial} = I-P_1\otimes P_2, \end{equation} where $I$ is the identity map. \end{lem} \begin{proof} First note that \begin{align*} \overline{\partial} S &= (\overline{\partial}_1\otimes I_2+\sigma_1\otimes\overline{\partial}_2) (K_1\otimes I_2 +\sigma_1 P_1\otimes K_2)\\ &=\overline{\partial}_1 K_1\otimes I_2 + \sigma_1 K_1\otimes \overline{\partial}_2+ P_1\otimes \overline{\partial}_2K_2, \end{align*} where one term vanishes because $\overline{\partial}_1P_1=0$. Similarly, since by the Hodge decomposition, $P_1\overline{\partial}_1=0$, we have, \begin{align*} S\overline{\partial}&=(K_1\otimes I_2 + \sigma_1 P_1\otimes K_2) (\overline{\partial}_1\otimes I_2+\sigma_1\otimes\overline{\partial}_2)\\ &=K_1\overline{\partial}_1\otimes I_2 +K_1\sigma_1\otimes \overline{\partial}_2+ P_1\otimes K_2\overline{\partial}_2\\ &=K_1\overline{\partial}_1\otimes I_2 -\sigma_1K_1\otimes \overline{\partial}_2+ P_1\otimes K_2\overline{\partial}_2, \end{align*} where we have used \eqref{eq-sigma} in the last line along with the fact that $K_1$ has degree $-1$. Combining the two expressions and canceling the middle terms we have \begin{align*} \overline{\partial} S + S\overline{\partial} &= (\overline{\partial}_1K_1+K_1\overline{\partial}_1) \otimes I_2 + P_1\otimes(\overline{\partial}_2 K_2 + K_2 \overline{\partial}_2)\\ &= (I_1-P_1)\otimes I_2 +P_1\otimes (I_2-P_2)\\ &= I_1\otimes I_2- P_1\otimes P_2, \end{align*} where we have used the homotopy formula \eqref{eq-homotopy} in each factor. The result follows. \end{proof} \subsection{Density results: Extension to $\boldmath{\dm(\overline{\partial})}$} In this section we use a density argument to extend the formulas of the last section. We first recall the following: \begin{lem}[{\cite[Lemma~4.3.2, part~(i)]{cs}}] \label{lem-fried}{\rm If $D$ is a Lipschitz domain, then the space $\mathcal{C}^\infty_*(\overline{D})$ of forms with $\mathcal{C}^\infty(\overline{D})$ coefficients is dense in the graph-norm in the domain $\dm(\overline{\partial})$ of the $L^2$ $\overline{\partial}$ operator on $D$.} \end{lem} Since $D$ is Lipschitz, it is locally star-shaped. This is a special case of Friedrichs' Lemma and follows from smoothing by convolution with a mollifier; see Section 1.2 in Chapter I in H\"ormander \cite{Hor1} or Part (i) of proof of the Density Lemma 4.3.2 in \cite{cs}. The following is now easy: \begin{lem}\label{lem-density} $\mathcal{C}^\infty(\overline{\Omega_1})\otimes\mathcal{C}^\infty(\overline{\Omega_2})$ is dense in the domain of $\overline{\partial}$ in the graph norm of the $\overline{\partial}$-operator on $\Omega=\Omega_1\times\Omega_2$. \end{lem} \begin{proof} Given a form $f\in\dm(\overline{\partial})$ on $\Omega$, by the Lemma~\ref{lem-fried}, we can approximate it in the graph norm by a form $\tilde{f}\in\mathcal{C}^\infty_*(\overline{\Omega})$. Note that it easily follows from Lemma~\ref{lem-cartan} that every form in $\mathcal{C}^\infty_*(\overline{\Omega})$ can be approximated in the $\mathcal{C}^k$ norm (where $0\leq k\leq \infty$) by forms in the algebraic tensor product $\mathcal{C}^\infty_*(\overline{\Omega_1})\otimes \mathcal{C}^\infty_*(\overline{\Omega_2})$. Therefore, approximating $\tilde{f}$ by a form in $\mathcal{C}^\infty_*(\overline{\Omega_1})\otimes \mathcal{C}^\infty_*(\overline{\Omega_2})$ in the $\mathcal{C}^1$ norm (which dominates the graph norm) our result follows. \end{proof} We now extend the formulas of the previous section from the space $\mathcal{C}^\infty_*(\overline{\Omega_1})\otimes\mathcal{C}^\infty_*(\overline{\Omega_2})$ of smooth decomposable forms (which is dense in the graph norm of $\overline{\partial}$) to $\dm(\overline{\partial})$. \begin{lem}\label{lem-ext} On the dense subspace $\dm(\overline{\partial})\subset L^2_*(\Omega)$ we have : \begin{equation}\label{eq-leibniz-csor} \overline{\partial}=\overline{\partial}_1{\widehat{\otimes}} I_2+ \sigma_1{\widehat{\otimes}} \overline{\partial}_2. \end{equation} The operator $S$ defined in \eqref{eq-kdef} can be extended to $L^2_*(\Omega)$ by the formula \begin{equation}\label{eq-kdef-csor} S= K_1{\widehat{\otimes}} I_2 +\sigma_1 P_1{\widehat{\otimes}} K_2, \end{equation} and on $\dm(\overline{\partial})$ the following homotopy formula holds: \begin{equation}\label{eq-khomotopy-csor} \overline{\partial} S+ S\overline{\partial} = I-P_1{\widehat{\otimes}} P_2. \end{equation} \end{lem} \begin{proof} All three formulas follow from the corresponding formulas for decomposable forms by taking limits, using Lemma~\ref{lem-density} for \eqref{eq-leibniz-csor} and \eqref{eq-khomotopy-csor}. \end{proof} \subsection{Consequences} Using the homotopy formula \eqref{eq-khomotopy-csor}, we can now prove: \begin{thm} Let $\Omega_1$ and $\Omega_2$ be two bounded domains in complex hermitian manifolds with Lipschitz boundaries. Suppose that the $\overline{\partial}$ operator has closed range in $L^2(\Omega_j)$ for all degrees, where $j=1,2$. Then $\overline{\partial}$ has closed range in $L^2(\Omega)$ for the product domain $\Omega=\Omega_1\times\Omega_2$. \end{thm} \begin{proof} We recall the result established in Lemma~\ref{lem-j}: if the map $\eta(f)=[f]$ is surjective, then $\overline{\partial}$ has closed range. In other words, we need to show that for every cohomology class $\alpha\in H^*_{L^2}(\Omega)$ there is a {\em harmonic form} $h\in \mathcal{H}_*(\Omega)$ such that $\alpha=[h]$. We will actually do better. We will show that there is such a $h$ in the tensor product $\mathcal{H}_*(\Omega_1){\widehat{\otimes}}\mathcal{H}_*(\Omega_2)\subset\mathcal{H}_*(\Omega)$. Note that this will also show that \begin{equation}\label{eq-harmonictensor} \mathcal{H}_*(\Omega_1){\widehat{\otimes}}\mathcal{H}_*(\Omega_2)=\mathcal{H}_*(\Omega). \end{equation} Indeed, let $f\in \ker(\overline{\partial})$ be a form representing the cohomology class $\alpha$, i.e. $\alpha=[f]$. Then, from the homotopy formula \eqref{eq-khomotopy-csor}, we have \[ f- \overline{\partial}(Kf)= (P_1{\widehat{\otimes}} P_2)f.\] Therefore, the form $(P_1{\widehat{\otimes}} P_2)f\in \mathcal{H}_*(\Omega_1){\widehat{\otimes}}\mathcal{H}_*(\Omega_2)$ also represents the same cohomology class $\alpha$, i.e. $[(P_1{\widehat{\otimes}} P_2)f]=\alpha$. Therefore every cohomology class in $H^*_{L^2}(\Omega)$ can be represented by a harmonic form in $\mathcal{H}_*(\Omega)$ (indeed by a harmonic form in the possibly smaller subspace $\mathcal{H}_*(\Omega_1){\widehat{\otimes}}\mathcal{H}_*(\Omega_2)$.) This shows that the map $\eta$ of Lemma~\ref{lem-j} is surjective. The equality \eqref{eq-harmonictensor} now follows from the fact that $\eta$ is injective. \end{proof} We now note a few important consequences of the above result: \begin{cor}(i) The $L^2$ K\"unneth formula holds for the Dolbeault cohomology with $L^2$ coefficients: \begin{equation}\label{eq-kunneth} H^*_{L^2}(\Omega)=H^*_{L^2}(\Omega_1){\widehat{\otimes}} H_{L^2}^*(\Omega_2) \end{equation} (ii) The harmonic projections satisfy $P=P_1{\widehat{\otimes}} P_2$ \end{cor} \begin{proof} Part (i) follows from the natural isomorphisms $H^*_{L^2}(\Omega)\cong\mathcal{H}_*(\Omega)$, $H^*_{L^2}(\Omega_1)\cong\mathcal{H}_*(\Omega_1)$ and $H^*_{L^2}(\Omega_2)\cong\mathcal{H}_*(\Omega_2)$ (note that the range of $\overline{\partial}$ is closed in each case.) Part (ii) follows from comparing the homotopy formulas \eqref{eq-khomotopy-csor} and \eqref{eq-homotopy}, or directly from \eqref{eq-harmonictensor}. \end{proof} We now come to the most significant consequence: \begin{thm}\label{prop-kiscansol} For $0\leq p\leq n$, the restriction of the map $S$ defined in \eqref{eq-kdef-csor} to the $\overline{\partial}$-closed $(p,1)$-forms coincides with the restriction of the canonical solution operator $\overline{\partial}^*\mathsf{N}$ to the same space. \end{thm} \begin{proof} From the Hodge decomposition, we have for $(p,q)$ forms that $\ker(\overline{\partial}_{p,q})= \im(\overline{\partial}_{p,q-1})\oplus \mathcal{H}_{p,q}(\Omega)$. If $q=0$, it follows that \begin{align*} \ker(\overline{\partial}_{p,0})&= \mathcal{H}_{p,0}(\Omega)\\ &= \bigoplus_{j+k=p}\mathcal{H}_{j,0}(\Omega_1){\widehat{\otimes}} \mathcal{H}_{k,0}(\Omega_2), \end{align*} by \eqref{eq-harmonictensor}. We claim that the range of $S_{p,1}$ is orthogonal to the space $\ker(\overline{\partial}_{p,0})$. By the computation above, it is sufficient to show that the range of $S_{p,1}$ is orthogonal to every form of the type $g_1\otimes g_2$, where $g_1$ and $g_2$ are harmonic forms of degrees $(j,0)$ and $(p-j,0)$, where $0\leq j\leq p$. Let $f_1, f_2$ be $L^2$ forms such that $f_1\otimes f_2$ is of bidegree $(p,1)$. Then, \begin{align*} (S(f_1\otimes f_2),g_1\otimes g_2)&= (K_1f_1\otimes f_2+\sigma_1 P_1f_1\otimes K_2f_2,g_1\otimes g_2)\\ &=(K_1f_1,g_1)(f_2,g_2)+(\sigma_1P_1f_1,g_1)(K_2f_2,g_2)\\ &= 0\cdot (f_2,g_2)+(\sigma_1 P_1f_1,g_1)\cdot 0& \\ &=0, \end{align*} where we have used the fact that $K_1,K_2$ being canonical solutions, have ranges orthogonal to $\overline{\partial}$-closed forms. If $f$ is a $\overline{\partial}$-closed $(p,1)$ form orthogonal to the harmonic forms, it follows from formula \eqref{eq-khomotopy-csor} that $\overline{\partial}(Sf)=f$. Since $Sf$ is orthogonal to $\ker(\overline{\partial})$, it follows that $Sf=\overline{\partial}^*\mathsf{N}f$. To complete the proof, we need to show that $S$ vanishes on the space of $(p,1)$ harmonic forms. By formula \eqref{eq-harmonictensor}, it follows that we only need to verify this on a harmonic form of the type $f\otimes g$, where $f,g$ are also harmonic forms. We have, \begin{align*} S(f\otimes g)&=K_1f\otimes g+\sigma_1P_1f\otimes K_2g\\ &= 0\otimes g+f\otimes 0\\ &=0, \end{align*} since $K_1$ and $K_2$ are the canonical solutions on the domains $\Omega_1$ and $\Omega_2$. \end{proof} {\em Remark:} For arbitrary degrees, the operator $S$ is not equal to the canonical solution operator $K=\overline{\partial}^*\mathsf{N}$. In fact, an examination of the proof of Lemma~\ref{lem-homotopy} shows that for the canonical solution $K$ on a domain, the ranges of the operators $\overline{\partial} K$ and $K\overline{\partial}$ are orthogonal. On the other hand, using the computations used in the proof of Lemma~\ref{lem-khomotopy}, we can check that \[ \left(\overline{\partial} S(f\otimes g), S\overline{\partial}(f\otimes g)\right)= - \norm{K_1 f}^2 \norm{\overline{\partial}_2 g}^2,\] so that $S$ is not the canonical solution on the product. Using a simple induction argument, we can extend the results of this section to $N$ factors. Further, as remarked above, all the arguments generalize to relatively compact domains in hermitian manifolds: \begin{thm}\label{thm-all}For $j=1,\dots, N$, let $M_j$ be a hermitian manifold and let $\Omega_j\Subset M_j$ be a Lipschitz domain. Suppose that the $L^2$ $\overline{\partial}$-operator on $\Omega_j$ (with weight $\phi_j$) has closed range for each $1\leq j\leq N$. Then we have the following: \begin{itemize} \item the $\overline{\partial}$-operator (with weight $\sum_{j=1}^N\phi_j$) has closed range on $\Omega$. \item the $L^2$ K\"{u}nneth formula holds: \[ H^*_{L^2}(\Omega)= H^*_{L^2}(\Omega_1){\widehat{\otimes}} \dots\ {\widehat{\otimes}} H^*_{L^2}(\Omega_N)\] \item the harmonic projection on $\Omega$ is given by \begin{equation}\label{eq-bergman} P=P_1{\widehat{\otimes}} \dots {\widehat{\otimes}} P_N.\end{equation} \item a solution operator for $\overline{\partial}$ on $\Omega$ is given by \begin{equation}\label{eq-tnj} S=\sum_{j=0}^{N-1} T_{N,j},\end{equation} with \[ T_{N,j}=\tau_j\mathsf{Q}_j{\widehat{\otimes}} K_{j+1}{\widehat{\otimes}} \mathsf{I}_j,\] where \begin{itemize} \item $\mathsf{Q}_j$ is the harmonic projection on the domain $U_j=\Omega_1\times\dots\times \Omega_j$, (the product of the first $j$ factors), \item $\tau_j$ is the map on $L^2_*(U_j)$ which multiplies forms of degree $d$ by $(-1)^d$, \item $\mathsf{I}_j$ is the identity map on forms on $\Omega_{j+2}\times\dots\times\Omega_{N}$, and \item it is understood that $T_{N,0}=K_1{\widehat{\otimes}} \mathsf{I}_0$ and $T_{N,N-1}=\tau_{N-1}\mathsf{Q}_{N-1}{\widehat{\otimes}} K_N$. \end{itemize} \item let $0\leq p \leq \sum_{j=1}^N\dim_{{\mathbb{C}}} M_j$; on the space of $\overline{\partial}$-closed $(p,1)$ forms on $\Omega$, the solution operator $S$ coincides with the canonical solution operator $\overline{\partial}^*\mathsf{N}$ of the $\overline{\partial}$-equation. \end{itemize} \end{thm} In particular, this proves Theorem~\ref{thm-main}. \section{Partial Sobolev spaces} \label{sec-partialsobolev} \subsection{Definitions} Recall that for a Lipschitz domain $D$ in ${\mathbb{R}}^n$, and an integer $k\geq 0$, the Sobolev space $W^k(D)$ is the Hilbert space obtained by completion of $\mathcal{C}^\infty(\overline{D})$ under the norm given by \[ \norm{f}_{W^k(D)}^2 = \sum_{\length{\alpha}\leq k} \norm{\mathsf{D}^\alpha f}_{L^2(D)}^2,\] where $\alpha=(\alpha_1,\dots,\alpha_n)$ is a multi-index, $\length{\alpha}=\alpha_1+\dots+\alpha_n$ is the length of multi-index, and $\mathsf{D}^\alpha$ is the partial derivative operator of order $\alpha$: \[ \mathsf{D}^\alpha= \frac{\partial^{\length{\alpha}}}{\partial^{\alpha_1} x_1\dots \partial^{\alpha_n}x_n}. \] We will obtain regularity estimates for the canonical solution on product domains in a generalized type of Sobolev space suited to the product structure of the domain. We will call these spaces {\em partial Sobolev spaces}. Such spaces are characterized by the fact that there are some values of the integer $l$ such that the norm controls only {\em some} distinguished partial derivatives of order $l$. For the usual Sobolev space $W^k(D)$, the norm controls either all or no derivatives of order $l$, depending on whether $l\leq k$ or $l>k$. For convenience of exposition, first consider a product domain $D\Subset{\mathbb{R}}^n$ represented as $D=D_1\times D_2$, where $D_1\Subset{\mathbb{R}}^{n_1}$ and $D_2\Subset{\mathbb{R}}^{n_2}$ are Lipschitz domains, with $n=n_1+n_2$. Let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a multi-index with $n$ components. We can write $\alpha=\alpha(1)+\alpha(2)$, where \[ \alpha(1)=(\alpha_1,\dots,\alpha_{n_1},\underbrace{0,\dots,0}_{n_2}),\] and \[ \alpha(2)=(\underbrace{0,\dots,0}_{n_1},\alpha_{n_1+1},\dots,\alpha_n).\] Then $\mathsf{D}^{\alpha(1)}$ acts only on the variables which come from $D_1$ and $\mathsf{D}^{\alpha(2)}$ acts only on the variables that come from $D_2$ in the product $D$, and we have $\mathsf{D}^\alpha=\mathsf{D}^{\alpha(1)}\mathsf{D}^{\alpha(2)}$. The $\widetilde{W}^k$-norm of a function $f\in\mathcal{C}^\infty(\overline{D})$ is defined to be \begin{equation}\label{eq-pssnorm} \norm{f}_{\widetilde{W}^k(D)}= \sum_{\substack{\length{\alpha(1)}\leq k\\\length{\alpha(2)}\leq k}} \norm{\mathsf{D}^\alpha f}^2_{L^2(D)}\end{equation} Note that the $\widetilde{W}^k$-norm dominates the ordinary $W^k$-norm on $D$, and is in turn dominated by the $W^{2k}$-norm. We now define the space $\widetilde{W}^k(D)$ to be the completion of $\mathcal{C}^\infty(\overline{D})$ under the norm \eqref{eq-pssnorm}. It is clear how to extend this definition to more than two factors: if $D=D_1\times\dots\times D_N$, then the $\widetilde{W}^k$-norm on $D$ is defined as \[ \norm{f}_{\widetilde{W}^k}^2(D) = \sum_{\substack{\length{\alpha(j)}\leq k\\1\leq j\leq N}} \norm{\mathsf{D}^\alpha f}^2_{L^2(D)}, \] where $\alpha(j)$ is the part of the multi-index $\alpha$ corresponding to the factor $D_j$, defined in analogy with the case $N=2$ considered above. \subsection{Basic Properties} We now summarize the basic properties of partial Sobolev space $\widetilde{W}^k(D)$, where $D=D_1\times\dots\times D_N$. From the definition, $\widetilde{W}^k(D)$ is a Hilbert space in the $\widetilde{W}^k$-norm. For $k=0$, the space $\widetilde{W}^0(D)$ coincides with $L^2(\Omega)$. In general, for each $k$, we have continuous inclusions: \begin{equation}\label{eq-inclusion} \mathcal{C}^{Nk}(\overline{D})\hookrightarrow W^{Nk}(D)\hookrightarrow \widetilde{W}^k(D)\hookrightarrow W^k(D)\hookrightarrow L^2(D). \end{equation} Since $\bigcap_{k\geq 0} W^{Nk}(D)=\bigcap_{k\geq 0} W^k(D)=\mathcal{C}^\infty(\overline{D})$, it follows that \begin{equation}\label{eq-cinfty} \bigcap_{k\geq 0} \widetilde{W}^k(D)=\mathcal{C}^\infty(\overline{D}).\end{equation} The significance of these spaces is explained by: \begin{lem} For $j=1,\dots,N$, let $\Omega_j\Subset{\mathbb{C}}^{n_j}$ be a Lipschitz domain, and denote the product by $\Omega=\Omega_1\times\dots\times\Omega_N$. Then we have an isometric equality of Hilbert spaces of forms on $\Omega$: \begin{equation}\label{eq-psstensor}\widetilde{W}^k_*(\Omega) = W^k_*(\Omega_1){\widehat{\otimes}}\dots{\widehat{\otimes}} W^k_*(\Omega_N).\end{equation} \end{lem} \begin{proof} For simplicity of exposition, we assume $N=2$. Thanks to the comments in $\S$\ref{sec-hilten}, in particular equation \eqref{eq-formhilbten}, it follows that we only need to show that \[ \widetilde{W}^k(\Omega) = W^k(\Omega_1){\widehat{\otimes}} W^k(\Omega_2).\] Thanks to Lemma~\ref{lem-cartan}, it follows easily by $\mathcal{C}^{2k}$ approximation, that $\mathcal{C}^\infty(\overline{\Omega_1})\otimes\mathcal{C}^\infty(\overline{\Omega_2})$ is dense on each side. Therefore, all it needs to prove isometric equality is to show that the $\widetilde{W}^k$ norm and the tensor product norm coincide on this subspace. A computation shows that we have $\norm{f\otimes g}_{\widetilde{W}^k(\Omega_1\times \Omega_2)}= \norm{f}_{W^k(\Omega_1)}\norm{g}_{ W^k(\Omega_2)}=\norm{f\otimes g}_{W^k(\Omega_1){\widehat{\otimes}} W^k(\Omega_2)}$ \end{proof} \subsection{Partial Sobolev Spaces on Manifolds} When for each $j$, the domain $\Omega_j$ is smoothly bounded in a hermitian manifold $M_j$, we can again define the partial Sobolev space $\widetilde{W}^k(\Omega)$ on the product. The simplest approach is to take \eqref{eq-psstensor} to be the definition and deduce the description in terms of distinguished derivatives from there. Alternatively, one can use a partition of unity to define $\widetilde{W}^k(\Omega)$ subordinate to a covering of $\overline{\Omega}$ by coordinate patches. \section{Regularity Results}\label{sec-regularity} We now prove some results regarding the regularity of the solution of the $\overline{\partial}$-equation on product domains. Our main tool is the operator $S$ defined in $\S$\ref{sec-dbarl2}. \subsection{Proof of Theorem~\ref{thm-reg}} By Theorem~\ref{prop-kiscansol}, the solution operator $S$ on the product $\Omega$ coincides with the canonical solution operator on $\overline{\partial}$-closed $(p,1)$-forms. Therefore, it is sufficient to show that $S$ is bounded from $\widetilde{W}^l_{p,1}(\Omega)$ to itself. In fact, it is easy to see that $S$ is bounded from $\widetilde{W}^l_*(\Omega)$ to itself. The regularity of the $\overline{\partial}$-Neumann operator on $W^k(\Omega_j)$ for each $k\geq 0$ implies that the canonical solution operator as well as the harmonic projection preserves the space of forms with $W^k$ coefficients for each $k$ (see \cite[Theorem~6.2.2 and Theorem~6.1.4]{cs}; note that in this reference (i) the hypothesis of pseudoconvexity is used only to deduce that the $\overline{\partial}$-Neumann operator is bounded in each Sobolev space, and (ii) although the arguments are stated only for domains in ${\mathbb{C}}^n$, they generalize easily to relatively compact domains in complex manifolds; for similar results on the Bergman projection, see \cite{BS2}.) Since $S$ is given by \eqref{eq-tnj}, in the notation of theorem~\ref{thm-all}, we have \begin{align*} T_{N,j}&=\tau_j\mathsf{Q}_j{\widehat{\otimes}} K_{j+1} {\widehat{\otimes}} \mathsf{I}_j\\ &= \tau_j P_1{\widehat{\otimes}}\dots P_j {\widehat{\otimes}} K_{j+1}{\widehat{\otimes}} I_{\Omega_{j+2}}\dots{\widehat{\otimes}} I_{\Omega_N}, \end{align*} where $P_\nu$ is the harmonic projection, $K_\nu$ is the canonical solution operator and $I_{\Omega_\nu}$ is the identity map on $L^2_*(\Omega_\nu)$. Therefore, the $\nu$-th factor in the tensor product representing $T_{N,j}$ is a bounded linear map on $W^k_*(\Omega_\nu)$. It follows (see $\S$\ref{sec-optensor}) that $T_{N,j}$ defines a bounded linear map from the tensor product $W^k_*(\Omega_1){\widehat{\otimes}}\dots{\widehat{\otimes}} W^k_*(\Omega_N)$ to itself, i.e., it is a bounded linear map from $\widetilde{W}^k_*(\Omega)$ to itself. The solution operator $S$ being the sum of the $T_{N,j}$'s is bounded on $\widetilde{W}^k_*(\Omega)$. The proof is complete. We note here that the hypothesis of Theorem~\ref{thm-reg} are not really necessary. All we need to know to conclude that the canonical solution has coefficients in $\widetilde{W}^l(\Omega)$, if the form $f$ has coefficients in $\widetilde{W}^l(\Omega)$ is the following: for each $j$, both the canonical solution and the harmonic projection on each factor $\Omega_j$ preserves the Sobolev space $W^l(\Omega)$. \subsection{Application to products of weakly pseudoconvex domains} We now consider the $\overline{\partial}$-equation on a product of smoothly bounded pseudoconvex domains: \begin{cor}\label{cor-main} For $j=1,...,N$, let $\Omega_j$ be a bounded pseudoconvex domain with smooth boundary in a Euclidean space ${\mathbb{C}}^{n_j}$. For $n=n_1+\dots+n_N$, let $\Omega\subset{\mathbb{C}}^n$ be the product domain $\Omega=\Omega_1\times\dots\times\Omega_N$. Then, for each $k\in \mathbb{N}$, there is an $C_k>0$ such that, if $t>C_k$, and we use the weight $\phi_t(z)=t\abs{z}^2$ on ${\mathbb{C}}^n$, we have \begin{itemize} \item for $1\leq q \leq n$, given a $\overline{\partial}$-closed form $f$ in the partial Sobolev space $\widetilde{W}^k_{0,q}(\Omega)$, the form $u=Sf$ is in $\widetilde{W}^k_{0,q-1}(\Omega)$. The form $u$ satisfies $\overline{\partial}{u}=f$, provided $f$ is orthogonal to the harmonic forms. \item if $q=1$, further we have that $u$ coincides with $ \overline{\partial}^*_t\mathsf{N}_t f$, the canonical solution with weight $t$. \end{itemize} \end{cor} \begin{proof} By the classical solution by Kohn of the weighted $\overline{\partial}$-Neumann problem (see \cite[Theorem~6.1.3]{cs}), for each $j=1,\dots,N$, given an integer $k\geq 0$, there is a $C_k^j>0$, such that if $t>C_k^j$, the $\overline{\partial}$-Neumann operator is bounded on $W^k(\Omega_j)$ provided the weight is taken to be the function $\phi^j_t$ on ${\mathbb{C}}^{n_j}$ given by $\phi^j_t(z)=t\abs{z}^2$. The result now follows using the same method as in Theorem~\ref{thm-reg}, on taking $C_k= \max_{1\leq j\leq N} C_k^j$ and noting that $\sum_{j=1}^N\phi^j_t =\phi_t$. \end{proof} Therefore, it is always possible to solve the $\overline{\partial}$-equation in a product of pseudoconvex domains, with estimates in $\widetilde{W}^k(\Omega)$ using the weight $\phi_t$. Using the inclusions \eqref{eq-inclusion} and standard results on interpolation, it follows that for each $s\geq 0$, and the operator $S$ maps forms with coefficients in $W^s(\Omega)$ to forms with coefficients in $W^{\frac{s}{N}}(\Omega)$. From this, using a standard ``Mittag-Leffler argument" (see \cite[pp. 127ff., {\em Proof of Theorem 6.1.1.}]{cs}), one can deduce the following from Corollary~\ref{cor-main}: \begin{cor}\label{cor-ms4} Under the same assumption as in Corollary~\ref{cor-main}, if $f\in\mathcal{C}^\infty_{p,q}(\overline{\Omega})$, is a $\overline{\partial}$-closed form, with $q\not=0$, then there exists $u\in \mathcal{C}^\infty_{p,q-1}(\overline{\Omega})$ such that $\overline{\partial} u=f$. \end{cor} For domains which are the intersection of a finite number of smoothly bounded pseudoconvex domains, such that the boundaries meet transversely at each point of intersection, the existence of a solution to the $\overline{\partial}$-equation smooth up to the boundary has been obtained before \cite{ms4} using integral kernels. This includes the result of Corollary~\ref{cor-ms4}, but our method here is simpler and also leads to estimates in Sobolev spaces. \subsection{Proof of Corollary~\ref{cor-strong}} Fix $1\leq j\leq N$ and let $0\leq p\leq n_j$. The canonical solution operator on $\Omega_j$ maps the space $\mathcal{C}^\infty_{p,1}(\overline{\Omega})$ of $(p,1)$-forms smooth up to the boundary to the space $\mathcal{C}^\infty_{p,0}(\overline{\Omega})$. Using formula \eqref{eq-homotopy} on $(p,0)$ forms, we see that the harmonic projection $P_j$ preserves the space $\mathcal{C}^\infty_{p,0}(\overline{\Omega_j}).$ By the Sobolev embedding theorem, the Sobolev norms $\norm{\cdot}_{W^k(\Omega_j)}$ form a system of seminorms which define the usual Fr\'{e}chet space structure on $\mathcal{C}^\infty(\overline{\Omega_j})$. Using a Fr\'{e}chet space version of the closed graph theorem (see e.g. \cite[Theorem~3 on p.~301]{hor}), we easily see that the map $K_j$ is continuous from $\mathcal{C}^\infty_{p,1}(\overline{\Omega_j})$ to $\mathcal{C}^\infty_{p,0}(\overline{\Omega_j})$ and $P_j$ is continuous from $\mathcal{C}^\infty_{p,0}(\overline{\Omega_j})$ to itself. Using the characterization of continuous linear maps between Fr\'{e}chet spaces (see \cite[Proposition~2 on p.~97]{hor}), we conclude that for each $l\in\mathbb{N}$, there is an $k=k(l,j,p)$ such that $K_j$ maps the Sobolev space $W^k_{p,1}(\Omega_j)$ continuously to the Sobolev space $W^l_{p,0}(\Omega_j)$ and $P_j$ maps the Sobolev space $W^k_{p,0}(\Omega_j)$ to the Sobolev space $W^l_{p,0}(\Omega_j)$. e can assume that for each $l$, the integer $k_l= k(l,j,p)$ has been chosen to be independent of $j$ and $p$. Also, since $P_j$ is a projection, it follows that $k_l\geq l$. Using the formula \eqref{eq-tnj}, the argument used in the proof of Theorem~\ref{thm-reg} shows that the operator $S$ maps the Partial Sobolev space $\widetilde{W}^{k_l}_{\boldsymbol{p},1}(\Omega)$ to $\widetilde{W}^l_{\boldsymbol{p},0}(\Omega)$ for each integer $l$. It follows from \eqref{eq-cinfty} that $S$ maps $\mathcal{C}^\infty_{\boldsymbol{p},1}(\overline{\Omega})$ to $\mathcal{C}^\infty_{\boldsymbol{p},0}(\overline{\Omega})$. Using Theorem~\ref{prop-kiscansol} the smoothness up to the boundary of $\overline{\partial}^* \mathsf{N}f$ follows whenever $\overline{\partial} f=0$ and the $(p,1)$-form $f$ is smooth up to the boundary. The statement regarding the Bergman projection now follows from the formula $B=I-\overline{\partial}^*\mathsf{N}\overline{\partial}= I- K\overline{\partial}$. \subsection{Some special product domains}\label{sec-chencao} We will apply our results to some special cases when the domain is not pseudoconvex or Stein. The first case is the product of an annulus between two pseudoconvex domain and a pseudoconvex domain. \begin{cor} Let $\Omega_1=D_2\setminus\overline D_1$ be the annulus between two pseudoconvex domains $D_1\subset \subset D_2\Subset {\mathbb{C}}^n$ with smooth boundary and let $\Omega_2$ be a bounded pseudoconvex domain in ${\mathbb{C}}^m$ with Lipschiz boundary. Let $\Omega$ be the product domain $\Omega=\Omega_1\times \Omega_2$. Then the $\overline{\partial}$ operator on $L^2(\Omega)$ has closed range. Furthermore, for $0\leq p\leq n+m$, we have \[ \dim H^{p,q}_{L^2}(\Omega)=\dim\mathcal{H}_{p,q}(\Omega)=\begin{cases} \infty, & \text{if $q=0$;}\\ 0, & \text{if $q\not=0$ or $q\not=n-1$;}\\ \infty & \text{if $q=n-1$.} \end{cases} \] \end{cor} This corollary follows easily from the fact that $\overline{\partial}$ has closed range on any bounded pseudoconvex domain in ${\mathbb{C}}^n$ by H\"ormander \cite{Hor1} (regardless of the smoothness of the boundary) and for the annulus between smooth pseudoconvex domains (see \cite{mcs1,mcs3}) for all degrees. It also follows from the H\"ormander's $L^2$ existence theorems, the harmonic space $\mathcal{H}_{p,q}(\Omega_2)$ on the pseudoconvex domain $\Omega_2$ vanishes unless $p=q=0$, when $\mathcal{H}_{0,0}$ is the space of $L^2$ holomorphic functions. For the annulus, we have that the cohomology $\mathcal{H}_{p,q}(\Omega_1)$ vanishes except for $q=0$ and $q=n-1$. Thus the corollary follows from the theorem above. For the annulus between two concentric balls $\Omega_1=\{z\in{\mathbb{C}}^n\colon 1<\abs{z}<2\}$, the nontrivial harmonic spaces $\mathcal H_{(p,n-1)}(\Omega_1)$ have been computed explicitly by H\"ormander (see Theorem 2.2 and equation (2.3) in \cite{Hor2}). We can apply the corollary to the case when $\Omega=\{z\in{\mathbb{C}}^n\colon 1<\abs{z}<2\}\times \{z\in{\mathbb{C}}^m\colon \abs{z}<1\}$ in ${\mathbb{C}}^{n+m}$. In this case, the closed range property for the ball follows from the work of Kohn \cite{Ko1}. For the annulus between two balls, the closure of the range of $\overline{\partial}$ in degree $(p,q)$ follows from \cite[pp. 57 ff.]{fk} for $q\not = n-1$ and from \cite{Hor2} if $q=n-1$. Thus $\overline{\partial}$ has closed range in the product domain $\Omega$. The harmonic space $\mathcal H_{(0,0)}$ on $\Omega$ is spanned by the monomials in ${\mathbb{C}}^{m+n}$. The other nontrivial harmonic spaces $\mathcal H_{(p,n-1)}(\Omega)$ can be expressed explicitly as the Hilbert tensor products of harmonic forms $\mathcal H_{(p,n-1)}(\Omega_1)$ with monomials in ${\mathbb{C}}^m$. We can therefore obtain a complete description of the harmonic forms in terms of Hilbert tensor products of spaces. Moreover, we have the following existence and regularity results for the $\overline{\partial}$-operator. \begin{cor} Let $\Omega=\{z\in{\mathbb{C}}^n\colon 1<\abs{z}<2\}\times \{z\in{\mathbb{C}}^m\colon \abs{z}<1\}=\Omega_1\times\Omega_2\Subset {\mathbb{C}}^{n+m}$, $n\ge 1$ and $m\ge 1$. Then the $\overline{\partial}$-Neumann operator $\mathsf{N}$ exists on $\Omega$. For any $(p,q)$-form $f$ with $\widetilde{W}^k(\Omega)$ (or $C^\infty(\overline\Omega)$) coefficients, where $k$ is any nonnegative integer and $1\leq q\leq n+m$, such that $\overline{\partial} f=0$ and $f\perp \mathcal{H}_{p,q}$, there exists a solution $u $ which has $\widetilde{W}^k(\Omega)$ (or $C^\infty(\overline \Omega)$) coefficients with $\overline{\partial} u=f$ in $\Omega$. If $q=1$, we can choose $u=\overline{\partial}^*\mathsf{N}f$ to be the canonical solution. \end{cor} This answers the question posed by X. Chen. Another interesting case is when one of the factors in the product is a compact manifold. In this case, the domain is pseudoconvex in the sense of Levi, but not Stein. Our theorem can also be applied to the following case. \begin{cor} Let $\Omega=\Omega_1\times M$ be the product of a bounded pseudoconvex domain $\Omega_1$ in ${\mathbb{C}}^n$ and let $M$ be a compact complex hermitian manifold. Then the $\overline{\partial}$ operator on $L^2(\Omega)$ has closed range and the Harmonic spaces satisfy the K\"{u}nneth formula \begin{equation} \mathcal{H}_*(\Omega_1)\otimes\mathcal{H}_*(M)=\mathcal{H}_*(\Omega). \end{equation} \end{cor} In this case, the space $ \mathcal{H}_*(M)$ is finite dimensional and the Hilbert Tensor product coincides with the algebraic tensor product. In particular, $\overline{\partial}$ has closed range on the product $\mathbb{D}\times {\mathbb{C}}\mathbb{P}^1$ of the disc and the Riemann sphere, each with its natural metric, thus answering a question raised by J. Cao.

proofpile-arXiv_069-15423

\section{Introduction} In recent years several attempts at controlling turbulence through a number of wall-based forcing methods have been reported \cite{gadelhak-2000}, often aimed at frictional drag reduction (DR). A large number of works, exploiting both numerical and experimental approaches, has been devoted to this goal. In this paper we focus on spanwise wall-based forcing, i.e. on a class of forcing methods designed to modify favorably the turbulent flow by introducing an external action directed in the spanwise direction. Early work addressing the modification of wall turbulence by creating a cross flow can be traced back to Bradshaw \& Pontikos \cite{bradshaw-pontikos-1985}. Spanwise forcing of turbulent flows has been reviewed by Karniadakis \& Choi \cite{karniadakis-choi-2003}. In 1992, Jung, Mangiavacchi \& Akhavan \cite{jung-mangiavacchi-akhavan-1992} introduced the spanwise-oscillating wall technique, where the wall of a fully turbulent channel flow is subject to an alternate harmonic motion in the spanwise direction. If $W$ indicates the spanwise velocity component at the wall, the law that defines this forcing method is: \begin{equation} \label{eq:time} W = A \sin \left( \frac{2\pi}{T} t \right) \end{equation} where $t$ is time, and $A$ and $T$ are the oscillation amplitude and period, respectively. Once $A$ and $T$ are set within the optimum range from the viewpoint of DR performance, the authors observed, by means of direct numerical simulations (DNS), that a strong suppression of turbulence occurs in the wall region, accompanied by a significant reduction of the mean friction. The analysis has been carried on in successive studies: we recall Quadrio \& Sibilla \cite{quadrio-sibilla-2000} for the pipe flow, and J.-I. Choi, Xu \& Sung \cite{choi-xu-sung-2002} and Quadrio \& Ricco \cite{quadrio-ricco-2003,quadrio-ricco-2004}, as DNS-based analyses which contributed new datasets and detailed descriptions of the flow. Laboratory experiments, due to Laadhari {\it et al.} \cite{laadhari-skandaji-morel-1994} and K. S. Choi \cite{choi-graham-1998,choi-2002}, complemented the numerical works and addressed the issue of dependency of DR on the value of the Reynolds number. Alltogether these works have contributed to demonstrating that the natural friction drag of the turbulent flow can be reduced (at least at moderate values of $Re$) up to 45\% for $A^+ \approx 25$ (quantities with the $^+$ superscript are made dimensionless with viscous wall units). An optimal value of the oscillation period exists, namely $T_{opt}^+ = 100 - 125$, that yields the maximum DR at all amplitudes. The net energy saving, that substracts from the reduced flow-driving pumping power the power expense required to move the wall against viscous resistance, has been addressed first by Baron and Quadrio \cite{baron-quadrio-1996}, and today it is recognized \cite{karniadakis-choi-2003} that it can reach up to 10\%. Crediting a commonly accepted qualitative explanation, the transverse oscillating boundary layer induced by the wall motion explains the drag reduction, since it produces a phase displacement between the wall-layer turbulence structures, capable of weakening the viscous wall cycle. When phase-averaged, this time-alternating layer in the turbulent regime has been found to coincide with the oscillating laminar Stokes layer, for which an analytic solution is known as a classic solution of the boundary layer equations \cite{schlichting-gersten-2000}. This has been recently exploited to determine a parameter \cite{quadrio-ricco-2004} that is capable of scaling linearly with DR and thus makes it possible to predict DR capabilities. Advantages and drawbacks of the oscillating-wall technique are obvious. It presents energetic performance that could make it worth of practical implementations, and thanks to its open-loop character it does not need distributed sensors or actuators, that would be still unpractical with the technology available today. On the other hand, by its very nature this technique requires moving parts, and thus does not lend itself to be implemented as a passive device, which is on the other hand the most appealing possibility application-wise. In this paper, we aim at extending the oscillating-wall technique, to take one further step towards the long-term objective of a successful practical implementation; in particular we want to translate the time-dependent forcing law expressed by Eq.(\ref{eq:time}) into a stationary formula. This goal can be achieved by exploiting the convective nature of wall-bounded flows. Though at the wall the mean velocity profile annihilates, it is well known that the convection velocity of a turbulent wall flow (or, more precisely, the convection velocity of turbulent fluctuations) resembles the mean velocity profile only in the bulk of the flow. Kim \& Hussain \cite{kim-hussain-1993} have shown some years ago that near the wall, say below $y^+ = 15$, the convection velocity becomes essentially independent upon the wall distance, and remains constant at the value $\mathcal{U}_w^+ \approx 10$. It is this near-wall value of the convection velocity that will enable us to translate the temporal forcing from Eq.(\ref{eq:time}) into a spatially-oscillating forcing, yielding the following forcing law that is expected to modify the turbulent flow in a similar manner to the oscillating wall: \begin{equation} \label{eq:space} W = A \sin \left( \frac{2\pi}{\lambda_x} x \right). \end{equation} \begin{figure} \centering \includegraphics[width=.7\columnwidth]{figure01.eps} \caption{Sketch of the wall spanwise forcing discussed in the present work.} \label{fig:sketch} \end{figure} Here $x$ denotes the streamwise coordinate, and $\lambda_x$ is the forcing wavelength. The resulting distribution of spanwise wall velocity is sketched in fig.\ref{fig:sketch}, where the coordinate system employed in this paper is also indicated. Though the control law (\ref{eq:space}) has never been considered in the literature, one paper where a similar space-time extension has been discussed in the past is the one by Berger and coworkers \cite{berger-etal-2000}. In a parametric DNS-based study a spanwise-oriented Lorentz volume force was simulated to obtain the following two forcing configurations: \begin{equation} \label{eq:berger-time} F_z = B \mathrm{e}^{-y/\Delta} \sin \left( \frac{2\pi}{T} t \right), \end{equation} \begin{equation} \label{eq:berger-space} F_z = B \mathrm{e}^{-y/\Delta}\sin \left( \frac{2\pi}{\lambda_x} x \right), \end{equation} where $\Delta$ is the penetration depth of the forcing, and $B$ its intensity. For certain values of the parameters, the streamwise-dependent law (\ref{eq:berger-space}) produced a DR comparable to that of the time-dependent law (\ref{eq:berger-time}), with the added benefit of an improved energetic balance. However, the difference between a body force and a wall-based forcing is substantial, as discussed for example by Zhao et al. \cite{zhao-wu-luo-2004}, and no definite conclusion can thus be drawn {\em a priori} from the study by Berger et al. \cite{berger-etal-2000} with respect to the oscillating velocity, which is strictly a wall-based forcing. Indeed, it will be shown later that some of the results described in Ref. \cite{berger-etal-2000} (for example the existence of an optimal wavelength that depends on the forcing intensity) do not apply at all to the type of forcing considered here. Schoppa \& Hussain \cite{schoppa-hussain-1998} showed that a significant amount of drag reduction can be achieved by introducing in a channel flow a spanwise velocity gradient, generated in the DNS by a large-scale, streamwise-aligned and $x$-independent rolls. Such velocity gradient weakens the near-wall cycle by suppressing the transient growth of streaks that would otherwise be stable according to normal-mode analysis. This mechanism is then addressed in a subsequent paper \cite{schoppa-hussain-2002}, where the role of streamwise-varying spanwise perturbations is highlighted in the context of the turbulence regeneration cycle. The dominant wavelength of the streak waviness is found to be very similar to the streamwise length of the coherent structures educed from conditional analysis of turbulent flow fields \cite{jeong-etal-1997}, namely about 300--400 viscous units. Aim of the present paper is to investigate the DR and energetic performance of the steady control law (\ref{eq:space}). Thanks to an accurate data set, purposely obtained by several DNS simulations, the $\lambda_x-A$ parameter space will be explored in detail. The value of the Reynolds number will be fixed at $Re_\tau=200$ (based on the friction velocity of the reference flow and half the channel gap); any dependence on this flow parameter is not discussed here. The effects of the forcing (\ref{eq:space}) on a laminar channel flow will be preliminarly studied. We will see that a laminar solution, although approximated, can be obtained in analytical form, and this solution will then be used to help understanding the DR properties of the forcing (\ref{eq:space}) when used on a turbulent flow. In the end, we will be able to connect this forcing to a physical application that does not necessarily involve a moving wall. The paper is organized as follows. Sec.\ref{sec:laminar} contains a theoretical discussion of the associated laminar flow, that is useful for predicting some of the energetic characteristics of the present technique (further analytical details are deferred to an Appendix). In Sec.\ref{sec:num} the numerical simulations of the turbulent case and their discretization parameters are described. Results about DR and energetic performances are reported and discussed in Sec.\ref{sec:results}, together with a visual and statistical description of the forced flow, compared to the reference unperturbed one. Some conclusive remarks, including examples of how the present forcing can be implemented in practice, are given in the concluding Sec.\ref{sec:conclusions}. \section{Laminar flow} \label{sec:laminar} We consider first the laminar flow in a plane channel subject to either the temporal boundary forcing (\ref{eq:time}) or the spatial boundary forcing (\ref{eq:space}). For the temporal case, it is easy to show that the incompressible momentum equation for the spanwise component $w$ decouples, so that the streamwise flow is described by the classic Poiseuille parabolic solution, and the entire flow consists in this parabolic profile plus a spanwise alternating motion, that is identical to the oscillating transversal boundary layer that develops in a still fluid bounded by a wall subject to harmonic oscillation (i.e. the so-called Stokes second problem, that possesses a classic analytical solution \cite{schlichting-gersten-2000}). This oscillating boundary layer will be called Temporal Stokes Layer (TSL) in the following. We address now the laminar flow subject to the space-varying boundary forcing (\ref{eq:space}), to define and discuss the spatial equivalent of the TSL, that will be called Spatial Stokes Layer (SSL). We leave some analytical details to Appendix \ref{sec:analytical}, but the main ideas are given here as follows. \begin{figure} \includegraphics[width=\columnwidth]{figure02.eps} \caption{Contour of the $w$ field for the laminar TSL and SSL. Top: temporal forcing $w(0,t)=\sin(\omega t)$. Bottom: spatial forcing $w(0,x)=\sin(\kappa x)$. Contours at $(-0.45,0.1,0.45)$, dashed lines indicate negative values.} \label{fig:wcont} \end{figure} In both the TSL and SSL, the field of the spanwise velocity component $w$ is a function of two independent variables, namely $t,y$ for TSL and $x,y$ for the SSL. We start by observing in fig.\ref{fig:wcont} how closely these $w$ fields for TSL and SSL resemble each other. The abscissa in fig.\ref{fig:wcont} is $t/T$ for the TSL, whereas it is changed to $x / \lambda_x$ for the SSL. Both fields are computed with the DNS code, although the analytical expression of the former is available \cite{schlichting-gersten-2000}: \begin{equation} \label{eq:solution-tsl} w(t,y) = C_t \Re \left[ \mathrm{e}^{i \omega t} \mathrm{e}^{-y / \delta_t} \right] , \end{equation} where $C_t$ is a (real) normalization constant, $\omega = 2 \pi / T$ is the angular frequency of the oscillation and $\delta_t$ is the thickness of the TSL, defined as: \begin{equation} \label{eq:deltat} \delta_t = \left( \frac{ \nu }{ \omega } \right)^{1/2}. \end{equation} Figure \ref{fig:wcont}, where the parameters $T$ and $\lambda_x$ are in the range of interest for turbulent DR purposes, illustrates the wall-normal structure of the TSL and the SSL. Within this parameter range, both Stokes layers are thin compared to the channel half-width $h$. The convection velocity being approximately constant over such a small wall-normal extension explains why the two contours in figure \ref{fig:wcont} look very similar. An analytical solution for the SSL can be arrived at under the small thickness approximation. After some analytical efforts, described in the Appendix, the $w(x,y)$ field of the laminar SSL can be shown to obey an expression, similar to Eq.(\ref{eq:solution-tsl}), that contains an Airy function instead of an exponential function: \begin{equation} \label{eq:solution-ssl} w(x,y)= C_x \Re \left[ \mathrm{e}^{i \kappa x} Ai \left( - \frac{i y}{\delta_x} \mathrm{e}^{-i 4/3 \pi} \right) \right] , \end{equation} where $C_x$ is a (real) normalization constant, $\kappa = 2 \pi / \lambda_x$ is the forcing wavenumber, $Ai$ is the Airy function of the first kind \cite{bender-orszag}, and $\delta_x$ is the thickness of the SSL, defined as: \begin{equation} \label{eq:deltax} \delta_x = \left( \frac{ \nu } { u_{y,0} \kappa } \right)^{1/3} . \end{equation} In this expression, $u_{y,0}$ represents the gradient of the streamwise mean velocity profile evaluated at the wall. The $w(x,y)$ laminar field computed by DNS and plotted in fig.\ref{fig:wcont} (bottom) is virtually indistinguishable from the same field as computed from the analytical solution (\ref{eq:solution-ssl}). Comparing the two expressions (\ref{eq:solution-tsl}) and (\ref{eq:solution-ssl}), a qualitative difference between the SSL and the classical TSL can be observed: the former is not decoupled from the longitudinal mean velocity profile $u(y)$, but it depends on $u_{y,0}$ through the thickness $\delta_x$ given by (\ref{eq:deltax}). This remains without consequences in the laminar case, where the streamwise flow is not affected by the wall forcing, whereas it will become important in the turbulent case. \subsection{Comparison with the mean spanwise turbulent flow} \begin{figure} \includegraphics[width=\columnwidth]{figure03.eps} \caption{Comparison between laminar and turbulent mean $w$ fields for temporal and spatial forcing. Left: analytical solution (lines) of the TSL and turbulent $\aver{w}_{xz}$ field (symbols) computed by DNS. Right: analytical solution (lines) of the SSL and turbulent $\aver{w}_{tz}$ field (symbols) computed by DNS. Profiles are taken at different (temporal and spatial) phases during the oscillation. The laminar solutions are linear in $A$, whereas the turbulent profiles are for $A^+=12$. Turbulent temporal case is at $Re_\tau=200$ and $T^+=125$. Turbulent spatial case is at $Re_\tau=200$ and $\lambda_x^+=1250$. To emphasize collapse with proper scaling, the SSL analytical solution is computed for a different wavelength, namely $\lambda_x^+=1900$, yielding a different $\delta_x$, so that the role of $y/\delta_x$ as similarity variable is highlighted.} \label{fig:w-lamturb} \end{figure} It is well documented \cite{quadrio-ricco-2003} for the time-oscillating wall that the phase-averaged $\aver{w}_{xz}(y)$ profile is identical to the laminar solution expressed by Eq.(\ref{eq:solution-tsl}), except for the initial transient where the oscillation is started from rest. (The operator $\aver{\cdot}_{xz}$ indicates averaging along the homogeneous directions $x$ and $z$.) This is well illustrated by fig.\ref{fig:w-lamturb} (left), where the wall-normal distribution of $w$ in the TSL after Eq.(\ref{eq:solution-tsl}) is plotted at various phases during the cycle, and compared with the turbulent $\aver{w}_{xz}$ field. The agreement, which has been related by Ricco \& Quadrio \cite{ricco-quadrio-2008} to the vanishing $y,z$ component of the Reynolds stresses tensor, is, as expected, excellent. The same result is shown in fig.\ref{fig:w-lamturb} (right) to hold true for the SSL case. Now, of course, the analytical solution (\ref{eq:solution-ssl}) has to be compared with the turbulent $\aver{w}_{tz}$ field. The role of $y/\delta_t$ as similarity variable is well known for the TSL. To emphasize that $y/\delta_x$ plays the same role for the SSL, the comparison between laminar and turbulent profiles is carried out for two cases with different $\kappa$ and thus, according to (\ref{eq:deltax}), with different $\delta_x$. The agreement between the two profiles is excellent. This confirms the analogy between TSL and SSL in terms of the connection between the laminar solution and the phase-averaged turbulent solution. This analogy can be used (see \ref{sec:results}) to increase our predictive capabilities when the spatial forcing (\ref{eq:space}) is applied to a turbulent flow. \section{Numerical method for DNS of turbulent flow} \label{sec:num} We turn now to the turbulent case, that will be dealt with by using the Direct Numerical Simulation (DNS) technique. The computer code used to solve the incompressible Navier--Stokes equations is a parallel DNS solver, based on mixed discretization (Fourier expansion in the homogeneous directions, and compact fourth-order accurate explicit compact finite difference schemes in the wall-normal direction), recently developed by Luchini and Quadrio \cite{luchini-quadrio-2006}. The turbulent flow in an indefinite plane channel is considered, with gap half-width given by $h$. One reference simulation without forcing has been carried out at $Re_\tau=200$, where $Re_\tau$ is the friction Reynolds number, defined based on $h$ and the friction velocity $u_\tau$, that will be used as reference velocity throughout the paper unless otherwise noted. The size of the computational domain is $L_x = 6 \pi h$, $L_y= 2 h$ and $L_z = 3 \pi h$. The total averaging time is rather large, to allow for well converged time-averaged results, and amounts to approximately $10^4$ viscous time units. The employed number of modes / points is given by $N_x=320$, $N_y=160$ and $N_z = 320$, that yields a standard spatial resolution. For the reference simulation a value of $C_f=7.94 \cdot 10^{-3}$ (defined based on the bulk velocity) is obtained, in agreement with the correlation $C_f = 0.0336 Re_\tau^{-0.273}$ given by Pope \cite{pope-2000}. Some 40 simulations with Eq.(\ref{eq:space}) used as boundary condition at the channel walls have been then carried out for different values of $\lambda_x$ and $A$. Owing to the periodic boundary conditions employed in the homogeneous directions, an integer number of forcing wavelength $\lambda_x$ must be contained in the domain length $L_x$. The systematic variation of $\lambda_x$ has thus required, mostly for the largest values of $\lambda_x$, slight adjustments of $L_x$. Compared to the reference value of $L_x=6 \pi h$, 3 cases had an actual value of $L_x = 7 \pi h$, and one was at $L_x = 5 \pi h$. Since such changes are of limited entity, the number of streamwise Fourier modes has been left unchanged, with the implied little change in streamwise spatial resolution for these few cases. Properly accounting for the initial transient, where the wall friction decreases from the unperturbed value to the reduced value given by the forcing, is enforced according to the procedure described by Quadrio \& Ricco \cite{quadrio-ricco-2004}, who reported an overall error in determining the friction coefficient below 1\%. We have also verified that the results are not affected by the chosen value of $L_x$. One simulation with $\lambda_x^+ = 1875$ and large DR has been repeated by doubling $L_x$ at $L_x = 12 \pi h$ (and of course doubling the number of streamwise Fourier modes). The value of $C_f$ measured in the simulation with the longer domain differs from the one computed by the simulation with standard domain length by much less than the error, mentioned above, related to the finite averaging time. The simulations were run on a computing system available in dedicated mode at the Universit\`a di Salerno, taking advantage of its large computing throughput to run several cases at a time. The system possesses 64 computing nodes, each of which is equipped with two dual-core AMD Opteron CPUs. The single computational case took about 10 days of wall-clock time when run in parallel by using 8 nodes. Up to 8 cases can be run simultaneously, so that the wall-clock time for the entire study was about 7 weeks. \section{Turbulent flow} \label{sec:results} The effectiveness of the steady spatial forcing (\ref{eq:space}) in reducing the frictional drag is assessed by examining the value of the skin-friction coefficient obtained in several different simulations, in which the amplitude $A$ and the wavelength $\lambda_x$ of the wall forcing are systematically varied. The measured coefficients are then compared to the value of the reference (unforced) flow. The savings in driving power will then be compared to the energetic cost of the wall velocity distribution, in order to asses the net power saving made possible by the spatial forcing. \subsection{Pumping power saved} \label{sec:Psav} Our simulations are performed at a fixed flow rate, so that a decrease of the frictional drag translates into a proportional decrease of the mean streamwise pressure gradient and of the power required to drive the flow. The power to drive the flow against viscous resistance is defined as: \[ \mathcal{P}_{dr} = \frac{U_bL_xL_z}{t_f-t_i} \int_{t_i}^{t_f} \left( \tau_{x,\ell} + \tau_{x,u} \right) \mathrm{d} t, \] where $\tau_{x,\ell}$ and $\tau_{x,u}$ are the space-averaged value of the streamwise component of the wall shear stress, evaluated at the lower and upper wall respectively, and $(t_f - t_i)$ is the time interval over which the time averaged is carried out, after discarding initial transients. The percentage saved power $P_{sav}$ is expressed as percentage of the power $\mathcal{P}_{dr,0}$ required to drive the flow in the reference case, and is defined as follows: \[ P_{sav} = 100 \frac{\mathcal{P}_{dr,0} - \mathcal{P}_{dr}}{\mathcal{P}_{dr,0}}. \] The quantity $P_{sav}$ exactly corresponds to the percentage of friction drag reduction, and is expected to be a function of $A$ and $\lambda_x$. \begin{figure} \centering \includegraphics[width=\columnwidth]{figure04.eps} \caption{Percentage drag reduction $P_{sav}$ as a function of the forcing wavelength. Empty symbols are temporal cases from Quadrio \& Ricco \cite{quadrio-ricco-2004}: for them $\lambda_x^+ = \mathcal{U}_w^+ T^+$ (see text).} \label{fig:Psav-lambda} \end{figure} In fig.\ref{fig:Psav-lambda} $P_{sav}$ is plotted first as a function of $\lambda_x^+$, for different values of the amplitude $A^+$. Available data \cite{quadrio-ricco-2004} obtained with temporal forcing are also plotted, with the oscillation period translated into an oscillation wavelength through the near-wall value of the convection velocity, namely $\mathcal{U}_w^+=10$. The capital information drawn from this plot is the validation of our working hypothesis: the present steady forcing indeed parallels the unsteady oscillating-wall technique when oscillation period and forcing wavelength are related through $\mathcal{U}_w$. In analogy to the oscillating wall, that yields the maximum DR at a well-defined oscillation period $T^+$, namely $T^+_{opt} = 100 - 125$, the steady forcing yields the maximum DR at a well-defined wavelength $\lambda_x^+$, namely $\lambda^+_{x,opt} = 1000 - 1250$. It is striking how well the prediction $\lambda^+_{x,opt}= \mathcal{U}_w^+ T^+_{opt}$ is confirmed by our simulations. DR is observed over a very wide range of wavelenghts, $200 < \lambda_x^+ < 8000$, analogously to the oscillating wall. The effects of the SSL on the turbulent flow are, however, only qualitatively similar to those of the TSL. Quantitative differences do exist, and in particular at a given wavelength the spatial forcing is observed to attain larger values of DR when compared to the temporal forcing at the same amplitude and equivalent period. \begin{figure} \centering \includegraphics[width=\columnwidth]{figure05.eps} \caption{Percentage drag reduction $P_{sav}$ as a function of forcing amplitude. As in fig.\ref{fig:Psav-lambda}, empty symbols are temporal cases from Quadrio \& Ricco \cite{quadrio-ricco-2004}: for comparison recall that $T^+ = \lambda_x^+ / \mathcal{U}_w^+$ (see text).} \label{fig:Psav-A} \end{figure} Fig.\ref{fig:Psav-A} shows $P_{sav}$ as a function of the amplitude $A^+$, for different values of $\lambda_x^+$. The plot contains data from spatial as well as from temporal forcing cases. The dependence of DR on the forcing amplitude is qualitatively very similar for TSL and SSL: DR monotonically grows with $A$, but the increase saturates to an apparently asymptotic behavior. Again, the space-dependent forcing appears to be more effective in reducing turbulent friction: for a given amplitude the spatial forcing yields a $20-30\%$ larger DR than the temporal forcing. A maximum DR of about $52\%$ is observed at $\lambda_x^+=1250$ with a forcing amplitude of $A^+=20$. \subsection{Power expended at the wall} \label{sec:Preq} In addition to $\mathcal{P}_{dr}$, the forced channel flow has an additional power input $\mathcal{P}_{req}$, that is required to enforce the wall motion against the spanwise shear stress. $\mathcal{P}_{req}$ is defined as: \begin{equation} \label{eq:preq} \mathcal{P}_{req} = \frac{L_z}{t_f-t_i} \int_{t_i}^{t_f} \int_0^{L_x} W \left( \tau_{z,\ell} + \tau_{z,u} \right) \mathrm{d} x \mathrm{d} t , \end{equation} where $\tau_z$ is the space-averaged value of the spanwise component of the wall shear stress, and $W$ is the spanwise velocity of the walls. \begin{figure} \centering \includegraphics[width=\columnwidth]{figure06.eps} \caption{Percentage required power $P_{req}$ as a function of the forcing wavelength. As in fig.\ref{fig:Psav-lambda}, empty symbols are temporal cases from Quadrio \& Ricco \cite{quadrio-ricco-2004}. The line-connected asterisks represent $P_{req}$ obtained from the laminar solution with $A^+=12$, from formula (\ref{eq:solution-tsl}) for the TSL and from formula (\ref{eq:solution-ssl}) for the SSL. \label{fig:Preq-lambda}} \end{figure} The required percentage power $P_{req}$ is defined in terms of the friction power $\mathcal{P}_{dr,0}$ of the reference flow as $P_{req} = \mathcal{P}_{req} / \mathcal{P}_{dr,0}$. $P_{req}$ is presented in fig.\ref{fig:Preq-lambda} as a function of the forcing wavelength. Again, spatial as well as temporal forcing data are included. Of course $P_{req}$ assumes negative values, i.e. work has to be done against the fluid viscosity. Comparing the two forcing methods, one can easily appreciate how the spatial forcing presents an energetic cost that is smaller than the cost of temporal forcing, by approximately a factor of 2. The line-connected points in fig.\ref{fig:Preq-lambda} represent the values $P_{req}$ that can be computed from laminar theory. TSL and SSL are considered at $A^+=12$, for which formulas (\ref{eq:solution-tsl}) and (\ref{eq:solution-ssl}) respectively are analytically integrated. The good agreement was expected, since it was already observed (see fig.\ref{fig:w-lamturb}) that laminar and turbulent mean profiles of spanwise velocity are coincident. \begin{figure} \centering \includegraphics[width=\columnwidth]{figure07.eps} \caption{Percentage net power saving $P_{net}$ as a function of the forcing wavelength. \label{fig:Pnet-lambda}} \end{figure} Lastly, a net percentage power saving $P_{net}$ is easily defined by comparing $P_{req}$ and $P_{sav}$, as follows: \[ P_{net}= P_{sav} + P_{req}. \] The net power saving $P_{net}$ is plotted in fig.\ref{fig:Pnet-lambda}. Since it has already been observed how SSL yields higher DR at lower energetic cost, of course here a significant difference in net gain is expected when comparing SSL and TSL. $P_{net}$ shows indeed an interesting maximum value of about $23\%$ at $A^+=6$, while remaining positive over a wide range of wavelengths. Moreover, positive $P_{net}$ are found for rather large amplitudes: at $A^+=12$ a net gain of about 5\% can still be observed, whereas the TSL at the same amplitude presents a net loss of about 30\%. It is worth noting that the search for the maximum of $P_{net}$ cannot be considered exhaustive, and thus the presently observed maximum value of 23\% at $A^+=6$ should be regarded as a starting point for a refined search. \subsection{Flow statistics} \label{sec:flowviz} \begin{figure} \centering \includegraphics[width=\columnwidth]{figure08-small.eps} \caption{Isosurfaces for the quantity $\lambda_2^+=-0.03$. Flow is from left to right. Top: reference flow; bottom: spatial forcing with $\lambda_x^+=1250$ and $A^+=12$.} \label{fig:ucont} \end{figure} Observing the main statistics of the flow and even a few instantaneous snapshots may help understanding how SSL affects turbulence. (Additional statistical quantities are presented elsewhere \cite{quadrio-viotti-luchini-2007}). In fig.\ref{fig:ucont} isosurfaces are visualized for the $\lambda_2$ quantity introduced by Jeong \& Hussain \cite{jeong-hussain-1995} and since then often used to identify turbulent vortical structures. The level is set at $\lambda_2^+=-0.03$. The top plot is for the reference flow, and the bottom plot is for the same flow subject to the effects of the spatial forcing, with $\lambda^+ = 1250$ and $A^+ = 12$. This is one of the cases with the highest drag reduction, i.e. 45\%. The SSL clearly appears to modify the near-wall turbulence and its structures. \begin{figure} \centering \includegraphics[width=\columnwidth]{figure09.eps} \caption{Streamwise mean velocity profile in law-of-the-wall form. Dotted line is the linear velocity profile $\aver{u}^+_{xzt} = y^+$.} \label{fig:uav} \end{figure} Similar considerations can be drawn from single realizations of the flow as well from its statistical description. Fig.\ref{fig:uav} reports the mean velocity profile in the law-of-the-wall form. The modification to the profile for the TSL and SSL cases are analogous when compared to the reference flow, but the effects are larger for the SSL. The drag reduction manifests itself through the thickening of the viscous sublayer, that results in the upward shift of the logarithmic portion of the velocity profile, as previously documented for other DR techniques, for example riblets \cite{choi-1989}. \begin{figure} \centering \includegraphics[width=\columnwidth]{figure10.eps} \caption{Root-mean-square value of the fluctuations for the streamwise velocity component.} \label{fig:urms} \end{figure} Another relevant statistical quantity that is modified by the action of the SSL is the turbulence intensity. Fig.\ref{fig:urms} presents the r.m.s. value of the fluctuations of the streamwise velocity component. To examine the effect of the lower Reynolds number implied by the strong drag reduction in a constant-flow-rate simulation, this plot is still scaled using inner variables, but both the friction velocity of the reference flow and the actual friction velocity of the drag-modified flow are used as velocity scale. The strong reduction of fluctuations is deemphasized with the latter form of scaling, and what remains is the structural effect of the SSL after subtracting the effect due to a smaller $Re_\tau$. The curves however remain significantly different; the main effect is a decrease of the fluctuation intensity, together with a displacement of the position of the maximum intensity further form the wall. Similar effects have been already documented for other DR methods. In particular the same observation has been put forward \cite{quadrio-ricco-2004} for the oscillating-wall. It may be useful to remind the reader that effective DR methods exist, for example the active opposition control \cite{choi-moin-kim-1994}, where $u_{rms}^+$ is unaffected when properly scaled; on the other hand, as discussed for example by Jim\'enez \cite{jimenez-2004}, experimental evidence exists that wall roughness may reduce the near-wall peak of turbulence intensities while increasing drag. \section{Discussion and conclusions} \label{sec:conclusions} This paper has studied a new form of boundary forcing for wall-bounded turbulent flows, that consists in imposing at the wall a steady distribution of spanwise velocity, modulated in the streamwise direction. In this study only sinusoidal modulation has been considered. Main motivation was to find a steady counterpart to the oscillating-wall technique. The link between the two kinds of forcing is the convection velocity of the turbulence fluctuations, that takes a well-defined non-zero value $\mathcal{U}_w$ at the wall and is thus capable of transforming a time scale into a length scale and viceversa. Thanks to a number of direct numerical simulations, the behavior of this new forcing in the parameter space has been determined, and DR up to $52\%$ has been observed for $A^+=20$ and $\lambda_x^+=1250$. For all amplitudes, the forcing wavelength that yields the maximum DR has been found to correspond to the optimal period of the oscillating wall converted in length through $\mathcal{U}_w$, thus confirming the validity of the analogy between temporal and spatial forcing. This analogy has been extended further by studying the laminar case: this was known to be relevant to the oscillating-wall technique, since in the turbulent case the spanwise profile after space-time averaging is identical to the laminar solution. The laminar solution, that can be written in terms of Airy functions, has been determined for the spatial case too, and it has been additionally verified that the turbulent spanwise flow when phase-averaged is identical to the laminar solution. This property can be leveraged to predict the power required for the control. We hope that a predictive quantity, similar to the parameter discussed by Quadrio \& Ricco \cite{quadrio-ricco-2004,ricco-quadrio-2008} and capable of describing the DR effects of the forcing, could be envisaged on the basis of this analytical solution. Together with qualitative analogies, there are quantitative differences between temporal and spatial forcing. The spatial forcing is more efficient in terms of DR, from the point of view of both absolute DR and net power saving. In particular, a net power saving as high as $23\%$ has been computed at $A^+=6$, with one unit of forcing power translating into 3 units of saved power. This is more than 3 times the largest net saving documented for the oscillating wall, and also significantly larger than the benefit obtained with passive devices like riblets, that are reportedly capable of a saving up to 8-10\% in laboratory conditions \cite{bechert-etal-1997}. The present form of spatial forcing is certainly realizable in principle, and an experimental setup is indeed under construction that will help shedding light on effects like the dependence of DR on $Re$. At the same time, we do not consider the present forcing, though steady, directly suitable as yet for practical applications. However, the successful design of a steady control law is one important step towards the realization of a passive drag-reducing device. In this framework, the sensitivity of the turbulence near-wall cycle to a well-defined streamwise lengthscale is a fundamental result, that paves the way to the search for an efficient drag-reducing type of roughness. In this context, we have recently applied for a patent \cite{quadrio-luchini-2008} concerning a design method of roughness distributions yielding improved benefits over conventional riblets. Such distributions may contain riblets with sinusoidally ondulated crests with wavelength $\lambda_{opt}$. A later paper by Peet {\em et al.} \cite{peet-sagaut-charron-2008} has confirmed by numerical simulations the potential success of similar sinusoidal riblets, reporting a 50\% improvement over conventional riblets. Since in \cite{peet-sagaut-charron-2008} the wavelength was in the optimum range, but only a single amplitude was tested, we are confident that even larger benefits can be attained by such a purely passive technique. \section{Acknowledgments} CV has been supported by Italian Ministry of University and Research through the grant PRIN 2005 on {\em Large scale structures in wall turbulence}. We acknowledge interesting discussions with Dr P.Ricco. Part of this work has been presented by CV in June 2007 at the XI European Turbulence Conference, Porto (P). \section{Appendix} \label{sec:analytical} In this Appendix an analytical, approximate solution of the Navier--Stokes equations for the laminar flow between indefinite plane walls is described, where the non-homogeneous boundary condition \begin{equation} \label{eq:bc-z} w(x,0,z,t) = A \cos (\kappa x) \end{equation} is imposed to the spanwise velocity component. The boundary forcing creates a layer of alternate spanwise motion, which develops close to the wall and resembles the temporal Stokes layer. We have referred to it in this paper as the Spatial Stokes Layer (SSL), in comparison to the conventional Temporal Stokes Layer (TSL). An analytical solution will be derived now for the velocity profile of the SSL, under the assumption that the wall-normal length scale characteristic of the layer is small in comparison to the channel width. The solution is steady, and thanks to the spanwise invariance of the differential system, including its boundary conditions, all the $z$ derivatives can be dropped from the momentum equations, which then read: \begin{subequations} \begin{equation} \label{eq:N-S4} u\pd{u}{x}+v\pd{u}{y} = -\frac{1}{\rho}\pd{p}{x} + \nu \left( \ppd{u}{x}+\ppd{u}{y} \right), \end{equation} \begin{equation} \label{eq:N-S5} u\pd{v}{x}+v\pd{v}{y} = -\frac{1}{\rho}\pd{p}{y} + \nu \left( \ppd{v}{x}+\ppd{v}{y} \right), \end{equation} \begin{equation} \label{eq:N-S6} u\pd{w}{x}+v\pd{w}{y} = \nu \left( \ppd{w}{x}+\ppd{w}{y} \right). \end{equation} \end{subequations} This highlights that Eqs.(\ref{eq:N-S4}) and (\ref{eq:N-S5}) decouple from Eq.(\ref{eq:N-S6}) to form an independent two-dimensional problem, unaffected by the inhomogeneous boundary condition (\ref{eq:bc-z}). Its solution is thus the classical laminar Poiseuille solution, that gives a parabolic longitudinal velocity profile, and predicts a wall-normal velocity $v \equiv 0$ everywhere. Thus Eq.(\ref{eq:N-S6}) can be further simplified as: \begin{equation} \label{eq:prob1} u\pd{w}{x}=\nu\ppd{w}{x}+\nu\ppd{w}{y} , \end{equation} In (\ref{eq:prob1}) $u=u(y)$ is the (known) parabolic Poiseuille profile. The PDE (\ref{eq:prob1}) is linear, and thus in the following we will consider $A=1$ without loss of generality. At this point a boundary-layer approximation is introduced. We suppose the SSL to be confined in a thin region close to the wall, and to vanish at a distance definitely smaller than $h$. If $\delta_x$ indicates the characteristic thickness of the SSL, we are requiring that $\delta_x \ll h$. Even before giving a precise definition of $\delta_x$, we have seen in fig.\ref{fig:wcont} that this requirement is satisfied when the flow parameters are set within the range of interest (which is where the spatial forcing achieves a substantial DR in the turbulent regime). For small $\delta_x / h$, $u(y)$ in Eq.(\ref{eq:prob1}) can be replaced with the first term in its Taylor expansion: \[ u(y) \approx u_{y,0} y . \] If $\lambda_x$ is comparable or larger than $h$, the boundary-layer approximation implies also that $\partial^2 w / \partial x^2$ in Eq.(\ref{eq:prob1}) is negligible compared to $\partial^2 w / \partial y^2$. We are thus left with the well-posed problem: \begin{equation} \label{eq:prob2} u_{y,0} y \pd{w}{x}=\nu\ppd{w}{y}, \end{equation} with boundary conditions: \begin{eqnarray} \label{eq:BC2} w(x,0) & = & \cos (\kappa x) \\ \nonumber \lim_{y\rightarrow\infty}w(x,y) & = & 0 \nonumber \end{eqnarray} to be solved in the domain $y \in (0,+\infty)$. Its general solution $w(x,y)$ has the form: \begin{equation} \label{eq:sobst} w(x,y) = \Re \left[ \mathrm{e}^{i \kappa x} F(y) \right] , \end{equation} where the function $F$ is complex valued, $F(y): \mathbf{R} \rightarrow \mathbf{C}$. By substituting the functional form (\ref{eq:sobst}) into Eq.(\ref{eq:prob2}) an ordinary differential equation for the unknown function $F$ is obtained: \begin{equation} \label{eq:prob3} i \nu^{-1} \kappa u_{y,0} y F(y) = \frac{\mathrm{d}^2 F(y)}{\mathrm{d} y^2} . \end{equation} Its boundary conditions follow directly from (\ref{eq:BC2}): \begin{equation} \label{eq:BC3} \Re [F(0)]=1, \quad \Im [F(0)]=0, \quad \lim_{y \rightarrow \infty}F(y) = 0 . \end{equation} To simplify notation, the factors $\nu^{-1} \kappa u_{y,0}$ in the l.h.s. of Eq.(\ref{eq:prob3}) are written in terms of a single parameter $\delta_x$, which has dimensions of a length: \begin{equation} \nu^{-1} \kappa u_{y,0} = \delta_x^{-3} , \end{equation} Eq.(\ref{eq:prob3}) then becomes: \begin{equation} \label{eq:prob4} i \delta_x^{-3} y F = \frac{\mathrm{d}^2 F}{\mathrm{d} y^2}. \end{equation} Introducing the change of variable $y= i \delta_x \tilde{y}$, and redefining the unknown function as $F(i \delta_x \tilde{y}) = \tilde{F}(\tilde{y})$ turns Eq.(\ref{eq:prob4}) into the following Airy equation: \begin{equation} \label{eq:airy} \tilde{y} \tilde{F} (\tilde{y}) = \frac{\mathrm{d}^2 \tilde{F}}{\mathrm{d} \tilde{y}^2} . \end{equation} Infinite solutions of an Airy equation exist for $\tilde{y}$ spanning the whole complex plane, when derivatives are considered in the sense of analytic functions. These solutions are linear combinations of the two special functions $Ai(\tilde{y})$ and $Bi(\tilde{y})$ \begin{equation} \label{eq:combination} \tilde{F}(\tilde{y})= \alpha Ai(\tilde{y}) + \beta Bi(\tilde{y}) , \end{equation} which are called Airy functions of the first and second kind respectively. The general solution (\ref{eq:combination}) has an alternate representation \cite{bender-orszag}, which turns out to be useful in our case: \begin{equation} \tilde{F}(\tilde{y})= \gamma Ai(\omega \tilde{y}) + \theta Ai(\omega^2 \tilde{y}) , \end{equation} where $\omega=\mathrm{e}^{-i2/3\pi}$. It can be shown \cite{bender-orszag} that among $Ai(\tilde{y})$, $Bi(\tilde{y})$, $Ai(\omega\tilde{y})$ and $Ai(\omega^2 \tilde{y})$, the only base function satisfying conditions (\ref{eq:BC3}) (up to a normalization factor) is $Ai(\omega^2 \tilde{y})$. As a consequence, the solution that satisfies the required boundary conditions is: \begin{equation} \tilde{F}(\tilde{y})= \frac{Ai(\omega^2 \tilde{y})}{Ai(0)}. \end{equation} By substituting this solution into (\ref{eq:sobst}), the expression for the unknown function $w(x,y)$ is eventually derived: \begin{equation} w(x,y)= C_x \Re \left[ \mathrm{e}^{i \kappa x} Ai \left( - \frac{i y}{\delta_x} \mathrm{e}^{-i4\pi/3} \right) \right] \end{equation} where $C_x = Ai(0)^{-1}$. \bibliographystyle{unsrt}

proofpile-arXiv_069-15698

\section{INTRODUCTION} Nuclear theory is important in the determination of reaction cross sections and rates for astrophysics in several respects. Firstly, a large number of reactions in astrophysical environments involve highly unstable nuclei which are unaccessible in laboratory measurements. This is especially true for the high temperature plasmas of stellar explosions. Secondly, despite of high plasma temperatures, the relevant particle interaction energy is low by nuclear physics standards. This is a challenge for experimentalists, especially when studying charged-particle reactions at or below the Coulomb barrier. Very small cross sections at low energy may even prevent a measurement and have to be predicted in this case. Thirdly, the nuclei occur in excited states because they are in thermal equilibrium with the stellar plasma. This modifies the reaction cross sections and has important consequences for the determination of stellar reaction rates. Laboratory measurements only account for a fraction of the possible transitions and the actual plasma corrections (stellar enhancement factors, SEF) can only be provided by theory. (Electron screening is also important but will not be discussed here.) Large-scale predictions of reaction cross sections across the chart of nuclei are required to provide the rates needed for reaction networks applied to nucleosynthesis. In astrophysical applications usually different aspects are emphasized than in pure nuclear physics investigations. Reliable predictions of the nuclear properties and optical model ingredients, like optical potentials for particle and $\alpha$ transmission coefficients \cite{kissprl,rausupp,yalcin,Moalpha,silver,raubranch}, nuclear level densities (NLD) \cite{rau97,mocelj}, resonance energies, and $\gamma$-transition strengths \cite{rau08,litvi}, are needed far from stability. Even at stability, there are still considerable uncertainties, especially in the optical potential for charged particles which is not well constrained at low energy. In addition to the well-known difficulty in predicting $\alpha$-potentials \cite{yalcin,mohr,somor,koehl,avri}, it was recently found that also the optical potential for protons may require special attention \cite{kissprl,rausupp,kissold,raujpconf}. \section{STELLAR ENHANCEMENT} The SEFs, given by the ratio of stellar rate to ground state rate $r^*/r_\mathrm{g.s.}$ and being a result of thermal population of nuclear excited states, are mainly important at the comparatively high temperatures of explosive burning ($r-$, $p-$, $rp-$, $\nu p$-processes), although they can play a role in the $s-$process in some cases, when low-lying (a few keV excitation energy) states are present. Because the number of possible transitions to the more strongly bound nucleus is larger, it is generally assumed that stellar effects are less pronounced in the direction of positive $Q-$value and measurements should preferrably be performed in this reaction direction to obtain rates closer to the stellar value. However, recently we showed that the general $Q-$value rule does not apply for a number of cases due to the suppression of low-energy transitions in the exit channel by an additional barrier (Coulomb or centrifugal) \cite{kissprl,rausupp}. effectively reducing the number of contributing transitions. The application of detailed balance implies that the populations of target and final nucleus are in equilibrium. It has to be noted, however, that the validity of detailed balance cannot be taken for granted in all cases. This is well known for nuclei having isomeric states but may also become problematic in nuclei with low NLD or for multi-particle emission channels. \section{$\gamma$-RAY ENERGIES} \begin{figure} \begin{minipage}{14pc} \includegraphics[width=14pc,angle=-90,clip]{fig1a.ps} \end{minipage}\hspace{5.5pc}% \begin{minipage}{14pc} \includegraphics[width=14pc,angle=-90,bb=50 100 554 770,clip=true]{fig1b.ps} \end{minipage} \caption{\label{fig:gamma}The maximally contributing $\gamma$ energies when capturing 60 keV neutrons on isotopes of Sn (left) and Pb (right) are compared to the neutron separation energies $S_\mathrm{n}$ in the compound nuclei \cite{rau08}. The mass number $A$ is the one of the final (compound) nucleus.} \end{figure} In particle capture reactions, $\gamma$ rays with energies in the range $0 < E_\gamma \leq S_\mathrm{particle}+E_\mathrm{particle}$ can be emitted. Due to the fact that the number of accessible levels is increasing exponentially with excitation energy $E_\mathrm{x}$ but the $\gamma$ strength is decreasing with $E_\mathrm{x}$ (because $E_\gamma=S_\mathrm{particle}+E_\mathrm{particle}-E_\mathrm{x}$), only a limited range of $E_\gamma$ will significantly contribute to the cross section, described by a peak around an energy of highest impact $E_\mathrm{sig}$ \cite{rau08}. This energy $E_\mathrm{sig}$ is shown in Fig.\ \ref{fig:gamma} for sequences of Sn and Pb isotopes. It is interesting to note that it remains mostly in the interval of $2-4$ MeV despite large differences in the neutron separation energy $S_\mathrm{n}$. (This is for 60 keV neutrons; $E_\mathrm{sig} \simeq (2-4) + E_\mathrm{n}$ MeV \cite{litvi}.) The exceptions are nuclei with very low NLD, where the transition to the ground state dominates and $E_\mathrm{sig}=S_\mathrm{particle}+E_\mathrm{particle}$. (Note, however, that in this case direct capture may be dominating; see below). For further details see \cite{rau08}. \section{DIRECT CAPTURE} \begin{figure} \begin{minipage}{14pc} \includegraphics[width=14pc,angle=-90,clip]{fig2a.ps} \end{minipage}\hspace{5.5pc}% \begin{minipage}{14pc} \includegraphics[width=14pc,angle=-90,clip]{fig2b.ps} \end{minipage} \caption{\label{fig:dc}Left: Relation between direct neutron capture and (unmodified) compound capture as function of neutron separation energy \cite{raujpg}. Right: Comparison between direct capture with energy-dependent spectroscopic factor, modified Hauser-Feshbach capture, and standard Hauser-Feshbach capture on even Sn isotopes (preliminary results).} \end{figure} Most astrophysically relevant reactions can be described in the Hauser-Feshbach statistical model (HF). Due to the low particle separation energies and/or NLDs encountered in very neutron- (or proton-) rich nuclei, direct reactions will also become important, even at very low energies (in the keV region) \cite{rau97,mocelj,chloup,data,rau00,descrau,rau98,bon07,raujpg}. Figure \ref{fig:dc} (left) compares direct capture (DC) with the regular HF \cite{raujpg}. It is clearly seen that the relative importance of DC increases with decreasing $S_\mathrm{n}$. This is because the number of final levels for $\gamma$ transitions from HF capture becomes lower. The Sn isotopes have the lowest NLD of the isotopes shown here and consequently the highest relative importance of DC. This comparison was made by simply comparing the DC and HF 30 keV Maxwellian Averaged Cross Section (MACS). Regular HF calculations assume a compound formation probability independent of the NLD at the compound formation energy. The availability of compound states and doorway states defines the applicability of the HF model \cite{rau97}. Relying on an average over resonances the HF model is not applicable with a low NLD at compound formation. On average it will then overpredict the resonant cross section (unless single resonances dominate) because it will overestimate the compound formation probability. This can be treated by introducing a modification of the formation cross section which includes the NLD dependence. For the parity dependence, this was discussed in \cite{loens}. A similar modification but also including the full NLD is available as an option in NON-SMOKER$^\mathrm{WEB}$ since version 4.0w \cite{websmoker}. The new code SMARAGD \cite{smaragd,raureview,cyburt} will have an improved version of this as default and thus implicitly account for low NLD at the compound formation energy. Preliminary results with this modification are shown in Fig.\ \ref{fig:dc} (right). The code SMARAGD will also include a global DC treatment using an averaged DC model and energy-dependent spectroscopic factors \cite{bon07,raujpg,raureview}. This average DC approach aims at providing robust predictions despite of considerable differences between microscopic predictions \cite{rau98}. Preliminary results for this DC are shown in Fig.\ \ref{fig:dc} (right), too. The final rate (or cross section) is the sum of the modified HF value and the DC one. Interestingly, for the isotopes shown here (except for $N=92$) this sum is approximated by the unmodified HF result within a factor of 10. This is in accordance with \cite{gordc} (see fig.\ 3 therein). This shows that it seems justified to use unmodified HF rates as crude estimate of the total rates for exotic nuclei.

proofpile-arXiv_069-15720

\section{Invariant and Non-invariant Gaussian random matrix ensembles} Any random matrix ensemble (RME) is determined through the probability distribution function (PDF) $P({\bf M})$ that depends on the matrix entries $M_{nm}$. An important special class of random matrix ensembles is given by the PDF which is invariant under rotation of basis ${\bf M}\rightarrow {\bf T}{\bf M}{\bf T}^{-1}$: \begin{equation} \label{inv} P({\bf M})\propto {\rm exp}\left[ -{\rm Tr}\,V({\bf M})\right], \end{equation} where $V({\bf M})$ is and arbitrary function of ${\bf M}$ analytic at ${\bf M}=0$. The invariance is ensured by the trace ${\rm Tr}$ in front of the matrix function $V({\b M})$. The RME with the PDF of the form Eq.(\ref{inv}) will be referred to as {\it invariant RME}. In general the PDF Eq.(\ref{inv}) corresponds to a non-trivial correlation between fluctuating matrix entries. However, there is one extremely important case when \\ (i) the matrix ${\bf M}$ is Hermitean ${\bf M}\equiv{\bf H}={\bf H}^{\dagger}$ and \\ (ii) $V({\bf H})=a{\bf H}^{2}$.\\ In this case $${\rm exp}\left[-{\rm Tr V({\bf H})}\right]={\rm exp}\left[- a\sum_{n,m} |H_{nm}|^{2}\right]=\prod_{n,m}{\rm exp}\left[ -a\,|H_{nm}|^{2}\right]$$ so that all matrix entries fluctuate independently around zero. This is the celebrated Gaussian random matrix ensemble of Wigner and Dyson (WD). Note that Gaussian random matrix ensembles can be also {\it non-invariant}. The generic non-invariant Gaussian RME is determined by the PDF of the form: \begin{equation} \label{NIE} P({\bf H})\propto {\rm exp}\left[-\sum_{n,m}\frac{|H_{nm}|^{2}}{A_{nm}} \right]. \end{equation} In this case each matrix entry fluctuate independently of the other but with the variance which depends on the indices $n,m$ that label the matrix entry. The simplest Gaussian non-invariant ensemble is the {\it Rosenzweig-Porter ensemble} \cite{RosPort} for which \begin{equation} \label{RP} A_{nm}=\left\{ \begin{matrix}a, & n\neq m\cr \Lambda\,a & n=m\cr \end{matrix}\right. \end{equation} It is remarkable that both the classic WD ensemble and the Rosenzweig-Porter ensemble allow for an exact solution \cite{Mehta, Kunz}. Physically, an invariant random matrix ensemble describes extended (but phase-randomized) states, where the localization effects are negligible. In contrast to that any non-invariant ensemble accounts for a sort of structure of eigenfunctions (e.g. localization) in a given basis which may be not the case in a different rotated basis (remember about the extended states in the tight-binding model which are the linear combinations of states localized at a given site). In particular the problem of localization in a quasi-1 wire can mapped onto the non-invariant {\it banded RME} with the variance matrix equal to: \begin{equation} \label{band} A_{nm}={\rm exp}\left[-|n-m|/B\right] \end{equation} This model can be efficiently mapped onto a nonlinear supersymmetric sigma model and solved by the transfer matrix method \cite{Mirlin-Fyod}. Finally, we mention a {\it critical power-law banded random matrix ensemble} (CPLB-RME) for which the variance matrix is of the Lorenzian form: \begin{equation} \label{crit} A_{nm}=\frac{1}{1+\frac{(n-m)^{2}}{B^{2}}}. \end{equation} This model (not yet solved) possesses a fascinating property of multifractality and is an extremely accurate model for describing the critical states at the Anderson localization transition point in dimensionality $d>2$. \section{Parametrization in terms of eigenvalues and eigenvectors} Consider a Hermitean $N\times N$matrix ${\bf H}={\bf H}^{\dagger}$. The physical meaning is mostly contained in the eigenvalues $E_{n}$ of this matrix and also in the unitary matrix ${\bf U}=({\bf U}^{\dagger})^{-1}$ whose $n$-th column is an $n$-th normalized eigenvector ${\bf \Psi}_{n}\equiv\{\Psi_{n}(r)\}$. Therefore it is sensible to parametrize the matrix ${\bf H}$ in the following way: \begin{equation} \label{param} {\bf H}={\bf U}\,{\bf E}\,{\bf U}^{\dagger}, \end{equation} where ${\bf E}={\rm diag}\{E_{n} \}$. Then instead of $N(N+1)/2$ independent entries of the Hermitean matrix ${\bf H}$ one will deal with $N(N-1)/2$ independent variables of the unitary matrix ${\bf U}$ plus $N$ eigenvalues. For invariant ensembles the PDF is independent of the eigenvector degrees of freedom and is determined only by the eigenvalues. In particular for the classic WD ensemble it reduces to: \begin{equation} \label{meas} P({\bf H})\propto {\rm exp}\left[-a\sum_{n}E_{n}^{2}\right]. \end{equation} However the change of variables involves also computing the Jacobian of the transformation Eq.(\ref{param}). The easiest way of computing it is to compute the form: \begin{equation} \label{form} {\rm Tr}(d{\bf H})^{2}={\bf Tr}\left(2d{\bf f}\,{\bf E}\,d{\bf f}\,{\bf E}-2{\bf E}^{2}\,(d{\bf f})^{2}+(d{\bf E})^{2} \right)=2\sum_{n>m}(E_{n}-E_{m})^{2}\,|df_{nm}|^{2}+\sum_{n}(dE_{n})^{2}, \end{equation} where $$ d{\bf f}={\bf U}^{\dagger}\,d{\bf U}=-d{\bf U}^{\dagger}\,{\bf U}=-d{\bf f}^{\dagger}.$$ The set of $f_{nm}$, ($n>m$), are $N(N-1)/2$ natural "coordinates" related to eigenvectors. Then the Jacobian of the transformation $d{\bf H}\rightarrow d{\bf f}\,d{\bf E}$ is given by \begin{equation} \label{J} J\propto \left\{\begin{matrix}\sqrt{D}, & {\bf H}&is&real\cr D,& {\bf H}& is & complex\end{matrix}\right.\;\;\;\;\;\;\;\;\;\;\;D=\Delta^{2}=\prod_{n>m}(E_{n}-E_{m})^{2}. \end{equation} \section{Joint probability distribution} According to Eqs.(\ref{meas}),(\ref{J}) the entire joint probability distribution function of eigenvalues and eigenvectors for an arbitrary invariant ensemble Eq.(\ref{inv}) takes the form: \begin{equation} \label{JPDF} d{\bf H}P({\bf H})=d{\bf f}d\{E_{n}\}\;{\rm exp}\left[ -\sum_{n=1}^{N} V(E_{n})\right]\, |\Delta|^{\beta}, \end{equation} where $\Delta$ is the Vandermond determinant: \begin{equation} \label{vandermond} \Delta_{N}=\left| \begin{matrix}1&1&...&1\cr E_{1}&E_{2}&...&E_{N}\cr E_{1}^{2}&E_{2}^{2}&...&E_{N}^{2}\cr .& .& ...&.\cr E_{1}^{N}&E_{2}^{N}&...&E_{N}^{N}\cr \end{matrix}\right|=\prod_{n>m} (E_{n}-E_{m}). \end{equation} A remarkable property of this distribution is that it is independent of the eigenvectors and depends only on eigenvalues. Since the Vandermond determinant is vanishing if any two eigenvalues coincide $E_{n}=E_{m}$ (two columns of the determinant are equal) the coincidence of two eigenvalues is statistically improbable. This is the basic property of the random matrix theory which is called {\it level repulsion}. \section{Level repulsion: poor man derivation} In order to understand the physical origin of level repulsion let us consider a situation where occasionally two levels are very close to each other $|E_{1}-E_{2}|\ll \Delta$, where $\Delta$ is the mean level separation. Then it is enough to consider only one block of the random matrix: $$ \left( \begin{matrix} \varepsilon_{1}& V\cr V^{*}& \varepsilon_{2}\cr\end{matrix}\right)$$ The true energy levels of this two-level system are well known: \begin{equation} \label{true} E_{1,2}=\frac{\varepsilon_{1}+\varepsilon_{2}}{2}\pm\frac{1}{2}\,\sqrt{(\varepsilon_{1}-\varepsilon_{2})^{2}+ |V|^{2}}. \end{equation} The two-level correlation function which is the probability density to find a level at a distance $\omega$ from the given one, is given by: \begin{equation} \label{TLCF} R(\omega)=\int d\varepsilon_{1}d\varepsilon_{2}{\cal D}V\, \left[ \delta(\omega-\sqrt{(\varepsilon_{1}-\varepsilon_{2})^{2}+ |V|^{2}})-\delta(\omega-\varepsilon_{1}+\varepsilon_{2})\right]\,P(\varepsilon_{1},\varepsilon_{2}, V), \end{equation} where \begin{equation} \label{DV} {\cal D}V=\left\{\begin{matrix}dV,&\beta=1\cr d\Im V d\Re V, & \beta=2 \cr \end{matrix} \right. \end{equation} Small energy difference $|E_{1}-E_{2}|\ll \Delta$ implies that {\it both} $\varepsilon_{1}-\varepsilon_{2}$ and $|V|$ are small. Then the PDF $P(\varepsilon_{1},\varepsilon_{2}, V)$ can be considered independent of $\varepsilon_{1}-\varepsilon_{2}$ and $|V|$. Thus integrating the $\delta$-functions over $\varepsilon_{1}-\varepsilon_{2}$ we arrive at: \begin{equation} \label{DV1} R(\omega)=\int {\cal D}V\,\left[ \sqrt{1-\frac{|V|^{2}}{\omega^{2}}}-1\right]\,\theta(\omega-|V|). \end{equation} Apparently this integral is convergent and the power counting immediately leads to: \begin{equation} \label{pc} R(\omega)\propto\left\{\begin{matrix}\omega, & \beta=1\cr \omega^{2},&\beta=2\cr \end{matrix} \right. \end{equation} The simple analysis above illustrates two important points. One --physical-- is that the level repulsion is nothing but the {\it avoided level crossing} which is well known in quantum mechanics and which is caused by the $|V|^{2}$ term in the square root in Eq.(\ref{true}). The other one -- formal-- is that the pseudo-gap in $R(\omega)$ near $\omega=0$ is the effect of the phase volume ${\cal D}V$ and the power of $\omega$ depends on the number of independent components of $V$ which is 1 in case $V$ is a real number and 2 if $V=\Re V + i \Im V$ is a complex number. It is known that the algebra of real and complex numbers allows only one further step of generalization. This is the algebra of quaternions: \begin{equation} \label{quat} \tau_{0}=1,\;\;\;\;\tau_{1,2,3}=i\sigma_{1,2,3}, \end{equation} where $\sigma_{1,2,3}$ are $2\times 2$ Pauli matrices. It appears that this generalization makes sense in the context of random matrices too. Namely, one can consider random matrices which entries are {\it real quaternions}, i.e.: \begin{equation} \label{V-quat} V=\sum_{i=0}^{3}\xi_{i}\,\tau_{i}, \end{equation} with real components $\xi_{i}$. This generalization corresponds to $\beta=4$ in Eqs.(\ref{Psi-q},\ref{CSM}). \section{Time-reversal symmetry and the Dyson symmetry classes} It turns out the the parameter $\beta$ is related with the time-reversal symmetry. In order to see this we note that the time-reversal operator ${\cal T}$ should obey a basic property $${\cal T}^{2}=\alpha\,1,\;\;\;\;\;\;\;|\alpha|=1$$ (time reversal applied twice leaves the wave function unchanged). As the time reversal operator should involve the complex conjugation of wave function one may write: \begin{equation} \label{cc} {\cal T}=K\,C, \end{equation} where $C$ is the complex conjugation operator and $K$ is an operator such that $$ K\,C\,K\,C=K\,K^{*}=\alpha\,1. $$ But $K$ must be a unitary operator (as the norm of the wave function must be conserved). That is why $$ \label{uni} K^{*}\,K^{T}=1. $$ From these two conditions one finds: \begin{equation} \label{al} K=\alpha\,K^{T}=\alpha\,(\alpha\,K^{T})^{T}=\alpha^{2}\,K. \end{equation} Thus we conclude that \begin{equation} \label{alph} \alpha^{2}=1\;\;\;\;\;\;\Rightarrow \alpha=\pm1. \end{equation} For spinless particles (or particles with even spin) we have: \begin{equation} \label{even} {\cal T}^{2}=1, \end{equation} and $K$ can always be chosen to be a unity operator $K=1$. However, for particles with half-integer spin \begin{equation} \label{odd} {\cal T}^{2}=-1, \end{equation} and $K$ is not an identity operator. In particular for spin-$\frac{1}{2}$ particles: \begin{equation} \label{K} K=\left(\begin{matrix}0 & 1\cr -1 & 0 \cr \end{matrix} \right) \end{equation} The physical meaning of this operator is very simple: it flips the spinor. The time-reversal symmetry $${\cal T}H{\cal T}^{\dagger}=H$$ in the cases Eq.(\ref{even}) and Eq.(\ref{odd}) implies, respectively \begin{equation} \label{even1} H=H^{*} \end{equation} and \begin{equation} \label{even1} H=-K\,H^{*}\,K. \end{equation} In the first case time reversal symmetry requires the Hamiltonian matrix ${\bf H}$ to be {\it real}, which corresponds to $\beta=1$. In the second case one can do a simple algebra exercise and show that the condition Eq.(\ref{even1}) is fulfilled if the Hamiltonian matrix ${\bf H}$ has entries of the form Eq.(\ref{V-quat}) with {\it real} coefficients $\xi_{i}$. As was already mentioned this case corresponds to $\beta=4$. Thus we see that the case $\beta=1$ (known as the {\it orthogonal ensemble}) corresponds to the particles with an even spin and a Hamiltonian that preserves the time reversal symmetry. The case $\beta=4$ (known as the {\it symplectic ensemble})corresponds to particles with an odd spin and a {\it spin-dependent} Hamiltonian that preserves time reversal symmetry. In particular it applies to a system with the {\it spin-orbit} interaction. The case $\beta=2$ (known as the {\it unitary ensemble})does not assume any definite relationship between $H$ and $H^{*}$, and thus the time-reversal symmetry must be broken (e.g. by magnetic field or magnetic impurities). According to the initial idea of Wigner and Dyson all systems with complex interactions do not possess any symmetry but possibly time-reversal symmetry and thus should be classified according to one of the three symmetry classes discussed above. \section{Level repulsion: classical and quantum analogy} \subsection{Classical plasma with logarithmic interaction} One that for the Gaussian invariant ensemble one can rewrite the JPDF Eq.(\ref{JPDF}) in the following way: \begin{equation} \label{Hamil} J\,P({\bf H})\propto {\rm exp}[-\beta\,{\cal L}],\;\;\;\;\;\;\;\;\;{\cal L}=-\sum_{n>m}\ln|E_{n}-E_{m}|+a'\sum_{n}E_{n}^{2}, \end{equation} where we introduces the Dyson symmetry parameter: \begin{equation} \label{beta} \beta=\left\{\begin{matrix}1, & for & real &symmetric& {\bf H}\cr 2, & for & complex & Hermitean & {\bf H}\cr 4, &for &real-quaternionic & Hermitean &{\bf H}\cr \end{matrix} \right. \end{equation} Note that by a proper choice of energy units the parameter $a'=a\beta^{-1}$ can be set equal to $\frac{1}{2}$ which will be always assumed throughout the lecture notes. Thus there is only one important parameter $\beta$ in the classic WD random matrix theory. Looking at Eq.(\ref{Hamil}) one concludes that the PDF in the $({\bf f},\,{\bf E})$ representation coincides with the partition function of classical particles repelling each other {\it logarithmically}, in a harmonic {\it confinement potential}. The Dyson symmetry parameter $\beta$ plays a role if an inverse temperature. The above derivation which lead to Eq.(\ref{Hamil}) can be repeated for an arbitrary invariant RME. The corresponding energy functional ${\cal L}$ of the logarithmically repelling particles will differ from Eq.(\ref{Hamil}) only by the confinement potential which will be no longer harmonic but rather $\beta^{-1}\,V(E_{n})$. The basic property of the PDF which dependends only on the set of eigenvalues $E_{n}$ (but not the eigenvector variables ${\bf f}$) is retained for all the invariant RME making the corresponding eigenfunction statistics trivial. This is no longer true once the invariance under basis rotation is broken. The latter circumstance is what makes non-invariant ensembles difficult to solve but at the same time having a rich variety of eigenfunction statistics. \subsection{Quantum analogy} Besides the analogy with logarithmically repelling {\it classical} particles at {\it finite temperature} $\beta^{-1}$ living in one dimension (1d) there is also an important analogy with the system of {\it quantum} particles in 1d. To facilitate this analogy let us remind that the Jacobian in Eq.(\ref{J}) can be expressed as the power of the Vandermond determinant Eq.(\ref{vandermond}) \begin{equation} \label{J-vandermond} J\propto |\Delta|^{\beta}. \end{equation} The property of the Vandermond determinant is that \begin{equation} \label{VdM-der} \sum_{n=1}^{N}\frac{\partial^{2}\Delta_{N}}{\partial E_{n}^{2}}=0. \end{equation} Another property is that it changes sign upon any permutation of two $E_{n}$ and $E_{m}$. These two properties imply that $\Delta_{N}$ can be considered as the many-body wave function $\Psi(\{ E_{n}\})$ of the system of $N$ free fermions with the Hamiltonian consisting only of kinetic energy: \begin{equation} \label{ham-ferm} {\cal H}_{K}=-\sum_{n=1}^{N}\frac{\partial^{2}}{\partial E_{n}^{2}} \end{equation} Moreover, as the energy of the corresponding many-body state is minimal possible for kinetic energy ${\cal E}=0$, this is a {\it ground state} of this free fermionic system. So we come to the statement that the Jacobian Eq.(\ref{J}) at $\beta=2$ is the probability density for the ground state of the free fermion system in the entire space. \begin{equation} \label{Psi-q} J\sim |\Psi_{0}(E_{n})|^{2}\propto\prod_{n>m}|E_{n}-E_{m}|^{\beta}. \end{equation} Note that the system of fermions in an infinite space is not well defined, as it expands indefinitely. Formally this is seen from the fact that the wave function Eq.(\ref{Psi-q}) is not normalizable. In order to fix this pathology one has to consider a full probability distribution function Eq.(\ref{JPDF}) which includes also the confinement potential $V(E_{n})$. For the harmonic confinement potential the property Eq.(\ref{VdM-der}) can be generalized in the following way: \begin{equation} \label{VdM-der-gen} \frac{1}{2m}\sum_{n=1}^{N}\frac{\partial^{2}}{\partial E_{n}^{2}}\left[\Delta_{N}\,e^{-\frac{m}{2}\sum_{n=1}^{N}E_{n}^{2}}\right] =\left[-{\cal E}_{N}+\frac{m}{2}\sum_{n=1}^{N}E_{n}^{2}\right]\,\Delta_{N}\, e^{-\frac{m}{2}\sum_{n=1}^{N}E_{n}^{2}}. \end{equation} Now we see that \begin{equation} \label{gen-wf} \Psi_{0}\propto \Delta_{N}\,e^{-\frac{m}{2}\sum_{n=1}^{N}E_{n}^{2}}, \end{equation} is an eigenfunction of the free fermions with mass $m$ in a harmonic confinement potential $V(E)=\frac{m}{2}E^{2}$. It corresponds to a certain positive energy ${\cal E}_{N}$ which arises due to confinement of fermions. Now suppose that this property is valid also for arbitrary $\beta$ and check that the wave function \begin{equation} \label{wwf} \Psi_{0,\beta}(\{E_{n}\})=\prod_{n>m}|E_{n}-E_{m}|^{\beta/2}\,{\rm sgn}(E_{n}-E_{m})\,e^{-\frac{m}{2}\sum_{n=1}^{N}E_{n}^{2}} \end{equation} is the eigenfunction of a certain Hamiltonian. Note that the coefficient $m$ can be done arbitrary small by a proper choice of $E_{n}$ units. So, for simplicity of further derivation we consider the case $m\rightarrow 0$. To this end we take the sum of second derivatives of the wave function applying the kinetic energy operator Eq.(\ref{ham-ferm}) to Eq.(\ref{wwf}) with $m\rightarrow 0$. The result appears to be proportional to $\Psi_{0,\beta}(\{E_{n}\})$: \begin{equation} \label{pot} -{\cal H}_{K}\,\Psi_{0,\beta}=\frac{\beta}{2}\left( \frac{\beta}{2}-1\right)\,|\Delta_{N}|^{-2}\sum_{n=1}^{N}\left( \frac{\partial\Delta_{N}}{\partial E_{n}}\right)^{2}\,\Psi_{0,\beta}. \end{equation} Thus at any $\beta\neq 2$ the system of fermions equivalent to an invariant random matrix theory is interacting with the interaction Hamiltonian: \begin{equation} \label{int-Ham} {\cal H}_{{\rm int}}=\frac{\beta}{2}\left( \frac{\beta}{2}-1\right)\,|\Delta_{N}|^{-2}\sum_{n=1}^{N}\left( \frac{\partial\Delta_{N}}{\partial E_{n}}\right)^{2}. \end{equation} Now if we use the property of the Vandermond determinant: \begin{equation} \sum_{n=1}^{N}\left(\frac{\partial \Delta_{N}}{\partial E_{n}}\right)^{2}=2|\Delta_{N}|^{2}\,\sum_{n=1}^{N}\frac{1}{(E_{n}-E_{m})^{2}} \end{equation} we finally obtain the total Hamiltonian of an equivalent system of fermions: \begin{equation} \label{CSM} \hat{H}=-\frac{1}{2}\sum_{n=1}^{N}\frac{\partial^{2}}{\partial E_{n}^{2}}+\frac{\beta}{2}\left(\frac{\beta}{2}-1 \right)\,\sum_{n>m}^{N}\frac{1}{(E_{n}-E_{m})^{2}}. \end{equation} This is the celebrated Calogero-Sutherland Hamiltonian \cite{CalSuth} with the inverse square interaction. For $\beta=2$ the interaction constant vanishes and the entire level repulsion is due to fermionic nature of the fictitious particles. For $\beta=1$ there is some attraction on top of the free fermionic mutual avoiding, while for $\beta=4$ the interaction is repelling. This additional interaction explains why the level repulsion for $\beta=4$ is stronger then for $\beta=2$ and for $\beta=1$ it is weaker than for $\beta=2$. \section{Plasma model and the Wigner semi-circle} The model of classical particles in one dimension with logarithmic repulsion Eq.(\ref{Hamil}) can be represented by a continuous energy functional: \begin{equation} \label{densityFun} {\cal L}=-\frac{1}{2}\,\int dE\int dE'\,\varrho(E)\,\varrho(E')\,\ln|E-E'|+\frac{1}{\beta}\int dE\;\varrho(E)\,V(E), \end{equation} expressed through the exact density \begin{equation} \label{ex-dens} \varrho(E)=\sum_{n}\delta(E-E_{n}). \end{equation} Now we make two {\it assumptions}:\\ (i) replace $\varrho(E)$ by an {\it ensemble average} value $\rho(E)$ and\\ (ii) neglect the thermal fluctuations by minimizing the energy functional Eq.(\ref{densityFun}) (with $\varrho$ replaced by $\rho$) instead of computing the partition function \begin{equation} \label{part} \sum_{{\rm config.}\{E_{n}\}}{\rm e}^{-\beta\,{\cal L}}. \end{equation} As a result one gets a kind of {\it mean field} approximation which is justified by the long-range, logarithmic nature of interaction. Minimizing Eq.(\ref{densityFun}) with respect to $\rho(E)$ and differentiating both sides with respect to $E$ one obtains: \begin{equation} \label{plasEq} \int_{-\infty}^{+\infty} \rho(E')\,\frac{dE'}{E-E'}=\frac{1}{\beta}\frac{dV}{dE}\equiv f(E). \end{equation} The physical meaning of this equation is very simple: the force acting upon the given "particle" from all other particles should be balanced by the confining force. This is the condition of the plasma equilibrium. From the mathematical viewpoint Eq.(\ref{plasEq}) is a {\it strongly singular} integral equation. Its solution is well known \cite{Muskh}. For an even function $V(E)=V(-E)$ it reads: \begin{equation} \label{Muskh} \rho_{0}(E)=\frac{1}{\pi^{2}}\,\sqrt{D^{2}-E^{2}}\int_{-D}^{D}\frac{f(E')}{\sqrt{D^{2}-E'^{2}}} \,\frac{dE'}{E'-E}, \end{equation} where the principle value of the integral is assumed in Eq.(\ref{plasEq}) and Eq.(\ref{Muskh}), namely \begin{equation} \label{valp} \frac{1}{E'-E}\rightarrow \frac{1}{2}\,\left(\frac{1}{E'-E-i0}+\frac{1}{E'-E+i0} \right). \end{equation} This definition allows to make an analytic continuation of Eq.(\ref{Muskh}) for $E$ in the complex plane with the cut along the real axis with $|E|>D$, where the bandwidth $D$ should be chosen from the condition that the total number of eigenvalues is equal to the size of matrix $N$: \begin{equation} \label{norm-cond} \int_{-D}^{D}\rho_{0}(E)\,dE = N. \end{equation} Namely, $\rho(E)$ can be represented as a sum of a function $\rho_{+}(E)$ which is regular in the upper half-plane $\Im E>0$ and a function $\rho_{-}(E)$ which is regular in the lower half-plane $\Im E<0$: \begin{equation} \label{anal-cont} \rho_{0}(E)=\rho_{+}(E)+\rho_{-}(E),\;\;\;\;\;\rho_{\pm}(E)= \frac{1}{2\pi^{2}}\,\sqrt{D^{2}-E^{2}}\int_{-D}^{D}\frac{f(E')}{\sqrt{D^{2}-E'^{2}}} \,\frac{dE'}{E'-E\mp i 0}. \end{equation} It is important that along the cut $|E|>D$ the analytic function $\sqrt{D^{2}-E^{2}}=\pm i\sqrt{E^{2}-D^{2}}$ has different signs just above and just below the cut. This means that for $|E|>D$ \begin{equation} \label{pm} \rho_{+}(E)+\rho_{-}(E)=-\frac{1}{\pi}\,\sqrt{E^{2}-D^{2}}\int_{-D}^{D}dE'\,\frac{f(E')}{\sqrt{D^{2}-E'^{2}}} \,\delta(E-E')=0. \end{equation} On the other hand, for real $E$ beyond the cut ($|E|<D$) one obtains: \begin{equation} \label{pm1} \rho_{+}(E)-\rho_{-}(E)=\frac{2\pi i}{2\pi^{2}}\,\sqrt{D^{2}-E^{2}}\int_{-D}^{D}\frac{f(E')}{\sqrt{D^{2}-E'^{2}}} \,\delta(E-E')=\frac{i}{\pi}\,f(E). \end{equation} Now we are in a position to check that Eq.(\ref{Muskh}) is really a solution of Eq.(\ref{plasEq}) for real $E$ beyond ($|E|<D$) the cut. Indeed, the integral over the real axis in Eq.(\ref{plasEq}) can be closed either through the upper complex half-plane of $E'$ or through the lower half-plane. We use the first option for the part containing $\rho_{+}(E')$ and the second option for the part containing $\rho_{-}(E')$. Each of the two contour integrals allows for the evaluation using the residue theorem. Then omitting the terms which do not have poles in the corresponding half-plane we obtain: $$ \int_{-\infty}^{+\infty} \rho_{0}(E')\,\frac{dE'}{E-E'}=\frac{1}{2}\int_{{\rm upper}}\frac{\rho_{+}(E')}{E-E'+i0}\,dE'+\frac{1}{2}\int_{{\rm lower}}\frac{\rho_{-}(E')}{E-E'-i0}\,dE'=-\pi i\,[\rho_{+}(E)-\rho_{-}(E)]=f(E). $$ This concludes the proof that Eq.(\ref{Muskh}) is indeed a solution of the integral equation Eq.(\ref{plasEq}). The beauty of the proof is that it is based only on the analytic properties of the solution. For the Gaussian ensemble where $f(E')=E'$ the integral in Eq.(\ref{Muskh}) is actually independent of $E$ ({\bf show this using the definition of the principle value of the integral}) \begin{equation} \label{int-ind} \int_{-D}^{D}\frac{E'}{\sqrt{D^{2}-E'^{2}}} \,\frac{dE'}{E'-E}=\int_{-D}^{D}\frac{1}{\sqrt{D^{2}-E'^{2}}} \,dE'=\pi. \end{equation} and the average density is the celebrated {\it semi-circle}: \begin{equation} \label{semi-circ} \rho_{0}(E)=\frac{1}{\pi}\,\sqrt{2N-E^{2}}. \end{equation} \section{Probability of having a hole in spectrum and the Wigner Surmise} One of the most popular statistics of eigenvalues of complex quantum systems is the {\it the level spacing distribution} $P(\omega)$: the probability density to have a level at a distance $\omega$ from a given level and no other levels between them. For $\omega$ much smaller than the mean level spacing $\Delta=\rho^{-1}$, it is improbable that in between of the two close levels there is yet another one or several levels. Then the requirement of having no levels in between of the two is unimportant and the leading term in $P(\omega)$ is the same as in the two-level correlation function $R(\omega)\propto \omega^{\beta}$ at $\omega\ll \Delta$. However, for $\omega\gg \Delta$ the two statistics dramatically differ: $R(\omega)$ tends to a constant whereas $P(\omega)$ is very small due to a small probability to have no levels in between of the two levels separated by a large distance. Basically the $P(\omega)$ for $\omega\gg \Delta$ is limited by the probability of having a hole of the size $\omega$ in the spectrum. Let us find this probability using the plasma analogy. As for any fluctuation, the probability of having a hole is given by the energy cost $\delta {\cal L}$ of this configuration relative to the equilibrium one: \begin{equation} \label{fluc} P(\omega)\propto {\rm exp}(-\beta\,\delta{\cal L}). \end{equation} One can cast the energy difference in the following way: \begin{equation} \label{deltaH} \Delta {\cal L}=\frac{1}{2}\int_{C}dE\int_{C}dE'\;\delta\rho(E)\,\delta\rho(E')\,\ln|E-E'|-\frac{1}{2} \int_{-\omega/2}^{\omega/2}dE\int_{-\omega/2}^{\omega/2}dE'\;\rho_{0}(E)\,\rho_{0}(E')\,\ln|E-E'|, \end{equation} where the integrals in the first term run over the real axis {\it outside the gap region} and in the second term they run {\it over the gap region}; $\rho_{0}(E)$ is the equilibrium density without the gap and $\delta\rho(E)=\rho_{\omega}(E)-\rho_{0}(E)$ with $\rho_{\omega}(E)$ being the solution of the integral equation Eq.(\ref{plasEq}) with the {\it additional condition} that there is a gap for $|E|<\omega/2$. The solution with the gap can also be constructed and and looks as follows: \begin{equation} \label{gapSol} \rho_{\omega}(E)=\frac{2|E|}{\pi^{2}\,\sqrt{E^{2}-(\omega/2)^{2}}}\,\sqrt{\bar{D}^{2}-E^{2}}\, \int_{\omega/2}^{\bar{D}} \frac{f(E')}{\sqrt{\bar{D}^{2}-E'^{2}}}\,\frac{\sqrt{E'^{2}-(\omega/2)^{2}}}{E'^{2}-E^{2}}\,dE'. \end{equation} It is important that for the {\it steep confinement} $V(E)\sim |E|^{\alpha}$ ($\alpha>1$) there is a scale separation, namely the integral in Eq.(\ref{gapSol}) varies slowly as a function of $E$ with the typical scale of $D\sim N^{1/\alpha}$. In the large $N$ limit one can disregard this dependence and consider \begin{equation} \label{DoS} \rho_{\omega}(E)=\rho_{0}\,\frac{|E|}{\sqrt{E^{2}-(\omega/2)^{2}}}. \end{equation} One can immediately recognize the gapped density of states with the square-root divergency near the gap edges similar to the one for a BCS superconductor. Now by making a re-scaling $E\rightarrow s x$, $E'\rightarrow s x'$ and observing that the double integral in the first term is convergent for $\rho_{\omega}(E)$ of the form Eq.(\ref{DoS}) we immediately obtain that $\Delta{\cal L}=\omega^{2}\,(a+b\ln \omega)$. More detailed inspection show that the coefficient $b=0$. The final result for $\Delta{\cal L}$ reads: \begin{equation} \label{delta-L} \Delta{\cal L}=\frac{\pi^{2}}{16}\,(\rho_{0}\,\omega)^{2}\approx 0.62\,(\rho_{0}\,\omega)^{2}. \end{equation} This implies that the spacing distribution function for large level separations $s=(\omega/\Delta)\gg 1$ is given by: \begin{equation} \label{asym-P} P(s)\propto {\rm exp}\left(-\frac{\pi^{2} \beta}{16}\,s^{2}\right). \end{equation} Note that the "Gaussian" form of $P(s)$ has nothing to do with the quadratic confinement potential (Gaussian invariant ensemble). In fact $P(s)$ has the same asymptotic form Eq.(\ref{asym-P})for all {\it steep} confinement potentials.\\ {\bf check that it has the same form for the confinement potential $V(E)=E^{4}$}. \\ Finally we mention a famous {\it interpolation formula} for $P(s)$ known as the $Wigner Surmise$: \begin{equation} \label{WS} P(s)=A(\beta)\,s^{\beta}\,{\rm exp}\left[-B(\beta)\,s^{2}\right],\;\;\;\;\;s=\frac{\omega}{\Delta}. \end{equation} The coefficients $A(\beta)$ and $B(\beta)$ are found from two conditions: the normalization to the total probability 1 and the condition that the mean level spacing in the units of $s$ is one: \begin{equation} \label{WigSur} \int_{0}^{\infty}P(s)\,ds=1,\;\;\;\;\;\;\int_{0}^{\infty}s\;P(s)\,ds=1. \end{equation} These conditions result in: \begin{equation} \label{A} A(\beta)=2B(\beta)^{\frac{\beta}{2}+1}\,\Gamma\left(\frac{\beta}{2}+1 \right),\;\;\;\;\;B(\beta)=\left\{\begin{matrix}\frac{\pi}{4}\approx 0.78\;(0.62)&\beta=1\cr \cr\frac{4}{\pi}\approx 1.27\;(1.24) & \beta=2\cr \cr\frac{64}{9\pi}\approx 2.26\,(2.48) & \beta=4 \cr \end{matrix} \right. \end{equation} For comparison we give in the brackets the exact values of $B_{{\rm exact}}(\beta)=\pi^{2}\beta/16$. One can see that they are rather close to the approximate values of the Wingner Surmise, especially for $\beta=2$. \section{Level compressibility, normalization sum rule and normalization anomaly} The {\it two-level correlation function} (TLCF) is formally defined as a correlation function of the exact density of states Eq.(\ref{ex-dens}): \begin{equation} \label{forTLCF} R_{N}(E,E')=\frac{\langle\varrho(E)\,\varrho(E') \rangle}{\rho(E)\,\rho(E')}\equiv \rho^{-1}(E)\delta(E-E')+1+Y_{N}(E,E'). \end{equation} The $\delta$-function in Eq.(\ref{forTLCF}) is the self-correlation coming from one and the same level $n$ in the sum in Eq.(\ref{ex-dens}). The 1 term gives the asymptotic value of TLCF at energy separations $|E-E'|\gg \Delta$ when the average of two densities of states can be decoupled. The function $Y(E,E')$ gives then a regular contribution to the TLCF which decreases to zero as $|E-E'|$ increase. There is an important {\it normalization sum rule} that applies to the TLCF. Indeed, consider \begin{equation} \label{step1} \int dE'\,\rho(E)\,\rho(E')\,R_{N}(E,E')=\int dE'\,\langle\varrho(E)\,\varrho(E') \rangle=\langle\varrho(E)\,\int dE'\,\varrho(E') \rangle \end{equation} The total number of states is equal to the number of degrees of freedom $N$ and {\it does not fluctuate}: \begin{equation} \label{int-N} \int dE'\,\varrho(E') =N. \end{equation} This normalization condition leads to $$ \int dE'\,\rho(E)\,\rho(E')\,R_{N}(E,E')=N\,\langle\varrho(E)\rangle=N\,\rho(E), $$ which implies that \begin{equation} \label{SR} \int_{-\infty}^{+\infty} dE'\,\rho(E')\,Y_{N}(E,E')=-1, \;\;\;\;\;\int_{-\infty}^{+\infty} dE'\,\rho(E')\,[R_{N}(E,E')-1]=0. \end{equation} This is the {\it normalization sum rule}. Note that the sum rule can only be proven if $N$ is finite and integration in Eq.(\ref{SR}) are extended over ${\it all}$ energies. Taking the limit $N\rightarrow\infty$ could be a dangerous procedure as in this case one has to worry about the commutativity of the limits $N\rightarrow\infty$ and {\it limits of integration}$\rightarrow\infty$. The sum rule is certainly satisfied if the limit $N\rightarrow\infty$ is done {\it after} doing the integral. A simple example below shows that it can be violated if the limit $N\rightarrow$ is taken {\it prior} of doing the integral. Consider an ensemble of diagonal random matrices with independently fluctuating random elements each having a distribution \begin{equation} \label{ddist} P(\varepsilon_{n})=\left\{ \begin{matrix} N^{-1}, & |\varepsilon_{n}|<N/2\cr 0 & |\varepsilon_{n}|>N/2\cr\end{matrix}\right. \end{equation} The TLCF for this ensemble can be computed straightforwardly: $$ \langle\varrho(E)\,\varrho(E') \rangle=\sum_{n\neq m}\int_{-N/2}^{N/2}\frac{d\varepsilon_{n}}{N}\int_{-N/2}^{N/2}\frac{d\varepsilon_{m}}{N}\, \delta(E-\varepsilon_{n})\,\delta(E'-\varepsilon_{m})+\sum_{n}\int_{-N/2}^{N/2}\frac{d\varepsilon_{n}}{N}\, \delta(E-\varepsilon_{n})\,\delta(E'-\varepsilon_{n}), $$ so that for $|E|<N/2$ and $|E'|<N/2$ one obtains: \begin{equation} \label{diag-TLCF} R(E,E')=\delta(E-E')+\frac{N^{2}-N}{N^{2}},\;\;\;\;\;\rho(E)=1,\;\;\;\;Y_{N}(E,E')=-\frac{1}{N}. \end{equation} At a finite $N$ the normalization sum rule Eq.(\ref{SR}) is obviously fulfilled: \begin{equation} \label{SR-d} \int_{-N/2}^{N/2}dE'\,\left(-\frac{1}{N} \right)=-1. \end{equation} However, if one takes the limit $N\rightarrow\infty$ in Eq.(\ref{diag-TLCF}) before integrating, one obtains $Y_{\infty}=0$, and the normalization sum rule will we violated: \begin{equation} \label{SR-d} \int_{-N/2}^{N/2}dE'\,\lim_{N\rightarrow\infty}Y_{N}(E,E')=0. \end{equation} This mechanism of violation of sum rules in the thermodynamic limit is called the {\it anomaly} and is well known in the field theory. A remarkable property of the Wigner-Dyson level statistics is that in this case (as well for all invariant RM ensembles, even with shallow confinement potentials) the normalization sum rule is not violated and the anomaly does not occur. Let us show how this property follows from the plasma model Eq.(\ref{densityFun}). To this end we note that the density-density correlation function is given by the variational derivative of the mean density with respect to the confinement potential: \begin{equation} \label{var} \langle\varrho(E)\,\varrho(E') \rangle-\langle\varrho(E)\rangle\,\langle\varrho(E') \rangle=-\frac{\delta\rho(E)}{\delta V(E')}\approx \rho_{0}^{2}\,[R_{\infty}(E-E')-1], \end{equation} where we assume that the mean density $\rho(E)\approx\rho_{0}$ does not change much at a scale of the mean level spacing $\Delta$. In this case one can approximately consider the TLCF as a function of the energy difference. Then integrating by parts in Eq.(\ref{Muskh}) and neglecting the energy dependence of the square roots we obtain: \begin{equation} \label{TLCF-as} -\frac{\delta\rho(E)}{\delta V(E')}=-\frac{1}{\pi^{2}\beta}\,\frac{1}{(E-E')^{2}}, \end{equation} where the regularization $$ (E-E')^{-2}\rightarrow \frac{1}{2}\,[(E-E'+i0)^{-2}+(E-E'-i0)^{-2}] $$ is assumed. Using this regularization one can immediately check that $$ \int_{-\infty}^{+\infty}[R_{\infty}(E)-1]\,dE=0, $$ as it is required by the sum rule Eq.(\ref{SR}) at $\rho(E)={\rm const}$. Thus we see that for the plasma model doing the limit $N\rightarrow\infty$ and doing the integral commute. The absence of the anomaly is related with the incompressible character of the system of logarithmically repelling particles. This is the reason why approximations made in deriving the plasma model Eq.(\ref{densityFun}) did not affect the regular fulfillment of the normalization sum rule. Below we define the {\it level compressibility} and show that it is zero if the normalization sum rule is not violated. To this end we define the {\it level number variance}: \begin{equation} \label{lnv} \Sigma(\bar{n})=\langle n^{2}\rangle-\bar{n}^{2}, \end{equation} where $n$ is the fluctuating number of levels in an energy interval $\delta_{E}$ that contains on the average $\bar{n}$ levels. Writing $$ n=\int_{E_{0}-\delta E/2}^{E_{0}+\delta E/2}\rho(E)\,dE\equiv\int_{\delta_{E}}\rho(E)\,dE $$ we obtain: $$ \Sigma(\bar{n})=\int_{\delta_{E}} \rho(E)\,dE \int_{\delta_{E}}\rho(E')\,[R_{N}(E,E')-1]\,dE'. $$ Now we assume that $N$ is large, the confinement is steep and thus $D\gg|\delta E|\gg \Delta$. This allows to consider $\rho(E)=\rho_{0}$ and $R_{N}(E,E')\approx R_{\infty}(E-E')$. Then one can integrate over $E$ at a fixed $E-E'$ and arrive at: \begin{equation} \label{Sig} \Sigma(\bar{n})=\bar{n}\,\int_{-\bar{n}}^{\bar{n}}[R_{\infty}(s)-1]\,ds -\int_{-\bar{n}}^{\bar{n}}|s|\,[R_{\infty}(s)-1]\,ds, \end{equation} where $s=(E-E')\rho_{0}$. The {\it level compressibility} is defined as \begin{equation} \label{compress} \chi(\bar{n})=\frac{d\Sigma}{d\bar{n}}=\int_{-\bar{n}}^{\bar{n}}[R_{\infty}(s)-1]\,ds. \end{equation} In the case where the anomaly does not occur we have $$ \lim_{\bar{n}\rightarrow\infty}\int_{-\bar{n}}^{\bar{n}}[R_{\infty}(s)-1]\,ds= \lim_{N\rightarrow\infty}\int_{-\infty}^{+\infty}[R_{N}(s)-1]\,ds, $$ which in view of the normalization sum rule Eq.(\ref{SR}) implies incompressible nature of the system of energy levels: \begin{equation} \label{chi0} \lim_{\bar{n}\rightarrow\infty}\chi(\bar{n})=0. \end{equation} The opposite is also true: if Eq.(\ref{chi0}) is fulfilled, the limits ${\bar{n}\rightarrow\infty}$ and $N\rightarrow\infty$ commute and there is no anomaly. Now we use the absence of the anomaly and the normalization sum rule to compute the level number variance at $\bar{n}\gg 1$. To this end we cast Eq.(\ref{Sig}) in the following way $$ \Sigma(\bar{n})=\bar{n}\,\int_{-\infty}^{+\infty}[R_{\infty}(s)-1]\,ds -2\bar{n}\int_{\bar{n}}^{\infty}[R_{\infty}(s)-1]\,ds -2\int_{0}^{\bar{n}}|s|\,[R_{\infty}(s)-1]\,ds, $$ The first integral vanishes because of the sum rule Eq.(\ref{SR}), the second term requires only the knowledge of TLCF at large distances $s\gg 1$ and can be computed using Eq.(\ref{TLCF-as}) and appears to be a constant of order 1. The third integral is logarithmic and this allows to compute the leading logarithmic term also using Eq.(\ref{TLCF-as}) which is valid at $s\gg 1$. Cutting the logarithmic divergency at $s\sim 1$ we obtain for $\bar{n}\gg 1$: \begin{equation} \label{Sigma} \Sigma(\bar{n})=\frac{2}{\pi^{2}\beta}\,\ln\bar{n}+O(1). \end{equation} This variance is considerably smaller than for independently fluctuating levels (diagonal RME) where it is distributed according to Poisson law: \begin{equation} \label{Poisson} \Sigma(\bar{n})=\bar{n}. \end{equation} \section{Orthogonal polynomials and energy level statistics for $\beta=2$.} As has been already mentioned level statistics in the Gaussian invariant RME with $\beta=2$ can be exactly mapped onto the system of non-interacting fermions in one dimension in the parabolic confinement potential $V(E)=E^{2}$. Let us check this statement. To this end we recall that the ground state {\it many-body} wavefunction of non-interacting fermions is the {\it Slatter determinant}: \begin{equation} \label{Slatter} \Psi(\{E_{n}\})=\left|\begin{matrix}\varphi_{0}(E_{1})&\varphi_{0}(E_{2})&...& \varphi_{0}(E_{N})\cr \varphi_{1}(E_{1})&\varphi_{1}(E_{2})&...& \varphi_{1}(E_{N})\cr...&...&...&...\cr \varphi_{N-1}(E_{1})&\varphi_{N-1}(E_{2})&...& \varphi_{N-1}(E_{N})\cr \end{matrix} \right|. \end{equation} The {\it one-particle} eigenfunctions $\varphi_{n}(E)=H_{n}(E)\,e^{-E^{2}/2}$ in the parabolic confinement potential $V(E)$ obeying the Schroedinger equation: \begin{equation} \label{Schred} -\frac{1}{2}\,\frac{\partial^{2}}{\partial E^{2}}\,\varphi_{n}(E)+\frac{1}{2}\,E^{2}\,\varphi_{n}(E)={\cal E}_{n}\,\varphi_{n}(E), \end{equation} are related to the Hermite {\it orthogonal polynomials} $H_{n}(E)$. These are the polynomials of the $n$-th order satisfying the orthogonality relation: \begin{equation} \label{ortho} \int_{-\infty}^{+\infty}{\rm exp}[-V(x)]\,H_{n}(x)\,H_{m}(x)=h_{n}\,\delta_{nm},\;\;\;\;\;\;V(x)=x^{2},\;\;\;\;h_{n}=1. \end{equation} It is important that the Hermite polynomials obey the {\it three-term} recursive relation: \begin{equation} \label{three-term} H_{n+1}(x)=2x\,H_{n}(x)-2n\,H_{n-1}(x). \end{equation} Using this relation one can show that the Slatter determinant Eq.(\ref{Slatter}) reduces to: \begin{equation} \label{Sl-VdM} \left|\begin{matrix}\varphi_{0}(E_{1})&\varphi_{0}(E_{2})&...& \varphi_{0}(E_{N})\cr \varphi_{1}(E_{1})&\varphi_{1}(E_{2})&...& \varphi_{1}(E_{N})\cr...&...&...&...\cr \varphi_{N-1}(E_{1})&\varphi_{N-1}(E_{2})&...& \varphi_{N-1}(E_{N})\cr \end{matrix} \right|={\rm const}\, \left|\begin{matrix}1&1&...&1\cr E_{1}&E_{2}&...& E_{N}\cr...&...&...&...\cr E_{1}^{N-1}&E_{2}^{N-1}&...& E_{N}^{N-1}\cr \end{matrix} \right|\,\rm exp\left[-\frac{1}{2}\,\sum_{n=1}^{N} E_{n}^{2}\right]. \end{equation} Indeed, the exponential factors $e^{-\frac{1}{2}E^{2}}$ in all the $\varphi_{n}(E)$ can be taken out of the determinant using the rule of multiplication of determinant by a factor which is equivalent to multiplication of all the elements in a column by this factor. So we obtain the exponential factor in the r.h.s. of Eq.(\ref{Sl-VdM}). Next choosing $H_{0}(E)=1$, $H_{1}(E)=x$ one can find all the other polynomials using the recursion relation Eq.(\ref{three-term}). In particular, $H_{2}=2x^{2}-2$. This polynomial should be plugged in the third line of the determinant in the l.h.s. of Eq.(\ref{Sl-VdM}). Note, however, that the constant term $-2$ can be omitted as its inclusion is equivalent to an addition of the first line to the third line in the determinant which according to the basic property of a determinant does not change its value. This process can be continued. For instance in $H_{3}=2x H_{2}-4H_{1}$ which stands in the fourth line of the determinant one can omit $-4H_{1}$, as $H_{1}$ stands in the second line. Then expressing $H_{2}=2x H_{1}-H_{0}=2x^{2}-1$ one can put in the fourth line $2x(2x^{2} -1)$ instead of $H_{3}$. Finally, observing that the term $-2x$ can be considered as a linear combination of $4x^{3}$ and the first line in the determinant and omitting this term we conclude that instead of $H_{3}$ on can put in the fourth line of the determinant just one term $4x^{3}$. The determinant in the r.h.s. is the famous {\it Vandermond determinant} which is equal to: \begin{equation} \label{VdM} \left|\begin{matrix}1&1&...&1\cr E_{1}&E_{2}&...& E_{N}\cr...&...&...&...\cr E_{1}^{N-1}&E_{2}^{N-1}&...& E_{N}^{N-1}\cr \end{matrix} \right|= \prod_{n>m}(E_{n}-E_{m}). \end{equation} Now we see that: \begin{equation} \label{psi-2} |\Psi(\{ E_{n}\})|^{2}={\rm const}\,{\rm exp}\left[-\sum_{n}E_{n}^{2}\right]\,\prod_{n>m}(E_{n}-E_{m})^{2}. \end{equation} This is exactly the probability distribution functions for the eigenvalues of the Gaussian RME with $\beta=2$. It turns out that the theory of orthogonal polynomials \cite{Sciego} is the powerful method to solve {\it any} orthogonal random matrix ensemble. What one has to do for that is to generate a set of orthogonal polynomials $p_{n}(x)$ obeying the orthogonality relation Eq.(\ref{ortho}) and to be able to compute the large-$N$ asymptotic behavior of the "wavefunctions": \begin{equation} \label{varphi-n} \varphi_{n}(E)=\varphi_{n}(E)=h_{n}^{-1/2}\,p_{n}(E)\,e^{-V(E)/2}. \end{equation} The generation of orthogonal polynomials is possible for any confinement potential $V(x)$ using the Gram-Schmidt orthogonalization procedure. According to this procedure one computes the Gram-Schmidt determinant: \begin{equation} \label{GSdet} G_{n}= \left|\begin{matrix}a_{0}&a_{1}&...&a_{n}\cr a_{1}&a_{2}&...& a_{n+1}\cr...&...&...&...\cr a_{n}&a_{n+1}&...& a_{2n}\cr \end{matrix} \right|, \end{equation} where $a_{n}$ are the moments: \begin{equation} \label{mom} a_{n}=\int_{-\infty}^{+\infty}{\rm exp}[-V(x)]\,x^{n}\,dx. \end{equation} Then the orthogonal polynomial of $n$-th power $p_{n}$ is given by: \begin{equation} \label{orpol} p_{n}(x)=\frac{q_{n}}{G_{n-1}}\,\left|\begin{matrix}a_{0}&a_{1}&...&a_{n}\cr a_{1}&a_{2}&...& a_{n+1}\cr...&...&...&...\cr a_{n-1}&a_{n}&...& a_{2n-1}\cr1&x&...&x^{n} \end{matrix} \right|, \end{equation} where $q_{n}$ is the coefficient in front of $x^{n}$ in $p_{n}$. It follows from this generic procedure that any set of orthogonal polynomials $p_{n}(x)$ should obey the three-term recursion relation similar to Eq.(\ref{three-term}). For the confinement potential $V(x)$ being an even function of $x$ and the choice $q_{n}=1$ (for the Hermite polynomials the standard definition corresponds to $q_{n}=2^{n}$) it reads: \begin{equation} \label{gen-3-term} p_{n+1}(x)=xp_{n}(x)-C_{n+1}\,p_{n-1}(x),\;\;\;\;\;\;C_{n+1}=\frac{G_{n}\,G_{n-2}}{G_{n-1}^{2}}, \;\;\;p_{0}(x)=1,\;\;\;\;p_{1}(x)=x. \end{equation} We note that this recursion relation generates orthogonal but not {\it ortho-normal} polynomials. The price of having $q_{n}=1$ is that the normalization constant $h_{n}$ in Eq.(\ref{ortho}) is not unity and is related to the coefficient $C_{n}$ as follows: \begin{equation} \label{h-C} h_{n}=h_{0}\,\prod_{m=2}^{n+1}C_{m},\;\;\;\;h_{0}=a_{0}. \end{equation} The recursive relation Eq.(\ref{gen-3-term}) appears to be the most convenient way of generating orthogonal polynomials for {\it any} confinement potential. It works also for non-classical polynomials for which there are no second-order differential equations (similar to the Schroedinger equation Eq.(\ref{Schred}) in the case of Hermite polynomials) which the "wavefunctions" Eq.(\ref{varphi-n}) should obey. The efficiency of the orthogonal polynomials in the problem of level statistics is largely due to the {\it Christoffel-Darboux formula}: \begin{equation} \label{Chr-Dar} K_{N}(x,y)=\sum_{n=0}^{N-1}\varphi_{n}(x)\,\varphi_{n}(y)=\sqrt{\frac{h_{N}}{h_{N-1}}}\,\, \frac{\varphi_{N-1}(x)\,\varphi_{N}(y)-\varphi_{N-1}(y)\,\varphi_{N}(x)}{y-x}. \end{equation} This formula can be proven by induction using the three term recursive relation Eq.(\ref{gen-3-term}) and a relation Eq.(\ref{h-C}) between $h_{n}$ and $C_{n}$. The Christoffel-Darboux formula is important because the mean density of states and the two level correlation function at $x\neq y$ are given by: \begin{equation} \label{mDoS} \rho(E)=\sum_{n=0}^{N-1}\varphi_{n}(x)^{2}=K_{N}(E,E), \end{equation} \begin{equation} \label{TLCF-K} \rho(E)\,\rho(E')\,R_{N}(E,E')= \sum_{n=0}^{N-1}\sum_{m=0}^{N-1}\left[\varphi^{2}_{n}(E)\,\varphi^{2}_{m}(E')- \varphi_{n}(x)\,\varphi_{n}(y)\,\varphi_{m}(y)\, \varphi_{m}(x)\right]= K_{N}(E,E)\,K_{N}(E',E')-K^{2}_{N}(E,E'). \end{equation} Eqs.(\ref{mDoS}),(\ref{TLCF-K}) can be proven formally without any reference to systems of non-interacting fermions. However, it is instructive to see how they follow from the fermionic second quantization formalism. Indeed, the density of non-interacting fermions and the density-density correlation function are given by: $$ \langle 0|\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)|0\rangle,\;\;\;\;\; \langle 0|\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)\,\hat{\Psi}^{\dagger}(y)\hat{\Psi}(y)|0\rangle, $$ where $\langle 0|...|0\rangle$ is the quantum-mechanical averaging over the ground state. According to the rules of second quantization the fermionic field operator $\hat{\Psi}(x)$ is given by the expansion over the single-particle wavefunctions: $$ \hat{\Psi}(x)=\sum_{n=0}^{N-1}\varphi_{n}(x)\,c_{n}, $$ where $c_{n}$ and $c_{n}^{\dagger}$ are the fermionic creation and annihilation operators obeying the anti-commutation relation $$ c_{n}^{\dagger}\,c_{m}+c_{m}\,c_{n}^{\dagger}=\delta_{nm}. $$ The averages over the ground state can be computed using the Wick theorem: \begin{equation} \label{Wick} \langle0| c_{n}^{\dagger}c_{m}|0\rangle=\delta_{nm},\;\;\;\;\;\;\langle0| \,c_{n_{1}}^{\dagger}c_{n_{2}} c_{n_{3}}^{\dagger}c_{n_{4}}|0\rangle=\delta_{n_{1}n_{2}}\,\delta_{n_{3}n_{4}}- \delta_{n_{1}n_{4}}\,\delta_{n_{2}n_{3}}. \end{equation} The two terms in Eq.(\ref{TLCF-K}) follow from the two terms in Eq.(\ref{Wick}) which correspond to two possible parings of $c^{\dagger}$ and $c$. Eq.(\ref{TLCF-K}) can be conveniently represented in terms of the determinant: \begin{equation} \label{TLCF-det} \rho(E)\,\rho(E')\,R_{N}(E,E')=\left|\begin{matrix}K_{N}(E,E)& K_{N}(E,E')\cr K_{N}(E',E)&K_{N}(E',E')\cr \end{matrix} \right|. \end{equation} Using the fermionic analogy and the Wick theorem one can prove that any multi-point level density correlation function can be represented in the form of a similar determinant: \begin{equation} \label{multi} \rho(E_{1})\rho(E_{2})...\rho(E_{n})\,R(E_{1},E_{2}...E_{n})=\left|\begin{matrix} K_{N}(E_{1},E_{1})&K_{N}(E_{1},E_{2})&...&K_{N}(E_{1},E_{n})\cr K_{N}(E_{2},E_{1})&K_{N}(E_{2},E_{2})&...& K_{N}(E_{2},E_{n})\cr...&...&...&...\cr K_{N}(E_{n},E_{1})&K_{N}(E_{n},E_{2})&...& K_{N}(E_{n},E_{n})\cr \end{matrix} \right|. \end{equation} We see that the analogy with non-interacting fermions allows to express any multi-point level density correlation function in terms of only one single kernel $K_{N}(x,y)$. The latter according to the Christoffel-Darboux theorem is a product of only two "wave functions" which require the knowledge of only two orthogonal polynomials $p_{N}(x)$ and $p_{N-1}(x)$. Thus the problem of energy level statistics is reduced to the problem of finding the asymptotic behavior of orthogonal polynomials of high order. \section{WKB quasi-classical approximation and the one-dimensional "Wigner crystal".} The semicircle law for the mean level density follows immediately and trivially from the free-fermion representation. Indeed, the density of one-dimensional fermions is directly related with the Fermi-momentum $p_{F}$ by: \begin{equation} \label{Fermi-np} \frac{2p_{F}}{2\pi}=\rho,\;\;\;\;\;\hbar=1. \end{equation} When the density varies slowly at a scale of the Fermi wavelength, one can apply Eq.(\ref{Fermi-np}) {\it locally} thus relating the local Fermi-momentum $p_{F}(x)$ with the local density $\rho(x)$. The local Fermi momentum corresponds to the momentum of the highest occupied state in a parabolic potential \begin{equation} \label{local pF} p_{F}=p_{N}(x)=\sqrt{2m E_{\rm kin}}=\sqrt{2{\cal E}_{n}-x^{2}}=\sqrt{2N+1 -x^{2}},\;\;\;\;\;m=1. \end{equation} Then the local density is obtained immediately from Eq.(\ref{Fermi-np}): \begin{equation} \label{semicircle2} \rho(x)=\frac{1}{\pi}\,p_{N}(x)=\frac{1}{\pi}\,\sqrt{2N+1 -x^{2}}. \end{equation} This is the celebrated {\it semicircle law}. In order to obtain the {\it density-density correlation function}, or the {\it two-level correlation function} one should work a little bit harder determining the large $N$ asymptotic behavior of $\varphi_{n}$ and applying Eq.(\ref{TLCF-K}). In the case of Hermite polynomials the problem of the large-$N$ asymptotic behavior can be solved quite easily. The reason is that there is the second-order differential equation (the Schroedinger equation) which the wavefunctions $\varphi_{N}(x)$ must obey. As is well known the solutions to the Schroediger equation corresponding to large quantum numbers bear the properties of classical motion in the corresponding potential. In quantum mechanics this corresponds to the "quasi-classical", or WKB approximation \cite{LL}. According to this approximation the wave function at $x^{2}< 2N$ is proportional to: \begin{equation} \label{WKB} \varphi_{N}(x)\propto[p_{N}(x)]^{-1/2}\,\left\{\begin{matrix}\cos\left[\int_{0}^{x} p_{N}(x')\,dx'\right]& N&is&even\cr& & & \cr\sin\left[\int_{0}^{x} p_{N}(x')\,dx'\right]&N&is&odd\cr \end{matrix}\right. \end{equation} At large $N$ we obtain: $$ p_{N}(x)\approx \sqrt{2N},\;\;\;\;\;\int_{0}^{x} p_{N}(x')\,dx'\approx x\,\sqrt{2N}. $$ Then the kernel $K_{N}(x,y)$ is easily calculated using the Christoffel-Darboux formula Eq.(\ref{Chr-Dar}): $$ \label{kernEL} K_{N}(x,y)={\rm const}\,\frac{\sin\left(\sqrt{2N}\,(x-y)\right)}{x-y},\;\;\;\;\;K_{N}(x,x)={\rm const}\,\sqrt{2N}. $$ The normalization constant ${\rm const}=1/\pi$ is most easily found from the comparison of $K_{N}(x,x)=\rho(x)$ and the semi-circle mean level density Eq.(\ref{semi-circ}) which at large $N$ reduces to $\rho(x)\approx \sqrt{2N}/\pi=\rho_{0}$. Now, introducing mean level spacing $\Delta=\rho^{-1}\approx \pi/\sqrt{2N}$ we arrive at: \begin{equation} \label{kernEL} K_{N}(x,y)\rightarrow K(s)=\rho_{0}\,\frac{\sin\left(\pi\,s\right)}{\pi s},\;\;\;\;\;s=\frac{x-y}{\Delta},\;\;\;\;\;\;\;N\rightarrow\infty. \end{equation} The two level correlation function Eq.(\ref{TLCF-K}) is then equal to: \begin{equation} \label{TLCF-beta2} R_{\infty}(x,y)=\delta(s)+1-\frac{\sin^{2}\left(\pi\,s\right)}{(\pi s)^{2}}=1-\frac{1}{2\pi^{2}\,s^{2}}+\frac{\cos(2\pi\, s)}{2\pi^{2}\,s^{2}}. \end{equation} One can see that the TLCF given by Eq.(\ref{TLCF-beta2}) has all the asymptotic limits right. It is proportional to $s^{2}$ at $s\ll 1$ and its envelope corresponds to Eq.(\ref{TLCF-as}) for $s\gg 1$. In addition to that it obeys the normalization sum rule Eq.(\ref{SR}). However, in Eq.(\ref{TLCF-beta2}) there is a term that oscillates with the period of the mean level spacing $\Delta$. This term evades the consideration based on the $2\times2$ matrix (small $s\ll 1$) and the effective continuous plasma model (large $s\gg 1$). Let us discuss the physical meaning of this term using the plasma model analogy but without the continuous approximation. It is well known that plasma of particles with the long-range repulsion in a confinement potential tend to develop a crystal order known as Wigner crystal. Such Wigner crystal of electrons have been observed on top of the helium surface. Our case is special, as it is one-dimensional. According to the Mermin theorem the crystal order cannot survive in one dimensions at a finite temperature because thermal fluctuations destroy the long-range order. However, local crystal order may exist. The last oscillating them in Eq.(\ref{TLCF-beta2}) reflects exactly this order. The short-range nature of this order manifests itself in the fast $s^{-2}$ decay of oscillations at large distances. So far in this section we have considered the $\beta=2$ case. As in the plasma analogy $\beta$ plays a role of inverse temperature, one would expect the oscillating term to decay slower for $\beta=4$ and faster for $\beta=1$. This expectation is in fact true. One can show using the more sophisticated application of the orthogonal polynomial machinery that in the limit $N\rightarrow\infty$ the two-level correlation functions for the orthogonal ($\beta=1$) ensemble $R_{\infty}^{{\rm orth}}$ and that for the symplectic ($\beta=4$) ensemble $R_{\infty}^{{\rm symp}}$ can also be expressed in terms of the kernel $K_{N}(s)$, Eq.(\ref{kernEL}): \begin{equation} \label{orth-R} Y_{\infty}^{{\rm orth}}(s)=-K^{2}(s)-\frac{dK(s)}{ds}\,\int_{s}^{\infty}K(x)\,dx, \end{equation} \begin{equation} \label{symp-R} Y_{\infty}^{{\rm symp}}(s/2)=-K^{2}(s)+\frac{dK(s)}{ds}\,\int_{0}^{s}K(x)\,dx. \end{equation} The $s/2$ in the argument of $Y_{\infty}^{{\rm symp}}(s/2)$ appears because of the Kramers degeneracy: for the same total number of levels $N$ the mean level spacing between doubly degenerate levels is two times longer. Accordingly, the $\delta(s)$ function in $R(s)$ enters with the pre-factor of 2. The asymptotic behavior of these functions for $s\gg1$ is the following: \begin{equation} \label{orth-asy} Y_{\infty}^{{\rm orth}}(s)=-\frac{1}{\pi^{2}\,s^{2}}+\frac{\cos(2\pi s)}{2\pi^{4}s^{4}}, \end{equation} \begin{equation} \label{symp-asy} Y_{\infty}^{{\rm symp}}(s)=-\frac{1}{4\pi^{2}\,s^{2}}+\frac{\cos(2\pi\, s)}{4\,s}+\frac{\cos(4\pi\,s)}{2(2\pi\,s)^{4}}. \end{equation} One can see that the leading oscillating term decrease as $s^{-4/\beta}$ as was expected. In addition to that, in the symplectic ensemble $\beta=4$, the sub-leading second-harmonic term appears which was absent to all orders in $1/s$ for $\beta=1,2$. The spectral correlations of a quantum system show up in the time-dependence of response to external time-dependent perturbations. For such applications one need to know the Fourier-transform $F(t)$ of the two-level correlation (cluster) function $Y_{\infty}(s)$. It appears to be amazingly simple for the unitary ensemble $\beta=2$: \begin{equation} \label{Funi} F^{{\rm unit}}=\left\{\begin{matrix}|t|-1,&|t|<1\cr 0,&|t|>1. \end{matrix}\right. \end{equation} with the jump of the first derivative at $t=1$ that leads to the oscillations with the period 1 which amplitude decreases as $1/s^{2}$. For the orthogonal ensemble $\beta=1$ there is a jump only in the third derivative: \begin{equation} \label{Fortho} F^{{\rm ortho}}=\left\{\begin{matrix}2|t|-1-|t|\,\ln(1+2|t|),&|t|<1\cr 1-|t|\,\ln\left( \frac{2|t|+1}{2|t|-1}\right),&|t|>1. \end{matrix}\right. \end{equation} For the symplectic ensemble $\beta=4$ there are {\it two} singular points: $|t|=1$ and $|t|=2$ which correspond to two oscillating terms in Eq.(\ref{symp-asy}): \begin{equation} \label{Fortho} F^{{\rm symp}}=\left\{\begin{matrix}\frac{1}{2}\,|t|-1-\frac{1}{4}\,|t|\,\ln|1-|t||,&|t|<2\cr 0,&|t|>2. \end{matrix}\right. \end{equation} \section{Wigner-Dyson level statistics and the Luttinger liquid.} The large-$s$ asymptotics of the two-level correlation function containing both the non-oscillating and the oscillating terms which decay as a certain power-law can be written in a compact form which involves only one single function $G(s)$: \begin{equation} \label{AA-u} Y^{{\rm unit}}_{\infty}(s)=-\frac{1}{4\pi^{2}}\,\frac{\partial^{2}G(s)}{\partial s^{2}}+\cos(2\pi s)\,e^{G(s)}, \end{equation} \begin{equation} \label{AA-o} Y^{{\rm ortho}}_{\infty}(s)=-\frac{1}{2\pi^{2}}\,\frac{\partial^{2}G(s)}{\partial s^{2}}+2\cos(2\pi s)\,e^{2G(s)}, \end{equation} \begin{equation} \label{AA-s} Y^{{\rm symp}}_{\infty}(s)=-\frac{1}{8\pi^{2}}\,\frac{\partial^{2}G(s)}{\partial s^{2}}+\frac{\pi}{\sqrt{8}}\,\cos(2\pi s)\,e^{G(s)/2}+\frac{1}{8}\,\cos(4\pi s)\,e^{2G(s)}, \end{equation} where \begin{equation} \label{G} G=-\ln(2\pi^{2}s^{2}). \end{equation} It turns out that the function $G(x)$ is proportional to the equal-time correlation function of a free bosonic field $\Phi(x,\tau)$ in the {\it two} dimensional space-(imaginary)time, which arises as a bosonized version of the Calogero-Sutherland model Eq.(\ref{CSM}) of interacted fermions. More generally, a great number of models of interacted electrons in one dimension fall into the universality class of {\it Luttinger liquid}\cite{Shurabook} which is characterized by a certain correlation functions at {\it large} separations in space and/or in time. All of them follow from the fact that the fermionic operator $\Psi(x,\tau)$ can be represented as \begin{equation} \label{psi-phi} \Psi(x,\tau)=R\,e^{ik_{F}x}+Le^{-ik_{F}x},\;\;\;\;\;R,\,L=\frac{1}{\sqrt{\pi}}\,{\rm exp}[\pm i\Phi_{R,L}(x,\tau)], \end{equation} where $ \Phi(x,\tau)=\frac{1}{2}\,[\Phi_{R}(x,\tau)+\Phi_{L}(x,\tau)]$ is the {\it free boson field} with the action: \begin{equation} \label{bos-act} S[\Phi]=\frac{1}{2\pi K}\int_{0}^{1/T}d\tau\,\int_{-\infty}^{+\infty}dx\,[(\partial_{x}\Phi)^{2}+ (\partial_{\tau}\Phi)^{2}]=\frac{L}{\pi K T}\sum_{k>0,\omega_{n}=2\pi T n}|A_{k,\omega_{n}}|^{2}\,(\omega_{n}^{2}+k^{2}), \end{equation} where \begin{equation} \label{kw} \Phi(x,\tau)=\sum_{k>0,\omega_{n}=2\pi T\,n}\left\{ A_{k,\omega}e^{i(\omega_{n}\tau+kx)}+c.c\right\}, \end{equation} $\omega_{n}=2\pi T n$ ($n$ are {\it all} integers) and $k=(2\pi/L)m >0$ to avoid double-counting. The action Eq.(\ref{bos-act}) corresponds to: \begin{equation} \label{var} \langle|A_{k,\omega}|^{2}\rangle = \frac{\pi K T}{L}\,\frac{1}{k^{2}+\omega_{n}^{2}}. \end{equation} It is remarkable that {\it interaction} of fermions is encoded in only one single parameter $K$ which is $K<1$ for repulsion, $K=1$ for the non-interacting fermions and $K>1$ for attraction. In other words, with respect to long-distance properties the system of interacted fermions in one dimensions ($1+1$ space-time) is equivalent to a system of free bosons. The physical meaning of this result is that for systems of the Luttinger-liquid universality class all the multitude of effects of electron interaction reduces to dynamics and thermodynamics of the plasmon collective modes. The density operator in this representation is given by: \begin{equation} \label{dens-LL} \rho(x,\tau)-\rho_{0}=\frac{1}{\pi}\,\partial_{x}\Phi(x,\tau)+A_{1}\,\cos[2k_{F}x+ 2\Phi(x,\tau)]+A_{2}\,\cos[4k_{F}x+4\Phi(x,\tau)]+..., \end{equation} where $A_{k}$ are the structural constants which are determined from the details of the system at small distances. The first term in Eq.(\ref{dens-LL}) comes from the combination $[R^{+}R+L^{+}L]$ in $\Psi^{\dagger}\Psi$. It is analogous to the $\delta\rho\propto \nabla{\bf u}$ term in hydrodynamics, where ${\bf u}(x)$ is the mass displacement at a point $x$. The correct evaluation of this contribution requires the regularization $R^{+}R=R^{+}(x+a,\tau)R(x,\tau)$ (and the similar regularization for $L$), where $a=1$ is the lattice constant which corresponds to $k_{F}=\pi$. Oscillating terms proportional to $e^{\pm 2ik_{F}x}$ arise from the cross combinations $R^{+}L+L^{+}R$ in $\Psi^{\dagger}\Psi$. Note that for interacting fermion system there is a vertex correction that involves momentum transfer far away from the Fermi points $\pm k_{F}$. As the result, the density operator expressed in terms of the field $\Psi(x,\tau)$ (which contains only momenta close to the Fermi points) is not simply equal to $\rho=\Psi^{\dagger}\Psi$ but may have higher order terms as well: \begin{equation} \label{high-or} \rho(x,\tau)=\Psi^{\dagger}\Psi+ c_{1}\,(\Psi^{\dagger}\Psi)^{2}+... \end{equation} It is these higher order terms that generate combinations containing higher harmonics like $e^{\pm 4ik_{F}x}$. Using Eq.(\ref{dens-LL}) and Eq.(\ref{bos-act}) one can express the density-density correlation function through the free bosonic correlation function. To this end we use the identity valid for any Gaussian field theory: $$ \langle e^{i n\Phi(x,\tau)}\,e^{-i n\Phi(0,0)}\rangle={\rm exp}\left\langle -\frac{n^{2}}{2}\,(\Phi(x,\tau)-\Phi(0,0))^{2}\right\rangle_{S}. $$ Now observe that for $s\gg 1$ and $T\rightarrow 0$ we have in the thermodynamic limit $L\rightarrow\infty$: $$ \langle \Phi(s,0)\Phi(0,0)\rangle_{S}-\langle \Phi(0,0)\Phi(0,0)\rangle_{S}=-\frac{K}{2}\int_{0}^{\pi}dq\,\frac{1-\cos(qs)} {q}\approx \frac{K}{4}\,G(s), $$ where $G(s)$ is given by Eq.(\ref{G}). Finally, for oscillating part of the density-density correlator we obtain: \begin{equation} \label{osc-ccor} \langle e^{i n\Phi(s,\tau)}\,e^{-i n\Phi(0,0)}\rangle={\rm exp}\left\{\frac{1}{4}K\,n^{2}\,G(s) \right\}. \end{equation} The non-oscillating part is expressed through the second derivative of the correlation function of free boson field: \begin{equation} \label{nosc-ccor} \frac{1}{\pi^{2}}\,\langle \partial_{s}\Phi(s,0)\partial_{s'}\Phi(s',0)\rangle_{S}=-\frac{K}{4\pi^{2}}\,\partial^{2}_{s}\,G(s). \end{equation} Eqs.(\ref{osc-ccor}),(\ref{nosc-ccor}) show that the phenomenology of the Luttinger liquid allows to relate the coefficient in front of the non-oscillating part of the density-density correlator with the coefficients in front of $G(s)$ in the exponent determining the amplitude of the oscillating terms. First of all we fix the interaction parameter $K$ from the amplitude $1/(2\pi^{2}\beta)$ of the non-oscillating part. Eq.(\ref{nosc-ccor}) suggests that: \begin{equation} \label{K-inter} K=\frac{2}{\beta}. \end{equation} Then a comparison of Eq.(\ref{osc-ccor}) with the oscillating terms in Eqs.(\ref{AA-u}),(\ref{AA-o}),(\ref{AA-s}) shows that all coefficients $\kappa$ in the amplitude of $n$-th harmonic $e^{\kappa\,G(s)}\,\cos(2\pi n\, x)$ are equal to $\kappa=n^{2}K$ with $K$ given by Eq.(\ref{K-inter}). This is exactly the parameter $K$ that corresponds to the Calogero-Sutherland model Eq.(\ref{CSM}). Thus we have demonstrated that the Wigner-Dyson level statistics at large level separations corresponds to the particle statistics of the Calogero-Sutherland model at zero temperature. One can ask a question: how the Wigner-Dyson ensemble should be deformed in order to retain this analogy with the Calogero-Sutherland model also for finite temperatures $T\neq 0$. The answer is \cite{KTs} that the proper deformation is given by the Gaussian non-invariant ensemble Eq.(\ref{crit}) at large values of the parameter $B$. The corresponding temperature of the Calogero-Sutherland model is \cite{KTs}: \begin{equation} \label{T-B} T=\frac{1}{4\beta B}. \end{equation} \section{Field theories for random matrix ensembles} In this section we derive the field theory for an arbitrary Gaussian random matrix ensemble. This formalism, known as nonlinear super-symmetric sigma-model \cite{Efet} has been first applied to the Wigner-Dyson random matrix ensemble. However, its real strength is in the possibility of extension to the non-invariant random matrix ensembles as well as to real disordered conductors with diffusive dynamics of particles. We start by writing down the expression for retarded ($G^{R}$) or advanced $G^{A}$ Green's function in terms of the functional integral over complex variables $\varphi_{n}$: \begin{equation} \label{GF} G_{nm}^{R/A}=\left([E_{\pm}-{\bf H}]^{-1}\right)_{nm}=\frac{\mp i}{Z}\,\int {\cal D}\varphi\,{\cal D}\varphi^{*}\,\,\varphi_{n}\,\varphi_{m}^{*}\,{\rm exp}\left[ iS_{\pm}[\varphi]\right], \end{equation} where \begin{equation} \label{Spm} S_{\pm}[\varphi]=\pm\sum_{i,j}\varphi_{i}^{*}\,\left[E_{\pm}\delta_{ij}-H_{ij} \right]\,\varphi_{j} \end{equation} and $E_{\pm}=E\pm(\omega/2 + i 0)$. The sign $\pm$ stands for $G^{R}$ or $G^{A}$, respectively. There would be no problem to average Eq.(\ref{GF}) over the Gaussian random entries $H_{nm}$ if not the normalization constant (partition function) $Z$: \begin{equation} \label{Z} Z=\int {\cal D}\varphi\,{\cal D}\varphi^{*}\,{\rm exp}\left[ iS_{\pm}[\varphi]\right]. \end{equation} which also depends on $H_{nm}$. With the $Z$ present one has a problem, the {\it problem of denominator}. There are different ways of overcoming this problem, e.g. the {\it replica trick}. However, here we use another trick, the {\it super-symmetry} method \cite{Efet}. In the core of this method is the calculus of anti-commuting (Grassmann) variables $\mu_{m}$: \begin{equation} \label{mu} \mu_{n}\,\mu_{m}=-\mu_{m}\,\mu_{n},\;\;\;\;\mu_{n}^{2}=0. \end{equation} One can define the Grassmann integral with the shortest table of integrals ever: \begin{equation} \label{int} \int \mu_{n}\,d\mu_{n}=\frac{1}{\sqrt{\pi}},\;\;\;\;\;\int d\mu_{n}=0. \end{equation} Since any function $f(\mu_{n})=f(0)+f'(0)\,\mu_{n}$, the Grassmann integration is essentially a differentiation. Now let us compute the integral \begin{equation} \label{int-Gr} \int \,\prod_{i}d\mu^{*}_{i}d\mu_{i}\, {\rm exp}\left[-\sum_{n}\mu_{n}^{*}\,a_{n}\,\mu_{n} \right]=\int\,\prod_{i}d\mu^{*}_{i}d\mu_{i}\,\mu_{i}^{*}\mu_{i}\,(-a_{i})= \int\,\prod_{i}\mu_{i}^{*}d\mu^{*}_{i}\,\mu_{i} d\mu_{i}\,a_{i}=\prod_{i}\frac{a_{i}}{\pi}. \end{equation} To accomplish this we expand the exponential function to leave only the term that contains a {\it complete} set $\mu_{1}^{*}\mu_{1}...\mu_{n}^{*}\mu_{n}$ of Grassmann variables and apply the table of integration Eq.(\ref{int}). The corresponding integral over the usual complex variables would give the following result: \begin{equation} \label{int-com} \int \,\prod_{i}d\varphi^{*}_{i}d\varphi_{i}\, {\rm exp}\left[-\sum_{n}\varphi_{n}^{*}\,a_{n}\,\varphi_{n} \right]=\prod_{i}\frac{\pi}{a_{i}}. \end{equation} We see a remarkable property: the product of the two integrals is equal to 1. This property remains true for any Gaussian integrals of commuting and anti-commuting variables. In particular, \begin{equation} \label{1-Z} Z^{-1}=\int {\cal D}\mu^{*}\,{\cal D}\mu\,{\rm exp}\left[ iS_{\pm}[\mu]\right]. \end{equation} Now the Green's functions can be represented without the denominator: \begin{equation} \label{GF1} G_{nm}^{R/A}= \mp i\,\int {\cal D}\psi\,\,\;\;\varphi_{n}\,\varphi_{m}^{*}\,{\rm exp}\left[ iS_{\pm}[\psi]\right], \end{equation} where \begin{equation} \label{sup-act} S_{\pm}[\psi]=S_{\pm}[\varphi]+S_{\pm}[\mu]= \pm\sum_{i,j}\psi_{i}^{\dagger}\,\left[E_{\pm}\delta_{ij}-H_{ij} \right]\,\psi_{j}. \end{equation} Here we introduced the {\it super-vectors} $\psi$ and $\psi^{\dagger}$: \begin{equation} \label{sup-vect} \psi^{\dagger}=(\varphi^{*},\mu^{*}),\;\;\;\;\;\psi=\left(\begin{matrix} \varphi\cr \mu\end{matrix} \right) \end{equation} and the super-measure: \begin{equation} \label{sup-measure} {\cal D}\psi={\cal D}\varphi^{*}\,{\cal D}\varphi\,{\cal D}\mu^{*}\,{\cal D}\mu. \end{equation} The action Eq.(\ref{sup-act}) and the integration measure Eq.(\ref{sup-measure}) are {\it super-symmetric}, i.e. the commuting and anti-commuting variables enter in a fully symmetric way. The super-symmetry is however broken in the {\it pre-exponent} in Eq.(\ref{GF1}), as it depends only on the commuting variables. Now when the problem of denominator is solved by the supersymmetry trick, the next step is to average over the Gaussian ensemble of $H_{ij}$. To this end we write: $$ \mp i\sum_{ij}\psi^{\dagger}_{i}\psi_{j}H_{ij} =\frac{\mp i}{2}\sum_{ij}\left(\psi^{\dagger}_{i}\psi_{j}H_{ij}+\psi^{\dagger}_{j}\psi_{i}H_{ji}\right) $$ Averaging of the r.h.s. is done independently for each pair of $i,j$ using the identity: $$ r.h.s.-\frac{1}{A_{ij}}\,|H_{ij}|^{2}=-\frac{1}{A_{ij}}\, \left(H_{ij}\pm\frac{i}{2}A_{ij}\,\psi_{i}^{\dagger}\psi_{j} \right)\,\left(H_{ji}\pm\frac{i}{2}A_{ij}\,\psi_{j}^{\dagger}\psi_{i} \right)-\frac{A_{ij}}{4}\,\psi^{\dagger}_{i}\psi_{j}\psi^{\dagger}_{j}\psi_{i}. $$ From now on for simplicity we will consider the case $\beta=2$. Then $$ \int dH_{ij}dH_{ij}^{*}\,{\rm exp}\left(r.h.s.-\frac{1}{A_{ij}}\,|H_{ij}|^{2} \right)=\int d\tilde{H_{ij}}d\tilde{H_{ij}^{*}}\,{\rm exp}\left( -\frac{1}{A_{ij}}\,|\tilde{H_{ij}}|^{2}\right)\;\;{\rm exp}\left( -\frac{A_{ij}}{4}\,\psi^{\dagger}_{i}\psi_{j}\psi^{\dagger}_{j}\psi_{i}\right), $$ where $$ \tilde{H_{ij}}=H_{ij}\pm\frac{i}{2}\psi_{i}^{\dagger}\psi_{j}. $$ The simplicity of the case $\beta=2$ is that $\tilde{H_{ij}}$ belongs to the same manifold of complex numbers as $H_{ij}$, so that one may replace in the integral over the entire manifold $\tilde{H_{ij}}\rightarrow H_{ij}$. Thus on the right hand side we obtain the normalization integral for the random matrix ensemble averaging. So we obtain for the disorder average: \begin{equation} \label{av} \left\langle{\rm exp}\left(\mp i\sum_{ij}\psi^{\dagger}_{i}\psi_{j}H_{ij} \right) \right\rangle={\rm exp}\left(-\frac{1}{4}\sum_{ij} A_{ij}\,\psi^{\dagger}_{i}\psi_{j}\psi^{\dagger}_{j}\psi_{i}\right). \end{equation} Now we define the super-matrix \begin{equation} \label{Q-til} \tilde{Q}_{i}=\psi_{i}\otimes\psi_{i}^{\dagger}=\left( \begin{matrix} \varphi_{i}\varphi_{i}^{*}&\varphi_{i}\mu_{i}^{*}\cr \mu_{i}\varphi_{i}^{*}& \mu_{i}\mu_{i}^{*}\cr \end{matrix}\right)\equiv\left( \begin{matrix} BB&BF\cr FB& FF\cr \end{matrix}\right) \end{equation} and the super-trace: \begin{equation} \label{st} {\rm STr}\left( \begin{matrix} BB&BF\cr FB& FF\cr \end{matrix}\right)=BB-FF. \end{equation} Then Eq.(\ref{av}) can be conveniently rewritten as $$ {\rm exp}\left(-\frac{1}{4}\sum_{ij}A_{ij}\,{\rm STr}[\tilde{Q}_{i}\tilde{Q}_{j}] \right). $$ Finally, the averaged Green's function can be represented as follows: \begin{equation} \label{GF2} \langle G_{nm}^{R/A}\rangle= \mp i\,\int {\cal D}\psi\,\,\;\;\varphi_{n}\,\varphi_{m}^{*}\,{\rm exp}\left[ -F[\tilde{Q}]\right],\;\;\;\;\;F[\tilde{Q}]=\mp iE_{\pm}\sum_{i}{\rm STr}[\tilde{Q}_{i}]+\frac{1}{4}\sum_{ij}A_{ij}\,{\rm STr}[\tilde{Q}_{i}\tilde{Q}_{j}],\;\;\;\;\tilde{Q}_{i}=\psi_{i}\otimes\psi_{i}^{\dagger} \end{equation} Thus we derived the {\it deterministic field theory} which is equivalent to the Gaussian random matrix ensemble and is suitable to compute the average Green's function. One can see that the matrix of variances $$ A_{ij}=\langle |H_{ij}|^{2}\rangle $$ plays a role of the coupling constant ("coupling matrix") in the corresponding action $F[\tilde{Q}]$. The filed-theory representation Eq.(\ref{GF2}) can be extended to consider the averaged product of $\langle G^{R}G^{A}\rangle $ which is necessary to be able to compute the {\it two-point} correlation functions. To this end, one introduce the double set of commuting and anti-commuting variables: one for $G^{R}$ and another for $G^{A}$ and then repeats with minor modifications all the above steps: \begin{equation} \label{R-A} \left\langle G_{nm}^{R}\left(E+\frac{\omega}{2}\right)G^{A}_{n'm'}\left(E-\frac{\omega}{2}\right)\right\rangle= \,\int {\cal D}\Psi\,\,\;\;\varphi^{R}_{n}\,\varphi^{R*}_{m}\,\varphi^{A}_{n'}\, \varphi^{A*}_{m'}\;\;{\rm exp}\left[ -F[\bar{Q}]\right], \end{equation} where the action $S[\bar{Q}]$ takes the form: \begin{equation} \label{SS} F[\bar{Q}]=-iE\sum_{i}{\rm STr}[\bar{Q}_{i}]-i\frac{\omega+i0}{2}\,\sum_{i}{\rm STr}[\Lambda \bar{Q}_{i}]+\frac{1}{4}\sum_{ij}A_{ij}\,{\rm STr}[\bar{Q}_{i}\bar{Q}_{j}],\;\;\;\;\bar{Q}_{i}=\Psi_{i}\otimes\bar{\Psi}_{i}, \end{equation} with \begin{equation} \label{Psi} \bar{\Psi}=\left(\begin{matrix}\varphi^{R*}&\mu^{R*}&-\varphi^{A*}&-\mu^{A*}\cr \end{matrix} \right),\;\;\;\;\Psi=\left(\begin{matrix}\varphi^{R}\cr\mu^{R}\cr\varphi^{A}\cr \mu^{A}\cr \end{matrix} \right),\;\;\;\;\Lambda=\Lambda_{2}\otimes {\bf 1}=\left(\begin{matrix}1&0&0&0\cr 0&1&0&0\cr0&0&-1&0\cr 0&0&0&-1\cr \end{matrix} \right). \end{equation} There is another, in some sense {\it dual}, field-theory representation similar to Eq.(\ref{R-A}). In contrast to Eq.(\ref{R-A}) it involves the {\it inverse coupling matrix} $(A^{-1})_{ij}$ rather than the variance matrix $A_{ij}$. In order to obtain this representation one makes the {\it Hubbard-Stratonovich transformation}: \begin{equation} \label{HS} {\rm exp}\left\{-\frac{1}{4}\sum_{ij}A_{ij}\,{\rm STr}[(\Psi_{i}\otimes\bar{\Psi}_{i})(\Psi_{j}\otimes\bar{\Psi}_{j})] \right\}=\int {\cal D}P\, {\rm exp}\left\{-\sum_{ij}A^{-1}_{ij}\,{\rm STr}[P_{i}P_{j}]+i\sum_{i}{\rm STr}[(\Psi_{i}\otimes\bar{\Psi}_{i})P_{i}]\right\}, \end{equation} where $P_{i}$ is a super-matrix field. If one substitutes the last term in Eq.(\ref{SS}) for Eq.(\ref{HS}) the remaining integral over the $\Psi$ fields is Gaussian which can be done using a generalization of Eqs.(\ref{int-Gr})-(\ref{int-com}) for the case where the super-matrix $K_{i}$ is not proportional to the unity matrix (for which ${\rm STr[\ln K_{i}]}=0$): \begin{equation} \label{Guss-int} \int {\cal D}\Psi\;\;{\rm exp}\left\{ \bar{\Psi}_{i}\,K_{i}\,\Psi_{j}\right\}={\rm exp}\left\{-\sum_{i}{\rm STr}[\ln K_{i}] \right\}. \end{equation} Now we are in a position to write down the full action of the dual representation: \begin{equation} \label{dual} F[P]=\sum_{ij}A^{-1}_{ij}\,{\rm STr}[P_{i}P_{j}]+\sum_{i}{\rm STr}[\ln(E-P_{i}+(\omega/2)\, \Lambda)]. \end{equation} In order to appreciate the possibilities this representation is offering and also for further simplifications of Eq.(\ref{dual}) we compute the inverse coupling matrix $A^{-1}_{ij}$ for the important case of the banded random matrix ensembles where the variance matrix $A_{ij}$ is given by Eq.(\ref{band}). The matrix element of a matrix inverse with respect to a matrix $A_{ij}=A(i-j)$ is given by: \begin{equation} \label{inV} A^{-1}_{ij}=\int_{-\pi}^{\pi}\frac{dk}{2\pi}\,G(k)\,e^{ik(i-j)},\;\;\;\; G(k)=\left[\sum_{m=-\infty}^{+\infty}\tilde{A}(k+2\pi\, m)\right]^{-1},\;\;\;\;\tilde{A}(k)=\int_{-\infty}^{+\infty} dr\,A(r)\,e^{-ik r}. \end{equation} Note that the summation over the reciprocal lattice vector $2\pi m$ is important. For the case of the exponential $A(r)=e^{-|r|/B}$, we have: $$ A(k)=\frac{2B}{1+B^{2}k^{2}},\;\;\;\;\sum_{m=-\infty}^{+\infty}\frac{2B}{1+B^{2}(k+2\pi m)^{2}}=\frac{\sinh\left(\frac{1}{B}\right)}{\cosh\left( \frac{1}{B}\right)-\cos k}. $$ One can see that for large $B\gg 1$ the inverse coupling matrix $A^{-1}_{ij}$ is extremely simple: \begin{equation} \label{lar-B} G(k)\approx B\,(1-\cos k),\;\;\;\;A^{-1}_{ij}\approx \frac{B}{2}\,(2\delta_{ij}-\delta_{i,j+1}-\delta_{i,j-1}),\;\;\;\;A_{0}^{-1}= \sum_{j}A^{-1}_{ij}\approx \frac{1}{2B}. \end{equation} Eq.(\ref{lar-B}) shows that locally it is just the lattice second derivative. However, the pre-factor in front of it is large $\sim B$ and it becomes infinite for the Wigner-Dyson ensemble. This means a large cost of variation of $P_{i}$ in the space. Let us consider $P_{i}=P_{0}$ being independent of $i$ to the first approximation. Then the action becomes: \begin{equation} \label{P0} N^{-1}\,F[P_{0}]=A_{0}^{-1}\,{\rm STr}P_{0}^{2}+{\rm STr}\ln(E-P_{0}). \end{equation} Here we also neglected a term proportional to $\omega$ which is legitimate as long as $\omega\ll \sqrt{B}$. Minimizing the action with respect to $P_{0}$ we obtain a saddle-point equation: \begin{equation} \label{spe} P_{0}(E-P_{0})=A_{0}/2. \end{equation} The solution to this saddle-point equation is {\it degenerate}: \begin{equation} \label{deg} P_{0}=\frac{1}{2}\,\left(E+iQ\,\sqrt{2A_{0}-E^{2}}\right), \end{equation} where $Q$ is a super-matrix obeying the constraint \begin{equation} \label{q21} Q^{2}=1. \end{equation} Another constraint comes from the requirement of super-symmetry: \begin{equation} \label{const2} {\rm STr}Q=0. \end{equation} Indeed, a super-matrix $Q$ obeying the constraint Eq.(\ref{q21}) after the diagonalization must contain only $\pm 1$ on the diagonal. The super-symmetry (the symmetry between commuting and anti-commuting variables) implies that it must have the same diagonal elements corresponding to fermionic and bosonic variables of the given type ($R$ or $A$). But then necessarily ${\rm STr}Q=0$. Now take into account (as the first order expansion in $\omega$) the term in Eq.(\ref{dual})proportional to the energy difference $\omega$ and allow for slow variations of the field $Q=Q_{i}$ in space which do not violate the saddle-point condition Eq.(\ref{q21}). Then we obtain plugging Eq.(\ref{deg}) into Eq.(\ref{dual}): \begin{equation} \label{sigma} F[Q]=-\frac{1}{4}\,(\pi\rho\,A_{0})^{2}\,\sum_{ij}A^{-1}_{ij}\,{\rm STr}[Q_{i}Q_{j}]-i\frac{\pi\rho\omega}{2}\sum_{i}{\rm STr}[\Lambda Q_{i}], \end{equation} where \begin{equation} \label{rrho} \rho=\rho_{0}(E)=\frac{\sqrt{2A_{0}-E^{2}}}{\pi A_{0}},\;\;\;\;A_{0}=\sum_{i}A_{ij}. \end{equation} This is the celebrated action of the {\it nonlinear} $\sigma$ {\it model} \cite{Efet, Mirlin-Fyod}. For the particular case of banded random matrices with $A_{ij}$ given by Eq.(\ref{band}) we obtain: \begin{equation} \label{sigma-band} F[Q]=-D\sum_{i}{\rm STr}[(Q_{i}-Q_{i+1})^{2}]-i\frac{\pi\rho\omega}{2}\sum_{i}{\rm STr}[\Lambda Q_{i}], \end{equation} where $D=\frac{1}{2}B\,\left(B-\frac{E^{2}}{4}\right)\sim B^{2}$. The continuous limit of this model is the diffusive nonlinear $\sigma$-model: \begin{equation} \label{sigma-band} F[Q]=-D\int dx\;{\rm STr}[(\nabla Q)^{2}]-i\frac{\pi\rho\omega}{2}\int dx\;{\rm STr}[\Lambda Q(x)], \end{equation} which was originally derived by Efetov \cite{Efet} to describe the crossover from the diffusive dynamics to the Anderson localization in the quasi-one dimensional multi-channel disordered wire. This demonstrates the isomorphism of the problem of quasi-one dimensional localization and the problem of banded random matrices \cite{Mirlin-Fyod}. Closing this chapter we note that the derivation of Eq.(\ref{sigma}) from the exact Eq.(\ref{dual}) requires the {\it saddle-point approximation} Eq.(\ref{spe}). Thus Eq.(\ref{sigma}) is justified only if the energy cost of space variations of $Q_{i}$ is high. This happens when the variance matrix $A_{ij}$ has a form of a banded matrix which is approximately constant at $|i-j|<B$, where $B\gg 1$. For the Wigner-Dyson ensemble $A_{ij}=1$ and the bandwidth is maximum possible $B\sim N$ (in particular, $A_{0}=N$). In the limit $N\rightarrow\infty$ all spacially varying configurations of the field $Q$ ({\it non-zero modes}) are strictly forbidden. Neglecting them we obtain the {\it zero-mode} nonlinear sigma-model \cite{Efet} which describes the statistics of energy levels in fully chaotic quantum systems of confined geometry ({\it quantum dots}): \begin{equation} \label{ZM} F_{WD}[Q]=-i\frac{\pi\,s}{2}\;{\rm STr}[\Lambda Q],\;\;\;\;s=\frac{\omega}{\Delta}. \end{equation} \section{How to compute observables: semi-circle law from the solution to a quadratic equation} Let us demonstrate how to compute observable quantities within the field theory using the simplest example of the mean density of states. It is given by \begin{equation} \label{dOs} \rho(E)=(-2\pi i)^{-1}\,(\langle G^{R}_{nn}(E) \rangle-\langle G^{R}_{nn}(E) \rangle)=\frac{1}{2\pi}\int {\cal D}\Psi\;(\varphi_{n}^{R*}\varphi_{n}^{R}+\varphi_{n}^{A*}\varphi_{n}^{A})\;e^{-F[\bar{Q}]}. \end{equation} One can check that the pre-exponent in Eq.(\ref{dOs}) can be represented as \begin{equation} \label{pre-exp-Pi} (\varphi_{n}^{R*}\varphi_{n}^{R}+\varphi_{n}^{A*}\varphi_{n}^{A})={\rm STr}[\Pi\,\Psi_{n}\otimes\bar{\Psi}_{n}], \end{equation} where \begin{equation} \label{Pi} \Pi=\Pi^{R}-\Pi^{A},\;\;\;\;\;\;\; \Pi^{R}=\left(\begin{matrix}1&0&0&0\cr0&0&0&0\cr0&0&0&0\cr0&0&0&0\cr \end{matrix}\right),\;\;\;\;\;\Pi^{A}=\left(\begin{matrix}0&0&0&0\cr0&0&0&0\cr0&0&1&0\cr0&0&0&0\cr \end{matrix}\right). \end{equation} Now we introduce an infinitesimal {\it source field} $h_{n}$ and add to the action Eq.(\ref{SS}) a term $$ \delta F[\bar{Q},h]=-i\sum_{i}h_{i}\,{\rm STr}[\Pi\,\bar{Q}_{i}]. $$ One can easily check that the density of states is given by a differentiation of the partition function with respect to the field $h$: \begin{equation} \label{derr} \rho(E)=\frac{1}{2\pi i}\,\left.\frac{\partial}{\partial h_{n}}\,\int {\cal D}\Psi\;e^{-F[\bar{Q},h]}\right|_{h\rightarrow 0},\;\;\;\;\;F[\bar{Q},h]=F[\bar{Q}]+\delta F[\bar{Q},h]. \end{equation} Note that the additional term in the action proportional to $h$ enters exactly like the term proportional to $\omega$, so that in the final action of the sigma-model Eq.(\ref{sigma}) one can simply substitute \begin{equation} \label{L-h} \frac{\omega}{2}\,\Lambda\rightarrow\frac{\omega}{2}\,\Lambda+h_{n}\,\Pi. \end{equation} Then Eq.(\ref{derr}) results in: \begin{equation} \label{rrrho} \rho(E)=\rho_{0}(E)\;\frac{1}{2}\int {\cal D}Q\,\;{\rm STr}[\Pi Q_{n}]\;e^{-F[Q]}, \end{equation} where $\rho_{0}(E)$ is given by Eq.(\ref{rrho}) and $F[Q]$ is given by Eq.(\ref{sigma}) at $\omega=0$. We see that the quantity $\rho_{0}(E)$ which appear in Eq.(\ref{sigma}) from the solution of the quadratic saddle-point equation Eq.(\ref{spe}) is not accidentally of the form of a semi-circle as the mean density of states is proportional to it. In the case of the Wigner-Dyson ensemble the functional $F[Q]$ at $\omega=0$ is simply zero and the integral in Eq.(\ref{rrrho}) is a constant independent of $E$. Thus we conclude that the semicircle law appears in this formalism from the solution of a quadratic saddle-point equation. Other statistics such as the two-point correlation functions can also be easily computed using the formalism of the nonlinear sigma model, however not so simply as the semi-circle law. For this one needs a proper parametrization of the matrix $Q$ which resolves the constraints Eqs.(\ref{q21}),(\ref{const2}). \section{Symmetry of super-matrices $\bar{Q}$ and $Q$.} Let us return back to the derivation of the functional representation in terms of $\bar{Q}$. It appears \cite{KOs} that by a change of variables: \begin{eqnarray} \label{ChoV} \varphi^{R/A}&=&\pm i\sqrt{\lambda_{1/2}}\,\,e^{\pm i\varphi/2+i\Omega}\,(1-\frac{1}{2}\chi^{*}_{R/A}\chi_{R/A})\\ \nonumber \mu^{R/A}&=& \pm i \sqrt{\lambda_{1/2}}\,\,e^{\pm i\varphi/2+i\Omega}\,\chi_{R/A}, \end{eqnarray} where $\lambda_{1/2}\geq 0$, $0\leq \varphi\leq 2\pi$, $0\leq\Omega\leq \pi$ and $\chi_{R/A}, \chi^{*}_{R/A}$ are the new anti-commuting variables, one can represent $\bar{Q}=\Psi\otimes\bar{\Psi}$ in the following form: \begin{equation} \label{param} \bar{Q}=U\,\bar{\Sigma}\;U^{-1}=\left(\begin{matrix}u_{R} & 0 \cr 0 & u_{A}\cr \end{matrix} \right)\;\left(\begin{matrix}\bar{\Sigma}_{RR} & \bar{\Sigma}_{RA} \cr \bar{ \Sigma}_{AR} & \bar{\Sigma}_{AA}\cr \end{matrix}\right)\,\left(\begin{matrix}u^{-1}_{R} & 0 \cr 0 & u^{-1}_{A}\cr \end{matrix}\right). \end{equation} The beauty of this form is that the commuting and anti-commuting variables are separated by factorization. Namely, the outer matrices $U,U^{-1}$ containing $2\times 2$ matrices $u_{R/A}$ and $u^{-1}_{R/A}$ \begin{equation} \label{U} u_{R/A}=\left(\begin{matrix}1-\frac{1}{2}\chi_{R/A}^{*}\chi_{R/A} & -\chi_{R/A}^{*} \cr \cr \chi_{R/A} & 1+\frac{1}{2}\chi_{R/A}^{*}\chi_{R/A}\cr \end{matrix}\right)_{BF},\;\;\;\;\;u^{-1}_{R/A}= \left(\begin{matrix}1-\frac{1}{2}\chi_{R/A}^{*}\chi_{R/A} & \chi_{R/A}^{*} \cr \cr -\chi_{R/A} & 1+\frac{1}{2}\chi_{R/A}^{*}\chi_{R/A}\cr \end{matrix}\right)_{BF} \end{equation} are made of the anti-commuting variables. The inner matrix $\bar{\Sigma}$ $$ \bar{\Sigma}=\left(\begin{matrix}\bar{\Sigma}_{RR} & \bar{\Sigma}_{RA} \cr \bar{\Sigma}_{AR} & \bar{\Sigma}_{AA}\cr \end{matrix}\right)_{RA}\equiv\left(\begin{matrix}\bar{\Sigma}_{BB} & 0 \cr 0 & \bar{\Sigma}_{FF}\cr \end{matrix}\right)_{BF}, $$ which is diagonal in the FB space, contains only commuting variables with only BB sector non-zero: \begin{equation} \label{Db} \bar{\Sigma}_{BB}=\left(\begin{matrix}\lambda_{1}& \sqrt{\lambda_{1}\lambda_{2}}\,e^{i\varphi} \cr -\sqrt{\lambda_{1}\lambda_{2}}\,e^{-i\varphi} & -\lambda_{2}\cr \end{matrix}\right)_{RA},\;\;\;\;\bar{\Sigma}_{FF}=0. \end{equation} One can show that the factorized form Eq.(\ref{param}) is common to both the field $\bar{Q}$ and the {\it dual field} $Q$, with matrices $U,U^{-1}$ being exactly the same. However the structure of the inner matrices $\bar{\Sigma}$ and $\Sigma$ are different. Efetov has shown \cite{Efet} that the constraints Eqs.(\ref{q21}),(\ref{const2}) give rise to the following structure of $\Sigma$: \begin{eqnarray} \label{DbEf} \Sigma_{BB}&=&\left(\begin{matrix}\lambda& \sqrt{\lambda^{2}-1}\,e^{i\varphi} \cr -\sqrt{\lambda^{2}-1}\,e^{-i\varphi} & -\lambda\cr \end{matrix}\right)_{RA},\\ \label{DbEf1} \Sigma_{FF}&=&\left(\begin{matrix}\lambda_{F}& \sqrt{1-\lambda^{2}_{F}}\,e^{i\varphi_{F}} \cr \sqrt{1-\lambda^{2}_{F}}\,e^{-i\varphi_{F}} & -\lambda_{F}\cr \end{matrix}\right)_{RA}, \end{eqnarray} where $\lambda\geq 1$, $-1\leq\lambda_{F}\leq 1$, and $\varphi, \varphi_{F}\in(0,2\pi)$. To make practical calculations possible we also give (without derivation) the expressions for the Jacobians of the transformation from original variables to the variables of the above paramerization. They are \begin{equation} \label{J-Os} J[\bar{Q}]=\frac{\pi}{4\;\lambda_{1}\lambda_{2}}, \end{equation} and \begin{equation} \label{J-Ef} J[Q]= \frac{1}{8(\lambda-\lambda_{F})^{2}}, \end{equation} for the theories with coupling matrices $A^{-1}_{ij}$ and $A_{ij}$, respectively. The matrices $\bar{\Sigma}_{BB}$ of the structure Eq.(\ref{Db}) as well as the matrices $\Sigma_{BB}$ of the structure Eq.(\ref{DbEf}) can be diagonalized by the {\it pseudo-unitary} rotation $R$: \begin{equation} \label{psunit} \bar{\Sigma}_{BB}=R\,\bar{D}\,R^{-1},\;\;\;\;\;\Sigma_{BB}=R\,D\,R^{-1},\;\;\;\;\; R^{-1}=\Lambda_{2}\,R^{\dagger}\,\Lambda_{2}, \end{equation} where \begin{equation} \label{psDia} \bar{D}=\left(\begin{matrix}|\lambda_{1}-\lambda_{2}|\;\theta(\lambda_{1}-\lambda_{2})&0\cr 0&-|\lambda_{1}-\lambda_{2}|\;\theta(\lambda_{2}-\lambda_{1})\cr \end{matrix} \right),\;\;\;\;\;D=\Lambda_{2}\equiv\left( \begin{matrix}1&0\cr 0&-1\cr \end{matrix} \right). \end{equation} It is clear that the rotation matrix $R$ can be multiplied by a diagonal matrix $$ \left( \begin{matrix}e^{i\Phi_{R}}&0\cr 0&e^{i\Phi_{A}}\cr \end{matrix} \right)\in U(1)\otimes U(1) $$ without violating the condition of pseudo-unitarity and without changing the matrix $\Sigma_{BB}$ or $\bar{\Sigma}_{BB}$. To eliminate the redundant degrees of freedom (which lead to the divergency of the functional integrals) the group of pseudo-unitary matrices $U(1,1)$ should be factorized as $R\;(U(1)\otimes U(1))$, where $R$ being a factor-group $\frac{U(1,1)}{U(1)\otimes U(1)}$. On top of that the diagonal matrix $\bar{D}$ has a free parameter $$ \lambda_{1}-\lambda_{2}\in {\bf R}. $$ The complete symmetry of the manifold of matrices $\bar{\Sigma}_{BB}$ and thus the complete symmetry of $\bar{Q}$ is: $$ \bar{Q}\in{\bf \frac{U(1,1)}{U(1)\otimes U(1)}\otimes} {\bf R}. $$ In contrast to that the symmetry of matrices $\Sigma_{BB}$ is simply $ \Sigma_{BB}\in \frac{U( 1,1)}{U(1)\otimes U(1)} $. Its counterpart $\Sigma_{FF}$ has the symmetry $ \Sigma_{FF}\in \frac{U(2)}{U(1)\otimes U(1)}$ as it can be diagonalized by the {\it unitary} rotation matrix $R_{F}$. The complete symmetry of the $Q$ field in the Efetov's nonlinear sigma-model is therefore: $$ Q\in{\bf \frac{U(1,1)}{U(1)\otimes U(1)}\otimes \frac{U(2)}{U(1)\otimes U(1)}}. $$ Note by passing that the number of independent variables in $\bar{Q}$ and $Q$ is different. While both have 4 anti-commuting variables, the number of commuting variables is $2+2=4$ for the filed $Q$ and $2+1=3$ or the field $\bar{Q}$. Thus we see that the duality transformation and the saddle-point approximation not only invert the coupling matrix $A_{ij}$ but also change the symmetry of the {\it target space}. Such type of duality is encountered in the string theory and is called $T-duality$. \section{Eigenfunction statistics} In this section we show how to compute eigenfunction statistics using the field theory formalism. As usual, the starting point is to express the physical quantity of interest in terms of the Green's functions. To this end we study the product: \begin{equation} \label{l-m} K_{l,m}=[G^{R}_{nn}]^{l}\,[G^{A}_{nn}]^{m}=\left( \sum_{i}\frac{|\Psi_{i}(n)|^{2}}{E-E_{i}+i\delta}\right)^{l}\;\left( \sum_{i'}\frac{|\Psi_{i'}(n)|^{2}}{E-E_{i'}-i\delta}\right)^{m}, \end{equation} where we used the representation of Green's functions in terms of exact eigenfunction $\Psi_{i}(n)$ and exact eigenvalues $E_{n}$ of a random matrix Hamiltonian. Let us multiply Eq.(\ref{l-m}) by an infinitesimal $\delta^{l+m-1}$ average over realizations of the random matrix ensemble and do the limit $\delta\rightarrow 0$. This trick singles out only one state of the double sum, the one that is accidentally at the energy $E$: \begin{equation} \label{one-state} \lim_{\delta\rightarrow 0}\delta^{l+m-1}\,K_{l,m}=\lim_{\delta\rightarrow 0}\left\langle \sum_{i}\frac{|\Psi_{i}(n)|^{2(l+m)}\;\delta^{l+m-1}}{(E-E_{i}+i\delta)^{l}(E-E_{i}-i\delta)^{m}}. \right\rangle \end{equation} The smallness of the interval $|E-E_{i}|\sim \delta$ is the reason why the power of $\delta$ is $l+m-1$ and not $l+m$. Indeed, let the joint probability distribution function for $\Psi_{i}(n)$ and $E_{i}$ be $P(\Psi_{i},E_{i})$. It is a smooth function of $E_{n}$ which does not change at a scale $\delta\rightarrow 0$. Then averaging in Eq.(\ref{one-state}) can be performed as follows: \begin{equation} \label{av-per} \sum_{i}\int d\Psi_{i}dE_{i}\,P(\Psi_{i},E_{i})\,\frac{|\Psi_{i}|^{2(l+m)}} {(E-E_{i}+i\delta)^{l}(E-E_{i}-i\delta)^{m}}\approx\sum_{i}\int d\Psi_{i}\,P(\Psi_{i},E)\,|\Psi_{i}|^{2(l+m)}\,C_{l,m}, \end{equation} where \begin{equation} \label{C-C} C_{l,m}=\int_{-\infty}^{+\infty} \frac{dE_{i}}{(E-E_{i}+i\delta)^{l}(E-E_{i}-i\delta)^{m}}=(2\delta)^{1-(l+m)}\;2\pi i^{m-l}\frac{(l+m-2)!}{(l-1)!(m-1)!}. \end{equation} Let us define also the moment of the $|\Psi_{i}(n)|^{2}$ at an energy $E$: \begin{equation} \label{mom-def} \langle |\Psi_{i}|^{2 p} \rangle_{E}=\frac{1}{\rho(E)}\,\left\langle \sum_{i}|\Psi_{i}|^{2 p}\;\delta(E-E_{i}) \right\rangle\equiv\frac{1}{\rho(E)}\,\sum_{i}\int d\Psi_{i}\int dE_{i}\;P(\Psi_{i},E_{i})\;|\Psi_{i}|^{2 p}\;\delta(E-E_{i}). \end{equation} Comparing Eqs.(\ref{mom-def}),(\ref{av-per}) we arrive at: \begin{equation} \label{mom-G} \langle |\Psi_{i}|^{2(l+m)}\rangle_{E} = \frac{1}{2\pi\rho(E)}\,i^{l-m}\;\frac{(l-1)!(m-1)!}{(l+m-2)!}\;\lim_{\delta\rightarrow+0} \left\{(2\delta)^{l+m-1}\langle [G^{R}_{nn}(E+i\delta)]^{l}[G^{A}_{nn}(E-i\delta)]^{m}\rangle\right\}. \end{equation} This is the expression of the eigenfunction moments in terms of the retarded and advanced Green's functions we were looking for. One can see that any non-trivial moment $m+l>1$ requires a non-trivial limiting procedure. The next standard step is to represent the average of the Green's functions in terms of the functional integral. It begins with the standard representation similar to Eq.(\ref{dOs}): \begin{equation} \label{stand} [G^{R}_{nn}(E+i\delta)]^{l}[G^{A}_{nn}(E-i\delta)]^{m}=\frac{i^{m-l}}{l!m!}\int{\cal D}\Psi\;(\varphi_{n}^{R*}\varphi_{n}^{R})^{l}\,(\varphi_{n}^{A*}\varphi_{n}^{A})^{m}\; e^{-F[\bar{Q}]}. \end{equation} Then the analogy with Eq.(\ref{dOs}) would suggest that we write $\varphi_{n}^{R*}\varphi_{n}^{R}={\rm STr}[\Pi^{R}\bar{Q}]$, raise the $h{\rm STr}[\Pi^{R}\bar{Q}]$ into the exponent with the help of the $l$-times differentiation with respect to the background field $h$ and then switch to the super-matrix filed $Q$ as in Eq.(\ref{rrrho}). However, in trying to do these "standard" steps we make a mistake. The reason is that the field $\bar{Q}_{n}$ is not slow-varying with $n$ and the background field $h_{n}$ should also contain fast space variations. This is what makes a difference compared to the case of the constant in space symmetry breaking field $\frac{1}{2}\omega \Lambda$ in Eq.(\ref{L-h}). One possible remedy \cite{Efet} is to single out the bi-linear combinations of $\varphi_{n}^{*}$ and $\varphi_{n}$ which do not contain fast space variations. We show how to do this for the product $\varphi_{n}^{R*}\varphi^{R}_{n}\varphi_{n}^{A*}\varphi^{A}_{n}$. As the result of averaging over random matrix ensemble should not depend on $n$ ({\it translational invariance on the average}) one can do the sum over $n$ and then divide the result by $N$. Switching to the Fourier-components $\varphi(p)$we can represent this sum as $$ \sum_{p_{1},p_{2},q}\varphi^{R*}(p_{1})\varphi^{R}(-p_{1}+q)\varphi^{A*}(p_{2}) \varphi^{A}(-p_{2}-q)=\sum_{p_{1},p_{2},q}\varphi^{R*}(p_{1})\varphi^{R}(-p_{2}+q) \varphi^{A*}(p_{2}) \varphi^{A}(-p_{1}-q). $$ Two sums in the above expression is a mere re-labeling of momenta, all what is really needed is that the sum of all momenta is zero. However, this re-labeling becomes a non-trivial operation if one assumes that the momentum $q$ is small. In assuming so we select a definite {\it domain of summation} such that the corresponding bi-linear combination of $\varphi$ is slow varying in space. Then {\it one single sum} can be presented as \begin{eqnarray} \label{select-slow} &&\sum_{p_{i}}\varphi^{R*}(p_{1})\varphi^{R}(p_{2})\varphi^{A*}(p_{3}) \varphi^{A}(p_{4})\;\delta(p_{1}+p_{2}+p_{3}+p_{4})=\\ \nonumber &&\sum_{p_{1},p_{2},q\ll 1}\varphi^{R*}(p_{1})\varphi^{R}(-p_{1}+q)\varphi^{A*}(p_{2}) \varphi^{A}(-p_{2}-q)+\sum_{p_{1},p_{2},q\ll 1}\varphi^{R*}(p_{1})\varphi^{R}(-p_{2}+q) \varphi^{A*}(p_{2}) \varphi^{A}(-p_{1}-q)+{\rm remainder}. \end{eqnarray} In the first term of Eq.(\ref{select-slow}) the bi-linear combinations $\varphi^{R*}(p_{1})\varphi^{R}(-p_{1}+q)$ and $\varphi^{A*}(p_{2})\varphi^{A}(-p_{2}-q)$ are slow, while in the second term slow are the combinations $\varphi^{R*}(p_{1})\varphi^{A}(-p_{1}-q)$ and $\varphi^{A*}(p_{2})\varphi^{R}(-p_{2}+q)$. In the remainder we collect all terms where there is no bi-linear slow combinations. The meaning of the above procedure of singling out the slow bi-linear combinations is that only such combinations lead to the divergent functional integral in the limit when $E_{+}-E_{-}=2i\delta$ tends to zero. The average of the remainder is not singular and can be neglected. Performing this procedure in Eq.(\ref{stand}) one obtains $(l+m)!$ possibilities to break the product $(\varphi_{n}^{R*}\varphi_{n}^{R})^{l}\,(\varphi_{n}^{A*}\varphi_{n}^{A})^{m}$ into the product of slow bi-linear combinations. All of them appear to make the same contribution to Eq.(\ref{stand}). Thus one can consider only one such term, do all the standard manipulations with the source fields as we explained above for the case of the mean density of states and multiply the result by $q!$. The final result for the simplest choice $m=1$ is: \begin{equation} \label{mom-fin-sigma} \langle |\Psi_{n}|^{2k}\rangle_{E}=-\frac{k}{2}\,\lim_{\delta\rightarrow 0}\left\{(2\pi\rho\delta)^{k-1}\;\int{\cal D}Q\;({\rm STr}[\Pi^{R}Q_{n}])^{k-1}\;{\rm STr}[\Pi^{A}Q_{n}]\;e^{-F[Q]}\right\}. \end{equation} One can see that it is $k!$ times larger than the one obtained by the "naive" manipulations with the background field. We spent some time to go into detail of this subtlety in order to show that sometimes "exact" manipulations with the background fields are dangerous if the fast varying components of the fields are treated improperly or simply omitted. This is the result of a saddle-point approximation used in the derivation of the nonlinear sigma-model. No such danger appear for the dual representation which did not involve any approximation: \begin{equation} \label{mom-fin-dual} \langle |\Psi_{n}|^{2k}\rangle_{E}=-\frac{1}{2\,(k-1)!}\,\lim_{\delta\rightarrow 0}\left\{(2\pi\rho\delta)^{k-1}\;\int{\cal D}\bar{Q}\;({\rm STr}[\Pi^{R}\bar{Q}_{n}])^{k-1}\;{\rm STr}[\Pi^{A}\bar{Q}_{n}]\;e^{-F[\bar{Q}]}\right\}. \end{equation} One can do one more step without specifying the functionals $F[Q]$ and $F[\bar{Q}]$ using the fact that the structure of dependence of the $Q$ and $\bar{Q}$ fields on the anti-commuting variables Eqs.(\ref{param}),(\ref{U}) is the same. To this end we define \cite{Mirlin2000}the {\it generating functions} $Y[Q]$ as the functional integral of $e^{-F[Q]}$ done over all the super-matrices $Q_{i}$, except the one at a space point $n$: \begin{equation} \label{generating} Y[Q_{n}]=\int_{Q_{i},i\neq n} {\cal D}Q\,e^{-F[Q]}. \end{equation} If the generating function is known the eigenfunction moments are given by the integral over one single super-matrix $Q_{n}$. One can show quite generally that this function does not depend on the anti-commuting variables. Then the integration of the anti-commuting variables is very simple as it involves only the pre-exponent in Eqs.(\ref{mom-fin-sigma}),(\ref{mom-fin-dual}). As the result of this integration the additional factor $(k-1)$ appears in these equations. However, the main thing is to understand how it comes that the infinitesimal factor $\delta^{k-1}$ is compensated by the integral over the super-matrix $Q_{n}$. There is only one scenario of for this to happen in the framework of the nonlinear sigma-model: this is to absorb $\delta$ into the variable $\lambda\rightarrow\delta\lambda$ which can take arbitrary large values. For this the pre-exponent in Eq.(\ref{mom-fin-sigma}) must be proportional to $\lambda^{k-2}$ in the limit of large $\lambda$ and also the generating function $Y(Q_{n})=Y(u)$ must be a function of one single variable $u=2\pi\rho\delta\lambda$. One can show that this is indeed the case: \begin{equation} \label{mom-fin} \langle |\Psi_{n}|^{2k}\rangle_{E}=\frac{k(k-1)}{N}\,\int_{0}^{\infty}du\,u^{k-2}\,Y(u). \end{equation} This is a remarkable formula, as it implies that the distribution function of $|\Psi|^{2}$ for any unitary ensemble $\beta=2$ is: \begin{equation} \label{Dfun} \left.{\cal P}(|\Psi|^{2})=N^{-1}\,\frac{\partial^{2}}{\partial u^{2}}\;Y(u)\right|_{u=|\Psi|^{2}}. \end{equation} For the dual theory Eq.(\ref{mom-fin-dual}), the generating function $\bar{Y}[\bar{Q}_{n}]$ defined similar to Eq.(\ref{generating}) may depend on the {\it two} variables. This is because there are not one but two {\it non-compact} variables $\lambda_{1}$ and $\lambda_{2}$ that may take arbitrary large values. Introducing new variables $$ s=(\lambda_{1}+\lambda_{2}),\;\;\;\;\;r=(\lambda_{1}-\lambda_{2}), $$ one obtains: \begin{equation} \label{mom-fin-dual1} \langle |\Psi_{n}|^{2k}\rangle_{E}=\frac{1}{4\pi\rho\,N}\,\frac{1}{(k-2)!}\,\int_{0}^{\infty}ds\;\int_{-\infty}^{+\infty}dr\; \,s^{k-2}\,\bar{Y}(s,r). \end{equation} Let us apply Eq.(\ref{Dfun}) to the simplest case of the eigenfunction statistics in the Wigner-Dyson random matrix theory. In this case the variables of the super-matrix $Q_{i}$ are locked to their values at $i=n$. Thus there is no integration in Eq.(\ref{generating}) whatsoever and we obtain: \begin{equation} \label{Y-WD} Y[Q]={\rm exp}\left[-\pi N\rho\delta\;{\rm STr}[\Lambda Q_{n}] \right]\rightarrow {\rm exp}(-2\pi\rho\delta\lambda\;N). \end{equation} Then Eq.(\ref{Dfun}) immediately gives for $\beta=2$ Wigner-Dyson RME the Gaussian eigenfunction distribution: \begin{equation} \label{Dfun-WD} {\cal P}(|\Psi|^{2})= N\;e^{-N|\Psi|^{2}}. \end{equation} Note that the Gaussian form of the distribution function is not a consequence of the Gaussian distribution of the entries of ${\bf H}$ but rather a consequence of the {\it central limit theorem} at any distribution of independently fluctuating entries which variance matrix $A_{ij}$ does not depend on $i-j$. One can show that for the orthogonal Gaussian ensemble $\beta=1$ it deviates from the Gaussian: \begin{equation} \label{Dfun-orth} {\cal P}(|\Psi|^{2})= \sqrt{\frac{N}{2\pi \,|\Psi|^{2}}}\;e^{-N|\Psi|^{2}/2}. \end{equation} The simplest non-trivial application \cite{KOs} of Eq.(\ref{mom-fin-dual1}) for the problem that cannot be treated by the nonlinear sigma-model is calculating the eigenfunction distribution function for the one-dimensional Anderson model. This model is described by the random matrix Hamiltonian $$ {\bf H}_{ii}=\varepsilon_{i},\;\;\;\;{\bf H}_{i,i\pm 1}=1, $$ where $\varepsilon_{i}$ is the Gaussian random variable with the variance $w\ll 1$ (weak disorder case). Outside the center of the band $E=0$ the result for the eigenfunction distribution function is amazingly simple: \begin{equation} \label{And-Dfun} {\cal P}(|\Psi|^{2})=\frac{V_{\rm loc}}{N}\;\frac{e^{-V_{\rm loc}|\Psi|^{2}}}{|\Psi|^{2}}, \end{equation} where $V_{\rm loc}$ is the localization radius. This distribution is not normalizable and should be cut a small values of $|\Psi|^{2}$. However, there is another way of normalizing it. This is the requirement that $\langle |\Psi_{n}|^{2}\rangle =N^{-1}$. The first moment of the distribution is perfectly well defined and gives the above pre-factor.

proofpile-arXiv_069-15747